Sample Solution Paper 2 - Math, Class 12

# Sample Solution Paper 2 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

``` Page 1

CBSE XII | Mathematics
Sample Paper – 2 Solution

CBSE Board
Class XII Mathematics
Sample Paper – 2  Solution

SECTION – A

1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A

2. A’ =
23
12
???
??
??

A =
21
32
???
??
??

B =
10
12
???
??
??

? ?
41
A 2B
56
???
??
??
??

? ?
45
A 2B '
16
???
??
??
??

OR

Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3

Page 2

CBSE XII | Mathematics
Sample Paper – 2 Solution

CBSE Board
Class XII Mathematics
Sample Paper – 2  Solution

SECTION – A

1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A

2. A’ =
23
12
???
??
??

A =
21
32
???
??
??

B =
10
12
???
??
??

? ?
41
A 2B
56
???
??
??
??

? ?
45
A 2B '
16
???
??
??
??

OR

Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3

CBSE XII | Mathematics
Sample Paper – 2 Solution

3. Projection of a vector a on another vector b is
b
a.
|b|

=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??

4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by

x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??

Section B

5.
11
3
sin sin cos x 1
5
??
??
??
??
??

Operating both sides by sin
-1
we get,

? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??

6.
??
?
5 x x 1
Given matrix is singular if 0
24

? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3

Page 3

CBSE XII | Mathematics
Sample Paper – 2 Solution

CBSE Board
Class XII Mathematics
Sample Paper – 2  Solution

SECTION – A

1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A

2. A’ =
23
12
???
??
??

A =
21
32
???
??
??

B =
10
12
???
??
??

? ?
41
A 2B
56
???
??
??
??

? ?
45
A 2B '
16
???
??
??
??

OR

Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3

CBSE XII | Mathematics
Sample Paper – 2 Solution

3. Projection of a vector a on another vector b is
b
a.
|b|

=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??

4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by

x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??

Section B

5.
11
3
sin sin cos x 1
5
??
??
??
??
??

Operating both sides by sin
-1
we get,

? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??

6.
??
?
5 x x 1
Given matrix is singular if 0
24

? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3

CBSE XII | Mathematics
Sample Paper – 2 Solution

7. ??
32
We have,  I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent  can ensure growth in the income of the doctor.

8. ? ? ? ? The distance of the plane  3x 4y 12z 3 0  from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144

?
?
?
?
3
=
169
3
13
3
13

Page 4

CBSE XII | Mathematics
Sample Paper – 2 Solution

CBSE Board
Class XII Mathematics
Sample Paper – 2  Solution

SECTION – A

1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A

2. A’ =
23
12
???
??
??

A =
21
32
???
??
??

B =
10
12
???
??
??

? ?
41
A 2B
56
???
??
??
??

? ?
45
A 2B '
16
???
??
??
??

OR

Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3

CBSE XII | Mathematics
Sample Paper – 2 Solution

3. Projection of a vector a on another vector b is
b
a.
|b|

=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??

4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by

x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??

Section B

5.
11
3
sin sin cos x 1
5
??
??
??
??
??

Operating both sides by sin
-1
we get,

? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??

6.
??
?
5 x x 1
Given matrix is singular if 0
24

? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3

CBSE XII | Mathematics
Sample Paper – 2 Solution

7. ??
32
We have,  I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent  can ensure growth in the income of the doctor.

8. ? ? ? ? The distance of the plane  3x 4y 12z 3 0  from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144

?
?
?
?
3
=
169
3
13
3
13

CBSE XII | Mathematics
Sample Paper – 2 Solution

OR

The shortest distance is given by,

? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2 1 1 2
12
21
12
2 1 1 2
22
12
A A . B B
BB
ˆˆ ˆ ˆ ˆ ˆ ˆ
A A i j 2k 4i j 3i 2k
ˆ ˆˆ
i j k
ˆˆ
B B 1 2 3 2i j
2 4 5
ˆ ˆ ˆ ˆ
A A . B B 3i 2k . 2i j 6
B B 2 1 5
66
so shortest distance between two lines units
55
??
?
?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ?
?
??

9. Let A
0 q r r s
r q 0 p q
s r q p 0
??
? ? ?
??

A is a skew symmetric matrix of order 3
?A’ = -A
?|A’| = |A| = -|A|
?|A| = -|A|
?2|A| = 0
?|A|= 0

10.  Let I =
1 cot x
dx
x log sin x
?
?
?

Put x + log sin x = t
?(1 + cot x) dx = dt
so integral I becomes
?
??
