CBSE Past Year Paper Session (2018) Solutions, Math Class 12 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : CBSE Past Year Paper Session (2018) Solutions, Math Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 Solution 
  
      
SECTION – A 
1. Given that the magnitude of each of the two vectors a and b have same magnitude. 
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
  
  
 
2.  
 
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
  
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 Solution 
  
      
SECTION – A 
1. Given that the magnitude of each of the two vectors a and b have same magnitude. 
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
  
  
 
2.  
 
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
  
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
3. Given that ? ab denotes the larger of ‘a’ and ‘b’. 
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
  
 
4.  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 3


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 Solution 
  
      
SECTION – A 
1. Given that the magnitude of each of the two vectors a and b have same magnitude. 
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
  
  
 
2.  
 
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
  
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
3. Given that ? ab denotes the larger of ‘a’ and ‘b’. 
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
  
 
4.  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 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
SECTION B 
5. A black and a red die are rolled together. 
? S = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,  
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 , 
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6   
  4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,  
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,       
  6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
 
We need to find the probability of obtaining the sum 8, given that red die resulted in a 
number less than 4. 
Let A be the event that a number on red die is less than 4. 
Let B be the event that the sum of the numbers is 8. 
 
A   = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 , 
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6        
??
??
??
??
??
??
??
  and  
B   =     {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}     
?A ? B = {(2, 6), (3, 5)}       
 
n (A) = 18, n (B) = 5, n (A ? B ) = 2,  and n(S) = 36 
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die 
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
 
 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 Solution 
  
      
SECTION – A 
1. Given that the magnitude of each of the two vectors a and b have same magnitude. 
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
  
  
 
2.  
 
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
  
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
3. Given that ? ab denotes the larger of ‘a’ and ‘b’. 
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
  
 
4.  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 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
SECTION B 
5. A black and a red die are rolled together. 
? S = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,  
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 , 
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6   
  4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,  
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,       
  6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
 
We need to find the probability of obtaining the sum 8, given that red die resulted in a 
number less than 4. 
Let A be the event that a number on red die is less than 4. 
Let B be the event that the sum of the numbers is 8. 
 
A   = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 , 
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6        
??
??
??
??
??
??
??
  and  
B   =     {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}     
?A ? B = {(2, 6), (3, 5)}       
 
n (A) = 18, n (B) = 5, n (A ? B ) = 2,  and n(S) = 36 
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die 
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
6. Given two vectors are ?? i 2j 3k and ?? 3i 2j k . 
 
? ? ? ? ? ?
?
? ? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
2 2 2
2 2 2 2 2 2
ab
sin .........(i)
ab
To find a b
i j k
a b 1 2 3
3 2 1
a b 2 6 i 1 9 j 2 6 k
a b 4i 8j 4k
a b 4 8 4 96 4 6............(i)
a 1 ( 2) 3 14 ........(ii) and b 3 ( 2) 3 14......
?
??
? ? ?
? ? ? ?
.....(iii)
ab
Since sin
ab
46
sin ......From (i), (ii) and (iii)
14 14
4 6 2 6
sin
7
14 14
 
 
7. Given 
?
?
bx 5
y ae  
Differentiating with y with respect to x, 
bx 5
bx 5
2
2
2
2
2
2
2
dy
ae b
dx
dy
by ....(i)( y ae )
dx
Diffrentiating (i) with respect to x,
d y dy
b
dx dx
d y 1 dy dy
.....from (i)
dx y dx dx
d y dy
y
dx dx
?
?
? ? ?
? ? ?
??
??
??
??
??
??
??
??
??
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
CBSE Board 
Class XII Mathematics 
Board Paper 2018 Solution 
  
      
SECTION – A 
1. Given that the magnitude of each of the two vectors a and b have same magnitude. 
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
  
  
 
2.  
 
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
  
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
3. Given that ? ab denotes the larger of ‘a’ and ‘b’. 
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
  
 
4.  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 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
SECTION B 
5. A black and a red die are rolled together. 
? S = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,  
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 , 
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6   
  4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,  
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,       
  6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
 
We need to find the probability of obtaining the sum 8, given that red die resulted in a 
number less than 4. 
Let A be the event that a number on red die is less than 4. 
Let B be the event that the sum of the numbers is 8. 
 
A   = 
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 , 
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6        
??
??
??
??
??
??
??
  and  
B   =     {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}     
?A ? B = {(2, 6), (3, 5)}       
 
n (A) = 18, n (B) = 5, n (A ? B ) = 2,  and n(S) = 36 
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die 
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
6. Given two vectors are ?? i 2j 3k and ?? 3i 2j k . 
 
? ? ? ? ? ?
?
? ? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
2 2 2
2 2 2 2 2 2
ab
sin .........(i)
ab
To find a b
i j k
a b 1 2 3
3 2 1
a b 2 6 i 1 9 j 2 6 k
a b 4i 8j 4k
a b 4 8 4 96 4 6............(i)
a 1 ( 2) 3 14 ........(ii) and b 3 ( 2) 3 14......
?
??
? ? ?
? ? ? ?
.....(iii)
ab
Since sin
ab
46
sin ......From (i), (ii) and (iii)
14 14
4 6 2 6
sin
7
14 14
 
 
7. Given 
?
?
bx 5
y ae  
Differentiating with y with respect to x, 
bx 5
bx 5
2
2
2
2
2
2
2
dy
ae b
dx
dy
by ....(i)( y ae )
dx
Diffrentiating (i) with respect to x,
d y dy
b
dx dx
d y 1 dy dy
.....from (i)
dx y dx dx
d y dy
y
dx dx
?
?
? ? ?
? ? ?
??
??
??
??
??
??
??
??
??
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2018 – Delhi Set 3 Solution 
 
     
8. 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
  
 
9. Given 
? ? ? ? ? ?
32
C x 0.005x 0.02x 30x 5000  
 Marginal cost is the rate of change of total cost with respect to output. 
??
? ? ? ? ? ? ?
?
? ? ? ? ? ?
??
22
2
dC
Marg inal cost
dx
Marg inal cost 0.005(3x ) 0.02(2x) 30 0.015x 0.04x 30
Marg inal cost at x 3
Marg inal cost 0.015 3 0.04 3 30
Marg inal cost 30.015
Therefore Marg inal cost is Rs. 30.015.
 
 
10.  Given 
?
???
?
??
??
1
1 cosx
y tan
sinx
  
? ?
2
2
1
1
1
1
x
x
1 cosx 2cos
2cos
2
2
y tan
xx
xx
2sin .cos
and sin x 2sin cos
22
22
x
y tan cot
2
x
y tan tan
22
x
y tan tan x x
22
Differentiating with respect to x,
dy 1 1
0
dx 2 2
?
?
?
?
??
??
??
??
??
?? ??
??
??
??
?
??
??
??
??
??
??
??
?? ???
? ? ?
?? ??
?? ??
?
?? ? ? ?
??
? ? ? ? ?
 
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