Class 12 Mathematics (Maths) Previous Year Paper - 2 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
 
     
 
     
Time: 3hrs    
General Instructions: 
1. All questions are compulsory.
2. The question paper consists of 26
comprises of 6 questions of 1 mar
Section C comprises of 7 questions of 6 marks each.
3. Use of calculators is not permitted
 
1 
Find the principal value of tan
 
2 
The 
-coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
Find its -coordinate. 
3 Let N N f ? : be defined by 
 
4 If ,
1 2
2 1
?
?
?
?
?
?
= A and ) (
2
= x x f
 
 
5 Find the value of the following 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
                                                                  
 
7 
Evaluate: dx x
?
- 2 1
) (sin 
 
 
8 
Show that the function g(
) = 
 
9 Write
?
?
?
+
?
?
?
?
?
?
+
- 2
1
2
1
1
2
sin
2
1
tan
x
x
10 Find the intervals in which the function f(x) = 2x
decreasing. Also find the points on which the tangents are para
 
11 A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability dist
How many times must a man toss a fair
head is more than 80% . 
 
 
12 If 


 
	

 




, prove that  
 
 
      
     
                                    
compulsory. 
he question paper consists of 26 questions divided into three sections A, B
questions of 1 mark each. Section B comprises of 13 questions of 4 marks e
questions of 6 marks each. 
se of calculators is not permitted. 
Section A 
.
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
be defined by . 4 ) ( x x f = Is function f  an onto function?G
, 3 2
2
- - x then find ). (A f 
Find the value of the following determinant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
are unit vectors along x, y, z-axis respectively, find the value of 
). 
                                                                 Section B 
) = |
  3|, 
 ? , is continuous but not differentiable at 
?
?
?
?
?
?
?
?
?
?
?
+
- - 2
2
1
1
1
cos
2
1
y
y
in simplest form. 
Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability distribution of number of tails. 
OR 
How many times must a man toss a fair coin, so that the probability of having at least 
prove that  
x
y
dx
dy 2
= . 
              Class: XII 
                                 M. M:100 
questions divided into three sections A, B and C. Section A 
questions of 4 marks each and 
 
1 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. 
1 
an onto function?Give reason. 1 
1 
 
 
1 
 
1 
 
4 
is continuous but not differentiable at 
 
 3. 
 
 
4 
 
4 
+ 36x + 1 is strictly increasing or 
axis. 
4 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
coin, so that the probability of having at least one 
4 
 
4 
 
Page 2


 
 
     
 
     
Time: 3hrs    
General Instructions: 
1. All questions are compulsory.
2. The question paper consists of 26
comprises of 6 questions of 1 mar
Section C comprises of 7 questions of 6 marks each.
3. Use of calculators is not permitted
 
1 
Find the principal value of tan
 
2 
The 
-coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
Find its -coordinate. 
3 Let N N f ? : be defined by 
 
4 If ,
1 2
2 1
?
?
?
?
?
?
= A and ) (
2
= x x f
 
 
5 Find the value of the following 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
                                                                  
 
7 
Evaluate: dx x
?
- 2 1
) (sin 
 
 
8 
Show that the function g(
) = 
 
9 Write
?
?
?
+
?
?
?
?
?
?
+
- 2
1
2
1
1
2
sin
2
1
tan
x
x
10 Find the intervals in which the function f(x) = 2x
decreasing. Also find the points on which the tangents are para
 
11 A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability dist
How many times must a man toss a fair
head is more than 80% . 
 
 
12 If 


 
	

 




, prove that  
 
 
      
     
                                    
compulsory. 
he question paper consists of 26 questions divided into three sections A, B
questions of 1 mark each. Section B comprises of 13 questions of 4 marks e
questions of 6 marks each. 
se of calculators is not permitted. 
Section A 
.
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
be defined by . 4 ) ( x x f = Is function f  an onto function?G
, 3 2
2
- - x then find ). (A f 
Find the value of the following determinant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
are unit vectors along x, y, z-axis respectively, find the value of 
). 
                                                                 Section B 
) = |
  3|, 
 ? , is continuous but not differentiable at 
?
?
?
?
?
?
?
?
?
?
?
+
- - 2
2
1
1
1
cos
2
1
y
y
in simplest form. 
Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability distribution of number of tails. 
OR 
How many times must a man toss a fair coin, so that the probability of having at least 
prove that  
x
y
dx
dy 2
= . 
              Class: XII 
                                 M. M:100 
questions divided into three sections A, B and C. Section A 
questions of 4 marks each and 
 
1 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. 
1 
an onto function?Give reason. 1 
1 
 
 
1 
 
1 
 
4 
is continuous but not differentiable at 
 
 3. 
 
