MATHEMATICS - WordPress.com...116 XII – Maths CLASS XII MATHEMATICS Units Weightage (Marks) (i)...

QUESTION BANK

DIRECTORATE OF EDUCATIONGOVT. OF NCT OF DELHI

CLASS XII

MATHEMATICS

123456789012345678901234567890121234567890123456789012345678901212345678901234567890123456789012123456789012345678901234567890112345678901234567890123456789012123456789012345678901234567890121234567890123456789012345678901212345678901234567890123456789011234567890123456789012345678901212345678901234567890123456789012123456789012345678901234567890121234567890123456789012345678901

123456789012345678901234567890121234567890123456789012345678901212345678901234567890123456789012123456789012345678901234567890112345678901234567890123456789012123456789012345678901234567890121234567890123456789012345678901212345678901234567890123456789011234567890123456789012345678901212345678901234567890123456789012123456789012345678901234567890121234567890123456789012345678901

SPONSORED BY :

DELHI BUREAU OF TEXT BOOKS

116 XII – Maths

CLASS XII

MATHEMATICS

Units Weightage (Marks)

(i) Relations and Functions 10

(ii) Algebra (Matrices and Determinants) 13

(iii) Calculus 44

(iv) Vector and Three dimensional Geometry. 17

(v) Linear Programming 06

(vi) Probability 10

Total : 100

Design

Type of Questions Weightage of Number of Total Markseach question questions

(i) Very short answer (VSA) 01 10 10

(ii) Short Answer (SA) 04 12 48

(iii) Long Answer (LA) 06 07 42

Internal Choice

There will be internal choice in 4 questions of short answer type and in 2 questions of Long answertype.

NOTEQuestions requiring Higher Order thinking skills (HOTS) have been added in every chapter. Suchquestions are marked with a star, and to help the students, hints to their solutions are given along withthe answers.

117 XII – Maths

CHAPTER 1

RELATIONS AND FUNCTIONS

POINTS TO REMEMBER

1. Empty relation is the relation R in X given by R = φ ⊂ X × X.

2. Universal relation is the relation R in X given by R = X × X.

3. Reflexive relation R in X is a relation with (a, a) ∈ R, ∀ a ∈ X.

4. Symmetric relation R in X is a relation satisfying (a, b) ∈ R ⇒ (b, a) ∈ R.

5. Transitive relation R in X is a relation satisfying

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.

6. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.

7. A function f = X → Y is one-one (or injective) if

f(x1) = f(x2) ⇒ x1 = x2, ∀ x1, x2 ∈ X

8. A function f : X → Y is onto (or surjective) if given any y ∈ Y, x X∃ ∈ such that f(x) = y.

9. A function f : X → Y is called bijective if it is one-one and onto.

10. For f : A → B and g : B → C, the function gof : A → C is given by (gof) (x) = g[f(x)] .x A∀ ∈

11. A function f : X → Y is invertible if :g Y X∃ → such that go f = Ix and fog = Iy.

12. A function f : X → Y is invertible if and only if f is one-one and onto.

13. A binary operation * on a set A is a function * : A × A → A.

14. An operation * on A is commutative if a * b = b * a, , .a b A∀ ∈

15. An operation * on A is associative if (a * b) * c = a * (b * c) , , .a b c A∀ ∈

16. An element e ∈ A, is the identity element for * : A × A → A if

a * e = a = e * a, , .a A∀ ∈

17. An element a ∈ A is invertible for * : A × A → A if there exists b ∈ A such that a * b = e = b

* a, where e is the identity for *. The element b is called inverse of a and is denoted by a–1.

118 XII – Maths

VERY SHORT ANSWER TYPE QUESTIONS

1. If A is the set of students of some boys school then write, which types of following relations are.(Universal, Empty or neither of the two).

R1 = {(a, b) : a, b are ages of students and |a – b| ≥ 0}

R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}

R3 = {(a, b) : a, b are weights of students and |a – b| > 0}

R4 = {(a, b) : a, b are students studying in same class}

R5 = {(a, b) : age of a is greater than age of b}

2. If A = {2, 3, 4, 5} then write whether each of the following relations on set A is a function or not?Give reasons also.

(i) {(2, 3), (3, 4), (4, 5), (5, 2)}

(ii) {(2, 4), (3, 4), (5, 4), (4, 4)}

(iii) {(2, 3), (2, 4), (5, 4)}

(iv) {(2, 3), (3, 5), (4, 5)}

(v) {(2, 2), (2, 3), (4, 4), (4, 5)}

*3. If f : R → R, g : R → R defined by

( ) ( ) += =3 – 7 8 7, then

8 3

x xf x g x

find (i) (fog) (7) =

(ii) (gof) (7) =

4. If f, g are the functions,

given by f = {(1, 2), (2, 3), (3, 7), (4, 6)}

g = {(0, 4), (1, 2), (2, 1)}

find fog.

5. If ( ) –11

xf x x

x= ∀ ≠

+

write (fof) (x)

6. If f : R → R defined by

( ) ( )–12 1, find ?

5

xf x f x

−= =

119 XII – Maths

7. Check the following functions for one-one. Also, give the reason for your answer.

(i) f : R → R s.t f(x) = x2 + 1 x R∀ ∈

(ii) f : R – {0} → R – {0} such that x · f(x) = 1

(iii) f : R → R such that f(x) = |x|.

(iv) f : R → R such that f(x) = x3.

(v) f : R → R such that f(x) = (x – 1) (x – 2) (x – 3)

(vi) f : R → R such that f(x) = [x] x R∀ ∈

where [ . ] denotes the greatest integer function.

(vii) f : R → R, f(x) = sin x x R∀ ∈

(viii) f : [0, π] → [–1, 1], f(x) = cos x [ ]0,x∀ ∈ π

(ix) ( ): – , , tan .2 2

f R f x xπ π → =� ��

8. Check whether the following functions are onto or not. Give one reason for your Answer.

(i) [ ] ( ) 1: 1, 1 , , sin .

2 2

−−π π − → =� ��

f f x x

(ii) ( ) −π π → − =� ��

1: , , tan .

2 2f R f x x

(iii) f : R → R, f (x) = x3

(iv) { } ( ) 1: – 0 , .f R R f x

x→ =

(v) f : N → N, f(x) = x3.

(vi) f : (0 ∞) → R, f(x) = x2.

(vii) { } ( ): –1 , .1

xf R R f x

x→ → =

+

(viii) [ ] ( ) [ ]2

sin: – 1, 1 , where

1

xf R f x

x

π→ =+

[ . ] denotes the greatest integer function.

9. If ‘*’ is a Binary operation defined on R then if

(i) a * b = a2 – b2, write 8 * (3 * 1)

(ii) ( )=* write 4 * 2 * 62

aba b

120 XII – Maths

(iii) ( )–* then write 10 * 7 * 3

3

a ba b =

(iv) ( )3

* find 5 * 3 * 1 .= aa b

b

SHORT ANSWER TYPE QUESTIONS

10. Check the following relations for each of (i) Reflexivity; (ii) Symmetricity; (iii) Transitivity;(iv) Equivalence Relation.

(a) R1 = {(A, B); |A| = |B|, A, B are line segments in the same plane}

(b) R2 = {(a, b), (b, b), (c, c), (a, c), (b, c)} in the set A = {a, b, c}

(c) R3 = {(a, b) : a ≥ b, a, b, ∈ R}

(d) R4 = {(a, b) : a divides b, a, b Î A } where A = {2, 3, 4, 5}

(e) R5 = {a, b), (b, a), (a, a)} in {a, b, c}.

(f) R6 = {(a, b) : a ≥ b, a, b ∈ N}

*(g) R7 = {(a, b) : a, b ∈ R, a ≤ b3}

(h) R8 = {(a, b) : a – b is multiple of 5, a, b, ∈ R}

(i) R9 = {(a, b) : b = 3a and a, b ∈ R}

(j) R10 = {(a, b) : a – b is an integer, a, b ∈ R}

11. Check the injectivity and surjectivity of the following functions.

(i) f : R → R, ( ) 3 – 7.

5

xf x =

(ii) f : N → N, defined by f(x) = x .x N∀ ∈

(iii) f : R → R, defined by f(x) = x2.

(iv) f : R → R, defined by f(x) = |x|.

*(v) f : N → N, defined by ( ) 1, if is odd

1, if is even.

x xf x

x – x

+= �

�

(vi) f : R → R, defined by f(x) = (x + 1)2 + 4.

(vii) ( ) −−π π → =� ��

1: , defined by tan 2 .`

4 4f R f x x

(viii) [ ] ( ): – 1, 1 defined by sin 2 .f R f x x→ =

121 XII – Maths

*(ix) ( )2

: , defined by .1

xf R R f x

x→ =

+(x) f : R → Z, defined by f(x) = [x], where [ . ] denotes the greatest integer function.

(xi) { } ( ): – – 1 , defined by .1

xf R R f x

x→ =

+

*(xii) f : R → R, defined by f(x) = (x – 1) (x – 2).

*12. If A = N × N and a Binary operation ‘*’ is defined on A as * : A × A → A such that (a, b) * (c,d) = (ac, bd). Check whether ‘*’ is commutative and Associative. Find the identity element for‘*’ on A if any.

13. Let * is a Binary operation defined on R by a * b = |a2 – b2|, then

(i) Is ‘*’ commutative?

(ii) Show that ‘*’ is not associative by giving one example for it.

14. If * is a Binary operation defined on R – {0} defined by 2

2* ,

aa b

b= then

(i) Is ‘*’ Commutative?

(ii) Is ‘*’ Associative?

15. Let * be a binary operation on Q – {1} such that a * b = a + b – ab.

(i) Prove that * is commutative and associative

(ii) Also find the identity element in Q–{1} (if any)

16. If f : R → R, defined by ( ) 3 – 1

4

xf x = find f–1(x) if after checking f(x) for one-one and onto.

17. If f : R → {–1, 1}, defined by f(x) = sin x invertible? If not, give reason. If yes, find f–1(x).

*18. If f : R → R, ( ) 2 – 1,

3

xf x = then find (fog) (x) and (gof) (x). Are they equal?

19. If f(x) = sin x, g(x) = x2 then find fog and gof. Are they equal?

122 XII – Maths

CHAPTER 2

INVERSE TRIGONOMETRIC FUNCTIONS

POINTS TO REMEMBER

Principal value branches of the branches of the inverse trigonometric function with their domainsand Ranges :

Function : Domain → Range

sin–1 : [ ] ––1, 1 ,

2 2

π π → � ��

.

cos–1 : [–1, 1] → [0 , π].

cosec–1 : ( ) { }–– – 1, 1 , 0 .

2 2R

π π → −� ��

sec–1 : ( ) [ ]– – 1, 1 0, .2

Rπ → π − � �

� �

tan–1 : –

, .2 2

Rπ π → � ��

cot–1 : R → (0, π)

Note : ( ) ( )–1–1 –11sin , sin sin etc.

sinx x x

x= ≠

VERY SHORT ANSWER TYPE QUESTIONS (1 Mark for Each Part)

1. Write the principal value of

(i) ( )–1sin – 3 2 (ii) ( )–1

sin 3 2 .

(iii) ( )–1cos – 3 2 (iv) ( )–1

cos 3 2 .

(v)–1 1

tan –3

� ��

(vi)–1 1

tan .3

� ��

(vii) cosec–1 (– 2). (viii) cosec–1 (2)

(ix)–1 1

cot –3

� ��

(x)–1 1

cot .3

� ��

(xi) sec–1 (– 2). (xii) sec–1 (2).

123 XII – Maths

(xiii) ( )− − − − − + + −� � � ��

1 1 13 1sin cos tan 1 3

2 2

2. Simplify each of following using principal value :

(i)–1 –11 2

tan – sec .3 3

� � � ��

(ii)–1 –11 3

sin – – cos .2 2

� ��

(iii) tan–1 (1) – cot–1 (–1). (iv)–1 –11 1

cos – sin – .2 2

+� � � ��

(v) ( )–1 –1 1tan 3 cot .

3

+ � �� (vi) ( ) ( )–1 –1

cosec 2 sec 2 .+

(vii) tan–1 (1) + cot–1 (1) + sin–1 (1). (viii) ( )–1 –1 1cot 3 sin – .

2

− � ��

(ix)–1 4

sin sin .5

π � ��

(x)–1 7

cos cos .5

π � ��

(xi)–1 5

tan tan .6

π � ��

(xii)–1 3

cosec cosec .4

π � ��

SHORT ANSWER TYPE QUESTIONS (4 Marks for Each Part)

3. Show that–1 1 cos 1 cos

tan .1 cos – 1 – cos 4 2

x x x

x x

+ + − π= +� �� +

4. Prove–1 1 1 2 2

sin sin sin 1 – 1 .x y x y y x− − + = + −� �

5. Prove–1 1 1 2 2

sin – sin sin 1 – 1 .x y x y y x− − = + −� �

6. Prove ( ) ( )–1 1 1 2 2cos cos cos – 1 – 1 – .x y xy x y

− − + = � ��

*7. Prove–1 –1cos – sin

tan tan – , tan 1 0.cos sin

a x b x a ax x

b x a x b b

= + >� � � �� +

8. Prove ( )1 1cos 1 costan cot 0, 2 .

1 sin 1 cos 4

− − + π − = ∈ π� �� − −x x

xx x

9. Prove2 2

–1 –1 –1

2 2tan sin cos .

–

x x a x

a aa x

− � �= =� � � ��

10. Prove –1 –1 –1 –1 –18 8 300cot 2 tan cos tan 2 tan sin tan .

17 17 161

� � � � � �+ =� � � ��

124 XII – Maths

11. Prove− −

+ + − π� � = +� �� + − −

2 21 1 2

2 2

1 1 1tan cos .

4 21 1

x xx

x x

12. Solve–1 –1

cot 2 cot 3 .4

x xπ+ =

13. Solve–1 –1

tan 2 tan 3 .4

x xπ+ =

14. Prove ( ) ( )–1 –1 –1 2 –1 21cot tan tan cos 1 2 cos 2 1 .x x x

x

� �+ + − + − = π� ��

15. (i) Prove –1 1 –1

tan tan tan 0, , , 0.1 1 1

a b b c c aa b c

ab bc ac

−− − − + + = >� � � � � �� − + +

(ii) Find the value of 1cot sin – .

2

− π � ��

*16. Prove that

(i) ( )–1 1 3 –1 13 sin sin 3 – 4 , , .

2 2x x x x

− = ∈ � ��

(ii) ( )–1 1 3 13 cos cos 4 – 3 , , 1 .

2x x x x

− = ∈ � ��

125 XII – Maths

CHAPTER 3 and 4

MATRICES AND DETERMINANTS

POINTS TO REMEMBER

Matrix : A matrix is an ordered rectangular array of numbers or functions. The numbers of functionsare called the elements of the matrix.

Order of Matrix : A matrix having ‘m’ rows and ‘n’ coloumns is called the matrix of order mxn.

Zero Matrix : A matrix having all the elements zero is called zero matrix or null matrix.

Diagonal Matrix : A square matrix is called a diagonal matrix if all its non diagonal elements arezero. The diagonal elements may or may not be zero.

Scalar Matrix : A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.

Identity Matrix : A scalar matrix in which each diagonal element is I, is called an identity matrixor a unit matrix.

∴ A = [aij]n × n

aij = 0 when i ≠ j

= 1 when i = j is a identity matrix.

Transpose of a Matrix : If A = [aij]m × n be an m × n matrix then the matrix obtained by interchang-ing the rows and columns of A is called the transpose of the matrix. If A = [aij]m × n. Then transposeA = A´ = [aij]n × m. Transpose of A is denoted by A´ or AT.

Symmetric Matrix : A square matrix A = [aij] is said by symmetric if A´ = A.

Skew symmetric Matrix : A square matrix A = [aij] is said to be a skew symmetric matrix ifA´ = –A.

Inverse of a Matrix : Inverse of matrix.

–1.

Adj AA

A=

where (Adj A) is the adjoint matrix which is the transpose of the cofactor matrix.

Singular Matrix : A square matrix is called singular if |A| = 0, otherwise it will be called anonsingular matrix.

Determinant : To every square matrix A = [aij] of order n × n, we can associate a number (realor complex) called determinant of A. It is denoted by det A or |A|.

If A is a nonsingular matrix then its inverse exists and A is called invertible matrix.

(AB)´ = B´A´

126 XII – Maths

(AB)–1 = B–1A–1

(A´)–1 = (A–1)´

If A is any non singular matrix of order n, then |adj A| = |A|n – 1

If A be any given square matrix of order n. Then A (adj A) = (adj. A) . A = |A|I.

Where I is the identity matrix of order n.

|A B| = |A||B| where A and B are square matrices of same order.

Area of triangle with vertices (x1, y1), (x2, y2) and (x3, y3) = 1 1

2 2

3 3

11

12 1

x y

x y

x y

∆ =

The points (x1, y1), (x2, y2), (x3, x3) are collinear if 1 1

2 2

3 3

1

1 0.

1

x y

x y

x y

=

VERY SHORT ANSWER TYPE QUESTIONS (1 Mark Each)

1. What is the matrix of order 2 × 2 whose general element aij is given by –

iji j if i j

ai j if i j

≥= � + <�

2. If the matrix P is the order 2 × 3 and the matrix Q is of order 3 × m, then what is the order ofthe matrix PQ?

3. If 1 1

0 1A

= � �

� � find A2.

4. If A = [1 3 2] and

6

2 ,

3

B � �= � ��

find AB.

5. What is the element a23 in the matrix A = λ[aij]3 × 3

where λ ∈ R and 2 – if

2 3 ifiji j i j

ai j i j

>= � + + ≥�

.

6. Let P and Q be two different matrices of order 3 × n and n × p then what is the order of thematrix 4Q – P, if it is defined.

7. Let A be a 5 × 7 type matrix, then what is the number of elements in the second column.

8. If 2 3 2

,5 3 5 5

a b

a b

+ =� � � �−� � � �

find the value of a.

9. Write the matrix X if −

− =� � � ��

8 2 7 53 .

6 0 0 0X

127 XII – Maths

*10. How many matrices of order 2 × 3 are possible with each entry 0 or 1?

11. Give an example of two non zero 3 × 3 matrices A and B such that AB = 0.

12. If 2 –3 0

,–1 4 5

A

= � ��

then find (3A)´.

13. If 2 3

1 0A P Q

= = +� �

� � where P is symmetric and Q is skew-symmetric matrix, then find the

matrix P.

14. If cos 20 sin 20

sin 70 cos 70A

° ° = � �° °� �

, what is |A|?

15. Find the value of the determinants .– –

a ib c id

c id a ib

+ ++

16. Find the value of xy if 3

3

3 8–4.

–4 4

x

y=

17. Write the cofactor of the element 5 in the determinants

2 –3 6

6 0 4 .

1 5 –7

18. Write the minor of the element b in the determinant .

a d g

b e h

c f i

*19. If 3 1 –1 1

,5 – 5 2

x

x= find the values (s) of x.

*20. If x ∈ R, 0 ≤ x ≤ ,2

π and

2 sin –1 3 0,

1 sin –4 sin

x

x x= then find the values of x.

21. If A = [aij] is 3 × 3 matrix and Aij is denote the co-factors of the corresponding elements aij’s,then what is the value of a21A11 + a22A12 + a23A13?

*22. If A is a square matrix of order 3 and |A| = – 2, find the value of |–3A|.

23. Find the area of triangle with vertices A(0, 2), B(0, 4), C(1, 2).

24. For what value(s) of λ, the points (λ, 0), (2, 0) and (4, 0) are colinear?

25. If A = [aij] is a 3 × 3 matrix and Mij’s denotes the minors of the corresponding elements aij’s then,write the expression for the value of |A| by expanding |A| by third column.

128 XII – Maths

26. If 02

xπ< < and the matrix

2 sin 3

1 2 sin

x

x � ��

is singular, find the value of x.

27. For what value of λ, the matrix –3 5

1 � �λ λ +� �

has no inverse?

*28. If A is a square matrix of order 3 such that |adj A| = 125, find |A|.

29. In the system of educations Ax = B, write the condition that the given system of educations hasinfinite solutions.

30. If 5 –3

,6 8

A

= � ��

find adj (adj A)

31. It A = 2B, where A and B are square matrices of order 3 × 3 and |B| = 5. What is |A|?

32. If the matrix sin cos

, find ´.– cos sin

A AAα α

= � �α α� �

33. If –2 –1 3 0

, .3 0 –1 2

B and C

= =� � � ��

Find 2B – 3C.

*34. Let A be a non singular matrix of order 3 × 3 such that |A| = 5. What is |adj A|?

35. Find a 2 × 2 matrix B such that.

6 5 11 0.

5 6 0 11B

=� � � �

� � � �

36. If 2

2 1 3 3 2, find and .

0 60 5

x y x yx y

y y

+ + + =� � � �

+ � ��

37. If

3 0 0

0 0 .

0 0 3

A x � �= � ��

For what value of x, A will be a scalar matrix.

*38. Find ∆ if .

a b b c c a

b c c a a b

c a a b b c

− − − � �∆ = − − −� �� − − −� �

39. Determine the value of x for which the matrix –2 4

6 3A

x

= � ��

is singular?

40. If 5 –2

,3 –2

A

= � ��

write the matrix A(adj A).

41. Write the adjoint of the matrix 2 3

.1 4

A

= � ��

129 XII – Maths

42. Find the value of

0 0

0 .

P

a q

b c r

� ��

43. If A is a 2 × 2 matrix and ( ) 12 0adj ,

0 12A A

= � �

� � what is |A|.

*44. If 0 0

,–3 0

A

= � ��

Find A20?

*45. If A, B, C are angles of triangle. Find the value of ∆ if

( )

( )

sin sin cos

– sin tan .

cos tan

A B C B C

B O A

A B A O

+ +∆ =

+ −

46. If .

1 1 1

x y y z z x

A z x y

+ + += Write the value of det A.

47. If 4 2

2 3 1

xA

x x

+ = � �− +� �

is symmetric matrix, then find x.

48. If 4 –2

,2 1

A

= � ��

find |5A|.

49. If 3 –4

.1 2

A

= � ��

Find 2|A|.

50. If sin – cos

, 0cos sin 2

x xA x

x x π= < <� ��

and A + A´ = I where I is unity matrix, find the value of x.

SHORT ANSWER TYPE QUESTIONS (4 Marks Each)

51. Construct a matrix A = [aij]3 × 4 whose entries are given by –

.iji j

ai j

=+

52. Find x, y, z and w if 2 –1 5

.2 3 0 13

x y x z

x y x w

− + =� � � �− +� � � �

53. Find A and B if 1 –2 3 3 0 1

2 38 and – 2 .2 0 –1 –1 6 2

A A B

+ = =� � � ��

54. Let

1 0 41 0 0 1 2

, and –2 1 0 ,2 3 3 2 1

3 2 6

A B C

� �= = =� � � � � ��

verify that (AB)C = A(BC).

130 XII – Maths

55. Find the matrix X so that 1 2 3 –7 8 9

.4 5 6 2 4 6

X− −

=� � � ��

56. If [ ]–1

2 and 2 1 4 ,

3

A B � �= = − − −� ��

verify that (AB)´ = B´A´.

57. Express the matrix 3 4

1 1A

− = � �−� �

as the sum of a symmetric and a skew symmetric matrix.

58. Express the matrix

3 3 1

2 2 1

4 5 2

P Q

− � �− − = +� �� − −� �

where P is a symmetric and Q is a skew symmetric

matrix.

59. Find the inverse of the following matrix by using elementary transformations 7 6

.2 2

� ��

60. Find the inverse of the matrix 1 1

2 3

− � ��

by using elementary transformations.

*61. If

0 – tan2

tan 02

A

θ � �� =

θ� ��

and I is the identity matrix of order 2, show that

( ) cos sin.

sin cosI A I A

θ − θ + = − = � �θ θ� �

62. Find the value of x such that [ ]1 3 2 1

1 1 2 5 1 2 0.

15 3 2

x

x

� � � � =� � � ��

63. If 4 3

,2 5

A

= � ��

find x and y such that A2 – xA + yI = 0.

*64. If cos sin

,– sin cos

Aθ θ

= � �θ θ� � then prove that

cos sin.

– sin cosn n n

A n Nn n

θ θ = ∈� �θ θ� �

65. If 3 –4

,1 –1

A

= � ��

then prove that 1 2 –4

,1 2

n n nA

n n

+ = � �−� �

where n is any positive integer.

66. Find A (adj A) without finding (adj A) if

1 2 3

3 1 2 .

1 0 3

A � �= � ��

131 XII – Maths

67. Given that 2 3

.4 7

A−

= � �−� � Compute A–1 and show that 9I – A = 2A–1.

68. Given that matrix 2 1

.3 2

A−

= � ��

Show that A2 – 4A + 7I = 0. Hence find A–1.

69. If 3 8

,2 1

A

= � ��

verify that 11 4– .

