SRI SANKARA SENIOR SECONDARY SCHOOL

MATHS REVISION ASSIGNMENT- CLASS- XII

1. Let f: Z → Z be defined by f(x) = x if x is even . Check whether f is one to one and onto.

x+1 if x is odd

2. Consider f: R + → [ -3, ∞ ) given by f(x) = 4x2 +6x -3. Check whether f is invertible : If it is invertible find its inverse.

3. A = QXQ , Q being the set of rational numbers . Let * be binary operation on A, defined by

(a, b)* (c,d) =(ac, ad+b). Check the commutativity, associativity. Find the identity element if it exists, and hence its

invertible elements.

4. Show that the relation S in the set R of real numbers defined as S = { (a,b) : a,b ∈ R such that modulus of a-b is a

multiple of 10 } is an equivalence relation.

5. Let A = { 1,2,3,4,5,6 } Let ‘ X7 ‘ be the operation on A defined as a X7 b = the least non negative remainder when

product of a and b is divided by 7. Construct the composition table for the binary operation. Find the identity element

and the inverse of 2 and 3.

6. Find the number of bijective functions from the set { 1,2,3,4,5,to itself.

7. A set S has 5 elements. Find the number of relations defined on S.

8. Find the value of (i) )1(sin 1 (ii) )2

3(cos 1 (iii) )

5

3(sinsin 1

(iv) )6

7tan(tan 1 (v) )1(tan 1

9. If zyx 111 tantantan , prove that x+y+z = xyz.

10. Solve for x: (i) cos-1 ( )1

12

2

x

x

+ tan-1 (

21

2

x

x

) =

3

2 (ii) 2 tan-1 (cosx) = tan-1 ( 2 cosec x).

11. Prove that 8

9 -

4

9sin-1 ( )

3

1 =

4

9sin-1 ( )

3

22

12. If A =

100

110

111

, prove by using principle of mathematical induction that An =

100

102

11

n

nnn

using elementary transformation, find the inverse of the following matrices.

i)

542

752

321

ii)

323

135

312

iii)

62

31

14. Construct a 3x4 matrix whose elements are given by aij = ½ ji 3

15. If A and B are skew matrices of the same order, then find out what type of matrix ABT – BAT is?

16. using properties of determinants, show that

i) 3

3610363

234232 a

cbabaa

cbabaa

cbabaa

ii) azyxa

zayx

zyax

zyxa

2

iii) 222 cbacba

cbaba

acbca

cbcba

17. Solve using matices: i) x+2y – 3z = –4 ii) x – y + 2z = 7 2x+3y+2z = 2 3x+4y – 5z = –5 3x – 3y – 4z = 11 2x – y + 3z = 12

18. A is a matrix of order 4x4. Find A2 if 3A .

19. Find a and b, given that the fn f(x) is continuous at x = 4 where f(x) =

44

124

44

3log

)4sin(

xwhenx

ebxwhena

xwhenx

x

x

20. Find K if f(x) =

2

2164

)162( 2

xfork

xforx

x

21. If y = asinx +bcosx, find 2

2

dx

ydin terms of y.

22. Verify Rolle’s Theorem for the function f(x)= log(x2+2) – log (x + 2) in [0,1]. 23. Verify L.M.V.T for the fn f(x) = x3 + x2 – 6x in [–1,4].

24. Differentiate: i) xxcosx + 1

12

2

x

x ii) sinx1/x + cosxx

25. if x = a(cost + sint) and y = a(sint - tcost), find 2

2

dt

xd,

2

2

dt

yd and

2

2

dx

yd

26. if siny = x sin (a+y), prove that dx

dy= sin2(a + y)/ sin a.