CBSE Past Year Paper Session (2016), Math Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2016  
All India   
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. Please check that this question paper contains 26 questions. 
3. Question 1 to 6 in Section A are very short – answer type questions carrying 1 
mark each. 
4. Questions 7 to 19 in Section B are long – answer I type question carrying 4 
mark each. 
5. Questions 20 to 26 in Section C are long – answer II type question carrying 6 
mark each. 
6. Please write down the serial number of the question before attempting it. 
 
SECTION – A 
Question numbers 1 to 6 carry 1 mark each. 
 
1. 
T
2
cos sin
If A , find  satisfying 0< when A A 2I ;
2 sin cos
?? ?? ?
? ? ? ? ? ?
??
? ? ?
??
 
T
where A is transpose of A.  
 
2. If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. 
 
3. For what values of k, the system of linear equations 
x + y + z = 2 
2x + y – z = 3 
3x + 2y + kz = 4 
has a unique solution? 
 
4. 
? ?
Write the sum of intercepts cut off by the plane r 2i j k 5 0 on the three axes. ? ? ? ? ?
 
5. Find ?  and µ if 
? ? ? ?
i 3j 9k 3i j k 0. ? ? ? ? ? ? ? ? 
 
 
6. Ifa 4i j k and c 2i 2j k, then find a unit vector parallel to the vector a b. ? ? ? ? ? ? ?
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2016  
All India   
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. Please check that this question paper contains 26 questions. 
3. Question 1 to 6 in Section A are very short – answer type questions carrying 1 
mark each. 
4. Questions 7 to 19 in Section B are long – answer I type question carrying 4 
mark each. 
5. Questions 20 to 26 in Section C are long – answer II type question carrying 6 
mark each. 
6. Please write down the serial number of the question before attempting it. 
 
SECTION – A 
Question numbers 1 to 6 carry 1 mark each. 
 
1. 
T
2
cos sin
If A , find  satisfying 0< when A A 2I ;
2 sin cos
?? ?? ?
? ? ? ? ? ?
??
? ? ?
??
 
T
where A is transpose of A.  
 
2. If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. 
 
3. For what values of k, the system of linear equations 
x + y + z = 2 
2x + y – z = 3 
3x + 2y + kz = 4 
has a unique solution? 
 
4. 
? ?
Write the sum of intercepts cut off by the plane r 2i j k 5 0 on the three axes. ? ? ? ? ?
 
5. Find ?  and µ if 
? ? ? ?
i 3j 9k 3i j k 0. ? ? ? ? ? ? ? ? 
 
 
6. Ifa 4i j k and c 2i 2j k, then find a unit vector parallel to the vector a b. ? ? ? ? ? ? ?
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
 
7. Solve for x : tan
-1
 (x - 1) + tan
-1
x + tan
-1
 (x + 1) = tan
-1
 3x. 
 
OR 
 
 
3
1 1 1
22
6x 8x 4x 1
Prove that tan tan tan 2x;2x .
3 1 12x 1 4x
? ? ?
??
?? ?
? ? ? ??
??
??
?? ??
??
 
  
8. A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges 
for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the 
charges of typing one English and one Hindi page separately. However typist 
charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How 
much less was charged from this poor boy? Which values are reflected in this 
problem?  
 
 
  
9. 
sin(a 1)x 2sinx
,x 0
x
If f(x)= 2 ,x 0
1 bx 1
,x 0
x
? ??
?
?
?
?
?
?
?
??
?
?
?
?
 
is continuous at x = 0, then find the values of a and b. 
 
  
  
10. 
2
dy cos (a y)
If x cos(a+y)= cosy then prove that = .
dx sina
?
 
2
2
d y dy
Hence show that sina +sin2(a+y) =0. 
dx
dx
 
 
OR 
 
    
2
1
dy 6x 4 1 4x
Find if y sin
dx 5
?
??
??
??
?
??
??
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2016  
All India   
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. Please check that this question paper contains 26 questions. 
3. Question 1 to 6 in Section A are very short – answer type questions carrying 1 
mark each. 
4. Questions 7 to 19 in Section B are long – answer I type question carrying 4 
mark each. 
5. Questions 20 to 26 in Section C are long – answer II type question carrying 6 
mark each. 
6. Please write down the serial number of the question before attempting it. 
 