13 13A I A

−=

70. Show that 2 3

3 4A

− = � �

� � satisfies the equation x2 – 6x + 17 = 0. Hence find A–1.

71. Prove that the product of two materices.

22

22

cos cos sincos cos sinand

cos sin sincos sin sin

φ φ φθ θ θ� �� φ φ φθ θ θ� � � �

is zero when θ and φ differ by an

odd multiple of .2

π

72. If A is any square matrix. Then show that (A – A´) is a skew symmetric matrix.

*73. If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of D if

log 10 1

log 13 1 .

log 15 1

x

y

z

∆ =

74. Show that :

( )( )( )

sin cos sin

sin cos sin 0.

sin cos sin

α α α + δβ β β + δ =γ γ γ + δ

75. Using the properties of determinant, prove the following questions (75 to 79).

2 2 2 2

2 2 2 2 2 2 2

2 2 2 2

4 .

b c a a

b c a b a b c

c c a c

+

+ =

+

76. 2 .

b c c a a b a b c

q r r p p q p q r

y z z x x y x y z

+ + ++ + + =+ + +

132 XII – Maths

77.

2 2

2 2 2 2 2

2 2

4 .

a bc ac c

a ab b ac a b c

ab b bc c

+

+ =

+

78. ( )2.

x a b c

a x b c x x a b c

a b x c

++ = + + +

+

79. Show that :

( ) ( ) ( ) ( )2 2 2.

x y z

x y z y z z x x y yz zx xy

yz zx xy

= − − − + +

80. (i) If the points (a, b) (a´, b´) and (a – a´, b – b´) are collinear. Show that ab´ = a´b.

(ii) If 2 5 4 3

and verity that .2 1 2 5

A B AB A B−

= = =� � � ��

81. Given

0 10 1 2

and 1 0 .2 2 0

1 1

A B

− � �= =� � � �−� � � ��

Find the product AB and also find (AB)–1.

82. (i) Using determinants find the area of the triangle whose vertices are (–3, 1), (2, 4) and(5, 1).

(ii) If 2 3

3,3 2

x

x x

− − =� �

� � find the value of x.

83. Solve the following equations for x.

0.

a x a x a x

a x a x a x

a x a x a x

+ − −− + − =− − +

84. Verify that (AB)–1 = B–1A–1 for the matrices 2 1 4 5

and .5 3 3 4

A B

= =� � � ��

85. Show that –11 tan 1 – tan cos 2 sin 2

.tan 1 tan 1 sin 2 cos 2

θ θ θ − θ =� � � � � �− θ θ θ θ� � � � � �

86. Using matrix method to solve the following system of equations : 5x – 7y = 2, 7x – 5y = 3.

133 XII – Maths

LONG ANSWER TYPE QUESTIONS (6 Marks Each)

*87. Let 2 3

1 2A

= � �−� �

and f(x) = x2 – 4x + 7. Show that f(A) = 0. Use this result to find A5.

88. If

cos sin 0

sin cos 0 ,

0 0 1

A

α − α � �= α α� ��

find adj A and verify that A . (adj A) = (adj A) A = |A| I3.

*89. Find the matrix X for which

3 2 1 1 2 1.

7 5 2 1 0 4X

− − =� � � � � �−� � � � � �

90. If

0 1 1

1 0 1 .

1 1 0

A � �= � ��

Show that A3 – 3A – 2I = 0. Hence find A–1.

91. Using elementary transformations, find the inverse of each of the matrices in Question 91 to 93.

2 3 3

2 2 3 .

3 2 2

− � �� −� �

92.

1 2 2

1 3 0 .

0 2 1

− � �−� �� −� �

93.

0 0 1

3 4 5 .

2 4 7

− � �� − − −� �

94. Show that

( )( )

( )( )

2

32

2

2 .

y z xy zx

xy x z yz xyz x y z

xz yz x y

+� �� ∆ = + = + +� �� +� �

95. By using properties of determinants prove that

( )2 2

32 2 2 2

2 2

1 2 2

2 1 2 1 .

2 2 1

a b ab b

ab a b a a b

b a a b

+ + −� �� − + = + +� �� − − −� �

96. Solve the system of linear equations by using matrix in equation 96 to 98.

2x – y + 4z = 1

3x – z = 2

134 XII – Maths

x – y – 2z = 3

97. 2x – y – z = 7

3x + y – z = 7

x + y – z = 3

98. x + y – 5z = 26

x + 2y + z = – 4

x + 3y + 6z = – 29

*99. If

3 2 1

4 1 2

7 3 –3

A � �= −� ��

find A–1 and hence solve the system of linear equations

3x + 4y + 7z = 14, 2x – y + 3z = 4, x + 2y – 3z = 0.

100. Find A–1, where

1 2 3

2 3 2

3 3 –4

A

− � �= � �� −� �

, hence solve the system of linear equations :

x + 2y – 3z = – 4

2x + 3y + 2z = 2

3x – 3y – 4z = 11

101. Solve by matrix method the following system of linear equations :

x – 2y = 10

2x + y + 3z = 8

–2y + z = 7

*102. The sun of three numbers is 2. If we subtract the second number from twice the first number,we get 3. By adding double the second number and the third number we get 0. Represent italgebraically and find the numbers using matrix method.

103. Compute the inverse of the matrix.

3 1 1

15 6 5

5 2 5

A

− � �= − −� �� −� �

and verify that A–1 A = I3.

*104. If the matrix –1

1 1 2 1 2 0

0 2 3 and 0 3 –1 ,

3 2 4 1 0 2

A B � � � �= − =� � � �� −� � � �

then compute (AB)–1.

135 XII – Maths

105. Determine the product

4 4 4 1 1 1

–7 1 3 1 2 –2

5 3 –1 2 1 3

− − � � � �−� � � �� −� � � �

and use it to solve the system of

equations.

x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

*106. Solve the following system of equations using matrix method.

2 3 104

x y z+ + =

4 6 5– 1

x y z+ =

6 9 20– 2.

x y z+ =

107. For the matrix

1 1 1

1 2 –3 .

2 –1 3

A � �= � ��

Show that A3 – 6A2 + 5A + 11I = 0 and hence find A–1.

136 XII – Maths

CHAPTER 5

DIFFERENTIATION

POINTS TO REMEMBER

� Continuity of a Function : A function f(x) is said to be continuous at x = c if ( ) ( )limx c

f x f c→

=

i.e., L.H.L. at x = c = R.H.L. at x = c = f(c).

f(x) is continuous in [a, b] iff :

f(x) is continuous at [ ], .x c c a b= ∀ ∈

� If f and g are two continuous function then f + g, f – g, f. g., cf, |f | are all continuous function.

�

f

g is continuous at x = a provided g(a) ≠ 0.

� Every polynomial function is a continuous function.

� f(x) is said to be derivable at x = c iff ( ) ( )–

limx c

f x f c

x c→ − exists and value of this limit is denoted

by f(c).

( ) 2

– .. . . , .

du dvv u

d dv du d u dx dxu v u vdx dx dx dx v v

= + =� ��

� If y is a function of u and ‘u’ is function of x them, dy dy du

dx du dx= × [chain rule].

� If x = φ1 (t), y = φ2 (t) then

( )( )

( )2

2

´

´

tdyg t

dx t

φ= =

φ say then ( )2

2´ . .

d y dtg t

dxdx

= � ��

� Rolle’s theorem : If f(x) is continuous in [a, b] and derivable in (a, b) and f(a) = f(b) then thereexists atleast one real no c ∈ (a, b) s.t.f´ (c) = 0.

� L.M.V.T. : If f(x) is continuous in [a, b] and derivable in (a, b) then ∃ atleast one point c Î

(a, b) s.t. ( ) ( ) ( )–´ .

f b f af c

b a=

−


1. At what point f(x) = |3x – 5| is not differentiable.

137 XII – Maths

*2. What is derivative of f(x) if f(x) = |x|, x ≠ 0.

3. At what point f(x) = | |x – 1| –2| is not differentiable.

4. Write the points of discontinuity of ( )2

2

1.

– 5 6

x xf x

x x

+ +=+

5. Write all the points of discoutinuity of f(x) = [x], where [x] is the greatest integer function.

*6. At what point, f(n) is discontinuous where, f(x) is signum function defined as

( ) 0.

0 0

xx

f x xx

≠� �=� �� =� �

7. Write the interval in which f(x) is continuous where f(x) = ex log x.

8. For what value of λ, ( )2

1 2

–1 2

x xf xx

λ + ≥= �<��

is continuous on R.

*9. Write the interval in which f(x) is continuous where ( ) log.

1 – 9

xf x

x=

2

*10. Write the value of K given that ( ) ≠�=�

=��

sin0

.

0

xx

f x xK x

*11. What is the derivative of x6 with respect to x3.

*12. What is the Derivative of f(log x) is f(x) = log x.

13. If Mean value theorem holds for f(x) = ex, x ∈ [0, 1], then for what value of x, M V T is verified.

*14. What is ( )− −+ − ≤ ≤1 1sin cos if 1 1.

dx x x

dx

15. Given g(0) = 2 and f(x) = x g(x) f´(x) and g´(x) exist then what is f´(0).

16. Write the derivative of the following function w.r.t. x.

(a)–1 0–

tan , .01

xa xaa x

>� � >� �+

(b)–1

2

1sec .

2 – 1x

� ��

(c) ( )−5log 3 5 .x *(d)−

� �� +

12

tan .1 12

x

x

(e)3

–12

3tan .

1 3

x x

x

−� �� −

(f)–1

tan 2.e x

(g) cos x3 . sin2 x5. (h) .x

e

138 XII – Maths

(i) ( )− ≤ ≤1sin 0 1.x x x (j) ( )−1

sin cos .x

SHORT ANSWER TYPE QUESTION (4 Marks)

Discuss the continuity of the following functions at indicated points.

17. ( ) 0at 0 .

0 0

xx

f x xxx

≠�= =�� =�

18. ( )–

0at 0 .

2 0

x xx

f x xxx

≠�= =�� =�

19. ( ) ( )– 1

0at 0 .log 1 3

5 0

xe

xf x xx

x

� ≠

= =� +� =�

20. ( ) � � ≠� � �� = =�� =

1sin 0

at 0 .

0 0

x xf x xx

x

21. f(x) = |x – 1| + |x + 1| at x = –1, x = 1.

22. ( ) <�= =�

+ ≥��

sin0

at 0 .

1 0

xx

f x xxx x

23. ( )1 – 1

0at 0 .

1 0

x xx

f x xm x

x

+ − ≠�= =δ�� =�

24. ( )

≠�= =�� =�

sin0

at 0 .

1 0

xx

f x xxx

25. For what value of K, ( )− < <

= �− ≤ <��

2

3 2 0 2

4 3 2 5

x xf x

x kx x is continuous in it’s domain.

26. For what values of a and b

( )

2if –2

2if –2

2if –2

2

xa x

xf x a b x

xb x

x

+ + <� +��= + =� +� + >� +�

is continuous at x = 2.

139 XII – Maths

*27. If ( ) 1

1f x

x=

− then find the point of discontinuity if any of f [f {f(x)].

28. Prove that f(x) = |x – 2| is continuous at x = 2 but not differentiable at x = 2.

29. For what value of K, ( )

− <� −��=�� − + <��

2

3 tan0

5 sin

= 0

13 4 0

2

x xx

x x

f x K x

x x x


30. Show that f(x) = x – [x] is discontinuous at x = 2. Also discuss the countinuity at 5,

2x = where

[ ] represents greatest integer function.

31. Check the differentiability of f(x) = |x – 1| + |x – 2| at x = 2.

*32. If ( ) � � ≠� � �� = �� =

1sin 0

0 0

px x

f x xx

is differentiable at x = 0, then find value of p.

33. For what value of a and b ( )5 if 2

if 2 10

21 if 0

n

f x ax b x

x

≤�= + < <�� ≥�

is continuous.

34. If y = (log x)x + xlog x then find .dy

dx

35. If –1 –1

2

1 2 1tan 2 tan find .

2 1

x dyy

x dxx

� �= +� �� −� �

36. If − −= � �

+� �

1 1sin 2 tan then find .

1

x dyy

x dx

37. If x2/3 + y2/3 = a2/3 then show that = − � ��

1

3 .dy y

dx x

38. If y = tan–1 x, show that ( )2

22

1 2 0.d y dy

x xdxdx

+ + =

39. If f(x) = log (xx + sec3 x), find f(x).

40. If 1 1 0x y y x+ + + = then prove that ( )2

–1, .

1

dyx y

dx x= ≠

+

140 XII – Maths

41. If 2 2

1 1x y y x a− + − = then prove that 2

2

1.

1

dy y

dx x

−=−

42. If (x + y)m + n = xm. yn then prove that .dy y

dx x=

43. If αx + αy = αx + y where α > 0 then prove that –

0.y xdy

dx+ α =

*44. If –1 1 sin 1 sin

cot1 sin 1 sin

x xy

x x

+ + −= � �+ − −� �

where π/2 < x < π, find .dy

dx

*45. If –1 2 2 –1 1

sin 2 1 find when ax .2 2

dyy ax a x

dx

= − ≤ ≤� �

46. Find the derivative of 1 1

2 2

2 2tan . . . sin .

1 1

x xw r t

x x

− − � � � �� − +

47. If = � ��

1sin logx y

a then show that ( )2 2

2 11 – 0.x y xy a y− − =

48. If –1 11 2

tan tan1 1 2

x xy

x x

−− + = +� � � �� + − then show that 0.

dy

dx=

49. If 2

2 1

1

xy f

x

− = � �� + and f 1 (x) = sin x2, then find .

dy

dx

50. If xy + yx + xx = ab, then find .dy

dx

51. If1

x a = θ +� �� θ

1–y a = θ� �� θ

then find .dy

dx

52. if x = a cos3 θ

y = a sin3 θ then find 2

2.

d y

dx

53. If x = aeθ (sin θ – cos θ)

y = aeθ (sin θ + cos θ), then show that at is 1.4

dyx

dx

π=

54. Find 2

2 2

2 1if , .

1 1

dy t tx y

dx t t

−= =+ +

141 XII – Maths

*55. If − = − − −� �

1 2sin 1 1 then find .

dyy x x x x

dx

56. If ( )2–1

2 3 2

1 tan, find .

sin 2x

x x dyy

dxe x−

−=

57. If y = a sin t, cos log tan , find .2

t dyx a t

dx

� �= +� ��

58. If –1 3 sin – 4 cos

cos , find .5

x x dyy

dx

= � ��

59. If 2

–1 12 5 1sin , find .

13

x x dyy

dx

+ −� �=� ��

60. If , find .x y dy

y xdx

=

61. If sin y = x sin (a + y) then show that ( )2

sin.

sin

a ydy

dx a

+=

62. If y = cos–1 x, find 2

2

d y

dx in terms of y.

63. If –1

sinsin ,

yx

m

= � �

� � then prove that ( )

22 2

21 0.

d y dyx x m y

dxdx− − + =

64. If 2 2

2 21

x y

a b+ = then show that

2 4

2 2 3

–.

d y b

dx a y=

65. If y3 – 3ax2 + x3 = 0 then prove that + =2 2 2

2 5

20.

d y a x

dx v

66. Find 2

2

d y

dx when

2

log .x

xy

e

= � �

� �

142 XII – Maths

CHAPTER 6

APPLICATIONS OF DERIVATIVES

POINTS TO REMEMBER

� Rate of Change : If x and y are connected by y = f(x) then dy

dx represents the rate of change

of y w.r.t. x.

� Equation of tangent to the curve y = f(x) at the point P(x1, y1) is given by ( )− = −1 1 .P

dyy y x x

dx

Similarly equation of normal is ( )− = − −1 11

.

P

y y x xdy

dx

The angle of intersection between two curves is the angle between the tangents to the curves

at the point of intersection. 1 2

1 2

–tan ,

1

m m

m mθ =

+ where m1, m2 are slopes of tangent at the point

of intersection P.

� A function f(x) is said to be strictly monotonic in (a, b) if it is either increasing or decreasing in(a, b).

� A function f(x) is said to be strictly increasing in (a, b) if 2,x x∀ in (a, b) s.t.

x1 < x2 ⇒ f(x1) < f(x2). Alternatively, f(x) is increasing in (a, b) if f´(x) > 0 ( ), .x a b∀ ∈

� A function f(x) is said to be strictly decreasing in (a, b) if 1 2,x x∀ in (a, b) s.t. x1 < x2 Þ f(x1)

> f(x2). Alternatively, f(x) is strictly decreasing in (a, b) if f ´(x) < 0 ∀ x ∈ (a, b).

� A function f(x) is said to have local maximum value at x = c, if there exists a neighbourhood(c – δ), (c + δ) of c, s.t. f(x) < f(c) ∀ x ∈ (c – δ, c + δ) x ≠ c. Similarly, local minimum valuecan be defined.

� Local maximum and local minimum values of f(x) may not be maximum and minimum value off(x).

� Critical Point : A point c is called critical point of y = f(x) if either f´(c) = 0 or f´(c) does not exist.

143 XII – Maths

Some useful results :

Figure Curved Total S.A. VolumeSurface area S.A.

Sphere 4πr2 4πr2 π 34

3r

Cone πr1 πr1 + πr2 π 21

3r h

Cylinder 2πrh 2πrh + 2πr2 πr2h


1. Write minimum value of f(x) = x2 + x + 1 in [0, 1].

*2. If Rolle’s theorem in applicable for the function f(x) = x2 – 3x + 1 in [–1, 4] then find the realno ‘c’ verifying Rolle’s theorem.

3. Find the interval where f(x) = cos x defined in [0, 2π] is decreasing.

4. Find the interval where f(x) = x2, x ∈ (– ∞, ∞) is decreasing.

*5. For what value (s) of λ, the function, f(x) = sin x – 3λx is always strictly increasing.

*6. Write the interval in which f(x) = xx is increasing (where x > 0).

7. Examine if f(x) = x9 + 2x5 + 3x3 + 1 is increasing or decreasing (0, ∞).

*8. Write the least value of ( ) ( )1, 0 .f x x x

x= + >

9. Write the maximum value of ( ) [ ]2

1in 0, 2 .

– 2 3f x

x x=

+

*10. Find the maximum and minimum value of f(x) = |2 sin 2x + 3|.

*11. On the curve ( ) 23,

2f x x= find the points at which tangent is parallel to the chord joining the

points 3

–1,2

A � ��

and B(2, 6).

*12. If the tangent to the curve at a point P is perpendicular to x–axis, then what is the value of dy

dx(if it exists) at the point P.

*13. If normal to the curve at a point P on y = f(x) is parallel to y–axis, then write the value of dy

dxat P.

14. What is the slope of the tangent to the curve y = x2 at (–1, 1).

144 XII – Maths

*15. If the tangent to the curve y = 2x2 – x at any point P is parallel to the line x – y = 0, then findthe coordinates of P.

16. If the tangent to the curve x = at2, y = 2at is perpendicular to x-axis then write the coordinatesof the point of contact of tangent.

*17. If curves y = 3e2x and y = be–2x cut each other orthogonally, then find b.

*18. At which point on y2 = 4x, the tangent makes an angle of 45° with the positive direction of .

*19. If kx + y = P is normal to the curve y2 = 12x at (3, 6) then what is value of k.

20. How many extreme values [maximum or minimum] are there of f(x) = x.

21. What is equation of normal to the curve y = sin x at origin.


22. Sand is pouring out from a pipe at the rate of 12 Cu cm/s. The falling sand forms a cone onethe ground in such a way that the height of the cone is always one sixth of the radius of thebase. How fast is the height of sand cone increasing when the height is 4 cm.

23. A particle moves along the curve y = x5 + 2. Find the points on the curve at which y co-ordinateis changing 5 times as fast as the x co-ordinate.

24. Find points of local maxima/minima for f(x). If f(x) = sin x – cos x where 0 < x < 2π. Also findthe local maximum or minimum values.

25. Find the intervals in which the function ( ) 4–

3

xf x x= is increasing or decreasing.

*26. If f(x) = x2 – 2x + 3 then using differentials, find the approximate value of f(1.9).

27. Find the value (s) of a for which :

(i) f(x) = x3 – ax is increasing on R.

(ii) g(x) = sin x + ax is increasing on R.

28. If radius of right circular cone is increasing at the rate of 10π cm3/sec, find the rate at which theheight of the cone is hanging at the instant when radius 5 cm and height 4 cm.

*29. Find the least value of the function. ( ) ( ), , 0 .b

f x ax a b xx

= + >

30. For the curve y = 2x3 – 3x2, find all the points on the curve at which the tangent passes throughthe origin.

31. Prove that the function :

f(x) = x50 + sin x – 1 is strictly increasing on . .2

π π� ��

145 XII – Maths

32. Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cosθ is at a constant distance from the origin.

33. Using errors and approximations find the approximate value of the following.

(i) ( )2

1;

2.002(ii) 0.082; (iii) 0.005.

34. Find the interval in which f(x) = sin x – cos x, 0 < x < 2p, is increasing or decreasing.

*35. If log 2 = 0.3010

and log10 e = 0.4343

find approximate value of log10 (2.2).

36. Find the interval (s) in which y = x2 (x – 3)2 increases.

37. Find the interval (s) in which function f(x) = 2x3 – 15x2 + 36x + 1 is strictly increasing ordecreasing.

38. Find the point of the curve y = 3x2 – 2x where the tangent is parallel to x–axis?

39. Find the equation of the tangent to the curve y = 2e–x/3 at the point where curve cuts y–axis.

40. If 8k2 = 1 then show that the curves y2 = x and xy = k cut at right angles.

41. Determine the interval in which function, f(x) = sin x + cos x in [0, 2π] is strictly increasing ordecreasing.

42. Find the maximum value of f(x) = 3 sin x + 4 cos x in [0, π/2].

43. Find the two positive numbers x and y such that their sum is 35 and product x2y3 is maximum.

44. Find the least value of ‘a’ such that the function f(x) = x2 + 2ax + 3 is strictly increasing on (3, 4).

45. Show that the acute angle of intersection between the curves xy = 6 and x2y = 12 is –1 3

tan . � �� π

46. Find approximate value of 0.037 using differentials.

47. Find all the points of the curve 2 2

– 19 16

x y = at which the tangent is

(i) Parallel to the x–axis; (ii) Parallel to y–axis.

*48. Find all values of a ∈ R such that the function f(x) = (a + 2) x3 – 3ax2 + 9ax – 1 decreases forall x ∈ R.

*49. Find the condition that the line x cos α + y sin α = p be a tangent to the curve 2 2

2 21.

x y

a b+ =

50. Find equation of tangent at 4

xπ= to the curve y = cot2 x – 2 cot x + 2.

146 XII – Maths

LONG ANSWER TYPE QUESTION (6 Marks Each)

51. Show that the point (1, 3) on y = x2 + 2 is nearest to the point (3, 2).

52. A point on the hypotenuse of a right triangle is at a distance ‘a’ and ‘b’ from the sides of thetriangle. Show that the minimum length of the hypotenuse is (a2/3 + b2/3)3/2.

53. If the length of three sides of a trapezium other than base are equal to 10cm, then find the areaof trapezium when it is maximum.

54. A given quantity of metal is to be cast into half cylinder with a rectangular base and semi-circularends. Show that when total surface areas is minimum, the ratio of length of cylinder to thediameter of its semi-circular ends is π : (π + 2).

55. Show that f(x) = sin4 x + cos4 x, x ∈ [0, π/2] is increasing on ,4 2

π π � ��

and decreasing on 0, .4

π � ��

56. Find the interval in which f(x) = 2 log (x – 2) – x2 + 4x + 1 is increasing or decreasing,

57. Find the equation of tangent to the curve y = x3 – 1) (x – 2) at the points where the curve cutsthe x–axis.

58. Show that the semi-verticle angle of a cone of maximum volume and given height is –1tan 2.

59. Prove that the radius of the right circular cylinder of greatest curved surface which can beinscribed in a given cone is half of that of the cone.

60. A rectangular sheet of tin 45 cm × 24 cm is to be made into a box without top by cutting offsquare from each corner and folding up the flaps. What should be the side of the square to becut off so that the volume of the box is maximum?

61. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a squareand the other into a circle. What should be the lengths of the two pieces so that the combinedarea of the square and the circle is minimum?

62. For a given curved surface of a right circular cone when volume is maximum, prove that semi-

vertical angle is –1 1

sin .3

� ��

63. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and

semi-vertical angle α is 3 24

tan .27

hπ α

64. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8

27

of the volume of the sphere.

65. A jet of an enemy is flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2).What is the nearest distance between the soldier and the jet?

147 XII – Maths

66. A rectangular window is surmounted by an equilateral triangle. Given that the perimeter is 16m,find the width of the window so that the maximum amount of light may enter.

67. Cost of fuel for running a car is proportional to the square of speed generated in km/hr. It costsRs. 48 per hour when the car is running the speed of 16 km/hr. What is the most economicalspeed if the fixed charges are Rs. 300 per hour over and above the running cost.

