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Page 1
CBSE XII | Mathematics
Sample Paper – 6 Solution
Mathematics
Class XII
Sample Paper – 6 Solution
SECTION – A
1. A matrix is of order m × n, then it has mn elements.
So, mn = 12
Which means we have to find values of m and n, such that it will satisfy the
above condition
So all the possible cases are
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1).
2.
Let f(x) = sin(2x
2
)
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
3.
Sol:
Order: 3
Degree: 1
4.
The Cartesian form of a line is given by
1 1 1
x x y y z z
a b c
? ? ?
??
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2
We get the equation as
x 2 y 1 z 4
1 1 2
? ? ?
??
?
Page 2
CBSE XII | Mathematics
Sample Paper – 6 Solution
Mathematics
Class XII
Sample Paper – 6 Solution
SECTION – A
1. A matrix is of order m × n, then it has mn elements.
So, mn = 12
Which means we have to find values of m and n, such that it will satisfy the
above condition
So all the possible cases are
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1).
2.
Let f(x) = sin(2x
2
)
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
3.
Sol:
Order: 3
Degree: 1
4.
The Cartesian form of a line is given by
1 1 1
x x y y z z
a b c
? ? ?
??
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2
We get the equation as
x 2 y 1 z 4
1 1 2
? ? ?
??
?
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
The equation can be written as
1
y
x 2 z 5
2
3 1 1
?
??
??
?
D.R.S. of this line is proportional to 3, 1,-1.
Also the line passes through (1,-1, 0)
x 1 y 1 z 0
3 1 1
? ? ?
??
?
SECTION – B
5.
(i) For all
ab
a,b N,a * b
2
Now,
b a a b
b * a a * b
22
Thus, the binary operation * is commutative.
(ii) Let a,b,c N
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
Thus, the binary operation * is not associative.
Page 3
CBSE XII | Mathematics
Sample Paper – 6 Solution
Mathematics
Class XII
Sample Paper – 6 Solution
SECTION – A
1. A matrix is of order m × n, then it has mn elements.
So, mn = 12
Which means we have to find values of m and n, such that it will satisfy the
above condition
So all the possible cases are
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1).
2.
Let f(x) = sin(2x
2
)
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
3.
Sol:
Order: 3
Degree: 1
4.
The Cartesian form of a line is given by
1 1 1
x x y y z z
a b c
? ? ?
??
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2
We get the equation as
x 2 y 1 z 4
1 1 2
? ? ?
??
?
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
The equation can be written as
1
y
x 2 z 5
2
3 1 1
?
??
??
?
D.R.S. of this line is proportional to 3, 1,-1.
Also the line passes through (1,-1, 0)
x 1 y 1 z 0
3 1 1
? ? ?
??
?
SECTION – B
5.
(i) For all
ab
a,b N,a * b
2
Now,
b a a b
b * a a * b
22
Thus, the binary operation * is commutative.
(ii) Let a,b,c N
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
Thus, the binary operation * is not associative.
CBSE XII | Mathematics
Sample Paper – 6 Solution
6.
The corresponding values of two equal matrices are equal
a + b = 6 and ab = 8
Therefore,
8
b
a
?
Substituting in first condition we get,
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
7.
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x
I ? ?
x
e f(x) f '(x) dx
?
??
So, I = e
x
f(x) + C = e
x
cot 2x + C , where C is a constant
Therefore,
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
Page 4
CBSE XII | Mathematics
Sample Paper – 6 Solution
Mathematics
Class XII
Sample Paper – 6 Solution
SECTION – A
1. A matrix is of order m × n, then it has mn elements.
So, mn = 12
Which means we have to find values of m and n, such that it will satisfy the
above condition
So all the possible cases are
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1).
2.
Let f(x) = sin(2x
2
)
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
3.
Sol:
Order: 3
Degree: 1
4.
The Cartesian form of a line is given by
1 1 1
x x y y z z
a b c
? ? ?
??
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2
We get the equation as
x 2 y 1 z 4
1 1 2
? ? ?
??
?
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
The equation can be written as
1
y
x 2 z 5
2
3 1 1
?
??
??
?
D.R.S. of this line is proportional to 3, 1,-1.
Also the line passes through (1,-1, 0)
x 1 y 1 z 0
3 1 1
? ? ?
??
?
SECTION – B
5.
(i) For all
ab
a,b N,a * b
2
Now,
b a a b
b * a a * b
22
Thus, the binary operation * is commutative.
(ii) Let a,b,c N
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
Thus, the binary operation * is not associative.
CBSE XII | Mathematics
Sample Paper – 6 Solution
6.
