Sample Question Paper 5 - Math, Class 12

# Sample Question Paper 5 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

``` Page 1

CBSE XII | Mathematics
Sample Paper -5

Mathematics
Class XII
Sample Paper -5
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. If A =
cos sin
sin cos
? ? ? ??
??
??
??
, then for what value of ? is A an identity matrix?

2. Determine the value of constant k, so that the function

2
x 25
f(x) x 5
x5
k x 5
?
??
?
??

is continuous at x=5

OR

Determine the value of ‘k’ for which the following function is continuous at x =3:
? ?
2
x 3 -36
, x 3
f(x)
x -3
k , x 3
?
?
?
?
?
?
?
?
?

3. Find
2
dx
9 25x ?
?

4. Find the equation of a line through (-2, 1, 3) and parallel to
x 3 y 4 z 8
3 5 6
? ? ?
?? .

Page 2

CBSE XII | Mathematics
Sample Paper -5

Mathematics
Class XII
Sample Paper -5
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. If A =
cos sin
sin cos
? ? ? ??
??
??
??
, then for what value of ? is A an identity matrix?

2. Determine the value of constant k, so that the function

2
x 25
f(x) x 5
x5
k x 5
?
??
?
??

is continuous at x=5

OR

Determine the value of ‘k’ for which the following function is continuous at x =3:
? ?
2
x 3 -36
, x 3
f(x)
x -3
k , x 3
?
?
?
?
?
?
?
?
?

3. Find
2
dx
9 25x ?
?

4. Find the equation of a line through (-2, 1, 3) and parallel to
x 3 y 4 z 8
3 5 6
? ? ?
?? .

CBSE XII | Mathematics
Sample Paper -5

Section-B
5. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

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

6. Find the point on the curve x
2
+ y
2
– 2x – 3 = 0,  at which the tangents are parallel  to x-
axis

7. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a point,
where t
2
?
?
.
8. A balloon, which always remains spherical on inflation, is being by inflated by
pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius
of the balloon increases when the radius is 15 cm.
OR
The total cost C(x) associated with the production of x units of an item is given by
C(x) = 0.005x
3
– 0.02x
2
+ 30x + 5000. Find the marginal cost when 3 units are
produced, where by marginal cost we mean the instantaneous rate of change of total
cost at any level of output.

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

10.  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.

11. Find ? if the vectors
??
? ? ? ? ? ? ? a i j 3k andb 4i 5j 2k are perpendicular to each other.
OR
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin
??
? ? ? ?

Page 3

CBSE XII | Mathematics
Sample Paper -5

Mathematics
Class XII
Sample Paper -5
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. If A =
cos sin
sin cos
? ? ? ??
??
??
??
, then for what value of ? is A an identity matrix?

2. Determine the value of constant k, so that the function

2
x 25
f(x) x 5
x5
k x 5
?
??
?
??

is continuous at x=5

OR

Determine the value of ‘k’ for which the following function is continuous at x =3:
? ?
2
x 3 -36
, x 3
f(x)
x -3
k , x 3
?
?
?
?
?
?
?
?
?

3. Find
2
dx
9 25x ?
?

4. Find the equation of a line through (-2, 1, 3) and parallel to
x 3 y 4 z 8
3 5 6
? ? ?
?? .

CBSE XII | Mathematics
Sample Paper -5

Section-B
5. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

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

6. Find the point on the curve x
2
+ y
2
– 2x – 3 = 0,  at which the tangents are parallel  to x-
axis

7. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a point,
where t
2
?
?
.
8. A balloon, which always remains spherical on inflation, is being by inflated by
pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius
of the balloon increases when the radius is 15 cm.
OR
The total cost C(x) associated with the production of x units of an item is given by
C(x) = 0.005x
3
– 0.02x
2
+ 30x + 5000. Find the marginal cost when 3 units are
produced, where by marginal cost we mean the instantaneous rate of change of total
cost at any level of output.

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

10.  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.

11. Find ? if the vectors
??
? ? ? ? ? ? ? a i j 3k andb 4i 5j 2k are perpendicular to each other.
OR
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin
??
? ? ? ?

CBSE XII | Mathematics
Sample Paper -5

12.  Evaluate:
? ?
? ?
?
?
?
cos x a
dx
sin x b

Section C

13.  Solve the following equation:
? ? ?
?? ? ? ? ? ? ?
??
? ? ? ? ? ?
??
? ? ? ? ? ?
