Page 1
CBSE XII | Mathematics
Board Paper 2014 – Set 3
CBSE Board
Class XII Mathematics
Board Paper 2014
Time: 3 hrs Total Marks: 100
Note:
? Please check that this question paper contains 5 printed pages.
? Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
? Please check that this question paper contains 29 questions.
? Please write down the Serial Number of the question before attempting it.
? 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question
paper only and will not write any answer on the answer-book during this period.
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A, B and C. Section
A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four
marks each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A. are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of four marks each and 2 questions of six marks each. You have to attempt only
one of the alternatives in all such questions.
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If A is a square matrix such that A
2
=A, then write the value of 7A - (I + A)
3
, where I is an
identity matrix.
2.
x-y z -1 4
If =
2x-y w 0 5
? ? ? ?
? ? ? ?
? ? ? ?
, find the value of x + y.
3.
?? 11
p
If tan x+tan y= ,xy<1
4
, then write the value of x + y + xy.
4. If
3x 7 8 7
=
-2 4 6 4
, find the value of x.
Page 2
CBSE XII | Mathematics
Board Paper 2014 – Set 3
CBSE Board
Class XII Mathematics
Board Paper 2014
Time: 3 hrs Total Marks: 100
Note:
? Please check that this question paper contains 5 printed pages.
? Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
? Please check that this question paper contains 29 questions.
? Please write down the Serial Number of the question before attempting it.
? 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question
paper only and will not write any answer on the answer-book during this period.
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A, B and C. Section
A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four
marks each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A. are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of four marks each and 2 questions of six marks each. You have to attempt only
one of the alternatives in all such questions.
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If A is a square matrix such that A
2
=A, then write the value of 7A - (I + A)
3
, where I is an
identity matrix.
2.
x-y z -1 4
If =
2x-y w 0 5
? ? ? ?
? ? ? ?
? ? ? ?
, find the value of x + y.
3.
?? 11
p
If tan x+tan y= ,xy<1
4
, then write the value of x + y + xy.
4. If
3x 7 8 7
=
-2 4 6 4
, find the value of x.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
5. If f(x) =
x
0
?
t sin t dt, write the value of f’(x).
6. Find the value of 'p' for which the vectors
ˆ ˆˆ
3i+2j+9k and
ˆ ˆˆ
i 2p j+3k ? are parallel.
7. If R = {(x, y): x + 2y = 8} is a relation on N, write the range of R.
8. If the cartesian equations of a line are ??
3-x y+4 2z-6
5 7 4
, write the vector equation for the
line.
9.
a
2
0
1 p
If dx=
4+x 8
?
find the value of a.
10. If a
?
and b
?
are perpendicular vectors, a b 13
??
?? and a5
?
? and find the value of
?
b.
SECTION – B
11. Solve the differential equation
-1
2 tan x
dy
(1 x ) y =e .
dx
??
12. Show that the four points A, B, C and D with position vectors
4i 5 j k , j k ,3i 9 j 4k
? ? ? ? ? ? ? ?
? ? ? ? ? ? and 4( i j k)
? ? ?
? ? ? respectively are coplanar.
OR
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 .
??
?
13. Evaluate:
p
2
0
4x sin x
dx
1+ cos x
?
OR
Evaluate:
2
x+2
dx
x +5x+6
?
Page 3
CBSE XII | Mathematics
Board Paper 2014 – Set 3
CBSE Board
Class XII Mathematics
Board Paper 2014
Time: 3 hrs Total Marks: 100
Note:
? Please check that this question paper contains 5 printed pages.
? Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
? Please check that this question paper contains 29 questions.
? Please write down the Serial Number of the question before attempting it.
? 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question
paper only and will not write any answer on the answer-book during this period.
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A, B and C. Section
A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four
marks each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A. are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of four marks each and 2 questions of six marks each. You have to attempt only
one of the alternatives in all such questions.
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If A is a square matrix such that A
2
=A, then write the value of 7A - (I + A)
3
, where I is an
identity matrix.
2.
x-y z -1 4
If =
2x-y w 0 5
? ? ? ?
? ? ? ?
? ? ? ?
, find the value of x + y.
3.
?? 11
p
If tan x+tan y= ,xy<1
4
, then write the value of x + y + xy.
4. If
3x 7 8 7
=
-2 4 6 4
, find the value of x.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
5. If f(x) =
x
0
?
t sin t dt, write the value of f’(x).
6. Find the value of 'p' for which the vectors
ˆ ˆˆ
3i+2j+9k and
ˆ ˆˆ
i 2p j+3k ? are parallel.
