CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 JEE Notes | EduRev

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
 
SECTION  A 
1.  
A is a square matrix of order 3 with |A| = 4    …. Given 
|–2A| = –2
3
|A| = –8 ? 4 = – 32  
 
2.  
y = sin
-1
x + cos
-1
x     …. given  
? ?
? ?
1
22
11
1
sin x  cos x
sin x co
dy d
dx dx
d d d dA dB
AB
dx dx dx dx dx
11
1 x 1 x
s x since
0
??
??
??
? ? ?
?
?
?
??
?
?
?
 
3.  
3
2
2
4
4
3
2
2
4
4
d y dy
x
dx dx
d y dy
x0
dx dx
??
??
??
????
??
??
?? ?? ??
??
??
??
??
? ? ? ??
??
??
?? ?? ??
??
 
Order = 4 and degree = 2  
 
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
 
SECTION  A 
1.  
A is a square matrix of order 3 with |A| = 4    …. Given 
|–2A| = –2
3
|A| = –8 ? 4 = – 32  
 
2.  
y = sin
-1
x + cos
-1
x     …. given  
? ?
? ?
1
22
11
1
sin x  cos x
sin x co
dy d
dx dx
d d d dA dB
AB
dx dx dx dx dx
11
1 x 1 x
s x since
0
??
??
??
? ? ?
?
?
?
??
?
?
?
 
3.  
3
2
2
4
4
3
2
2
4
4
d y dy
x
dx dx
d y dy
x0
dx dx
??
??
??
????
??
??
?? ?? ??
??
??
??
??
? ? ? ??
??
??
?? ?? ??
??
 
Order = 4 and degree = 2  
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
4.  
Let a, b and c be direction ratios of a line l, m and n be the direction cosines 
of the line. Then 
2 2 2 2 2 2 2 2 2
a b c
l , m , n
a b c a b c a b c
? ? ?
? ? ? ? ? ?
 
Here a = –18, b = 12 and c = – 4 
2 2 2 2 2 2
a b c ( 18) (12) ( 4)
324 144 16
484 22
? ? ? ? ? ? ? ?
? ? ?
??
 
2 2 2 2 2 2
2 2 2
a 18 9 b 12 6
l , m and
22 11 22 11
a b c a b c
c 4 2
n
22 11
a b c
??
? ? ? ? ? ? ?
? ? ? ?
??
? ? ?
??
 
OR 
Find the Cartesian equation of the line which passes through the point 
 (–2, 4, –5) and is parallel to the line
x 3 4 y z 8
3 5 6
? ? ?
?? . 
Cartesian equation of a line passing through (x1, y1, z1) and parallel to a line 
having direction ratios a, b, c is given by, 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Since the line passes through the (–2, 4, –5), 
? x1 = –2,  y1 = 4  and z1 = –5 
Given that the line is parallel to
x 3 4 y z 8
3 5 6
? ? ?
?? , 
? a = 3, b = -5 and c = 6 
Therefore the Cartesian equation of the line is given by 
 
x ( 2) y 4 z ( 5)
3 5 6
x 2 4 y z 5
3 5 6
? ? ? ? ?
??
?
? ? ?
? ? ?
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
 
SECTION  A 
1.  
A is a square matrix of order 3 with |A| = 4    …. Given 
|–2A| = –2
3
|A| = –8 ? 4 = – 32  
 
2.  
y = sin
-1
x + cos
-1
x     …. given  
? ?
? ?
1
22
11
1
sin x  cos x
sin x co
dy d
dx dx
d d d dA dB
AB
dx dx dx dx dx
11
1 x 1 x
s x since
0
??
??
??
? ? ?
?
?
?
??
?
?
?
 
3.  
3
2
2
4
4
3
2
2
4
4
d y dy
x
dx dx
d y dy
x0
dx dx
??
??
??
????
??
??
?? ?? ??
??
??
??
??
? ? ? ??
??
??
?? ?? ??
??
 
Order = 4 and degree = 2  
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
4.  
Let a, b and c be direction ratios of a line l, m and n be the direction cosines 
of the line. Then 
2 2 2 2 2 2 2 2 2
a b c
l , m , n
a b c a b c a b c
? ? ?
? ? ? ? ? ?
 