68. Two sides of a triangle are of lengths a and b and angle between them is θ. What value of θwill maximize the area of triangle? Also find the maximum area.

148 XII – Maths

CHAPTER 7

INTEGRATION

POINTS TO REMEMBER

Integration is inverse process of Differentiation.

STANDARD FORMULAE

1.

1

11

log –1

n

nx

c nx dx n

x c n

+� + ≠ −= � +� + =�

∫

2. ( )

( ) 1

11

1log –1

n

n

ax bc n

nax b dx

ax b c na

+ + + ≠ −�� ++ = �� + + =��

∫

3. sin – cos .x dx x c= +∫ 4. cos sin .x dx x c= +∫

5. tan . – log cos log sec .x dx x c x c= + = +∫

6. cot log sin .x dx x c= +∫ 7.2

sec . tan .x dx x c= +∫

8.2

cosec . – cot .x dx x c= +∫ 9. sec . tan . sec .x x dx x c= +∫

10. cosec cot – cosec .x x dx c= +∫ 11. sec log sec tan .x dx x x c= + +∫

12. cosec log cosec – cot .x dx x x c= +∫ 13. –1

2

1sin , 1 .

1dx x c x

x= + <

−∫

14.–1

2

1tan .

1dx x c

x= +

+∫ 15.–1

2

1sec , 1.

1dx x c x

x x= + >

−∫

16.2 2

1 1log .

2

a xdx c

a a xa x

+= +−−∫ 17.

2 2

1 1log .

2

x adx c

a x ax a

−= ++−∫

18.–1

2 2

1 1tan .

xdx c

a aa x= +

+∫ 19. –1

2 2

1sin .

–

xdx c

aa x= +∫

149 XII – Maths

20.2 2

2 2

1log .dx x a x c

a x= + + +

+∫ 21.

2 2

2 2

1log .

–dx x x a c

x a= + − +∫

22.2

2 2 2 2 –1sin .

2 2

x a xa x dx a x c

a− = − + +∫

23.2

2 2 2 2 2 2log .

2 2

x aa x dx a x x a x c+ = + + + + +∫

24.2

2 2 2 2 2 2log .

2 2

x ax a dx x a x x a c− = − − + − +∫

25. .x x

e dx e c= +∫

26.1

. .log

x xa dx a c

a= +∫

INTEGRATION BY SUBSTITUTION

1.( )( )

( )´log .

f xdx f x c

f x= +∫

2. ( )[ ] ( ) ( )[ ] 1

´ .1

nn f x

f x f x dx cn

+

= ++∫

3.( )

( )[ ]( )( ) 1

´.

– 1

n

n

f x f xdx c

nf x

− +

= ++∫

INTEGRATION BY PARTS

( ) ( ) ( ) ( ) ( ) ( ). . – ´ . .f x g x dx f x g x dx f x g x dx dx =� � � �� ∫ ∫ ∫ ∫

PROPERTIES OF DEFINITE INTEGRALS

( ) ( ) ( ) ( ) ( ), where .b

a

f x dx F b F a F x f x dx= − =∫ ∫

1. ( ) ( )– .b a

a b

f x dx f x dx=∫ ∫ 2. ( ) ( ) .b b

a a

f x dx f t dt=∫ ∫

150 XII – Maths

3. ( ) ( ) ( ) .b c b

a a c

f x dx f x dx f x dx= +∫ ∫ ∫ 4. ( ) ( ) .b b

a a

f x dx f a b x dx= + −∫ ∫

5. ( ) ( )

–

0; if is odd function.a

a

f x f x=∫

6. ( ) ( ) ( ) ( )2

0 0

2 , if 2 .a a

f x dx f x dx f a x f x= − =∫ ∫

= 0 if f(2a – x) = – f(x).

Integral as limit of sum :

( ) ( ) ( ) ( ) ( )→

= + + + + + + + −� �∫ 0lim 2 ..... 1

b

ha

f x dx h f a f a h f a h f a n h

where .b a

hh

−=


1. Evaluate the following integrals

(i)1

.x dxx

+� �� ∫ (ii)2

1.

1 sindx

x−∫

(iii)2

cos.

1 cos

xdx

x−∫ (iv)2

21

1.

1dx

x x −∫

(v)

π++∫

2

0

4 3 sinlog .

4 3 cos

xdx

x(vi)

( )1

.log log log

dxx x x∫

(vii) ( )cosec cosec cot .x x x dx+∫ (viii)8 8

8 .8

x xx dx

x

+ + +� �� ∫

(ix) ( )–1.

sin cos

dx

x∫ (x)

199 4

1

cos .x x dx∫2. Evaluate the following integrals.

(i)2

1.

x x

x

+ −∫ (ii)

21– .ax dx

ax

� �� ∫

151 XII – Maths

(iii) 4 2sin 3 3 sec .

2

x xx e dx − +� �� ∫ (iv)

1 cos 2.

1 cos 2

xdx

x

+−∫

(v) .2 1

xdx

x +∫ (vi)sec cosec

.log tan

x xdx

x∫

(vii)2

2

cos 2 2 sin.

cos

x xdx

x

+∫ (viii) ( )log log

.a x x a

e e dx+∫

(ix) 1 sin , .2

x dx xπ − < < π� �� ∫ (x)

( )1

.2 3 log

dxx x+∫

(xi)sin

.x

dxx∫ (xii)

2 1 3

6

tan.

1

x xdx

x

−

+∫

(xiii) 2 log .x dx∫ (xiv)2

.2

ax bdx

ax bx c

+

+ +∫

(xv)sin

.cos

xdx

a b x+∫ (xvi) ( ) .x c

c x dx+∫

(xvii)1

.3 log

dxx x x+∫ (xviii) 2

1.

16 25dx

x+∫

(xix)2 2

.x x

dx

e e−+∫ (xx)

2

1.

9 4dx

x −∫

(xxi)2

1.

16 25dx

x−∫ (xxii)2

1.

4 9dx

x −∫

(xxiii)2 2 2

1.

–dx

a b x∫

3. Evaluate the following definite integrals :

(i)

π

+∫2 3 2

3 2 3 2

0

sin.

sin cos

xdx

x x(ii)

1

–1.

xe dx∫

(iii)2

7

–

2

sin .x dx

π

π∫ (iv)

2 3

4 4–

2

sin.

sin cos

xdx

x x

π

π +∫

152 XII – Maths

(v)1

2

0

1.

1dx

x+∫ (vi)2

2

0

sin.

1 cos

x

x

π

+∫

(vii)1 –1

2

0

tan.

1

xdx

x+∫ (viii)1

2

0

.1

x

x

edx

e+∫

*(ix)5 log 4 log

3 log 2 log.

x x

x x

e edx

e e

−

−∫ (x)1

0

.x

x e dx∫


4. Evaluate the following integrals :

(i)( )–1 2

4

cosec tan.

1

x xdx

x+∫ (ii)1 1

.1 1

x xdx

x x

+ − −+ + −∫

(iii)( ) ( )

1.

sin sindx

x a x b− −∫ (iv)( )( )

cos.

cos

x adx

x a

+−∫

(v) cos cos 2 cos 3 .x x x dx∫ (vi)5

cos .x dx∫

(vii)2 4

sin cos .x x dx∫ (viii) 5 4sin cos .x x dx∫

(ix)3 4

cot cosec .x x dx∫ (x)2 2 2 2

sin cos.

sin cos

x xdx

a x b x+∫

(xi)( )3

1.

cos cosdx

x x a+∫ (xii)

6 6

2 2

sin cos.

sin cos

x xdx

x x

+∫

*(xiii) ( )1

6 2 21x x x x dx+∫ (xiv)sin cos

.sin 2

x xdx

x

+∫

(xv) ( )4

1.

1dx

x x +∫ *(xvi)2

.1 3 sin

dx

x+∫

(xvii)sin

.sin 3

xdx

x∫ *(xviii)5 5

5 5 5 .x x

xdx∫

5. Evaluate :

(i)4 2

.1

xdx

x x+ +∫ *(ii)( )2

1.

6 log 7 log 2dx

x x x + +� �∫

153 XII – Maths

(iii)2

.1 3

dx

x x+ −∫ (iv)2

1.

9 8 2dx

x x+ −∫

(v)2

1.

2 3 2dx

x x+ −∫ (vi)

( ) ( )1

.dxx a x b− −∫

(vii)( )( )

sin.

sin

xdx

x a

− α+∫ (viii)

2

5 2.

3 2 1

xdx

x x

−

+ +∫

(ix)2

2.

6 12

xdx

x x+ +∫ (x)2

2.

4

xdx

x x

+

−∫

(xi)2

1 – .x x x dx+∫ (xii) ( ) 23 2 1 .x x x dx− + +∫

*(xiii) sec 1 .x dx+∫ (xiv)2

2 sin 2 cos.

6 cos 4 sindx

θ − θ

− θ − θ∫6. Evaluate :

(i) ( )7.

1

dx

x x +∫ *(ii) ( ) ( )2 2 2 2.

xdx

x a x b− −∫

(iii)( ) ( )

sin.

1 cos 2 3 cos

xdx

x x+ +∫ (iv)3 2

1.

6

xdx

x x x

−

+ −∫

(v)( ) ( )

21

.1 cos 2 3 cos

xdx

x x

++ +∫ (vi)

( ) ( ) ( )1

.1 2 3

xdx

x x x

−+ − +∫

(vii)( ) ( )

22

.2 1

x xdx

x x

+ +− −∫ (viii)

( ) ( )( ) ( )

2 2

3 2

1 2.

3 4

x xdx

x x

+ +

+ +∫

*(ix)4

4.

16

xdx

x −∫ (x)( ) ( )2

.2 1 4

dx

x x+ +∫

(xi)( )

.sin 1 2 cos

dx

x x−∫ *(xii)( )

.cos 2 sin

dx

x x−∫

(xiii)sin

.sin 4

xdx

x∫ (xiv)2

4 2

1.

1

xdx

x x

−

+ +∫

(xv) tan .x dx∫ (xvi)2

9.

81

xdx

x4

+

+∫

*(xvii)( )−

∫ 32

log.

1

x xdx

x

154 XII – Maths

*(xviii) � �∫2

2

1

,x dx where [x] is greatest integer funbction

7. Evaluate :

(i)5 3

sin .x x dx∫ (ii)3

sec .x dx∫

(iii) ( )cos .ax

e bx c dx+∫ *(iv) –12

6sin .

1 9

xdx

x+∫

(v) cos .x dx∫ (vi) ( )2log .

nx x dx∫

*(vii)( )

2

2.

sin cos

x

x x x+∫ (viii) ( )1 –1sin cos .x x dx

− −∫

*(ix)–1

sin .x

dxa x+∫ (x)

3 –1tan .x x dx∫

(xi)( )2 2

4

1 log 1 – 2 log.

x x xdx

x

+ +� �∫ (xii)

2 1 sin 2.

1 cos 2

x xe dx

x

+ � �� +∫

(xiii)2

1.

2

x xe dx

x

− � �� ∫ (xiv)

2

2

1.

1

x xe dx

x

− � �� +∫

(xv)( )( )

2

2

1.

1

x xe dx

x

+

+∫ (xvi)( )( )2 sin 2

.1 cos 2

x xe dx

x

++∫

(xvii)( )2

1 1.

log logdx

x x

� �−� ��

∫ *(xviii) ( )( )2

1log log .

logx dx

x

� �+� ��

∫

(xix) ( ) ( )sin log cos log .x x dx+∫8. Evaluate the following definite integrals :

(i)

4

0

sin cos.

9 16 sin 2

x xdx

x

π

++∫ (ii)

2

0

cos 2 log sin .x x dx

Π

∫

*(iii)1 2

2

0

1.

1

xx dx

x

−

+∫ (iv)( )

1 2 1

3 220

sin.

1

xdx

x

−

−∫

(v)( ) ( ) ( )

2

0

cos.

1 sin 2 sin 3 sin

xdx

x x x

π

+ + +∫ (vi)2

4 4

0

sin 2.

sin cos

xdx

x x

π

+∫

155 XII – Maths

(vii)2

4

0

2 sin .x dx

π

∫ (viii)

1

2

21

4

.dx

dxx x−

∫

(ix)2 2

2

1

5.

4 3

xdx

x x+ +∫ *(x)2

0

sin.

1 cos

x xdx

x

π

++∫

9. Evaluate :

(i) { }3

1

1 2 3 .x x x dx− + − + −∫ *(ii) ( )2

2

sin cos .x x dx

π

−π

−∫

(iii)

0

.1 sin

xdx

x

π

+∫ (iv) ( )4

0

log 1 tan .x dx

π

+∫

(v)2

0

log sin .x dx

π

∫ (vi) ( )2

0

sin.

1 cos

x xdx

x

π

+∫

(vii) ( ) ( )

3

23

2

2 when 2 1

where 3 2 when 1 1

3 2 when 1 2.−

− − ≤ <��= − + − ≤ <�

− ≤ <��∫

x x x

f x dx f x x x x

x x

(viii)2

4 4

0

sin cos.

sin cos

x x xdx

x x

π

+∫ (ix)2 2 2 2

0

.cos sin

xdx

a x b x

π

+∫

10. Evaluate the following integrals as limit of a sum

(i)3

2

1

.x dx∫ (ii)1

–1

.x

e dx∫

(iii)1

0

.xdx∫ (iv)2

–

0

.x

e dx∫

(v) ( )2

0

2 3 .x dx+∫ (vi) ( )2

–1

7 5 .x dx−∫

156 XII – Maths

11. Evaluate the following integrals.

(i)3

3

tan tan.

1 tan

x xdx

x

+

+∫ (ii)4 4

tan sec.

x xdx

x∫

(iii)3

2

1

2 .x x dx−∫ (iv)1

–12

0

2sin .

1

xdx

x

� �� +∫

*(v)1

–1

1 sinlog .

1 sin

xdx

x

+ � �� −∫ *(vi)

12

–1

log 1 .x x dx+ +∫

(vii)8

2

10.

10

xdx

x x

−+ −∫ (viii)

cos

cos cos

0

.x

x x

edx

e e

π

−+∫

*(ix)3

–3

1.

1x

dxa+∫ (x)

( )

21

.1 2

xdx

x x

−−∫

LONG ANSWER TYPE QUESTIONS (6 MARKS EACH)

12. Evaluate the following integrals :

(i)5

5

4.

xdx

x x

+

−(ii)

( ) ( )221 4

dxdx

x x− +∫

(iii)( ) ( )

3

2

2

1 3

xdx

x x+ −∫ (iv)4

4– 16

xdx

x∫

(v) ( )2

0

tan cot .x x dx

π

+∫ (vi)4

1.

1dx

x +∫

(vii)( )

–1

220

tan.

1

x xdx

x

∞

+∫ (viii) ( )

1–1 2

0

cot 1 .x x dx− +∫

*(ix)( )

( )1

2

0

log 1.

1

xdx

x

+

+∫ *(x)( )5 2

1 cos.

1 cos

xdx

x

+

−∫

*(xi)( ) ( )

.sin 2 cos 2 sin cos

dx

x x x x− +∫ (xii)1 –1

–1 –1

sin cos.

sin cos

x xdx

x x

− −

+∫

157 XII – Maths

(xiii)1

.1

xdx

x

++∫ (xiv)

3 2

–1

sin .x x dxπ∫

13. Evaluate the following integrals as limit of sums :

(i) ( )4

2

2 1 .x dx+∫ (ii) ( )2

2

0

3 .x dx+∫

(iii) ( )3

2

1

3 2 4 .x x dx− +∫ (iv) ( )4

2 2

0

3 .x

x e dx+∫

(v) ( )5

2

2

3 .x x dx+∫ (vi) ( )3

2

0

2 3 5 .x x dx+ +∫

158 XII – Maths

CHAPTER 8

APPLICATIONS OF THE INTEGRALS

POINTS TO REMEMBER

AREA OF BOUNDED REGION

1. Area bounded by the curve y = f(x), the x axis and between the ordinate. a+ x = a and x = bis given by

( )Area =b

a

f x dx∫

yy f x= ( )

a bO

y

a bO

y f x= ( )

x x

2. Area bounded by the curve x = f(y) the y-axis and between absussa a+ y = c and y = d is givenby

( )Area =d d

c c

x dy f y dy=∫ ∫

y

x f y= ( )

c

Ox

d

y

x f y= ( )

c

Ox

d

3. Area bounded by two curves y = f(x) and y = g(x) such that 0 ≤ g(x) ≤ f(x) for all x ∈ [ab] andbetween the ordinate at x = a and x = b is given by

( ) ( )[ ]Area = –b

a

f x g x dx∫

159 XII – Maths

yy f x= ( )

cOxd

y gm=

4. If the curve y = f(x) interest the axis (x-axis) then the area of shaded region is given by

1 2Area = .A A+

ca

A 1

A 2

( ) ( )Area = .c b

a c

f x dx f x dx∫ ∫


1. Find the area enclosed by circle x2 + y2 = a2.

2. Find the area of region bounded by y2 = 4x, x = 1, x = 4 and x – axis in first quadrant.

3. Find the area enclosed by the ellipse

2 2

2 21 .

x ya b

a b+ = >

4. Find the area of region in the first quadrant enclosed by x–axis the line y = x and the circlex2 + y2 = 32.

5. Find the area of region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9}

6. Prove that the curve y = x2 and, x = y2 divide the square bounded by x = 0, y = 0, x = 1,y = 1 into three equal parts.

7. Find smaller of the two areas enclosed between the ellipse 2 2

2 21

x y

a b+ = and the line

bx + ay = ab.


8. Find the common area bounded by the circles x2 + y2 = 4 and (x – 2)2 + y2 = 4.

9. Using integration find area of region bounded by the triangle whose vertices are

(a) (–1, 0), (1, 3) and (3, 2)

160 XII – Maths

(b) (–2, 2) (0, 5) and (3, 2)

10 Using integration find the area bounded by the lines.

(i) x + 2y = 2, y – x = 1 and 2x + y – 7 = 0

(ii) y = 4x + 5, y = 5 – x and 4y – x = 5.

11. Find the area of the region {(x, y) : x2 + y2 ≤ 1 ≤ x + y}.

12. Find the area of the region bounded by

y = |x – 1| and y = 1.

*13. Using integration find the area enclosed by the curve y = cos x, y = sin x and x–axis in theinterval (0, π/2).

*14. Sketch the graph y = |x – 5|. Evaluate −∫6

05 .x dx

*15. Find the area enclosed by the curve

y = |x – 1| and y = 3 – |x|

16. Find the area enclosed by the curve y = sin x between x = 0 and 3

2x

π= and x-axis.

17. Find the area bounded by semi circle 2

25y x= − and x-axis.

18. Find area of region given by {(x, y) : x2 ≤ y ≤ |x|}.

19. Find area of smaller region bounded by ellipse 2 2

19 4

x y+ = and straight line 2x + 3y = 6.

20. Find the area of region bounded by the curve x2 = 4y and line x = 4y – 2.

*21. Find the area bounded by the ellipse 2 2

2 21

x y

a b+ = and ordinate x = ae and x = 0, where b2

= a2 (1 – e2) and e < 1.

22. Find the area enclosed by parabola y2 = 4ax and the line y = mx.

23. Find the area of region bounded by y = x2 and the line y = x.

24. Using integration find the area of region in first quadrant enclosed by x-axis the line 3x y=and the circle x2 + y2 = 4.

25. Draw a sketch of the region {(x, y) : x2 + y2 ≤ 4 ≤ x + y} and find its area.

26. Find the area enclosed between the y-axis the line y = 1 and the curve x2 + y2 – 2x = 0.

27. Find smaller of two areas bounded by the curve y = |x| and x2 + y2 = 8.

161 XII – Maths

CHAPTER 9

DIFFERENTIAL EQUATION

POINTS TO REMEMBER

� Differential Equation : Equation containing derivatives of a dependant variable with respect toan independent variable is called differential equation.

� Order of a Differential Equation : The order of a differential equation is defined to be the orderof the highest order derivative occurring in the differential equation.

� Degree of a Differential Equation : The degree of differential equation is defined to be thedegree of highest order derivative occurring in it after the equation has been made free fromradicals and fractions. Solving a differential equation.

(i) Type ( ) ( )= .dy

f x g ydx

: Variable separable method separate the variables and get f(x)

dx = h(y) dy. The ( ) ( )f x dx h y dy c= +∫ ∫ is the required solution.

(ii) Homogenous differential equation : A differential equation of the form ( )( )

,

,

f x ydy

dx g x y=

where f(x, y) and g(x, y) are both homogeneous functions of the same degree in x and

y i.e., of the form dy y

Fdx x

= � �� is called a homogeneous differential equation. Substituting

y = vx and then ,dy dv

v xdx dx

= + we get variable separable form.

(ii) Linear differential equation : Type I : dy

py qdx

+ = where p and q are functions of

x. Its solution is y . (I. F.) = ∫q(I. F.)dx where . . .p dx

I F e= ∫

VERY SHORT ANSWER TYPE QUESTIONS

1. Write the order and degree of the following differential equations.

(i) cos 0.dy

ydx

+ = (ii)22

23 4.

dy d y

dx dx

+ =� ��

(iii)

54 2

4 2sin .

d y d yx

dx dx

+ = � �

� �(iv)

5

5log 0.

d y dy

dxdx

+ =� ��

162 XII – Maths

*(v)

1 32

21 .

dy d y

dx dx

+ = � �

� �(vi)

3 2 22

21 .

dy d yK

dx dx

� �+ = � ��

(vii)

2 33 2

3 2sin .

d y d yx

dx dx

+ =� � � �

� � � �

2. Write the general solution of following differential equations.

(i)5 2 2

.dy

x xdx x

= + − (ii) (ex + e–x) dy = (ex – e–x)dx

(iii)3

.x edy

x e xdx

= + + (iv) 5 .x ydy

dx

+=

(v)1 cos 2

.1 cos 2

dy x

dx y

−=+

(vi)1 2

.3 1

dy y

dx x

−=+

(vii)2

sec .dy

x xdx

= (viii) .log

dy y

dx x x=

3. What is the integrating factor in each of the following linear differential equations.

(i) cos sin .dy

y x xdx

+ = (ii)2

sin cos .cos

dy yx x

dx x+ =

(iii)2 2

cos .dy

x y x xdx

+ = (iv) log . tan . .xdy

x x y x edx

+ =

(v)3

– . log .dy

y xdx x

= (vi) ( ) 2tan sec .

dxy x y

dy+ =

(vii)2

1. sin .

1

dxx y

dy y+ =

+(viii) – 3 2.

dxx y

dy= +


4. (i) Verify that –1

sinm xy e= is a solution of ( )

22 2

21 0.

d y dyx x m y

dxdx− − − =

(ii) Show that y = sin (sin x) is a solution of diff. equation

22

2tan – cos .

d y dyx y x

dxdx+ =

(iii) Show that = + By Ax

x is a solution of

163 XII – Maths

22

2. – 0.d y dy

x x ydxdx

+ =

(iv) Show that function y = a cos (log x) + b sin (log x) is the solution of

22

2. 0.d y dy

x x ydxdx

+ + =

(v) Verity that y2 = 4a (x + a) is a solution of 2

1 – 2 .dy dy

y xdx dx

� �= = � ��

(vi) Show that 2 2x x x xy ae be ae be

− −= + + + is a solution of 2

2– – 2 0.

d y dyy

dxdx=

*(vii) Verify that ( )2 2logy x x a= + + satisfies the diff. equation.

( )2

2 22

0.d y dy

a x xdxdx

+ + =

*(viii) Find the differential equation of family of all circles having centres on x-axis and radius2 units.

(ix) Find the differential equation of the family of curves y = ex (A cos x + B sin x), where

A and B are arbitrary constants.

(x) Find the differential equation of an ellipse with major and minor axes 2a and 2brespectively.

(xi) Find the differential equation corresponding to the family of curve ay2 = (x – c)3, wherec is an arbitrary constant.

(xii) By eliminating the constants ‘a’ and ‘b’ obtain the differential equation for which xy = aex

+ be–x + x2 is a solution.

*(xiii) Form the differential equation corresponding to the family of curves y = c(x – c)2.

(xiv) Form the differential equation representing the family of curves (y – b)2 = 4(x – a).

5. Solve the following diff. equations.

*(i) ( )– 2 1.dy

x ydx

+ = (ii) cot sin 2 .dy

y x xdx

+ =

(iii)2

2 log .dy

x y x xdx

+ = *(iv) (tan–1 y – x) dy = (1 + y2)dx

(v) ( ) ( )–12 tan

1 0.dy

y x e ydx

+ + − = (vi)1 sin

. cos , 0.dx x

y x xdy x x

+ = + >

164 XII – Maths

(vii)–

1 .ydx

x edy

+ = + (viii)3

cos cos sin .dy

x x xdx

+ =

(ix) y ey dx = (x3 + 2xey) dy.

6. Solve each of the following differential equations :

(i)2

2 .dy dy

y x ydx dx

− = +� �� (ii) cos y dx + (1 + 2e–x) sin y dy = 0.