7. If R = {(x, y): x + 2y = 8} is a relation on N, write the range of R.
8. If the cartesian equations of a line are ??
3-x y+4 2z-6
5 7 4
, write the vector equation for the
line.
9.
a
2
0
1 p
If dx=
4+x 8
?
find the value of a.
10. If a
?
and b
?
are perpendicular vectors, a b 13
??
?? and a5
?
? and find the value of
?
b.
SECTION – B
11. Solve the differential equation
-1
2 tan x
dy
(1 x ) y =e .
dx
??
12. Show that the four points A, B, C and D with position vectors
4i 5 j k , j k ,3i 9 j 4k
? ? ? ? ? ? ? ?
? ? ? ? ? ? and 4( i j k)
? ? ?
? ? ? respectively are coplanar.
OR
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 .
??
?
13. Evaluate:
p
2
0
4x sin x
dx
1+ cos x
?
OR
Evaluate:
2
x+2
dx
x +5x+6
?
CBSE XII | Mathematics
Board Paper 2014 – Set 3
14. Find the value(s) of x for which y = [x (x - 2)]
2
is an increasing function.
OR
Find the equations of the tangent and normal to the curve ??
22
22
xy
1
ab
at the point
? ?
2 a,b .
15. If the function ? f : R R be given by f (x) = x
2
+ 2 and ? g: R R be given by
? ?
x
g x ,x 1,
x-1
?? find fog and gof and hence find fog (2) and gof ( -3).
16. Prove that
11
1 x 1 x p 1 - 1
tan cos x, x 1
42
1+x 1 x 2
??
??
? ? ?
? ? ? ?
??
??
??
OR
-1 -1
x-2 x+2 p
If tan +tan
x-4 x+4 4
??
??
?
?? ??
??
??
, find the value of x.
17. An experiment succeeds thrice as often as it fails. Find the probability that in the next five
trials, there will be at least 3 successes.
18. If y = Pe
ax
+ Q e
hx
, show that
2
2
d y dy
(a+b) + aby=0
dx dx
?
19. Using properties of determinants, prove that:
?
?
?
1 a 1 1
1 1+b 1 abc + bc + ca + ab
1 1 1 c
20. If x = cost (3 - 2 cos
2
t) and y = sin t (3 - 2 sin
2
t), find the value of
dy p
at t = .
dx 4
21. Find the particular solution of the differential equation log
??
?
??
??
dy
3x + 4y,
dx
given that y =
0 when x = 0.
Page 4
CBSE XII | Mathematics
Board Paper 2014 – Set 3
CBSE Board
Class XII Mathematics
Board Paper 2014
Time: 3 hrs Total Marks: 100
Note:
? Please check that this question paper contains 5 printed pages.
? Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
? Please check that this question paper contains 29 questions.
? Please write down the Serial Number of the question before attempting it.
? 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question
paper only and will not write any answer on the answer-book during this period.
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A, B and C. Section
A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four
marks each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A. are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of four marks each and 2 questions of six marks each. You have to attempt only
one of the alternatives in all such questions.
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If A is a square matrix such that A
2
=A, then write the value of 7A - (I + A)
3
, where I is an
identity matrix.
2.
x-y z -1 4
If =
2x-y w 0 5
? ? ? ?
? ? ? ?
? ? ? ?
, find the value of x + y.
3.
?? 11
p
If tan x+tan y= ,xy<1
4
, then write the value of x + y + xy.
4. If
3x 7 8 7
=
-2 4 6 4
, find the value of x.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
5. If f(x) =
x
0
?
t sin t dt, write the value of f’(x).
6. Find the value of 'p' for which the vectors
ˆ ˆˆ
3i+2j+9k and
ˆ ˆˆ
i 2p j+3k ? are parallel.
7. If R = {(x, y): x + 2y = 8} is a relation on N, write the range of R.
8. If the cartesian equations of a line are ??
3-x y+4 2z-6
5 7 4
, write the vector equation for the
line.
9.
a
2
0
1 p
If dx=
4+x 8
?
find the value of a.
10. If a
?
and b
?
are perpendicular vectors, a b 13
??
?? and a5
?
? and find the value of
?
b.
SECTION – B
11. Solve the differential equation
-1
2 tan x
dy
(1 x ) y =e .
dx
??
12. Show that the four points A, B, C and D with position vectors
4i 5 j k , j k ,3i 9 j 4k
? ? ? ? ? ? ? ?
? ? ? ? ? ? and 4( i j k)
? ? ?
? ? ? respectively are coplanar.
OR
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 .