 
4 
 
4 
+ 36x + 1 is strictly increasing or 
axis. 
4 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
coin, so that the probability of having at least one 
4 
 
4 
 
XII A 2 of 3 
 
 
 
13 Evaluate:
?
+ + ) cos( ) cos( b x a x
dx
 
OR 
Evaluate:
?
+ + + 1
2 3
x x x
dx
 
 
 
 
4 
14 Form the differential equation of family of ellipses having foci on y axis and centre at 
origin. 
OR 
Solve the differential equation: x x y
dx
dy
x log 2
2
= + 
 
4 
 
15 Show that the lines 
7
5
5
3
3
1 +
=
+
=
+ z y x
and 
5
6
3
4
1
2 - =
- =
- z y x
intersect each other. 
Find the point of intersection also. 
 
 
4 
 
16 Solve  that  differential equation: , 0 ) ( ) 3 (
2 2
= + + + dy xy x dx y xy  
 
 1,  
 1 
4 
17 Let N be a set of all natural numbers and let R be a relation on N×N, defined by  
(a, b) R (c, d) ? ad = bc for all (a, b), (c, d) ?N×N. Show that R is an equivalence relation 
on N×N. 
 
4 
 
18 Using properties of determinants, show that: . 4
2 2 2
2
2
2
c b a
ca ab a
c bc ca
bc b ab
=
- - - 
 
OR 
Find the matrix X such that 
?
?
?
?
?
?
?
?
?
? - - - =
?
?
?
?
?
?
?
?
?
?
- - 10 20 10
0 4 3
10 8 1
4 2
1 0
1 2
X 
 
 
 
4 
 
19 
b a, and c
r
are three vectors such that . , a c b c b a
r r
r
r
r
r
= × = × Prove that , ,b a c
r
are mutually at 
right angles and . , 1 a c b
r r
r
= = 
 
 
4 
 
Section C 
 
20 Find the equation of the line passing through the point (2,-3,-5) and perpendicular to the 
plane 6
  3  5  2 
 0. Also, find the point of intersection of this line and the plane. 
6 
 
 
21 
 
 
In answering a question on a MCQ test with 4 choices per question, a student knows the 
answer, guesses or copies the answer. Let 1/2be the probability that he knows the answer, 
1/4 be the probability that he guesses and ¼ that he copies it. Assuming that a student, who 
copies the answer, will be correct with the probability3/4, what is the probability that 
student knows the answer, given that he answered it correctly? 
Ram does not know the answer to one of the questions in the test. The evaluation process 
has negative marking. Which value would Ram violate if he resorts to unfair means? How 
would an act like the above hamper his character development in the coming years?   
 
 
6 
Page 3


 
 
     
 
     
Time: 3hrs    
General Instructions: 
1. All questions are compulsory.
2. The question paper consists of 26
comprises of 6 questions of 1 mar
Section C comprises of 7 questions of 6 marks each.
3. Use of calculators is not permitted
 
1 
Find the principal value of tan
 
2 
The 
-coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
Find its -coordinate. 
3 Let N N f ? : be defined by 
 
4 If ,
1 2
2 1
?
?
?
?
?
?
= A and ) (
2
= x x f
 
 
5 Find the value of the following 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
                                                                  
 
7 
Evaluate: dx x
?
- 2 1
) (sin 
 
 
8 
Show that the function g(
) = 
 
9 Write
?
?
?
+
?
?
?
?
?
?
+
- 2
1
2
1
1
2
sin
2
1
tan
x
x
10 Find the intervals in which the function f(x) = 2x
decreasing. Also find the points on which the tangents are para
 
11 A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability dist
How many times must a man toss a fair
head is more than 80% . 
 