SECTION – A 
Question numbers 1 to 6 carry 1 mark each. 
 
1. 
T
2
cos sin
If A , find  satisfying 0< when A A 2I ;
2 sin cos
?? ?? ?
? ? ? ? ? ?
??
? ? ?
??
 
T
where A is transpose of A.  
 
2. If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. 
 
3. For what values of k, the system of linear equations 
x + y + z = 2 
2x + y – z = 3 
3x + 2y + kz = 4 
has a unique solution? 
 
4. 
? ?
Write the sum of intercepts cut off by the plane r 2i j k 5 0 on the three axes. ? ? ? ? ?
 
5. Find ?  and µ if 
? ? ? ?
i 3j 9k 3i j k 0. ? ? ? ? ? ? ? ? 
 
 
6. Ifa 4i j k and c 2i 2j k, then find a unit vector parallel to the vector a b. ? ? ? ? ? ? ?
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
 
7. Solve for x : tan
-1
 (x - 1) + tan
-1
x + tan
-1
 (x + 1) = tan
-1
 3x. 
 
OR 
 
 
3
1 1 1
22
6x 8x 4x 1
Prove that tan tan tan 2x;2x .
3 1 12x 1 4x
? ? ?
??
?? ?
? ? ? ??
??
??
?? ??
??
 
  
8. A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges 
for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the 
charges of typing one English and one Hindi page separately. However typist 
charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How 
much less was charged from this poor boy? Which values are reflected in this 
problem?  
 
 
  
9. 
sin(a 1)x 2sinx
,x 0
x
If f(x)= 2 ,x 0
1 bx 1
,x 0
x
? ??
?
?
?
?
?
?
?
??
?
?
?
?
 
is continuous at x = 0, then find the values of a and b. 
 
  
  
10. 
2
dy cos (a y)
If x cos(a+y)= cosy then prove that = .
dx sina
?
 
2
2
d y dy
Hence show that sina +sin2(a+y) =0. 
dx
dx
 
 
OR 
 
    
2
1
dy 6x 4 1 4x
Find if y sin
dx 5
?
??
??
??
?
??
??
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
11.Find the equation of tangents to the curve y= x
3
 + 2x – 4, which are 
perpendicular to line x + 14y + 3 = 0.
  
 
12. Find : 
2x
3
(2x 5)e
dx
(2x 3)
?
?
?
 
OR 
    Find : 
2
2
x x 1
dx
(x 1)(x 2)
??
?
??
 
 
13. Evaluate : 
2
2
x
2
x
dx.
15 ?
?
?
 
 
14. Find : 
2
(x 3) 3 4x x dx. ? ? ?
?
 
  
  
15. Find the particular solution of differential equation:  
     
dy x ycosx
given that y 1 when x 0.
dx 1 sinx
?
? ? ? ?
?
 
  
16. Find the particular solution of the differential equation  
2y e
x/y
 dx + (y ? 2x e
x/y
) dy = 0 
 given that x = 0 when y = 1. 
 
17. Show that the four points A(4,5,1), B(0, ?1, ?1), C(3,9,4) and D( ?4,4,4) 
are coplanar. 
 
18. Find the coordinates of the foot of perpendicular drawn from the point A     
   ( ?1,8,4) to the line joining the points B(0, ?1,3) and C(2, ?3, ?1). Hence find  
the image of the point A in the line BC. 
 
19. A bag X contains 4 white balls and 2 black balls, while another bag Y contains    
  3 white balls and 3 black balls. Two balls are drawn (without replacement) at   
        random from one of the bags and were found to be one white and one 
  black. Find the probability that the balls were drawn from bag Y. 
 
