Sample Question Paper 6 - Math, Class 12

# Sample Question Paper 6 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

``` Page 1

CBSE XII | Mathematics
Sample Paper – 6

Mathematics
Class XII
Sample Paper – 6
Time: 3 hours                    Total Marks: 100

1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B, C and D.
Section A comprises of 4 questions of one mark each, section B comprises of 8
questions of two marks each, section C comprises of  11 questions of four marks
each and section D comprises of 6 questions of six marks each.
3. Use of calculators is not permitted.

SECTION – A

1. A matrix has 12 elements. What are the possible orders it can have?

2. Differentiate sin(2x
2
) w.r.t. x

3. Determine the order and degree of the following differential equation:
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??

4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has
direction ratios proportional to 1, 1, -2.
OR
Find the equation of a line passing through (1,-1, 0) and parallel to the line
x 2 2y 1 5 z
3 2 1
? ? ?
??

SECTION – B

5. (i) Is the binary operation *, defined on set N, given by

ab
a * b for all a,b N,
2
commutative?
(ii) Is the above binary operation * associative?

Page 2

CBSE XII | Mathematics
Sample Paper – 6

Mathematics
Class XII
Sample Paper – 6
Time: 3 hours                    Total Marks: 100

1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B, C and D.
Section A comprises of 4 questions of one mark each, section B comprises of 8
questions of two marks each, section C comprises of  11 questions of four marks
each and section D comprises of 6 questions of six marks each.
3. Use of calculators is not permitted.

SECTION – A

1. A matrix has 12 elements. What are the possible orders it can have?

2. Differentiate sin(2x
2
) w.r.t. x

3. Determine the order and degree of the following differential equation:
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??

4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has
direction ratios proportional to 1, 1, -2.
OR
Find the equation of a line passing through (1,-1, 0) and parallel to the line
x 2 2y 1 5 z
3 2 1
? ? ?
??

SECTION – B

5. (i) Is the binary operation *, defined on set N, given by

ab
a * b for all a,b N,
2
commutative?
(ii) Is the above binary operation * associative?

CBSE XII | Mathematics
Sample Paper – 6

6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b.

7. Evaluate:
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx
OR

Evaluate:
? ?
2
1x
x 1 2x
?
?
?
dx
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??

9. Form differential equations of the family of curves represented by c( y + c )
2
= x
3
,
where c is a parameter

10. Find the angle between and ab.
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ?

OR

Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?

11. A die is tossed thrice. Find the probability of getting an odd number at least once.
OR
A random variable X has the following probability distribution. Find

X 0 1 2 3 4 5
P(X) 0.1 K 0.2 2K 0.3 K
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3)

Page 3

CBSE XII | Mathematics
Sample Paper – 6

Mathematics
Class XII
Sample Paper – 6
Time: 3 hours                    Total Marks: 100

1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B, C and D.
Section A comprises of 4 questions of one mark each, section B comprises of 8
questions of two marks each, section C comprises of  11 questions of four marks
each and section D comprises of 6 questions of six marks each.
3. Use of calculators is not permitted.

SECTION – A

1. A matrix has 12 elements. What are the possible orders it can have?

2. Differentiate sin(2x
2
) w.r.t. x

3. Determine the order and degree of the following differential equation:
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??

4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has
direction ratios proportional to 1, 1, -2.
OR
Find the equation of a line passing through (1,-1, 0) and parallel to the line
x 2 2y 1 5 z
3 2 1
? ? ?
??

SECTION – B

5. (i) Is the binary operation *, defined on set N, given by

ab
a * b for all a,b N,
2
commutative?
(ii) Is the above binary operation * associative?

CBSE XII | Mathematics
Sample Paper – 6

6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b.

7. Evaluate:
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx
OR

Evaluate:
? ?
2
1x
x 1 2x
?
?
?
dx
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??

9. Form differential equations of the family of curves represented by c( y + c )
2
= x
3
,
where c is a parameter

10. Find the angle between and ab.
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ?

OR

Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?

11. A die is tossed thrice. Find the probability of getting an odd number at least once.
OR
A random variable X has the following probability distribution. Find

X 0 1 2 3 4 5
P(X) 0.1 K 0.2 2K 0.3 K
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3)

CBSE XII | Mathematics
Sample Paper – 6

12. Two cards are drawn successively with replacement from a well shuffled pack of
52 cards. Find the probability distribution of the number of aces.

SECTION – C

13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then
find
(i) The identify element of * in A.
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and
??
??
??
1
,4
2
.
