Past Year Paper, Mathematics (Set - 1), 2015, Class 12, Maths | Additional Study Material for JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2015  
All India Set – 1 
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 – 6 in Section A are very short – answer type questions carrying 1 mark 
each. 
4. Questions 7 – 19 in Section B are long – answer I type question carrying 4 marks each. 
5. Questions 20 – 26 in Section B are long – answer II type question carrying 6 marks 
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. ? ? ? ? ? ? ? If a 2i j 3k and b 3i 5j 2k,thenfind a b .  
2. Find the angle between the vectors  ?? i jand j k. 
 
3. Find the distance of a point (2, 5, -3) from the plane
? ?
? ? ? r. 6i 3j 2k 4. 
 
4. Write the element a 12 of the matrix A = [a ij] 2 × 2, whose elements a ij are given by a ij = e
2ix
 
sin jx. 
 
5. Find the differential equation of the family of lines passing through the origin. 
 
 
6. Find the integrating factor for the following differential equation: ??
dy
x log x y 2log x
dx
 
 
 
 
 
 
 
 
 
 
Page 2


  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2015  
All India Set – 1 
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 – 6 in Section A are very short – answer type questions carrying 1 mark 
each. 
4. Questions 7 – 19 in Section B are long – answer I type question carrying 4 marks each. 
5. Questions 20 – 26 in Section B are long – answer II type question carrying 6 marks 
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. ? ? ? ? ? ? ? If a 2i j 3k and b 3i 5j 2k,thenfind a b .  
2. Find the angle between the vectors  ?? i jand j k. 
 
3. Find the distance of a point (2, 5, -3) from the plane
? ?
? ? ? r. 6i 3j 2k 4. 
 
4. Write the element a 12 of the matrix A = [a ij] 2 × 2, whose elements a ij are given by a ij = e
2ix
 
sin jx. 
 
5. Find the differential equation of the family of lines passing through the origin. 
 
 
6. Find the integrating factor for the following differential equation: ??
dy
x log x y 2log x
dx
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
7. If
1 2 2
A 2 1 2 ,
2 2 1
 then show that A
2
 – 4A – 5I = O, and hence find A
-1
 
OR 
 If 
2 0 1
A 5 1 0 ,
0 1 3
 
then find A
-1
 using elementary row operations.
 
  
8. Using the properties of determinants, solve the following for x: 
 
x 2 x 6 x 1
x 6 x 1 x 2 0
x 1 x 2 x 6
 
  
9. Evaluate:
2
/2
0
sin x
dx
sinx cosx
.  
OR 
 Evaluate 
2
3x
1
e 7x 5 dx  as a limit of sums. 
  
10. Evaluate: 
 
2
42
x
dx
x x 2
 
 
11.In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at 
random from this set and tossed five times. If all the five times, the result was heads, 
find the probability that the selected coin had heads on both the sides.
  
OR 
 How many times must a fair coin be tossed so that the probability of getting at least one 
head is more than 80%? 
 
12. Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are 
coplanar. 
 
Page 3


  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2015  
All India Set – 1 
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 – 6 in Section A are very short – answer type questions carrying 1 mark 
each. 
4. Questions 7 – 19 in Section B are long – answer I type question carrying 4 marks each. 
5. Questions 20 – 26 in Section B are long – answer II type question carrying 6 marks 
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. ? ? ? ? ? ? ? If a 2i j 3k and b 3i 5j 2k,thenfind a b .  
2. Find the angle between the vectors  ?? i jand j k. 
 
3. Find the distance of a point (2, 5, -3) from the plane
? ?
? ? ? r. 6i 3j 2k 4. 
 
4. Write the element a 12 of the matrix A = [a ij] 2 × 2, whose elements a ij are given by a ij = e
2ix
 
sin jx. 
 