Here a = –18, b = 12 and c = – 4 
2 2 2 2 2 2
a b c ( 18) (12) ( 4)
324 144 16
484 22
? ? ? ? ? ? ? ?
? ? ?
??
 
2 2 2 2 2 2
2 2 2
a 18 9 b 12 6
l , m and
22 11 22 11
a b c a b c
c 4 2
n
22 11
a b c
??
? ? ? ? ? ? ?
? ? ? ?
??
? ? ?
??
 
OR 
Find the Cartesian equation of the line which passes through the point 
 (–2, 4, –5) and is parallel to the line
x 3 4 y z 8
3 5 6
? ? ?
?? . 
Cartesian equation of a line passing through (x1, y1, z1) and parallel to a line 
having direction ratios a, b, c is given by, 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Since the line passes through the (–2, 4, –5), 
? x1 = –2,  y1 = 4  and z1 = –5 
Given that the line is parallel to
x 3 4 y z 8
3 5 6
? ? ?
?? , 
? a = 3, b = -5 and c = 6 
Therefore the Cartesian equation of the line is given by 
 
x ( 2) y 4 z ( 5)
3 5 6
x 2 4 y z 5
3 5 6
? ? ? ? ?
??
?
? ? ?
? ? ?
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
 
5.  
? is defined on the set R of real numbers as  
a b a b ? ? ?
22
……. (i) 
Let ‘e’ be the identity element in R with respect to ? 
a e a e a ? ? ? ? ? 
Using the definition of ?, we have 
22
22
a* e = a + e
a + e = a ?
 
Taking square on both the sides, 
 
a e a
e a a
e
? ? ?
? ? ? ?
??
2 2 2
2 2 2
0
0
 
Hence, 0 is the identity element in R with respect to *. 
 
6.  
02
A
34
??
?
??
?
??
 
0 2k
kA
3k 4k
??
??
??
?
??
 
It is given that 
0 3a
kA
2b 24
??
?
??
??
 
0 2k 0 3a
3k 4k 2b 24
? ? ? ?
?
? ? ? ?
?
? ? ? ?
 
Using equality of matrices. 
-4k = 24 ?k = -6 
2k = 3a 
?
3a = -12 
?
a = -4 
3k = 2b 
?
2b = -18 
?
b=-9 
?k = -6, a = -4 and b = -9 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
 
SECTION  A 
1.  
A is a square matrix of order 3 with |A| = 4    …. Given 
|–2A| = –2
3
|A| = –8 ? 4 = – 32  
 
2.  
y = sin
-1
x + cos
-1
x     …. given  
? ?
? ?
1
22
11
1
sin x  cos x
sin x co
dy d
dx dx
d d d dA dB
AB
dx dx dx dx dx
11
1 x 1 x
s x since
0
??
??
??
? ? ?
?
?
?
??
?
?
?
 
3.  
3
2
2
4
4
3
2
2
4
4
d y dy
x
dx dx
d y dy
x0
dx dx
??
??
??
????
??
??
?? ?? ??
??
??
??
??
? ? ? ??
??
??
?? ?? ??
??
 
Order = 4 and degree = 2  
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
4.  
Let a, b and c be direction ratios of a line l, m and n be the direction cosines 
of the line. Then 
2 2 2 2 2 2 2 2 2
a b c
l , m , n
a b c a b c a b c
? ? ?
? ? ? ? ? ?
 
Here a = –18, b = 12 and c = – 4 
2 2 2 2 2 2
a b c ( 18) (12) ( 4)
324 144 16
484 22
? ? ? ? ? ? ? ?
? ? ?
??
 
2 2 2 2 2 2
2 2 2
a 18 9 b 12 6
l , m and
22 11 22 11
a b c a b c
c 4 2
n
22 11
a b c
??
? ? ? ? ? ? ?
? ? ? ?
??
? ? ?
??
 
