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91 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 Marks

each 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 answer

type.

N O T E

Questions requiring Higher Order thinking skills (HOTS) have been added in every chapter.

92 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 functions 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.

93 XII – Maths

VERY SHORT ANSWER T YPE QUESTIONS

1. If A is the set of students of a school then write, which 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 students studying in same class}

R4 = {(a, b) : a, b are age of students and a > b}

2. Is the relation R in the set A = {1, 2, 3, 4, 5} defined as R = {(a, b) : b = a + 1} reflexive?

3. If R, be a relation in set N given by

R = {(a, b) : a = b – 3, b > 5}

Does elements (5, 7) ∈ R?

4. If f : {1, 3} → {1, 2, 5} and g : {1, 2, 5} → {1, 2, 3, 4} be given by

f = {(1, 2,), (3, 5)}, g = {(1, 3), (2, 3), (5, 1)}

Write down gof.

5. Let g, f : R → R be defined by

( ) ( )x 2g x , f x 3x 2.

3

+= = − Write fog.

6. If f : R → R defined by

( ) 2 1

5

−=

xf x

be an invertible function, write f –1(x).

7. If ( ) 1,1

= ∀ ≠ −+x

f x xx

Write fo f(x).

8. Let * is a Binary operation defined on R, then if

(i) a * b = a + b + ab, write 3 * 2

(ii)( )2

*3

+=

a ba b , Write (2 * 3) * 4.

(iii) a * b = 4a – 9b2, Write (1 * 2) * 3.

94 XII – Maths

9. What is the number of bijective function from a set A to B, when A and B have same number of

elements.

10. If f, g : R → R be defined by

( ) ( )3 7 8 7, , then

8 3

− += =

x xf x g x

What is fog (7).

SHORT ANSWER TYPE QUESTIONS (4 MARKS)

11. Determine whether each of the following relations are

(i) Reflexive (ii) symmetric (iii) Transitive (iv) Equivalence relation..

(a) R1 = {(a, b) : a ≥ b, a, b ∈ R}

(b) R2 = {(a, b) : (a – b) is a multiple of 3, a, b ∈ R}

(c) R3 = {(a, b) : (a – b) is an even integer, a, b ∈ R}

(d) R4 = {(a, b) : 3a – b = 0, a, b ∈ R}

(e) R5 = {(a, b) : a ≤ b3, a, b ∈ R}

(f) R6 = {(a, b) : a = b + 2, a, b ∈ R}

12. Check the following functions for one-one and onto.

(a) ( ) −→ =

3 2: ,

5

xf R R f x

(b) ( ) 2: , 4→ = +f R R f x x

(c) ( ): , 3→ = −f R R f x x

(d) ( ): , ,1

→ =+x

f A R f xx

where A = R – {–1}.

13. Let f : R → R be a function defined by ( ) 2 3.

7

−=

xf x Show that f is invertible and hence find

f –1.

14. Let 4 4

:3 3

− − → −

f R R be a function given by ( ) 4

.3 4

=+x

f xx

Show that f is invertible with ( )1 4.

4 3

− =−

xf x

x

95 XII – Maths

15. Show that function f : A → B defined as ( ) 3 4

5 7

+=

−x

f xx

where 7 3

,5 5

= − = −

A R B R

is invertible and hence find f –1.

16. Let * be a binary operation on Q. Such that a * b = a + b – ab.

(i) Prove that * is Commutative and associative.

(ii) Find identify element of * in Q (if exists).

17. If * is a binary operation defined on R – {0} defined by 2

2* ,=

aa b

b then check * for cummutativity

and associativity.

18. If A = N × N and binary operation * is defined on A as (a, b) * (c, d) = (ac, bd).

(i) Check * for commutativity and associativity.

(ii) Find the identity element for * in A (If exists).

19. Show that the relation R defined by (a, b) R(c, d) ⇔ a + d = b + c on the set N × N is an

equivalence relation.

20. Let * be a binary operation on set Q defined by * ,4

=ab

a b show that

(i) 4 is the identity element of * on Q.

(ii) Every non zero element of Q is invertible with

{ }1 16, 0 .− = ∈ −a a Q

a

H.O.T.S.

VERY SHORT ANSWER TYPE QUESTIONS (1 Mark)

21. Let f : A → B defined as ( ) ( )1

4 4= −f x a x where A, B ⊂ R, what is fof (x).

22. A relation R in the set R of real numbers is defined as

R = {(a, b) : a ≤ b2}. Is R reflexive?

SHORT ANSWER TYPE QUESTIONS (4 Marks)

23. Consider f : R+ → [–5, ∞] as f(x) = 9x2 + 6x – 5. Show that f is invertible and Hence find f –1.

96 XII – Maths

A N S W E R S

1. R1 – Universal relation.

R2 – Empty relation

R3 – Neither empty nor universal.

R4 – Neither empty nor universal.

2. Not reflexive.

3. (5, 7) ∉ R

4. gof = {(1, 3), (3, 1)}.

5. fog = IR i.e. fog (x) = x.

6. ( )1 5 1

2

− +=

xf x

7. ( )2 1

=+

xfof x

x

8. (i) 11 (ii)1369

27(iii) –209

9. n !

10. 7.

11. (a) Reflexive and Transitive but not symmetric

(b) Equivalence relation

(c) Equivalence relation

(d) Neither reflexive nor symmetric nor transitive

(e) Neither reflexive nor symmetric nor transitive

(f) Neither reflexive nor symmetrix nor transitive.

12. (a) Bijective function (i.e., one-one and onto).

(b) Neither one-one nor onto.

(c) Neither one-one nor onto.

(d) one-one but not onto.

13. ( )1 7 3.

2

− +=

xf x

97 XII – Maths

15. ( )1 7 4.

5 3

− +=

−y

f xy

16. (ii) 0 is the identity element of * in Q.

17. (i) * is not commutative.

(ii) * is not associative.

18. (i) * is commutative as well as associative

(ii) (iii) is identity element.

ANSWERS OF HOTS

21. x

22. It is not reflexive.

23. ( )− + −=1 6 1

.3

xf x

98 XII – Maths

CHAPTER 2

INVERSE TRIGONOMETRIC FUNCTIONS

POINTS TO REMEMBER

1. sin–1 x, cos–1 x, ... etc., are angles.

2. sin–1 (sin x) = x and sin (sin–1 y) [ ]−π π = ∈ ∈ −

, , , 1, 1 .2 2

y x y

3.1 1 1 1 11 1 1

cosec sin ; sec cos ; cot tan− − − − − = = =

x xx x x

4. sin–1 (–x) = – sin–1 x; tan–1(–x) = –tan–1 x; cosec–1 (–x) = – cosec–1 x.

5. cos–1 (–x) = π – cos–1 x; sec–1 (–x) = π – sec–1 x; cot–1 (– x) = π – cot–1 x.

6.1 1 1 1sin cos or cos sin .

2 2

− − − −π π+ = = −x x x x

7.1 1 1 1tan cot or cot tan .

2 2

− − − −π π+ = = −x x x x

8.1 1 1 1sec cosec or cosec sec .

2 2

− − − −π π+ = = −x x x x

9.1 1 1tan tan tan ; 1.

1

− − − + + = < −

x yx y xy

xy

10.− − − −

− = > − + 1 1 1tan tan tan ; 1.

1

x yx y xy

xy

11. Infact every formula in trigonometry can be written in the language of inverse trigonometric

functions.

Function Domain Range/Principal Value Branch Graph

Inverse sine function [–1, +1] ,2 2

π π −

1

sin–1 x = y ⇔ x = sin y

Inverse cosine function [–1, +1] [0, π]

1

cos–1 x = y ⇔ x = cos y

99 XII – Maths

Inverse tangent function R ,2 2

π π −

Yπ2

–π2

⇔ tan–1 x = y

Inverse cosecant function ] ] [ [, 1 1,−∞ − ∪ + + ∞ [ ], 0 0,2 2

π π − ∪

y = cosec–1 x iff

x = cosec y

Inverse secant function ]–∞, –1] ∪ [+1, +∞[ [ ] { }ππ −0,

2

y = sec–1 x

⇒ x = sec y

Inverse cotangent function R ]0, π[

y = cot–1 x

⇒ x = cot y

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

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

100 XII – Maths

(xi) sec–1 (– 2). (xii) sec–1 (2).

(xiii) ( )− − − − − + + −

1 1 13 1sin cos tan 1 3

2 2

2. What is value of the following functions (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 –1cosec 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


3. Show that –1 1 cos 1 cos

tan .1 cos – 1 – cos 4 2

x x x

x x

+ + − π= +

+

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

1 sin 1 cos 4

− − + π − = ∈ π − −

x xx

x x

5. Prove

2 2–1 –1 –1

2 2tan sin cos .

–

x x a x

a aa x

− = =

6. Prove–1 –1 –1 –1 –18 8 300

cot 2 tan cos tan 2 tan sin tan .17 17 161

+ =

7. Prove− −

+ + − π = + + − −

2 21 1 2

2 2

1 1 1tan cos .

4 21 1

x xx

x x

8. Solve–1 –1

cot 2 cot 3 .4

x xπ

+ =

101 XII – Maths

9. Solveπ

=–1 –1

tan 2 – tan 3 .4

x x

10. Prove2

1 1

2

1 2tan tan .

2 21

− − − π + = −

x x

x x

11. Prove1 1tan tan .

4

− − − π− =

+

m m n

n m n

12. Prove − − − +

+ = −+ +

21 1

2 2

1 2 1 1tan sin cos .

2 2 11 1

x y x y

xyx y

13. Solve for x

21 1

2 2

1 1 2 2cos tan .

2 31 1

− − − π+ = + −

x x

x x

H.O.T.S.


14. − −− = − +

1 1cos sintan tan

cos sin

a x b x ax

b x a x b if tan 1 0.+ >

ax

b

15. Prove ( ) ( )1 1 1 2 1 21cot tan tan cos 1 2 cos 2 1 .− − − −

+ + − + − = π x x x

x

16. Prove 1 1 1tan tan tan 0.1 1 1

− − −− − − + + = + + +

a b b c c a

ab bc ac If a, b, c > 0

17. Find the value of 1

cot sin – .2

− π

18. Find value of x for

2 tan–1 (cos x) = tan–1 (2 cosec x)

19. Express the following in simplest form

1 2sin 1 1 .− − − − x x x x

20. If tan–1 a + tan–1 b + tan–1 c = π then prove that a + b + c = abc.

102 XII – Maths


21. What is value of x if

1 11sin sin cos .

5 2

− − π + =

x

22. If f(x) = cos–1 (log x) then what is value of f(1) + f(e)

23. What is range of y = sin–1 [x], where [ ] is greatest integer function.

24. What is value of 1 1 1cot sec sin .− −

+ x

x

A N S W E R S

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)–

3

π(x)

3

π(xi)

2

3

π(xii)

3

π

(xiii) .2

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

8. x = 1. 9.1

.6

=x

13. √3 . 17. .4

−π

18. 0 or .4

π=x 19. sin–1 x – sin–1 √x.

21. x = 1. 22. .2

π

23. { }– , 0, .2 2

π π24. 0.

103 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 or functions

are called the elements of the matrix.

Order of Matrix : A matrix having ‘m’ rows and ‘n’ columns 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 are

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 matrix or

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 interchanging

the rows and columns of A is called the transpose of the matrix. If A = [aij]m × n. Then transpose

A = 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.. or aij = aji ∀ & .i j

Skew Symmetric Matrix : A square matrix A = [aij] is said to be a skew symmetric matrix if

A´ = –A. or aij = – aji ∀ & .i j

Inverse of a Matrix : Inverse of a square matrix.

1, provided 0.

−= ≠

Adj AA A

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 a nonsingular

matrix.

Determinant : To every square matrix A = [aij] of order n × n, we can associate a number (real or

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)1 = B1A1 Adj (AB) = (Adj B) (Adj A)

104 XII – Maths

(AB)–1 = B–1A–1

(A´)–1 = (A–1)´ Adj A´ = (Adj A´)

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

1 1

2 2

3 3

1

1 0.

1

⇔ =

x y

x y

x y

VERY SHORT ANSWER TYPE QUESTIONS (1 Mark)

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

ij

i j if i ja

i 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 of

the 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 ifij

i j i ja

i 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 the matrix

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. Write the matrix X if −

− =

8 2 7 53 .

6 0 0 0X

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

105 XII – Maths

10. If 2 3

1 0A P Q

= = +

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

matrix P.

11. If cos 20 sin 20

sin 70 cos 70A

° ° = ° °

, what is |A|?

12. Find the value of the determinants .– –

a ib c id

c id a ib

+ +

+

13. Find the value of xy if

3

3

3 8–4.

–4 4

x

y=

14. Write the cofactor of the element 5 in the determinant

2 –3 6

6 0 4 .

1 5 –7

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

a d g

b e h

c f i

16. If 3 1 –1 1

,5 – 5 2

x

x= find the values (s) of x.

17. 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?

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

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

20. If 02

xπ

< < and the matrix 2 sin 3

1 2 sin

x

x

is singular, find the value of x.

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

1

λ λ +

has no inverse?

22. If 5 –3

,6 8

A

=

find adj (adj A)

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

24. If the matrix sin cos

, find ´.– cos sin

A AAα α

= α α

106 XII – Maths

25. If –2 –1 3 0

, .3 0 –1 2

B and C

= =

Find 2B – 3C.

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

27. Find a 2 × 2 matrix B such that.

6 5 11 0.

5 6 0 11B

=

28. If 2

2 1 3 3 2, find and .

0 60 5

x y x yx y

y y

+ + + =

+

29. If

3 0 0

0 0 .

0 0 3

A x

=

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

30. Find ∆ if .

a b b c c a

b c c a a b

c a a b b c

− − − ∆ = − − − − − −

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

6 3A

x

=

is singular?

32. If 5 –2

,3 –2

A

=

write the matrix A(adj A).

33. Find the value of

0 0

0 .

P

a q

b c r

34. If A is a 2 × 2 matrix and ( )12 0

adj ,0 12

A A

=

what is |A|.

35. If .

1 1 1

x y y z z x

A z x y

+ + +

= Write the value of det A.

36. If 4 2

2 3 1

xA

x x

+ = − +

is symmetric matrix, then find x.


