CBSE Past Year Paper Session (2017) Solutions, Math Class 12 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : CBSE Past Year Paper Session (2017) Solutions, Math Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper 2017 Solution 
All India   
      
SECTION – A 
1. ? A(adjA) | A|I 
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8
 
 
 
2.   ? Since f(x) is continuous at x 3. 
?
?
?
?
?
??
??
??
?
? ? ? ?
??
?
??
??
?
? ? ? ? ? ?
? ? ?
??
x3
2
x3
x3
x3
x3
lim f(x) f(3)
(x 3) 36
lim k
x3
(x 3 6)(x 3 6)
lim k
x3
(x 9)(x 3)
lim k
x3
lim (x 9) k ( x 3, x 3 0)
3 9 k
k 12
 
  
3.  
?
?
22
sin x cos x
dx
sinxcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log|sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper 2017 Solution 
All India   
      
SECTION – A 
1. ? A(adjA) | A|I 
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8
 
 
 
2.   ? Since f(x) is continuous at x 3. 
?
?
?
?
?
??
??
??
?
? ? ? ?
??
?
??
??
?
? ? ? ? ? ?
? ? ?
??
x3
2
x3
x3
x3
x3
lim f(x) f(3)
(x 3) 36
lim k
x3
(x 3 6)(x 3 6)
lim k
x3
(x 9)(x 3)
lim k
x3
lim (x 9) k ( x 3, x 3 0)
3 9 k
k 12
 
  
3.  
?
?
22
sin x cos x
dx
sinxcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log|sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
4.  
Since the ratio of the coefficients are the same
 
 
?
??
?
? ? ?
? ? ?
?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
1
1
2
2 1 2
that is, 
5 2.5 5
2 10 2
5 25 5
222
555
the planes are parallel.
2x y +2z = 5
Comparing with ax + by + cz + d = 0,
a = 2, b = -1, c = 2, d = -5
Consider, 5x 2.5y 5z 20
5
5x y 5z 20
2
10x 5y 10z 40
2x y 2z 8
Comparing with ax + by + cz + d = 0
? ? ? ? ?
?
?
? ? ?
? ? ?
?
??
?
?
?
2
12
2 2 2
,
a = 2, b = -1, c = 2, d = -8
Let, d be the distance between the given planes.
d = Length of the perpendicular from 2x y +2z = 5 to 2x y 2z 8,
|d d |
d=
2 ( 1) 2
5 ( 8)
d=
4 1 4
3
d=
9
3
d=
3
d = 1
Distance between the given planes is 1 unit.
 
 
 
 
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper 2017 Solution 
All India   
      
SECTION – A 
1. ? A(adjA) | A|I 
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8
 
 
 
2.   ? Since f(x) is continuous at x 3. 
?
?
?
?
?
??
??
??
?
? ? ? ?
??
?
??
??
?
? ? ? ? ? ?
? ? ?
??
x3
2
x3
x3
x3
x3
lim f(x) f(3)
(x 3) 36
lim k
x3
(x 3 6)(x 3 6)
lim k
x3
(x 9)(x 3)
lim k
x3
lim (x 9) k ( x 3, x 3 0)
3 9 k
k 12
 
  
3.  
?
?
22
sin x cos x
dx
sinxcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log|sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
4.  
Since the ratio of the coefficients are the same
 
 
?
??
?
? ? ?
? ? ?
?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
1
1
2
2 1 2
that is, 
5 2.5 5
2 10 2
5 25 5
222
555
the planes are parallel.
2x y +2z = 5
Comparing with ax + by + cz + d = 0,
a = 2, b = -1, c = 2, d = -5
Consider, 5x 2.5y 5z 20
5
5x y 5z 20
2
10x 5y 10z 40
2x y 2z 8
Comparing with ax + by + cz + d = 0
? ? ? ? ?
?
?
? ? ?
? ? ?
?
??
?
?
?
2
12
2 2 2
,
a = 2, b = -1, c = 2, d = -8
Let, d be the distance between the given planes.
d = Length of the perpendicular from 2x y +2z = 5 to 2x y 2z 8,
|d d |
d=
2 ( 1) 2
5 ( 8)
d=
4 1 4
3
d=
9
3
d=
3
d = 1
Distance between the given planes is 1 unit.
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
 
SECTION – B 
5. ? Since A is skew symmetric matrix. 
??
? ? ?
? ? ?
? ? ?
? ? ? ? ?
??
??
T
T
T3
T
T
Therefore,A A
AA
A ( 1) A
AA
A A ......(Since A A )
2 A 0
A0
 
 
 
