Page 1
CBSE XII | Mathematics
Sample Paper – 10
Mathematics
Class XII
Sample Paper – 10
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. Find the value of x, if
5 2 3 x 8 10
6 1 2 5 8 6
? ? ? ? ? ?
??
? ? ? ? ? ?
? ? ? ? ? ?
2. Differentiate cos x w.r.t. x
3. Is the differential equation given by
32
32
d y d y dy
y sin y 0
dx dx dx
? ? ? ? , linear or nonlinear.
Give reason.
4. Find the angle between following pairs of line
5 x y 3 1 z x 1 y z 5
and
2 1 3 3 2 1
? ? ? ? ?
? ? ? ?
? ? ?
OR
Find the angle between following pairs of line
x 2 y 1 z 3 x 2 2y 8 z 5
, and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ?
? ? ?
SECTION – B
5. For the power set of all subsets of a non-empty set, a relation ARB is defined if and
only if A ? B. Is R an equivalence relation on the Power set?
Page 2
CBSE XII | Mathematics
Sample Paper – 10
Mathematics
Class XII
Sample Paper – 10
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. Find the value of x, if
5 2 3 x 8 10
6 1 2 5 8 6
? ? ? ? ? ?
??
? ? ? ? ? ?
? ? ? ? ? ?
2. Differentiate cos x w.r.t. x
3. Is the differential equation given by
32
32
d y d y dy
y sin y 0
dx dx dx
? ? ? ? , linear or nonlinear.
Give reason.
4. Find the angle between following pairs of line
5 x y 3 1 z x 1 y z 5
and
2 1 3 3 2 1
? ? ? ? ?
? ? ? ?
? ? ?
OR
Find the angle between following pairs of line
x 2 y 1 z 3 x 2 2y 8 z 5
, and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ?
? ? ?
SECTION – B
5. For the power set of all subsets of a non-empty set, a relation ARB is defined if and
only if A ? B. Is R an equivalence relation on the Power set?
CBSE XII | Mathematics
Sample Paper – 10
6. Find a matrix A such that 2A – 3B + 5C = O, where B =
2 2 0
3 1 4
???
??
??
and
C =
2 0 2
7 1 6
? ??
??
??
7. Find :
22
sin x - cos x
dx
sin x cos x
?
8.
2
Find
dx
5 - 8x - x
?
OR
Evaluate:
2
4
x1
dx.
x1
?
?
?
9. Form the differential equation of parabolas having vertex at the origin and axis
along positive y axis
10. If a is a unit vector and (x a).(x a) 15 ? ? ? then find x .
OR
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
11. A company has two plants to manufacturing scooters. Plant I manufactures 70% of
the scooters and plant II manufactures 30%. At plant I, 30% of the scooters are
rated of standard quality and at plant II, 90% of the scooters are rated of standard
quality. A scooter is chosen at random and is found to be of standard quality. Find
the probability that it is manufactured by plant II.
12. In a game Rohan wins a rupee for a six and loses a rupee for any other number.
Rohan has been asked to throw the dice thrice but he has to quit when he get six.
Find his expected earnings.
Page 3
CBSE XII | Mathematics
Sample Paper – 10
Mathematics
Class XII
Sample Paper – 10
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. Find the value of x, if
5 2 3 x 8 10
6 1 2 5 8 6
? ? ? ? ? ?
??
? ? ? ? ? ?
? ? ? ? ? ?
2. Differentiate cos x w.r.t. x
3. Is the differential equation given by
32
32
d y d y dy
y sin y 0
dx dx dx
? ? ? ? , linear or nonlinear.
Give reason.
4. Find the angle between following pairs of line
5 x y 3 1 z x 1 y z 5
and
2 1 3 3 2 1
? ? ? ? ?
? ? ? ?
? ? ?
OR
Find the angle between following pairs of line
x 2 y 1 z 3 x 2 2y 8 z 5
, and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ?
? ? ?
SECTION – B
5. For the power set of all subsets of a non-empty set, a relation ARB is defined if and
only if A ? B. Is R an equivalence relation on the Power set?