? ? ?
dt
log| t | C
t
log| x logsin x| C

Page 5

CBSE XII | Mathematics
Sample Paper – 2 Solution

CBSE Board
Class XII Mathematics
Sample Paper – 2  Solution

SECTION – A

1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A

2. A’ =
23
12
???
??
??

A =
21
32
???
??
??

B =
10
12
???
??
??

? ?
41
A 2B
56
???
??
??
??

? ?
45
A 2B '
16
???
??
??
??

OR

Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3

CBSE XII | Mathematics
Sample Paper – 2 Solution

3. Projection of a vector a on another vector b is
b
a.
|b|

=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??

4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by

x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??

Section B

5.
11
3
sin sin cos x 1
5
??
??
??
??
??

Operating both sides by sin
-1
we get,

? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??

6.
??
?
5 x x 1
Given matrix is singular if 0
24

? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3

CBSE XII | Mathematics
Sample Paper – 2 Solution

7. ??
32
We have,  I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent  can ensure growth in the income of the doctor.

8. ? ? ? ? The distance of the plane  3x 4y 12z 3 0  from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144

?
?
?
?
3
=
169
3
13
3
13

CBSE XII | Mathematics
Sample Paper – 2 Solution

OR

The shortest distance is given by,

? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2 1 1 2
12
21
12
2 1 1 2
22
12
A A . B B
BB
ˆˆ ˆ ˆ ˆ ˆ ˆ
A A i j 2k 4i j 3i 2k
ˆ ˆˆ
i j k
ˆˆ
B B 1 2 3 2i j
2 4 5
ˆ ˆ ˆ ˆ
A A . B B 3i 2k . 2i j 6
B B 2 1 5
66
so shortest distance between two lines units
55
??
?
?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ?
?
??

9. Let A
0 q r r s
r q 0 p q
s r q p 0
??
? ? ?
??

A is a skew symmetric matrix of order 3
?A’ = -A
?|A’| = |A| = -|A|
?|A| = -|A|
?2|A| = 0
?|A|= 0

10.  Let I =
1 cot x
dx
x log sin x
?
?
?

Put x + log sin x = t
?(1 + cot x) dx = dt
so integral I becomes
?
??
? ? ?
dt
log| t | C
t
log| x logsin x| C

CBSE XII | Mathematics
Sample Paper – 2 Solution

OR

Given
?
?
?
2
2
cos2x 2sin x
I dx
cos x

? ?
? ?
?
?
??
? ? ?
?
?
? ? ?
?
??
?
?
?
?
?
2
2
2 2 2
22
2
22
2
22
2
2
cos2x 2sin x
I dx
cos x
cos x sin x 2sin x
I dx ..... cos2x cos x sin x
cos x
cos x sin x
I dx
cos x
1
I dx .... cos x sin x 1
cos x
I sec x dx
I tan x c

11. ? ? ? ? Let a 2i j 5k, then
? ? ? ? ?
? ? ? ?
??
? ? ?
? ? ? ? ? ?
? ? ? ?
?
2 2 2
a ( 2) (1) ( 5)
a 4 1 25
a 30
a 2i j 5k 2 1 5
Now, a i j k
30 30 30 30 a
21
Thus, the direction cosines of the vector 2i j 5k are ,
30 30
5
and .
30

12.  A =
a 0 0
0 a 0
0 0 a
??
??
??
??
??
then |A| = a³
Since, a is a nonzero real number so |A| ? 0
If A is an invertible matrix of order n, then |adj(A)| = |A|
n -1

So |adj A| = |A|² = (a³)
²
= a
6

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Solution Paper 2 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. What are the main topics covered in the Math Class 12 exam?
Ans. The main topics covered in the Math Class 12 exam are algebra, calculus, coordinate geometry, mathematical reasoning, statistics, and probability.
 2. How should I prepare for the Math Class 12 exam?
Ans. To prepare for the Math Class 12 exam, it is important to understand the concepts thoroughly, practice a variety of problems, solve previous year question papers, and seek help from teachers or tutors if needed.
 3. Are calculators allowed in the Math Class 12 exam?
Ans. Yes, calculators are generally allowed in the Math Class 12 exam. However, it is important to check the specific guidelines provided by the exam board to ensure which types of calculators are permitted.
 4. Can I use graph paper during the Math Class 12 exam?
Ans. Yes, in most cases, graph paper is allowed in the Math Class 12 exam. It is advisable to check the exam guidelines to confirm the specific requirements regarding the use of graph paper.
 5. How can I improve my problem-solving skills for the Math Class 12 exam?
Ans. To improve problem-solving skills for the Math Class 12 exam, it is recommended to practice solving a wide range of problems, understand different problem-solving techniques, and analyze the solutions to identify patterns and strategies. Additionally, seeking guidance from teachers or attending problem-solving workshops can also be beneficial.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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