5. Find the differential equation of the family of lines passing through the origin. 
 
 
6. Find the integrating factor for the following differential equation: ??
dy
x log x y 2log x
dx
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
7. If
1 2 2
A 2 1 2 ,
2 2 1
 then show that A
2
 – 4A – 5I = O, and hence find A
-1
 
OR 
 If 
2 0 1
A 5 1 0 ,
0 1 3
 
then find A
-1
 using elementary row operations.
 
  
8. Using the properties of determinants, solve the following for x: 
 
x 2 x 6 x 1
x 6 x 1 x 2 0
x 1 x 2 x 6
 
  
9. Evaluate:
2
/2
0
sin x
dx
sinx cosx
.  
OR 
 Evaluate 
2
3x
1
e 7x 5 dx  as a limit of sums. 
  
10. Evaluate: 
 
2
42
x
dx
x x 2
 
 
11.In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at 
random from this set and tossed five times. If all the five times, the result was heads, 
find the probability that the selected coin had heads on both the sides.
  
OR 
 How many times must a fair coin be tossed so that the probability of getting at least one 
head is more than 80%? 
 
12. Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are 
coplanar. 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
13. A line passing through the point A with position vector a 4i 2j 2k is parallel to the 
vector b 2i 3j 6k . Find the length of the perpendicular drawn on this line from a 
point P with vector 
1
r i 2j 3k . 
 
 
 
 
14. Solve the following for x: 
sin
-1
 (1 - x) – 2 sin
-1
 x = 
?
2
  
OR 
 Show that: 
 
11
3 17
2sin tan
5 31 4
 
  
15. If y = e
ax
. cos bx, then prove that  
 
2
22
2
d y dy
2a a b y 0
dx
dx
 
  
16. If x
x
 + x
y
 + y
x
 = a
b
 , then find 
dy
.
dx
 
 
17. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find 
dy
dx
 at t.
4
 
 
18. Evaluate: 
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
19.  Three schools X, Y, and Z organized a fete (mela)  for collecting funds for flood victims 
in which they sold hand-helds fans, mats and toys made from recycled material, the sale 
price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the 
number of articles of each type sold: 
 
School SchoolX SchoolY SchoolZ
Article
Hand - held fans 30 40 35
Mats 12 15 20
Toys 70 55 75
 
Using matrices, find the funds collected by each school by selling the above articles and 
the total funds collected. Also write any one value generated by the above situation. 
Page 4


  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2015  
All India Set – 1 
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 – 6 in Section A are very short – answer type questions carrying 1 mark 
each. 
4. Questions 7 – 19 in Section B are long – answer I type question carrying 4 marks each. 
5. Questions 20 – 26 in Section B are long – answer II type question carrying 6 marks 
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. ? ? ? ? ? ? ? If a 2i j 3k and b 3i 5j 2k,thenfind a b .  
2. Find the angle between the vectors  ?? i jand j k. 
 
3. Find the distance of a point (2, 5, -3) from the plane
? ?
? ? ? r. 6i 3j 2k 4. 
 
4. Write the element a 12 of the matrix A = [a ij] 2 × 2, whose elements a ij are given by a ij = e
2ix
 
sin jx. 
 
5. Find the differential equation of the family of lines passing through the origin. 
 
 
6. Find the integrating factor for the following differential equation: ??
dy
x log x y 2log x
dx
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
7. If
1 2 2
A 2 1 2 ,
2 2 1
 then show that A
2
 – 4A – 5I = O, and hence find A
-1
 
OR 
 If 
2 0 1
A 5 1 0 ,
0 1 3
 
then find A
-1
 using elementary row operations.
 
  
8. Using the properties of determinants, solve the following for x: 
 
x 2 x 6 x 1
x 6 x 1 x 2 0
x 1 x 2 x 6
 
  
9. Evaluate:
2
/2
0
sin x
dx
sinx cosx
.  
OR 
 Evaluate 
2
3x
1
e 7x 5 dx  as a limit of sums. 
  
10. Evaluate: 
 
2
42
x
dx
x x 2
 
 
11.In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at 
random from this set and tossed five times. If all the five times, the result was heads, 
find the probability that the selected coin had heads on both the sides.
  