37. Find x, y, z and w if 2 –1 5

.2 3 0 13

x y x z

x y x w

− + = − +

107 XII – Maths

38. Find A and B if 1 –2 3 3 0 1

2 38 and – 2 .2 0 –1 –1 6 2

A A B

+ = =

39. 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).

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

.4 5 6 2 4 6

X− −

=

41. If [ ]–1

2 and 2 1 4 ,

3

A B

= = − − −

verify that (AB)´ = B´A´.

42. 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.

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

.2 2

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

1 1 2 5 1 2 0.

15 3 2

x

x

=

45. If 4 3

,2 5

A

=

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

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

1 2 3

3 1 2 .

1 0 3

A

=

47. Given that 2 3

.4 7

A−

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

48. Given that matrix 2 1

.3 2

A−

=

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

49. Show that 2 3

3 4A

− =

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

108 XII – Maths

50. Prove that the product of two matrices.

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

51. Show that :

( )

( )( )

sin cos sin

sin cos sin 0.

sin cos sin

α α α + δ

β β β + δ =

γ γ γ + δ

52. Using the properties of determinant, prove the following questions 52 to 56.

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 b

53. 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

+ + +

+ + + =

+ + +

54.

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

+

+ =

+

55. ( )2.

x a b c

a x b c x x a b c

a b x c

+

+ = + + +

+

56. Show that :

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

x y z

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

yz zx xy

= − − − + +

57. (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−

= = =

109 XII – Maths

58. Given

0 10 1 2

and 1 0 .2 2 0

1 1

A B

− = = −

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

59. 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

+ − −

− + − =

− − +

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

and .5 3 3 4

A B

= =

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

LONG ANSWER TYPE QUESTIONS (6 MARKS)

62. Let 2 3

1 2A

= −

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

63. 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.

64. Find the matrix X for which

3 2 1 1 2 1.

7 5 2 1 0 4X

− − = −

65. Using elementary transformations, find the inverse of the matrices.

2 3 3

2 2 3 .

3 2 2

− −

66. 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

67. Solve the system of linear equations by using matrix in equations.

2x – y + 4z = 1

3x – z = 2

x – y – 2z = 3

110 XII – Maths

68. 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

69. The sum 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 it

algebraically and find the numbers using matrix method.

70. Compute the inverse of the matrix.

3 1 1

15 6 5

5 2 5

A

− = − − −

and verify that A–1 A = I3.

71. 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.

72. 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.

73. 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+ =

74. 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.

111 XII – Maths

H.O.T.S.


75. How many matrices of order 2 × 3 are possible with each entry as 0 or 1.

76. If 2 sin x -1 3 0

, 0 , and1 sin x 4 sin2

π∈ ≤ ≤ =

−x R x

x

Then find the value of x

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

78. If 0 0

,3 0

=−

A find A20.


79. If

0 tan2

tan 02

θ−

=θ

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

I + A = (I – A) θ − θ

θ θ

cos sin.

sin cos

80. Using properties of determinants, show that

( )

( )

( )

( ) ( ) ( ) ( ) ( )

2 2

2 2 2 2 2

2 2

.

+

+ = − − − + + + +

+

b c a bc

c a b ca a b b c c a a b c a b c

a b c ab

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

log 10 1

log 13 1 .

log 15 1

∆ =

x

y

z


82. Show that

( )( )

( )( )

2

32

2

2 .

+

+ = + +

+

y z xy zx

xy x z yz xyz x y z

xz yz x y

112 XII – Maths

83. If

3 2 1

4 1 2 ,

7 3 3

= −

−

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

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

A N S W E R S

1.0 3

.1 0

2. 2xm.

3.1 2

.0 1

4. [18]

5. 10λ. 6. 3 × 3.

7. 5. 8.5 1

.2 0

x

=

9.1 0 0 0

, .0 0 0 1

A B

= =

10.2 2

.2 0

11. 0. 12. a2 + b2 + c2 + d2.

13. (–3)1/3. 14. 28.

15. id – fg. 16.2

.3

± [Hint. : – 3x2 – 5 = – 7].

17. 0. 18. 54. [Hint. : order 3 ⇒ |–3A| = (–3)3 |A|].

19. x = any real number. 20. .3

π

21.3

– .8

λ = 22.5 –3

.6 8

23. 40. 24. I2.

25.13 2

.9 6

− −

26. 25.

27.6 5

.5 6

− −

28. x = 2, y = 1.

29. 3. 30. 0; [Hint. : [R1 → R

1 + R

2 + R

3]]

113 XII – Maths

31. x = – 4. 32.4 0

.0 4

− −

33. pqr. 34. 12.

35. 0. 36. 5.

37. x = 1, y = 2, z = 3, w = 4.

38.

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

− −

= =

− −

40.1 2

.2 0

X−

=

43.

1 3

.71

2

− −

44. x = – 2 or x = – 14. 45. x = 9, y = 14.

46. – 14I3 47.–1 7 31

4 22A

=

48.–1 2 11

.3 27

A

= − 49.

–1 4 31.

3 217A

= −

58. ( )–11 2 2 21, .

2 2 2 16AB AB

− = = − −

61.11 1

, .24 24

x y= =

62.5 118 93

31 118A

− − = −

. [Hint. : A2 – 4A + 7I = 0, A2 = 4A – 7I, A3 = 4 (4A – 7I – 7A)

64.16 3

.24 5

X−

= − [Hint. : if A × B = P, X = A–1 P B–1]

65.

2 0 31

1 1 0 .5

2 1 2

− − −

67.10 –31 –8

, , .19 19 19

x y z= = =

68.–1

6 17 13–1

14 5 8 .67

15 9 1

A

− = − − −

x = 3, y = – 2, z = 1.

114 XII – Maths

69. x = 1, y = – 2, z = 2. [Hint. : Suppose three numbers as x, y, z]

70.–1

2 0 1

5 1 0 .

0 1 3

A

− =

71. ( )− = − −

1

16 12 11

21 11 7 .19

10 2 3

AB [Hint. : (AB)–1 = B–1A–1]

72. x = 3, y = – 2, z = – 1.

73. x = 2, y = 3, z = 5. [Hint. : Let 1 1 1

, , .u v wx y z

= = = ]

74.–1

3 4 51

9 1 4 .11

5 3 1

A

− = − − − −

75. 64.

76. , .6 2

π π

77. +5√5

78. 0

81. 0.

83. x = 1, y = 1, z = 1.

115 XII – Maths

CHAPTER 5

CONTINUITY AND 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., ( ) ( ) ( )– +→ →

= =x c x cLt f x Lt f x f c

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

� f(x) is continuous in [a, b] iff it is continuous in (a, b) and ( ) ( )lim+→

=x a

f x f a

( ) ( )–

lim→

=x b

f x f b

� f(x) and g(x) are continuous functions at x = c and d is constant then,

f + g, f – g, df, f · g, f + d, |f | are all continuous at x = c.

�

( )( )

f x

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

� Every polynomial function is continuous on R.

� Every trigonometric function, Exponential function and logarithmic function are continuous in their

respective domain.

� f(x) is derivable at x = c iff

( ) ( )lim→

−−x c

f x f c

x c exists and value of this limit is called the derivative at x = c and is

denoted by f´(c).

� ( ). .= +d dv du

u v u vdx dx dx

�

−

= 2

dv dvv u

d u dx dxdx v v

·

� If y = f(u), x = g(u) then

( )( )

´.

´=

dy f u

dx g u

116 XII – Maths

� If y is a function of t and t is a function of x then, .= ×dy dy dt

dx dt dx

� 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 there

exists 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 these exists atleast one point

c ∈ (a, b) such that ( )( ) ( )–

´ .f b f a

f cb a

=−


1. Write for what value of x, f(x) = |3x + 1| is not derivable.

2. Write the set of points of discontinuity of the function, g(x) = ||x – 1||

3. What is derivative of f(x) = |x – 1| at x = 1.

4. What are the points of discontinuity of the function ( )2

3

1.

+ +=

−

x xf x

x x

5. Write all the points of discontinuity of the function f(x) = [x] in [–1, 3], where [x] denotes the

greatest integer function.

6. At what point ( ) ( ) , 0sgn

0, 0

≠

= = =

xx

f x x x

x

is discontinuous.

7. Is the function e–x sin x is continuous on R?

8. If ( )2

1, 1

3 1

3 1

λ + < = = >

x x

f x x

x x

then for which value of λ, f(x) is continuous on R.

9. Write the value of k, for which ( )

sin 3, 0

2

, 02

≠

= =

xx

xf x

kx

is continuous .∀ ∈x R

10. What is the derivative of x6 w.r.t. x2.

11. Given that g(0) = 7 and f(x) = x g(x). Also f´(x) and g´(x) exist, then write value of f´(0).

12. Write the derivatives of the following functions.

(i) log2 (2x – 1) (ii) e3 log x

117 XII – Maths

(iii) 2 x (iv) 1 2 1 2tan cot− −+x x

(v) ( )− ≤ ≤1sin 0 1.x x x


13. Discuss the continuity of following functions at the indicated points.

(i) ( ) , 0at 0.

2, 0

−≠

= = =

x xx

f x xx

x

(ii) ( )tan 3

, 0at 0.5

5 3 0

≠

= = =

xx

f x xx

x

(iii) ( ) ( )2 sin 1 0at 0.

0 0

≠ = =

=

x x xf x x

x

(iv) f(x) = |x| + |x – 1| at x = 1.

(v) ( )sin

, 0at 0.

1 0

≠

= = =

xx

f x xx

x

(vi) ( )

< = =

+ ≥

sin 2, 0

at 0.

2 0

xx

f x xx

x x

(vii) ( )[ ] − ≠

= = =

, 0́at 1.

0 0́

x x xf x x

x

14. 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.

15. For what values of a and b

( )

+ + <

+ = + =

+ + > +

2if –2

2

if –2

22 if –2

2

xa x

x

f x a b x

xb x

x

is continuous at x = 2.

16. Prove that f(x) = |x + 3| is continuous at x = –3 but not derivable at x = –3.

118 XII – Maths

17. If ( )1

sin , 0

0 0

≠ =

=

px xf x x

x

is derivable at x = 0 the find the value of p.

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

dx

19. If − −

= + > −

1 1

2

1 2 1tan 2 tan , 0, find .

2 1

x dyy x

x dxx

20. If 1 1

sin 2 tan then ?1

− −= =

+

x dyy

x dx

21. If 3x + 3y = 3x + y then Prove that –3 .−= y xdy

dx

22. If y = tan–1 x then show that ( )2

2

21 2 0.+ + =

d y dyx x

dxdx

23. If 2 21 1− + − =x y y x a then show that 2

2

1.

1

−= −

−

dy y

dx x

24. If ( )2 21 1− + − = −x y a x y then show that 2

2

1.

1

−= +

−

dy y

dx x

25. If (x + y)r + s = xr + ys then prove that .=dy y

dx x

26. If 1 1 sin 1 sintan where

21 sin 1 sin

− + − − π= < < π + + −

x xy x

x x find .

dy

dx

27. Find the derivative of 1

2

2tan

1

− −

x

x with respect to 1

2

2sin .

1

− +

x

x

28. Find derivative of log (sin x) w.r.t. log x.

29. If 1

sin log

= x y

a then show that (1 – x2) y´´ – xy´ – a2y = 0.

30. If xy + yx + xx = ab then find ?=dy

dx

31. If x = a cos3 θ, y = a sin3 θ then find

2

2.

d y

dx

32. If x = aeθ (sin θ – cos θ)

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

π=

dyx

dx

119 XII – Maths

33. If 1 2sin 1 1− = − − − y x x x x then find .dy

dx

34. If ( )( )

32 1

4

sin,

tan

−

=xe x

yx x

find .dy

dx

H.O.T.S.

35. If 2

1 12 5 1sin ,

13

− + − =

x xy ?=

dy

dx

36. If yx = xy, find .dy

dx

37. If sin y = x sin (a + y) then show that ( ) { }

2sin,

sin

+= ≠ π ∈

a ydyx n n z

dx a

38. If y = sin–1 x, find 2

2

d y

dx in terms of y.

39. If 2 2

2 21+ =

x y

a b then show that

2 4

2 2 3.

−=

d y b

dx a y

40. If y3 – 3ax2 + x3 = 0 then Prove that 2 2 2

2 5

2.

−=

d y a x

dx y

41. If ax2 + 2bxy + by2 = 1 the prove that ( )

−=

+

2 2

2 3.

d y h ab

dx hx by

42. Write the points of discontinuity of f(x) = [x] in [3, 9].

43. Write the critical points of ( )2

log .2

= −x

f x x

44. Evaluate [ ]( )3

lim→

−x

x x , where [ . ] denotes the greatest integer function.

45. If for a function f(x),

f´(x) = (x – 2)2 (x + 1) then write the interval in which f(x) is increasing or decreasing.

46. If ( )3 7, 1

2 5 1

10 1

+ <

= − = >

ax x

f x bx x

x x

is continuous for all values of x, then find the value of a and b.

A N S W E R S

1. x = –1/3 2. φ

3. Derivative does not exist. 4. –1, 0, 1.

120 XII – Maths

5. 0, 1, 2, 3 6. x = 0

7. Yes 8. λ = 2

9. k = 3 10. 3x4

11. f´(0) = 7

12. (i) 2

2log

2 1e

x −(ii) 3x2 (iii)

2log 2

2

x

ex

(iv) 0

(v)− 3

3

2 1

x

x

13. (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous

(v) Discontinuous (vi) Continuous (vii) Discontinuous

14.17

.6

K = 15. a = 0, b = –1

17. p > 1.

18. ( ) ( ) ( )− − + + 1 log 1log 1 log log log 2 loge

x xe e e e ex x x x x

19. 0 20. 21

x

x

−

−

26. –1/2 27. 1

28. x cot x. 30.( )1

1

– . log 1 log

log

y x x

y x

y x y y x xdy

dx x x xy

−

−

+ + + =+

31.

24

2

1sec cosec

3

d y

adx= θ θ 33. 2 2

1 1–

1 2

dy

dx x x x=

− −

34. 2 1

3 42 – – 2 cosec 2

1 siny x

xx x−

+

−

35. 2

1

1 x

−

−36.

( )( )

log.

log

y y x y

x x y x

−

−

38. sec2 y tan y

42. 4, 5, 6, 7, 8, 9. 43. –1, 0, 1

44. Limit Does not exist. 45. decreasing in (–∞, –1] and increasing in {–1, ∞).

46.15

1, .2

a b= − =

121 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 1

1.