 
12 If 


 
	

 




, prove that  
 
 
      
     
                                    
compulsory. 
he question paper consists of 26 questions divided into three sections A, B
questions of 1 mark each. Section B comprises of 13 questions of 4 marks e
questions of 6 marks each. 
se of calculators is not permitted. 
Section A 
.
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
be defined by . 4 ) ( x x f = Is function f  an onto function?G
, 3 2
2
- - x then find ). (A f 
Find the value of the following determinant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
are unit vectors along x, y, z-axis respectively, find the value of 
). 
                                                                 Section B 
) = |
  3|, 
 ? , is continuous but not differentiable at 
?
?
?
?
?
?
?
?
?
?
?
+
- - 2
2
1
1
1
cos
2
1
y
y
in simplest form. 
Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability distribution of number of tails. 
OR 
How many times must a man toss a fair coin, so that the probability of having at least 
prove that  
x
y
dx
dy 2
= . 
              Class: XII 
                                 M. M:100 
questions divided into three sections A, B and C. Section A 
questions of 4 marks each and 
 
1 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. 
1 
an onto function?Give reason. 1 
1 
 
 
1 
 
1 
 
4 
is continuous but not differentiable at 
 
 3. 
 
 
4 
 
4 
+ 36x + 1 is strictly increasing or 
axis. 
4 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
coin, so that the probability of having at least one 
4 
 
4 
 
XII A 2 of 3 
 
 
 
13 Evaluate:
?
+ + ) cos( ) cos( b x a x
dx
 
OR 
Evaluate:
?
+ + + 1
2 3
x x x
dx
 
 
 
 
4 
14 Form the differential equation of family of ellipses having foci on y axis and centre at 
origin. 
OR 
Solve the differential equation: x x y
dx
dy
x log 2
2
= + 
 
4 
 
15 Show that the lines 
7
5
5
3
3
1 +
=
+
=
+ z y x
and 
5
6
3
4
1
2 - =
- =
- z y x
intersect each other. 
Find the point of intersection also. 
 
 
4 
 
16 Solve  that  differential equation: , 0 ) ( ) 3 (
2 2
= + + + dy xy x dx y xy  
 
 1,  
 1 
4 
17 Let N be a set of all natural numbers and let R be a relation on N×N, defined by  
(a, b) R (c, d) ? ad = bc for all (a, b), (c, d) ?N×N. Show that R is an equivalence relation 
on N×N. 
 
4 
 
18 Using properties of determinants, show that: . 4
2 2 2
2
2
2
c b a
ca ab a
c bc ca
bc b ab
=
- - - 
 
OR 
Find the matrix X such that 
?
?
?
?
?
?
?
?
?
? - - - =
?
?
?
?
?
?
?
?
?
?
- - 10 20 10
0 4 3
10 8 1
4 2
1 0
1 2
X 
 
 
 
4 
 
19 
b a, and c
r
are three vectors such that . , a c b c b a
r r
r
r
r
r
= × = × Prove that , ,b a c
r
are mutually at 
right angles and . , 1 a c b
r r
r
= = 
 
 
4 
 
Section C 
 
20 Find the equation of the line passing through the point (2,-3,-5) and perpendicular to the 
plane 6
  3  5  2 
 0. Also, find the point of intersection of this line and the plane. 
6 
 
 
21 
 
 
In answering a question on a MCQ test with 4 choices per question, a student knows the 
answer, guesses or copies the answer. Let 1/2be the probability that he knows the answer, 
1/4 be the probability that he guesses and ¼ that he copies it. Assuming that a student, who 
copies the answer, will be correct with the probability3/4, what is the probability that 
student knows the answer, given that he answered it correctly? 
Ram does not know the answer to one of the questions in the test. The evaluation process 
has negative marking. Which value would Ram violate if he resorts to unfair means? How 
would an act like the above hamper his character development in the coming years?   
 
 
6 
XII A 3 of 3 
 
 
 
 
22 
A point on the hypotenuse of a right angled triangle is at distances a and b from the sides. 
Show that the minimum length of the hypotenuse is ( )
3 / 2 3 / 2
b a +
2 / 3
. 
OR 
If the sum of the lengths of the hypotenuse and a side of a triangle is given, show that the 
area of the triangle is maximum when the angle between them is 


. 
 
6 
23 Make a rough sketch of the region given below and find its area using methods of 
integration :
,  ; 0 =  = 


 3, 0 =  = 2
  3 , 0 = 
 = 3". 
OR 
Find the area of  region bounded by the curve y=
2
5 x - and y= . 1 - x 
 
6 
 
24 
Given two matrices
?
?
?
?
?
?
?
?
?
? - =
2 1 0
4 3 2
0 1 1
A and 
?
?
?
?
?
?
?
?
?
?
- - - - =
5 1 2
4 2 4
4 2 2
B verify that BA = 6I. Use the 
result to solve the system: , 3 = - y x , 17 4 3 2 = + + z y x . 7 2 = + z y 
 
6 
 
 
25 
Evaluate : #
$%$
&'()
*
$+,-'
*
$

.
 