??
?
13. Evaluate:
p
2
0
4x sin x
dx
1+ cos x
?
OR
Evaluate:
2
x+2
dx
x +5x+6
?
CBSE XII | Mathematics
Board Paper 2014 – Set 3
14. Find the value(s) of x for which y = [x (x - 2)]
2
is an increasing function.
OR
Find the equations of the tangent and normal to the curve ??
22
22
xy
1
ab
at the point
? ?
2 a,b .
15. If the function ? f : R R be given by f (x) = x
2
+ 2 and ? g: R R be given by
? ?
x
g x ,x 1,
x-1
?? find fog and gof and hence find fog (2) and gof ( -3).
16. Prove that
11
1 x 1 x p 1 - 1
tan cos x, x 1
42
1+x 1 x 2
??
??
? ? ?
? ? ? ?
??
??
??
OR
-1 -1
x-2 x+2 p
If tan +tan
x-4 x+4 4
??
??
?
?? ??
??
??
, find the value of x.
17. An experiment succeeds thrice as often as it fails. Find the probability that in the next five
trials, there will be at least 3 successes.
18. If y = Pe
ax
+ Q e
hx
, show that
2
2
d y dy
(a+b) + aby=0
dx dx
?
19. Using properties of determinants, prove that:
?
?
?
1 a 1 1
1 1+b 1 abc + bc + ca + ab
1 1 1 c
20. If x = cost (3 - 2 cos
2
t) and y = sin t (3 - 2 sin
2
t), find the value of
dy p
at t = .
dx 4
21. Find the particular solution of the differential equation log
??
?
??
??
dy
3x + 4y,
dx
given that y =
0 when x = 0.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
22. Find the value of p, so that the lines
? ? ?
??
1
1 x 7y 14 3
:
3 p 2
z
l and
2
7 7x y 5 6
:
3p 1 5
? ? ?
??
z
l
are perpendicular to each other. Also find the equations of a line passing through a point (3,
2, - 4) and parallel to line l 1.
SECTION – C
23. 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.
? ? ? ?
??
? ? ?
??
??
24. Using integration, find the area of the region bounded by the triangle whose
vertices are (-1, 2), (1, 5) and (3, 4).
25. A manufacturing company makes two types of teaching aids A and B of Mathematics for
class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for
finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours
for finishing. For fabricating and finishing, the maximum labour hours available per week
are 180 and 30 respectively. The company makes a profit of 80 on each piece of type A and
120 on each piece of type B. How many pieces of type A and type B should be manufactured
per week to get a maximum profit? Make it as an LPP and solve graphically. What is the
maximum profit per week?
26. There are three coins. One is a two-headed coin (having head on both faces), another is a
biased coin that comes up heads 75% of the times and third is also a biased coin that
comes up tails 40% of the times. One of The three coins is chosen at random and tossed,
and it shows heads. What is the probability that it was the two-headed coin?
OR
Two numbers are selected at random (without replacement) from the first six positive
integers. Let X denote the larger of the two numbers obtained. Find the probability
distribution of the random variable X, and hence find the mean of the distribution.
Page 5
CBSE XII | Mathematics
Board Paper 2014 – Set 3
CBSE Board
Class XII Mathematics
Board Paper 2014
Time: 3 hrs Total Marks: 100
Note:
? Please check that this question paper contains 5 printed pages.
? Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
? Please check that this question paper contains 29 questions.
? Please write down the Serial Number of the question before attempting it.
? 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question
paper only and will not write any answer on the answer-book during this period.
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A, B and C. Section
A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four
marks each and Section C comprises of 7 questions of six marks each.
3. All questions in Section A. are to be answered in one word, one sentence or as per the exact
requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of four marks each and 2 questions of six marks each. You have to attempt only
one of the alternatives in all such questions.
5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.
SECTION – A
1. If A is a square matrix such that A
2
=A, then write the value of 7A - (I + A)
3
, where I is an
identity matrix.
2.
x-y z -1 4
If =
2x-y w 0 5
? ? ? ?
? ? ? ?
? ? ? ?
, find the value of x + y.
3.
?? 11
p
If tan x+tan y= ,xy<1
4
, then write the value of x + y + xy.
4. If
3x 7 8 7
=
-2 4 6 4
, find the value of x.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
5. If f(x) =
x
0
?
t sin t dt, write the value of f’(x).
6. Find the value of 'p' for which the vectors
ˆ ˆˆ
3i+2j+9k and
ˆ ˆˆ
i 2p j+3k ? are parallel.
