Page 1
CBSE XII | Mathematics
Sample Paper – 3
Mathematics
Class XII
Sample Paper 3
Time: 3 hour 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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.
2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.
3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.
4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?
Men workers Women workers
Factory I 40 15
Factory II 35 40
Factory III 72 64
Page 2
CBSE XII | Mathematics
Sample Paper – 3
Mathematics
Class XII
Sample Paper 3
Time: 3 hour 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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.
2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.
3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.
4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?
Men workers Women workers
Factory I 40 15
Factory II 35 40
Factory III 72 64
CBSE XII | Mathematics
Sample Paper – 3
OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.
SECTION – B
5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear?
8.
?
??
??
??
1
25
Write A for A = .
13
OR
If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
OR
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?
Page 3
CBSE XII | Mathematics
Sample Paper – 3
Mathematics
Class XII
Sample Paper 3
Time: 3 hour 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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.
2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.
3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.
4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?
Men workers Women workers
Factory I 40 15
Factory II 35 40
Factory III 72 64
CBSE XII | Mathematics
Sample Paper – 3
OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.
SECTION – B
5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear?
8.
?
??
??
??
1
25
Write A for A = .
13
OR
If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
OR
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?
CBSE XII | Mathematics
Sample Paper – 3
11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).
12. ?
2 1 1
If P(A)= , P(B)= , P(A B)= , then find P(A /B).
5 3 5
OR
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.
SECTION – C
13.
? ?
?
??
3
5
3
x 5 x
Differentiate , wrt x
7 3x 8 5x
OR
? ? ?
2 '' '
If y = acos(log x) + bsin(log x), prove that x y xy y 0,
14.
? ? ? ? Let f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
15.
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
16.
2 2 2 2
A plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
Page 4
CBSE XII | Mathematics
Sample Paper – 3
Mathematics
Class XII
Sample Paper 3
Time: 3 hour 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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.
2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.
3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.
4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?
Men workers Women workers
Factory I 40 15
Factory II 35 40
Factory III 72 64
CBSE XII | Mathematics
Sample Paper – 3
OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.
SECTION – B
5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear?
8.
?
??
??
??
1
25
Write A for A = .
13
OR
If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
OR
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?
CBSE XII | Mathematics
Sample Paper – 3
11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).
12. ?
2 1 1
If P(A)= , P(B)= , P(A B)= , then find P(A /B).
5 3 5
OR
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.
SECTION – C
13.
? ?
?
??
3
5
3
x 5 x
Differentiate , wrt x
7 3x 8 5x
OR
? ? ?
2 '' '
If y = acos(log x) + bsin(log x), prove that x y xy y 0,
14.
? ? ? ? Let f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
15.
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
16.
2 2 2 2
A plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
CBSE XII | Mathematics
Sample Paper – 3
17. 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?
18.
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
??
19. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at
a point, where t
2
?
?
.
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?
21. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??
OR
?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1
22. ? Show that the function f defined by f(x) = 1-x + x , x R is continuous.
OR
Show that a logarithmic function is continuous at every point in its domain.
23.
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin
Page 5
CBSE XII | Mathematics
Sample Paper – 3
Mathematics
Class XII
Sample Paper 3
Time: 3 hour 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. This graph does not represent a function. Make the required changes in this graph, and
draw the graph, so that it represents a function.
2. Find the value of ‘ ?’ for which ?(î + j +
ˆ
k ) is a unit vector.
3. Find the slope of the tangent to the curve y = x
3
– x + 1 at the point where the curve cuts
the y-axis.
4. This 3 x 2 matrix gives information about the number of men and women workers in
three factories I, II and III who lost their jobs in the last 2 months. What do you infer
from the entry in the third row and second column of this matrix?
Men workers Women workers
Factory I 40 15
Factory II 35 40
Factory III 72 64
CBSE XII | Mathematics
Sample Paper – 3
OR
If for any 2 x 2 square matrix A, A (adj A) =
80
80
??
??
??