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 ∀ 1 2,x x in (a, b) s.t.

x1 < x

2 ⇒ f(x

1) < f(x

2). 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 < x

2 ⇒ f(x

1)

> 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 value can

be defined.

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

f(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.

� If f(x) is defined in [a, b] and f´(x) = 0 gives x = x1, x2, x3, .... xn then

Max. {f(a), f(x1), f(x2), ..... f(xn), f(b)} is called global maximum value. Similarly, Global minimum

value can be defined.

122 XII – Maths


1. If f(x) = cos x, x ∈ [0, 2π] then write the interval in which f(x) is decreasing.

2. Write the interval in which f(x) = x2 is decreasing.

3. For what value of λ, f(x) = sin 2x – 3λx is strictly increasing.

4. Write the maximum value of ( )2

1

2 3f x

x x=

− + in [0, 2].

5. Find the max. and min. value of ( ) 3 sin 5 .f x x= +

6. Write the slope of the normal to the curve f(x) = 3x2 – 7x + 1 at x = 1.

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

at P.

8. On the curve ( ) 23,

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

points ( )31, and 2, 6 .

2A B

−

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

x= + >

10. y = 2x – 1 is normal to the curve y = x2 at which point?

11. If y = 3e2x and y = b e–2x cut each other at right angles, find the value of b.

12. If the tangent to y = 3x – x2 is parallel to the line y = 0 then find the point of contact of tangent

with the curve.

13. At which point on = +215,

2 3y x tangent makes an angle of 30° with the +ve directions of

x-axis.

14. In which interval f(x) = tan x – x is increasing?

15. If the radius of the circle is decreasing at the rate of 3 cm/sec. then, write the rate at which area

is changing when r = 5 cm.

16. If length and breadth of a rectangle are increasing at the rate of 5 cm/sec. and 2 cm/sec.

respectively. Find the rate at which area is increasing if length = 12 cm and breadth = 10 cm.


17. Sand is pouring out from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the

ground in such a way that the height of the cone is always one sixth of the radius of the base.

How fast is the height of sand cone increasing when the height is 4 cm.

123 XII – Maths

18. Find the points of local maxima/minima for f(x). If f(x) = sin x – cos x where 0 < x < 2π. Also find

the local maximum and minimum values.

19. Find the interval(s) in which ( )2

4

32

xf x x= − is increasing or decreasing?

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

21. For the curve y = 2x3 – 3x2, find the points on the curve at which the tangent passes through

(0, 0).

22. Prove that the function : f(x) = x50 + sin x – 1 is strictly increasing on , .2

π π

23. Show that ( ) cos 24

f x xπ

= + is an increasing function in

3 5, .

8 8

π π

24. Find the intervals in which f(x) = x3 + 3x2 – 105x + 25 is increasing or decreasing?

25. Separate the interval [0, π] into the interval in which f(x) = x – sin 2x is increasing or decreasing?

26. Find the point on the curve y = x3 – 3x2 + 9x + 6 at which slope of the tangent to the curve is

minimum. Also, find the minimum slope.

27. Find the absolute maximum value of f(x) = x + sin 2x in [0, π].

28. Show that the surface area of a closed cuboid with a square base and given volume is minimum

when it is a cube.

29. Show that the equation of the tangent to ( )2 2

1 11 12 2 2 2

1 at , is 1.xx yyx y

P x ya b a b

+ = + =

30. Find the equation of tangent to y2 = 4ax at 2

2, .

a a

mm

31. Show that 12 3

x y+ = touches the curve

–23

x

y e= at the point where the curve crosses

y-axis.

32. If x = y2 and xy = r cut each other at right angles then find the value of r2.

LONG ANSWER TYPE QUESTION (6 MARKS)

33. A point on the hypotenuse of a right triangle is at a distance ‘a’ and ‘b’ from the sides of the

triangle. Show that the minimum length of the hypotenuse is (a2/3 + b2/3)3/2.

34. If the length of three sides of a trapezium other than base are equal to 10 cm, then find the area

of trapezium when it is maximum.

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

π π

and decreasing on

[0, π/4].

124 XII – Maths

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

the x–axis.

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

38. Prove that the radius of the right circular cylinder of greatest curved surface which can be

inscribed in a given cone is half of that of the cone.

39. A rectangular sheet of tin 45 cm × 24 cm is to be made into a box without top by cutting off square

from each corner and folding up the flaps. What should be the side of the square to be cut off

so that the volume of the box is maximum?

40. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square

and the other into a circle. What should be the lengths of the two pieces so that the combined

area of the square and the circle is minimum?

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

vertical angle is –1 1

sin .3

42. 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

43. 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.

44. 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?

H.O.T.S.

45. Show that ( )2

1f x

x= is neither increasing nor decreasing in (–∞, ∞).

46. Find the least value of a such that f(x) = x2 + ax + 1 is increasing on [1, 2].

47. In which interval f(x) = 5x3/2 – 3x5/2, x ≥ 0 is decreasing?

48. Find the point on curve y = (x + 1)2 where tangent is parallel to the chord joining the points

A(–3, 7) and B(2, 5).

49. Verify mean value theorem for

( )2

2 1 2

3 7 2

x xf x

x x x

+ ≥ =

− + < in [0, 4] (if applicable).

50. Using differentials, find the approximate value of 0.037.

125 XII – Maths

51. If y = sin x and x changes from 22

to .14 2

π Find the approximate change in the value of y.

52. 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.

53. A particular moves along the curve 6y = x3 + 2. Find the points on the curve at which y-coordinate

is changing 8 times as fast as x-coordinate.

A N S W E R S

1. [0, π] 2. (– ∞, 0]

3.2

– ,3

∞ − 4. 1/2

5. Max value = 8, Min. Value = 2. 6. 1

7. 0 8.1 3

,2 8

9. 2 10.1 1

– ,4 16

11.1

.12

12.3 9

,2 4

13.1

1, 52 3

+ 14. (–∞, ∞)

15. Decreasing at the rate of 30π cm2/sec. 16. 74 cm2/sec

SHORT ANSWER T YPE QUESTIONS

17.1

cm sec48π

18. Local max. value = √2 at 3

4x

π=

Local min. value = –√2 at 7

.4

xπ

=

19. Decreasing in 1 1

, 0,8 8

−∞ − ∪

Increasing in 1 1

, 0 , .8 8

− ∪ ∞

20. Increasing in (2, 3) and decreasing in (3, ∞).

126 XII – Maths

21. ( )3 –27, , 0, 0

4 32

24. Increasing in (–∞, –7) ∪ (5, ∞), Decreasing in (–7, 5).

25. Decreasing in 5

0, , 2 .6 6

π π ∪ π

Increasing in 5

, .6 6

π π

26. (1, 13), Minimum slope = 6. 27. π.

30.a

y mxm

= + 32. 1/8.

34. 75 3 sq. units. 36. 3x + y – 3 = 0, 7x – y – 14 = 0.

39. 5 cm. 40.112 28

, .4 4

m mπ + π +

44. √5 46. –2

47. [1, ∞) 48.6 1

– ,5 25

49. M.V. Theorem Not applicable at x = 2, because f(x) is not derivable.

50. 0.1925 51. 0

52.16

.6 – 3

m 53. ( ) –314, 11 , –4, .

3

127 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 .

1

dx x c x

x

= + <−

14.–1

2

1tan .

1dx x c

x= +

+ 15.–1

2

1sec , 1.

1

dx 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

= +

128 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.

22 2 2 2 –1

sin .2 2

x a xa x dx a x c

a− = − + +

23.

22 2 2 2 2 2

log .2 2

x aa x dx a x x a x c+ = + + + + +

24.

22 2 2 2 2 2

log .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= +

INTE GRATION BY SUBSTI TU TION

1.( )

( )( )´

log .f x

dx f x cf 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

− +

= ++

INTE GRATION 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=

129 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 :

( ) ( ) ( ) ( ) ( )→

= + + + + + + + − 0

lim 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

2

1

1.

1

dx

x x −

(v)

π+

+ 2

0

4 3 sinlog .

4 3 cos

xdx

x(vi) ( )cosec cosec cot .x x x dx+

(vii)( )–1

.sin cos

dx

x

2. Evaluate the following integrals.

(i)

21

.x x

x

+ − (ii)

21

– .ax dxax

(iii)4 2

sin 3 3 sec .2

x xx e dx

− + (iv)

1 cos 2.

1 cos 2

xdx

x

+

−

130 XII – Maths

(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 logdx

x x+

(xi)sin

.x

dxx (xii) 2 log .x dx

(xiii)2

.2

ax bdx

ax bx c

+

+ + (xiv)sin

.cos

xdx

a b x+

(xv) ( ) .x c

c x dx+ (xvi)1

.3 log

dxx x x+

(xvii)2

1.

16 25dx

x+ (xviii)2

1.

9 4dx

x −

(xix)2

1.

16 25dx

x− (xx)2

1.

4 9

dx

x −

3. Evaluate the following definite integrals :

(i)

π

+ 2 3 2

3 2 3 2

0

sin.

sin cos

xdx

x x(ii)

27

–

2

sin .x dx

π

π

(iii)

1

2

0

1.

1dx

x+ (iv)

π

+ 2

2

0

sin.

1 cos

xdx

x

(v)

1 –1

2

0

tan.

1

xdx

x+ (vi)

1

2

0

.1

x

x

edx

e+


4. Evaluate the following integrals :

(i)( )–1 2

4

cosec tan.

1

x xdx

x+ (ii)1 1

.1 1

x xdx

x x

+ − −

+ + −

131 XII – Maths

(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)3 4

cot cosec .x x dx

(ix)2 2 2 2

sin cos.

sin cos

x xdx

a x b x+ (x)

( )3

1.

cos cos

dx

x x a+

(xi)

6 6

2 2

sin cos.

sin cos

x xdx

x x

+ (xii)

sin cos.

sin 2

x xdx

x

+

5. Evaluate :

(i)4 2

.1

xdx

x x+ + *(ii)( )2

1.

6 log 7 log 2dx

x x x + +

(iii)2

.1

dx

x x+ − (iv)2

1.

9 8

dx

x x+ −

(v)( ) ( )

1.dx

x a x b− − (vi)( )

( )sin

.sin

xdx

x a

− α

+

(vii)2

5 2.

3 2 1

xdx

x x

−

+ + (viii)

2

2.

6 12

xdx

x x+ +

(ix)2

2.

4

xdx

x x

+

− (x)

21 – .x x x dx+

(xii) ( ) 23 2 1 .x x x dx− + + (xiii) sec 1 .x dx+

6. Evaluate :

(i)( )7

.1

dx

x x + (ii)

( ) ( )sin

.1 cos 2 3 cos

xdx

x x+ +

(iii)( ) ( )

21

.1 cos 2 3 cos

xdx

x x

+

+ + (iv)( ) ( ) ( )

1.

1 2 3

xdx

x x x

−

+ − +

(v)( ) ( )

22

.2 1

x xdx

x x

+ +

− − (vi)( ) ( )( ) ( )

2 2

3 2

1 2.

3 4

x xdx

x x

+ +

+ +

(vii)( ) ( )2

.2 1 4

dx

x x+ + (viii)

( ).

sin 1 2 cos

dx

x x−

132 XII – Maths

(ix)sin

.sin 4

xdx

x (x)

2

4 2

1.

1

xdx

x x

−

+ +

(xi) tan .x dx (xii)

29

.81

xdx

x4

+

+ 7. Evaluate :

(i)5 3

sin .x x dx (ii)3

sec .x dx

(iii) ( )cos .ax

e bx c dx+ (iv)–1

2

6sin .

1 9

xdx

x+

(v) cos .x dx (vi)3 –1

tan .x x dx

(vii)2 1 sin 2

.1 cos 2

x xe dx

x

+ + (viii)

2

1.

2

x xe dx

x

−

(ix)

2

2

1.

1

x xe dx

x

− + (x)

( )( )

2

2

1.

1

x xe dx

x

+

+

(xi)( )

( )2 sin 2

.1 cos 2

x xe dx

x

+

+ (xii) ( )( )2

1log log .

logx dx

x

+

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 22

0

sin.

1

xdx

x

−

−

(v)

2

4 4

0

sin 2.

sin cos

xdx

x x

π

+ (vi)

2 2

2

1

5.

4 3

xdx

x x+ +

(vii)

2

0

sin.

1 cos

x xdx

x

π

+

+

9. Evaluate :

(i) { }3

1

1 2 3 .x x x dx− + − + − (ii)

0

.1 sin

xdx

x

π

+

133 XII – Maths

(iii) ( )4

0

log 1 tan .x dx

π

+ (iv)

2

0

log sin .x dx

π

(v)( )2

0

sin.

1 cos

x xdx

x

π

+

(vi) ( ) ( )

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

(vii)

2

4 4

0

sin cos.

sin cos

x x xdx

x x

π

+ (viii)2 2 2 2

0

.cos sin

xdx

a x b x

π

+

10. Evaluate the following integrals as limit of a sum

(i)

32

1

.x dx (ii)

1

–1

.x

e dx

(iii)

2–

0

.x

e dx (iv) ( )2

0

2 3 .x dx+ 11. Evaluate the following integrals.

(i)

32

1

2 .x x dx− (ii)

1–1

2

0

2sin .

1

xdx

x

+

(iii)

1

–1

1 sinlog .

1 sin

xdx

x

+ − (iv)

cos

cos cos

0

.

x

x x

edx

e e

π

−+



(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

134 XII – Maths

(v) ( )2

0

tan cot .x x dx

π

+ (vi)4

1.

1dx

x +

(vii)

( )

–1

220

tan.

1

x xdx

x

∞

+

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+

H.O.T.S.

