Page 1
CBSE XII | Mathematics
Sample Paper – 8 Solution
Mathematics
Class XII
Sample Paper – 8 Solution
SECTION – A
1. The element ‘6’ lies on 3
rd
row and 3
rd
column
So,
a33 = 6
2. 2x + 3y = cos x
Differentiating w.r.t. x, we get,
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
3. DE:
2
2
2
d y dy
s sy s
dx dx
??
It is nonlinear, since we have product of dependent variable and differential
coefficient
dy
y
dx
4.
? ? y4
x 5 z 6
3 7 2
??
??
??
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2.
So its vector equation is
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ?
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
Page 2
CBSE XII | Mathematics
Sample Paper – 8 Solution
Mathematics
Class XII
Sample Paper – 8 Solution
SECTION – A
1. The element ‘6’ lies on 3
rd
row and 3
rd
column
So,
a33 = 6
2. 2x + 3y = cos x
Differentiating w.r.t. x, we get,
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
3. DE:
2
2
2
d y dy
s sy s
dx dx
??
It is nonlinear, since we have product of dependent variable and differential
coefficient
dy
y
dx
4.
? ? y4
x 5 z 6
3 7 2
??
??
??
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2.
So its vector equation is
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ?
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 8 Solution
SECTION – B
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b}
The operation table for the given operation * on the given set is as follows
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5
6. We have,
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
Page 3
CBSE XII | Mathematics
Sample Paper – 8 Solution
Mathematics
Class XII
Sample Paper – 8 Solution
SECTION – A
1. The element ‘6’ lies on 3
rd
row and 3
rd
column
So,
a33 = 6
2. 2x + 3y = cos x
Differentiating w.r.t. x, we get,
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
3. DE:
2
2
2
d y dy
s sy s
dx dx
??
It is nonlinear, since we have product of dependent variable and differential
coefficient
dy
y
dx
4.
? ? y4
x 5 z 6
3 7 2
??
??
??
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2.
So its vector equation is
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ?
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 8 Solution
SECTION – B
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b}
The operation table for the given operation * on the given set is as follows
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5
6. We have,
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
CBSE XII | Mathematics
Sample Paper – 8 Solution
7.
2
5x 2
dx
1 2x 3x
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
Page 4
CBSE XII | Mathematics
Sample Paper – 8 Solution
Mathematics
Class XII
Sample Paper – 8 Solution
SECTION – A
1. The element ‘6’ lies on 3
rd
row and 3
rd
column
So,
a33 = 6
2. 2x + 3y = cos x
Differentiating w.r.t. x, we get,
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
3. DE:
2
2
2
d y dy
s sy s
dx dx
??
It is nonlinear, since we have product of dependent variable and differential
coefficient
dy
y
dx
4.
? ? y4
x 5 z 6
3 7 2
??
??
??
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2.
So its vector equation is
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ?
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 8 Solution
SECTION – B
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b}
The operation table for the given operation * on the given set is as follows
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5
6. We have,
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
CBSE XII | Mathematics
Sample Paper – 8 Solution
7.
2
5x 2
dx
1 2x 3x
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
CBSE XII | Mathematics
Sample Paper – 8 Solution
8.
2
Let x y
2
22
x y A B
y 4 y 9 y 4 y 9
x 4 x 9
y A(y 9) B(y 4)
Comparing both sides,
A B 1 and 9A 4B 0
49
Solving, we get A and B
55
22
11
11
49
I dx
5 x 4 5 x 9
4 1 x 9 1 x
tan tan C
5 2 2 5 3 3
2 x 3 x
tan tan C
5 2 5 3
OR
x
3
x
3
x
33
x
23
(x 3)e
dx
(x 5)
(x 5 2)e
dx
(x 5)
(x 5) 2
e . dx
(x 5) (x 5)
12
e . dx
(x 5) (x 5)
?
?
??
?
?
?? ?
??
??
??
??
??
??
??
??
??
?
?
?
?
? ? ?
??
??
??
??
??
??
?
?
?
xx
x
23
x
2
This is of the form
e [f(x) f '(x)]dx e f(x) C
12
e .dx
(x 5) (x 5)
e
C
(x 5)
Page 5
CBSE XII | Mathematics
Sample Paper – 8 Solution
Mathematics
Class XII
Sample Paper – 8 Solution
SECTION – A
1. The element ‘6’ lies on 3
rd
row and 3
rd
column
So,
a33 = 6
2. 2x + 3y = cos x
Differentiating w.r.t. x, we get,
? ?
dd
2x 3y cosx
dx dx
dy
2 3 sin x
dx
dy sin x 2
dx 3
??
