Sample Question Paper 3 - Math, Class 12

# Sample Question Paper 3 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

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CBSE XII | Mathematics
Sample Paper – 3

Mathematics
Class XII
Sample Paper 3
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.

2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.

3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.

4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?

Men workers                    Women workers
Factory I                    40                                         15

Factory II                  35                                          40

Factory III                 72                                          64

Page 2

CBSE XII | Mathematics
Sample Paper – 3

Mathematics
Class XII
Sample Paper 3
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.

2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.

3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.

4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?

Men workers                    Women workers
Factory I                    40                                         15

Factory II                  35                                          40

Factory III                 72                                          64

CBSE XII | Mathematics
Sample Paper – 3

OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.

SECTION – B

5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??

6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?

7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear?

8.
?
??
??
??
1
25
Write A for A = .
13

OR

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1

10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x

OR

The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?

Page 3

CBSE XII | Mathematics
Sample Paper – 3

Mathematics
Class XII
Sample Paper 3
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.

2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.

3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.

4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?

Men workers                    Women workers
Factory I                    40                                         15

Factory II                  35                                          40

Factory III                 72                                          64

CBSE XII | Mathematics
Sample Paper – 3

OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.

SECTION – B

5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??

6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?

7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear?

8.
?
??
??
??
1
25
Write A for A = .
13

OR

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1

10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x

OR

The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?

CBSE XII | Mathematics
Sample Paper – 3

11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).

12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5

OR
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the
event “number obtained is even” and B be the event “Number obtained is red.” Find if A
and B are independent events.

SECTION – C

13.
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x

OR

? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,

14.

? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function

15.

Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0

16.

2 2 2 2
A  plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?

Page 4

CBSE XII | Mathematics
Sample Paper – 3

Mathematics
Class XII
Sample Paper 3
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.

2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.

3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.

4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?

Men workers                    Women workers
Factory I                    40                                         15

Factory II                  35                                          40

Factory III                 72                                          64

CBSE XII | Mathematics
Sample Paper – 3

OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.

SECTION – B

5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??

6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?

7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear?

8.
?
??
??
??
1
25
Write A for A = .
13

OR

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1

10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x

OR

The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?

CBSE XII | Mathematics
Sample Paper – 3

11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).

12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5

OR
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the
event “number obtained is even” and B be the event “Number obtained is red.” Find if A
and B are independent events.

SECTION – C

13.
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x

OR

? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,

14.

? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function

15.

Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0

16.

2 2 2 2
A  plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?

CBSE XII | Mathematics
Sample Paper – 3

17. Four cards are drawn successively with replacement from a well shuffled deck of 52
cards. What is the probability that
(i) All the four cards are spades?
(ii) Only 3 cards are spades?
(iii) None is a spade?

18.
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
??

19. Find the equation of a tangent to  the curve given by
33
x asin t , y bcos t ?? at
a point, where t
2
?
?
.
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?

21. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1

22. ? Show that the function f defined by f(x)  =  1-x + x , x   R is continuous.
OR
Show that a logarithmic function is continuous at every point in its domain.

23.
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin

Page 5

CBSE XII | Mathematics
Sample Paper – 3

Mathematics
Class XII
Sample Paper 3
Time: 3 hour                    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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.

2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.

3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.

4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?

Men workers                    Women workers
Factory I                    40                                         15

Factory II                  35                                          40

Factory III                 72                                          64

CBSE XII | Mathematics
Sample Paper – 3

OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.

SECTION – B

5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??

6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?

7. ? ? ? ? For what value of 'a' the vectors  2i 3j 4k  and  ai 6j 8k are collinear?

8.
?
??
??
??
1
25
Write A for A = .
13

OR

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1

10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x

OR

The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?

CBSE XII | Mathematics
Sample Paper – 3

11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).

12. ?
2 1 1
If P(A)= , P(B)= ,    P(A B)= , then find P(A /B).
5 3 5

OR
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the
event “number obtained is even” and B be the event “Number obtained is red.” Find if A
and B are independent events.

SECTION – C

13.
? ?
?
??
3
5
3
x 5 x
Differentiate   ,  wrt  x
7 3x 8 5x

OR

? ? ?
2 '' '
If  y  =  acos(log x) + bsin(log x), prove that  x y xy y 0,

14.

? ? ? ? Let  f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function

15.

Find the distance between the parallel planes
r. 2i 1j 3k 4 and  r. 6i 3j 9k 13 0

16.

