Page 1
CBSE XII | Mathematics
Sample Paper – 9 Solution
Mathematics
Class XII
Sample Paper – 9 Solution
SECTION – A
1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10
2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
It is linear, since x is independent variable.
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
Page 2
CBSE XII | Mathematics
Sample Paper – 9 Solution
Mathematics
Class XII
Sample Paper – 9 Solution
SECTION – A
1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10
2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
It is linear, since x is independent variable.
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 9 Solution
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
SECTION – B
5. f: R+ ? [4, 8) defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?
Page 3
CBSE XII | Mathematics
Sample Paper – 9 Solution
Mathematics
Class XII
Sample Paper – 9 Solution
SECTION – A
1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10
2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
It is linear, since x is independent variable.
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 9 Solution
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
SECTION – B
5. f: R+ ? [4, 8) defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
7.
x
x 2x
e
I dx
5 4e e
?
?
??
??
xx
Let e t e dx dt
Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
Page 4
CBSE XII | Mathematics
Sample Paper – 9 Solution
Mathematics
Class XII
Sample Paper – 9 Solution
SECTION – A
1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10
2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
It is linear, since x is independent variable.
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 9 Solution
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
SECTION – B
5. f: R+ ? [4, 8) defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
7.
x
x 2x
e
I dx
5 4e e
?
?
??
??
xx
Let e t e dx dt
Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
8.
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?
x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form,
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?
OR
2
3
x
dx
1x
?
?
Let 1 + x
3
= t
? 0 + 3x
2
dx = dt
?
2
dt
x dx
3
?
2
3
3
dt
x
3
dx
t
1x
1 dt
3t
1
log t c
3
1
log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?
Page 5
CBSE XII | Mathematics
Sample Paper – 9 Solution
Mathematics
Class XII
Sample Paper – 9 Solution
SECTION – A
1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10
2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??
3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??
It is linear, since x is independent variable.
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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 9 Solution
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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??
SECTION – B
5. f: R+ ? [4, 8) defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)
22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?
? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2
7.
x
x 2x
e
I dx
5 4e e
?
?
??
??
xx
Let e t e dx dt
Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?
2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??
1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
8.
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?
x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form,
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?
OR
2
3
x
dx
1x
?
?
Let 1 + x
3
= t
? 0 + 3x
2
dx = dt
?
2
dt
x dx
3
?
2
3
3
dt
x
3
dx
t
1x
1 dt
3t
1
log t c
3
1
log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?
CBSE XII | Mathematics
Sample Paper – 9 Solution
9. We have to differentiate it w.r.t. x two times
? ?
? ?
? ?
2
2
2
2
2
2
Differentiating
dy
2y m 2x
dx
dy
y mx............... 1
dx
differentiating again
d y dy
ym
dx dx
from 1
d y dy y dy
y
dx dx x dx
which is the required differential equation
??
??
??
? ? ?
??
??
??
??
??
??
10.
r. 3i 4j 12k 13 0
22
2 2 2 2
2
22
3x 4y 12z 13 0
Distance of point (1, 1, p) from the plane , is given by
3 1 4 1 12 p 13 20 12p
3 4 12 3 4 12
Distance of point ( 3, 0, 1) from the plane , is given by
3 3 4 0 12 1 13
8
3 4 12 3
2
22
4 12
22
2 2 2 2
The two distances are equal
20 12p 8
3 4 12 3 4 12
20 12p 8
20 12p 8
7
p 1,
3
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