Sample Question Paper 7 - Math, Class 12

# Sample Question Paper 7 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

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``` Page 1

CBSE XII | Mathematics
Sample Paper – 7

Mathematics
Class XII
Sample Paper – 7
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 =[aij], such that
ij
i 2j
a
2
?
? , find the value of element at 3
rd
column and 2
nd
row.
2. Find
dy
dx
, if y + sin y = cos x
3. Determine the order and degree of the following differential equation:
2
dy 1
2
dy
dx
dx
??
??
??
??

4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the
vector
ˆ ˆˆ
3i 2j 8k ?? .
OR

Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6).
SECTION – B

5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is
even}, is an equivalence relation.
6. Find matrix A if
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
.
7. sin x sin 2x sin 3x dx
Page 2

CBSE XII | Mathematics
Sample Paper – 7

Mathematics
Class XII
Sample Paper – 7
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 =[aij], such that
ij
i 2j
a
2
?
? , find the value of element at 3
rd
column and 2
nd
row.
2. Find
dy
dx
, if y + sin y = cos x
3. Determine the order and degree of the following differential equation:
2
dy 1
2
dy
dx
dx
??
??
??
??

4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the
vector
ˆ ˆˆ
3i 2j 8k ?? .
OR

Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6).
SECTION – B

5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is
even}, is an equivalence relation.
6. Find matrix A if
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
.
7. sin x sin 2x sin 3x dx

CBSE XII | Mathematics
Sample Paper – 7

8. Evaluate:
2
2
dx
1 x 1 x

OR
Evaluate:
sin x a
dx
sin x a

9. Form a differential equation corresponding to y
2
= a(b – x)(b + x), by eliminating
parameters a and b
10. If a 5, b 13, a b 60, find a b ? ? ? ? ?
OR
Find the value of ? which makes the vectors a, b, and c coplanar, where
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k .
11. Find the probability distribution of the number of Kings drawn when two cards are
drawn one by one, without replacement, from a pack of 52 playing cards
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls
respectively. One bag is selected at random and from the bag selected, a ball is
drawn. Find the probability that the ball is red.
OR
A company has two factories to manufacture machinery. Factory I manufactures
70% of the machinery and factory II manufactures 30% of the machinery. At factory
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of
the machinery are rated to be of a standard quality. A machine is chosen at random
and is found to be of a standard quality. What is the probability that it came from
factory II?
SECTION – C

13. Let A = R × R and * be a binary operation on A defined by
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A.

Page 3

CBSE XII | Mathematics
Sample Paper – 7

Mathematics
Class XII
Sample Paper – 7
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 =[aij], such that
ij
i 2j
a
2
?
? , find the value of element at 3
rd
column and 2
nd
row.
2. Find
dy
dx
, if y + sin y = cos x
3. Determine the order and degree of the following differential equation:
2
dy 1
2
dy
dx
dx
??
??
??
??

4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the
vector
ˆ ˆˆ
3i 2j 8k ?? .
OR

Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6).
SECTION – B

5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is
even}, is an equivalence relation.
6. Find matrix A if
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
.
7. sin x sin 2x sin 3x dx

CBSE XII | Mathematics
Sample Paper – 7

8. Evaluate:
2
2
dx
1 x 1 x

OR
Evaluate:
sin x a
dx
sin x a

9. Form a differential equation corresponding to y
2
= a(b – x)(b + x), by eliminating
parameters a and b
10. If a 5, b 13, a b 60, find a b ? ? ? ? ?
OR
Find the value of ? which makes the vectors a, b, and c coplanar, where
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k .
11. Find the probability distribution of the number of Kings drawn when two cards are
drawn one by one, without replacement, from a pack of 52 playing cards
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls
respectively. One bag is selected at random and from the bag selected, a ball is
drawn. Find the probability that the ball is red.
OR
A company has two factories to manufacture machinery. Factory I manufactures
70% of the machinery and factory II manufactures 30% of the machinery. At factory
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of
the machinery are rated to be of a standard quality. A machine is chosen at random
and is found to be of a standard quality. What is the probability that it came from
factory II?
SECTION – C

13. Let A = R × R and * be a binary operation on A defined by
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A.

CBSE XII | Mathematics
Sample Paper – 7

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
? ? ? ?
?
? ? ? ?
? ? ? ?