2 2 2 2
A  plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?

CBSE XII | Mathematics
Sample Paper – 3

17. Four cards are drawn successively with replacement from a well shuffled deck of 52
cards. What is the probability that
(i) All the four cards are spades?
(ii) Only 3 cards are spades?
(iii) None is a spade?

18.
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
??

19. Find the equation of a tangent to  the curve given by
33
x asin t , y bcos t ?? at
a point, where t
2
?
?
.
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?

21. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1

22. ? Show that the function f defined by f(x)  =  1-x + x , x   R is continuous.
OR
Show that a logarithmic function is continuous at every point in its domain.

23.
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin

CBSE XII | Mathematics
Sample Paper – 3

SECTION - D

24. Show that the right circular cone of least curved surface and given volume has an
altitude equal to 2 times the radius of the base.

OR

43
Find the points at which the function f given by f(x) = (x - 2 ) (x 1) is minimum. ?

25. Obtain the inverse of the following matrix using elementary operations.
0 1 2
A= 1 2 3
3 1 1
??
??
??
??
??

26.

?
?
22
22
22
22
Calculate the area
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a
ab
xy
(ii) Triangle AOB is in the first quadrant  of the ellipse  + 1,where OA = a and OB = b.
ab
Find the area enclosed between the chord AB and the arc AB of the ellipse
(iii) Find the ratio of the two areas found.

OR

??
??
2 2 2
2
Find the smaller of the two areas in which the circle x y 2a
is divided by the parabola y ax, a 0

27. Find the equation of a plane that is parallel to the x-axis and passes through the line
common to two intersectiing planes r. i+j+k 1 0 and r. 2i+3j-k 4

28. Two trainee carpenters A and B earn Rs. 150 and Rs. 200 per day respectively. A can
make 6 frames and 4 stools per day while B can make 10 frames and 4 stools per day.
How many days shall each work, if it is desired to produce atleast 60 frames and 32
stools at a minimum labour cost? Solve the problem graphically.

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Question Paper 3 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. What are the important topics to study for the Math Class 12 exam?
Ans. The important topics to study for the Math Class 12 exam include calculus, algebra, coordinate geometry, probability, vectors, and matrices. These topics are crucial as they form the foundation for various mathematical concepts and problem-solving techniques.
 2. How should I prepare for the Math Class 12 exam effectively?
Ans. To prepare effectively for the Math Class 12 exam, it is important to create a study schedule and allocate specific time for each topic. Practice solving a variety of problems from each topic, refer to textbooks and study guides, and make use of online resources such as video tutorials and practice tests. Additionally, reviewing previous exam papers can help understand the exam pattern and identify areas that require more focus.
 3. Are there any tips for solving calculus problems in the Math Class 12 exam?
Ans. Yes, here are some tips for solving calculus problems in the Math Class 12 exam: - Understand the concepts thoroughly and practice solving different types of calculus problems regularly. - Pay attention to the given conditions and constraints in the problem statement. - Break down complex problems into smaller steps and solve them systematically. - Check your answers for accuracy and review the solution to ensure it aligns with the problem requirements. - Practice time management to ensure you can attempt all the calculus problems within the given exam duration.
 4. How can I improve my problem-solving skills for the Math Class 12 exam?
Ans. Improving problem-solving skills for the Math Class 12 exam requires practice and a systematic approach. Here are some tips: - Understand the problem statement thoroughly before attempting to solve it. - Identify the key information and relevant concepts required to solve the problem. - Break down complex problems into smaller, manageable steps. - Use diagrams, graphs, and equations to represent the problem and its solution. - Practice solving a variety of problems from each topic, gradually increasing the difficulty level. - Seek help from teachers, classmates, or online resources if you encounter difficulties. - Regularly review and analyze your mistakes to learn from them and improve your problem-solving strategies.
 5. How important is time management during the Math Class 12 exam?
Ans. Time management is crucial during the Math Class 12 exam to ensure that you can attempt all the questions within the given duration. Here are some tips for effective time management: - Read the entire question paper carefully before starting to solve any problem. - Allocate time for each section or topic based on the marks distribution. - Prioritize easy or familiar questions to solve them quickly and gain confidence. - If you get stuck on a difficult problem, move on to the next one and come back to it later if time permits. - Keep track of time while solving each problem and make sure you leave enough time for reviewing your answers. - Avoid spending too much time on a single question; if you're unsure, make an educated guess and move forward.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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