6. Since a polynomial function is continuous and differentiable everywhere, 
? ? ? ?
? ? ? ?
??
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
?
3
3
therefore, f(x) is continuous on [ 3,0] and differentiable on ( 3,0).
Also, f( 3) 3 3 3 3 3 3 3 0
and f(0) 0 3 0 0
f( 3) f(0)
Thus, all the three conditions of Rolle's theorem are satisfied.
So, there exists a c ?
??
? ? ?
?
? ? ?
? ? ?
? ? ?
??
? ? ?
??
? ? ?
3
2
2
2
2
2
( 3,0) such that f '(c)= 0.
We have,
f(x) x 3x
f '(x) 3x 3
So, f '(x) 0
3x 3 0
3(x 1) 0
(x 1) 0
x1
x1
Clearly, only x = 1 lies in the interval ( 3,0).
Thus, c = 1 ( 3,0).
 
 
 
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper 2017 Solution 
All India   
      
SECTION – A 
1. ? A(adjA) | A|I 
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8
 
 
 
2.   ? Since f(x) is continuous at x 3. 
?
?
?
?
?
??
??
??
?
? ? ? ?
??
?
??
??
?
? ? ? ? ? ?
? ? ?
??
x3
2
x3
x3
x3
x3
lim f(x) f(3)
(x 3) 36
lim k
x3
(x 3 6)(x 3 6)
lim k
x3
(x 9)(x 3)
lim k
x3
lim (x 9) k ( x 3, x 3 0)
3 9 k
k 12
 
  
3.  
?
?
22
sin x cos x
dx
sinxcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log|sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
4.  
Since the ratio of the coefficients are the same
 
 
?
??
?
? ? ?
? ? ?
?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
1
1
2
2 1 2
that is, 
5 2.5 5
2 10 2
5 25 5
222
555
the planes are parallel.
2x y +2z = 5
Comparing with ax + by + cz + d = 0,
a = 2, b = -1, c = 2, d = -5
Consider, 5x 2.5y 5z 20
5
5x y 5z 20
2
10x 5y 10z 40
2x y 2z 8
Comparing with ax + by + cz + d = 0
? ? ? ? ?
?
?
? ? ?
? ? ?
?
??
?
?
?
2
12
2 2 2
,
a = 2, b = -1, c = 2, d = -8
Let, d be the distance between the given planes.
d = Length of the perpendicular from 2x y +2z = 5 to 2x y 2z 8,
|d d |
d=
2 ( 1) 2
5 ( 8)
d=
4 1 4
3
d=
9
3
d=
3
d = 1
Distance between the given planes is 1 unit.
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
 
SECTION – B 
5. ? Since A is skew symmetric matrix. 
??
? ? ?
? ? ?
? ? ?
? ? ? ? ?
??
??
T
T
T3
T
T
Therefore,A A
AA
A ( 1) A
AA
A A ......(Since A A )
2 A 0
A0
 
 
 
6. Since a polynomial function is continuous and differentiable everywhere, 
? ? ? ?
? ? ? ?
??
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
?
3
3
therefore, f(x) is continuous on [ 3,0] and differentiable on ( 3,0).
Also, f( 3) 3 3 3 3 3 3 3 0
and f(0) 0 3 0 0
f( 3) f(0)
Thus, all the three conditions of Rolle's theorem are satisfied.
So, there exists a c ?
??
? ? ?
?
? ? ?
? ? ?
? ? ?
??
? ? ?
??
? ? ?
3
2
2
2
2
2
( 3,0) such that f '(c)= 0.
We have,
f(x) x 3x
f '(x) 3x 3
So, f '(x) 0
3x 3 0
3(x 1) 0
(x 1) 0
x1
x1
Clearly, only x = 1 lies in the interval ( 3,0).
Thus, c = 1 ( 3,0).
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
 
7. Let x be the length of an edge of the cube, V be the volume and S be the surface area 
at any time t. 
 
?
??
?
??
??
??
??
??
??
??
??
??
??
? ? ?
??
??
32
3
3
2
2
2
2
2
x 10
Then,V x and S 6x .
It is given that,
dV
9 cm / sec
dt
d
(x ) 9 
dt
dx
3x 9 
dt
dx 3
 
dt
x
Now, S = 6x
dS dx
12x
dt dt
dS 3
12x
dt
x
dS 36
dt x
dS 36
3.6 cm / sec
dt 10
 
 
 
8. We have,  
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
?
32
2
2
2
2
f(x) x 3x 6x 100
f '(x) 3x 6x 6
f '(x) 3(x 2x 2)
f '(x) 3(x 2x 1 1)
f '(x) 3 [(x 1) 1) 0 for all real numbers
So, f(x) is increasing for all x R
 
 
 
 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper 2017 Solution 
All India   
      
SECTION – A 
1. ? A(adjA) | A|I 
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8
 
 
 