CBSE XII | Mathematics
Sample Paper – 10
6. Find a matrix A such that 2A – 3B + 5C = O, where B =
2 2 0
3 1 4
???
??
??
and
C =
2 0 2
7 1 6
? ??
??
??
7. Find :
22
sin x - cos x
dx
sin x cos x
?
8.
2
Find
dx
5 - 8x - x
?
OR
Evaluate:
2
4
x1
dx.
x1
?
?
?
9. Form the differential equation of parabolas having vertex at the origin and axis
along positive y axis
10. If a is a unit vector and (x a).(x a) 15 ? ? ? then find x .
OR
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
11. A company has two plants to manufacturing scooters. Plant I manufactures 70% of
the scooters and plant II manufactures 30%. At plant I, 30% of the scooters are
rated of standard quality and at plant II, 90% of the scooters are rated of standard
quality. A scooter is chosen at random and is found to be of standard quality. Find
the probability that it is manufactured by plant II.
12. In a game Rohan wins a rupee for a six and loses a rupee for any other number.
Rohan has been asked to throw the dice thrice but he has to quit when he get six.
Find his expected earnings.
CBSE XII | Mathematics
Sample Paper – 10
OR
How many times must a fair coin be tossed so that the probability of getting atleast
one head is more than 90%.
SECTION – C
13. A relation R on the set of complex numbers is defined by
12
12
12
zz
z Rz
zz
?
?
?
Show that R is an equivalence relation.
OR
Let f: N ? N be defined by
f(n) =
if n is odd
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
Find whether the function f is bijective or not.
14. Prove the following equation.
1 1 1
1 5 2 1
2tan sec 2tan
5 7 8 4
? ? ?
?
? ? ?
15.
? ? ? ?
2
x+4 2x 2x
Prove that 2x x+4 2x 5x 4 4 x
2x 2x x+4
? ? ?
16. Differentiate
22
1 1 2
22
1 x 1 x
tan w.r.t. cos x
1 x 1 x
??
??
? ? ? ??
??
? ? ?
??
??
OR
If
33
sin t cos t dy
x , y , find
dx
cos2t cos2t
??
Page 4
CBSE XII | Mathematics
Sample Paper – 10
Mathematics
Class XII
Sample Paper – 10
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. Find the value of x, if
5 2 3 x 8 10
6 1 2 5 8 6
? ? ? ? ? ?
??
? ? ? ? ? ?
? ? ? ? ? ?
2. Differentiate cos x w.r.t. x
3. Is the differential equation given by
32
32
d y d y dy
y sin y 0
dx dx dx
? ? ? ? , linear or nonlinear.
Give reason.
4. Find the angle between following pairs of line
5 x y 3 1 z x 1 y z 5
and
2 1 3 3 2 1
? ? ? ? ?
? ? ? ?
? ? ?
OR
Find the angle between following pairs of line
x 2 y 1 z 3 x 2 2y 8 z 5
, and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ?
? ? ?
SECTION – B
5. For the power set of all subsets of a non-empty set, a relation ARB is defined if and
only if A ? B. Is R an equivalence relation on the Power set?
CBSE XII | Mathematics
Sample Paper – 10
6. Find a matrix A such that 2A – 3B + 5C = O, where B =
2 2 0
3 1 4
???
??
??
and
C =
2 0 2
7 1 6
? ??
??
??
7. Find :
22
sin x - cos x
dx
sin x cos x
?
8.
2
Find
dx
5 - 8x - x
?
OR
Evaluate:
2
4
x1
dx.
x1
?
?
?
9. Form the differential equation of parabolas having vertex at the origin and axis
along positive y axis
10. If a is a unit vector and (x a).(x a) 15 ? ? ? then find x .
OR
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
11. A company has two plants to manufacturing scooters. Plant I manufactures 70% of
the scooters and plant II manufactures 30%. At plant I, 30% of the scooters are
rated of standard quality and at plant II, 90% of the scooters are rated of standard
quality. A scooter is chosen at random and is found to be of standard quality. Find
the probability that it is manufactured by plant II.
12. In a game Rohan wins a rupee for a six and loses a rupee for any other number.
Rohan has been asked to throw the dice thrice but he has to quit when he get six.