OR 
Find the Cartesian equation of the line which passes through the point 
 (–2, 4, –5) and is parallel to the line
x 3 4 y z 8
3 5 6
? ? ?
?? . 
Cartesian equation of a line passing through (x1, y1, z1) and parallel to a line 
having direction ratios a, b, c is given by, 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Since the line passes through the (–2, 4, –5), 
? x1 = –2,  y1 = 4  and z1 = –5 
Given that the line is parallel to
x 3 4 y z 8
3 5 6
? ? ?
?? , 
? a = 3, b = -5 and c = 6 
Therefore the Cartesian equation of the line is given by 
 
x ( 2) y 4 z ( 5)
3 5 6
x 2 4 y z 5
3 5 6
? ? ? ? ?
??
?
? ? ?
? ? ?
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
 
5.  
? is defined on the set R of real numbers as  
a b a b ? ? ?
22
……. (i) 
Let ‘e’ be the identity element in R with respect to ? 
a e a e a ? ? ? ? ? 
Using the definition of ?, we have 
22
22
a* e = a + e
a + e = a ?
 
Taking square on both the sides, 
 
a e a
e a a
e
? ? ?
? ? ? ?
??
2 2 2
2 2 2
0
0
 
Hence, 0 is the identity element in R with respect to *. 
 
6.  
02
A
34
??
?
??
?
??
 
0 2k
kA
3k 4k
??
??
??
?
??
 
It is given that 
0 3a
kA
2b 24
??
?
??
??
 
0 2k 0 3a
3k 4k 2b 24
? ? ? ?
?
? ? ? ?
?
? ? ? ?
 
Using equality of matrices. 
-4k = 24 ?k = -6 
2k = 3a 
?
3a = -12 
?
a = -4 
3k = 2b 
?
2b = -18 
?
b=-9 
?k = -6, a = -4 and b = -9 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
7. 
sinx cosx
I dx
1 sin2x
?
?
?
?
 
Consider, 1 + sin2x = cos
2
 x + sin
2
 x + 2sin x cos x = (cosx + sinx)
2 
?
? ?
2
sinx cosx
I dx
cosx sinx
?
?
?
?
 
?
cosx sinx
I 1 dx
cosx sinx
?
??
?
?
 
Consider, f(x) = cos x + sin x then f’(x) = cos x – sin x 
? I 1ln cos x sinx c ? ? ? ?               
? ?
? ?
? ?
f' x
I dx ln f x c
fx
? ? ?
?
 
 
8. 
? ?
? ?
sin x a
I dx
sin x a
?
?
?
?
 
? ?
? ?
sin x a 2a
I dx
sin x a
??
??
?
?
 
? ? ? ?
? ?
sin x a cos2a cos x a sin2a
I dx
sin x a
? ? ?
??
?
?
 
? ?
? ?
? ?
? ?
sin x a cos2a cos x a sin2a
I dx
sin x a sin x a
?? ??
? ? ? ??
??
??
??
?
 
? ? ? ?
I cos2a sin2acot x a dx ? ? ? ?
?
 
? ?
I x cos2a sin2a ln sin x a c ? ? ? ? ? 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2019  
All India Set – 1 
Time: 3 hrs  Total Marks: 100 
      
 
SECTION  A 
1.  
A is a square matrix of order 3 with |A| = 4    …. Given 
|–2A| = –2
3
|A| = –8 ? 4 = – 32  
 
2.  
y = sin
-1
x + cos
-1
x     …. given  
? ?
? ?
1
22
11
1
sin x  cos x
sin x co
dy d
dx dx
d d d dA dB
AB
dx dx dx dx dx
11
1 x 1 x
s x since
0
??
??
??
? ? ?
?
?
?
??
?
?
?
 
3.  
3
2
2
4
4
3
2
2
4
4
d y dy
x
dx dx
d y dy
x0
dx dx
??
??
??
????
??
??
?? ?? ??
??
??
??
??
? ? ? ??
??
??
?? ?? ??
??
 
Order = 4 and degree = 2  
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
4.  
Let a, b and c be direction ratios of a line l, m and n be the direction cosines 
of the line. Then 
2 2 2 2 2 2 2 2 2
a b c
l , m , n
a b c a b c a b c
? ? ?
? ? ? ? ? ?
 
Here a = –18, b = 12 and c = – 4 
2 2 2 2 2 2
a b c ( 18) (12) ( 4)
324 144 16
484 22
? ? ? ? ? ? ? ?
? ? ?
??
 