OR 
 
 A and B throw a pair of dice alternately, till one of them gets a total of 10 and   
 wins the game. Find their respective probabilities of winning, if A starts first. 
 
 
 
 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2016  
All India   
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. Please check that this question paper contains 26 questions. 
3. Question 1 to 6 in Section A are very short – answer type questions carrying 1 
mark each. 
4. Questions 7 to 19 in Section B are long – answer I type question carrying 4 
mark each. 
5. Questions 20 to 26 in Section C are long – answer II type question carrying 6 
mark each. 
6. Please write down the serial number of the question before attempting it. 
 
SECTION – A 
Question numbers 1 to 6 carry 1 mark each. 
 
1. 
T
2
cos sin
If A , find  satisfying 0< when A A 2I ;
2 sin cos
?? ?? ?
? ? ? ? ? ?
??
? ? ?
??
 
T
where A is transpose of A.  
 
2. If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. 
 
3. For what values of k, the system of linear equations 
x + y + z = 2 
2x + y – z = 3 
3x + 2y + kz = 4 
has a unique solution? 
 
4. 
? ?
Write the sum of intercepts cut off by the plane r 2i j k 5 0 on the three axes. ? ? ? ? ?
 
5. Find ?  and µ if 
? ? ? ?
i 3j 9k 3i j k 0. ? ? ? ? ? ? ? ? 
 
 
6. Ifa 4i j k and c 2i 2j k, then find a unit vector parallel to the vector a b. ? ? ? ? ? ? ?
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
 
7. Solve for x : tan
-1
 (x - 1) + tan
-1
x + tan
-1
 (x + 1) = tan
-1
 3x. 
 
OR 
 
 
3
1 1 1
22
6x 8x 4x 1
Prove that tan tan tan 2x;2x .
3 1 12x 1 4x
? ? ?
??
?? ?
? ? ? ??
??
??
?? ??
??
 
  
8. A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges 
for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the 
charges of typing one English and one Hindi page separately. However typist 
charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How 
much less was charged from this poor boy? Which values are reflected in this 
problem?  
 
 
  
9. 
sin(a 1)x 2sinx
,x 0
x
If f(x)= 2 ,x 0
1 bx 1
,x 0
x
? ??
?
?
?
?
?
?
?
??
?
?
?
?
 
is continuous at x = 0, then find the values of a and b. 
 
  
  
10. 
2
dy cos (a y)
If x cos(a+y)= cosy then prove that = .
dx sina
?
 
2
2
d y dy
Hence show that sina +sin2(a+y) =0. 
dx
dx
 
 
OR 
 
    
2
1
dy 6x 4 1 4x
Find if y sin
dx 5
?
??
??
??
?
??
??
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
11.Find the equation of tangents to the curve y= x
3
 + 2x – 4, which are 
perpendicular to line x + 14y + 3 = 0.
  
 
12. Find : 
2x
3
(2x 5)e
dx
(2x 3)
?
?
?
 
OR 
    Find : 
2
2
x x 1
dx
(x 1)(x 2)
??
?
??
 
 
13. Evaluate : 
2
2
x
2
x
dx.
15 ?
?
?
 
 
14. Find : 
2
(x 3) 3 4x x dx. ? ? ?
?
 
  
  
15. Find the particular solution of differential equation:  
     
dy x ycosx
given that y 1 when x 0.
dx 1 sinx
?
? ? ? ?
?
 
  
16. Find the particular solution of the differential equation  
2y e
x/y
 dx + (y ? 2x e
x/y
) dy = 0 
 given that x = 0 when y = 1. 
 
17. Show that the four points A(4,5,1), B(0, ?1, ?1), C(3,9,4) and D( ?4,4,4) 
are coplanar. 
 
18. Find the coordinates of the foot of perpendicular drawn from the point A     
   ( ?1,8,4) to the line joining the points B(0, ?1,3) and C(2, ?3, ?1). Hence find  
the image of the point A in the line BC. 
 