OR 
 How many times must a fair coin be tossed so that the probability of getting at least one 
head is more than 80%? 
 
12. Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are 
coplanar. 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
13. A line passing through the point A with position vector a 4i 2j 2k is parallel to the 
vector b 2i 3j 6k . Find the length of the perpendicular drawn on this line from a 
point P with vector 
1
r i 2j 3k . 
 
 
 
 
14. Solve the following for x: 
sin
-1
 (1 - x) – 2 sin
-1
 x = 
?
2
  
OR 
 Show that: 
 
11
3 17
2sin tan
5 31 4
 
  
15. If y = e
ax
. cos bx, then prove that  
 
2
22
2
d y dy
2a a b y 0
dx
dx
 
  
16. If x
x
 + x
y
 + y
x
 = a
b
 , then find 
dy
.
dx
 
 
17. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find 
dy
dx
 at t.
4
 
 
18. Evaluate: 
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
19.  Three schools X, Y, and Z organized a fete (mela)  for collecting funds for flood victims 
in which they sold hand-helds fans, mats and toys made from recycled material, the sale 
price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the 
number of articles of each type sold: 
 
School SchoolX SchoolY SchoolZ
Article
Hand - held fans 30 40 35
Mats 12 15 20
Toys 70 55 75
 
Using matrices, find the funds collected by each school by selling the above articles and 
the total funds collected. Also write any one value generated by the above situation. 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – C 
 
Question numbers 20 to 26 carry 6 marks each. 
 
20.  Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary opearation 
on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then find 
 (i) The identify element of * in A. 
 (ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 
??
??
??
1
,4
2
. 
OR 
Let f : W ? W be defined as 
n 1, if n is odd
f n 
n 1, if n is even
 Show that f is invertible and find the inverse of f. Here, W is the set of all whole 
numbers. 
 
21. Sketch the region bounded by the curves  ??
2
y 5 x and ?? y x 1 and find its area 
using intergration  . 
 
22. Find the particular solution of the differential equation x
2
dy = (2xy + y
2
) dx, given that y 
= 1 when x = 1.
  
 
OR 
Find the particular solution of the differential equation 
? ?
?
??
? ? ?
??
??
1
2 m tan x
dy
1 x e y ,
dx
 
given that y =1 when x = 0. 
 
23. Find the absolute maximum and absolute minimum values of the function f given by f(x) 
= sin
2
x – cos x, x ? (0, p) 
 
24. Show that lines: 
? ?
? ?
r i j k i j k
r 4j 2 k 2j j 3k are coplanar.
? ? ? ? ? ? ?
? ? ? ? ? ?
 
 Also, find the equation of the plane containing these lines. 
  
 
 
 
 
  
Page 5


  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
CBSE Board 
Class XII Mathematics 
Board Paper – 2015  
All India Set – 1 
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 – 6 in Section A are very short – answer type questions carrying 1 mark 
each. 
4. Questions 7 – 19 in Section B are long – answer I type question carrying 4 marks each. 
5. Questions 20 – 26 in Section B are long – answer II type question carrying 6 marks 
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. ? ? ? ? ? ? ? If a 2i j 3k and b 3i 5j 2k,thenfind a b .  
2. Find the angle between the vectors  ?? i jand j k. 
 
3. Find the distance of a point (2, 5, -3) from the plane
? ?
? ? ? r. 6i 3j 2k 4. 
 
4. Write the element a 12 of the matrix A = [a ij] 2 × 2, whose elements a ij are given by a ij = e
2ix
 
sin jx. 
 
5. Find the differential equation of the family of lines passing through the origin. 
 
 
6. Find the integrating factor for the following differential equation: ??
dy
x log x y 2log x
dx
 
 
 
 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – B 
 
Question numbers 7 to 19 carry 4 marks each. 
 
7. If
1 2 2
A 2 1 2 ,
2 2 1
 then show that A
2
 – 4A – 5I = O, and hence find A
-1
 
OR 
 If 
2 0 1
A 5 1 0 ,
0 1 3
 
then find A
-1
 using elementary row operations.
 