1 1 1
x 1 2x 1 23
tan tan tan
x 1 2x 1 36

14. Using elementary transformations, find the inverse of the matrix
1 3 2
3 0 1
2 1 0
? ??
??
??
??
??
??

OR
Using matrices solve the following system of linear equations:

x y 2z 7
3x 4y 5z 5
2x y 3z 12

15. Using properties of determinants prove the following:

3 3 3
1 1 1
a b c
a b c
= (a – b) (b – c) (c – a) (a + b + c)
OR

Find the area bounded by the curve y = 2 x – x
2
and the line y = - x.

16.  Evaluate: ? ?
2
0
2log sin x log sin2x dx
?
?
?

17.  Evaluate:
2
44
0
xsinxcosx
dx
sin x cos
?
?
?

OR
Page 4

CBSE XII | Mathematics
Sample Paper -5

Mathematics
Class XII
Sample Paper -5
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. If A =
cos sin
sin cos
? ? ? ??
??
??
??
, then for what value of ? is A an identity matrix?

2. Determine the value of constant k, so that the function

2
x 25
f(x) x 5
x5
k x 5
?
??
?
??

is continuous at x=5

OR

Determine the value of ‘k’ for which the following function is continuous at x =3:
? ?
2
x 3 -36
, x 3
f(x)
x -3
k , x 3
?
?
?
?
?
?
?
?
?

3. Find
2
dx
9 25x ?
?

4. Find the equation of a line through (-2, 1, 3) and parallel to
x 3 y 4 z 8
3 5 6
? ? ?
?? .

CBSE XII | Mathematics
Sample Paper -5

Section-B
5. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

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

6. Find the point on the curve x
2
+ y
2
– 2x – 3 = 0,  at which the tangents are parallel  to x-
axis

7. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a point,
where t
2
?
?
.
8. A balloon, which always remains spherical on inflation, is being by inflated by
pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius
of the balloon increases when the radius is 15 cm.
OR
The total cost C(x) associated with the production of x units of an item is given by
C(x) = 0.005x
3
– 0.02x
2
+ 30x + 5000. Find the marginal cost when 3 units are
produced, where by marginal cost we mean the instantaneous rate of change of total
cost at any level of output.

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

10.  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.

11. Find ? if the vectors
??
? ? ? ? ? ? ? a i j 3k andb 4i 5j 2k are perpendicular to each other.
OR
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin
??
? ? ? ?

CBSE XII | Mathematics
Sample Paper -5

12.  Evaluate:
? ?
? ?
?
?
?
cos x a
dx
sin x b

Section C

13.  Solve the following equation:
? ? ?
?? ? ? ? ? ? ?
??
? ? ? ? ? ?
??
? ? ? ? ? ?
1 1 1
x 1 2x 1 23
tan tan tan
x 1 2x 1 36

14. Using elementary transformations, find the inverse of the matrix
1 3 2
3 0 1
2 1 0
? ??
??
??
??
??
??

OR
Using matrices solve the following system of linear equations:

x y 2z 7
3x 4y 5z 5
2x y 3z 12

15. Using properties of determinants prove the following:

3 3 3
1 1 1
a b c
a b c
= (a – b) (b – c) (c – a) (a + b + c)
OR

Find the area bounded by the curve y = 2 x – x
2
and the line y = - x.

16.  Evaluate: ? ?
2
0
2log sin x log sin2x dx
?
?
?

17.  Evaluate:
2
44
0
xsinxcosx
dx
sin x cos
?
?
?

OR

CBSE XII | Mathematics
Sample Paper -5

Find
? ?
4
2
x dx
x 1)(x 1) ??
?

18.

?
? ? ?
y
dy
Solve the given differential equation: (x + 1 ) 2e 1if y(0) 0
dx

19.

The vector equations of two lines are:
? ?
ˆˆ ˆ ˆ ˆ ˆ
r = i + 2j +3k + ? i - 3 j + 2 k and
? ?
ˆˆ ˆ ˆ ˆ ˆ
r = 4i +5j + 6k + µ 2 i - 3 j + k
Find the shortest distance between the above lines.

20.  Find the vector and Cartesian equation of the plane through
ˆ ˆˆ
3i j 2k ?? and parallel to
the lines
? ?
? ?
ˆˆ ˆ ˆ ˆ
r j 3k 2i 5j k
ˆ ˆ ˆ ˆ ˆ
r i 3j k µ 5i 4j
? ? ? ? ? ? ?