 
 
6 
 
 
 
26 A factory owner purchases two types of machines A and B for his factory. The 
requirements and the limitations for the machines are as follows: 
 
Machine 
Area 
Occupied 
Labour 
Force 
Daily input    
(in units) 
A 1000 m² 12 men 60 
B 1200 m² 8 men 40 
 
He has maximum area of 9000 m² available, and 72 skilled labourers who can operate both 
the machines. How many machines of each type should he buy to maximize the daily 
output? 
 
6 
 
_______________________________________________________________________________________ 
 
Page 4


 
 
     
 
     
Time: 3hrs    
General Instructions: 
1. All questions are compulsory.
2. The question paper consists of 26
comprises of 6 questions of 1 mar
Section C comprises of 7 questions of 6 marks each.
3. Use of calculators is not permitted
 
1 
Find the principal value of tan
 
2 
The 
-coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
Find its -coordinate. 
3 Let N N f ? : be defined by 
 
4 If ,
1 2
2 1
?
?
?
?
?
?
= A and ) (
2
= x x f
 
 
5 Find the value of the following 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
                                                                  
 
7 
Evaluate: dx x
?
- 2 1
) (sin 
 
 
8 
Show that the function g(
) = 
 
9 Write
?
?
?
+
?
?
?
?
?
?
+
- 2
1
2
1
1
2
sin
2
1
tan
x
x
10 Find the intervals in which the function f(x) = 2x
decreasing. Also find the points on which the tangents are para
 
11 A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability dist
How many times must a man toss a fair
head is more than 80% . 
 
 
12 If 


 
	

 




, prove that  
 
 
      
     
                                    
compulsory. 
he question paper consists of 26 questions divided into three sections A, B
questions of 1 mark each. Section B comprises of 13 questions of 4 marks e
questions of 6 marks each. 
se of calculators is not permitted. 
Section A 
.
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
be defined by . 4 ) ( x x f = Is function f  an onto function?G
, 3 2
2
- - x then find ). (A f 
Find the value of the following determinant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
are unit vectors along x, y, z-axis respectively, find the value of 
). 
                                                                 Section B 
) = |
  3|, 
 ? , is continuous but not differentiable at 
?
?
?
?
?
?
?
?
?
?
?
+
- - 2
2
1
1
1
cos
2
1
y
y
in simplest form. 
Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability distribution of number of tails. 
OR 
How many times must a man toss a fair coin, so that the probability of having at least 
prove that  
x
y
dx
dy 2
= . 
              Class: XII 
                                 M. M:100 
questions divided into three sections A, B and C. Section A 
questions of 4 marks each and 
 
1 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. 
1 
an onto function?Give reason. 1 
1 
 
 
1 
 
1 
 
4 
is continuous but not differentiable at 
 
 3. 
 
 
4 
 
4 
+ 36x + 1 is strictly increasing or 
axis. 
4 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
coin, so that the probability of having at least one 
4 
 
4 
 
XII A 2 of 3 
 
 
 
13 Evaluate:
?
+ + ) cos( ) cos( b x a x
dx
 
OR 
Evaluate:
?
+ + + 1
2 3
x x x
dx
 
 
 
 
4 
14 Form the differential equation of family of ellipses having foci on y axis and centre at 
origin. 
OR 
Solve the differential equation: x x y
dx
dy
x log 2
2
= + 
 
4 
 
15 Show that the lines 
7
5
5
3
3
1 +
=
+
=
+ z y x
and 
5
6
3
4
1
2 - =
- =
- z y x
intersect each other. 
Find the point of intersection also. 
 
 
4 
 
16 Solve  that  differential equation: , 0 ) ( ) 3 (
2 2
= + + + dy xy x dx y xy  
 
 1,  
 1 
4 
17 Let N be a set of all natural numbers and let R be a relation on N×N, defined by  
(a, b) R (c, d) ? ad = bc for all (a, b), (c, d) ?N×N. Show that R is an equivalence relation 
on N×N. 
 