OR

Let f : W ? W
be defined as
n 1, if n is odd
f n
n 1, if n is even

Show that f is invertible and find the inverse of f. Here, W is the set of all whole
numbers.

14. Solve the Equation:

11
1x
1+x
x,(x 0)
1
tan = tan
2
??
?
?
??
??
??

15. Show that x = 2 is a root of the equation formed by the following determinant

x -6 -1
2 -3x x-3 0
-3 2x x+2
?

Hence, solve the equation.

Page 4

CBSE XII | Mathematics
Sample Paper – 6

Mathematics
Class XII
Sample Paper – 6
Time: 3 hours                    Total Marks: 100

1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B, C and D.
Section A comprises of 4 questions of one mark each, section B comprises of 8
questions of two marks each, section C comprises of  11 questions of four marks
each and section D comprises of 6 questions of six marks each.
3. Use of calculators is not permitted.

SECTION – A

1. A matrix has 12 elements. What are the possible orders it can have?

2. Differentiate sin(2x
2
) w.r.t. x

3. Determine the order and degree of the following differential equation:
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??

4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has
direction ratios proportional to 1, 1, -2.
OR
Find the equation of a line passing through (1,-1, 0) and parallel to the line
x 2 2y 1 5 z
3 2 1
? ? ?
??

SECTION – B

5. (i) Is the binary operation *, defined on set N, given by

ab
a * b for all a,b N,
2
commutative?
(ii) Is the above binary operation * associative?

CBSE XII | Mathematics
Sample Paper – 6

6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b.

7. Evaluate:
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx
OR

Evaluate:
? ?
2
1x
x 1 2x
?
?
?
dx
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??

9. Form differential equations of the family of curves represented by c( y + c )
2
= x
3
,
where c is a parameter

10. Find the angle between and ab.
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ?

OR

Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?

11. A die is tossed thrice. Find the probability of getting an odd number at least once.
OR
A random variable X has the following probability distribution. Find

X 0 1 2 3 4 5
P(X) 0.1 K 0.2 2K 0.3 K
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3)

CBSE XII | Mathematics
Sample Paper – 6

12. Two cards are drawn successively with replacement from a well shuffled pack of
52 cards. Find the probability distribution of the number of aces.

SECTION – C

13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then
find
(i) The identify element of * in A.
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and
??
??
??
1
,4
2
.
OR

Let f : W ? W
be defined as
n 1, if n is odd
f n
n 1, if n is even

Show that f is invertible and find the inverse of f. Here, W is the set of all whole
numbers.

14. Solve the Equation:

11
1x
1+x
x,(x 0)
1
tan = tan
2
??
?
?
??
??
??

15. Show that x = 2 is a root of the equation formed by the following determinant

x -6 -1
2 -3x x-3 0
-3 2x x+2
?

Hence, solve the equation.

CBSE XII | Mathematics
Sample Paper – 6

16. If
? ?
2
1 x dy
y ,prove that 1 x y 0
1 x dx
?
? ? ? ?
?

OR
Differentiate w.r.t. x
log10x + logx10 + logx x + log1010
17. Differentiate w.r.t. x
12
cos 1 x
ye
?
?
?

18. Find the equation of tangent and normal to the curve
5x
y 3e ?? where it crosses
the y-axis.

19. Evaluate:
2
5x 3
dx
x 4x 10
?
?
??

20. Evaluate:
? ?
2
2
x +2x +1 dx
0
?
as limit of sum.

21. Solve the differential equation (1 + e
2x
) dy +( 1 + y
2
)e
x
dx = 0
given that when x = 0, y = 1.
OR

Solve the differential equation x (1 + y
2
) dx - y (1 + x
2
)dy = 0,
given that y = 0, when x = 1.

22. If a i j k and b j k , find a vector c such that a c b and a.c 3

23. A variable plane is at a constant distance p from the origin and meet the
coordinate axes in A, B, C. Show that the locus of the centroid of the tetrahedron
OABC is
x
-2
+y
-2
+z
-2
=16p
-2

Page 5

CBSE XII | Mathematics
Sample Paper – 6

Mathematics
Class XII
Sample Paper – 6
Time: 3 hours                    Total Marks: 100

1. All questions are compulsory.
2. The question paper consist of 29 questions divided into three sections A, B, C and D.
Section A comprises of 4 questions of one mark each, section B comprises of 8
questions of two marks each, section C comprises of  11 questions of four marks
each and section D comprises of 6 questions of six marks each.
