Summer vacations Holiday Home Work 2019-20 Class-XII Maths 1. Give the example of a relation, which is transitive but neither reflexive nor symmetric. 2. Find the values of unknown quantities if. 3 4 1( ) , the values of x & y7 4 0 4x y yi find

x

+ − = −

3 1 2( ) 2 3 5 3x y yi

y x

+ − = − −

3. If ( )2 21 1x y a x y− + − = − then prove that 2211dy y

dx x

−=

− (hots) 4. Write the function in the simplifies from 1 Cos( ) tan 1 Sinxix

− + 2 21 2 21 1( ) 1 1x x

ii Cot

x x

− + + − + + − 21 21 1) Cos 2 1 x

iii

x

− + +

+ 5. Find the derivative of Sin CosSin Cosx x x

x x x

−+

6. Evaluate 1 15 13( ) tan tan Cos Cos6 6n ni − − +

1 1 11 1( ) tan tan Sin 23 3ii Cotπ− − − − − + +

1 17 7( ) Cos Cos tan t an6 6iii

π π− − +

( )1 13( )Sin Sin Sec 25iv

π− − + −

7. Prove that 1 1 14 5 16Sin Sin Sin5 13 65 2π− − − + + = (hots) 8. Find dy

dx if 1 1Sin Cosy x x− −= + 9. Let Z be the set of integers, m be a positive integer and R be the relation on Z defined by

( ){ }, : , , R x y x y Z x y is divisible by m= ∈ − Prove that R is an equivalence relation. 10. Prove that 1 1 11 1 4102 tan tan tan5 8 7− − − + =

11. Solve for 21 12 21 1 2 2: Cos tan2 31 1x xx

x x

π− −−+ =

+ − (hots)

12. If 23A R = −

and a function :f A A→ is defined by ( ) 4 36 4xf x

x

+=

−Find 1f − if f is invertible. 13. Show that 11 3 4 7tan Sin2 4 3− − =

14. Find the value of 21 12 211 2tan Sin Cos2 1 1 yx

x y

− − −+

+ + 15. Let S be any non empty set. (i) Prove that union and intersection are the binary operations on P(S). (ii) Prove that both these operations are commutative and associative. (iii) Find the identity elements for these operations. (iv) Find the invertible elements and their inverses in P(S) for both the operations. 16. Find dy

dx if ( )2 2logy x x a= + + . 17. If ( )2 1 m

y x x= + + then show that ( )22 221 0d y dy

x x m ydxdx

− + − = .. 18. Verify Lagrange’s mean value theorem for the function ( ) [ ]3 25 3 , 1,3f x x x x= − − ∈ 19. Let ( )

( )( )

321 Sin , 23Cos, 21 Sin ,2 2xx

x

f x a x

b xx

x

π

π

ππ

−<

= = −

>−

find the values of a and b so that the given function is continuous at 2xπ

= . (hots) 20. Determine the values of a,b,c for which the function ( )

( )2 3Sin 1 Sin ; 0; 0 ; 0x a x xx

x

f x c x

xbx xx

b x

+ + <

= =

− >

May be continuous at 0x = 21. If ( )Sin xxy x x= + then find dy

dx 22. Find dy

dxif ( ) ( )Cos Siny x

x y= 23. If 2 2 1log 2 tan yx y

x

−+ = then show that dy x y

dx x y

+=

− 24. Find 22d y

dxif ( ) ( )Cos Sin ; Sin Cosx a t t t y a t t t= + = − .

25. For the function ( ) 3 26 ,f x x x ax b= − + + it is given that ( ) ( )1 3 0f f= = . Find the values of a & b. Verify the Rolle’s Theorem on [1,3]. 26. Find dydx

if 21 3 4 1Cos 5x xy −

+ − = (hots) 27. If 1Sin cos 1, ,tx a y a t

− −= = Show that dy y

dx x= − . 28. Find dy

dxif 1 2Sin 1 1y x x x x

− = − − − . 29. If y x yx e −= , then show that

( )2loglogdy x

dx ex= (hots) 30. If 1tan log ,x y

a

=

then show that ( ) ( )22 21 2 0d y dy

x x adxdx

+ + − = . 31. Find dydx

if 11 2 3Sin 1 36x x

xy

+−=

+. (hots) 32. If Sin & Cos log tan 2tx a t y a t

= = +

,Find 22d y

dx 33. If 2 2logy x x a = + +

show that ( )22 2 2 0d y dy

x a xdxdx

+ + = 34. Differentiate 21 11 1tan . . tanxw r t x

x

− −+ − 35. Show that the function f defined as: ( ) 23 2, 0 . 12 1 25 4 2x x

f x x x x

x x

− < ≤

= − < ≤ − >

is continuous aat x=2 but not differentiable. 36. It is given that the function . ( ) [ ]3 2 5; 1,3f x x bx ax x= + + + ∈ . Rolle’s Theorem holds with 12 3c = + find the values of a &b. (hots) 37. Find the value of k so that the function defined by ( )

Cos ;2 23; 2k xx

xf x

x

ππ

π

≠ −= =

is continuous at 2xπ

= . 38. Find the value of k so that the function defined by ( )( )3Sin 1 ; 12tan Sin : 0k x x

f xx x

xx

π + ≤= − ≥

is continuous at x=0. 39. Find the 2nd order derivative if Sin(log )y x=

40. If (i) 1 1 11 2 32 1 3A

= − −

Show that 3 2 36 5 11 0A A A I− + + = and hence obtain 1.A− Also verify your result. (ii) 3 17 5A