(iii)2 2

1 1 0.x y dy y x dx− + − = (iv) ( ) ( )2 21 1 – 0.x y dy xy dx− + =

(v) (xy2 + x) dx + (yx2 + y) dy = 0; y(0) = 1.

(vi)3 3

sin cos .xdy

y x x xy edx

= +

(vii) tan x tan y dx + sec2 x sec2 y dy = 0

(viii) ( ) ( )tan 2 sin 3 2 sin 3 2 .dy

y x y x ydx

= + + −

(ix)2 – 2 2 2 2

.x y ydy

e x edx

+ −= +

(x) ( ) ( ) ( )2 21 1 0, 0 1.

x xe dy y e dx y+ + + = =

7. Solve the following differential equations :

(i) x2 y dx – (x3 + y3) dy = 0. (ii)2 2 2

.dy

x x xy ydx

= + +

(iii) ( ) ( )2 22 0, 1 1.x y dx xy dy y− + = =

*(iv) 1 1 0.

x x

y y xe dx e dy

y

� � � �+ = − =

� ��

(v) sin sin .x x

y dx x y dyy y

= −� � � ��

(vi) tan .dy y y

dx x x

= + � ��

(vii) 2 2

2dy xy

dx x y=

+(viii)

2.

x y ydye x e

dx

+= +

*(ix) ( )– log – log – 1 .dy

x y x ydx

= (x) 1 – .dy

x x y xydx

= − +

(xi)2

2sin .

xd yxe x x

dx= + (xii)

2

2

1 –.

1

dy y

dx x=

−

165 XII – Maths

8. (i) Form the differential equation of the family of circles touching y-axis at (0, 0).

*(ii) Form the differential equation of family of parabolas having vertex at (0, 0) and axisalong the (i) positive y-axis (ii) +ve x-axis.

(iii) Form the differential equation of the family of hyperbols centred at (0, 0) and axes alongthe coordinate axis.

*(iv) Form differential equation of all circles passing through origin and whose centre lie onx-axis.

(v) Form the differential equation of family of curves, y = a sin (bx + c), a, b, c are arbitraryconstants.

9. Show that the differential equation : ( ) ( )+ = −cos sin .x x

y x dy y dx x y dx x dyy y

is homogeneous and solve it.

10. Show that the differential equation 2

2

dy x y

dx x y

+=−

is homogeneous and solve it.

11. Show that the differential equation :

(x2 + 2xy – y2) dx + (y2 + 2xy – x2) dy = 0 is homogeneous and solve it.


(i) 2 cos 3 .dy

y xdx

− =

(ii)2

sin cos 2 sin cosdy

x y x x xdx

+ = given that y = 1 when .2

xπ=



(i) (x3 + y3) dx = (x2y + xy2)dy. (ii) 2 2– .x dy y dx x y dx= +

(iii) cos sin – sin cos 0.y y y y

y x y dx x y x dyx x x x

� � � � � � � �+ − =� � � ��

(iv) x2dy + y(x + y) dx = 0 given that y = 1 when x = 1.

(v) 0

y

x dyxe y x

dx− + = given that y = 0 when x = e.

(vi) (x3 – 3xy2) dx = (y3 – 3x2y)dy.

166 XII – Maths

CHAPTER 10–11

VECTORS AND THREEDIMENSIONAL GEOMETRY

POINTS TO REMEMBER

� Vector : A directed line segment represents a vector.

� Addition of vectors : If two vectors are taken as two sides of a triangle taken in order then theirsum is the vector represented by the third side of triangle taken in opposite order (triangle law).

� Multiple of a vector by a scalar : a��

is any vector and λ ∈ R then aλ��

is vector of magnitude

aλ��

in a direction parallel to a��

.

� If 0a ≠��

then a

a

��

�� is unit vector in direction .a��

� Scalar Product : . cosa b a b= θ��

where θ is the angle between a��

and .b��

� Projection of a��

along b��

is ..

a b

b

��

��

�

2.a a a=

��

� Vectors a��

and b��

are perpendicular iff . 0.a b =��

� Cross Product : �sina b a b n× = θ��

where �n is a unit vector perpendicular to a��

and

b��

, and θ is the angle between and .a b��

� Unit vector perpendicular to plane of anda b��

is .a b

a b

×±×

��

��

� Vector anda b��

are collinear if × =��

0 .a b

�

�

1 2 3

1 2 3

i j k

a b a a a

b b b

× =

� �

��

where �1 2 3 anda a i a j a k= + +

��

�1 2 3b b i b j b k= + +

��

167 XII – Maths

� Area of a triangle whose two sides are 1

and .2

a b a b= ×��

� Area of a parallelogram whose adjacent sides are and .a b a b= ×��

� If ,a b��

represents the two diagonals of a parallelogram, then area of parallelogram

1.

2a b= ×��

THREE DIMENSIONAL GEOMETRY

� Distance between P(x1, y1, z1) and Q(x2, y2, z2) is

( ) ( ) ( )= − + − + −�� 2 2 2

2 1 2 1 2 1 .P Q x x y y z z

� The coordinates of point R which divides line segment PQ where P(x1, y1, z1) and Q(x2, y2, z2)

in ratio m : n are 2 1 2 1 2 1, , .mx nx my ny mz nz

m n m n m n

+ + + � �� + + +

� If α, β, γ are the angles made by any line with coordinate axes respectively then l, m, n. Wherel = cos α, m = cos β, n = cos γ are called the, direction cosines of the line and l2 + m2 + n2 =1. If a, b, c are the direction ratios then direction cosines are

2 2 2 2 2 2 2 2 21 , , .

a b cm n

a b c a b c a b c= ± = ± = ±

+ + + + + +

� Direction ratios of a line joining (x1, y1, z1) and (x2, y2, z2) are x2 – x1 : y2 – y1 : z2 – z1.

� Vector equation of straight line :

(i) Through a point A ( )a��

and parallel to vector b��

is � .r a b= + λ��

(ii) Passing through two points ( )A a��

and ( )B b��

is ( )= + λ −��

.r a b a

(iii) Line passing through two given points (x1, y1, z1) and (x2, y2, z2) is

1 1 1

2 1 2 1 2 1

– – –,

– – –

x x y y z z

x x y y z z= = in cartesian form.

� Angle θ between two lines with DC’s l1, m1, n1 and l2, m2, n2 is given by

cos θ = l1 l2 + m1m2 + n1 n2

OR

with D.R’s a1b1c1 or a2, b2, c2

1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

cos .a a b b c c

a b c a b c

+ +θ =

+ + + +

168 XII – Maths

If lines are 1 1 2 2and .r a b r a µ b= + λ = +��

then, 1 2

1 2

.cos .

b b

b bθ =

��

��

� Equation of plane :

(i) Passing through ( )A a��

and perpendicular to ( )n��

is ( )– . 0r a n =��

Or

.r n d=��

where =��

. .a n d

(ii) Passing through three given points is

1 1 1

2 1 2 1 2 1

3 1 3 1 3 1

– – –

– – – 0.

– – –

x x y y z z

x x y y z z

x x y y z z

=

(iii) Having intercepts a, b, c on coordinate axes is 1.x y z

a b c+ + =

� Angle between two planes 1 1 2 2. and .r n d r n d= =��

is 1 2

1 2

.cos .

n n

n nθ =

��

��

� Distance of a point (x1, y1, z1) from a plane ax + by + cz + d = 0 is 1 1 1

2 2 2.

ax by cz d

a b c

+ + +

+ +

� Equation of plane passing through intersection of two planes a1x + b1y + c1z + d1 = 0 and a2x+ b2y + c2z + d2 = 0 is (a1x + b1y + c1z + d1) + λ(a2x + b2y + c2z + d2) = 0.

� Equation of plane passing through intersection of two planes 1 1 2 2. and .r n d r n d= =��

is ( )1 2 1 2. .r n n d d+ λ = + λ��

� Angle between a plane .r n d=��

and a line = + λ θ =��

��

��

.is sin .

m nr a m

m n


*1. What is the horizontal and vertical components of a vector a��

of magnitude 5 making an angle

of 150° with the direction of x-axis.

2. What is a ∈ R such that 1,a x =��

where �– 2 2 ?x i j k= +� �

3. Write when .x y x y+ = +��

4. What is the area of a parallelogram whose sides are given by �2 – and 5 ?i j i k+� � �

*5. What is the angle between and ,a b��

If . 3 and 3 3.a b a b= × =��

169 XII – Maths

6. If A is the point (4, 5) and vector AB��

has components 2 and 6 along x-axis and y-axisrespectively then write point B.

7. What is the point of trisection of PQ nearer to P if position of P and Q are �3 3 – 4i j k+� � and�9 8 – 10 .i j k+� �

8. What is the vector in the direction of �2 3 2 3 ,i j k+ +� � whose magnitude is 10 units?

*9. What are the direction cosines of a vector equiangular with co-ordinate axes?

10. What are the angles which �3 – 6 2i j k+� � makes with coordinate axes.

11. Write a unit vector perpendicular to both the vectors � �3 – 2 and – 2 – 2 .i j k i j k+ +� � � �

12. What is the projection of the vector –i j� � on the vector ?i j+� �

13. If 2, 2 3a b= =��

and ,a b⊥��

what is the value of ?a b+��

14. For what value of λ, �4a i j k= λ + +��

� � is perpendicular to �2 6 3 ?b i j k= + +��

� �

15. What is ( ) ( ), if . – 3a a b a b+ =��

and 2 ?b a=��

16. What is the angle between a��

and b��

, if – ?a b a b= +��

*17. In a parallelogram ABCD, �2 4AB i j k= − +��

� � and �4 .AC i j k= + +��

� � What is the length of

side BC?

18. What is the area of a parallelogram whose diagonals are given by vectors �2 2i j k+ −� � and�− +� 2 ?i k

19. Find x��

if for a unit vector � �( ) �( ), – . 12a x a x a+ =��

.

20. If 2 2 2

,a b a b+ = +��

then what is the angle between a��

and .b��

21. If a��

and b��

are two unit vectors and a b+��

is also a unit vector then what is the angle

between a��

and ��

?b

22. If �, ,i j k� � are the usual three mutually perpendicular unit vectors then what is the value of�( ) �( ) ( ). . . ?i j k j i k k j i× + × + ×

��

23. What is the angle between x��

and y��

if . ?x y x y= ×��

24. Write a unit vector in xy-plane, making an angle of 30° with the +ve direction of x–axis.

*25. Two adjacent sides of a parallelogram are � �− + − −� � � �2 4 5 and 2 3 .i j k i j k Find a unit vector

parallel to the diagonal, which is coinitial with and .a b��

170 XII – Maths

26. If , anda b c��

are unit vectors with 0 ,a b c+ + =��

then what is the value of

. . . ?a b b c c a+ +��

*27. If – 1,a b a b= = =��

then what is the value of ?a b+��

*28. For any vector ,a��

what is the value of ( ) ( ) � �( )× + × + ×��

� � � �. . . .i a j j a i k a k

29. If a��

and b��

are unit vectors such that ( )2a b+��

is perpendicular to ( )5 4 ,a b−��

then

what is the angle between a��

and b��

?

30. Write a unit vector which makes an angle of 4

π with x–axis and

3

π with z-axis and an acute

angle with y-axis.

31. What is the ratio in which xy plane divides the line segment joining the points (–1, 3, 4) and(2, –5, 6)?

32. If x coordinate of the point P on the join of Q(2, 2, 1) and R(–5, 1, –2) is 4, then in what ratioP divides QR.

33. What is the distance of a point P(a, b, c) from x-axis?

34. Write the equation of a line passing through (1, –1, 2) and perpendicular to plane 2x – 3y + 4z= 7.

35. What is the angle between the lines 2x = 3y = – z and 6x = – y = – 4z?

*36. If a line makes angle α, β, and γ with co-ordinate axes, then what is the value of

sin2 α + sin2 β + sin2 γ?

37. What is the perpendicular distance of plane 2x – y + 3z = 10 from origin?

38. What is the y-intercept of the plane x – 5y + 7z = 10?

39. Write the value of λ, so that the lines given below are perpendicular to each other

1 – 2 1 – 1 2 – 5 3and .

3 4 4 2 5

x y z x y z− − −= = = =λ

40. A (3, 2, 0), B(5, 3, 2) and C(5, 8, – 10) are the vertices of ∆ABC. D and E are mid points ofAB and AC respectively. What are the direction cosines of DE?

41. What is the equation of the line, which passes through the point (–2, 4, –5) and parallel to3 – 4 8

?5 5 –6

x y z+ += =

*42. What is the equation of a line passing through point (–1, 2, 3) and equally inclined to the axes?

*43. The foot of perpendicular from point (1, 6, 3) on line – 1 2

1 2 3

x y z −= = is (1, 3, q); what is

the value of q?

171 XII – Maths

44. What is the angle between the straight lines :

1 2 3 1 2 3, ?

2 2 4 1 2 –3

x y z x y z+ − + − + −= = = =

45. If the direction ratios of a line are proportional to 1, –3, 2 then what are the direction cosinesof the line?

46. If a line makes angles and 2 4

π π with x-axis and y-axis respectively then what is the acute angle

made by the line with z axis?

47. What is the acute angle between the planes 2x + 2y – z + 2 = 0 and 4x + 4y – 2z + 5 = 0?

48. What is the distance between the planes 2x + 2y – z + 2 = 0 and 4x + 4y – 2z + 5 = 0.

49. What is the equation of the plane which cuts off equal intercepts of unit length on the coordinateaxes.

50. Are the planes x + y – 2z + 4 = 0 and 3x + 3y – 6z + 5 = 0 intersecting?

51. What is the equation of the plane through the point (1, 4, – 2) and parallel to the plane– 2x + y – 3z = 7?

52. Write the vector equation of the plane which is at a distance of 8 units from the origin and is

normal to the vector �( )2 2 .i j k+ +� �

53. What is equation of the plane if the foot of perpendicular from origin to this plane is (2, 3, 4)?

54. What is the angle between the line 1 2 1 2 –

3 4 –4

x y z+ −= = and the plane 2x + y – 2z +

4 = 0?

55. If O is origin OP = 3 with direction ratios proportional to –1, 2, – 2 then what are the coordinatesof P?

56. What is the distance between the line � �( )2 – 2 3 4r i j k i j k= + + λ + +��

� � � � from the plane�( ). – 5 – 5 0.r i j k+ + =

��

*57. What is the distance between the line �( )– 2 –r i j i j k= + λ +��

� � � � from the plane�( ). 3.r i j k+ + =

��


58. If ABCDEF is a regular hexagon then using triangle law of addition prove that :

3 6AB AC AD AE AF AD AO+ + + + = =��

O being the centre of hexagon.

172 XII – Maths

59. Points L, M, N divides the sides BC, CA, AB of a ∆ABC in the ratios 1 : 4, 3 : 2, 3 : 7

respectively. Prove that AL BM CN+ +��

is a vector parallel to CK��

where K divides ABin ratio 1 : 3.

60. If PQR and P´Q´R´ are two triangles and G, G´ are their centroids, then prove that

´ ´ ´ 3 ´ .PP QQ RR GG+ + =��

61. PQRS is parallelogram. L and M are mid points of QR and RS. Express PL��

and PM��

in terms

of PQ��

and .PS��

Also prove that 3

.2

PL PM PR+ =��

62. For any two vectors, show that .a b a b+ ≤ +��

*63. Evaluate ( ) ( ) �( )2 2 2.a i a j a k+ + × + ×

��

64. If �a and �b are unit vector inclined at an angle θ than prove that :

(i) � �1sin – .

2 2a b

θ = (ii) � �

� �

–tan .

2

a b

a b

θ =+

65. The scalar product of vector �i j k+ +� � with a unit vector along the sum of the vector

� �2 4 – 5 2 3i j k and i j k+ λ + +� � � � is equal to 1. Find the value of λ.

66. , anda b c��

are three mutually perpendicular vectors of equal magnitude. Show that

+a b c+��

makes equal angles with , anda b c��

with each angle as –1 1cos .

3

� ��

67. For any two vectors, show that ( )22 2– . .a b a b a b× =

��

68. If �3 – and 2 – 3i j b i j kα = = +��

� � � � then express β��

in the form of 1 2,β = β + β��

where

1β��

is parallel to 2andα β��

is perpendicular to .α��

69. If , ,a b c��

are three vectors such that 0a b c+ + =��

then prove that a b× =��

.b c c a× = ×��

70. If 3, 5, 7 and 0 ,a b c a b c= = = + + =��

find the angle between and .a b��

*71. � ( ) �, 2 and – 2 – .a i j k b i j j c x i x j k= + + = − + = +��

� � � � � � � If c��

lies in the plane of

and ,a b��

then find the value of x.

72. Let � � �, 3 – and 7 – ,a i j b j k c i k= − = =��

� � � find a vector ��

d which is perpendicular to

and and . 1.a b c d =��

173 XII – Maths

73. Prove that the normal vector to the plane containing three points with position vectors

, anda b c��

lies in the direction of vector .b c c a a b× + × + ×��

74. If , ,a b c��

are position vectors of the vertices A, B, C of a triangle ABC then show that the

area of ∆ABC is 1

.2

a b b c c a× + × + ×��

75. If , ,a b c��

are three vectors forming a DABC then prove that area of 1

2ABC a b∆ = ×

��

1 1.

2 2b c c a= × = ×��

76. If � �, –a i j k c j k= + + =��

� � � are the given vectors then find a vector b��

satisfying theequation.

, . 3.a b c a b× = =��

77. Let , 10 2 and and OA a OB a b OC b O= = + =��

is the origin. Let S denotes the

area of the quadrilateral OABC and T denotes the area of parallelogram with OA and OC asadjacent sides. Prove that S = 6T.

78. Find a unit vector perpendicular to plane ABC where. Position vector A, B, C are �3 – 2 ,i j k+� �

� �− − − +� � � �3 and 4 3i j k i j k respectively.

*79. Find the values of a for which the vector ( ) ( ) �2 24 2 – 9r a i j a k= − + +

�� makes acute

angle with coordinate axes.

*80. Let � � �a b c be unit vectors such that � � � �. . 0a b a c= = and the angle between � �and b c is ,6

π

then prove that � � �( )= ± 2 .a b c×

81. Prove that angle between any two diagonals of cube is cos–1(1/3).

82. Find the image of the point (3, – 2, 1) in the plane 3x – y + 4z = 2.

83. The line – 4 2 4

1 2 2

x y k z− −= =−

lies exactly in the plane 2x – 4y + z = 7. Find the value of K.

84. A l1, m1, n1, and l2, m2, n2 are direction casines of two mutually perpendicular lines, show thatthe direction cosines of line perpendicular to both of them are

m1n2 – n1m2, n1l2 – l1n2, l1m2 – m1l2.

85. Find vector and cartesian equation of a line passing through a point with position vectors�2 –i j k+� � and which is parallel to the line joining the points with position vectors�– 4 and 2i j k i j+ + +� � � � �2 .k+

174 XII – Maths

*86. The certesian equations of a line is 6x – 2 = 3y + 1 = 2z – 2, find direction ratios of the line.Also find cartesian and vector equations of a line parallel to this line and passing through(2, – 1, – 1).

87. Find image (Reflection) of the point (7, 4, – 3) in the line – 1 2

.1 2 3

x y z −= =

88. Find equations of a plane passing through the points (2, –1, 0) and (3, –4, 5) and parallel tothe line 2x = 3y = 4z.

89. Find distance of the point (– 1, – 5, – 10) from the point of intersection of line

− + −= =2 1 2

3 4 12

x y z and the plane x – y + z = 5.

90. Find equation of the plane passing through the point (2, 3, – 4) and (1, –1, 3) and parallel tothe x–axis.

91. Find equation of the plane which bisects the line joining the points (– 1, 2, 3) and (3, – 5, 6)at right angle.

92. What is the equations of plane : ( ) �( ) �( )2 3r i j i j k u i j k= − + λ + + + − +��

� � � � � � in scalarproduct.

93. Find the distance of the point (1, –2, 3) from the plane x – y + z = 5, measured parallel to the

line = =−

.2 3 6

x y z

94. Find the equation of the plane passing through the intersection of two plane 3x – 4y + 5z = 10,2x + 2y – 3z = 4 and parallel to the line x = 2y = 3z.

95. Show that the lines x = ay + b, z = cy + d and x = a´y + b´, z = c´y + d´ are perpendicular ifaa´ + cc´ + 1 = 0.

96. Find the equations of the planes parallel to the x – 2y + 2z – 3 = 0 whose perpendicular distancefrom the point (1, 2, 3) is 1 unit.

97. A variable plane moves in such a way that the sum of the reciprocals of it’s intercepts on thethree coordinate axis is constant. Show that the plane passes through a fixed point.

98. Find equation of the plane passing through the point (3, 4, 2) and (7, 0, 6) and is perpendicularto the plans 2x – 5y = 15.

99. Find cartesian as well as vector equation of the plane through the intersection of the plane

( ) �( )2 6 12 0 and . 3 4 0r i j r i j k= + + = − + =��

� � � � which is at a unit distance from origin.

100. Find equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and whichcontain the line of intersection of the plane x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

101. Find equation of the plane containing the points (0, – 1, – 1) (– 4, 4, 4), (4, 5, 1). Also showthat (3, 9, 4) lies on the required plane.

175 XII – Maths


102. The vector equations of two lines are :

� �( ) � �( )2 – 2 and 2 2 2r i j k i j k r i j k i j k= + + + λ + = − − + +��

� � � � � � � � . Find the shortestdistance between them.

103. Check the coplanarity of lines

�( ) �( )–3 5 –3 5r i j k i j k= + + + λ + +��

� � � �

�( ) �( )– 2 5 – 2 5r i j k µ i j k= + + + + +��

� � � �

If they are coplanar, find equation of the plane containing the lines.

104. Find shortest distance between the lines :

8 19 10 15 29 5and

3 16 7 3 8 5

x y z x y z− + − − − −= = = =− −

105. Show that the lines 1 2 3 4 1

and2 3 4 5 2

x y z x yz

− − − − −= = = = intersect. Also find

the point of intersection.

106. Find shortest distance between the lines whose vector equations are :

( ) ( ) ( ) � ( ) ( ) ( ) �= − + − + − = + + − + +��

� � � �1 2 3 2 and 1 2 1 2 1r t i t j t k r s i s j s k

107. Find the equations of the two lines through the origin such that each line is intersecting the line

3 3

2 1 1

x y z− −= = at an angle of .3

π

108. A plane passes through (1, – 2, 1) and is perpendicular to the planes 2x – 2y + 2z = 0 and x– y + 2z = 4. Find the distance of that plane from origin.

109. Find the equation of the plane passing through the intersection of planes 2x + 3y – z = – 1 andx + y – 2z + 3 = 0 and perpendicular to the plane 3x – y – 2z = 4. Also find the inclination ofthis plane with xy–plane.

110. Find the shortest distance and the vector equation of line of shortest distance between the linesgiven by

�( ) �( )3 8 3 3r i j k i j k= + + + λ − +��

� � � �

�( ) �( )–3 7 6 –3 2 4r i j k µ i j k= − + + − +��

� � � �

111. Show that the lines joining the points (7, 0, 6) and (2, 5, 1) intersects the line joining the points(2, 2, – 1) (3, 4, 2). Also find the point of intersection.

112. Find the equations of two planes through the points (4, 2, 1) and (2, 1, – 1) and making an angle

of 4

π with the plane x – 4y + z – 9 = 0,

176 XII – Maths

113. A variable plane is at a constant distance 3p from the origin and meet the coordinate axes inA, B, C. Show that the locus of centroid of ∆ABC is x–2 + y–2 + z–2 = p–2.

114. A vector n��

of magnitude 8 units inclined to x–axis at 45°, y axis at 60° and an acute angle

with z-axis. If a plane passes through a point ( )2, –1,1 and is normal to n��

, find its equationin vector form.

115. Find the foot of perpendicular from the point �2 5i j k− +� � on the line �( )11 2 8r i j k= − −��

� �

�( )10 4 11i j k+ λ − −� � . Also find the length of the perpendicular.

116. Three vectors of magnitude a, 2a, 3a meet in a point and their directions are along the diagonalsof the adjacent faces of a cube, determine their resultant.

117. A line makes angles α, β, γ, δ with the four diagonals of a cube. Prove that

2 2 2 2 4cos cos cos cos .

3α + β + γ + δ =

177 XII – Maths

CHAPTER 12

LINEAR PROGRAMMING


1. A man has Rs. 1500 to purchase two types of shares of two different companies S1 and S2.Market price of one share S1 is Rs 180 and S2 is Rs. 120. He wishes to purchase a maximumto ten shares only. If one share of type S1 gives a yield of Rs. 11 and of type S2 Rs. 8 thenhow much shares of each type must be purchased to get maximum profit? And what will be themaximum profit?