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK EACH)


(i) ( )1 1sin cosx x dx− −+ (ii) ( )1tan cot x dx−

(iii) ( )2

1

cos 1 logdx

x x+ (iv) 2x

xdx

e

(v)

1

1

xe dx

− (vi)

43 4

4

sinx x dx

π

π−

(vii)

2

0

1

1 tandx

x

π

+

135 XII – Maths

SHORT ANSWER TYPE QUESTIONS (4 MARKS EACH)

15. Evaluate

(i) [ ]1 1

1 1

sin cos, 0, 1

sin cos

x xdx x

x x

− −

− −

−∈

+ (ii)1

1

xdx

x

−

+

(iii)( ) + + −

2 2

4

1 log 1 2 logx x xdx

x(iv)

( )

2

2sin cos

xdx

x x x+

(v) 1sinx

dxa x

−

+ (vi)

3

6

sin cos

sin 2

x xdx

x

π

π

+

(vii) ( )2

–2

sin cosx x dx

π

π

− (viii) 2 2 2 2

0cos sin

x dx

a x b x

π

+

(ix) [ ]2

1

,x dx

π

where [x] is greatest integer function

(x)

3

2

1

sin .x x dx

−

π


16. Evaluate

(i)( )

1

1 2

0

cot 1 x x dx− − + (ii) ( ) ( )sin 2 cos 2 sin cos

dx

x x x x− +

(iii)( )1

2

0

log 1

1

xdx

x

+

+ (iv) ( )2

0

2 log sin – log sin 2 .x x dx

π

A N S W E R S

1. (i)3 22

2 .3

x x c+ + (ii) tan x + c.

(iii) – cosec x + c (iv) .3

π

136 XII – Maths

(v) 0; (vii) – cot x – cosec x + C

(ix) sin–1 x + c

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 2

1 21 2 1– 2 1

2 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) 2x (log x – 1) + c.

(xiii)21

log 22

ax bx c c+ + + (xiv)1

– log cos .a b x cb

+ +

(xv)

1

1.log 1

x cc x

cc c

+

+ ++

(xvi) log |3 + log x| + c.

(xvii)–11 5

tan .20 4

x c+ (xviii)1 3 2

log .12 3 2

xc

x

−+

+

(xix)1 4 5

log .40 4 5

xc

x

++

−(xx)

21log 2 4 9 .

2x x c+ − +

3. (i) .4

π(ii) 0;

(iii) .4

π(iv) .

4

π

(v)

3 2

.12

π(vi)

–1tan .

4e

π−

4. (i)( )–1 2

2

1 1log cosec tan .

2x c

x

− +

(ii) ( )2 2 21 11 log 1 .

2 2x x x x x c− − + + − +

137 XII – Maths

(iii) ( )

( )

( )1 sin

logsin sin

x ac

a b x b

−+

− −

(iv) x cos 2a – sin 2a log |sec (x – a)| + c.

(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

+ − − +

(viii)

6 4cot cot

.6 4

x xc

− + +

(ix) ( )2 2 2 2 2 2

1.

sin cos

c

a b a x b x

+− +

[Hint. : put a2 sin2 x + b2 cos2 x = t]

(x) –2 cosec cos tan . sin .a a x a c− + [Hint. : Take sec2 x as numerator]

(xi) tan x – cot x – 3x + c.

(xii) sin–1 (sin x – cos x) + c.

5. (i)

2–11 2 1

tan .3 3

xc

++

[Hint : put x2 = t]

(ii)2 log 1

log3 log 2

++

+

xC

x [Hint : put log x = t]

(iii)1 5 – 1 2

log5 5 1 2

xc

x

++

+ −(iv)

1 4sin .

5

xc− −

+

(v) 2 log x a x b c− + − +

(vi)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

(vii)−− +

+ + + +

2 15 11 3 1log 3 2 1 tan

6 3 2 2

xx x c

138 XII – Maths

(viii)− +

− + + + +

2 1 33 log 6 12 2 3 tan

3

xx x x c

(ix)2 1 2

4 4 sin2

xx x c

− − − − + +

(x) ( ) ( ) −− − + − + − + − + +

32 2 12

1 1 5 2 11 2 1 1 sin

3 8 16 5

xx x x x x c

(xi) ( )322 2 27 1 3 1

1 1 log 12 2 8 2

x x x x x x x x c + + − + + + + + + + + +

(xii)21

log cos cos cos2

x x x c− + + + + [Hint. : Multiply and divide by sec 1x + ]

6. (i)

7

7

1log

7 1

xc

x+

+

(ii)1 cos

log2 3 cos

xc

x

++

+

(iii)( )

3 1 5log 1 log 3

8 2 1 8x x c

x− − + + +

−

(iv)9 4 1

log 3 log 2 110 15 6

x x x c+ + − − + +

(v)( )22

4 log1

xx c

x

−+ +

−

(vi)1 12

tan 3 tan3 3 2

x xx c

− − + − +

[Hint. : put x2 = t]

(vii)2 12 1 1

log 2 1 log 4 tan17 17 34 2

xx x c

−+ − + + +

(viii)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]

(ix)1 1 sin 1 1 2 sin

log log8 1 sin 4 2 1 2 sin

x xc

x x

− + ++ +

− −

(x)2

2

1 1log

2 1

x xc

x x

− ++

+ +

139 XII – Maths

(xi)− − − +

+ + + +

11 tan 1 1 tan 2 tan 1tan log

2 2 tan 2 2 tan 2 tan 1

x x xc

x x x

(xii)2

11 9tan

3 2 3 2

xc

− −+

7. (i)3 3 31

cos sin3

x 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 21

2 tan 3 log 1 93

x x x c−

− + + [Hint. : put 3x = tan θ]

(v) [ ]2 sin cosx x x c+ +

(vi)

4 311

tan – .4 12 4

x x xx c

− −+ +

(vii)21

tan .2

xe x c+

(viii) .2

xe

cx

+

(ix) 2.

1

xe

cx

++

(x)1

.1

x xe c

x

− + +

(xi) ex tan x + c.

(xii) log log – .log

xx x c

x+ [Hint. : put log x = t ⇒ x = et]

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

140 XII – Maths

(v) .2

π(vi)

15 25 65 – 10 log log .

8 2 5

+

(vii) π/2.sin

. .1 cos 1 cos

+ + +

Hint :x x

dxx x

9. (i) 8. (ii) π.

(iii) log 2.8

π(iv)

–log 2.

2

π

(v)21.

4π

(vi) 95/12. ( ) ( ) ( ) ( )−

− − −

= + +

2 1 1 2

2 2 1 1

f x dx f x dx f x dx f x dxHint. :

(vii)

2

.16

π

(viii)

2

.2ab

π( ) ( )

0 0

Use

a a

f x f a x = −

Hint. :

10. (i)26

.3

(ii)1

.ee

−

(iii) 2

11 .

e− (iv) 10.

11. (i) 2. (ii) log 2.2

π−

(iii) 0. (iv) π/2.

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 :

141 XII – Maths

(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.

13. (i) 14. (ii)26

.3

(iii) 26. (iv)( )81127 .

2e+

(v)141

.2

14. (i)2

x cπ

+ (ii)2

2 2

xx c

π− +

(iii) ( )tan 1 log x c+ + (iv) 2

1

2 xc

e− +

(v) 2e – 2 (vi) 0

(vii) .4

π

15. (i)( ) 2

12 2 1 2sin

x x xx x c−− −

+ − +π π

(ii) 1 2–2 1 cosx x x x c−− + + − +

(iii)

− + + − +

3 2

2 2

1 1 1 21 log 1

3 3c

x x

(iv)sin cos

sin cos

x x xc

x x x

−+

+

(v) ( ) 1tanx

x a ax ca

−+ − + (vi)1 3 1

2 sin2

− −

142 XII – Maths

(vii) 0 (viii)2

2ab

π

(ix) 2 3 5− + (x) 2

3 1.+

π π

16. (i) log 22

π− (ii)

1 tanlog

5 2 tan 1

x xc

x

−− +

+

(iii) log 2.8

π(iv)

1log .

2 2

π

143 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 ordinats, x = a and x = b is given

by

( )Area =

b

a

f x dx

y

y f x= ( )

a bO

y

a b

O

y f x= ( )

x x

2. Area bounded by the curve x = f(y) the y-axis and between abcissae, y = c and y = d is given by

( )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] and

between the ordinate at x = a and x = b is given by

( ) ( )[ ]Area = –

b

a

f x g x dx

144 XII – Maths

yy f x= ( )

cOx

d

y g x= ( )

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

A1

A2

b

( ) ( )+ 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 circle

x2 + 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) (b) (–2, 2) (0, 5) and (3, 2)

145 XII – Maths

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

2x

π= and x-axis.

14. Find the area bounded by semi circle 2

25y x= − and x-axis.

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

16. Find area of smaller region bounded by ellipse

2 2

19 4

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

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

18. Using integration find the area of region in first quadrant enclosed by x-axis the line 3x y=

and the circle x2 + y2 = 4.

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

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

H.O.T.S.

21. Find the area lying above x-axis and included between the circle x2 + y2 = 8x and the parabola

y2 = 4x.

22. Using integration find the area enclosed by the curve y = cos x, y = sin x and x-axis in the interval

0, .2

π

23. Sketch the graph y = |x – 5|. Evaluate 6

05 .x dx−

A N S W E R S

1. πa2 sq. units. 2.28

3 sq. units

3. πab sq. units 4. 4π – 8 sq. units

146 XII – Maths

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. 3 sq. units 14.25

2 π sq. units

15.1

3 sq. units 16. ( )π −

32 sq. units

2

17.9

sq. units8

18.π

sq. unit3

19. ( )π − 2 sq. unit 20. 2π sq. unit.

21. ( )48 3 sq. units

3+ π 22. ( )2 2 sq. units.−

23. 5 sq. units.

147 XII – Maths

CHAPTER 9

DIFFERENTIAL EQUATION

POINTS TO REMEMBER

� Differential Equation : Equation containing derivatives of a dependant variable with respect to

an independent variable is called differential equation.

� Order of a Differential Equation : The order of a differential equation is defined to be the order

of the highest order derivative occurring in the differential equation.

� Degree of a Differential Equation : The degree of differential equation is defined to be the

degree of highest order derivative occurring in it after the equation has been made free from

radicals 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=


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

+ =

148 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

−=

+

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+ =


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 = +B

y Axx

is a solution of

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

+ + =

149 XII – Maths

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

and B are arbitrary constants.

(vi) Find the differential equation of an ellipse with major and minor axes 2a and 2b respectively.

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

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

5. Solve the following diff. equations.

(i) cot sin 2 .dy

y x xdx

+ = (ii)2

2 log .dy

x y x xdx

+ =

(iii)1 sin

. cos , 0.dx x

y x xdy x x

+ = + >

(iv)3

cos cos sin .dy

x x xdx

+ = (v) 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

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) sin sin .x x

y dx x y dyy y

= −

(v) tan .

dy y y

dx x x

= +

(vi)2 2

2dy xy

dx x y=

+(vii)

2.

x y ydye x e

dx

+= +

(xii)

2

2

1 –.

1

dy y

dx x=

−

150 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 axis along

the (i) positive y-axis (ii) +ve x-axis.

(iii) Form differential equation of all circles passing through origin and whose centre lie on

x-axis.

9. Show that the differential equation 2

2

dy x y

dx x y

+=

− is homogeneous and solve it.

10. 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.

HIGHER ORDER THINKING SKILLS (HOTS)

VERY SHORT ANSWER TYPE QUESTIONS (1 MARKS)

13. (i) Write the order and degree of the differential equation tan 0.dy dy

dx dx

+ =

151 XII – Maths

(ii) What will be the order of the differential equation, corresponding to the family of curves

y = a cos (x + b), where a is arbitrary constant.

(iii) What will be the order of the differential education y = a + bex + c where a, b, c are arbitrary

constant.

(iv) Find the integrating factor for solving the differential education tan cos .dy

y x xdx

+ =

(v) Find the integrating factor for solving the differential equation 2

1sin .

1

dxx y

dy y+ =

+

14. (i) Form the differential equation of the family of circles in the first quadrant and touching

the coordinate axes.

(ii) Verify that ( )2 2logy x x a= + + satisfies the differential education

( )2

2 2

20.

d y dya x x

dxdx÷ + =

(iii) Show that the general solution of the differential equation

2

2

10

1

dy y y

dx x x

+ ++ =

+ + is given

by (x + y + 1) = A(1 – x – y – 2xy). Write A is parameter.

15. Solving the following differential equation

(i) (tan–1 y – x) dx = (1 + y2) dx.

(ii) ( )2 1dy

x ydx

− + = .

(iii) )1 1 0.

x x

y y xe dx e dy

y

+ + − =

(iv) (x – sin y) dy + tan y dx = 0, y(0) = 0.

LONG ANSWER TYPE QUESTIONS (6 MARKS EACH)

16. Solve the following differential equation

(i) ( ) ( )sin cosy y

x dy y dx y y dx x dy xx x

− = +

(ii) 3ex tan y dx + (1 – ex) sec2 y dy = 0 given that ,4

yπ

= when x = 1.

(iii) 2cot 2 cotdy

y x x x xdx

+ = + given that y(0) = 0.

152 XII – Maths

A N S W E R S

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)

−= + +log

x xey 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.

3.

(i) esin x (ii) etan x

(iii) e–1/x (iv)( )2log

2

xe

(v) 3

1

x(vi) sec y

4.

(v) − + =2

22 2 0

d y dyy

dxdx[Hint : find

2

2, and eliminate A and B.

dy d y

dx dx]

(vi)

+

22

2=

dy d y dyx xy y

dx dxdx

(vii)

= × −

3

4 2dy dy

y x ydx dx

[ divide by and find .]dy

y cdx

Hint :

(viii)

+ =

2 3

22 0

d y dy

dxdx

5.

(i) = +3

2 sinsin

3

xy x c (ii)

( )−= +

2

2

4 log 1

16

ex x cy

x

153 XII – Maths

(iii) = + >sin , 0c

y x xx

(iv) y = tan x – 1 + ce–tan x

(v) x = – y2e–y + cy2

6.

(i) ( ) ( )= + −2 1 2cy x y (ii) ( )+ =2 secx

e y c

(iii) − + − =2 2

1 1x y c

(iv)− −

= − − − +− +

22 2

2

1 11log 1 1

2 1 1

yx y c

y

(v) ( ) ( )+ + =2 2

1 1 2x y

(vi) = − + + − +3 51 1

log cos cos3 5

x xy x x xe e c

(vii) − =cos 2

log tanx

y cy

7.

(i)−

+ =3

3log

3

xy c

y

(ii)−

= +

1tan log

yx c

x[Hint. : Homogeneous Equation]

(iii) x2 + y2 = 2x

(iv)( )

=cos x y

y ce [Hint. : Put =x

vy

] (v)

= sin

ycx

x

(vi) ( )− =2 2

c x y y

(vii)−

− = + +3

3

y x xe e c [Hint. : Factorise R.H.S.]