The corresponding values of two equal matrices are equal
a + b = 6 and ab = 8
Therefore,
8
b
a
?
Substituting in first condition we get,
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
7.
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x
I ? ?
x
e f(x) f '(x) dx
?
??
So, I = e
x
f(x) + C = e
x
cot 2x + C , where C is a constant
Therefore,
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
? ?
2
1x
dx
x 1 2x
?
?
?
Here
? ?
2
1x
x 1 2x
?
?
is an improper rational fraction.
Reducing it to proper rational fraction gives
? ? ? ?
2
1 x 1 1 2 x
x 1 2x 2 2 x 1 2x
??
??
??
??
??
??
??
…………(1)
Now, let
? ?
2 x A B
x 1 2x x (1 2x)
?
??
??
? ? ? ?
? ?
2 x A(1 2x) Bx
2 x A x(2A B)
x 1 2x x 1 2x
Equating the coefficientsweget,A 2 and B 3
2 x 2 3
So,
x 1 2x x (1 2x)
? ? ?
? ? ? ? ? ? ?
??
??
?
??
??
Substituting in equation (1), we get
? ?
2
1 x 1 1 2 3
x 1 2x 2 2 x (1 2x)
?? ?
? ? ?
??
??
??
? ?
2
1 x 1 1 2 3
i.e dx dx
x 1 2x 2 2 x (1 2x)
??
?? ?? ?
? ? ?
?? ??
??
?? ??
dx dx 3 dx x 3 1
log x log 1 2x C
2 x 2 (1 2x) 2 2 ( 2)
x3
log x log 1 2x C
24
? ? ?
? ? ? ? ? ? ? ? ?
??
? ? ? ? ?
Page 5
CBSE XII | Mathematics
Sample Paper – 6 Solution
Mathematics
Class XII
Sample Paper – 6 Solution
SECTION – A
1. A matrix is of order m × n, then it has mn elements.
So, mn = 12
Which means we have to find values of m and n, such that it will satisfy the
above condition
So all the possible cases are
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1).
2.
Let f(x) = sin(2x
2
)
? ? ? ? ? ? ? ?
? ? ? ?
? ?
22
2
2
d d d
f x sin 2x 2x
dx dx dx
cos 2x 4x
4xcos 2x
?
?
?
3.
Sol:
Order: 3
Degree: 1
4.
The Cartesian form of a line is given by
1 1 1
x x y y z z
a b c
? ? ?
??
Substituting point (2, -1, 4) and d. r. s. 1, 1, -2
We get the equation as
x 2 y 1 z 4
1 1 2
? ? ?
??
?
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
The equation can be written as
1
y
x 2 z 5
2
3 1 1
?
??
??
?
D.R.S. of this line is proportional to 3, 1,-1.
Also the line passes through (1,-1, 0)
x 1 y 1 z 0
3 1 1
? ? ?
??
?
SECTION – B
5.
(i) For all
ab
a,b N,a * b
2
Now,
b a a b
b * a a * b
22
Thus, the binary operation * is commutative.
(ii) Let a,b,c N
bc
a
b c 2a b c
2
a * b * c a *
2 2 4
ab
c
a b a b 2c
2
a * b * c * c
2 2 4
a * b * c a * b * c
Thus, the binary operation * is not associative.
CBSE XII | Mathematics
Sample Paper – 6 Solution
6.
The corresponding values of two equal matrices are equal
a + b = 6 and ab = 8
Therefore,
8
b
a
?
Substituting in first condition we get,
? ? ? ?
2
8
a6
a
a 6a 8 0
a 4 a 2 0
a 4 or a 2 and
b 2 or b 4 respectively.
??
? ? ?
? ? ?
??
??
7.
x
sin4x 4
Let I e dx
1 cos4x
?
? ??
?
??
?
??
x
x2
2
sin2(2x) 4
e dx
1 cos2(2x)
2sin2xcos2x 4
e dx [Using,sin2x 2sin x.cosxand2sin x 1 cos(2x)
2sin (2x)
?
?
?? ?
?
??
?
??
??
?
? ? ? ?
??
??
??
? ?
xx
2 2 2
x2
2(sin(2x)cos(2x) 4 sin(2x)cos(2x) 2
e dx e dx
2sin 2x sin 2x sin 2x
e cot(2x) 2cosec 2x dx
??
?
? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
Now, let f(x) = cot(2x) then f’(x) = -2cosec
2
2x
I ? ?
x
e f(x) f '(x) dx
?
??
So, I = e
x
f(x) + C = e
x
cot 2x + C , where C is a constant
Therefore,
? ?
xx
sin4x 4
e dx e cot 2x + C
1 cos4x
?