2. Daily wages of two tailors A and B is respectively Rs. 150 and Rs. 200. ‘A’ can stitch 6 shirtsand 4 pairs of trousers per day, while B can stitch 10 shirts and 4 pairs of trousers per day.How many days should each of them work if it desired to produce at least 60 shirts and 32 pairsof trousers at a minimum labour cost?

3. A dealer wishes to purchase two types of dinner sets, type I and II. He has only Rs. 5760 toinvest and has space for at most 20 dinner sets. Cost of type A and B dinner sets is Rs. 360and Rs. 240 respectively. He earns a profit of Rs. 44 and Rs. 36 respectively on type I and IItype of dinner sets respectively. Assuming that he can sell all the dinner sets he buys, howshould he invest the money in order to maximize the profit? Find maximum profit also.

4. A company manufacture two types of lamps say A and B. Both lamps go through a cutter andthen a finisher. Lamp A requires 2 hours of the cutter’s time and 1 hours of the finisher’s time.

Lamp B requires 1 hour of cutter’s and 2 hours of finisher’s time. The cutter has 100 hours andfinishers has 80 hours of time available each month. Profit on one lamp A is Rs. 7.00 and onone lamp B is Rs. 13.00. Assuming that he can sell all that he produces, how many of each typeof lamps should be manufactured to obtain maximum profit?

5. A catering agency has two kitchens to prepare food for two places A and B. From these placesmidday meal is to be supplied to three different schools situated at P, Q and R. The monthlyrequirement of the schools are respectively 40, 40 and 50 food packets. A packet contains lunchfor 1000 students. Preparing capacity of kitchen A and B are 60 and 70 packets per monthrespectively. The transportation cost per packet for kitchens to schools is given below :

Transportation Cost per Packet (in rupees)

To From

A B

P 5 4

Q 4 2

R 3 5

How many packets from each kitchen should be transported to schools so that the cost oftransportation is minimum. Also find the minimum cost. Convert the above problem to an L.P.P.and solve graphically.

178 XII – Maths

6. A farmer has a supply of chemical fertilizer of type I which contains 10% nitrogen and 6%phosphoric acid and type II contains 5% nitrogen and 10% phosphoric acid. After testing the soilconditions of a field, it is found that at least 14 kg of nitrogen and 14kg of phosphoric acid isrequired for a good crop. The fertilizer I costs Rs. 2 per kg and type II costs Rs. 3 per kg. Howmany kilograms of each fertilizer should be used to meet the requirement at the minimum cost.Solve graphically as L.P.P.

7. An airline agree to charter planes for a group. The group needs at least 160 first class seatsand at least 300 tourist class seats. The airline must use at least two of its model 314 planeswhich have 20 first class and 30 tourist class seats. The airline will also use some of its model535 planes which have 30 first class seats and 60 tourist class seats. Each flight of a model 314plane costs the company Rs. 1 lakh, and each flight of a model 535 plane costs Rs. 1.5 lakh.How many of each type of planes should be used to minimise the flight cost? Solve this as aLPP.

8. A firm manufactures two type of products A and B, each of which must be processed on twomachines M1 and M2. The hourly requirements per unit for each product in each machine, theweekly capacity of each machine, selling price per unit, labour cost per unit, and raw materialcost per unit are as follows :

Product A Product B Weekly capacity

Machine M1 3 hrs. 2 hrs. 130 hrs.

Machine M2 4 hrs. 6 hrs. 260 hrs.

Selling price per unit Rs. 25 Rs. 30

Labour cost per unit Rs. 16 Rs. 20

Raw material cost Rs. 4 Rs. 4 per unit.

How many units of each type should be produced to maximise profit?

9. Kellogg is a new cereal formed by a mixture of bran and rice that contains at least 88 gramsof protein at least 56 milligrams at iron. Knowing that bran contains 80 grams of protein and 40milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligramsof iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 perkg and rice costs Rs. 4 per kg. Solve graphically as L.P.P.

10. A dietician mixes together two kinds of foods in such a way that the mixture contains at least6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C, and 9 units of vitamin D. Thevitamin contents of 1 kg of food X and 1kg of food Y are given below :

Vitamin A Vitamin B Vitamin C Vitamin D

Food X 1 1 1 2

Food Y 2 1 3 1

One kg of food X cost Rs. 5, whereas one kg of food Y costs Rs. 8. Find the least cost of mixturewhich will produce the desired diet, graphically.

11. If a young man drives his motorcycle at 25 km per hour, he has to spend Rs. 2 per kilometreon petrol. If he drives it at a faster speed of 40 km per hour, the petrol cost increases to Rs.5 per kilometre. He has Rs. 100 to spend on petrol and wishes to find the maximum distancehe can travel within one hour. Express this as linear programming problem and then solve it.

179 XII – Maths

12. Mr. Jain wants to invest Rs. 1,00,000 in Public Provident Fund (PPF) and in National Bonds. Hehas to invest at least Rs. 40,000 in P.P.F. and at least Rs. 20,000 in bonds. If rate of intereston PPF is 9% per annum and that on bonds is 6% per annum, how should he invest the moneyto earn maximum annual income? Also find the maximum annual income.

13. A company has factories located at each of two places P and Q. From these locations a certaincommodity is delivered to each of three depots situated at A, B and C. The weekly requirementsof the depots are respectively 8, 8 and 9 units of the commodity while the weekly productioncapacities of the factories at P and Q are respectively 10 and 15 units. The cost of transportationfrom P to A, B, C is Rs. 10, Rs. 12, Rs. 15 respectively per unit and from Q to A, B, C is Rs.15, Rs. 13, Rs. 10 respectively per unit. How many units should be transported from eachfactory to each depot in order that the transportation cost is minimum. Formulate the above LPPmathematically and then solve it.

14. Two godowns X and Y have a grain storage capacity of 100 quintals and 50 quintals respectively.Their supply goes to three ration shops A, B and C whose requirements are 60, 50 and 40quintals respectively. The costs of transportation per quintal from the godowns to the shops aregiven in the following table :

Cost of transportation (in Rs. per quintal)

To Form X Y

A 6.00 4.00

B 3.00 2.00

C 2.50 3.00

How should the supplies be transported in order that the transportation cost in minimum?

15. Find maximum and minimum values of z = 2x + y subject to the constraints :

x ≥ 0, y ≥ 0

x + 3y ≥ 6

x – 3y ≤ 3

3x + 4y ≤ 24

– 3x + 2y ≤ 6

5x + y ≥ 5

16. Solve the following LPP problem graphically :

Maximize and minimize z = 3x + 5y

Subject : 3x – 4y + 12 ≥ 0

2x – 4y + 2 ≥ 0

2x – 3y – 12 ≥ 0

0 ≤ x ≤ 4

y ≥ 2

180 XII – Maths

17. (i) Maximize z = 3x, + 3x2. If possible graphically subject to constraints :

x1 – x2 ≤ – 1

– x1 + x2 ≤ 0

x1 ≥ 0, x2 ≥ 0

(ii) Maximize z = 3x + 3y, if possible graphically subject to constraints :

x – y ≤ 1

x + y ≤ 3

x, y ≥ 0

18. A farmer owns a field of area 1000m2. He wants to plant fruit trees in it. He has sum of Rs. 2400to purchase young trees. He has the choice of two types of trees. Type A requires 10m2 ofground per tree and cost Rs. 30 per tree and type B requires 20m2 of ground per tree and costsRs. 40 per tree. When full grown a type – A tree produces an average of 20 kg of fruit whichcan be sold at a profit of Rs. 12 per kg and a type – B tree produces an average of 35 kg offruit which can be sold at a profit of Rs. 10 per kg. How many of each type should be plantedto achieve maximum profit when trees are fully grown? What is the maximum profit?

19. Find the maximum and minimum values of 5x + 2y subject to constraints :

x ≥ 0, y ≥ 0

–2x – 3y ≤ – 6

x – 2y ≤ 2

6x + 4y ≤ 24

–3x + 2y ≤ 3

20. A company produces two types of belts, A and B. Profits on these belts are Rs. 2 and Rs. 1.5each belt, respectively. A belt of type A requires twice as much time as belt of type B. Thecompany can produce at the most 1000 belts of type B perday. Material for 800 belts per dayin available. At the most 400 buckles for belts of type A and 700 for those of type B are availableper day. How many belts of each type should the company produce so as to maximize theprofit?

181 XII – Maths

CHAPTER 13

PROBABILITY

POINTS TO REMEMBER

� Conditional Probability : If event B has already occurred then probability of event

( )( )

.A P A B

A PB P B

∩ = =� ��

� Multiplication Rule of Probability :

(i) ( ) ( ) ( ). . .A B

P A B P B P P A PB A

∩ = =� � � ��

(ii) ( ) ( ) . . .B C

P A B C P A P PA AB

∩ ∩ = � � � ��

� If (i) A and B are independent events then P (A ∩ B) = P(A) . P(B)

(ii) A, B and C are independent events then.

P (A ∩ B) = P(A).P(B), P(B ∩ C) = P(B).P(C ∩ A) = P(C).P(A).

P(A ∩ B ∩ C) = P(A).P(B).P(C)

� If A and B are Independent the

(i) A and cB / are independent

(ii)c

A / and B are independent

(iii) andc c

A B/ / are independent.

� Baye’s Theorem. If E1, E2 _ _ _ _ _ _ , En are mutually exclusive and exhaustive events andA be any event on sample space S, s.t P(A) ≠ 0, If A has already occurred then

( )

( )=

� �� =� ��

� �� ∑

1

.

.

iin

iii

AP Ei P

EEP

A AP E P

E

i = 1, 2, _ _ _ _ _ _ n

182 XII – Maths

� Probability distribution of a random variable. Let a random variable x assume x1, x2, _ _ _ _ _xn with corresponding probabilities p1, p2, _ _ _ _ pn. Then different values of a random variabletogether with their corresponding probabilities form a probability distribution.

Mean of probability distribution, 1

1

n

i i

in

i

i

p x

µ

p

=

=

=∑

∑Variance σ2 = Σ(xi – µ)2 pi

OR

σ2 = Σpixi2 — µ2.

VERY SHORT ANSWER TYPE Q QUESTIONS (1 Mark Each)

1. If ( )4 9, ,

9 13

AP P B

B

= =� �� write value of P(A ∩ B).

2. A fair dice is tossed and if events E = {2, 4, 6}, F {2, 3}, then write the value of

(i) E

PF

� ��

(ii) .F

PE

� ��

3. If A and B are independent events such that ( ) ( )1 1, ,

4 13P A P B= = write the value of

(i) P(A ∩ B´) (ii) P(A´ ∩ B).

*4. If A and B are independent events and if ( ) 4,

5P A B∩ = write the value of P(A´ ∪ B´).

5. If ( ) ( )1, 0,

2P A P B= = then write the value of P(A/B). (If exist).

6. A problem in mathematics is given to two students whose chances of solving it are 1

and2 3

1

respectively. What is the probability that both students fail to solve it?

7. The probability of student A passing an examination is 3

5 and student B passing is

4.

5 Find the

probability that only A passes the examination.

*8. The probability distribution of a random variable X is given as :

X 0 1 2

P(X)144

169

24

169

1

169

183 XII – Maths

Write the value of mean of this probability distribution.

9. The probability distribution of X is :

X 0 1 2 3

P(X) 0.2 k k 2k

Write the value of k and P(X = 3).

*10. If X has a Binomial distribution 1

4,3

B � ��

write :

(i) P(X = 3) (ii) P(X = 1)

11. Find the mean of the binomial distribution 1

2, .2

B � ��

12. Find P(X = 0) of the Binomial distribution 1

10, .3

B � ��

13. Find P(X = 1) of the Binomial distribution 1

3, .6

B � ��

14. A dice is thrown 3 times. If getting a ‘six’ is considered a success, find the probability of 3successes.

15. What is the probability of the die coming up with a number less than 5?

16. In a probability distribution of X, what is the sum of all the probabilities.

17. What is the probability that a leap year will have 53 Sundays?

18. In a single throw of three dice, determine the probability of getting a total of 5.

19. A police-man fires four bullets on a dacoit. The probability that the dacoit will be killed by onebullet is 0.6. What is the probability that the dacoit is still alive?

*20. Find the expectation of the number of heads in 15 tosses of a coin.

SHORT ANSWER TYPE QUESTIONS (4 Mark Each)

21. Assuming a family has two children :

(i) Write sample space.

(ii) What is the probability that both the children are boys given that at least one of themis boy.

(iii) What is the probability that both the children are girls given that at least one of them is girl.

(iv) What is the probability that second is not a boy given that first is a boy.

184 XII – Maths

22. Fifteen horses numbered 1 to 15 participated in Race. If it is known that the winner horse is oddnumbered.

(i) What is probability that the number of winner horse is more that 10.

(ii) What is the probability that the number is prime.

23. A die is thrown twice and both numbers observed to be odd. Find the probability that :

(i) Both are same.

(ii) Their sum is 4.

(iii) Their sum is more than 10.

24. Three coins are tossed once. If :

E : Head appears on first coin.

A : Exactly two tails appear.

B : Exactly one tail appears.

C : No tail appears.

Find : , , .A B C

P P PE E E

� � � � � ��

25. An anti aircraft gun can take a maximum of four shots at an enemy plane moving away fromit. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2and 0.1 respectively. What is the probability that gun strikes the plane?

26. A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls. Onebag is selected at random and form the selected bag one ball is drawn. Find the probability thatthe ball drawn is red.

27. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards aredrawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

28. Suppose that reliability of a HIV test is specified as follows :

Of people having HIV, 90% of the test detected but 10% go undetected. Of people free of HIV,99% of the test are judged HIV negative but 1% are diagnosed as showing HIV positive. Froma large population of which only 0.1% have HIV, one person is selected at random, given theHIV test, and the pathologist reports him/her as HIV negative. What is the probability that theperson is actually has HIV.

29. A candidate has to reach the examination centre in time. Probability of his going by bus or

scooter or by other means of transport is 3 1 3

, ,10 10 5

respectively. The probability that he will be

late will be 1 1

and4 3

respectively, if he travels by bus or scooter. But he reaches in time if he

uses any other mode of transport. He reached late at the centre. Find the probability that hetravelled by bus.

185 XII – Maths

30. Find the probability distribution of the number of green balls drawn when 3 balls are drawn oneby one without replacement from a bag containing 3 green and 5 white balls.

31. Three cards are drawn successively with replacement from a well shuffled deck of 52 cards. Arandom variable X denote the number of cards of heart in three cards drawn. Determine theprobability distribution of X.

*32. Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in single die is considereda success, find the probability of at most three successes.

*33. The probability of a shooter hitting a target is 3

.4

How many minimum number of times must

he/she should fire so that the probability of hitting the target at least once is more than 0.99?

34. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fairdie is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Findthe expected value of the amount he wins/loses.

35. If a fair coin is tossed 10 times, find the probability of

(i) at least six heads.

(ii) at most six heads.

36. On a multiple choice examination with three possible answers for each of the five questions.What is the probability that a candidate would get four or more correct answers just by guessing?

*37. Find the probability distribution of number of doublets in three throws of a pair of dice.

38. The sum and the product of the mean and variance of a binomial distribution are 24 and 128respectively. Find the distribution.

LONG ANSWER TYPE QUESTIONS (6 Mark Each)

39. In an office 50% females likes to wear suits, 20% saree and 30% jeans. If 10% likes to wearboth suit and saree, 15% wear suit and jeans, 6% wear saree and jeans and only 3% who likesto wear all the three. Then find whether the following events are independent.

(i) A and B (ii) B and C

(iii) and A (iv) A, B and C.

Where A : female wear suit

B : female wear saree

C : Female wear jeans

40. In a colony their are 60 persons who have cars or motorbike or scooter. 28 persons have cars,28 has motorbike, 25 has scooter. If 8 person have cars and bike, 9 have bike and scooter, 7have cars and scooter. 3 of them have all the three vehicles, and

If C : Persons having car.

186 XII – Maths

M : Persons having motorbike.

S : Persons having scooter.

Find if the following events are independent.

(i) C and M (ii) M and S; (iii) C and S; (iv) C, M and S.

41. In a competition students of a school of class VIII painted pictures using Red, Blue colouraccording as the data shown below :

Painting Only in Red Only in Blue Red and Blue both

VIII A 12 3 15

VIII B 15 5 20

VIII C 10 2 13

If the painting that was awarded was having blue colour in it, find the probability that :

(i) It was from VIII B (ii) It was from VIII A (iii) It was from VIII C

42. In attempting to solve a question, probability that the student knows the answer is 3

10, copies

is 1

,5

guesses is 1

,10

and do not attempt is 2

.5

The probability that his answer is correct given

that he copied is 1

4 and the probability of giving correct answer by guess is

1.

5 If it is given

that his answer is correct find the probability that was done by guessing.

43. Probability of attempting solving a problem by A, B, C is 1 1 1

, ,2 3 6

respectively. And probability

that A, B, C will solve it correctly is 50%, 60% and 80% respectively. If the problem is solvedcorrectly, what is the probability that it was solved by (i) A (ii) B (iii) C.

44. From a well shuffled pack of playing cards, three cards are drawn without replacement.

(i) Find the probability distribution of the number of kings.

(ii) Find its mean and variance.

45. From a well shuffled pack of playing cards, five cards are drawn one by one with replacement.

(i) Find the probability distribution of the number of Aces.

(ii) Find its mean and variance.

46. Find the probability distribution of number of sixes while throwing a dice 5 times. Also find itsmean and variance.

47. Three tailors A, B and C stitched 500, 300 and 200 shirts respectively. The probability ofstitching a defective shirt by them is 0.01%, 0.15% and 0.2% respectively. If a shirt stitched isgiven to be defective, find the probability that it was stitched by (i) A (ii) B (iii) A or C (iv) B or C.

187 XII – Maths

48. Suppose 15% of men and 36% of women have grey hair. If probability of dying hair by men is21% and by women is 63%. A dyed hair person is selected at random, what is the probabilitythat this person is a (i) Male (ii) Female ?

49. A random variable X has the following probability distribution.

X 1 2 3 4 5 6

P(X) O k 2k k2 2k2 6k2

Determine (i) k (ii) P(X < 5) (iii) P(X > 3).

Find also mean and variance of the distribution.

50. An unbiased coin is tossed six times. What is the probability of getting.

(i) at most ‘3’ heads

(ii) at least 2 heads

(iii) Also find mean and variance of number of heads.

51. A pair of dice is thrown 7 times. If getting a total of 7 is considered a success, what is theprobability of

(i) no success; (ii) 6 successes;

(iii) at least 6 successes (iv) at most 6 successes.

188 XII – Maths

CBSE QUESTIONS PAPER – 2008

MATHEMATICS

xf.krTime allowed : 3 hours Maximum marks : 100

fu/kkZfjr le; % 3 ?k.Vs vf/kdre vad % 100

General Instructions

1. All question are compulsory.

2. The question paper consists of 29 questions divided into three sections A, B abd C. Section Acomprises of 10 questions of one mark each, Section B comprises of 12 questions of four markseach and Section C comprises of 7 questions of six marks each.

3. All questions in Section A are to be answered in one word, one sentence or as per the exactrequirement of the question.

4. There is no overall choice. However, internal choice has been provided in 4 questions of fourmarks each and 2 questions of six marks each. You have to attempt only one of the alternativesin all such questions.

5. Use of calculators is not permitted.

lkekU; funsZ'k %

1. lHkh iz'u vfuok;Z gSaA

2. bl iz'ui=k esa 29 iz'u gSa tks rhu [k.Mksa esa foHkkftr gSa] v] c rFkk lA [k.M v esa 10 iz'u gSa ftuesa ls izR;sd,d vad dk gSA [k.M c esa 12 iz'u gSa ftuesa ls izR;sd pkj vad dk gSA [k.M l esa 7 iz'u gSa ftuesa ls izR;sdN% vad dk gSA

3. [k.M v esa lHkh iz'uksa ds mÙkj ,d 'kCn] ,d okD; vFkok iz'u dh vko';drk vuqlkj fn;s tk ldrs gSaA

4. iw.kZ iz'ui=k esa fodYi ugha gSA fQj Hkh pkj vadksa okys 4 iz'uksa esa rFkk 6 vadksa okys 2 iz'uksa esa vkUrfjd fodYigSaA ,sls lHkh iz'uksa esa ls vkidks ,d gh fodYi djuk gSA

5. dSydqysVj ds iz;ksx dh vuqefr ugha gSA

SECTION A

[k.M ^v*

1. If f(x) = x + 7 and g(x) = x – 7, x ∈ R, find (fog) (7).

;fn f(x) = x + 7 rFkk g(x) = x – 7, x ∈ R, rks (fog) (7) Kkr dhft,A

189 XII – Maths

2. Evaluate : –1 1

sin – sin – .3 2

π � ��

eku Kku dhft, % –1 1sin – sin – .

3 2

π � ��

3. Find the value of x and y if : 1 3 0 5 6

2 .0 1 2 1 8

y

x

+ =� � � � � ��

x rFkk y ds eku Kkr dhft, ;fn % 1 3 0 5 6

2 .0 1 2 1 8

y

x

+ =� � � � � ��

4. Evaluate : + +

� �− + −� �

a ib c id

c id a ib .

eku Kkr dhft, % + +

� �− + −� �

a ib c id

c id a ib .

5. Find the co-factor of a12 in the following :

2 3 5

6 0 4 .

1 5 7

−

−

fuEu esa a12 dk lg[kaM Kkr dhft, %

2 3 5

6 0 4 .

1 5 7

−

−

6. Evaluate : 2

3.

1

xdx

x+∫

eku Kkr dhft, % 2

3.

1

xdx

x+∫

7. Evaluate : 1

2

0

.1

dx

x+∫

eku Kkr dhft, % 1

2

0

.1

dx

x+∫

8. Find a unit vector in the direction of �3 – 2 6 .a i j k= +��

� �

lfn'k �3 – 2 6 .a i j k= +��

� � ds vuqfn'k ,d ek=kd lfn'k Kkr dhft,A

190 XII – Maths

9. Find the angle between the vectors � �– and .a i j k b i j k= + = + −��

� � � �

lfn'kksa �–a i j k= +��

� � rFkk �b i j k= + −��

� � ds chp dk dks.k Kkr dhft,A

10. For what value of λ are the vectors � �2 and 2 3a i j k b i j k= + λ + = − +��

� � � � perpendicular to

each other?

λ ds fdl eku ds fy, lfn'k �2a i j k= + λ +��

� � rFkk lfn'k �2 3b i j k= − +��

� � ijLij yEcor~ gSa\

SECTION B

[k.M ^c*

11. (i) Is the binary operation *, defined on set Q, given by *2

a ba b

+= for all a, b ∈ Q,

commutative?

(ii) Is the above binary operation * associative?

(i) D;k] f}vk/kkjh lafØ;k *, tks leqPp; Q ij ifjHkkf"kr gS rFkk lHkh a, b, ∈ Q, ds fy,

*2

a ba b

+= }kjk iznÙk gS] Øe fofues;dkjh gS\

(ii) D;k mijksDr f}vk/kkjh lafØ;k * lkgp;Zdkjh gSa\

12. Prove the following

1 –1 –1 –11 1 1 1tan tan tan tan .

3 5 7 8 4

− π+ + + =

fuEu fl) dhft, %

1 –1 –1 –11 1 1 1tan tan tan tan .

3 5 7 8 4

− π+ + + =

13. Let

3 2 5

4 1 3 .

0 6 7

A � �= � ��

Express A as sum of two materics such that one is symmetric and the

other is skew symmetric.

OR

If

1 2 2

2 1 2 ,

2 2 1

A � �= � ��

verify that A2 – 4A – 5I = 0.

191 XII – Maths

ekuk 3 2 5

4 1 3 .

0 6 7

A � �= � ��

A dks ,sls nks vkO;wgksa ds ;ksx ds :i esa fyf[k, ftuesa ls ,d lefer vkO;wg rFkk

nwljk fo"ke lefer vkO;wg gSA

vFkok

;fn 1 2 2

2 1 2 ,

2 2 1

A � �= � ��

gS] rks lR;kfir dhft, fd A2 – 4A – 5I = 0.

14. For what value of k is the following function continuous at x = 2?

( )2 1 ; 2

; 2

3 1 ; 2

x x

f x k x

x x

+ <�= =�� − >�

k ds fdl eku ds fy, fuEu Qyu x = 2 ij lrr gS\

( )2 1 ; 2

; 2

3 1 ; 2

x x

f x k x

x x

+ <�= =�� − >�

15. Differentiate the following with respect to x :

–1 1 – 1tan .

1 1

x x

x x

+ −� �� + + −

fuEu dk x ds lkis{k vodyu dhft, %

–1 1 – 1tan .

1 1

x x

x x

+ −� �� + + −

16. Find the equation of tangent to the curve x = sin 3t, y = cos 2t, at t = π/4.

oØ x = sin 3t, y = cos 2t, ds fcUnq t = π/4 ij Li'kZ js[kk dk lehdj.k Kkr dhft,A

17. Evaluate : 2

0

sin.

1 cos

x xdx

x

π

+∫

eku Kkr dhft, % 2

0

sin.