(viii) − −= +

1 1sin siny x c

8.

(i) − + =2 2

2 0dy

x y xydx

[Hint. : The family of circles is, + + =2 2

2 0x y gx ]

154 XII – Maths

(ii) = =2 , 2dy dy

y x y xdx dx

(iii) − + =2 2

2 0dy

x y xydx

9.− +

+ + = +

2 2 1 2log 2 3 tan

3

x yx xy y c

x

10. ( )= ++

3

2 2

x cx y

xx y

11.

(i) = − +23 sin 3 2 cos 3

13 13

xx xy ce

(ii) = +22 1

sin cosec3 3

y x x

12.

(i) ( ){ }− = −logy x c x y (ii) = + +2 2 2

cx y x y

(iii)

= cos

yxy c

x[Hint. : Put y = vx]

(iv) = +2

3 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

13. (i) Order = 1, Degree = not define

(ii) Order = 1 (iii) Order = 2

(iv) sec x (v)1tane y

−

14. (i) (x – y)2 {1 + y12} = (x + y y´)2

15. (i) x = tan–1 y – 1 + c ·−1tan ye

(ii) x = y – 1 + cey (iii)x

yx ye c+ =

(iv) y = sin–1 2x

16. (i) secy

C xyx

=

(ii) (1 – e)3 tan y = (1 – ex)3

(iii) y = x2.

155 XII – Maths

CHAPTER 10–11

VECTORS AND THREE

DIMENSIONAL 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 their

sum is the vector represented by the third side of triangle taken in opposite order (triangle law).

� Multiple of a vector by a scalar : auuur

is any vector and λ ∈ R then aλuuur

is vector of magnitude

aλuuur

in a direction parallel to auuur

.

� If 0a ≠uuur

then a

a

uuur

uuur is unit vector in direction .auuur

� Scalar Product : . cosa b a b= θuuur uuur uuur uuur

where θ is the angle between auuur

and .buuur

� Projection of auuur

along buuur

is .

.a b

b

uuur uuur

uuur

�2

.a a a=uuur uuur uuur

� Vectors auuur

and buuur

are perpendicular iff . 0.a b =uuur uuur

� Cross Product : $sina b a b n× = θuuur uuur uuur uuur

where $n is a unit vector perpendicular to auuur

and

buuur

, and θ is the angle between and .a buuur uuur

� Unit vector perpendicular to plane of anda buuur uuur

is .a b

a b

×±

×

uuur uuur

uuur uuur

� Vector anda buuur uuur

are collinear if × =uuur uuur uuur

0 .a b

�

$

1 2 3

1 2 3

i j k

a b a a a

b b b

× =

$ $

uuur uuur

where $1 2 3 anda a i a j a k= + +

uuur$ $

$1 2 3b b i b j b k= + +

uuur$ $

156 XII – Maths

� Area of a triangle whose two sides are ×uuur uuur uuur uuur1

and is .2

a b a b

� Area of a parallelogram whose adjacent sides are ×uuur uuur uuur uuur

and is .a b a b

� If ,a buuur uuur

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

1.

2a b= ×uuur uuur

T HREE DIMENSIONAL GEOME TR Y

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

( ) ( ) ( )= − + − + −uuuur 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. Where

l = 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 2, , .

a b cl m 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 ( )auuur

and parallel to vector buuur

is = + λuur uuur uuur

.r a b

(ii) Passing through two points ( )A auuur

and ( )B buuur

is ( )= + λ −uur uuur uuur uuur

.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 <a1, b1, c1> 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 =

+ + + +

157 XII – Maths

If lines are 1 1 2 2and .r a b r a µ b= + λ = +uur uuur uuur uur uuuur uuuur

then, 1 2

1 2

.cos .

b b

b bθ =

uuur uuuur

uuur uuuur

� Equation of plane :

(i) Passing through ( )A auuur

and perpendicular to ( )nuuur

is ( )– . 0r a n =uur uuur uuur

Or .r n d=uur uuur

where =uuur uuur

. .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= =

uur uuur uur uuuur is

1 2

1 2

.cos .

n n

n nθ =

uuur uuuur

uuur uuuur

� 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 + b

1y + c

1z + d

1 = 0 and a

2x

+ b2y + c

2z + d

2 = 0 is (a

1x + b

1y + c

1z + d

1) + λ(a

2x + b

2y + c

2z + d

2) = 0.

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

uur uuur uur uuuur is

( )+ λ = + λuur uuur uuuur

1 2 1 2. .r n n d d

� Angle between a plane .r n d=uur uuur

and a line = + λ θ =

uuur uuuruur uuur uuur

uuur uuur.

is sin .m n

r a mm n


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

of magnitude 5 making an angle

of 150° with the direction of x-axis.

2. What is a ∈ R such that 1,a x =uuur

where $– 2 2 ?x i j k= +$ $

3. Write when .x y x y+ = +uuur uuur uuur uuur

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

5. If A is the point (4, 5) and vector ABuuuuur

has components 2 and 6 along x-axis and y-axis

respectively then write point B.

158 XII – Maths

6. What is the point of trisection of PQ nearer to P if position of P and Q are $3 3 – 4i j k+$ $ and

$9 8 – 10 .i j k+$ $

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

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

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

10. What is the projection of the vector –i j$ $ on the vector ?i j+$ $

11. If 2, 2 3a b= =uuur uuur

and ,a b⊥uuur uuur

what is the value of ?a b+uuur uuur

12. For what value of λ, $4a i j k= λ + +uuur

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

$ $

13. What is ( ) ( ), if . – 3a a b a b+ =uuur uuur uuur uuur uuur

and 2 ?b a=uuur uuur

14. What is the angle between auuur

and buuur

, if – ?a b a b= +uuur uuur uuur uuur

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

$− +$ 2 ?i k

16. Find xuuur

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

.

17. If 2 2 2

,a b a b+ = +uuur uuur uuur uuur

then what is the angle between auuur

and .buuur

18. If auuur

and buuur

are two unit vectors and a b+uuur uuur

is also a unit vector then what is the angle

between auuur

and uuur

?b

19. 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× + × + ×ur

$ $ $ $ $ $

20. What is the angle between xuuur

and yuuur

if . ?x y x y= ×uuur uuur uuur uuur

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

22. If , anda b cuuur uuur uur

are unit vectors with 0 ,a b c+ + =uuur uuur uur uuur

then what is the value of

. . . ?a b b c c a+ +uuur uuur uuur uur uur uuur

23. If auuur

and buuur

are unit vectors such that ( )2a b+uuur uuur

is perpendicular to ( )5 4 ,a b−uuur uuur

then

what is the angle between auuur

and buuur

?

24. 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.

159 XII – Maths

25. What is the ratio in which xy plane divides the line segment joining the points (–1, 3, 4) and

(2, –5, 6)?

26. 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 ratio

P divides QR.

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

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

= 7.

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

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

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

32. 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

33. A (3, 2, 0), B(5, 3, 2) and C(5, 8, – 10) are the vertices of ∆ABC. D and E are mid points of AB

and AC respectively. What are the direction cosines of DE?

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

3 – 4 8?

5 5 –6

x y z+ += =

35. 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+ − + − + −= = = =

36. If the direction ratios of a line are proportional to 1, –3, 2 then what are the direction cosines of

the line?

37. 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?

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

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

40. What is the equation of the plane which cuts off equal intercepts of unit length on the coordinate

axes.

41. What is the equation of the plane through the point (1, 4, – 2) and parallel to the plane

– 2x + y – 3z = 7?

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

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

160 XII – Maths

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

44. 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?

45. If O is origin OP = 3 with direction ratios proportional to –1, 2, – 2 then what are the coordinates

of P?

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

$ $ $ $ from the plane

$( ). – 5 – 5 0.r i j k+ + =uur

$ $


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

3 6AB AC AD AE AF AD AO+ + + + = =uuuuur uuuuur uuuuur uuuuur uuuuur uuuuur uuuuur

O being the centre of hexagon.

48. 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+ +uuuuur uuuuur uuuuur

is a vector parallel to CKuuuuur

where K divides AB in ratio 1 : 3.

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

´ ´ ´ 3 ´ .PP QQ RR GG+ + =uuuuuur uuuuuur uuuuuur uuuuuuuur

50. PQRS is parallelogram. L and M are mid points of QR and RS. Express PLuuuur

and PMuuuuur

in terms

of PQuuuuur

and .PSuuuuur

Also prove that 3

.2

PL PM PR+ =uuuur uuuuur uuuuur

51. The scalar product of vector $i j k+ +$ $ with a unit vector along the sum of the vectors

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

52. , anda b cuuur uuur uur

are three mutually perpendicular vectors of equal magnitude. Show that

+a b c+uuur uuur uur

makes equal angles with , anda b cuuur uuur uur

with each angle as –1 1

cos .3

53. If $3 – and 2 – 3i j b i j kα = = +uuur uuur

$ $ $ $ then express βuur

in the form of 1 2,β = β + βuur uur uur

where

1βuur

is parallel to 2andα βuuur uur

is perpendicular to .αuuur

54. If , ,a b cuuur uuur uur

are three vectors such that 0a b c+ + =uuur uuur uur uuur

then prove that a b× =uuur uuur

.b c c a× = ×uuur uur uur uuur

55. If 3, 5, 7 and 0 ,a b c a b c= = = + + =uuur uuur uur uuur uuur uur uuur

find the angle between and .a buuur uuur

161 XII – Maths

56. Let � $ $, 3 – and 7 – ,a i j b j k c i k= − = =uuur uuur uur

$ $ $ find a vector uuurd which is perpendicular to

and and . 1.a b c d =uuur uuur uur uuur

57. If $ $, –a i j k c j k= + + =uuur uur

$ $ $ are the given vectors then find a vector buuur

satisfying the equation.

, . 3.a b c a b× = =uuur uuur uur uuur uuur

58. Find a unit vector perpendicular to plane ABC. Position vectors of A, B, C are $3 – 2 ,i j k+$ $

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

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

60. 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.

61. 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+

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

.1 2 3

x y z −= =

63. Find equations of a plane passing through the points (2, –1, 0) and (3, –4, 5) and parallel to the

line 2x = 3y = 4z.

64. 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.

65. Find equation of the plane passing through the point (2, 3, – 4) and (1, –1, 3) and parallel to the

x–axis.

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

right angle.

67. 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

68. 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.

69. Find the equations of the planes parallel to the plane x – 2y + 2z – 3 = 0 whose perpendicular

distance from the point (1, 2, 3) is 1 unit.

70. Find equation of the plane passing through the point (3, 4, 2) and (7, 0, 6) and is perpendicular

to the plane 2x – 5y = 15.

162 XII – Maths

71. 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= + + = − + =uur uur

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

72. Find equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which

contain the line of intersection of the plane x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

73. Find equation of the plane containing the points (0, – 1, – 1) (– 4, 4, 4), (4, 5, 1). Also show that

(3, 9, 4) lies on the required plane.


74. 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 + = − − + +uur uur

$ $ $ $ $ $ $ $ . Find the shortest

distance between them.

75. Check the coplanarity of lines

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

$ $ $ $

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

$ $ $ $

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

76. Find shortest distance between the lines :

8 19 10 15 29 5and

3 16 7 3 8 5

x y z x y z− + − − − −= = = =

− −

77. 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.

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

( ) ( ) ( ) $ ( ) ( ) ( ) $= − + − + − = + + − + +uur uur

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

79. 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

80. 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.

81. Find the equation of the plane passing through the intersection of planes 2x + 3y – z = – 1 and

x + y – 2z + 3 = 0 and perpendicular to the plane 3x – y – 2z = 4. Also find the inclination of

this plane with xy–plane.

82. Find the shortest distance and the vector equation of line of shortest distance between the lines

given by

163 XII – Maths

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

$ $ $ $

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

$ $ $ $

83. 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.

84. 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,

85. A variable plane is at a constant distance 3p from the origin and meet the coordinate axes in A,

B, C. Show that the locus of centroid of ∆ABC is x–2 + y–2 + z–2 = p–2.

86. A vector nuuur

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 nuuur

, find its equation in

vector form.

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

$ $

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

SHORT ANSWER TYPE QUESTION (4 MARKS)

88. Evaluate ( ) ( ) $( )2 2 2

.a i a j a k× + × + ×uuur uuur uuur

$ $

89. $ $ ( ) $; 2 and 2 .a i j k b i j k c xi x j k= + + = − + = + − −uuur uuur uuur

$ $ $ $ $ $ If c lies in the plane of auuur

and

buuur

then find value of x.

90. Find the value of a for which the vector ( ) ( ) $2 24 2 9r a i j a k= − + + −uur

$ $ makes acute angle

with coordinate axes.

91. Let $ $ $, ,a b c be unit vectors such that $ $ $ $ 0a b a c= =· · and the angle between $b and $c is 6

π

then prove that ( )2 .a b c= ± ×uuur uuur uuur

92. Prove that angle between any two diagnals of a cube is cos–1(1/3).

93. The cartesian equations of a line is 6x – 2 = 3y + 1 = 2z – 2. Find direction ratios of the line.

Also find vector and cartesian equations of a line parallel to this line and passing through

(2, –1, –1).

LONG ANSWERS TYPE QUESTIONS (6 MARKS)

94. Three vectors of magnitude a, 2a and 3a meet in a point and their directions are along the

diagonals of the adjacent faces of a cube determine their resultant.

164 XII – Maths

95. A line makes angle α, β, γ and δ with four diagonals of a cube prove that

2 2 2 2 4cos cos cos cos .

3α + β + γ + δ =

96. Show that the lines x = ay + b, z = cy + d and x = a´y + b´, z = c´y + d´ are perpendicular if

aa´ + cc´ = –1.

HIGHER ORDER THINKING SKILL QUESTIONS (HOTS)


97. What is the angle between auuur

and buuur

if 3 and 3 3.a b a b= × =·uuur uuur uuur uuur

98. What are the direction cosines of a vector equiangular with co-ordinate axes.

99. In a parallelogram ABCD $ $2 – 4 and 4 .AB i j k AC i j k= + = + +uuuuur uuuuur

$ $ $ $ what is the length of side BC.