? ??
?
??
?
??
CBSE XII | Mathematics
Sample Paper – 6 Solution
OR
? ?
2
1x
dx
x 1 2x
?
?
?
Here
? ?
2
1x
x 1 2x
?
?
is an improper rational fraction.
Reducing it to proper rational fraction gives
? ? ? ?
2
1 x 1 1 2 x
x 1 2x 2 2 x 1 2x
??
??
??
??
??
??
??
…………(1)
Now, let
? ?
2 x A B
x 1 2x x (1 2x)
?
??
??
? ? ? ?
? ?
2 x A(1 2x) Bx
2 x A x(2A B)
x 1 2x x 1 2x
Equating the coefficientsweget,A 2 and B 3
2 x 2 3
So,
x 1 2x x (1 2x)
? ? ?
? ? ? ? ? ? ?
??
??
?
??
??
Substituting in equation (1), we get
? ?
2
1 x 1 1 2 3
x 1 2x 2 2 x (1 2x)
?? ?
? ? ?
??
??
??
? ?
2
1 x 1 1 2 3
i.e dx dx
x 1 2x 2 2 x (1 2x)
??
?? ?? ?
? ? ?
?? ??
??
?? ??
dx dx 3 dx x 3 1
log x log 1 2x C
2 x 2 (1 2x) 2 2 ( 2)
x3
log x log 1 2x C
24
? ? ?
? ? ? ? ? ? ? ? ?
??
? ? ? ? ?
CBSE XII | Mathematics
Sample Paper – 6 Solution
8.
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
22
2
2x
I dx
x 1 x 3
Let x z
2xdx dz
dz
I
z 1 z 3
By partialfraction,
1 A B
z 1 z 3 z 1 z 3
1 A z 3 B z 1
?
?
?
??
?
??
??
??
??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
22
22
2
2
Putting z 3, we obtain :
1 2B
1
B
2
1
A
2
1
1
1 2
2
z 1 z 3 z 1 z 3
dz 1 dz dz
z 1 z 3 2 z 1 z 3
11
log z 1 log z 3 C
22
2xdx 1 1
log x 1 log x 3 C
22
x 1 x 3
1 x 1
log C
2
x3
? ? ?
?
??
??
??
??
??
?
??
??
? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
??
?
??
?
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FAQs on Sample Solution Paper 6 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What are the important topics to study for the Class 12 Math exam? |  |
Ans. The important topics to study for the Class 12 Math exam include calculus, algebra, probability, matrices, vectors, and differential equations. It is crucial to have a strong understanding of these topics as they form the foundation for many advanced mathematical concepts.
| 2. How can I improve my problem-solving skills in Math for the Class 12 exam? | |
Ans. To improve your problem-solving skills in Math for the Class 12 exam, it is essential to practice regularly. Solve a variety of problems from different sources, such as textbooks, previous year question papers, and online resources. Additionally, try to understand the concepts behind the problems, rather than just memorizing formulas, as this will help you approach different types of problems with more clarity.
| 3. What are some useful tips to manage time effectively during the Class 12 Math exam? | |
Ans. Managing time effectively during the Class 12 Math exam is crucial to ensure that you can attempt all the questions within the given time limit. Some useful tips to manage time effectively include:
- Read the entire question paper thoroughly before starting to solve any questions.
- Allocate a specific amount of time to each section or topic based on the marks assigned to them.
- Start with the questions you find easier and leave the tougher ones for later.
- Avoid spending too much time on any particular question. If you're stuck, move on and come back to it later if you have time.
| 4. How can I improve my understanding of complex concepts in Math for the Class 12 exam? | |
Ans. Improving your understanding of complex concepts in Math for the Class 12 exam requires regular practice and a strategic approach. Some tips to improve your understanding include:
- Break down complex concepts into smaller, more manageable parts.
- Seek help from your teacher or classmates if you're struggling with a particular concept.
- Use visual aids or diagrams to visualize the concepts.
- Solve a variety of problems related to the complex concept to gain a deeper understanding of its applications.
| 5. Are there any specific strategies to score well in the Class 12 Math exam? | |
Ans. Yes, there are specific strategies to score well in the Class 12 Math exam. Some effective strategies include:
- Practice solving previous year question papers to get an idea of the exam pattern and types of questions asked.
- Focus on understanding the concepts rather than rote memorization.
- Revise regularly and make concise notes for quick revision.
- Solve sample papers and mock tests to improve your speed and accuracy.
- Seek help from your teacher or join study groups to clarify doubts and learn from others.
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Nov 25, 2024 Last updated |
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