1 cos

x xdx

x

π

+∫

18. Solve the following differential equation :

(x2 – y2) dx + 2xy dy = 0

given that y = 1, when x = 1.

192 XII – Maths

OR

Solve the following differential equation :

( )( )2

.2

x y xdy

dx x y x

−=

+ if y = 1, when x = 1.

fuEu vody lehdj.k dks gy dhft, %

(x2 – y2) dx + 2xy dy = 0

fn;k gS fd tc x = 1 gSa rks y = 1.

vFkok


( )( )2

.2

x y xdy

dx x y x

−=

+ ;fn y = 1 tc x = 1.

19. Solve the following differential equation :

2cos tan .

dyx y x

dx+ =


2cos tan .

dyx y x

dx+ =

20. If � �and – ,a i j k b j k= + + =��

� � � find a vector c��

such that a c b× =��

and . 3.a c =��

OR

If + + = = = =��

0 and 3, 5 and 7,a b c a b c show that the angle between a��

and b��

is 60°.

;fn � �– ,a i j k b j k= + + =��

� � �rFkk tks lfn'k c��

Kkr dhft, ;fn a c b× =��

rFkk

. 3.a c =��

vFkok

;fn + + = = = =��

0 3, 5 7,a b c a b crFkk rFkk rks n'kkZb, fd a��

rFkk b��

dschp dk dks.k 60° dk gSA

21. Find the shortest distance between the following lines :

3 5 7 1 1 1and .

1 2 1 7 6 1

x y z x y z− − − + + += = = =− −

193 XII – Maths

OR

Find the point on the line 2 1 3

3 2 2

x y z+ + −= = at a distance from 3 2 from the point

(1, 2, 3).

fuEu js[kkvksa ds chp U;wure nwjh Kkr dhft, %

3 5 7 1 1 1 .

1 2 1 7 6 1

x y z x y z− − − + + += = = =− −

rFkk

vFkok

js[kk 2 1 3

3 2 2

x y z+ + −= = ij og fcUnq Kkr dhft, tks fcUnq (1, 2, 3) ls 3 2 dh nwjh ij gksA

22. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probabilitydistribution of number of successes.

iklksa dk ,d ;qXe pkj ckj Qsadk x;kA ;fn nksuksa iklksa ij ,d gh la[;k vkuk lQyrk ekuk tk, tkslQyrkvksa dh la[;k dk izkf;drk caVu Kkr dhft,A

SECTION C

[k.M ^l*23. Using properties of determinants, prove tha following :

( ) ( ) ( ) ( )α β γ

α β γ = α − β β − γ γ − α α + β + γβ + γ γ + α α + β

2 2 2.

lkjf.kdksa ds xq.k/keks± dk iz;ksx dj fuEu fl) dhft, %

( ) ( ) ( ) ( )α β γ

α β γ = α − β β − γ γ − α α + β + γβ + γ γ + α α + β

2 2 2.

24. Show that the rectangle of maximum area that can be inscribed in a circle is a square.

OR

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of

height h is 1

.3

h

fl) dhft, fd fdlh o r ds vUrxZr fufeZr vf/kdre {ks=kQy dk vk;kr ,d oxZ gksxkA

194 XII – Maths

vFkok

fl) dhft, fd fdlh 'kadq ftldh Å¡pkbZ h gS] ds vUrxZr vf/kdre vk;ru okys fufeZr csyu dh Å¡pkbZ 1

3h

gksxhA

25. Using integration find the area of the region bounded by the parabola y2 = 4x and the circle4x2 + 4y2 = 9.

lekdyu dk iz;ksx djrs gq, ijoy; y2 = 4x rFkk o Ùk 4x2 + 4y2 = 9 ds e/;orhZ {ks=k dk {ks=kQy Kkrdhft,A

26. Evaluate : –

a

a

a xdx

a x

−+∫

eku Kkr dhft, % –

a

a

a xdx

a x

−+∫

27. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to eachof the following planes :

2x + 3y – 3z = 2 and 5x – 4y + z = 6.

OR

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to

the line 3 3 2

.2 7 5

x y z+ − −= =

ml lery dk lehdj.k Kkr dhft, tks fd fcUnq (–1, –1, 2) ls gksdj tkrk gS rFkk fuEufyf[kd nksuksa ryksaij yEcor gS % 2x + 3y – 3z = 2 rFkk 5x – 4y + z = 16.

vFkok

ml ry dk lehdj.k Kkr dhft, tks fcUnqvksa (3, 4, 1) vkSj (0, 1, 0) ls gksdj tkrk gS rFkk ljy js[kk3 3 2

2 7 5

x y z+ − −= = ds lekUrj gSA

28. A factory owner purchases two types of machines, A and B for his factory. The requirementsand the limitations for the machines are as follows :

Machine Area occupied Labour force Daily output (in units)

A 1000 m2 12 men 60

B 1200 m2 8 men 40

He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both themachines. How many machines of each type should he buy to maximise the daily output?

,d QSDVªh ekfyd A vkSj B nks izdkj dh e'khusa viuh QSDVªh ds fy, Ø; djrk gSA bu e'khuksa dh vko';drk,¡rFkk lhek,¡ fuEuor~ gSa %

195 XII – Maths

e'khu {ks=kQy ftruk Jfedksa dh izfrfnu mRiknuLFkku ?ksjrh gSA vko';drk ¼bdkbZ la[;k½

A 1,000 oxZ eh- 12 O;fDr 60

B 1,200 oxZ eh- 8 O;fDr 40

QSDVªh ekfyd ds ikl vf/kdre 9000 oxZ eh {ks=k rFkk 72 dq'ky Jfed miyC/k gSa tks nksuksa izdkj dh e'khusapyk ldrs gSaA mls izR;sd izdkj dh fdruh&fdruh e'khusa [kjhnuh pkfg, ftlls fd vf/kdre mRiknu laHkogks lds\

29. An insurance company insured 2,000 scooter drives, 4,000 car drivers and 6,000 truck drivers.The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15respectively. One of the insured persons meets with an accident. What is the probability that heis a scooter driver.

,d chek dEiuh 2000 LDwVj pkydksa] 4000 dkj pkydksa rFkk 6000 Vªd pkydksa dk chek djrh gSA ,dLdwVj] dkj rFkk Vªd ds nq?kZVukxzLr gksus dh izkf;drk,a Øe'k% 0.01, 0.03, rFkk 0.15 gSaA chekÑr O;fDr;ksa¼pkydksa½ esa ls ,d nq?kZVukxzLr gks tkrk gSA ml O;fDr ds LdwVj pkyd gksus dh izkf;drk D;k gS\

196 XII – Maths

MARKING SCHEME

SECTION A

1. 7; 2. 1; 3. x = 3, y = 3; 4. a2 + b2 + c2 + d2; 5. 46;

6. ( )31. log 1 ;

3x c+ + 7. ;

4

π(8) �3 2 6

– ;7 7 7

i j k+� �

9.–1 1

cos – .3

θ = � �� 10.

5.

2λ =

11. (i) If the candidate verifies that the operation is commutative but not associative, full creditmay be given. 2 + 2

12.–1 –1 –1 –1 –1 –11 1 1 1 8 15 15 56

tan tan tan tan tan tan .3 5 7 8 14 15 55 56

LHS = + + + = +� � � �� (1½ + 1½)

1 –1 14 3tan tan tan 1 4 .

7 11RHS

− −= + = = π = 1

13. For � �=� ��

3 4 0

´ 2 1 6 .

5 3 7

A 1

∴3 3 5 2

´3 1 9 2 which is symmetric.

2 5 2 9 2 7

A A

+ � �=� ��

1

0 1 5 2´

1 0 –3 2 which is skew symmetric.2 –5 2 3 2 0

A A−

− � �=� ��

1

∴3 2 5 3 3 5 2 0 1 5 2

4 1 3 3 1 9 2 1 0 –3 2

0 6 7 5 2 9 2 7 –5 2 3 2 0

− � � � � � �= +� � � � � ��

1

OR

For 29 8 8

8 9 8

8 8 9

A � �=� ��

2

4 8 8 5 0 0

4 8 4 8 , 5 0 5 0

8 8 4 0 0 5

A I � � � �= =� � � ��

1

197 XII – Maths

For verifying A2 – 4A – 5l = 0. 1

14. Getting LHS = 5, RHL = 5 1½ × 2 = 3

⇒ K = 5. 1

15. Let cos 2 , 1 2 cos , 1 2 sin .x x x= θ + = θ − = θ 1

∴ 1 –11 tantan tan . tan –

1 tan 4y

− − θ π = = θ� �� + θ� �1 + ½ = 1½

–11– – cos

4 4 2x

π π= θ = ½

∴2

1.

2 1

dy

dx x=

−1

16. At1

4 , , 0.2

t x y= π = = 1

−= = − ∴ = 2 sin 23 cos 3 , 2 sin 2

3 cos 3

dx dy dy tt t

dt dt dx t1½

∴4

2 2

3t

dy

dx = π

=� �� ½

Equation of tangent 3 2 2 2 or 3 – 2 2 2 0y x y x= − + = 1

17.( )( )2 2

0 0

sin sin

1 cos 1 cos

x x x xI dx dx

x x

π ππ −= =

+ +∫ ∫ 1

∴2

0

sin2

1 cos

xI dx

x

π

= π+∫ 1

∴ ( ) ( )221

0– tan cos – – 4 .4

I xπ− π = π = π π =� � 1 + ½ + ½ = 2

18. Writing 2 2

, which is homogeneous.2

dy y x

dx xy

−= ½

Putting .dy dv

y vx v xdx dx

= ⇒ = + ½

∴ getting 2

2–

1

v dxdv

xv=

+1

198 XII – Maths

∴ + = − + =2log 1 log log log

cv x c

x1

⇒ x2 + y2 = cx ⇒ c = 2 when x = 1, y = 1. 1

∴ x2 + y2 = 2x.

OR

Writing2

2

dy y x

dx y x

−=+ ½

dy dvy vx v x

dx dx= ⇒ = + ½

Getting 2

2 1–

2 1

v dxdv

xv v

+ =− +

1

∴ 2 2

1 4 1 3–

1 12 42 12 2

v dv dxdv

xv v v v

− + =− + − + 1

Getting2 2

12

1

1 2 3 4log tan – log

2 7 71 3 3

log 2 tan2 7 7

y xy x y xx c

xx

where c

−

−

− + −�+ = +��= + ��

1

19. Writing 2 2

sec . tan . secdy

x y x xdx

+ = ½

I.F. = etan x 1

∴ Solution is, = +∫tan tan 2

. . tan . sec .x x

y e e x x dx c ½

∴( )

( )

tan tan

tan

. tan 1

tan 1 .

x x

x

y e e x c

or y x c e

= − + ��= − + �

2

20. Let �c xi y j zk= + +��

� �

∴ ( ) ( ) � ( ) �× = − − − + − = + −��

� � � �0a c i z x j z x k y x i j k 1

⇒ y = z = x – 1 1

. 3 3a c x y z= ⇒ + + =��

½

∴ �2 5 2 25 3 ,

3 3 3 3x y z c i j k= = = ⇒ = + +

�� 1 + ½

199 XII – Maths

OR

0 –a b c a b c+ + = ⇒ + =��

1

∴2 2 2

2 .a b a b c+ + =��

1

9 + 25 + 2ab cos θ = 49 1

∴1

cos 3 60 .2

θ = + ∴ θ = π = ° 1

21. � �

� �

1 1

2 2

3 5 7 , 2

, 7 6

a i j k b i j k

a i j k b i j k

= + + = − + ��

= − − − = − + ��

��

��

1

( ) ( )2 1 1 2

1 2

– .5 .

a a b bD

b b

×=

×

��

�� 1

� �2 1 1 2– –4 – 6 – 8 , 4 6 8a a i j k b b i j k= × = + +

�� ½ + 1 = 1½

∴ . . 116.S D = ½

OR

Getting x = 3λ – 2, y = 2λ – 1, z = 2λ + 3 1

Distance D from ( )1, 2, 3 3 2.=

∴ ( ) ( ) ( ) ( )2 2 3 23 2 3 2 2 3 2= λ − + λ − + λ 1

⇒ λ = 0 or 30

.17

λ = 1

∴ Points are ( )56 43 111, , or –2, – 1, 3

17 17 17

� �� 1

2241 5 1 5

, Prob. dist is 6 6 6 6

p q = = ∴ +� �� 1

x 0 1 2 3 4 ½

P(X)625

1296

500

1296

150

1296

20

1296

1

12962½

200 XII – Maths

SECTION C

23. ( )α β γ α β γ

→ + ⇒ ∆ = α β γ = α + β + γ α β γα + β + γ α + β + γ α + β + γ

2 2 2 2 2 23 3 1

1 1 1

R R R

1½ + ½ = 2

c1 → c1 – c2 and c2 → c2 – c3

⇒ ( )α − β β − γ

∆ = α + β + γ α − β β − γ γ2 2 2 2 2

0 0 1

x

1 + 1 = 2

( ) ( ) ( )γ

= α + β + γ α − β β − γ α + β β + γ γ 2

1 1

0 0 11

= (α + β + γ) (α – β) (β – γ) (γ – α) 1

24. x = 2r cos θ, y = 2r . sin θ 1

θ

r0 y

C

B

D

A x

1

Area A = x . y = 4r2 . sin θ cos θ = 2r2 sin 2θ 1

∴ 24 . cos 2 ; 0 4

dA dAr

d d= θ = ⇒ θ = π

θ θ1

Showing 2

24

0d A

d θ = π

<� �

� �θ1

∴ Area is maximum at θ = π/4 ⇒ x = y ⇒ square. 1

B

C

D

rE

A

R

h h´

1

201 XII – Maths

V = πr2h´ 1

22 2

2´ 2 ´ ´

Rh h hh h

h = π + +� � 1

0 ´´ 3

dv hh

dh= ⇒ = 1

Showing 2

20

´

dv

dh< 1

∴ V is maximum, when height of cylinder 1

.3

h= 1

25.

I II

Y

O 32

12

X

1

Points of intersection,1

2x = 1

Required area = + −∫ ∫1 2 3 2

2

0 1 2

92 2 2

4x dx x dx 2

–12 2 2 9 9 1– – sin

3 2 8 4 3

π= + 2

–19 2 9 1Note : Full credit for – – sin

8 6 4 3A

π= +� ��

26. 1 22 2 2 2

a a a

a a a

a x a dx x dxI dx I I

a x a x a x− − −

−= = − = ++ − −

∫ ∫ ∫ 1 + 1 = 2

In, I1 function is even and in I2, function is odd 1m

: I2 = 0 1m

∴2 2

0

2 2 . 2 . .a

dxI a a a

a x= = π = π

−∫ 2

202 XII – Maths

27. Equation of plane through (–1, –1, 2) is

a(x + 1) + b (y + 1) + c (z – 2) = 0 (i) 1

∴ 2a + 3b – 3c = 0 and 5a – 4b + c = 0 2

Solving to get a : b : c = 9 : 17 : 23 2

∴ Equation of plane is 9x + 17y + 23z = 20 1

OR

Equation of plane through (3, 4, 1) is

a(x – 3) + b(y – 4) + c(z – 1) = 0 1

∴ we get 3a + 3b + c = 0 and 2a + 7b + 5c = 0 2

Solving to get a : b : c = 8 : – 13 : 15 2

∴ Equation of plane is 8x – 13y + 15z + 13 = 0 1

28. Getting the constraints as

5x + 6y ≤ 45, 3x + 2y ≤ 18, x ≥ 0, y ≥ 0 2

Total out put P = 60x + 40y 1

Correct graph 2

Vertices of feasible region are

( ) ( ) ( )9 450, 0 , 6, 0 , , 0, 15 2

4 8

� ��

∴ P at (0, 0) = 0 P at (6, 0) = 360 1

P at (0, 15/2) = 300m9 45

at , 360.4 8

P =� ��

∴ Max. P = 360. 1

29. ( ) ( ) ( )1 2 31 1 1

, ,6 3 2

P E P E P E= = = 1½

( ) ( ) ( )1 2 31 3 15

, ,100 100 100

P A E P A E P A E= = = 1½

( ) ( ) ( )

( ) ( )1 1

1 3

1

.

i i

P E P A EP E A

P E P A E

=

∑ 1

1.

52= 1

��

203 XII – Maths

PRACTICE QUESTION PAPER–1

Time : 3 hours Total Marks = 100


1. Question paper has three sections, Section A, contains 10 questions of 1 mark each, SectionB contains 12 questions of 4 marks each and Section C contains 7 questions of 6 marks each.

2. All questions are compulsory.

3. Internal choices are given in some questions, where one part is to be attempted out of two.

4. Calculators are not allowed.

SECTION A

1. If 5 1

4 1A

− = � ��

what is adj A?

2. What is the principal value of –1 3

cos – ?2

� ��

3. If A is a matrix of order 2 × 2 and |A| = – 3, what is the value of |3A|?

4. Let f : R → R be defined as ( ) 2 – 3,

5

xf x = what is f–1?

5. What is the differential equation representing the family of straight lines y = mx + c, m, c beingarbitrary constants?

6. Write the general solution of the differential equation in .dy

xydx

=

7. If a line makes angle 90°, 150° and 60° with the positive direction of x, y and z-axis respectivily,what are its direction cosines?

8. Write the vector equation of the line 2 4 6

.3 5 2

x y z+ + −= =

9. Give example of two non zero 2 × 2 matrices A and B such that A.B. = 0.

10. If � �; ;a i j b j k c k= + = + =��

� � � what will be the unit vector in the direction of

?a b c+ +��

11. Let X be a nonempty set and P(X) be its powerset. Let * be the binary operation defined in P(X)by A * B = A ∩ B, ( ), .A B P X∀ ∈

(i) show that * is commutative.

204 XII – Maths

(ii) * is associative

(iii) Find the identity element in P(X) w.r.t. *

(iv) Find the invertible elements of P(X)

OR

Show that the relation R in Set A = {x : x ∈ Z, 0 ≤ x ≤ 15} given b R = {(a, b) : (a – b)is a multiple of 3}, is an equivalence relation. Also find the set of all elements related to 4.

12. Find x, if –1 –1 1 –1 15 5

tan 4 cot 2 tan cos tan 2 tan sin .13 13

x− − � � � �= +� � � ��

13. If a, b, c are all positive and distinct, then prove that :

a b c

b c a

c a b

∆ = is negative.

14. Determine the values of m, n, p for which the function :

( )

( )

2

3 2

sin sin 1for 0

, for 0 is continuous at 0.

for 0

x m xx

xf x n x x

x px xx

px

+ + <��= < =�� + −� >��

15. Find the intervals in which the function f(x) = (x + 3)3 . (x – 2)3 is strictly increasing or decreasing.

16. For a > 0 find ,dy

dx if

1t

ty a−

= and 1 a

x tt

= −� ��

If–1 1

sin cos, , for 0.

t tx a y a a

−

= = >

Show that – .dy y

dx x=

17. Evaluate : ( ) ( )

1

sin – cos –dx

x xα β∫OR

Evaluate : ( ) ( )

1dx

x a x b− −∫

18. Evaluate ( )5 2

23 2x x dx−∫ by the method of limit a sum.

205 XII – Maths

OR

Using properties of definite integrals evaluate :

0log 1 cos x dx

π+∫

19. Evaluate ( ) ( )2 2

2 3.

1 1

xdx

x x

+

− +∫

20. Using vectors show that the angle between any two diagonals of a cube is –1 1cos .

3

� ��

OR

Find the angle between the lines whose direction cosines (l, m, n) are given by the equations.3l + m + 5n = 0, 6mn – 2nl + 5lm = 0.

21. Let � � �4 2 , 3 2 7 and 2 4 .a i j k b i j k c i j k= + + = − + = − +��

� � � � � � Find a vector d��

which

perpendicular to both and , and . 15.a b c d =��

22. The probability that a person will buy a shirt is 0.2, the probability that be will buy a trouser is0.3 and the probability that he will buy a shirt given that he buys a trouser, is 0.4. Find theprobability that he will buy a shirt and a trouser. Find also the probability that he will buy atrouser given that he buys a shirt.

23. Obtain the inverse of the matrix

3 1 1

1 2 3

0 1 2

A � �=� �

using elementary transformations and verify

your result.

OR

The sum of three numbers is 2. If twice the second number is added to the sum of first and third,the sum is 1. By adding second and third number to five times the first number, we get 6. Findthe three numbers using matrices.

24. The cost of fuel for running a bus is proportional to the square of the speed generated in km/hr. It costs Rs. 48 per hour when the bus is moving at a speed of 20 km/hr. What is the mosteconomical speed if the fixed charges are Rs. 108 for an hour, over and above the runningcharges?

25. Find the area of region common to the circle x2 + y2 = 16 and the parabola x2 = 6y, usingintegration.

26. Solve the differential equation : (x3 – 3xy2)dx = (y3 – 3x2y) dy

OR

Form the differential equation of the family of circle in the second quadrant which touch thecoordinate axes.

206 XII – Maths

27. Find the shortest distance between the pair of lines whose cartesian equations are :

1 1 1 2and ; 2.

2 3 3 1

x y x yz z

− + + −= = = =

28. A fair coin is tossed 6 times. Find the probability of getting :

(i) exactly 5 heads

(ii) at least 5 heads

(iii) at most 5 heads

Find also the mean and variance of the number of heads.

29. A diet is to contain at least 60 units of vitamin A and 45 units of minerals. Two foods F1 andF2 are available. Food F1 costs Rs. 4 per unit and food F2 costs Rs. 6 per unit. One unit of foodF1 contains 3 unit of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units ofVitamin A and 3 units of minerals. Formulate this as a linear programming problem to find theminimum cost for diet that consists of mixture of these two foods and also meets the minimalnutritional requirements and solve it graphically.

207 XII – Maths

PRACTICE QUESTION PAPER–2

Time : 3 hours Total Marks = 100


1. Question paper has three sections, Section A, contains 10 questions of 1 mark each, SectionB contains 12 questions of 4 marks each and Section C contains 7 questions of 6 marks each.

2. All questions are compulsory.

3. Internal choices are given in some questions, where one part is to be attempted out of two.

4. Calculators are not allowed.

SECTION A

1. If f(x) = sin–1 x, g(x) = x3.

Then what is gof (x), if it is defined.

2. If the points (2, –3), (λ, – 1) and (0, 4) are collinear, find the value (s) of ‘λ’.

3. Write the principal value of 1 –1

tan .3

− � ��

4. Write the no. of all possible matrices of order 2 × 2 with each entry 4 to 5.

5. If A be a square matrix of order 3 × 3 and |A| = 10, then write the value of |4A|.

6. Write the order and degree of the differential equation

22 3

25 sin .

d y dyx

dxdx

+ =� � � �

7. Write the differential equation representing the family of parabolas y2 = 4bx.

8. If the position vector a��

of point P(3, n) is such that 5,a =��

then write the value of n.

9. Find the angle between the lines, 1 1 2

and .2 3 4 1 3 0

x y z x y z− −= = = =− −

10. Find the distance between the planes.

3x + 4y – 7 = 0 and 3x + 4y + 3 = 0.

SECTION B

11. If cos–1 x + cos–1 y + cos–1 z = π then show that x2 + y2 + z2 + 2xyz = 1.

Or

208 XII – Maths

Prove that1 1 7

tan 2 tan .5 4 17

− π − − =� ��

12. Express 3 4

2 7A

− = � �

� � as the sum of a symmetric and a skew-symmetric matrix.

13. Show that f : R → R defined by f(x) = x3 – 8, is bijective. Also find f –1 (19).

14. For what values of ‘a’ and ‘b’

( )

2

3

5 if 1

8 if 1

3 if 1

x ax x

f x x

bx x x

− <��= =��

− >��


15. If x = a cos3 θ, y = sin3 θ, find 2

2.

d y

dx

OR

If y = (sin x)x + (log x)cos x, find .dy

dx

16. If f(x) = x2 – 3x + 1 then using differential find the approximate value of f (1.09).

OR

Find the equation of normal to the ellipse, 3x2 + 6y2 = 6 at P(x1, y1) where y1 = 1.

17. Evaluate : sin

.sin 4

xdx

x∫

18. Evaluate : 2

.3 1

x x

dx

e e+ +∫

19. Evaluate :

3

6

sin cos.

sin 2

x xdx

x

Π

Π

+∫

OR

Evaluate : 2 1 sin 2

.1 cos 2

x xe dx

x

+ � �� +∫

20. Prove that, for any two vectors and .a b��

( )22 2 2– . .a b a b a b× =

��

209 XII – Maths

21. Find the equation of the plane passing through the intersection of planes x + 2y + 3z = 4 and2x + y – z = – 5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0.

22. Assume that the probability that a bomb dropped from an aeroplane will strike a certain targetis 0.3. If 5 bombs are dropped. Find the probability that.

(i) Exactly 4 will strike the target. (ii) Atleast one will hit the target.