7. If R = {(x, y): x + 2y = 8} is a relation on N, write the range of R.
8. If the cartesian equations of a line are ??
3-x y+4 2z-6
5 7 4
, write the vector equation for the
line.
9.
a
2
0
1 p
If dx=
4+x 8
?
find the value of a.
10. If a
?
and b
?
are perpendicular vectors, a b 13
??
?? and a5
?
? and find the value of
?
b.
SECTION – B
11. Solve the differential equation
-1
2 tan x
dy
(1 x ) y =e .
dx
??
12. Show that the four points A, B, C and D with position vectors
4i 5 j k , j k ,3i 9 j 4k
? ? ? ? ? ? ? ?
? ? ? ? ? ? and 4( i j k)
? ? ?
? ? ? respectively are coplanar.
OR
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 .
??
?
13. Evaluate:
p
2
0
4x sin x
dx
1+ cos x
?
OR
Evaluate:
2
x+2
dx
x +5x+6
?
CBSE XII | Mathematics
Board Paper 2014 – Set 3
14. Find the value(s) of x for which y = [x (x - 2)]
2
is an increasing function.
OR
Find the equations of the tangent and normal to the curve ??
22
22
xy
1
ab
at the point
? ?
2 a,b .
15. If the function ? f : R R be given by f (x) = x
2
+ 2 and ? g: R R be given by
? ?
x
g x ,x 1,
x-1
?? find fog and gof and hence find fog (2) and gof ( -3).
16. Prove that
11
1 x 1 x p 1 - 1
tan cos x, x 1
42
1+x 1 x 2
??
??
? ? ?
? ? ? ?
??
??
??
OR
-1 -1
x-2 x+2 p
If tan +tan
x-4 x+4 4
??
??
?
?? ??
??
??
, find the value of x.
17. An experiment succeeds thrice as often as it fails. Find the probability that in the next five
trials, there will be at least 3 successes.
18. If y = Pe
ax
+ Q e
hx
, show that
2
2
d y dy
(a+b) + aby=0
dx dx
?
19. Using properties of determinants, prove that:
?
?
?
1 a 1 1
1 1+b 1 abc + bc + ca + ab
1 1 1 c
20. If x = cost (3 - 2 cos
2
t) and y = sin t (3 - 2 sin
2
t), find the value of
dy p
at t = .
dx 4
21. Find the particular solution of the differential equation log
??
?
??
??
dy
3x + 4y,
dx
given that y =
0 when x = 0.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
22. Find the value of p, so that the lines
? ? ?
??
1
1 x 7y 14 3
:
3 p 2
z
l and
2
7 7x y 5 6
:
3p 1 5
? ? ?
??
z
l
are perpendicular to each other. Also find the equations of a line passing through a point (3,
2, - 4) and parallel to line l 1.
SECTION – C
23. 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.
? ? ? ?
??
? ? ?
??
??
24. Using integration, find the area of the region bounded by the triangle whose
vertices are (-1, 2), (1, 5) and (3, 4).
25. A manufacturing company makes two types of teaching aids A and B of Mathematics for
class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for
finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours
for finishing. For fabricating and finishing, the maximum labour hours available per week
are 180 and 30 respectively. The company makes a profit of 80 on each piece of type A and
120 on each piece of type B. How many pieces of type A and type B should be manufactured
per week to get a maximum profit? Make it as an LPP and solve graphically. What is the
maximum profit per week?
26. There are three coins. One is a two-headed coin (having head on both faces), another is a
biased coin that comes up heads 75% of the times and third is also a biased coin that
comes up tails 40% of the times. One of The three coins is chosen at random and tossed,
and it shows heads. What is the probability that it was the two-headed coin?
OR
Two numbers are selected at random (without replacement) from the first six positive
integers. Let X denote the larger of the two numbers obtained. Find the probability
distribution of the random variable X, and hence find the mean of the distribution.
CBSE XII | Mathematics
Board Paper 2014 – Set 3
27. Two schools A and B want to award their selected students on the values of sincerity,
truthfulness and helpfulness. The school A wants to award x each, y each and z each for
the three respective values to 3, 2 and 1 students respectively with a total award money
of 1,600. School B wants to spend 2,300 to award its 4, 1 and 3 students on the
respective values (by giving the same award money to the three values as before). If the
total amount for one prize on each value is 900, using matrices, find the award money for
each value. Apart from these three values, suggest one more value which should be
considered for award.
28. If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that
the area of the triangle is maximum, when the angle between them is 60°.
29. Evaluate:
4 2 2 4
1
dx
sin x+sin xcos x+cos x
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