100. Two adjacent sides of a parallelogram are $ $2 – 4 5 and 2 3 .i j k i j k+ − −$ $ $ $ What is a unit vector

parallel to the diagonal which is co-initial with and .a buuur uuur

101. If = – 1a b a b= =uuur uuur uuur uuur

then what is value of .a b+uuur uuur

102. For any vector auuur

what is the value of ( ) ( ) $ $( ) ?i a j j a i k a k× + × + ×· · ·uuur uuur uuur

$ $ $ $

103. If a line makes α, β and γ angles with co-ordinate axes then what is value of δm2 γ + δm2β+ δm2γ ?

104. What is the equation of a line passing through point (–1, 2, 3) and equally include to the axes.

105. The foot of perpendicular from (1, 6, 3) on the line ( )1 11, 3,

1 2 3

x y zis q

− −= = what is the

value of q.

106. What is the distance between the line $( )2r i j i j k= − + λ − +uur

$ $ $ $ from the plane

$( ) 3.r i j k+ + =·uur

$ $

A N S W E R S

1. 5 3 5, .

2 22.

1

3±

3. and are collinear.x yuuur uuur

4. 126 sq. units.

5. (6, 11) 6. (5, 4, –6)

165 XII – Maths

7. $4 6 4 3+ +i j k$ $ 8.1 1 13 6 2

, ,7 7 7

− − −−Cos Cos Cos

9. $3 4 1

26 26 26+ −i j k$ $ 10. 0

11. 4 12. λ = – 9

13. 2=auuur

14. 90°

15.3

sq. units.2

16. 13

17. 90° 18. 120°

19. –1 20.4

π

21.3 1

.2 2

+i j$ $22.

−3.

2

23. 60° 24. $+ +$ $1 1 1.

2 2 2i j k

25. 2 : 3 enternally 26. 2 : 5

27. 2 2+b c 28.

1 1 2.

2 3 4

− + −= =

−

x y z

29. 90° 30.10

14

31. –2 32.8

5

−

33.5 12

0, , .13 13

−34. 2 4 5

.5 5 6

+ − += =

−

x y z

35.1 2 3

1 1 1

x y z+ − += = 36. ± ±m

1 3 2, , .

14 14 14

37. 60° 38. 60°

39.1

units.6

40. x + y + z = 1.

41. 2x – y + 3z = – 8. 42. $( )+ + =uur

$ $. 2 2 24.r i j k

43. 2x + 3y + 5z = 29. 44. 0°

166 XII – Maths

45. (–1, 2, –2) 46.10

3 3

50.

1

2

1

2

= +

= +

PL PQ PS

PM PQ PS

uuur uuur uuur

uuuur uuur uuur51. –6.

53.

( )$( )

β = −

β = + −

$ $

$ $

1

2

13

2

13 6

2

i j

i j k

55.1 11

.14

−

Cos

56. $1 1 3.

4 4 4+ +i j k$ $

57. $1 4 4.

3 3 3+ +i j k$ $ 58. $( )1

10 7 4 .165

+ −i j k$ $

59. (0, –1, –3) 60. 7.

61. $( ) $( )= − + + λ − +uur

$ $ $ $2 2 2 2 2 .r i j k i j k 62. (– 9, 2, 1).

63. 29x – 27y – 22z = 85. 64. 13 units.

65. 7y + 4z – 5 = 0.

66. 4x – 7y + 3z – 28 = 0. 67. 1 unit

68. x – 20y + 27z = 14. 69. x – 2y + 2z = 0.

70. 5x + 2y – 3z = 17.

71. $( ). 2 2 3 0, 2 2 3 0.+ + + = + + + =r i j k x y zuur

$ $

Or

$( ). 2 2 3 0, 2 2 3 0.− + − + = − + − + =r i j k x y zuur

$ $

72. 51x + 15y – 50z + 173 = 0. 73. 5x – 7y + 11 z + 4 = 0.

74.3 3

275. $( ), 2 1− + =r i j k

uur$ $

76. 14 units. 77. (–1, –1, –1).

79. 8

2979. ; and .

1 2 1 1 1 2

x y z x y z= = = =

− − −

167 XII – Maths

80. 2 2

81.−

+ + − =

1 47 13 4 9 0, sin

234x y z

82. $( ) $( ). 3 30, 3 8 3 6 15 3 .= = + + + µ − − +S D r i j k i j k$ $ $ $ $

83. (3, 4, 2). 84. x + 2y – 2z = 6, 2x – 2y – z = 3.

86. $( )+ + =uur

$ $. 2 2.r i j k 87. $2 3 , 14.+ +i j k$ $

88.2

2 auuur

89. x = –2

90. a ∈ (–3, –2) ∪ (2, 3)

93.$( ) $( )− + +

= = = − + + λ − +uur

$ $ $ $2 1 1, 2 2 2 2

1 2 3

x y zr i j k i j k

94.$( )4 3 5

2

ai j k+ +$ $

97. 60°

98.1 1 1

, ,3 3 3

99. 3 units.

100.$3 6 2

–7 7 7

i j k+$ $101. √3 .

102. 0. 103. 2.

104.1 2 3

.1 1 1

x y z+ − += = 105. q = 5.

106. 0.

168 XII – Maths

CHAPTER 12

LINEAR PROGRAMMING

POINTS TO REMEMBER

� Linear Programming is the method used to obtain minimum or maximum value of the objective

function restricted to some linear constraints.

� Constraints : The linear inequation or restrictions on the variables of a linear-programming

problem are called constraints.

� Objective Functions : A linear programming problem is one that is concerned with finding the

optimal value (maximum or minimum) of a linear function of several variables. This linear function

is called objective function Z, Z = ax + by, a and b are constraints.

Types of linear programming problems are

(i) Diet Problems

(ii) Manufacturing Problems

(iii) Transportation problems etc.

� Feasible Region : It is the common region determined by all the constraints of a linear programming

problem.

� To Find Feasible Region : Draw the graph of all the linear inequations and shade common region

determined by all the constraints.

� Feasible Solutions : Points within and on the boundary of the feasible region represents feasible

solutions of the constraints.

� Optimal Feasible Solution : It occurs at the corner point and select the point which optimizes

Z according to question.


1. A man has Rs. 1500 to purchase two types of shares of two different companies S1 and S2.

Market price of one share of S1 is Rs 180 and S2 is Rs. 120. He wishes to purchase a maximum

to ten shares only. If one share of type S1 gives a yield of Rs. 11 and of type S2 Rs. 8 then how

much shares of each type must be purchased to get maximum profit? And what will be the

maximum profit?

2. A company manufacture two types of lamps say A and B. Both lamps go through a cutter and

then 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 and

finishers has 80 hours of time available each month. Profit on one lamp A is Rs. 7.00 and on one

169 XII – Maths

lamp B is Rs. 13.00. Assuming that he can sell all that he produces, how many of each type of

lamps should be manufactured to obtain maximum profit?

3. Solve the following LPP problem graphically :

Maximise and Minimize Z = 3x + 5y

subject to 3x – 4y + 12 ≥ 0

2x – 4y + 12 ≥ 0

2x – 3y – 12 ≥ 0

0 ≤ x ≤ 4

y ≥ 2

4. A company produces two types of belts A and B. Profits on these belts are Rs. 2 and Rs. 1.50

per belt respectively. A belt of type A requires twice as much time as belt of type B. The company

can produce almost 1000 belts of type B per day. Material for 800 belts per day is available.

Almost 400 buckles for belts of type A and 700 for type B are available per day. How much belts

of each type should the company produce so as to maximize the profit?

5. Minimize Z = 3x + 3y, if possible, subject to the constraints

x – y ≤ 1; x + y ≤ 3, x ≥ 0; y ≥ 0.

6. To Godowns X and Y have a grain storage capacity of 100 quintals and 50 quintals respectively.

Their supply goes to three ration shop A, B and C whose requirements are 60, 50 and 40 quintals

respectively. The cost of transportation per quintals from the godowns to the shops are given in

following table :

To Cost of transportation (in Rs. per quintal

From X Y

A 6.00 4.00

B 3.00 2.00

C 2.50 3.00

How should the supplies be transported to minimize the transportation cost?

7. An Aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 in made on each first

class ticket and a profit of Rs. 300 is made on each second class ticket. The airline reserves at

least 20 seats for first class. However atleast four times as many passengers prefer to travel by

second class than by first class. Determine, how many tickets of each type must be sold to

maximize profit for the airline.

8. A diet for a sick person must contain atleast 4000 units of vitamins, 50 units of minerals and 1400

units of calories. Two foods A and B are available at a cost of Rs. 5 and Rs. 4 per unit respectively.

One unit of food A contains 200 unit of vitamins, 1 unit of minerals and 40 units of calories

whereas one unit of food B contains 100 units of vitamins, 2 units of minerals and 40 units of

calories. Find what combination of the food A and B should be used to have least cost but it must

satisfy the requirements of the sick person.

170 XII – Maths

9. A dealor wishes to purchase a number of fans and sewing machines. He has only Rs. 5760 to

invest and has space for almost 20 items. A fan and sewing machine cost Rs. 360 and Rs. 240

respectively. He can sell a fan at a profit of Rs. 22 and sewing machine at a profit of Rs. 18.

Assuming that he can sell whatever he buys, how should he invest his money to maximise his

profit?

10. If a young man rides his motorcycle at 25 km/h, he has to spend Rs. 2 per km on petrol. If he

rides at a faster speed of 40 km/h, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to

spend on petrol and wishes to find the maximum distance he can travel within one hour. Express

this as L.P.P. and then solve it graphically.

11. A producer has 20 and 10 units of labour and capital respectively which he can use to produce

two kinds of goods X and Y. To produce one unit of X, 2 units of capital and 1 unit of labour is

required. To produce one unit of Y, 3 units of labour and one unit of capital is required. If X and

Y are priced at Rs. 80 and Rs. 100 per unit respectively, how should the producer use his

resources to maximise the total revenue?

12. A farmer has a supply of chemical fertilizers of type A which contains 10% nitrogen and 6%

phosphoric acid and type B which contains 5% nitrogen and 10% phosphoric acid. After soil

testing, it is found that at least 7 kg of nitrogen and the same quantity of phosphoric acid is

required for a good crop. Fertilizer of type A costs Rs. 5 per kg and type B costs Rs. 8 per kg.

How many kilograms of each type of fertilizers should be bought to meet the requirement at the

minimum cost.

13. A factory owner purchases two types of machines A and B for his factory. The requirements and

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 the

machines. How many machines of each type should he buy to maximise the daily output.

14. A manufacturer makes two types of cups A and B. There machines are required to manufacture

the cups and the time in minute required by each in as given below :

Types of Cup Machine

I II III

A 12 18 6

B 6 0 9

Each machine is available for a maximum period of 6 hours per day. If the profit on each cup A

is 75 paise and on B is 50 paise, find how many cups of each type should be manufactured to

maximise the profit per day.

15. Using graphical method, solve the following L.P.P.

Maximise Z = 4x + 5y

171 XII – Maths

Subject to constraints

2x + y ≤ 30

x + 2y ≤ 24

x ≥ 3

y ≤ 9

y ≥ 0

A N S W E R S

1. Maximum Profit = Rs. 95 with 5 shares of each type.

2. Lamps of type A = 40, Lamps of type B = 20.

3. Max. value of Z = 42 at x = 4 and y = 6.

Min. value of Z = 19 at x = 3 and y = 2.

4. Maximum Profit Rs. 1300, No. of belts of type A = 200 and type B = 600.

5. Maximum value is infinity as solution is unbounded.

6. From X to A, B and C 10 quintals, 50 quintals and 40 quintals respectively.

From Y to A, B, C 50 quintals, NIL and NIL respectively.

7. No. of first class tickets = 40, No. of 2nd class tickets = 160.

8. Food A : 5 units, Food B : 30 units.

9. Fan : 8; Sewing machine : 12, Max. Profit = Rs. 392.

10. At 25 km/h he should travel 50/3 km, At 40 km/h, 40/3 km. Max. distance 30 km in 1 hr.

11. X : 2 units; Y : 6 units; Maximum revenue Rs. 760.

12. Type A : 50 kg, Type B : 40 kg Minimum cost Rs. 570.

13. Type A : 6; Type B : 0

14. Cup A : 15; Cup B : 30

15. x = 12, y = 6 Zmax = 96.

172 XII – Maths

CHAPTER 13

PROBABILITY

POINTS TO REMEMBER

� Conditional Probability : Probability of event A given that event B has already occurred

( )

( ).

A P A BP

B 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 ∩ C) = P(A) P(B) P(C);

P (A ∩ B) = P(A).P(B),

P(B ∩ C) = P(B).P(C) and

P(C ∩ A) = P(C).P(A)

� If A and B are Independent then

(i) A and c

B/ are independent

(ii)c

A/ and B are independent

(iii) andc c

A B/ /

are independent.

� Baye’s Theorem : If E1, E

2 _ _ _ _ _ _ , E

n are mutually exclusive and exhaustive events and

A be any event on sample space S, such that P(A) ≠ 0, If A has already occurred then

( )

( )=

=

1

.

.

iin

i

ii

AP Ei P

EEP

A AP E P

E

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

173 XII – Maths

� Probability Distribution : Let a random variable. Let a random variable x assumes values

x1, x

2, x

3, _ _ _ _ _ x

n with corresponding probabilities p

1, p

2, p

3, _ _ _ _ p

n. Then different values

of the random variable alongwith their corresponding probabilities form a probability distribution.

Mean of a probability distribution,

1

n

i i

i

µ p x

=

≡ =

Variance2 2 2

1

– .

n

i i

i

p x µ

=

≡ σ =


1. Find P(A ∩ B)if ( ) ( ) 6 22 and .

13 5

AP A P B P

B

= = =

2. Given that the numbers appearing on throwing a pair of dice together are different. Find the

probability that the sum of numbers appearing on the dice is 6.

3. A and B are independent events. If ( ) ( ) ( )1 1and . Find .

2 6P A P A B P B= ∩ =

4. A problem is given to three students whose chances of solving it are 1 1 1

, and2 3 4

respectively.

What is the probability that the problem is solved?

5. If 2

5

AP

B

=

find P(A). It is given that A and B are independent events.

