Sample Solution Paper 2 - Math, Class 12 JEE Notes | EduRev

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JEE : Sample Solution Paper 2 - Math, Class 12 JEE Notes | EduRev

 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
 
 
 
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