? ? ? ? ? ?

21. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The probability of an accident involving a scooter, a car and a truck are 0.01,
0.03 and 0.15 respectively. One of the insured persons meets with an accident. What
is the probability that he is a scooter driver?

22. Two numbers are selected at random (without replacement) from the first six positive
integers. Let X denote the larger of the two numbers obtained.  Find the probability
distribution of the random variable X, and hence find the mean of the distribution.

23. A small firm manufactures gold rings and chains. The total number of rings and chains
manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to
make a chain. The maximum number of hours available per day is 16. If the profit on a
ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that
should be manufactured per day, so as to earn the maximum profit. Make it as an
L.P.P. and solve it graphically.

Section D

24. A school wants to award its students for the values of Honesty, Regularity and Hard
work with a total cash award of Rs 6,000. Three times the award money for Hard
work added to that given for honesty amounts to Rs 11,000. The award money given
for Honesty and Hard work together is double the one given for Regularity. Represent
the above situation algebraically and find the award money for each value, using
matrix method. Apart from these values, namely, Honesty, Regularity and Hard work,
suggest one more value which the school must include for awards.
Page 5

CBSE XII | Mathematics
Sample Paper -5

Mathematics
Class XII
Sample Paper -5
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. If A =
cos sin
sin cos
? ? ? ??
??
??
??
, then for what value of ? is A an identity matrix?

2. Determine the value of constant k, so that the function

2
x 25
f(x) x 5
x5
k x 5
?
??
?
??

is continuous at x=5

OR

Determine the value of ‘k’ for which the following function is continuous at x =3:
? ?
2
x 3 -36
, x 3
f(x)
x -3
k , x 3
?
?
?
?
?
?
?
?
?

3. Find
2
dx
9 25x ?
?

4. Find the equation of a line through (-2, 1, 3) and parallel to
x 3 y 4 z 8
3 5 6
? ? ?
?? .

CBSE XII | Mathematics
Sample Paper -5

Section-B
5. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??

OR

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

6. Find the point on the curve x
2
+ y
2
– 2x – 3 = 0,  at which the tangents are parallel  to x-
axis

7. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a point,
where t
2
?
?
.
8. A balloon, which always remains spherical on inflation, is being by inflated by
pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius
of the balloon increases when the radius is 15 cm.
OR
The total cost C(x) associated with the production of x units of an item is given by
C(x) = 0.005x
3
– 0.02x
2
+ 30x + 5000. Find the marginal cost when 3 units are
produced, where by marginal cost we mean the instantaneous rate of change of total
cost at any level of output.

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

10.  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.

11. Find ? if the vectors
??
? ? ? ? ? ? ? a i j 3k andb 4i 5j 2k are perpendicular to each other.
OR
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin
??
? ? ? ?

CBSE XII | Mathematics
Sample Paper -5

12.  Evaluate:
? ?
? ?
?
?
?
cos x a
dx
sin x b

Section C

13.  Solve the following equation:
? ? ?
?? ? ? ? ? ? ?
??
? ? ? ? ? ?
??
? ? ? ? ? ?
1 1 1
x 1 2x 1 23
tan tan tan
x 1 2x 1 36

14. Using elementary transformations, find the inverse of the matrix
1 3 2
3 0 1
2 1 0
? ??
??
??
??
??
??

OR
Using matrices solve the following system of linear equations:

x y 2z 7
3x 4y 5z 5
2x y 3z 12

15. Using properties of determinants prove the following:

3 3 3
1 1 1
a b c
a b c
= (a – b) (b – c) (c – a) (a + b + c)
OR

Find the area bounded by the curve y = 2 x – x
2
and the line y = - x.

16.  Evaluate: ? ?
2
0
2log sin x log sin2x dx
?
?
?

17.  Evaluate:
2
44
0
xsinxcosx
dx
sin x cos
?
?
?

OR

CBSE XII | Mathematics
Sample Paper -5

Find
? ?
4
2
x dx
x 1)(x 1) ??
?

18.

?
? ? ?
y
dy
Solve the given differential equation: (x + 1 ) 2e 1if y(0) 0
dx

19.

The vector equations of two lines are:
? ?
ˆˆ ˆ ˆ ˆ ˆ
r = i + 2j +3k + ? i - 3 j + 2 k and
? ?
ˆˆ ˆ ˆ ˆ ˆ
r = 4i +5j + 6k + µ 2 i - 3 j + k
Find the shortest distance between the above lines.