2 2 2 2 2 2
2 2 2
a 18 9 b 12 6
l , m and
22 11 22 11
a b c a b c
c 4 2
n
22 11
a b c
??
? ? ? ? ? ? ?
? ? ? ?
??
? ? ?
??
 
OR 
Find the Cartesian equation of the line which passes through the point 
 (–2, 4, –5) and is parallel to the line
x 3 4 y z 8
3 5 6
? ? ?
?? . 
Cartesian equation of a line passing through (x1, y1, z1) and parallel to a line 
having direction ratios a, b, c is given by, 
1 1 1
x x y y z z
a b c
? ? ?
?? 
Since the line passes through the (–2, 4, –5), 
? x1 = –2,  y1 = 4  and z1 = –5 
Given that the line is parallel to
x 3 4 y z 8
3 5 6
? ? ?
?? , 
? a = 3, b = -5 and c = 6 
Therefore the Cartesian equation of the line is given by 
 
x ( 2) y 4 z ( 5)
3 5 6
x 2 4 y z 5
3 5 6
? ? ? ? ?
??
?
? ? ?
? ? ?
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
SECTION B 
 
5.  
? is defined on the set R of real numbers as  
a b a b ? ? ?
22
……. (i) 
Let ‘e’ be the identity element in R with respect to ? 
a e a e a ? ? ? ? ? 
Using the definition of ?, we have 
22
22
a* e = a + e
a + e = a ?
 
Taking square on both the sides, 
 
a e a
e a a
e
? ? ?
? ? ? ?
??
2 2 2
2 2 2
0
0
 
Hence, 0 is the identity element in R with respect to *. 
 
6.  
02
A
34
??
?
??
?
??
 
0 2k
kA
3k 4k
??
??
??
?
??
 
It is given that 
0 3a
kA
2b 24
??
?
??
??
 
0 2k 0 3a
3k 4k 2b 24
? ? ? ?
?
? ? ? ?
?
? ? ? ?
 
Using equality of matrices. 
-4k = 24 ?k = -6 
2k = 3a 
?
3a = -12 
?
a = -4 
3k = 2b 
?
2b = -18 
?
b=-9 
?k = -6, a = -4 and b = -9 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
7. 
sinx cosx
I dx
1 sin2x
?
?
?
?
 
Consider, 1 + sin2x = cos
2
 x + sin
2
 x + 2sin x cos x = (cosx + sinx)
2 
?
? ?
2
sinx cosx
I dx
cosx sinx
?
?
?
?
 
?
cosx sinx
I 1 dx
cosx sinx
?
??
?
?
 
Consider, f(x) = cos x + sin x then f’(x) = cos x – sin x 
? I 1ln cos x sinx c ? ? ? ?               
? ?
? ?
? ?
f' x
I dx ln f x c
fx
? ? ?
?
 
 
8. 
? ?
? ?
sin x a
I dx
sin x a
?
?
?
?
 
? ?
? ?
sin x a 2a
I dx
sin x a
??
??
?
?
 
? ? ? ?
? ?
sin x a cos2a cos x a sin2a
I dx
sin x a
? ? ?
??
?
?
 
? ?
? ?
? ?
? ?
sin x a cos2a cos x a sin2a
I dx
sin x a sin x a
?? ??
? ? ? ??
??
??
??
?
 
? ? ? ?
I cos2a sin2acot x a dx ? ? ? ?
?
 
? ?
I x cos2a sin2a ln sin x a c ? ? ? ? ? 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2019 – All India Set – 1 
 
  
OR 
 
? ?
2
I logx dx ?
?
 
log x = t ? x = e
t 
dx = e
t
 dt 
2
t
I t e dx ??
?
 
? ?
2 t t
I t e dt 2t e dt dt ? ? ?
? ? ?
 
2 t t
I t e 2 te dt ? ? ?
?
 
? ?
2 t t t
I t e 2 t e dt 1 e dt dt
??
? ? ? ? ?
??
??
? ? ?
 
2 t t t
I t e 2 te e
??
? ? ? ?
??
 
? ?
2 t t t
I t e 2 te e ? ? ? ? 
? ? ? ?
2
I x logx 2 xlogx x c ? ? ? ? ?
 
 
 
 
 
 
 
 
 
 
 
 
 
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