Page 1
CBSE XII | Mathematics
Sample Paper – 8
Mathematics
Class XII
Sample Paper – 8
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. Write the position of the element 6 in the given matrix, and denote it as aij.
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
2. Find
dy
dx
, if 2x + 3y = cos x
3. Is the differential equation given by
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give
reason.
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for
the line.
OR
Find the angle between following pairs of line
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ?
SECTION – B
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}.
Write the operation table of the operation *.
6. Solve the matrix equation
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
7. Evaluate:
2
5x 2
dx
1 2x 3x
Page 2
CBSE XII | Mathematics
Sample Paper – 8
Mathematics
Class XII
Sample Paper – 8
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. Write the position of the element 6 in the given matrix, and denote it as aij.
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
2. Find
dy
dx
, if 2x + 3y = cos x
3. Is the differential equation given by
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give
reason.
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for
the line.
OR
Find the angle between following pairs of line
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ?
SECTION – B
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}.
Write the operation table of the operation *.
6. Solve the matrix equation
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
7. Evaluate:
2
5x 2
dx
1 2x 3x
CBSE XII | Mathematics
Sample Paper – 8
8. Evaluate:
2
22
x
dx
x 4 x 9
OR
Evaluate:
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being
parameters.
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
OR
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k
and c 2i j 4k .
11. 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?
12. A random variable X has the following probability distribution :
X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k
2
2k
2
7k
2
+k
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3)
OR
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find
their respective probabilities of winning, if A starts first.
SECTION – C
13. Let A = Q x Q and let * be a binary operation on A defined by
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative
and associative. Then, with respect to * on A
(i) Find the identify element in A.
(ii) Find the invertible elements of A.
Page 3
CBSE XII | Mathematics
Sample Paper – 8
Mathematics
Class XII
Sample Paper – 8
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. Write the position of the element 6 in the given matrix, and denote it as aij.
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
2. Find
dy
dx
, if 2x + 3y = cos x
3. Is the differential equation given by
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give
reason.
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for
the line.
OR
Find the angle between following pairs of line
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ?
SECTION – B
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}.
Write the operation table of the operation *.
6. Solve the matrix equation
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
7. Evaluate:
2
5x 2
dx
1 2x 3x
CBSE XII | Mathematics
Sample Paper – 8
8. Evaluate:
2
22
x
dx
x 4 x 9
OR
Evaluate:
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being
parameters.
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
OR
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k
and c 2i j 4k .
11. 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?
12. A random variable X has the following probability distribution :
X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k
2
2k
2
7k
2
+k
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3)
OR
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find
their respective probabilities of winning, if A starts first.
SECTION – C
13. Let A = Q x Q and let * be a binary operation on A defined by
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative
and associative. Then, with respect to * on A
(i) Find the identify element in A.
(ii) Find the invertible elements of A.
CBSE XII | Mathematics
Sample Paper – 8
OR
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined
by (a, b) * (c, d) = (ac, ad + b). Show that
(i) ‘*’ is not commutative
(ii) ‘*’ is associative
(iii) The identity element with respect to ‘*’ is (1, 0)
14. Write in the simplest form:
12
y cot 1 x x
?
? ? ?
??
??
??
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative
c a b
16. If sin y = x sin (a + y), prove that
? ?
2
sin a y
dy
dx sina
?
?
OR
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
17. If
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
??
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post
which is 6 m high. Find the rate at which the length of his shadow increases.
19. Evaluate:
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums.
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0.
OR
Solve: ? ?
2
2
dy
x y a
dx
??
Page 4
CBSE XII | Mathematics
Sample Paper – 8
Mathematics
Class XII
Sample Paper – 8
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. Write the position of the element 6 in the given matrix, and denote it as aij.
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
2. Find
dy
dx
, if 2x + 3y = cos x
3. Is the differential equation given by
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give
reason.
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for
the line.
OR
Find the angle between following pairs of line
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ?
SECTION – B
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}.
Write the operation table of the operation *.
