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.
Read More