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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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FAQs on CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What are the benefits of solving CBSE past year papers for the Mathematics Class 12 exam? |  |
Ans. Solving CBSE past year papers for the Mathematics Class 12 exam has several benefits. It helps students understand the exam pattern, marking scheme, and types of questions that can be expected in the actual exam. It also allows students to practice time management and improve their speed in solving problems. Additionally, solving past year papers helps in revising important concepts and identifying areas that need more focus, leading to better preparation and higher chances of scoring well in the exam.
| 2. How can I use the CBSE past year paper solutions for Mathematics Class 12 effectively? | |
Ans. To use the CBSE past year paper solutions for Mathematics Class 12 effectively, start by attempting the exam paper without referring to the solutions. Once you complete the paper, compare your answers with the solutions provided to identify any mistakes or areas where improvement is needed. Analyze the solutions thoroughly to understand the correct approach and concepts behind each question. Make notes of important formulas or techniques used in the solutions. Practice solving similar questions from other resources using the knowledge gained from the solutions to reinforce your understanding and improve your problem-solving skills.
| 3. How can solving CBSE past year papers help in reducing exam stress for the Mathematics Class 12 students? | |
Ans. Solving CBSE past year papers can significantly reduce exam stress for Mathematics Class 12 students. By practicing with these papers, students become familiar with the exam format and gain confidence in their preparation. They get accustomed to the types of questions asked in the exam, reducing the element of surprise during the actual test. This familiarity and confidence help in managing exam anxiety and stress. Additionally, solving past year papers allows students to identify their strengths and weaknesses, giving them a clear direction for focused revision and reducing the fear of the unknown.
| 4. Are the CBSE past year paper solutions for Mathematics Class 12 available online? | |
Ans. Yes, the CBSE past year paper solutions for Mathematics Class 12 are available online. There are various websites and educational platforms that provide these solutions for free or for a nominal fee. Students can access these solutions in PDF or online format, making it convenient for them to study and practice. It is advisable to choose reliable sources for accessing the solutions to ensure accuracy and quality. Additionally, some schools and coaching institutes also provide the past year paper solutions to their students for better preparation.
| 5. How can I make the most of the CBSE past year paper solutions for Mathematics Class 12 while self-studying? | |
Ans. While self-studying, you can make the most of the CBSE past year paper solutions for Mathematics Class 12 by following a systematic approach. Start by solving the papers without referring to the solutions, just like in an actual exam. Once you complete a paper, check your answers using the solutions provided. Analyze the solutions to understand the correct approach and concepts used. Make a note of any mistakes or areas where improvement is needed. Focus on understanding the step-by-step solutions and practice similar questions from other resources using the knowledge gained. Regularly solve past year papers and track your progress to assess your preparation level and make necessary adjustments in your study plan.
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CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 Notes
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CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 JEE Questions
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The content of CBSE Past Year Paper Session (2019) Set- 1 Solutions, Mathematics Class 12 is prepared as per the latest JEE syllabus.
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