14. Prove that
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?

15. If a, b, and c are in A.P., find the value of the determinant
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?

16. If
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
.
OR

If
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??

17. If
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??

18. 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.
19. Evaluate:
2
x+2
dx
x +5x+6
?

20. Prove that,
0
x
dx
1 sin x
?
?
??
?

21. Solve the differential equation:
1
dy
sin x y
dx
?
??
??
??
??

Page 4

CBSE XII | Mathematics
Sample Paper – 7

Mathematics
Class XII
Sample Paper – 7
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 =[aij], such that
ij
i 2j
a
2
?
? , find the value of element at 3
rd
column and 2
nd
row.
2. Find
dy
dx
, if y + sin y = cos x
3. Determine the order and degree of the following differential equation:
2
dy 1
2
dy
dx
dx
??
??
??
??

4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the
vector
ˆ ˆˆ
3i 2j 8k ?? .
OR

Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6).
SECTION – B

5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is
even}, is an equivalence relation.
6. Find matrix A if
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
.
7. sin x sin 2x sin 3x dx

CBSE XII | Mathematics
Sample Paper – 7

8. Evaluate:
2
2
dx
1 x 1 x

OR
Evaluate:
sin x a
dx
sin x a

9. Form a differential equation corresponding to y
2
= a(b – x)(b + x), by eliminating
parameters a and b
10. If a 5, b 13, a b 60, find a b ? ? ? ? ?
OR
Find the value of ? which makes the vectors a, b, and c coplanar, where
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k .
11. Find the probability distribution of the number of Kings drawn when two cards are
drawn one by one, without replacement, from a pack of 52 playing cards
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls
respectively. One bag is selected at random and from the bag selected, a ball is
drawn. Find the probability that the ball is red.
OR
A company has two factories to manufacture machinery. Factory I manufactures
70% of the machinery and factory II manufactures 30% of the machinery. At factory
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of
the machinery are rated to be of a standard quality. A machine is chosen at random
and is found to be of a standard quality. What is the probability that it came from
factory II?
SECTION – C

13. Let A = R × R and * be a binary operation on A defined by
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A.

CBSE XII | Mathematics
Sample Paper – 7

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
? ? ? ?
?
? ? ? ?
? ? ? ?

14. Prove that
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?

15. If a, b, and c are in A.P., find the value of the determinant
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?

16. If
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
.
OR

If
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??

17. If
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??

18. 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.
19. Evaluate:
2
x+2
dx
x +5x+6
?

20. Prove that,
0
x
dx
1 sin x
?
?
??
?

21. Solve the differential equation:
1
dy
sin x y
dx
?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 7

OR
Solve the differential equation:
? ?
2 dy
4x y 1
dx
? ? ?

22. The scalar product of the vector a i j k
? ? ? ?
? ? ? with a unit vector along the sum of vectors
b 2i 4 j 5k
? ? ? ?
? ? ? and c ? i 2 j 3 k
? ? ? ?
? ? ?  is equal to one. Find the value of ? and hence find
the unit vector along b c .
??
?

23. Find the shortest distance between the lines given by
x 3 y 5 z 7
1 2 1
and
x 1 y 1 z 1
7 6 1

SECTION – D

24. If
111
A 1 1 1
111
??
??
?
??
??
??
,then prove that
n 1 n 1 n 1
n n 1 n 1 n 1
n 1 n 1 n 1
333
A 3 3 3
333
???
???
???
??
??
?
??
??
??
for every positive integer n.
OR
Let
x
0 tan
2
A
x
tan 0
2
?? ??
?
?? ??
??
??
?
??
??
?? ??
????
and I be the identity matrix of order 2.
Show that: I + A = (I – A)
cosx sinx
sinx cosx
? ??
??
??

25. A square piece of tin with side 18 cm is to be made into a box without a top by
cutting a square piece from each corner and folding up the flaps. What should be the
side of the square to be cut off, so that the volume of the box is the largest? Also find
the maximum volume of the box.