19. A bag X contains 4 white balls and 2 black balls, while another bag Y contains    
  3 white balls and 3 black balls. Two balls are drawn (without replacement) at   
        random from one of the bags and were found to be one white and one 
  black. Find the probability that the balls were drawn from bag Y. 
 
OR 
 
 A and B throw a pair of dice alternately, till one of them gets a total of 10 and   
 wins the game. Find their respective probabilities of winning, if A starts first. 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
 
 
SECTION – C 
Question numbers 20 to 26 carry 6 marks each. 
20. Three numbers are selected at random (without replacement) from first six 
positive integers. Let X denote the largest of the three numbers obtained. 
Find the probability distribution of X.Also, find the mean and variance of the 
distribution. 
 
21. LetA= R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a+c, b+d) 
Show that * is commutative and associative. Find the identity element for *  
on A. Also find the inverse of every element (a, b) ? A. 
 
22. 
4sin
Prove that y is an increasing function of  on 0,
2 cos 2
?? ??
? ? ? ?
??
??
??
  
OR 
 
Show that semi-vertical angle of a cone of maximum volume and given 
slant height is cos
-1 
1
3
??
??
??
  
 
23. Using the method of integration, find the area of the triangular region 
whose vertices are (2, -2), (4, 3) and (1, 2). 
 
24. Find the equation of the plane which contains the line of intersection of the 
planes 
ˆ ˆˆ
r.(i 2j 3k) 4 0 and
ˆ ˆˆ
r.( 2i j k) 5 0
? ? ? ?
? ? ? ? ?
  
and whose intercept on x-axis is equal to that of on y-axis. 
 
25. A retired person wants to invest an amount of Rs. 50, 000. His broker 
recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 
9% return respectively on the invested amount. He decides to invest at 
least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also 
wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear 
programming problem graphically to maximise his returns. 
 
 
 
 
 
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2016  
All India   
Time: 3 hrs  Total Marks: 100 
      
General Instructions:  
1. All questions are compulsory. 
2. Please check that this question paper contains 26 questions. 
3. Question 1 to 6 in Section A are very short – answer type questions carrying 1 
mark each. 
4. Questions 7 to 19 in Section B are long – answer I type question carrying 4 
mark each. 
5. Questions 20 to 26 in Section C are long – answer II type question carrying 6 
mark each. 
6. Please write down the serial number of the question before attempting it. 
 
SECTION – A 
Question numbers 1 to 6 carry 1 mark each. 
 
1. 
T
2
cos sin
If A , find  satisfying 0< when A A 2I ;
2 sin cos
?? ?? ?
? ? ? ? ? ?
??
? ? ?
??
 
T
where A is transpose of A.  
 
2. If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. 
 
3. For what values of k, the system of linear equations 
x + y + z = 2 
2x + y – z = 3 
3x + 2y + kz = 4 
has a unique solution? 
 
4. 
? ?
Write the sum of intercepts cut off by the plane r 2i j k 5 0 on the three axes. ? ? ? ? ?
 
5. Find ?  and µ if 
? ? ? ?
i 3j 9k 3i j k 0. ? ? ? ? ? ? ? ? 
 
 
6. Ifa 4i j k and c 2i 2j k, then find a unit vector parallel to the vector a b. ? ? ? ? ? ? ?
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
 
7. Solve for x : tan
-1
 (x - 1) + tan
-1
x + tan
-1
 (x + 1) = tan
-1
 3x. 
 
OR 
 
 
3
1 1 1
22
6x 8x 4x 1
Prove that tan tan tan 2x;2x .
3 1 12x 1 4x
? ? ?
??
?? ?
? ? ? ??
??
??
?? ??
??
 
  
8. A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges 
for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the 
charges of typing one English and one Hindi page separately. However typist 
charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How 
much less was charged from this poor boy? Which values are reflected in this 
problem?  
 
 
  
9. 
sin(a 1)x 2sinx
,x 0
x
If f(x)= 2 ,x 0
1 bx 1
,x 0
x
? ??
?
?
?
?
?
?
?
??
?
?
?
?
 
is continuous at x = 0, then find the values of a and b. 
 
