summer vacations holiday home work 2017-18 class-xii...
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Summer vacations Holiday Home Work 2017-18
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 & y
7 4 0 4
x y yi find
x
3 1 2( )
2 3 5 3
x y yi
y x
3. If 2 21 1x y a x y then prove that 2
2
1
1
dy y
dx x
4. Write the function in the simplifies from
1 Cos( ) tan
1 Sin
xi
x
2 21
2 2
1 1( )
1 1
x xii Cot
x x
21
2
1 1) Cos
2 1
xiii
x
5. Find the derivative of Sin Cos
Sin Cos
x x x
x x x
6. Evaluate
1 15 13( ) tan tan Cos Cos
6 6
n ni
1 1 11 1( ) tan tan Sin
23 3ii Cot
1 17 7( )Cos Cos tan t an
6 6iii
1 13( )Sin Sin Sec 2
5iv
7. Prove that 1 1 14 5 16Sin Sin Sin
5 13 65 2
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 tan
5 8 7
11. Solve for 2
1 1
2 2
1 1 2 2:Cos tan
2 31 1
x xx
x x
12. If 2
3A R
and a function :f A A is defined by
4 3
6 4
xf x
x
Find 1f if f is
invertible.
13. Show that 11 3 4 7tan Sin
2 4 3
14. Find the value of 2
1 1
2 2
11 2tan Sin Cos
2 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 1m
y x x then show that 2
2 2
21 0
d y dyx x m y
dxdx ..
18. Verify Lagrange’s mean value theorem for the function 3 25 3 , 1,3f x x x x
19. Let
3
2
1 Sin,
23Cos
,2
1 Sin,
2 2
xx
x
f x a x
b xx
x
find the values of a and b so that the given function is
continuous at 2
x
.
20. Determine the values of a,b,c for which the function
2
3
Sin 1 Sin; 0
; 0
; 0
x a x xx
x
f x c x
xbx xx
b x
May be continuous at 0x
21. If Sinxxy x x then find
dy
dx
22. Find dy
dxif Cos Sin
y xx y
23. If 2 2 1log 2 tany
x yx
then show that dy x y
dx x y
24. Find 2
2
d 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 dy
dxif
21 3 4 1
Cos5
x xy
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
2
log
log
dy x
dx ex
30. If 1
tan log ,x ya
then show that
22
21 2 0
d y dyx x a
dxdx .
31. Find dy
dxif
11 2 3
Sin1 36
x x
xy
.
32. If Sin & Cos log tan2
tx a t y a t
,Find
2
2
d y
dx
33. If 2 2logy x x a
show that 2
2 2
20
d y dyx a x
dxdx
34. Differentiate 2
1 11 1tan . . tan
xw r t x
x
35. Show that the function f defined as: 2
3 2, 0 . 1
2 1 2
5 4 2
x 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.
37. Find the value of k so that the function defined by
Cos;
2 2
3;2
k xx
xf x
x
is continuous
at 2
x
.
38. Find the value of k so that the function defined by
3
Sin 1 ; 12
tan Sin: 0
k x x
f xx x
xx
is
continuous at x=0.
39. Find the 2nd order derivative if Sin(log )y x
40. Find the inverse of
1 3 2
3 0 1
2 1 0
A
using elementary transformations.
41. Use properties to prove.
(i)
2
2 2 2 2
2
1
1 1
1
a ab ac
ba b bc a b c
ca cb c
–1–
CHAPTER 5: CONTINUITY AND DIFFERENTIATION
Find the value of k so that the function defined by
0:sintan
0:)1(2
sin)(
3x
x
xx
xxkxf
is continuous.
Q2 Test the continuity of the function
0:1
)31log(
0:2
3
0:2tan
3sin
)(
2x
e
x
x
xx
x
xf
x
Q3 Let
2:
)2(
)sin1(2
:
2:
cos3
sin1
)(
2
2
3
xx
xb
xa
xx
x
xf
Find the value of a and b so that the given function is continuous at 2
x
Q4 If the function f defined by
0:
0:
0:sin)1sin(
)(
2
3
2
x
bx
xbxx
xc
xx
xa
xf
Q5 Differentiate x
x21 1
tan with respect to
21 12cos xx
Q6 If m
xxy
12 then show that 0)1( 2
2
22 ym
dx
dyx
dx
ydx
Q7 If ptytx sin;sin then prove that 0)1( 2
2
22 yp
dx
dyx
dx
ydx
Q8 Finddx
dy if
1
1sin
1
1sec 11
x
x
x
xy
Q9 Differentiate if xx xxy sincos )(cos)(sin
Q10 Differentiate xy yx )(sin)(cos
Q11 If tayt
tax sin;2
tanlogcos
find
dx
dy
Q12 Ifx
yyx 122 tan2)log( then show that
yx
yx
dx
dy
Q13 If ayx
yx
22
221tan then prove that
x
y
dx
dy
Q14 Verify Rolle’s Theorem
2,0:cossin)( 44
xxxxf
Q15 Verify Mean Value Theorem for the function ,0:2sinsin)( xxxxf
Mathematics Assignments Class XII 2017 – 18
Q1
–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