  
8. Using the properties of determinants, solve the following for x: 
 
x 2 x 6 x 1
x 6 x 1 x 2 0
x 1 x 2 x 6
 
  
9. Evaluate:
2
/2
0
sin x
dx
sinx cosx
.  
OR 
 Evaluate 
2
3x
1
e 7x 5 dx  as a limit of sums. 
  
10. Evaluate: 
 
2
42
x
dx
x x 2
 
 
11.In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at 
random from this set and tossed five times. If all the five times, the result was heads, 
find the probability that the selected coin had heads on both the sides.
  
OR 
 How many times must a fair coin be tossed so that the probability of getting at least one 
head is more than 80%? 
 
12. Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are 
coplanar. 
 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
13. A line passing through the point A with position vector a 4i 2j 2k is parallel to the 
vector b 2i 3j 6k . Find the length of the perpendicular drawn on this line from a 
point P with vector 
1
r i 2j 3k . 
 
 
 
 
14. Solve the following for x: 
sin
-1
 (1 - x) – 2 sin
-1
 x = 
?
2
  
OR 
 Show that: 
 
11
3 17
2sin tan
5 31 4
 
  
15. If y = e
ax
. cos bx, then prove that  
 
2
22
2
d y dy
2a a b y 0
dx
dx
 
  
16. If x
x
 + x
y
 + y
x
 = a
b
 , then find 
dy
.
dx
 
 
17. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find 
dy
dx
 at t.
4
 
 
18. Evaluate: 
? ?
? ?
?
?
?
x
3
x 3 e
dx
x5
 
 
19.  Three schools X, Y, and Z organized a fete (mela)  for collecting funds for flood victims 
in which they sold hand-helds fans, mats and toys made from recycled material, the sale 
price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the 
number of articles of each type sold: 
 
School SchoolX SchoolY SchoolZ
Article
Hand - held fans 30 40 35
Mats 12 15 20
Toys 70 55 75
 
Using matrices, find the funds collected by each school by selling the above articles and 
the total funds collected. Also write any one value generated by the above situation. 
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
SECTION – C 
 
Question numbers 20 to 26 carry 6 marks each. 
 
20.  Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary opearation 
on A defined by (a, b) * (c, d) = (ac, b + ad) for (a. b), (c, d) ? A. Then find 
 (i) The identify element of * in A. 
 (ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 
??
??
??
1
,4
2
. 
OR 
Let f : W ? W be defined as 
n 1, if n is odd
f n 
n 1, if n is even
 Show that f is invertible and find the inverse of f. Here, W is the set of all whole 
numbers. 
 
21. Sketch the region bounded by the curves  ??
2
y 5 x and ?? y x 1 and find its area 
using intergration  . 
 
22. Find the particular solution of the differential equation x
2
dy = (2xy + y
2
) dx, given that y 
= 1 when x = 1.
  
 
OR 
Find the particular solution of the differential equation 
? ?
?
??
? ? ?
??
??
1
2 m tan x
dy
1 x e y ,
dx
 
given that y =1 when x = 0. 
 
23. Find the absolute maximum and absolute minimum values of the function f given by f(x) 
= sin
2
x – cos x, x ? (0, p) 
 
24. Show that lines: 
? ?
? ?
r i j k i j k
r 4j 2 k 2j j 3k are coplanar.
? ? ? ? ? ? ?
? ? ? ? ? ?
 
 Also, find the equation of the plane containing these lines. 
  
 
 
 
 
  
  
 
CBSE XII | Mathematics 
Board Paper 2015 – All India Set – 1 
 
  
25. Minimum and maximum z = 5x + 2y subject to the following constraints:  
x – 2y = 2   
3x + 2y = 12 
-3x + 2y = 3 
x = 0, y = 0 
 
26. Two the numbers are selected at random (without replacement) from first six positive 
integers. Let X denote the larger of the two numbers obtained. Find the probability 
distribution of X. Find the mean and variance of this distribution.