6. Solve the matrix equation
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
7. Evaluate:
2
5x 2
dx
1 2x 3x
CBSE XII | Mathematics
Sample Paper – 8
8. Evaluate:
2
22
x
dx
x 4 x 9
OR
Evaluate:
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being
parameters.
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
OR
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k
and c 2i j 4k .
11. 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?
12. A random variable X has the following probability distribution :
X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k
2
2k
2
7k
2
+k
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3)
OR
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find
their respective probabilities of winning, if A starts first.
SECTION – C
13. Let A = Q x Q and let * be a binary operation on A defined by
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative
and associative. Then, with respect to * on A
(i) Find the identify element in A.
(ii) Find the invertible elements of A.
CBSE XII | Mathematics
Sample Paper – 8
OR
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined
by (a, b) * (c, d) = (ac, ad + b). Show that
(i) ‘*’ is not commutative
(ii) ‘*’ is associative
(iii) The identity element with respect to ‘*’ is (1, 0)
14. Write in the simplest form:
12
y cot 1 x x
?
? ? ?
??
??
??
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative
c a b
16. If sin y = x sin (a + y), prove that
? ?
2
sin a y
dy
dx sina
?
?
OR
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
17. If
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
??
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post
which is 6 m high. Find the rate at which the length of his shadow increases.
19. Evaluate:
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums.
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0.
OR
Solve: ? ?
2
2
dy
x y a
dx
??
CBSE XII | Mathematics
Sample Paper – 8
22.
a) If
ˆ ˆˆ
i . j.k represents the right handed system of mutually perpendicular vectors
and
ˆ ˆ ˆ ˆ ˆ
3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of
12
? ? ? ? ? where
1
? is
parallel to
2
and ?? is perpendicular to ? .
b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If
each of these is perpendicular to the sum of the other two vectors, find a b c ?? .
23.
Find the coordinates of the point, where the line
x 2 y 1 z 2
3 4 2
intersects the
plane x – y + z – 5 = 0. Also find the angle between the line and the plane.
SECTION – D
24. If
1 0 2
A 0 2 2
203
??
??
?
??
??
??
, then show that A is a root of polynomials f(x) = x
3
– 6x
2
+ 7x + 2
OR
If
1 1 0
A 0 1 1
2 3 4
?? ??
??
??
??
??
??
and
1 2 3
B 0 1 0
1 1 0
??
??
?
??
??
??
, show that AB?BA
25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius
r.
26. Find the area of the region
{(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2}
OR
Calculate the area
?
22
22
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a
ab
22
22
xy
(ii) Triangle AOB is in the first quadrant of the ellipse + 1,
ab
Where OA = a and OB = b.
Find the area enclosed between the chord AB and the arc AB of the ellipse
?
(iii) Find the ratio of the two areas found.
Page 5
CBSE XII | Mathematics
Sample Paper – 8
Mathematics
Class XII
Sample Paper – 8
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. Write the position of the element 6 in the given matrix, and denote it as aij.
1 16 8 9
7 5 3 2
4 10 6 11
??
??
??
??
??
2. Find
dy
dx
, if 2x + 3y = cos x
3. Is the differential equation given by
2
2
2
d y dy
s sy s
dx dx
?? , linear or nonlinear. Give
reason.
4. The Cartesian equations of a line are
x 5 y 4 z 6
3 7 2
? ? ?
?? . Find a vector equation for
the line.
OR
Find the angle between following pairs of line
x 4 y 1 z 3 x 1 y 4 z 5
and
3 5 4 1 1 2
? ? ? ? ? ?
? ? ? ?
SECTION – B
5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}.
Write the operation table of the operation *.
6. Solve the matrix equation
2
2
x2 x
3
2y 9 y
? ?? ? ? ? ?
??
?? ? ? ? ?
? ? ? ? ??
7. Evaluate:
2
5x 2
dx
1 2x 3x
CBSE XII | Mathematics
Sample Paper – 8
8. Evaluate:
2
22
x
dx
x 4 x 9
OR
Evaluate:
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
9. Form differential equation of the family of curves y = a sin (bx + c), a and c being
parameters.
10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k
Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD
OR
Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k
and c 2i j 4k .