20.  Find the vector and Cartesian equation of the plane through
ˆ ˆˆ
3i j 2k ?? and parallel to
the lines
? ?
? ?
ˆˆ ˆ ˆ ˆ
r j 3k 2i 5j k
ˆ ˆ ˆ ˆ ˆ
r i 3j k µ 5i 4j
? ? ? ? ? ? ?
? ? ? ? ? ?

21. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The probability of an accident involving a scooter, a car and a truck are 0.01,
0.03 and 0.15 respectively. One of the insured persons meets with an accident. What
is the probability that he is a scooter driver?

22. Two numbers are selected at random (without replacement) from the first six positive
integers. Let X denote the larger of the two numbers obtained.  Find the probability
distribution of the random variable X, and hence find the mean of the distribution.

23. A small firm manufactures gold rings and chains. The total number of rings and chains
manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to
make a chain. The maximum number of hours available per day is 16. If the profit on a
ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that
should be manufactured per day, so as to earn the maximum profit. Make it as an
L.P.P. and solve it graphically.

Section D

24. A school wants to award its students for the values of Honesty, Regularity and Hard
work with a total cash award of Rs 6,000. Three times the award money for Hard
work added to that given for honesty amounts to Rs 11,000. The award money given
for Honesty and Hard work together is double the one given for Regularity. Represent
the above situation algebraically and find the award money for each value, using
matrix method. Apart from these values, namely, Honesty, Regularity and Hard work,
suggest one more value which the school must include for awards.

CBSE XII | Mathematics
Sample Paper -5

25. Obtain the differential equation of all the circles touching the x-axis at the origin.

26.  A nutritionist has to develop a special diet using two foods P and Q. Each packet
(containing 30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of
cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q
contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of
vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and
at most 300 units of cholesterol. How many packet of each food should be used to
minimize the amount of vitamin A in the diet? What is the minimum amount of
vitamin A?

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.
?
? ? ?
y
dy
Solve the given differential equation: (x + 1 ) 2e 1if y(0) 0
dx

28. Find  the  equation of the  plane  through the  line  of intersection of the planes  x + y + z =
1 and   2x + 3y + 4z = 5  which is perpendicular to the plane  x- y + z  = 0. Also find the
distance of the plane obtained above, from the origin.

OR
Find the angle between the lines whose direction cosines are given by the equations:
3l + m + 5n = 0; 6mn - 2nl + 5lm = 0

29.
? ? ? ? ? ? ? ?
? ?
Let N be the set of all natural numbers and let R be the relation on N N
defined by a,b R c,d ad bc for all a,b , c,d N N.
Show that R is an equivalence relation on N N. Also find the equivalence class 2,6.
?
? ? ? ?
?? ?
??

OR
-1 -1
4 4 4x+3
Consider f : R - - R -  given by f(x) =  Show that f is bijective.
3 3 3x+4
Find the inverse of f and hence find f (0) and x such that f (x)=2
? ? ? ?
?
? ? ? ?
? ? ? ?

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Question Paper 5 - 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, coordinate geometry, trigonometry, and vectors. It is essential to have a thorough understanding of these topics to perform well in the exam.
 2. How can I improve my problem-solving skills in Math?
Ans. To improve problem-solving skills in Math, practice is key. Solve a variety of mathematical problems from different sources, such as textbooks, previous years' question papers, and online resources. Additionally, seek guidance from teachers or tutors if you face any difficulties in understanding or solving specific problems.
 3. Are there any specific tips for preparing for the Math exam?
Ans. Yes, here are a few tips for preparing for the Math exam: - Understand the concepts thoroughly and make sure to clarify any doubts. - Practice regularly and solve different types of problems. - Create a study schedule and allocate specific time for each topic. - Revise regularly and solve previous years' question papers to familiarize yourself with the exam pattern. - Stay calm and confident during the exam.
 4. Where can I find additional study materials for Class 12 Math?
Ans. Additional study materials for Class 12 Math can be found in various sources such as textbooks, reference books, online educational platforms, and websites dedicated to Math learning. Additionally, you can also seek guidance from teachers, join study groups, or participate in online forums to gain more resources and insights.
 5. How can I manage my time effectively during the Math exam?
Ans. To manage time effectively during the Math exam, it is crucial to practice solving problems within a limited time frame. While practicing, set a timer and try to complete the problems within the given time. This will help you develop a sense of time management. During the exam, read the questions carefully, prioritize the ones you are confident about, and allocate time accordingly. Avoid spending too much time on a single question and move on to the next if you are unsure.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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