  
  
10. 
2
dy cos (a y)
If x cos(a+y)= cosy then prove that = .
dx sina
?
 
2
2
d y dy
Hence show that sina +sin2(a+y) =0. 
dx
dx
 
 
OR 
 
    
2
1
dy 6x 4 1 4x
Find if y sin
dx 5
?
??
??
??
?
??
??
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
11.Find the equation of tangents to the curve y= x
3
 + 2x – 4, which are 
perpendicular to line x + 14y + 3 = 0.
  
 
12. Find : 
2x
3
(2x 5)e
dx
(2x 3)
?
?
?
 
OR 
    Find : 
2
2
x x 1
dx
(x 1)(x 2)
??
?
??
 
 
13. Evaluate : 
2
2
x
2
x
dx.
15 ?
?
?
 
 
14. Find : 
2
(x 3) 3 4x x dx. ? ? ?
?
 
  
  
15. Find the particular solution of differential equation:  
     
dy x ycosx
given that y 1 when x 0.
dx 1 sinx
?
? ? ? ?
?
 
  
16. Find the particular solution of the differential equation  
2y e
x/y
 dx + (y ? 2x e
x/y
) dy = 0 
 given that x = 0 when y = 1. 
 
17. Show that the four points A(4,5,1), B(0, ?1, ?1), C(3,9,4) and D( ?4,4,4) 
are coplanar. 
 
18. Find the coordinates of the foot of perpendicular drawn from the point A     
   ( ?1,8,4) to the line joining the points B(0, ?1,3) and C(2, ?3, ?1). Hence find  
the image of the point A in the line BC. 
 
19. A bag X contains 4 white balls and 2 black balls, while another bag Y contains    
  3 white balls and 3 black balls. Two balls are drawn (without replacement) at   
        random from one of the bags and were found to be one white and one 
  black. Find the probability that the balls were drawn from bag Y. 
 
OR 
 
 A and B throw a pair of dice alternately, till one of them gets a total of 10 and   
 wins the game. Find their respective probabilities of winning, if A starts first. 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
 
 
SECTION – C 
Question numbers 20 to 26 carry 6 marks each. 
20. Three numbers are selected at random (without replacement) from first six 
positive integers. Let X denote the largest of the three numbers obtained. 
Find the probability distribution of X.Also, find the mean and variance of the 
distribution. 
 
21. LetA= R × R and * be a binary operation on A defined by 
(a, b) * (c, d) = (a+c, b+d) 
Show that * is commutative and associative. Find the identity element for *  
on A. Also find the inverse of every element (a, b) ? A. 
 
22. 
4sin
Prove that y is an increasing function of  on 0,
2 cos 2
?? ??
? ? ? ?
??
??
??
  
OR 
 
Show that semi-vertical angle of a cone of maximum volume and given 
slant height is cos
-1 
1
3
??
??
??
  
 
23. Using the method of integration, find the area of the triangular region 
whose vertices are (2, -2), (4, 3) and (1, 2). 
 
24. Find the equation of the plane which contains the line of intersection of the 
planes 
ˆ ˆˆ
r.(i 2j 3k) 4 0 and
ˆ ˆˆ
r.( 2i j k) 5 0
? ? ? ?
? ? ? ? ?
  
and whose intercept on x-axis is equal to that of on y-axis. 
 
25. A retired person wants to invest an amount of Rs. 50, 000. His broker 
recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 
9% return respectively on the invested amount. He decides to invest at 
least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also 
wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear 
programming problem graphically to maximise his returns. 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2016 – All India Set – 1 
 
  
26. Using properties of determinants, prove that 
2
23
2
(x y)    zx       zy
  zx     (z y)   xy 2xyz(x y z)
  zy        xy    (z x)
?
? ? ? ?
?
  
 
OR 
 
32
3
1  0  2
If A= 0  2  1 andA - 6A +7A + kI = O find k.
2  0  3
??
??
??
??
??