SECTION C

23. For the matrix

3 2 1

5 0 4 ,

1 6 3

A

− � �=� ��

find A–1 and use it to solve, system of equations.

3x + 5y + z = 5

–2x + 6z = 10

x + 4y + 3z = 15

24. Show that the semi-vertical angle of a right circular cone of given surface area and maximum

volume is 1 1

sin .3

− � ��

25. Find the area of the region bounded by y2 ≥ 4x and 4x2 + 4y2 ≤ 9.

OR

Evaluate : ( )2

2

0

3 2x x dx−∫ as a limit of a sum.

26. Solve initial value problem : . sin sin 0,y x y dy y

x e y xx dx x

− + =� � � �� given that y(1) = 0.

27. Find the shortest distance between the two skew lines :

1 2 2 1and

1 2 3 1 2 3

x y z x y z− + − −= = = =−

28. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on eachexecutive class ticket and a profit of Rs. 600 is made on each economy class ticket. The Airlinereserve at least 20 seats for executive class. However at least 4 times as many passengersprefer to travel by economy class, than by the executive class. Determine how many tickets ofeach type must be sold in order to maximise the profit for the Airline. What is the maximum profit?

29. In a test, an examine either guesses or copies or knows the answer to a multiple choice question

with four choices. The probability that he makes a guess is 2

3 and the probability that he copies

the answer is 1

.8

The probability that his answer is correct, given that he copied it, is 1

.8

Find

the probability that he knew the answer to the question, given that he correctly answered it.

210 XII – Maths

HINTS AND ANSWERS

CHAPTER 1

RELATION AND FUNCTIONS

1. R1 – Universal Relation

R2 – Empty Relation.

R3 – Neither Empty Nor Universal.

R4 – Neither Empty nor Universal.

R5 – Neither Empty nor Universal

2. (i) Yes. Because every element in set A has different image in A.

(ii) Yes, (Constant function)

(iii) No, Because 2 have two images.

(iv) No, Because 5 has no image.

(v) No, Because 2 does not have a unique image.

3. (Fog) (7) = 7

(gof) (7) = 7

[Hint. : Use (fog) (x) = f [g(x)]

4. fog = {(0, 6), (1, 3), (2, 2)}

5. ( ) ( ) .2 1

xfof x

x=

+

6. ( )–1 5 1.

2

xf x

+=

7. (i) Not one-one Hint. : f(2) = f(–2)

(ii) One-one

(iii) Not one-one

(iv) One-one

(v) Not one-one Hint. : f(1) = f(2) = f(3) = 0

(vi) Not one-one Hint. : f(1.3) = f(1.4) = 1

(vii) Not one-one because f(0) = f(π) = 0.

211 XII – Maths

(viii) One-one because x1 ≠ x2 in [0, π] ⇒ f(x1) ≠ f(x2)

(ix) One-one.

8. (i) Onto (function) [Hint. : Range = Codmain.]

(ii) onto (Surjective function).

(iii) onto

(iv) Not onto Because Rf ≠ R

(v) Not onto, Because Rf = {1, 8, 27, ...} ≠ N

(vi) Not onto, Because Rf = (0, ∞) ≠ R

(vii) Not onto, because 1 has no preimage.

(viii) Not onto, as f(x) is constant function Rf = {0}.

9. (i) 0 (ii) 12

(iii) 26/9 (iv) 5/27

10. (a) Equivalence Relation

(b) Transitive Relation/

(c) Reflexive and Transitive.

(d) Reflexive and Transitive.

(e) Symmetric and Transitive.

(f) Reflexive and Transitive.

(g) Neither symmetric nor reflexive nor transitive.

(h) Equivalence Relation.

(i) Neither reflexive nor symmetric nor transitive.

(j) Equivalence Relation.

11. (i) Bijective (one-one and onto)

(ii) Bijective

(iii) Neither injective nor surjective.

(iv) Neither injective nor surjective.

(v) Bijective

(vi) Neither one-one nor onto.

(vii) Bijective

212 XII – Maths

(viii) Bijective

(ix) Injective but not surjective.

(x) Bijective

(xi) Injective but not surjective

(xii) Neither injective nor surjective.

12. Commutative and associative. (1, 1) is identity element.

13. (i) Yes, ‘*’ is commutative

(ii) 2 * (3 * 4) = 2 * |32 – 42| = 2 * 7 = |4 – 49| = 45

(2 * 3) * 4 = |22 – 32| * 4 = 5 * 4 = |25 – 16| = 9.

∴ ‘*’ is not associative.

14. (i) No, ‘*’ is not commutative

(ii) Not associative.

15. 0 is the identity element in Q – {1}.

16. f(x) is bijective and ( )–1 4 1.

3

xf x

+=

17. No, it is not invertible. Because, f is not one-one function : f(0) = f(π) = 0

18. (fog) (x) = x.

(gof) (x) = x. Hence they are equal.

19. No, They are not equal.

213 XII – Maths

CHAPTER 2

INVERSE TRIGONOMETRIC FUNCTIONS

1. (i)–

3

π(ii)

3

π(iii)

5

6

π

(iv)6

π(v)

–

6

π(vi)

6

π

(vii)–

6

π(viii)

6

π(ix)

2

3

π

(x)3

π(xi)

2

3

π(xii)

3

π

(xiii) .6

π

2. (i) 0 (ii)–

3

π(iii)

–

2

π

(iv)2

π(v)

2

3

π(vi)

2

π

(vii) π (viii)3

π(ix)

5

π

(x)3

5

π(xi)

–

6

π(xii) .

4

π

7. Hint. : Divide Num. and Den. By b cos x.

12. x = 1 [Hint. : Other value of x does not satisfy.]

13. x = 1/6.

15. (ii)3

.4

π

214 XII – Maths

CHAPTER 3

MATRICES AND DETERMINANTS

1.0 3

.1 0

� ��

2. 2xm.

3.1 2

.0 1

� ��

4. [18]

5. 10λ. 6. 3 × 3.

7. 5. 8. 2.

9.5 1

.2 0

x

= � ��

10. 64. [Hint. : 26 = 64]

11.1 0 0 0

, .0 0 0 1

A B

= =� � � ��

12.

6 3

9 12 .

0 15

− � �−� ��

13.2 2

.2 0

� ��

14. 0.

15. a2 + b2 + c2 + d2. 16. (–3)1/3.

17. 28. 18. id – fg.

19.2

.3

± [Hint. : – 3x2 – 5 = – 7]. 20. , .6 2

π π[Hint. : 2 sin2 x + 1 = 3 sin x].

21. 0. 22. 54. [Hint. : order 3 ⇒ |–3A| = (–3)3 |A|].

23. 1 Sq. unit. 24. x = any real number.

25. a13,M13 – a23M23 + a33M33 26. .3

π

27.3

– .8

λ = 28. 5 5.± [Hint. : |Adj A| = |A|n–1]

29. |A| = 0 and (adj A)B = 0 30.5 –3

.6 8

� ��

31. 40. 32. I2.

215 XII – Maths

33.13 2

.9 6

− � �−� �

34. 25. [Hint. : |Adj A| = |A|n – 1]

35.6 5

.5 6

� �− −� �

36. x = 2, y = 1.

37. 3. 38. 0; [Hint. : [R1 → R1 + R2 + R3]]

39. x = – 4. 40.4 0

.0 4

− � �−� �

41.4 3

.1 2

− � �−� �

42. pqr.

43. 12. 44. 0. [Hint. : A2 = 0]

45. 0. [Hint. : A + B + C = π] 46. 0.

47. 5. 48. 200.

49. 20. 50. .6

π

51.

1 1 30 –

3 2 51 1 1

0 .3 5 31 1 1

02 5 7

A

− −� ��

−� �= −� ��

−� ��

52. x = 1, y = 2, z = 3, w = 4.

53.

11 9 9 5 2 1–

7 7 7 7 7 7, .1 18 4 4 12 5

7 7 7 7 7 7

A B

− −� � � �� = =� � � �− −� � � ��

54.1 2

.2 0

X−

= � ��

59.

1 3.7

12

− � �� −� ��

60.

3 1

5 5 .2 1

–5 5

� ��

61. [Hint. : Use − θ θθ = θ =+ θ + θ

2

2 2

1 tan 2 2 tan 2cos and sin

1 tan 2 1 tan 2

216 XII – Maths

62. x = – 2 or x = – 14.

63. x = 9, y = 14. 64. [Hint. : Use principle of mathematical induction]

66. – 14I3 67.–1 7 31

4 22A

= � �

� �

68. –1 2 11.

3 27A

= � �−� �

70. –1 4 31.

3 217A

= � �−� �

73. [Hint. : x = AR10–1 where A = first term, R = common ratio]

81. ( )–11 2 2 21, .

2 2 2 16AB AB

− = =� � � �− −� � � �

82. (i) 20 sq. unit. (ii) 1

–3, .2

x x= =

86.11 1

, .24 24

x y= =

87.5 118 93

31 118A

− − = � �−� �

. [Hint. : A2 – 4A + 7I = 0, A2 = 4A – 7I, A3 = 4 (4A – 7I – 7A)

89.16 3

.24 5

X−

= � �−� �[Hint. : if A × B = P, X = A–1 P B–1]

90.–1

1 1 11

1 1 1 .2 1 1 1

A

− � �= −� �� −� �

91.

2 0 31

1 1 0 .5 2 1 2

− � �−� �� −� �

92.

3 2 6

1 1 2 .

2 2 5

� ��

93.

2 1 1

11 –1 3.

4 2 41 0 0

− � �−� �� −� �

96.10 –31 –8

, , .19 19 19

x y z= = =

97. x = 2, y = –1, z = – 2.

217 XII – Maths

98. x = 1, y = 0, z = – 5.

99. x = 1, y = 1, z = 1. [Hint. : equations are A´x = B, so X = (A´)–1 B = (A)–1 B]

100. –16 17 13

–114 5 8 .

67 15 9 1

A

− � �= −� �� − −� �

x = 3, y = – 2, z = 1.

101. x = 4, y = –3, z = 1.

102. x = 1, y = – 2, z = 2. [Hint. : Suppose three numbers as x, y, z]

103.–1

2 0 1

5 1 0 .

0 1 3

A

− � �= � ��

104. ( )− � �= −� �� −� �

116 12 1

121 11 7 .

19 10 2 3

AB [Hint. : (AB)–1 = B–1A–1]

105. x = 3, y = – 2, z = – 1.

106. x = 2, y = 3, z = 5. [Hint. : Let 1 1 1

, , .u v wx y z

= = = ]

107.–1

3 4 51

9 1 4 .11 5 3 1

A

− � �= − −� �� − −� �

218 XII – Maths

CHAPTER 5

DIFFERENTIATION

1.5

.3

x =

2. .x

x[Hint. ( ) ( )2

2

1, ´ 2 .

2

xf x x f x X x

xx= = = ]

3. x = 3. 4. x = 2, x = 3.

5. All integer

6. Discontinuous at x = 0. [Hint. : Limit dis intenist as LHL = – 1, RHL = 1]

7. (0, ∞). 8.–1

.2

λ =

9.1

0,3

� ��

[Hint. : log x is continuous in (0, ∞), 2

1 9x− is continuous in 1 1

– ,3 3

� ��

, common is 1

03

� ��

.

10. 1. [Hint. : As 0

sin1, so 1

→= =

x

xLt k

x]

11. 2x3. ( )23 33

. 2d

x xdx

=� �

� �Hint :

12.1

.logx x [Hint. : f(log x) = log (log x)]

13. loge (e – 1). [Hint. : ( )1 0

–, – 1, log – 1 .

1 – 0

x xe

e ee e e x e= ⇒ = ⇒ =

14. 0.− − π π � �+ = =� ��

1 1. sin cos , so 0

2 2

dx x

dxHint :

15. 2.

16. (a) ( )–1

2 1x x+(b) 2

–2

1 x−

(c) −

53

log3 5

ex

219 XII – Maths

(d) 2 2

4 3–

1 16 1 9x x+ +

–1 4 3tan

1 4 . 3

x x

x x

− � �� +� �Hint. :

(e) 2

3

1 x+[Hint. : put x = tan θ]

(f) –1 2

tan4

2.

1

x xe

x+

(g) 10x4 sin x5 cos x5 cos x3 – 3x2 sin x3 sin x5

(h) 4

x

e

xe(i) 2

3

2 1

x

x−

(j) −

−sin

2 1 cos cos

x

x x

17. Discontinuous.

18. Discontinuous.

19. Discontinuous.

20. Continuous.

21. Continuous. ( )− ≤ −

� ��= − ≤ <� �� ≥��

2 if 1

2 if 1 1

2 if 1

x x

f x x

x x

Hint. :

22. Continuous.

23. Continuous

( ) ( )[ ]( ) ( )→ →

+ − − /= = =� �×� �+ + − + −/

� ��

0 0

1 1 2 21

sin sin 1 21 1 . 1 1x x

x x xLt Lt

x xx x x x x x

x x

Hint. :

24. Continuous.

25.17

.6

26. a = 1, b = –1.

27. 0, 1. [Hint. : As ( ) 1f x

x= so discontinuity at x = 1

As ( )[ ] 1,

xf f x

x

−= so discontinuity at x = 0.]

28.1

.2

k =

220 XII – Maths

30. Discontinuous 5

.2

x =

31. Not differentiable.

32. ( )1, .p ∈ ∞ ( ) −

→ →

� �−� �� = = − >� � � � � ��

1

0 0

1sin 0

1. 0 . sin, , for which 1 0

p

p

x x

xxf Lt Lt x p

x xHint :

33. a = 2, b = 1.

34. ( ) ( ) − + +� ��

log 11log log log 2 log .

log

xxx x x xx

[Hint. : take log on both sides]

35. 0.

36. 2

–.

1

x

x−

39.( ) 3

2

1 log sec tan.

sec

x

x

x x x x

x x

+ +

+

44.1

– .2

. 1 sin cos sin2 2

+ = +� �� Hint :

x xx

45.2 2

2.

1

a

a x−[Hint. : put ax = sin θ]

46. 1.

49.( )

2

22

12 .

1

x x

x

+ −� �� +� �

50.( )

( )1

1 –1

– log 1 log.

log

x y x

x y

y y y x x x

x y x x

−

−

+ + +� �

+

51.2

2

1.

1

θ +

θ −

52.θ θ4

1.

3 cos sina

54.2

2.

1

t

t −

221 XII – Maths

55.2 2

1 1.

1 2x x x−

− −[Hint. : put x = sin, θ √x = sin φ]

56.( ) ( )2 1

1 22 4 cot 2

2 1 1 tany x

x x x−

+ − −� �− +� �

57. tan t.

58. –1. = α = α� ��

3 4. put sin , cos

5 5Hint :

59.2

–1.

1 x−

60.( ) ( )− −

2 2

2or .

log 1 log

y y

x x y x x y

62. – cot y cosec2 y.

66.2

–2.

x

222 XII – Maths

CHAPTER 6

APPLICATIONS OF DERIVATIVES

1. 1

2.3

.2

C = [f´(x) = 2x – 3 = 0, x = 3/2]

3. (0, π).

4. (– ∞, 0).

5.1

– , – .3

∞� �� [Hint. : f´(x) = cos x – 3λ, cos x – 3λ > 0,

1–

3λ < ]

6.1

,e

∞� �� [Hint. : f´(x) = xx (log x + 1) ⇒ log x + 1 > 0, x > e–1]

7. Increasing

8. 2 [Hint. : f(x) is minimum at x = 1]

9.1

.2

10. Maximum value of f(x) = 5, Minimum value of f(x) = 1.

[Hint. : –1 ≤ sin 2x ≤ 1, –2 ≤ 2 sin 2x ≤ 2, 1 ≤ 2 sin 2x + 3 ≤ 5]

11.1 3

, .2 8

� ��

[Hint. : Slope of chord dy

dx= = slope of tangent to the curve]

12. Not defined = = ∞� ��

tan 90dy

dxHint. :

13. 0.

–1 1, 0

0

dydy dxdx

= =� ��

14. – 2.

15.1

, 0 .2

� ��

slope of line = 1dy

dx

= � �Hint. :

16. (0, 0).

17.1

.12

[Hint. : Product of slopes of tangents at intersecting point = – 1]

223 XII – Maths

18. (1, 2). tan 45dy

dx

=� �� Hint. :

19. 1. [Hint. : Slope of Normal = – K, Slope of Normal ( )

1

3, 6dy dx= Find and equate]

20. No extremum value. [Hint. : f´(x) = 0]

21. x – y = 0.

22. cm/s.48

π

23. (1, 3) (–1, 1)

24. Maximum 2 at 3

,4

xπ= Minimum is – 1 at

7.

4x

π=

25. Increasing in 1

,4

∞� �� and decreasing in 1

– , .4

∞� ��

26. 2; 8. [Hint. : Let x = 2, x = ∆x = 1.9 , ∆n = – 0.1, f(x) + ∆xf´(x) = f(x + ∆x)].

27. (i) (–∞, 0] (ii) [1, ∞).

28. Decreasing at the rate of 3.6 cm/sec.

29. –2 .ab

( ) ( )( ) = − = � = ± = − = −� ��

2´ 0 , 2 2

bf x a x b a f b a ab f b a ab

xHint. :

30. ( ) 3 –270, 0 , , .

4 32

� ��

31. (i) 0.2495; (ii) 0.2867 (iii) 0.0707

34. Increasing in (0, 3π/4) U (7π/4, 2π) and decreasing (3π/4, 7π/4)

35. 0.3444. [Hint. : Use f(x + h) = f(x) + h f´(h)]

36. ( )30 3, .

2

∪ ∞� ��

37. Increasing in (– ∞, 2) ∪ (3, ∞) and decreasing in (2, 3).

38.1 1

, – .3 3

� ��

39. 2x + 3y = 6.

224 XII – Maths

41. Increasing in 50, , 2

4 4

π π ∪ π� � � �� and decreasing in

5, .

4 4

π π � ��

42. Maximum value = 5.

43. x = 14, y = 21.

44. a = – 3.

46. 0.19235.

47. (i) No point at which tangent is parallel to x axis.

(ii) (3, 0), (– 3, 0).

48. a ≤ – 3. [Hint. : f´(x) = 3(a + 2)x2 – 6ax + 9a < 0, ( )

( )± − × + ×= <

+

26 36 4 3 2 9

06 2

a a a ax

a

49. a2 cos2 α + b2 sin2 α = p2.

[Hint. : Find ( )

� ��

1 1

,x y

dy

dx Then equation of tangent at (x1, y1).

The eliminate the value of x1 and y1]

50. y – 1 = 0.

53. 75 3 sq. unit.

56. Increasing in (2, 3), decreasing in (3, ∞).

57. 3x + y – 3 = 0

7x – y – 14 = 0

60. 5 cm.

61.28 112

, .4 4

m mπ + π +

62. 5 unit.

66.16

.6 – 3

68.1

maximum Area .2 2

abπθ = =

225 XII – Maths

CHAPTER 7

INTEGRATION

1. (i)3 22

2 .3

x x c+ + (ii) tan x + c.

(iii) – cosec x + c (iv) .3

π

(v) 0; (vi) ( )log log log .x c+

(vii) – cot x – cosec x + C (viii)78 1

8 8 log .log 8 8

x

x x c+ + + +

(ix) sin–1 x + c (x) 0.

( ) ( )− − = −� �1 1 2

sin cos sin sin 1x xHint. :

2. (i)

5 3 1

2 2 22 2– 2 .

5 3x x x c+ + (ii)

2 1log 2 .

2

ax x x c

a+ − +

(iii)4cos 3

2 tan .3 4 2

xx xe c

− − + + (iv) – cosec x + c.

(v)( ) ( )

3 21 21 2 1

– 2 12 3

xx c

+ + +� ��

(vi) log [log(tan x)] + c.

(vii) tan x + c. (viii)

1

.1 log

a xx a

ca a

+

+ ++

(ix) 2 – cos – sin .2 2

x xc +� �

� �(x) ( )1

log 2 3 log .3

x c+ +

(xi) –2 cos x c+ (xii) ( )2–1 31tan .

6x c+

(xiii) 2x (log x – 1) + c. (xiv)21

log 22

ax bx c c+ + +

(xv)1

– log cos .a b x cb

+ + (xvi)

1

1.log 1

x cc x

cc c

+

+ ++

226 XII – Maths

(xvii) log |3 + log x| + c. (xviii)–11 5

tan .20 4

x c+

(xix) ( )1 21tan .

2

xe c

− + (xx)1 3 2

log .12 3 2

xc

x

− ++

(xxi)1 4 5

log .40 4 5

xc

x

+ +−

(xxii)21

log 2 4 9 .2

x x c+ − +

(xxiii)1

log .2 –

a bxc

ab a bx

+ +

3. (i) .4

π(ii) 2e – 2. [Hint. : e|x| is even function]

(iii) 0; (iv) 0

(v) .4

π(vi) .

4

π

(vii)3 2

.12

π(viii)

–1tan .

4e

π−

(ix)3

.3

xc+ (x) 1 [Hint. : ea log x = xa.]

4. (i) ( )–1 22

1 1log cosec tan .

2x c

x

− +� ��

(ii) ( )2 2 21 11 log 1 .

2 2x x x x x c− − + + − +

[Hint. : Rationalize numerator and denominator.]

(iii) ( )( )( )

1 sinlog

sin sin

x ac

a b x b

− +− −

[Hint. : sin (a – b) = sin {(x – b) – (x – a)}].

(iv) x cos 2a – sin 2a log |sec (x – a)| + c. [Hint. : put xa = t]

(v) [ ]112 6 sin 2 3 sin 4 2 sin 6 .

48x x x x c+ + + +

(vi)3 52 1

sin sin sin .3 5

x x x c− + +

(vii)1 1 1 1

2 sin 2 sin 4 sin 6 .32 2 2 6

x x x x c + − − +� ��

227 XII – Maths

(viii)9

7 5cos 2 1cos cos .

9 7 5

xx x c− + − +

(ix)6 4

cot cot.

6 4

x xc

− + +� �

� �

(x) ( )2 2 2 2 2 2

1.

sin cosc

a b a x b x+

− +[Hint. : put a2 sin2 x + b2 cos2 x = t]

(xi) –2 cosec cos tan . sin .a a x a c− + [Hint. : Take sec2 x as numerator]

(xii) tan x – cot x – 3x + c.

(xiii) ( ) ( ) ( )3 2 25 2 5 2 5 22 2 4 21 1 1 .

5 7 5 3x x x c

+ + − + + +� �

� �[Hint. : put 1 + x5/2 = t]

(xiv) sin–1 (sin x – cos x) + c. (xv)41

log – log 1 .4

x x c+ +

(xvi) ( )− +11tan 2 tan .

2x c (xvii)

+ +−

1 3 tanlog .

2 3 3 tan

xc

(xviii) ( )+

x55

3

5.

log 5c [Hint. : put 5x = t]

5. (i)2

–11 2 1tan .

3 3

xc

+ +� ��

[Hint : put x2 = t]

(ii)2 log 1

log3 log 2

+ ++

xC

x [Hint : put log x = t]

(iii)− − +� ��

1 2 3sin .

13

xc

(iv) 2 log x a x b c− + − +

(vii)1 2 2cos

cos sin sin . log sin sin sincos

xx x c

− − α − α + − α +� �� α

( )( )

( )2 2

sin sin:

sin sin sin

x x

x x

− α − α=� �� + α − α� �Hint

(viii)−− + + + + +� ��

2 15 11 3 1log 3 2 1 tan

6 3 2 2

xx x c

228 XII – Maths

(ix)− + − + + + +� ��

2 1 33 log 6 12 2 3 tan

3

xx x x c

(x)2 1 2

4 4 sin2

xx x c

− − − − + +� ��

(xi) ( ) ( ) −− − + − + − + − + +� ��

32 2 121 1 5 2 1

1 2 1 1 sin3 8 16 5

xx x x x x c

(xii) ( )322 2 27 1 3 1

1 1 log 12 2 8 2

x x x x x x x x c � �+ + − + + + + + + + + +� ��

(xiii)21

log cos cos cos2

x x x c− + + + + [Hint. : multiply and divide by sec 1x + ]

(xiv) ( )2 12 log sin 4 sin 5 7 tan sin 2 c

−θ − θ + + θ − +

6. (i)7

7

1log

7 1

xc

x+

+

(ii) ( )2 2

2 22 2

1log

2

x bc

x ba b

− +−−

[Hint. : put x2 = t]

(iii)1 cos

log2 3 cos

xc

x

+ ++

(iv)1 3 2

log log 2 log 36 10 15

x x x c− + − − + +

(v)( )

3 1 5log 1 log 3

8 2 1 8x x c

x− − + + +

−

(vi)9 4 1

log 3 log 2 110 15 6

x x x c+ + − − + +

(vii)( )22

4 log1

xx c

x

−+ +−

(viii)1 12

tan 3 tan3 3 2

x xx c

− − + − +� �� [Hint. : put x2 = t]

(ix)11 2

log tan2 2 2

x xx c

x

−− + − +� �� +[divide numerator by denominator]

(x)2 12 1 1

log 2 1 log 4 tan17 17 34 2

xx x c

−+ − + + +

229 XII – Maths

(xi)1 1 2

log 1 cos log 1 cos log 1 2 cos2 6 3

x x x c− − − + + − +

[Hint. : multiply Nr and Dr by sin x and put cos x = t]

(xii)1 1 1

log 1 sin log 1 sin log 2 sin6 2 3

x x x c− − − − + − +

[Hint. : ( ) ( )

=− −∫ ∫ 2

1 cos

cos 2 sin cos 2 sin

xdx dx

x x x x]

[Hint. : Multiply numerator and denominator by cos x and put sin x = t]

(xiii)1 1 sin 1 1 2 sin

log log8 1 sin 4 2 1 2 sin

x xc

x x

− + ++ +− −

(xiv)2

2

1 1log

2 1

x xc

x x

− + ++ +

(xv)− − − + + +� �� + +

11 tan 1 1 tan 2 tan 1tan log

2 2 tan 2 2 tan 2 tan 1

x x xc

x x x

(xvi)2

11 9tan

3 2 3 2

xc

− − +� ��

(xvii) 1

2

logsec

1

xx c

x

−− + +−

[Hint. : Take log x as I function.