2.   ? Since f(x) is continuous at x 3. 
?
?
?
?
?
??
??
??
?
? ? ? ?
??
?
??
??
?
? ? ? ? ? ?
? ? ?
??
x3
2
x3
x3
x3
x3
lim f(x) f(3)
(x 3) 36
lim k
x3
(x 3 6)(x 3 6)
lim k
x3
(x 9)(x 3)
lim k
x3
lim (x 9) k ( x 3, x 3 0)
3 9 k
k 12
 
  
3.  
?
?
22
sin x cos x
dx
sinxcosx
 
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log|sin2x| C .......( dx log f(x) c)
f(x)
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
4.  
Since the ratio of the coefficients are the same
 
 
?
??
?
? ? ?
? ? ?
?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
1
1
2
2 1 2
that is, 
5 2.5 5
2 10 2
5 25 5
222
555
the planes are parallel.
2x y +2z = 5
Comparing with ax + by + cz + d = 0,
a = 2, b = -1, c = 2, d = -5
Consider, 5x 2.5y 5z 20
5
5x y 5z 20
2
10x 5y 10z 40
2x y 2z 8
Comparing with ax + by + cz + d = 0
? ? ? ? ?
?
?
? ? ?
? ? ?
?
??
?
?
?
2
12
2 2 2
,
a = 2, b = -1, c = 2, d = -8
Let, d be the distance between the given planes.
d = Length of the perpendicular from 2x y +2z = 5 to 2x y 2z 8,
|d d |
d=
2 ( 1) 2
5 ( 8)
d=
4 1 4
3
d=
9
3
d=
3
d = 1
Distance between the given planes is 1 unit.
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
 
SECTION – B 
5. ? Since A is skew symmetric matrix. 
??
? ? ?
? ? ?
? ? ?
? ? ? ? ?
??
??
T
T
T3
T
T
Therefore,A A
AA
A ( 1) A
AA
A A ......(Since A A )
2 A 0
A0
 
 
 
6. Since a polynomial function is continuous and differentiable everywhere, 
? ? ? ?
? ? ? ?
??
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
?
3
3
therefore, f(x) is continuous on [ 3,0] and differentiable on ( 3,0).
Also, f( 3) 3 3 3 3 3 3 3 0
and f(0) 0 3 0 0
f( 3) f(0)
Thus, all the three conditions of Rolle's theorem are satisfied.
So, there exists a c ?
??
? ? ?
?
? ? ?
? ? ?
? ? ?
??
? ? ?
??
? ? ?
3
2
2
2
2
2
( 3,0) such that f '(c)= 0.
We have,
f(x) x 3x
f '(x) 3x 3
So, f '(x) 0
3x 3 0
3(x 1) 0
(x 1) 0
x1
x1
Clearly, only x = 1 lies in the interval ( 3,0).
Thus, c = 1 ( 3,0).
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
 
7. Let x be the length of an edge of the cube, V be the volume and S be the surface area 
at any time t. 
 
?
??
?
??
??
??
??
??
??
??
??
??
??
? ? ?
??
??
32
3
3
2
2
2
2
2
x 10
Then,V x and S 6x .
It is given that,
dV
9 cm / sec
dt
d
(x ) 9 
dt
dx
3x 9 
dt
dx 3
 
dt
x
Now, S = 6x
dS dx
12x
dt dt
dS 3
12x
dt
x
dS 36
dt x
dS 36
3.6 cm / sec
dt 10
 
 
 
8. We have,  
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
?
32
2
2
2
2
f(x) x 3x 6x 100
f '(x) 3x 6x 6
f '(x) 3(x 2x 2)
f '(x) 3(x 2x 1 1)
f '(x) 3 [(x 1) 1) 0 for all real numbers
So, f(x) is increasing for all x R
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2017 – All India Set 1 Solution 
 
  
9. ? ? ? ? Given that line joining the point s P 2,2,1 and Q 5,1, 1 then ? 
       
Equation of the line is,
x 2 y 2 z 1
5 2 1 2 2 1
x 2 y 2 z 1
3 1 3
x2
3
x coordinate is 4.
42
3
2
3
Hence, z coordinate is,
z1
3
z 1 2
33
z 1 2
z1
Coordinate of z is 1.
? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
??
?
? ? ?
?
?
? ? ?
??
?
?
??
?
?
?
?
? ? ?
??
?
 
 
 
10. It is given that 
? ? ? ?
?
? ? ?
3 1 3 1
P(A) and P(B)
6 2 6 2
and P(A B)= P(Numbers that are even as well as red)
= P(Number appearing is 2)
1
=
6
Clearly, P(A B) P(A) P(B)
Hence, A and B are not independent events.
 
 
 
 
 
 
 
 
 
 
 
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