3. Use of calculators is not permitted.

SECTION – A

1. A matrix has 12 elements. What are the possible orders it can have?

2. Differentiate sin(2x
2
) w.r.t. x

3. Determine the order and degree of the following differential equation:
2
32
t
32
d y d y dy
e
dx dx dx
??
? ? ?
??
??

4. Write the Cartesian equation of line passing through a point (2, -1, 4) and has
direction ratios proportional to 1, 1, -2.
OR
Find the equation of a line passing through (1,-1, 0) and parallel to the line
x 2 2y 1 5 z
3 2 1
? ? ?
??

SECTION – B

5. (i) Is the binary operation *, defined on set N, given by

ab
a * b for all a,b N,
2
commutative?
(ii) Is the above binary operation * associative?

CBSE XII | Mathematics
Sample Paper – 6

6. If
a b 2 6 2
5 ab 5 8
? ? ? ? ?
?
? ? ? ?
? ? ? ?
, then find values of a and b.

7. Evaluate:
x
sin 4x 4
e
1 cos 4x
?
?? ?
??
?
??
dx
OR

Evaluate:
? ?
2
1x
x 1 2x
?
?
?
dx
8. Evaluate:
? ? ? ?
22
2x
dx
x 1 x 3
?
??

9. Form differential equations of the family of curves represented by c( y + c )
2
= x
3
,
where c is a parameter

10. Find the angle between and ab.
If 0 ? ? ? a b c and a 3, b 5 & c 7 ? ? ?

OR

Find   if the vectors ?
ˆˆ ˆ ˆ ˆ ˆ
a i j 3k and b 4i 5j 2k are perpendicular to each other. ? ? ? ? ? ? ?

11. A die is tossed thrice. Find the probability of getting an odd number at least once.
OR
A random variable X has the following probability distribution. Find

X 0 1 2 3 4 5
P(X) 0.1 K 0.2 2K 0.3 K
(i) The value of K (ii) P(X ? 1)   (iii) P(X > 3)

CBSE XII | Mathematics
Sample Paper – 6

12. Two cards are drawn successively with replacement from a well shuffled pack of
52 cards. Find the probability distribution of the number of aces.

SECTION – C

13. Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary
operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then
find
(i) The identify element of * in A.
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and
??
??
??
1
,4
2
.
OR

Let f : W ? W
be defined as
n 1, if n is odd
f n
n 1, if n is even

Show that f is invertible and find the inverse of f. Here, W is the set of all whole
numbers.

14. Solve the Equation:

11
1x
1+x
x,(x 0)
1
tan = tan
2
??
?
?
??
??
??

15. Show that x = 2 is a root of the equation formed by the following determinant

x -6 -1
2 -3x x-3 0
-3 2x x+2
?

Hence, solve the equation.

CBSE XII | Mathematics
Sample Paper – 6

16. If
? ?
2
1 x dy
y ,prove that 1 x y 0
1 x dx
?
? ? ? ?
?

OR
Differentiate w.r.t. x
log10x + logx10 + logx x + log1010
17. Differentiate w.r.t. x
12
cos 1 x
ye
?
?
?

18. Find the equation of tangent and normal to the curve
5x
y 3e ?? where it crosses
the y-axis.

19. Evaluate:
2
5x 3
dx
x 4x 10
?
?
??

20. Evaluate:
? ?
2
2
x +2x +1 dx
0
?
as limit of sum.

21. Solve the differential equation (1 + e
2x
) dy +( 1 + y
2
)e
x
dx = 0
given that when x = 0, y = 1.
OR

Solve the differential equation x (1 + y
2
) dx - y (1 + x
2
)dy = 0,
given that y = 0, when x = 1.

22. If a i j k and b j k , find a vector c such that a c b and a.c 3

23. A variable plane is at a constant distance p from the origin and meet the
coordinate axes in A, B, C. Show that the locus of the centroid of the tetrahedron
OABC is
x
-2
+y
-2
+z
-2
=16p
-2

CBSE XII | Mathematics
Sample Paper – 6

SECTION – D

24. Let A =
23
12
??
??
?
??
and f(x) = x
2
– 4x + 7. Show that f(A) = O. Use this result to find
A
5
.
OR
Let f(x) = x
2
– 5x + 6. Find f(A), if A =
2 0 1
2 1 3
1 1 0
??
??
??
?? ?
??

25. A window is in the form of a rectangle surmounted by a semicircular opening.
The total perimeter of the window is 10 m. Find the dimensions of the window
to admit maximum light through the whole opening.