=

, Find x and y such that 2 .A xl yA+ = Hence find 1.A− . (iii) 2 31 2A

=

, find the values of a & b so that 2 :A al bA I+ + is the second order unit matrix. (iv) 0 00 00 0a

A a

a

=

, Find nA . 41. Find the inverse of 1 3 23 0 12 1 0A

− = − −

using elementary transformations. 42. Use properties to prove. (i) 2 2 2 2 221 1 11a ab ac

ba b bc a b c

ca cb c

+

+ = + + +

+

(ii) ( )( )

( )( )

2 2 22 32 2 22 2 2b c a a

b c a b abc a b c

c c a b

+

+ = + +

+

(iii) ( )( )

( )( )

2 2 32 2b c ab ac

ba c a bc abc a b c

ca cb a b

+

+ = + +

+

(iv) ( )

2 2 32 22 2bc b bc c bc

a ac ac c ca ab bc ca

a ab b ab ab

− + +

+ − + = + +

+ + −

(v) 2 2 2 2 2 2 22 2 4b c ab ac

ba c a bc a b c

ca cb a b

+

+ =

+

43. Find the matrix A satisfying the equation 2 1 3 2 1 03 2 5 3 0 1A−

= − (hots)

44. For 1a b

A bcc

a

= +

show that ( )1 2 1A a bc I aA− = + + − . 45. Use properties to prove that ( ) ( ) ( ) ( )

2 32 32 311 11x x px

y y py pxyz x y y z z x

z z pz

+

+ = + − − −

+

and hence. deduce that, if , ,x y z are all different then 2 32 32 311 01x x x

y y y

z z z

+

+ =

+

only if 1xyz = − . 46. (i) Find a,b,c and d if 5 2 34 7ab c

d

−

−

is a symmetric matrix. (ii) Find x,y,z if 2 51 0 47x y

x y z

− −

+

is a symmetric matrix. (iii) Find x, if 0 1 21 0 33 0x

−

−

−

is a skew symmetric matix. 47. Form a 2 2× matrix ;ij ij

iA a a

j = = find 12a . 48. Show that B’AB is a symmetric or a skew symmetric according as A is symmetric or skew symmetric. 49. If A and B are square matrices of the same order such that AB=BA then by using induction prove that AB”=B”A. Further prove that ( )” ” ”AB A B n N= ∀ ∈ . 50. Two Schools A and B decided to award prizes to their students for three values honesty punctuality and obedience. School A decided to award a total of Rs. 11000 for three values to 5, 4 and 3 students while school B decides to award Rs. 10700 for these values to 4,3 and 5 students respectively. The three prizes together amount to Rs. 2700. Represent the above situation by a matrix equation and solve. Which value do you prefer and why?

–1–

CHAPTER 2: INVERSE TRIGONOMETRY

Q1 Express in simplest form:

a)

2

2

12

111cos

x

x b)

x

x

sin1

costan 1

Q2 Prove 2

211

2

1)))(sin(cotcos(tan

x

xx

Q3 Evaluate )]tan(cot)(tan[costan2 111 xxec

Q4 Solve the equation: 42

1tan

2

1tan 11

x

x

x

x

Q5 Solve: 2

sin2)1(sin 11 xx

Q6 Solve: 8

5)(cot)(tan

22121 xx

Q7 Solve: 2

1sintan 2121 xxxx

Q8 Prove that a

b

b

a

b

a 2cos

2

1

4tancos

2

1

4tan 11

Q9 If xxy costancoscot 11 then prove that 2

tansin 2 xy

Q10 Prove that 3

74

4

3sin

2

1tan 1

Mathematics Assignments Class XII 2017 – 18

–1–

CHAPTER 3 & 4 – MATRICES AND DETERMINANTS

Q1 Find x and y if

40

14

47

3

x

yyx

Q2 Find x. 2

0

x such that A + A’=I where

xx

xxA

sincos

cossin

Q3 Express

211

523

423

as the sum of a symmetric and skew symmetric matrices.

Q4 Find a, b and c if AA’=I

cba

cba

cb

A

20

Q5 If

302

120

201

A then prove that A3 – 6A2 + 7A + 2I = 0 hence find A–1.

Q6 Find the inverse of

012

103

231

A using elementary transformations.

Q7 If

10

baA then use P.M.I. show that Nna

abaA

nn

n

101

)1(

Q8 Find the matrix X so that

642

987

654

321X

Q9 If

100

0cossin

0sincos

)( xx

xx

xf Find f(x), f(–x).

Q10 If

11

0A and

15

01B find the values of such that A2 = B

Q11 Find (AB)–1 if

225

5615

113

A and

120

031

221

B

Q12 Prove that 3

222

222

222

)(2

)(

)(

)(

cbaabc

bacc

bacb

aacb

Q13 Prove that 3

22

22

22

)( cabcab

ababbaba

cacacaca

bccbcbbc

Q14 Prove that ))((3

3

3

3

cabcabcba

cbcac

cbbab

cabaa

Q15 Using matrices find A–1 where

433

232

321

A hence solve the system of linear equations

11433and2232432 zyxzyxzyx

Mathematics Assignments Class XII 2017 – 18