26. Prove that the curves y² = 4x and x² = 4y divide the area of the square bonded
by x = 0, x = 4, y = 4, and y = 0 into three equal parts.
Page 5

CBSE XII | Mathematics
Sample Paper – 7

Mathematics
Class XII
Sample Paper – 7
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 =[aij], such that
ij
i 2j
a
2
?
? , find the value of element at 3
rd
column and 2
nd
row.
2. Find
dy
dx
, if y + sin y = cos x
3. Determine the order and degree of the following differential equation:
2
dy 1
2
dy
dx
dx
??
??
??
??

4. Find the vector equation of line through point (5, 2, -4) and which is parallel to the
vector
ˆ ˆˆ
3i 2j 8k ?? .
OR

Find the vector equation of a line passing through the points (-1, 0, 2) and (3, 4, 6).
SECTION – B

5. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is
even}, is an equivalence relation.
6. Find matrix A if
2 3 3 6
A
1 4 3 8
? ? ? ? ?
??
? ? ? ?
??
? ? ? ?
.
7. sin x sin 2x sin 3x dx

CBSE XII | Mathematics
Sample Paper – 7

8. Evaluate:
2
2
dx
1 x 1 x

OR
Evaluate:
sin x a
dx
sin x a

9. Form a differential equation corresponding to y
2
= a(b – x)(b + x), by eliminating
parameters a and b
10. If a 5, b 13, a b 60, find a b ? ? ? ? ?
OR
Find the value of ? which makes the vectors a, b, and c coplanar, where
a =2i j+k, b=i+2j-3k
ˆ ˆ ˆ ˆ ˆ ˆ
? and
ˆ ˆˆ
c =3i - ? j + 5 k .
11. Find the probability distribution of the number of Kings drawn when two cards are
drawn one by one, without replacement, from a pack of 52 playing cards
12. Two bags I and II contain 4 red, 3 black balls and 2 red and 4 black balls
respectively. One bag is selected at random and from the bag selected, a ball is
drawn. Find the probability that the ball is red.
OR
A company has two factories to manufacture machinery. Factory I manufactures
70% of the machinery and factory II manufactures 30% of the machinery. At factory
I, 80% of the machinery are rated to be of a standard quality and at factory II 90% of
the machinery are rated to be of a standard quality. A machine is chosen at random
and is found to be of a standard quality. What is the probability that it came from
factory II?
SECTION – C

13. Let A = R × R and * be a binary operation on A defined by
(a, b) * (c, d) = (a + c, b + d) Show that * is commutative and associative.
Find the identity element for * on A. Also find the inverse of every element (a, b) ? A.

CBSE XII | Mathematics
Sample Paper – 7

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
? ? ? ?
?
? ? ? ?
? ? ? ?

14. Prove that
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?

15. If a, b, and c are in A.P., find the value of the determinant
? =
x 1 x 2 x a
x 2 x 3 x b
x 3 x 4 x c
? ? ?
? ? ?
? ? ?

16. If
22
1
22
1 x 1 x dy
y tan , 1 x 1,x 0, find
dx
1 x 1 x
?
??
? ? ? ??
? ? ? ? ?
??
? ? ?
??
??
.
OR

If
? ?
2 2 1
y dy x y
log x y 2tan ,show that
x dx x y
?
? ??
? ? ?
??
?
??

17. If
22
11
22
x y dy y
cos tan a, prove that .
x y dx x
??
?? ?
??
??
?
??

18. 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.
19. Evaluate:
2
x+2
dx
x +5x+6
?

20. Prove that,
0
x
dx
1 sin x
?
?
??
?

21. Solve the differential equation:
1
dy
sin x y
dx
?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 7

OR
Solve the differential equation:
? ?
2 dy
4x y 1
dx
? ? ?

22. The scalar product of the vector a i j k
? ? ? ?
? ? ? with a unit vector along the sum of vectors
b 2i 4 j 5k
? ? ? ?
? ? ? and c ? i 2 j 3 k
? ? ? ?
? ? ?  is equal to one. Find the value of ? and hence find
the unit vector along b c .
??
?