26. Using integration, find the area of the circle x
2
+ y
2
= 16 which is exterior to the
parabola y
2
= 6x.
OR
Find the area of the smaller region bounded by the ellipse
22
22
x y x y
1 and the line 1
ab
ab

27. Find the equation of the plane passing through the point (-1, 3, 2) and
perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

OR
Find the equation of the plane which contains the line of intersection of the
planes
? ?
r. i 2j 3k 4 0 ? ? ? ? ,
? ?
r. 2i j k 5 0 ? ? ? ? and which is perpendicular to
the plane
? ?
ˆ
r. 5i 3j 6k 8 0 ? ? ? ? .

28. A firm makes items A and B and the total number of items it can make in a day is
24. It takes one hour to make item A and only half an hour to make item B. The
maximum time available per day is 16 hours. The profit per item of A is Rs. 300
and Rs. 160 on one item of B. How many items of each type should be produced
to maximize the profit? Solve the problem graphically.
```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Question Paper 6 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. How can I prepare for the Class 12 Math exam effectively?
Ans. To prepare for the Class 12 Math exam effectively, you can follow these steps: 1. Understand the syllabus and exam pattern: Familiarize yourself with the topics and weightage of each chapter in the syllabus. Also, understand the marking scheme and question paper pattern. 2. Create a study schedule: Make a study schedule that allows you to allocate sufficient time to each topic. Divide your study sessions into smaller, manageable chunks. 3. Practice previous years' question papers: Solve previous years' question papers to get an idea of the exam pattern and the type of questions asked. This will also help you in time management. 4. Revise regularly: Regular revision is crucial to retain the concepts and formulas. Set aside dedicated time for revision and make use of revision notes or flashcards. 5. Seek help when needed: If you come across any difficult topics or concepts, don't hesitate to seek help from your teacher, classmates, or online resources.
 2. How can I improve my problem-solving skills in Math?
 3. How can I manage my time effectively during the Class 12 Math exam?
Ans. Time management is crucial during the Class 12 Math exam to ensure that you can attempt all the questions within the given duration. Here are some tips to manage your time effectively: 1. Familiarize yourself with the question paper: Before you start solving the questions, quickly go through the entire question paper to get an idea of the difficulty level and mark questions that you find easier to solve. 2. Allocate time to each question: Divide the total time you have by the number of questions to determine how much time you can spend on each question. This will help you prioritize and avoid spending too much time on a single question. 3. Start with easier questions: Begin the exam by solving the questions you find easier. This will boost your confidence and save time for the more challenging ones. 4. Skip and come back: If you encounter a difficult question, don't get stuck on it. Skip it and move on to the next one. You can come back to it later if you have time left. 5. Practice time-bound mock tests: Regularly practice solving Math problems within a time limit to improve your speed and efficiency. This will train you to manage your time better during the actual exam.
 4. How can I overcome exam anxiety for the Class 12 Math exam?
Ans. Exam anxiety can be overwhelming, but there are strategies to overcome it and perform your best in the Class 12 Math exam. Here are some tips: 1. Prepare well: Confidence comes from preparation. Make sure you have thoroughly studied the syllabus, practiced enough problems, and revised the important topics. 2. Positive self-talk: Replace negative thoughts with positive affirmations. Remind yourself that you have prepared well and that you are capable of handling the exam. 3. Relaxation techniques: Practice relaxation techniques such as deep breathing, meditation, or listening to calming music to relax your mind and body before the exam. 4. Time management: Plan your time effectively during the exam to avoid feeling rushed or overwhelmed. Follow a systematic approach and allocate time to each question. 5. Seek support: Talk to your teachers, friends, or family members about your exam anxiety. They can provide guidance, support, and reassurance during this time.
 5. How can I score well in the Class 12 Math exam?
Ans. Scoring well in the Class 12 Math exam requires a combination of understanding concepts, practicing problem-solving, and effective exam strategies. Here are some tips to score well: 1. Understand the concepts: Focus on understanding the underlying concepts rather than rote memorization. This will help you apply the concepts to solve different types of problems. 2. Practice regularly: Practice solving a variety of Math problems regularly to improve your problem-solving skills and speed. Solve previous years' question papers and sample papers to get accustomed to the exam pattern. 3. Focus on important topics: Identify the important topics based on their weightage in the syllabus and previous years' exams. Allocate more time to studying and practicing these topics. 4. Create concise notes: Prepare concise and organized notes that summarize the key formulas, concepts, and problem-solving techniques. These notes will come in handy during revision. 5. Review and revise: Regularly review and revise the topics you have studied to reinforce your understanding. Dedicate dedicated time for revision before the exam to ensure retention of concepts.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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