6. A and B are two independent events such that ( ) ( )1 1and

4 3P A P B= = . Find P(A ∩ B´ )

7. If P(A) = 0.25, P(B) = 0.50 and P(A ∩ B) = 0.14. Find P (neither A nor B).

8. In a three letter word of English, find the probability that all the three letters are repeated.

9. Is the following probability distribution valid?

x 0 1 2 3

p(x) 0.3 –0.1 0.4 0.4

10. Find k from the following probability distribution :

x –1 0 1 2

p(x) k 2k k 0.04

11. A boy throws a coin. If he throws a head, he gets Rs. 10. If he throws a tail, he gets Rs. 2. If he

throws the coin only once, find his expectation.

174 XII – Maths

12. Four bad eggs are accidently mixed with 10 good ones. Three eggs are drawn at random without

replacements. Find the probability that all the three are bag eggs.

13. What is the probability that a non-leap year has 53 Sundays.

14. A policeman fires three shots on a dacoit. The probability that the dacoit will be killed by one fire

is 0.6. What is the probability that the dacoit is still alive.

15. Find P(x = 1) of the Binomial distribution 1

3, .6

B


16. A pair of dice is thrown and the sum of the numbers is observed to be even. What is the

probability that both dice have come up with even numbers?

17. The probability that a students selected at random will pass in mathematics is 4

5 and the probability

that he will pas in mathematics and computer science is 1

.2

What is the probability that he will

pass in computer science given that he has passed in Mathematics.

18. A bag contains 5 white, 3 red and 2 blue balls. Four balls are drawn at random one by one without

replacement. Find the probability of drawing atleast one white ball.

19. A can hit a larget 4 times in 5 shots, B 3 times in 4 shots and C twice in 3 shots. They fire a

volley. What is the probability that atleast two shots hit?

20. A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls. A bag

is selected at random and a ball is drawn from it. Find the probability that the ball drawn in red.

21. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two card are drawn

alrandom and found to be spade each. Find the probability that the lost cord was also spade.

22. A candidate may use bus, scooter or some other means to reach the examination centre with the

probability 3 1 3

, and10 10 5

respectively. The probability that he will be late are 1 1

and4 3

respectively

if he travels by bus or scooter but will reach in time if he uses any other means. If he reached

late at the centre, find the probability that he travelled by bus.

23. Find the probability distribution of the number of green balls drawn when 3 balls are drawn one

by one without replacement from a bag containing 3 green and 5 white balls.

24. Three cards are drawn successively with replacement from a pack of well shuffted 52 cards.

Determine the probability distribution of X if X denotes the number of cards of heart in three drawn

cards.

25. In a game, a man wins a rupee for a six and looses a rupee for any other number when a fair

die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find

the expected value of amount he wins/looses.

175 XII – Maths

26. A fair coin is tossed 10 times. Find the probability of almost 6 heads.

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

28. In 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?

29. Five dice are thrown simultaneously. If getting 3, 4 or 5 on a single die is considered a success

then find the probability of almost three successes.

30. Assuming that a family has two children :

(i) Write the Sample space

(ii) What is the probability that both the children are boys given that atleast one of them is

a boy.


31. In solving a question, probability that the students knows the answer is 3

,10

copies with probability

1,

5 guesses with probability

1.

10 Probability that the student does not attempt the question is

2.

5

Probability that the answer is correct given that he copied is 1

4 and the probability of giving

correct answer by guessing is 1

.5

Given that his answer is correct find the probability that he did

it by guessing.

32. Probability of attempting a problem by A, B and C is 1 1 1

, and2 3 6

respectively. Probability that

A, B and C will solve it correctly is 1 3 4

, and2 5 5

respectively. If the problem is solved correctly,

what is the probability that it was solved by (i) A; (ii) B; (iii) C?

33. Find the mean and variance of the number of kings if three cards are drawn at random without

replacement from a pack of well shuffled 52 cards.

34. Find the probability distribution of number of sixes in throwing a die five times. Also find its mean

and variance.

35. Suppose 15% of men and 36% of women have grey hair. Probability of dying hair by men is 21%

and by women is 63%. A dyed hair person is selected at random. What is the probability that the

selected person is (i) Male (ii) Female?

36. An unbiased coin is tossed six times. Find the probability of getting (i) almost 3 heads; (ii) atleast

2 heads; (iii) Mean and variance of no. of heads.

37. A pair of dice is thrown 7 times. Getting a total of 7 is considered success. Find the probability

of (i) No success; (ii) 6 successes; (iii) atleast 6 successes; (iv) Almost 6 successes.

176 XII – Maths

38. Three person review a book. Odds in favour of the book are 5:2; 4:3 and 3:4 respectively. Find

the probability that majority are in favour of book.

39. A man takes steps forward with probability 0.4 and backward with probability 0.6. Find the probability

that at the end of eleven steps, he is one step away from the starting point.

40. Four cards are drawn from a pack of well shuffled 52 cards. Find the probability that they are all

of different suits.

A N S W E R S


1.6

652.

2

15

3.1

34.

3

4

5.2

56.

1

6

7. 0.39 8.1

676

9. No. 10. k = 0.24

11. Rs. 6 12.1

91

13.1

714. 0.064

15.75

216

SHORT ANSWER T YPE QUESTIONS

16.1

217.

5

8

18.41

4219.

5

6

20.19

4221.

11

50

22.9

13

177 XII – Maths

23.

X : 0 1 2 3

P(X) :5

28

15

28

15

56

1

56

24.

X : 0 1 2 3

P(X) :27

64

27

64

9

64

1

64

25. Expected to loose Rs. 91

5426.

53

64

27.

X : 0 1 2 3

P(X) :125

216

75

216

15

216

1

216

28.11

24329.

13

16

30. (i) S = {bb, bg, gb, gg} (ii)1

3

LON G ANSWER T YPE QUESTIONS

31.2

37

16. (i)3

7(ii)

12

35(iii)

8

35

33.

X : 0 1 2 3

P(X) :4324

5525

1128

5525

72

5525

1

5525

34.

X : 0 1 2 3 4 5 Mean 5

6=

P(X) :3125

7776

3125

7776

1250

7776

250

7776

250

7776

1

7776Variance

275

36=

178 XII – Maths

35. (i)5

41(ii)

36

41

36. (i)21

32(ii)

11

32(iii) Mean = 3; Variance = 1.5

37. (i)

75

6

(ii)

71

356

(iii)

51

6

(iv)

71

1 –6

38.209

34339. 462 (0.24)5

40.2197

.20825

179 XII – Maths

PRACTICE QUESTIONS PAPER – I

MATHEMATICS

Time allowed : 3 hours Maximum marks : 100

General Instructions

1. All question are compulsory.

2. The question paper has three sections. Section A contains 10 questions of one mark each,

Section B contains 12 questions of 4 marks each and Section C contains 7 questions of six marks

each.

3. All questions are compulsory.

4. Interval choice are given in some questions, where one part is to be attempted out of two.

5. Calculators are not allowed.

SECTION A

1. What are the direction cosines of the vector $– 2 .a i j k= +uuur

$ $

2. If $ $– and –a i j k b i j k= + = +uuur uuur

$ $ $ $ find . .a buuur uuur

3. If 3, 2 and . 3,a b a b= = =uuur uuur uuur uuur

find the angle between and .a buuur uuur

4. Write the value of x and y if

1 3 0 5 62

0 1 2 1 8

y

x

+ =

5. If 1 2

4 2A

then find the value of x if |2A| = x |A|.

6. Write the principal value of –1 3

sin .2

7. Evaluate

( )

2

23 2

3 4 5.

2 5 1

x xdx

x x x

+ −

+ − +

8. Find the derivative of sin2 (2x + 3) w.r.t. x.

9. If the binary operation *, defined on Q is defined as

a * b = 2a + b – ab, for all a, b ∈ Q, find the value of 5 * 4.

10. Write a 2 × 2 matrix A = [aij] whose elements are aij = i – j.

180 XII – Maths

SECTION B

11. Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1,T2) : T1 is

congruent to T2}.

Show that R is an equivalence relation.

12. Prove that 1 1 11 2 1 3

tan tan cos4 9 2 5

− − − + =

OR

Solve the following for x.

21 1

2 2

2 1 2tan cos .

31 1

x x

x x

− − − π + = − +

13. Using the properties of determinent prove that ( )2.

a l m n

l a m n a a l m n

l m a n

+

+ = + + +

+

OR

If

1 2 2

2 1 2 .

2 2 1

A =

Verify that A2 – 4A – 5I = 0

14. For what value of k, is the function

( ) 2

1 cos 4, 0

8

0

xx

f x x

k x

− ≠ =

=

continuous at x = 0?

15. If

12

2

sinlog 1

1

x xy x

x

−

= + −−

then show that ( )

1

3 22

sin.

1

dy x

dx x

−

=

−

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

tπ

=

17. Evaluate ( )20

sin.

1 cos

x xdx

x

π

+

181 XII – Maths

18. Solve the differential equation

( )2

2

21 2 .

1

dyx xy

dx x− + =

−

OR

Solve the differential equation. (x2 – y2) dx + 2xy dy = 0.

19. Solve the following differential equation

2cos tan .

dyx y x

dx+ =

20. If , ,a b cuuur uuur uuur

are three vectors such that and .a b a c a b a c= × = ×· ·uuur uuur uuur uuur uuur uuur uuur uuur

0,a ≠uuur

then show that .b c=uuur uuur

21. Find the angle between the line 2 2 5 3

3 4 6

x y z− − −= =

− and the plane x + 2y + 2z – 5 = 0.

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).

22. Five dice are thrown simultaneously. If the occurrence of an even number on a single die is

considered a success. Find the probability of getting almost 3 successes.

SECTION C

23. If

2 1 1 3 1 1

1 2 1 and 1 3 1

1 1 2 1 1 3

A B

− − = − − = − −

Find AB use the result to solve the following system of linear equation

2x – y + z = –1

–x + 2y – z = 4

x – y + 2z = –3

24. A window is in the form of a rectangle Surmounted by a semi-circular opening. The total perimeter

of the window is 10 m. Find the dimensions of the window to admit maximum light through the

whole opening.

OR

Find the interval in which the function f(x) = (x + 1)3 (x – 3)3 is (i) increasing (ii) decreasing.

182 XII – Maths

25. Evaluate 4 4

1.

sin cosdx

x x+ 26. Find the area of the region

( ){ }2 2, : 25 9 225 and 5 3 15 .x y x y x y+ ≤ + ≥

27. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to each

of 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+ − −= =

28. There are two bags I and II containing 3 red and 4 white balls and 2 red and 3 white balls

respectively. A bag is selected at random and a ball is drawn from it. If it is found to be a red ball.

Find the probability that it is drawn from the first bag.

29. One kind of Cake requires 200 gm of flour and 25 gm of fat, and another kind of cake requires

100 gm of flour and 50 gm of fat. Find the maximum number of Cakes which can be made from

5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in

making the Cakes. Formulate the above as a linear programming problem and solve graphically.

183 XII – Maths

MARKING SCHEME FOR PAPER – I

SECTION A

1.−1 1 2

, , .6 6 6

2. . 1.a b =uuur uuur

3.3

π

4. x = 3, y = 3.

5. 4.

6.3

π

7.3 2

1

2 5 1c

x x x− +

+ − +

8. 2 sin (4x + 6).

9. –6

10.0 1

.1 0

−

SECTION B

11. (i) Reflexive

As every triangle is congruent to itself

i.e., (T1, T

1) ∈ R

∴ R is reflexive. 1

(ii) Symmetric

Now ( )1 2 1 2 2 1,T T R T T T T∈

∴ (T2, T1) ∈ R

∴ R is symmetric 1

184 XII – Maths

(iii) Transitive

Now (T1, T

2), (T

2, T

3) ∈ R

1 2 2 3andT T T T≅ ≅

⇒ 1 3T T≅

∴ (T1, T3) ∈ R

∴ R is transitive.

From (i), (ii) and (iii), R is an equivalence relation 2

12.− − − −

+

= + = = −

1 1 1 1

1 2

1 2 14 9. . . tan tan tan tan1 24 9 21 ,4 9

L H S 2

( )( )

− − − − = = = +

21 1 1

2

1 1 21 1 1 3tan cos cos . . .

2 2 2 51 1 2R H S 1+1 = 2

OR

21

2

1cos 2 tan

1

xx

x

− −= π −

+1

⇒1 1 2

2 tan 2 tan3

x x− − π

π − − = 1

⇒1 1

4 tan tan3 12

x x− −π π

= 1

⇒ tan .12

xπ

= 1

13. Let

a l m n

l a m n

l m a n

+

∆ = +

+

Applying c1 → c1 + (c2 + c3) we get

a l m n m n

a l m n a m n

a l m n m a n

+ + +

∆ = + + + +

+ + + +

1

( )1

1

1

m n

a l m n a m n

m a n

= + + + +

+

185 XII – Maths

Appllying R2 → R

2 – R

1 and R

3 → R

3 – R

1 we get

( )1

0 0

0 0

m n

a l m n a

a

∆ = + + + 2

= (a + l + m + n) [a(a – 0)] = a2 (a + l + m + n) 1

OR

2

9 8 9

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

For verifying A2 – 4A – 5I = 0. 1

14. Here f(0) = k 1/2

( )2

0 0

1 cos 4lim lim

8x x

xf x

x→ →

−=

→ →= =

2 2

2 20 0

2 sin 2 sin 2lim Lt

2 4 4x x

x x

x x·1

2

0

sin 2Lt

2x

x

x→

=

1

= (1)2 = 1 1/2

For continuity at x = 0

( ) ( )0

lim 0x

f x f k→

= = 1/2

∴ k = 1.

15.

12

2

sinlog 1

1

x xy x

x

−

= + −−

( ) ( )2 1 1

3 2 22 22

sin sin

11 11

dy x x x x x

dx xx xx

− −

= + + −−− −−

2

( )

2 1 1

3 2 22

sin sin

11

x x x

xx

− −

= +−−

186 XII – Maths

1 2

22

sin1

11

x x

xx

− = + −−

1

1 2 2

22

sin 1

11

x x x

xx

− + − = −−

1/2

( )

1

3 22

sin

1

x

x

−

=

−1/2

16. At1

, , 04 2

t x yπ

= = = 1

= =3 cos 3 , –2 sin 2dx dy

t tdt dt

1/2 + 1/2

∴2 sin 2

3 cos 3

dy t

dx t

−= 1/2

∴

4

2 2.