(xviii) ( )2 3 5− − + [Hint. : � � � � � �= +∫ ∫ ∫322

2 2 2

1 1 2

x dx x x dx ]

75. (i) 3 3 31cos sin

3x x x c − + +� �

(ii) [ ]1sec tan log sec tan

2x x x x c+ + +

[Hint. : Write sec3x = sec x . sec2 x and take sec x as first function]

(iii) ( ) ( )[ ] 12 2cos sin

axe

a bx c b bx c ca b

+ + + ++

(iv) 1 212 tan 3 log 1 9

3x x x c

− − + + [Hint. : put 3x = tan θ]

(v) [ ]2 sin cosx x x c+ +

(vi)( )

( ) ( )

21

2 3

log 2 log 2

1 1 1

n x xx c

n n n

+ � �− + +� �+ + +� �

230 XII – Maths

(vii)sin cos

sin cos

x x xc

x x x

− ++

[Hint. : ( )

( )( )

2

2

cossec

sin cos sin cos

x x xx x

x x x x x x=

+ +]

(viii) ( ) 11 2 sin

2x x x x x c

2 − −π− − − + 1 1cos – sin

2x x

− −π =� �� Hint. :

(xi)1 1

tan – tan .x x

x ax a ca a

− −+ + [Hint. : put x = a tan2 θ]

(x)4 3

11tan – .

4 12 4

x x xx c

− − + +� ��

(xi)

3

22 2

1 1 1 21 log 1 .

3 3c

x x

� � � �− + + − +� ��

(xii)21

tan .2

xe x c+

(xiii) .2

xe

cx

+

(xiv) 2.

1

xe

cx

++

(xv)1

.1

x xe c

x

− +� �� +

(xvi) ex tan x + c.

(xvii) + .log

xC

x

(xviii) log log – .log

xx x c

x+ [Hint. : put log x = t ⇒ x = et]

(xix) x sin (log x) + c.

8. (i)1

log 3.20

(ii) –π/2

(iii)1

– .4 2

π[Hint. : put x2 = t] (iv)

1– log 2.

4 2

π

(v)5 3

log 2 log 3.2 2

(vi) .2

π

231 XII – Maths

(vii)3

.8

π(viii) .

2

π

(ix)15 25 6

5 – 10 log log .8 2 5

+ � ��

(x) π/2.sin

. .1 cos 1 cos

� �+� �� + +� �Hint :

x xdx

x x

9. (i) 8. (ii) 0.

(iii) π. (iv) log 2.8

π

(v)–

log 2.2

π(vi)

21.

4π

(vii) 95/12. ( ) ( ) ( ) ( )−

− − −

� �= + +� ��

∫ ∫ ∫ ∫2 1 1 2

2 2 1 1

f x dx f x dx f x dx f x dxHint. :

(viii)2

.16

π

(ix)2

.2ab

π ( ) ( )

0 0

Use a a

f x f a x � �= −� ��

∫ ∫Hint. :

10. (i)26

.3

(ii)1

.ee

−

(iii)1

.2

(iv) 2

11 .

e−

(v) 10. (vi) – 9/2.

11. (i)2 11 1 2 tan 1 1

log tan tan 1 tan – log 1 tan6 3 3 3

− − − + + + +� ��

xx x x c

(ii) +52tan .

5x c (iii) 2.

(iv) log 2.2

π − (v) 0.

(vi) 0. (vii) 3.

(viii) π/2. (ix) 3.

232 XII – Maths

(x)3

log – log 1 22 4

xx x c+ − + ( ) ( )

0 0

. 1a a

xf x ax f a x put a t

� �= − + =� ��

∫ ∫Hint. :

12. (i)2 –15 3 1

4 log log 1 log 1 log 1 tan .4 4 2

x x x x x x c− + − + + + + − +

( ) ( ) ( )5

5 2

4 4. 1

1 1 1

x x

x x x x x x

+ +� �= +� �− − + +� �Hint :

(ii)( )

( )2 –12 1 1 3log 1 – log 4 tan .

25 5 1 25 50 2

xx x c

x

− −− + + +−

( ) ( ) ( )2 222

1.

1 1 41 4

A B cx D

x x xx x

+ = + +� �− − +− +� �Hint :

(iii)( )

1 81 272 log 1 log 3 .

8 8 2 3x x x c

x− + + − − +

−

(iv)11 2

log tan .2 2 2

x xx c

x

−− + − +� �� +

(v) 2.π

(vi)( )− − − +− +

+ λ +

2 21

2

1 1 1 2 1tan log

2 2 2 4 2 2 1

x x xc

x x

(vii) π/8.

(viii) log 2.2

π −

(ix) 8 log 2.π

(x)( )2

1.

2 1 cosc

x− +

−

(xi)− +

+1 tan

log .5 2 tan 1

x xc

x Divide Num. and Den. by cos2 x.

(xii)( ) 2

12 2 1 2sin .

x x xx x c

−− −+ − +π π

(xiii)−− − + + +1 2

2 1 cos – .x x x x c

(xiv) 2

3 1.+

π π

233 XII – Maths

13. (i) 14. (ii)26

.3

(iii) 26. (iv) ( )81127 .

2e+

(v)141

.2

(vi) 93/2.

234 XII – Maths

CHAPTER 8

APPLICATIONS OF INTEGRALS

1. πa2 sq. units. 2.28

3 sq. units

3. πab sq. units 4. 4π – 8 sq. units

5.−π + − � ��

12 9 9 1sin sq. units

6 8 8 37.

( )π − 2sq. units

4

ab

8. π −� ��

82 3 sq. units

39. (a) 4 sq. units (b) 2 sq. units

10. (a) 6 sq. unit [Hint. Coordinate of verties are (0, 1) (2, 3) (4, – 1)]

(b) 15

2 sq. [Hint. Coordinate of verties are (– 1, 1) (0, 5) (3, 2)]

11.π −� ��

1sq. units

4 212. 1 sq. units

13. ( )−2 2 sq. units [Hint. Required Area π π

π

+� �

� �∫ ∫4 2

0 4sin cosdx ndn ]

14. 5 sq. units

15. 4 sq. unit

[Hint.

y

y

x

= 3 – y

x

= 3 – y

x = – 1

yx

= – 1

(–1, 0) (2, 0)10x

16. 3 sq. units

17.25

2 π sq. units 18.

1

3 sq. units

19. ( )π −32 sq. units

220.

9sq. units

8

235 XII – Maths

21. ( )−− +2 11 sin sq. unitab e e e 22.

2

3

8sq. units

a

m

23.1

sq. unit6

24.π

sq. unit3

25. ( )π − 2 sq. unit 26. ( )− π1 4 sq. units

27. 2π sq. unit.

236 XII – Maths

CHAPTER 9

DIFFERENTIAL EQUATIONS

1.

(i) order = 1, degree = 1 (ii) order = 2, degree = 1

(iii) order = 4, degree = 1 (iv) order = 5, degree not defined.

(v) order = 2, degree = 2 (vi) order = 2, degree = 2

(vii) order = 3, degree = 2

2.

(i) = + − +6 3

2 log6 6

x xy x c (ii)

−= + +logx x

ey e e c

(iii)+

= + + ++

4 1

.4 1

exx x

y e ce

(iv) 5x + 5 –y = c

(v) 2(y – x) + sin 2y + sin 2x = c. (vi) 2 log |3x + 1| + loge |1 – 2y| = c.

(vii) y = x tan x – log |sec x| + c. (viii) y = c loge x.

3.

(i) esin x (ii) etan x

(iii) e–1/x (iv)( )2log

2

xe

(v) 3

1

x(vi) sec y

(vii)1

tan ye

−

(viii) e–y.

4.

(viii) [Hint : Family of circles having centre on x-axis and radius = 2 units is of the type (x – h)2 +y2 = 4]

Ans. � �+ =� ��

22

1 4.dy

ydx

(ix) − + =2

22 2 0

d y dyy

dxdx[Hint : find

2

2, and eliminate A and B.

dy d y

dx dx]

237 XII – Maths

(x) +� ��

22

2=

dy d y dyx xy y

dx dxdx(xi) =� ��

3

8 27dy

a ydx

(xii) + − + − =2

22

2 2 0.d y dy

x xy xdxdx

(xiii) = × −� � � ��

3

4 2dy dy

y x ydx dx

[ divide by and find .]dy

y cdx

Hint :

(xiv) + =� ��

2 3

22 0

d y dy

dxdx

5. (a) x = y – 1 + cey. [Hint : − = −2dx

x ydy

which is (L.D.equation)]

(b) = +3

2 sinsin

3

xy x c (c)

( )−= +

2

2

4 log 1

16

ex x cy

x

(d) ( ) −− −= − +1

1 tantan 1 .

yx y c e (e)

− −= +1

tan 1tan

yxe c y

(f) = + >sin , 0c

y x xx

(g) xey = ey + y + c

(h) y = tan x – 1 + ce–tan x (i) x = – y2e–y + cy2

6.

(i) ( ) ( )= + −2 1 2cy x y (ii) ( )+ =2 secx

e y c

(iii) − + − =2 21 1x y c

(iv)− −

= − − − +− +

22 2

2

1 11log 1 1

2 1 1

yx y c

y

(v) ( ) ( )+ + =2 21 1 2x y

(vi) = − + + − +3 51 1log cos cos

3 5

x xy x x xe e c

(vii) − =cos 2log tan

xy c

y(viii) 3 sec 2y + 4 cos 3x = c

(ix)( )− = + +2 1 3

3 6 2y x

e e x c (x) y = e–x

238 XII – Maths

7.

(i)− + =

3

3log

3

xy c

y

(ii)− = +� ��

1tan log

yx c

x[Hint. : Homogeneous Equation]

(iii) x2 + y2 = 2x (iv) c = x + y ex/y [Hint. : Put = .x

vy

]

(v)( )= cos x y

y ce [Hint. : Put =xv

y] (vi)

=� �� sin

ycx

x

(vii) ( )− =2 2c x y y

(viii)−− = + +

3

3

y x xe e c [Hint. : Factorise R.H.S.]

(ix) =� �� log

ycx

x[Hint. : Homogeneous diff. Equation]

(x) ( )+ = − +2

log 12

xy x c [Hint. : Factorise R.H.S.]

(xi) ( )= − − − + +1 22 sin 2 cosx

y x e x x x c x c where c1, c2 are arbitrary constant[Hint. : Integrate Twice]

(xii) − −= +1 1sin siny x c

8.

(i) − + =2 22 0

dyx y xy

dx[Hint. : The family of circles is, + + =2 2

2 0x y gx ]

(ii) = =2 , 2dy dy

y x y xdx dx

(iii) + =� ��

2 2

2

d y dy dyxy x y

dx dxdx

(iv) − + =2 22 0

dyx y xy

dx

(v) = � ��

3 2

3 2

d y dy d yy

dxdx dx[Hint. : Differentiate thrice and Eliminate a, b, c]

239 XII – Maths

9. = =� ��

sec Putx x

cxy vy y

10. − + + + = +� ��

2 2 1 2log 2 3 tan

3

x yx xy y c

x

11. ( )= ++

3

2 2

x cx y

xx y12.

(i) = − + 23 sin 3 2 cos 3

13 13

xx xy ce

(ii) = +22 1sin cosec

3 3y x x

13.

(i) ( ){ }− = −logy x c x y (ii) = + +2 2 2cx y x y

(iii) =� �� cos

yxy c

x[Hint. : Put y = vx]

(iv) = +23 2x y y x [Hint. : Put y = vx]

(v) ( )= − ≠log log , 0y x x x (vi) ( )+ = −2 2 2 2.c x y x y

240 XII – Maths

CHAPTER 10, 11

VECTOR AND 3-D

1. 5 3 5, .

2 22.

1

3±

3. and are collinear.x y��

4. 126 sq. units.

5. 60° 6. (6, 11)

7. (5, 4, –6) 8. �4 6 4 3+ +i j k� �

9. 1 1 1,

3 3 310.

1 1 13 6 2, ,

7 7 7

− − −−Cos Cos Cos

11. �3 4 1

26 26 26+ −i j k� � 12. 0

13. 4 14. λ = – 9

15. 2=a��

16. 90°

17. 3 units. 18.3

sq. units.2

19. 13 20. 90°

21. 120° 22. –1

23.4

Π24.

3 1.

2 2+i j� �

25. �− +� �3 6 2.

7 7 7i j k 26.

−3.

2

27. 3 28. 0

29. 60° 30. �+ +� �1 1 1.

2 2 2i j k

31. 2 : 3 enternally 32. 2 : 5

33. 2 2+b c 34.1 1 2

.2 3 4

− + −= =−

x y z

35. 90° 36. 2

241 XII – Maths

37.10

1438. –2

39.8

5

−40.

−5 12, , .13 13

O

41. 2 4 5.

5 5 6

+ − += =−

x y z 42. 1 2 3.

1 1 1

+ − += =x y z

43. q = 5 44. 90°

45. ± ±∓1 3 2

, , .14 14 14

46. 60°

47. 60° 48.1

units.6

49. x + y + z = 1. 50. No

51. 2x – y + 3z = – 8. 52. �( )+ + =��

� �. 2 2 24.r i j k

53. 2x + 3y + 5z = 29. 54. 0°

55. (–1, 2, –2) 56. 10

3 3

57. 0 61.

1

21

2

= +� �� = +� ��

PL PQ PS

PM PQ PS

��

��

63.2

2 a��

65. –6.

68.( )

�( )1

2

13

21

3 62

= −� �� = + −� ��

B i j

B i j k

��

��

70.1 11

.14

− � ��

Cos

71. x = – 2. 72. �1 1 3.

4 4 4+ +i j k� �

76. �1 4 4.

3 3 3+ +i j k� � 78. �( )1

10 7 4 .165

+ −i j k� �

79. a∈ ( – 3, –2) U (2, 3). 82. (0, –1, –3)

83. 7. 85. �( ) �( )2 2 2 2 .= − + + λ − +r i j k i j k��

� � � �

242 XII – Maths

86. �( ) �( )2 1 1, 2 2 2 2 .

1 2 3

− + += = = − + + λ − +x y zr i j k i j k��

� � � �

87. (– 9, 2, 1). 88. 29x – 27y – 22z = 85.

89. 13 units. 90. 7y + 4z – 5 = 0.

91. 4x – 7y + 3z – 28 = 0. 92. �( ). 5 2 3 7.− − =r i j k��

� �

93. 1 units. 94. x – 20y + 27z = 14.

96. x – 2y + 2z = 0. 98. 5x + 2y – 3z = 17.

99. �( ). 2 2 3 0, 2 2 3 0.+ + + = + + + =r i j k x y z��

� �

Or

�( ). 2 2 3 0, 2 2 3 0.− + − + = − + − + =r i j k x y z��

� �

100. 51x + 15y – 50z + 173 = 0.

101. 5x – 7y + 11 z + 4 = 0.

102. 3 2

2

103. �( ), 2 1− + =r i j k��

� �

104. 14 units.

105. (–1, –1, –1).

106. 8

29

107. = = = =− − −

; .1 2 1 1 1 2

x y z x y z

108. 2 2

109.−

+ + − = � ��

1 47 13 4 9 0, sin

234x y z

110. �( ) �( ). 3 30, 3 8 3 6 15 3 .= = + + + µ − − +S D r i j k i j k� � � � �

111. (3, 4, 2).

112. x + 2y – 2z = 6, 2x – 2y – z = 3.

114. �( )+ + =��

� �. 2 2.r i j k

243 XII – Maths

115. �2 3 , 14.+ +i j k� �

116. �( )4 3 52

+ +ai j k� �

HINTS FOR SELECTED QUESTIONS

1. Hz component = 2 cos (150°), Vertical component = 5 sin 150°.

5. 1tan .

.

− ×� �=� ��

a bQ

a b

��

��

9. l = m = n and l2 + m2 + m2 = 1.

17. = −BC AC AB��

25. Diagonal coinitial with = +a b is a b��

27.2 2 2 2

2 . + + − = +� �

a b a b a b��

28. is a vector to both and .× ⊥a i a i��

� �

36. Cos2 + cos2β + Cos2r = 1

42. d r’s are 1, 1, 1.

43. (1, 3, q) lies on the line.

57. Zero, as line is not parallel to plane.

63. ( ) � ( ) � �( ) �× = α × = β × = γ α��

� �2 22 2 22 2 2 2

sin , sin , sin and cosa i a a j a a k a + cos2β

+ cos2γ = 1.

71. ( )⊥ ×c a b��

79. �. 0, . 0, . 0.> > >r i r j r k��

� �

80. ( )is parallel to so .× = λ ×a b c a b c��

86. Change the line in symmetric form.

244 XII – Maths

CHAPTER 12

LINEAR PROGRAMMING

1. Maximum profit = Rs. 95 with 5 shares of each type.

2. Minimum cost = Rs. 1350 No. of days for which A wark 5 days

No. of days for which B wark = 3 days.

= � ��

3. Maximum profit = Rs. 784 Dinner sets of type I = 8

Dinner sets of type II = 12 � ��

4. For minimum profits, Lamps off type A = 40

Lamps of type B = 20 � ��

5. Minimum transportation cost = Rs. 300.

No. of pactets transported from A to P = 10, A to Q = Nil, A to R = 50

No. of packets transported from B top = 30, B to Q = 40, B → R = 10.

6. Minimum cost = Rs. 440 Fertilizer of type I = 100 kg.

Fertilizer of type II = 80 kg. � ��

7. No of flights of model 314 type = 6.

No of flights of model 535 type = 2.

8. For maximum profit, No. of units of type A = 26

No. of units of type B = 26

9. Minimum cost = Rs. 7 14

Quantity of bran =10

Quantity of rice = Nil

10. Minimum cost = Rs. 41 Food of type A = 5 kg.

Food of type B = 2kg.

11. Maximum distance = 30 km, when

50Distance covered with a speed of 25 km/hr = km

3

40Distance covered with a speed of 40 km/hr = km.

3

12. Maximum income = Rs. 8400, when

245 XII – Maths

Investment in P.P. F = Rs. 80,000

Investment in bonds = Rs. Rs. 20,000

13. From factory P to A, B, C are 8, 2, 0 respectively

From factory Q to A, B, C are 0, 6, 9 respectively.

14. From X to A, B, C are 50q, 40q, Nil respectively.

From Y to A, B, C are 50q, Nil, Nil respectively.

15. Manimum value 1 84 15

14 at ,3 13 13

� ��

.

Minimum value is 1 9 25

3 at , .14 14 14

� ��

16. Minimum value of z = 19 at x = 3, y = 2

Maximum value of z = 42 at x = 4, y = 6.

17. (i) Maximum of this problem does not exist.

(ii) Manimum value is infinity or solution is unbounded.

18. Maximum profit = Rs. 20,100 when 40 trees of type A and 30 trees of type B are planted.

19. Maximum value is 19 at 7 3

,2 4

� ��

Minimum value is 63 3 24

at ,13 13 13

� � .

20. Manimum profit = Rs. 1300.

No. of belts of type A = 200

No. of belts of type B = 600.

246 XII – Maths

CHAPTER 13

PROBABILITY

1. 4.

13

2. (i) 1

2

(ii) 1

3

3. (i) 3

13

(ii) 3

52

4. 1

5 [Hint : P(A´ ∪B´) = P (Α ∩B)´]

5. Not defined.

6. 1

3

7. 3

25

8. 2 144 24 1: 0 1 2 .

3 169 169 169

= + × + ×� ��

Hint u x

9. K = 0.2, P (x = 3) = 0.4.

10. (i) 8 1 2: 4,

81 3 3

= = ∴ =� ��

Hint n p q

(ii) ( )3

332 1 2

3 4 .81 3 3

= = � ��

P x c

11. 1 Hint.

12.10

2

3

� ��

13. 75

216

247 XII – Maths

14. 1

216

15. 2

3

16. 1

17. 2

7

18. 1

36

19. 0.0256

20. 7.5 [Hint : Expectation = E(x) = np]

21. (i) S = {bb, bg, gb, gg}. (ii)1

3

(iii) 1

3(iv) 1

2.

22. (i) 3

8(ii)

2.

8

23. (i) 1

3(ii) 2

9

(iii) 0

24.1

4

=� ��

AP

E

1

4

=� ��

BP

E

1

4

=� ��

CP

E

25. 0.6976.

26. 19

42

27. 11

50

28. 0.083

248 XII – Maths

29. 9

13

30. ( ) ( ) ( ) ( )5 15 15 10 , 1 , 2 , 3 .

28 28 56 56= = = =P P P P

31. ( ) ( ) ( ) ( )27 27 9 10 , 1 , 2 , 3 .

64 64 64 64= = = =P P P P

32. 13

16[Hint : Required probability = 1 – P (X > 3)

= 1 – {P(X = 4) + P (X = 5)}

33. 4 [Hint : P (X ≥ · 1) > 0.99

1– P (X = 0) > 0.99

01

1 . 0.99]4

− >ncn

34. Expected to lose Rs. 91.

54

35. (i) 193

512(ii) 53

64

36. 11

243

37. ( ) ( ) ( ) ( )125 75 15 10 , 1 , 2 , 3 .

216 216 216 216= = = =P P P P

{Hint : Possible doublets are (1, 1), (2, 2) , (3, 3) (4, 4), (5, 5) (6, 6)}

38. ( )32

1 132 ; 0, 1, 2..., 32

2 2

− = = =� � � �� r

r r

cP X r r .

{Hint : np + npq = 24 and np × npq = 128}

39. (i) A and B are independent.

(ii) B and C are independent.

(iii) A and C are independent.

(iv) A, B and C are independent. [Hint : Draw veen diagram.]

40. (i) C and M are not independent.

(ii) M and S not independent.

(iii) S and M are not independent.

249 XII – Maths

(iv) If (i), (ii), (iii) are not independent C, M and S are also not independent.

41. (i) 25

58

(ii) 9

29

(iii) 15

58

42. 2

37[Hint : E : doing question correctly

K : he knows

C : he copies

G : he guesses

L : he leaves the question]

( ) ( ) ( ) ( )3 1 1 2, , ,

10 5 10 5= = = =P K P C P G P L

( ) ( ) ( ) ( )1 1/ 1, / , / , / 0

4 5= = = =P E K P E C P E G P E L

Now use Baye’s theorem.

43. 3

7

44. X 0 1 2 3

P(X)4324

5525

1128

552572

5525

1

5525

3 588Mean , Variance

13 2873= =

Hint : ( ) ( )Mean µ = ∑ i iX P X

( )2 2 2Variance σ = ∑ − µi iX P

45. X 0 1 2 3

P(X)1728

2197

432

2197

36

2197

1

2197

250 XII – Maths

505Mean , Variance 0.2661

2197= =

46. X 0 1 2 3 4 5

P(X) 3125

7776

3125

7776

1250

7776

250

7776

250

7776

1

7776

( ) ( )25 275Mean , Variance

6 36µ = σ =

47. (i) 1

18(ii) 1

2

(iii)1

2(iv) 17

18

48. (i) 5

41(ii) 36

41

49. (i) 1

7=K (ii) ( ) 3

37

> =P X

(iii) ( ) 13

7> =P X

50. (i) 21

32(ii) 11

32

(iii) Mean = 3, Variance = 1.5

51. (i)7

5

6

� ��

(ii)7

135

6

� ��

(iii)5

1

6

� ��

(iv)7

11

6

− � ��

( ) 6 1: Here

36 6

− = = = =� �

� �Hint

r

r n rcP X r n p q P

MATHEMATICS - WordPress.com...116 XII – Maths CLASS XII MATHEMATICS Units Weightage (Marks) (i)...

Documents

Transcript of MATHEMATICS - WordPress.com...116 XII – Maths CLASS XII MATHEMATICS Units Weightage (Marks) (i)...