? ? ?
??
?
3. DE:
2
2
2
d y dy
s sy s
dx dx
??
It is nonlinear, since we have product of dependent variable and differential
coefficient
dy
y
dx
4.
? ? y4
x 5 z 6
3 7 2
??
??
??
Clearly, it passes through (5, -4, 6) and has a direction ratios proportional to 3, 7, 2.
So its vector equation is
? ? ? ?
ˆˆ ˆ ˆ ˆ
r 5i 4j 6k 3i 7j 2k ? ? ? ? ? ? ?
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
8
cos
53
??
??
? ? ? ?
??
??
??
??
CBSE XII | Mathematics
Sample Paper – 8 Solution
SECTION – B
5. The binary operation * on the set {1, 2, 3, 4, 5} is defined by a * b = min {a, b}
The operation table for the given operation * on the given set is as follows
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5
6. We have,
? ? ? ?
2
2
2
2
2
2
2
2
x2 x
3
2y 9 y
2 x 3x
9 y 6y
x 3x 2
y 6y 9
x 3x 2 0
x 2 x 1 0
x 2 or x 1
y 6y 9 0
6 36 36
y 3 3 2
2
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
? ?? ? ??
?
?? ??
?
?? ??
? ? ?
??
? ? ?
? ? ?
??
? ? ?
??
? ? ?
CBSE XII | Mathematics
Sample Paper – 8 Solution
7.
2
5x 2
dx
1 2x 3x
2
2
2
2
2
x
5
5 dx
1 2x 3x
12
6x
5
5
dx
6 1 2x 3x
12
6x 2 2
5
5
dx
6 1 2x 3x
22
6x 2
5
5
dx
6 1 2x 3x
2 2
2
2
5 6x 2 5 22 1
dx dx
6 6 5 1 2x 3x
12
3x
39
5 11 1
log 1 2x 3x dx
69
12
x
39
21
21
1
x
5 11 3 3
log 1 2x 3x tan C
69
22
3
5 11 3x 1
log 1 2x 3x tan C
6
3 2 2
CBSE XII | Mathematics
Sample Paper – 8 Solution
8.
2
Let x y
2
22
x y A B
y 4 y 9 y 4 y 9
x 4 x 9
y A(y 9) B(y 4)
Comparing both sides,
A B 1 and 9A 4B 0
49
Solving, we get A and B
55
22
11
11
49
I dx
5 x 4 5 x 9
4 1 x 9 1 x
tan tan C
5 2 2 5 3 3
2 x 3 x
tan tan C
5 2 5 3
OR
x
3
x
3
x
33
x
23
(x 3)e
dx
(x 5)
(x 5 2)e
dx
(x 5)
(x 5) 2
e . dx
(x 5) (x 5)
12
e . dx
(x 5) (x 5)
?
?
??
?
?
?? ?
??
??
??
??
??
??
??
??
??
?
?
?
?
? ? ?
??
??
??
??
??
??
?
?
?
xx
x
23
x
2
This is of the form
e [f(x) f '(x)]dx e f(x) C
12
e .dx
(x 5) (x 5)
e
C
(x 5)
CBSE XII | Mathematics
Sample Paper – 8 Solution
9. We have to differentiate it w.r.t. x two times
? ?
? ?
? ? ? ?
2
2
2
2
2
2
differentiating
dy
abcos bx c .......(1)
dx
differentiating again
dy
ab sin bx c .........(2)
dx
dy
b y............. y asin bx c
dx
which is the required differential equation
??
? ? ?
? ? ? ?
10. ABCD is a parallelogram with,
AB 2i 4j 5k;AD i 2j 3k
Using the parallelogramlaw of vector addition, diagonal is given by
AC AB AD 2i 4j 5k i 2j 3k 3i 6j 2k
2 2 2
Unit vector parallel to diagonal AC
AC 3i 6j 2k 3i 6j 2k
3i 6j 2k AC 3 ( 6) (2)
3i 6j 2k 1
3i 6j 2k
7 49
Area of the parallelogram
2 2 2
ABCD= AB AD
i j k
2 4 5
1 2 3
i(12 10) j( 6 5) k( 4 4) i(22) j ( 11) k(0) i(22) j (11)
(22) (11) 0 11 5 sq units
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