CBSE Past Year Paper Session (2014), Math Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 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
?