3t

dy

dx π=

=

1/2

Equation of tangecil 3 2 2 2 or 3 – 2 2 2 0.y x y x= − + =

17.2

0

sin.

1 cos

x xI dx

x

π=

+

Getting2

0

sin2

1 cos

xI dx

x

π π=

+ 1½

Put cos x = t ⇒ – sin dx = dt ⇒ x = 0, t = 1, x = π ⇒ π = –1

∴ 20

21

dtI

t

π= π

+ 1½

2I = π2/2

2

.4

Iπ

= 1

18. Writing the differential equation as

( )2 22

2 2.

1 1

dy xy

dx x x

+ =− −

1/2

Integrating factor −= 2

2

1

xdx

xe

187 XII – Maths

∴ The solution is 1

( )2

2

2· 1

1y x dx

x− =

− 1

⇒ ( )2 11 log

1

xy x c

x

−− = +

+· 1½

OR

Writing

2 2

,2

dy y x

dx xy

−= which is homogeneous 1/2

Puttingdy dv

y vx v xdx dx

= 1/2

∴ getting 2

2

1

v dxdv

xv

−=

+1

∴ log |1 + v2| = – log |x| + log |c|

logc

x= 1

⇒ x2 + y2 = cx

⇒ c = 2 when x = 1 and y = 1 1

∴ x2 + y2 = 2x

19. Writing2 2

sec tan . secdy

xy x xdx

+ = 1/2

I.F. = etan x. 1

∴ Solution is tan tan 2

. . tan . secx x

y e e x x dx c= + 1/2

( )tan tan. tan 1

x xy e e x c= − +

or y = (tan x – 1) + c . e–tan x. 2

20. ( ). . 0a b a c a b c= uuur uuur uuur uuur uuur uuur uuur

⇒ either ( )0 or 0 ora b c a b c= − = ⊥ −uuur uuur uuur uuur uuur uuur

...(i) 1½

( )– 0a b a c a b c× = × uuur uuur uuur uuur uuur uuur uuur

⇒ either ( )0 or 0 or –a b c a b c= − =uuur uuur uuur uuur uuur uuur

...(ii) 1½

188 XII – Maths

Give that 0a ≠uuur

and a vector can not be parallel or perpendicular to the same vector

∴ 0b c− =uuur uuur

[from (i) and (ii)]

⇒ b c=uuur uuur

. 1

21. Equation of line is

2 5 2 3

3 2 6

x y z− − −= =

∴ DR´s at line are 3, 2,6 1

DR's of the normal to the plane x + 2y + 2z – 5 = 0

are 1, 2, 2 1

Let θ be the angle between line and plane then

∴1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

sina a b b c c

a b c a b c

+ +θ =

+ + + + +1/2

+ +=

3.1 2.2 6.2

49 . 91

19

21=

∴ Angle between line and plane 1 19

sin .21

−θ = 1/2

OR

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

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

∴ ( ) ( ) ( )2 22 2

3 2 3 3 2 3 2= λ − + λ − + λ 1

18 = 17λ2 – 30λ + 18

⇒ λ(17λ – 30) = 0

⇒30

0 or .17

λ = λ = 1

∴ points one ( )56 43 111, , or 2, 1, 3 .

17 17 17

− −

1

22. P(sucess) = P(even number)

= P(2, 4, 6)

189 XII – Maths

3 1

6 2= = 1

∴ P(Failure) 1 1

12 2

= − = 1/2

P(At most 3 sucesses) = P(x ≤ 3)

= 1 – P(x > 3) 1/2

= 1 – P(4 or 5 successes)

= 1 – [P(4) + P(5)] 1/2

4 5

4 5

1 1 11 5 5

2 2 2c c

= − +

1/2

= − +

5 11

32 32

61

32= −

26 13or .

32 16= 1

23.

2 1 1 3 1 1

1 2 1 1 3 1

1 1 2 1 1 3

AB

− − = − − − −

0 0 0

4 0 1 0 4

0 0 1

I

= =

1½

i.e.,1

.4

A B I=

∴–1

3 1 11 1

1 3 14 4

1 1 3

A B

− = = −

½ + ½

The given system of equation is

2x – y + z = –1

–x + 2y – z = 4

x – y + 2z = –3

190 XII – Maths

In matrix form

2 1 1 1

1 2 1 4

1 1 2 3

x

y

z

− − − − = − −

1

∴ Ax = C

or X = A–1 C 1/2

or

3 1 1 11

1 3 1 44

1 1 3 3

x

y

z

− − = = − −

4 11

8 24

4 1

= = − −

1½

∴ x = 1, y = 2, z = –1. 1/2

24. Let r be the radius of the semi circle.

∴ Breadth of rectangle portion = 2r

Let 'l' be the length of the rectangular portion

∴ Perimeter of window = 2r + l + πr + l

= 2r + 2l + πr 1/2

As perimeter of window is 10 m.

∴ 2r + 2l + πr = 10

2l = 10 – 2r – πr ...(i) 1/2

Area of window A = 2r × l + 1/2πr2

2 2110 2

2r r r= − − π

10 – 4dA

r rdr

= − π 1

Fox maximum Area dA/dr = 0

= 10 – 4r – πr = 0

10

4r =

+ π

and d 2A/dr 2 = –4 – π < 0 1

2r

2r

ll

A B

D C

For correct figure one mark.

191 XII – Maths

∴ A is maximum at =+ π

10

4r m

From (i) ( ) 10 202 10 2

4 4l m= − + π =

+ π + π

∴10

4l =

+ π1/2

Hence dimentions one

10 20length .; breadth .

4 4mt mt= =

+ π + π1/2

OR

We are given f(x) = (x + 1)3 (x – 3)2

Getting f 1(x) = 6(x + 1)2 . (x – 3)2 (x – 1) ...(i) 2

(i) For f(x) to be increasing f´(x) > 0 1/2

or 6(x + 1)2 (x – 3)2 (x – 1) > 0

or x – 1 > 0

x > 1 1

∴ f(x) is increasing function in (1, ∞). 1/2

(ii) For f(x) to be decreasing f´(x) < 0

i.e., 6(x + 1)2 (x – 3)2 (x – 1) < 0 1/2

i.e., x – 1 < 0

or x < 1 1

∴ f(x) is decreasing function in (– ∞, 1).

25. Let4 4

sin cos

dxI

x x=

+

( )2 2

4

sec 1 tan

tan 1

x xdx

x

+=

+ 1

Let tan x = t ⇒ sec2 x dx = dt

∴

2

4

1

1

tI dt

t

+=

+

192 XII – Maths

2

2

11

12

t dt

tt

+

= − +

2

Let 1

t zt

− = then 2

11 dt dz

t

+ =

( )222

dzI

z

=+

11tan

2 2

zc

− = +

211 tan 1

tan .2 2 tan

nc

x

− −= +

2

26. 25x2 + 9y2 = 225 ⇒2 2

19 25

x y+ =

5

4

2

2

1

1

2

3

4

5

–3 –2 –1 1 2 3

CXX´

Y´

Y

A

Correct Fig. 1

Finding the point of intersection x = 0, x = 3

( )3 3

2

0 0

1 1Area 225 25 – 15 5

3 3x dx x dx= − − 2

( )33 2

2 1

0 0

5 9 5 39 sin

3 2 2 3 3 2

x x xx

− − = − + +

1

193 XII – Maths

( )15 15 152 sq. unit.

4 2 4

π= − = π − 1

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

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 8x – 13y + 15z = 0 1

28. Let event E1 : the first bag is selected

event E2 : The second bag is selected/

and event A : A red ball is drawn.

then ( ) ( )1 2

1 1and

2 2P E P E= = 1

( ) ( )1 2

3 2and .

7 5P A E P A E= = 2

( ) ( ) ( )( ) ( ) ( ) ( )

1 11

1 1 2 2

.

. .

P E P A EP E A

P E P A E P E P A E=

+ 1

1 3.

2 71 3 1 2

. .2 7 2 5

=+

3 3

1514 14 .3 1 29 29

14 15 70

= = =+

2

29. Let number of cakes of first kind be x and that of second kind be y.

L.P.P. is maximize Z = x + y.

194 XII – Maths

Subject to 200x + 100y ≤ 5000

or 2x + y ≤ 50 ...(i)

25x + 50y ≤ 1000

or x + 2y ≤ 40 ...(ii)

x ≥ 0, y ≥ 0 ...(iii)

For (i), (ii), (iii) 2

Correct Graph 2 marks

Maximum at B(20, 10)

∴ Maximum Number of Cakes = 20 + 10 = 30.

50

40

30

20

10

Y

10 20 30 40 50XX´

Y´

0

(24%)

(0, 20)

C

B(20, 10)

2 + = 50x y

x y + 3 = 40

195 XII – Maths

PRACTICE QUESTIONS PAPER – II

MATHEMATICS

Time Allowed : 3 hours Maximum Marks : 100

General Instructions

1. Question paper has three sections. Section A contains 10 questions of one mark each, Section

B contains 12 questions of 4 marks each and Section C contains 7 questions of six marks each.

2. All questions are compulsory.

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

4. Calculators are not allowed.

SECTION A

1. Write the order and degree of the differential equation

322

2

21 .

dy d yx

dx dx

+ =

2. Find the integrating factor of the differential equation

3 cot sin 2 .dy

y x xdx

− =

3. Evaluate 2

cos.

sin

a b xdx

x

−

4. The radius of a circle is increasing at the rate of 2 cm/s, find the rate at which its circumference

is increasing?

5. If 5 2 9

0, find .2 5 3

xx

x

+=

+

6. If 3 0 2

,7 5 1 5

x x y

x

+ − = −

find the values of x and y.

7. Find the cofactor of '4' in the following determinants.

2 3 5

6 0 4

1 5 7

−

−

8. If the sum of two unit vectors $ $anda b is a unit vector, then what is the angle between $ $anda b .

196 XII – Maths

9. If the projection of buuur

on auuur

is .a buuur uuur

, then what is the value of .auuur

10. If the vectors $ $–2 3 and 6 2i j yk xi j k+ + − +$ $ $ $ are collinear, then find the values of x and y.

SECTION B

11. Prove that 1 11 1 1 1

tan cos , 1.1 1 4 2 2

x xx x

x x

− − + − − π= − − ≤ ≤ + + −

OR

Solve for x : ( ) ( )1 1 1 8tan 1 tan 1 tan .

31x x

− − −+ + − =

12. If

1 2 3 4 1 5

´ 5 7 9 and 1 2 0 ,

2 1 1 1 3 1

A B

− − − = = −

then find (A + 2B)´.

13. Find the derivative of the function given by f(x) = (1 + x) (1 + x2) (1 + x4) (1 + x8), and hence

find f ´(1).

14. If

21 4 3 1

sin , find .5

x x dyy

dx

− + − =

15. Find the equation of normal to the curve x2/3 + y2/3 = 8 at the point x = 8.

OR

The length x of a rectangle is decreasing at the rate of 4cm/minute and the width y is increasing

or the rate of 5 cm/minute. When x = 8 cm and y = 6cm, find the rate of change of the area of

the rectangle.

16. Evaluate : ( ) ( )2 2

2.

1 3

xdx

x x+ +

17. Evaluate :21 sin 2

.1 cos 2

xxe dx

x

+

+ . .

18. Evaluate :2

4 40

sin cos.

sin cos

x x xdx

x x

π

+ 19. Solve the differential equation : (x – y) (dx + dy) = dx – dy, given that y = –1 when x = 0.

OR

Solve the differential equation :

2yex/y dx + (y – 2x . ex/y) dy = 0, given that when x = 0, y = 1.

197 XII – Maths

20. If $3 – , 2 – 3i j i j kα = β = +uuur uur

$ $ $ $ , express βuur

in the form 1 2β + β

uuur uuur where

1βuuur

is parallel to

αuuur

and 2βuuur

is perpendicular to .αuuur

OR

If auuur

and buuur

are unit vector inclined at an angle θ, then prove that 1

cos .2 2

a bθ

= +uuur uuur

21. Find the vector equation of a line passing through (1, 2, 3) and parallel to the planes

$( ) $( )– 2 5 and 3 6.r i j k r i j k+ = + + =. .uur uur

$ $ $ $

22. Twelve cards numbered 1 to 12 are placed in a box, mixed up thoroughly and then one card is

drawn randomly. If it is known that the number on the card drawn is more than 4, what is the

probability that it is an even number.

SECTION C

23. Consider the binary operation * : R × R → R and 0 : R × R → R defined as a * b = |a – b| and

a o b = a, ∀ a, b ∈ R.

Show that * is commutative but not associative, o is associative but not commutative. Further

show that a * (b o c) = (a * b) o (a * c), ∀ a, b, c ∈ R.

Also verify if a o (b * c) = (a o b) * (a o c) or not.

24. If

2 1 1

8 1 2 ,

5 1 1

A

= −

find A–1. Using A–1, solve the system of linear equations :

2x + 8y + 5z = 5, x + y + z = – 2, x + 2y – z = 2.

If

2 1 1

1 2 1 ,

1 1 2

A

− = − − −

verify that A3 – 6A2 + 9A – 4I = 0. Where I is the identity matrix of order

3. Hence find A–1.

25. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that

its depth is 2m and volume is 8m3. If building of tank costs Rs. 70 per sq. meter for the base and

Rs. 45 per sq. meter for sides. What is the cost of least expansive tank?

OR

Find the intervals in which the function given by

f(x) = tan–1 (sin x + cos x), 0 < x < 2π.

is (a) strictly increasing (b) strictly decreasing.

198 XII – Maths

26. Find the area lying above x-axis and included between the circle x2 + y2 = 6x and the parabola

y2 = 3x.

27. Two cards are drawn simultaneously (without replacement) from a well shuffled pack of 52 cards.

Find the mean, variance and standard deviation of the number of spades.

28. Show that the point A(1, 1, 1), B(2, 2, –6), C(3, 1, –5) and D(–1, 1, 7) are coplaner.

29. A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B

and C. Production of one chair requires 2 hours on machine A, 1 hour each on machine A and

B and 3 hours on machine C. The profit on selling one chair is Rs. 30, while by selling one table

the profit is Rs. 60. The total time available per week on machine A is 70 hours, on B is 40 hours

and on machine C is 90 hours. How many chairs and tables should be made per week so as to

maximise profit? Formulate the problem as a L.P.P. and solve it graphically.

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