Find his expected earnings.
CBSE XII | Mathematics
Sample Paper – 10
OR
How many times must a fair coin be tossed so that the probability of getting atleast
one head is more than 90%.
SECTION – C
13. A relation R on the set of complex numbers is defined by
12
12
12
zz
z Rz
zz
?
?
?
Show that R is an equivalence relation.
OR
Let f: N ? N be defined by
f(n) =
if n is odd
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
Find whether the function f is bijective or not.
14. Prove the following equation.
1 1 1
1 5 2 1
2tan sec 2tan
5 7 8 4
? ? ?
?
? ? ?
15.
? ? ? ?
2
x+4 2x 2x
Prove that 2x x+4 2x 5x 4 4 x
2x 2x x+4
? ? ?
16. Differentiate
22
1 1 2
22
1 x 1 x
tan w.r.t. cos x
1 x 1 x
??
??
? ? ? ??
??
? ? ?
??
??
OR
If
33
sin t cos t dy
x , y , find
dx
cos2t cos2t
??
CBSE XII | Mathematics
Sample Paper – 10
17. Differentiate
2
1
1 x 1
tan
x
?
??
??
??
??
??
w.r.t.
1
tan x
?
, x?0
18. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a
point, where t
2
?
?
.
19. Find
22
cos ?
d ?
(4 + sin ? ) ( 5 - 4 c o s ? )
?
20. Evaluate:
2
44
0
x sinxcosx
dx
sin x cos x
?
?
?
21. Solve:
22
xdy ydx x y dx ? ? ?
OR
Solve:
xx
yy
2ye dx y 2xe dy 0
??
? ? ? ??
??
??
22. Find the shortest distance between the lines
12
r and r whose vector equations are
? ? 1
ˆ ˆ ˆ ˆ ˆ
r i j 2i j k ? ? ? ? ? ?
? ? 2
ˆˆ ˆ ˆ ˆ ˆ
r 2i j k µ 3i 5j 2k ? ? ? ? ? ?
23. Find the equation of the plane determined by the point A(3, - 1, 2), B(5, 2, 4) and
C(-1, -1, 6) and hence find the distance between the plane and the point P(6, 5, 9).
SECTION – D
24. Obtain the inverse of following matrix using elementary operations
Page 5
CBSE XII | Mathematics
Sample Paper – 10
Mathematics
Class XII
Sample Paper – 10
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. Find the value of x, if
5 2 3 x 8 10
6 1 2 5 8 6
? ? ? ? ? ?
??
? ? ? ? ? ?
? ? ? ? ? ?
2. Differentiate cos x w.r.t. x
3. Is the differential equation given by
32
32
d y d y dy
y sin y 0
dx dx dx
? ? ? ? , linear or nonlinear.
Give reason.
4. Find the angle between following pairs of line
5 x y 3 1 z x 1 y z 5
and
2 1 3 3 2 1
? ? ? ? ?
? ? ? ?
? ? ?
OR
Find the angle between following pairs of line
x 2 y 1 z 3 x 2 2y 8 z 5
, and
2 7 3 1 4 4
? ? ? ? ? ? ?
? ? ?
? ? ?
SECTION – B
5. For the power set of all subsets of a non-empty set, a relation ARB is defined if and
only if A ? B. Is R an equivalence relation on the Power set?
CBSE XII | Mathematics
Sample Paper – 10
6. Find a matrix A such that 2A – 3B + 5C = O, where B =
2 2 0
3 1 4
???
??
??
and
C =
2 0 2
7 1 6
? ??
??
??
7. Find :
22
sin x - cos x
dx
sin x cos x
?
8.
2
Find
dx
5 - 8x - x
?
OR
Evaluate:
2
4
x1
dx.
x1
?
?
?
9. Form the differential equation of parabolas having vertex at the origin and axis
along positive y axis
10. If a is a unit vector and (x a).(x a) 15 ? ? ? then find x .
OR
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
11. A company has two plants to manufacturing scooters. Plant I manufactures 70% of
the scooters and plant II manufactures 30%. At plant I, 30% of the scooters are
rated of standard quality and at plant II, 90% of the scooters are rated of standard
quality. A scooter is chosen at random and is found to be of standard quality. Find
the probability that it is manufactured by plant II.
