Page 1
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
Page 2
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
Page 3
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
Page 4
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
CBSE XII | Mathematics
Sample Paper – 10 Solution
7.
?
?
22
sin x cos x
dx
sin xcosx
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
8.
?
??
2
dx
5 8x x
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
Page 5
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
CBSE XII | Mathematics
Sample Paper – 10 Solution
7.
?
?
22
sin x cos x
dx
sin xcosx
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
8.
?
??
2
dx
5 8x x
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
CBSE XII | Mathematics
Sample Paper – 10 Solution
OR
Let I =
2
2
4
2
2
1
1
x1
x
dx dx
1
x1
x
x
??
?
?
?
?
?
(Dividing numerator and denominator by x
2
)
I
2
2
1
1
x
dx
1
x2
x
?
?
?
??
??
??
??
Substituting x -
1
x
= t,
2
1
1 dx dt
x
??
? ? ?
??
??
we get,
? ?
22
2
1
dt dt
I
t2
t2
1t
tan
22
?
??
??
?
?
?
1
2
1
1
x
1
x
tan
22
1 x 1
tan c
2 2x
?
?
??
?
??
?
??
??
??
??
?
??
??
??
??
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