, then write the value of
|A|.
SECTION – B
5. Evaluate:
1
1
sin sin
32
?
?? ? ??
??
?? ??
?? ??
6. Evaluate:
1
1
2x
log dx
2x
?
? ??
??
?
??
?
7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear?
8.
?
??
??
??
1
25
Write A for A = .
13
OR
If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
9.
?
?
?
?
1
2
1
Simplify cot for x < 1.
x1
10.
?
? ? ? ?
? ? ?
1
2 2 2
5x 1 1 dy 2 3
If y = tan , x < , then prove that .
dx 66 1 6x 1 4x 1 9x
OR
The volume of a cube is increasing at the rate of 9 cm
3
/s. How fast is its surface area
increasing when the length of an edge is 10 cm ?
CBSE XII | Mathematics
Sample Paper – 3
11. Obtain the differential equation of the family of circles passing through the points (a,0) and
(-a,0).
12. ?
2 1 1
If P(A)= , P(B)= , P(A B)= , then find P(A /B).
5 3 5
OR
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.
SECTION – C
13.
? ?
?
??
3
5
3
x 5 x
Differentiate , wrt x
7 3x 8 5x
OR
? ? ?
2 '' '
If y = acos(log x) + bsin(log x), prove that x y xy y 0,
14.
? ? ? ? Let f(x) x 3, g(x) = x 3; x N,
Show that (i) f is not an onto function (ii) gof is an onto function
15.
Find the distance between the parallel planes
r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0
16.
2 2 2 2
A plane is at a distance of p units from the origin.
It makes an intercept of a,b,c with the x , y and z axis repectively.
Show that it satisfies the equation:
1 1 1 1
a b c p
? ? ?
CBSE XII | Mathematics
Sample Paper – 3
17. 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?
18.
2
1 2 1 2
5
Solve the equation: (tan x) (cot x)
8
??
?
??
19. Find the equation of a tangent to the curve given by
33
x asin t , y bcos t ?? at
a point, where t
2
?
?
.
20. Evaluate:
4
0
sinx cosx
dx
9 16sin2x
?
?
?
?
21. If
cos sin
A
sin cos
?? ??
?
??
? ? ?
??
then prove that
n
cosn sinn
A ,n N
sinn cosn
?? ??
??
??
? ? ?
??
OR
?
??
? ? ? ?
??
2
2
2
If is one of the cube roots of unity, evaluate the given determinant
1
1
1
22. ? Show that the function f defined by f(x) = 1-x + x , x R is continuous.
OR
Show that a logarithmic function is continuous at every point in its domain.
23.
? ? ?
?
? ? ? ?
? 2
(3 sin 2)cos
Evaluate: d
5 cos 4sin
CBSE XII | Mathematics
Sample Paper – 3
SECTION - D
24. Show that the right circular cone of least curved surface and given volume has an
altitude equal to 2 times the radius of the base.
OR
43
Find the points at which the function f given by f(x) = (x - 2 ) (x 1) is minimum. ?
25. Obtain the inverse of the following matrix using elementary operations.
0 1 2
A= 1 2 3
3 1 1
??
??
??
??
??
26.
?
?
22
22
22
22
Calculate the area
xy
(i) between the curves + 1,and the x-axis between x = 0 to x = a
ab
xy
(ii) Triangle AOB is in the first quadrant of the ellipse + 1,where OA = a and OB = b.
ab
Find the area enclosed between the chord AB and the arc AB of the ellipse
(iii) Find the ratio of the two areas found.
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. Find the equation of a plane that is parallel to the x-axis and passes through the line
common to two intersectiing planes r. i+j+k 1 0 and r. 2i+3j-k 4
28. Two trainee carpenters A and B earn Rs. 150 and Rs. 200 per day respectively. A can
make 6 frames and 4 stools per day while B can make 10 frames and 4 stools per day.
How many days shall each work, if it is desired to produce atleast 60 frames and 32
stools at a minimum labour cost? Solve the problem graphically.
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