11. 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?
12. A random variable X has the following probability distribution :
X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k
2
2k
2
7k
2
+k
Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3)
OR
A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find
their respective probabilities of winning, if A starts first.
SECTION – C
13. Let A = Q x Q and let * be a binary operation on A defined by
(a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative
and associative. Then, with respect to * on A
(i) Find the identify element in A.
(ii) Find the invertible elements of A.
CBSE XII | Mathematics
Sample Paper – 8
OR
Let A = Q × Q, Q being the set of rational. Let ‘*’ be a binary operation on A, defined
by (a, b) * (c, d) = (ac, ad + b). Show that
(i) ‘*’ is not commutative
(ii) ‘*’ is associative
(iii) The identity element with respect to ‘*’ is (1, 0)
14. Write in the simplest form:
12
y cot 1 x x
?
? ? ?
??
??
??
15. Let a, b, and c be positive numbers , but not equal and not all are zero.
a b c
Show that the value of the determinant b c a is negative
c a b
16. If sin y = x sin (a + y), prove that
? ?
2
sin a y
dy
dx sina
?
?
OR
If
11
x y dy
y b tan tan , find
a x dx
??
??
??
??
??
17. If
1
3
cos3x dy 3
y cos , then show that
cos x dx cosx cos3x
?
??
18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post
which is 6 m high. Find the rate at which the length of his shadow increases.
19. Evaluate:
4 2 2 4
1
dx
sin x+sin xcos x+cos x
?
20. Evaluate: ? ?
2
1
7x 5 dx,
?
?
? as a limit of sums.
21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0.
OR
Solve: ? ?
2
2
dy
x y a
dx
??
CBSE XII | Mathematics
Sample Paper – 8
22.
a) If
ˆ ˆˆ
i . j.k represents the right handed system of mutually perpendicular vectors
and
ˆ ˆ ˆ ˆ ˆ
3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of
12
? ? ? ? ? where
1
? is
parallel to
2
and ?? is perpendicular to ? .
b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If
each of these is perpendicular to the sum of the other two vectors, find a b c ?? .
23.
Find the coordinates of the point, where the line
x 2 y 1 z 2
3 4 2
intersects the
plane x – y + z – 5 = 0. Also find the angle between the line and the plane.
SECTION – D
24. If
1 0 2
A 0 2 2
203
??
??
?
??
??
??
, then show that A is a root of polynomials f(x) = x
3
– 6x
2
+ 7x + 2
OR
If
1 1 0
A 0 1 1
2 3 4
?? ??
??
??
??
??
??
and
1 2 3
B 0 1 0
1 1 0
??
??
?
??
??
??
, show that AB?BA
25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius
r.
26. Find the area of the region
{(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2}
OR
Calculate the area
?
22
22
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a
ab
22
22
xy
(ii) Triangle AOB is in the first quadrant of the ellipse + 1,
ab
Where OA = a and OB = b.
Find the area enclosed between the chord AB and the arc AB of the ellipse
?
(iii) Find the ratio of the two areas found.
CBSE XII | Mathematics
Sample Paper – 8
27. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and
B(3, -1, 2) and parallel to the line
x 4 y 3 z 1
1 4 7
? ? ?
??
?
OR
Find the equation of the line passing through the point (-1,3,-2) and perpendicular
to the lines
x y z x 2 y 1 z 1
and
1 2 3 3 2 5
.
28. A manufacturing company makes two models A and B of a product. Each piece of
Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each
piece of Model B requires 12 labour hours for fabricating and 3 labour hours for
finishing. For fabricating and finishing, the maximum labour hours available are 180
and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model
A and Rs. 12000 on each piece of Model B. How many pieces of Model A and Model B
should be manufactured per week to realize a maximum profit? What is the
maximum profit per week?
29. Two bags A and B contain 3 red and 4 black balls, and 4 red and 5 black balls
respectively. From bag A, one ball is transferred to bag B and then a ball is drawn from
bag B. The ball is found to be red in colour. Find the probability that
(a)The transferred ball is black ?
(b) The transferred ball is red ?
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