23. Find the shortest distance between the lines given by
x 3 y 5 z 7
1 2 1
and
x 1 y 1 z 1
7 6 1

SECTION – D

24. If
111
A 1 1 1
111
??
??
?
??
??
??
,then prove that
n 1 n 1 n 1
n n 1 n 1 n 1
n 1 n 1 n 1
333
A 3 3 3
333
???
???
???
??
??
?
??
??
??
for every positive integer n.
OR
Let
x
0 tan
2
A
x
tan 0
2
?? ??
?
?? ??
??
??
?
??
??
?? ??
????
and I be the identity matrix of order 2.
Show that: I + A = (I – A)
cosx sinx
sinx cosx
? ??
??
??

25. A square piece of tin with side 18 cm is to be made into a box without a top by
cutting a square piece from each corner and folding up the flaps. What should be the
side of the square to be cut off, so that the volume of the box is the largest? Also find
the maximum volume of the box.

26. Prove that the curves y² = 4x and x² = 4y divide the area of the square bonded
by x = 0, x = 4, y = 4, and y = 0 into three equal parts.

CBSE XII | Mathematics
Sample Paper – 7

OR
Find the area of the smaller region bounded by the ellipse
22
xy
1
94
?? and the
straight line
xy
1
32
?? .
27. Find the angle between the following pair of lines:
x 2 y 1 z 3 x 2 2y 8 z 5
and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ? ?
? ? ?

And check whether the lines are parallel or perpendicular.

OR
Write the vector equations of the following lines and hence determine the distance
between them:
x 1 y 2 z 4 x 3 y 3 z 5
;
2 3 6 4 6 12
? ? ? ? ? ?
? ? ? ?

28. A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000
bricks respectively. He receives orders from three builders P, Q and R for 15,000,
20,000 and 15,000 bricks respectively. The cost in rupees for transporting 1000
bricks to the builders from the depots are given below:
To
From
P Q R
A 40 20 30
B 20 60 40

How should the manufacturer fulfil the orders so as to keep the cost of
transportation minimum?

29. A factory manufactures screws. Machines X, Y and Z manufacture respectively 1000,
2000, and 3000 screws, of which 1%, 1.5% and 2 % of their outputs are respectively
defective. A screw is drawn at random from the product and is found to be defective.
What is the probability that it is manufactured by machine X?

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Question Paper 7 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. What topics are covered in the Class 12 Math exam?
Ans. The Class 12 Math exam covers various topics such as algebra, calculus, geometry, probability, and statistics. Students are expected to have a thorough understanding of these topics to perform well in the exam.
 2. How can I prepare effectively for the Class 12 Math exam?
Ans. To prepare effectively for the Class 12 Math exam, it is important to have a clear understanding of the concepts and practice regularly. Students can refer to textbooks, study guides, and previous year question papers. Additionally, seeking help from teachers or joining a study group can also be beneficial.
 3. Are there any important formulas that I should remember for the Class 12 Math exam?
Ans. Yes, there are several important formulas that students should remember for the Class 12 Math exam. Some of them include the quadratic formula, trigonometric identities, differentiation and integration formulas, and the formula for the area and volume of various geometric shapes. It is advisable to make a list of these formulas and revise them regularly.
 4. How can I improve my problem-solving skills for the Class 12 Math exam?
Ans. Improving problem-solving skills for the Class 12 Math exam requires regular practice. Start by solving a variety of problems from different topics and difficulty levels. Analyze the solutions and identify the areas where you face difficulties. Seek help from teachers or classmates if needed. Additionally, solving previous year question papers and mock tests can also help in enhancing problem-solving abilities.
 5. Are there any online resources or apps available for Class 12 Math exam preparation?
Ans. Yes, there are several online resources and apps available for Class 12 Math exam preparation. Some popular ones include Khan Academy, BYJU'S, Vedantu, and Meritnation. These platforms offer video lessons, practice exercises, and mock tests to help students prepare for their exams. It is important to choose a reliable and reputable resource that aligns with the curriculum and syllabus.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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