12. In a game Rohan wins a rupee for a six and loses a rupee for any other number.
Rohan has been asked to throw the dice thrice but he has to quit when he get six.
Find his expected earnings.
CBSE XII | Mathematics
Sample Paper – 10
OR
How many times must a fair coin be tossed so that the probability of getting atleast
one head is more than 90%.
SECTION – C
13. A relation R on the set of complex numbers is defined by
12
12
12
zz
z Rz
zz
?
?
?
Show that R is an equivalence relation.
OR
Let f: N ? N be defined by
f(n) =
if n is odd
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
Find whether the function f is bijective or not.
14. Prove the following equation.
1 1 1
1 5 2 1
2tan sec 2tan
5 7 8 4
? ? ?
?
? ? ?
15.
? ? ? ?
2
x+4 2x 2x
Prove that 2x x+4 2x 5x 4 4 x
2x 2x x+4
? ? ?
16. Differentiate
22
1 1 2
22
1 x 1 x
tan w.r.t. cos x
1 x 1 x
??
??
? ? ? ??
??
? ? ?
??
??
OR
If
33
sin t cos t dy
x , y , find
dx
cos2t cos2t
??
CBSE XII | Mathematics
Sample Paper – 10
17. Differentiate
2
1
1 x 1
tan
x
?
??
??
??
??
??
w.r.t.
1
tan x
?
, x?0
18. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at a
point, where t
2
?
?
.
19. Find
22
cos ?
d ?
(4 + sin ? ) ( 5 - 4 c o s ? )
?
20. Evaluate:
2
44
0
x sinxcosx
dx
sin x cos x
?
?
?
21. Solve:
22
xdy ydx x y dx ? ? ?
OR
Solve:
xx
yy
2ye dx y 2xe dy 0
??
? ? ? ??
??
??
22. Find the shortest distance between the lines
12
r and r whose vector equations are
? ? 1
ˆ ˆ ˆ ˆ ˆ
r i j 2i j k ? ? ? ? ? ?
? ? 2
ˆˆ ˆ ˆ ˆ ˆ
r 2i j k µ 3i 5j 2k ? ? ? ? ? ?
23. Find the equation of the plane determined by the point A(3, - 1, 2), B(5, 2, 4) and
C(-1, -1, 6) and hence find the distance between the plane and the point P(6, 5, 9).
SECTION – D
24. Obtain the inverse of following matrix using elementary operations
CBSE XII | Mathematics
Sample Paper – 10
0 1 2
A 1 2 3
311
??
??
?
??
??
??
OR
If
1 2 3
A 3 2 1
4 2 1
??
??
??
??
??
??
, then show that A is a root of polynomials f(x) = x
3
– 23x – 40
25.
43
Find the points of maxima and minima of the function f given by f(x) = (x - 2 ) (x 1) ?
26. Find the area of the smaller region bounded by the curves x
2
+ y
2
= 4
and y
2
= 3(2x – 1).
OR
Find the area of the region bounded by lines
5 3 3
y x 5;x y 9 0;y x
2 4 2
? ? ? ? ? ? ?
27. 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 distance of the point (2, 12, 5) from the point of intersection of the line
r 2i 4 j 2k ? 3 i 4 j 2 k
? ? ? ? ? ? ?
??
? ? ? ? ? ?
??
??
and the plane r . i 2 j k 0.
? ? ? ?
??
? ? ?
??
??
28. A co-operative society of farmers has 50 hectares of land to grow two crops X and Y.
The profit from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000
respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at
rates of Rs. 20 lakh and Rs. 10 lakh per hectare. Further, no more than Rs. 800 lakh
of herbicide should be used in order to protect fish and wild life using a pond which
collects drainage from this land. How much land should be allocated to each crop so
as to maximize the total profit of the society?
29. Two dice are rolled twice. Find the probability distribution of the random variable X,
which denotes the number of doublets. Find its mean and variance.
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