4 
 
18 Using properties of determinants, show that: . 4
2 2 2
2
2
2
c b a
ca ab a
c bc ca
bc b ab
=
- - - 
 
OR 
Find the matrix X such that 
?
?
?
?
?
?
?
?
?
? - - - =
?
?
?
?
?
?
?
?
?
?
- - 10 20 10
0 4 3
10 8 1
4 2
1 0
1 2
X 
 
 
 
4 
 
19 
b a, and c
r
are three vectors such that . , a c b c b a
r r
r
r
r
r
= × = × Prove that , ,b a c
r
are mutually at 
right angles and . , 1 a c b
r r
r
= = 
 
 
4 
 
Section C 
 
20 Find the equation of the line passing through the point (2,-3,-5) and perpendicular to the 
plane 6
  3  5  2 
 0. Also, find the point of intersection of this line and the plane. 
6 
 
 
21 
 
 
In answering a question on a MCQ test with 4 choices per question, a student knows the 
answer, guesses or copies the answer. Let 1/2be the probability that he knows the answer, 
1/4 be the probability that he guesses and ¼ that he copies it. Assuming that a student, who 
copies the answer, will be correct with the probability3/4, what is the probability that 
student knows the answer, given that he answered it correctly? 
Ram does not know the answer to one of the questions in the test. The evaluation process 
has negative marking. Which value would Ram violate if he resorts to unfair means? How 
would an act like the above hamper his character development in the coming years?   
 
 
6 
XII A 3 of 3 
 
 
 
 
22 
A point on the hypotenuse of a right angled triangle is at distances a and b from the sides. 
Show that the minimum length of the hypotenuse is ( )
3 / 2 3 / 2
b a +
2 / 3
. 
OR 
If the sum of the lengths of the hypotenuse and a side of a triangle is given, show that the 
area of the triangle is maximum when the angle between them is 


. 
 
6 
23 Make a rough sketch of the region given below and find its area using methods of 
integration :
,  ; 0 =  = 


 3, 0 =  = 2
  3 , 0 = 
 = 3". 
OR 
Find the area of  region bounded by the curve y=
2
5 x - and y= . 1 - x 
 
6 
 
24 
Given two matrices
?
?
?
?
?
?
?
?
?
? - =
2 1 0
4 3 2
0 1 1
A and 
?
?
?
?
?
?
?
?
?
?
- - - - =
5 1 2
4 2 4
4 2 2
B verify that BA = 6I. Use the 
result to solve the system: , 3 = - y x , 17 4 3 2 = + + z y x . 7 2 = + z y 
 
6 
 
 
25 
Evaluate : #
$%$
&'()
*
$+,-'
*
$

.
 
 
 
6 
 
 
 
26 A factory owner purchases two types of machines A and B for his factory. The 
requirements and the limitations for the machines are as follows: 
 
Machine 
Area 
Occupied 
Labour 
Force 
Daily input    
(in units) 
A 1000 m² 12 men 60 
B 1200 m² 8 men 40 
 
He has maximum area of 9000 m² available, and 72 skilled labourers who can operate both 
the machines. How many machines of each type should he buy to maximize the daily 
output? 
 
6 
 
_______________________________________________________________________________________ 
 
 
                                                  Mathematics (Set - A) 
Date:                         Class: XII 
 
1 
Find the principal value of .
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 


8
 
1 
2 
The -coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. Find its -
coordinate. 
 
-1 
1 
3 
Let N N f ? : be defined by . 4 ) ( x x f = Is function f  an onto function. 
No , 1 has no preimage. 
1 
4 
If ,
1 2
2 1
?
?
?
?
?
?
= A show that , 3 2 ) (
2
- - = x x x f find ). (A f
 
,
0 0
0 0
?
?
?
?
?
?
= A 
1 
5 
Find the value of the following dererminant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
0 
 
1 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along x, y, z-axis respectively, find the value of 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
1
 
 
1 
                                                                  Section B  
 
7 
Evaluate dx x
?
- 2 1
) (sin
 
Sol:  
dx x
?
- 2 1
) (sin
 
Put x=sin? 
. cos ) sin (sin
2 1
? ? ? d
?
- 
? ? ? d
?
cos ) (
2
                                                         (2)  
Apply Integration by parts.

 
c x x x x x + - - +
- - 2 sin 1 2 ) (sin
1 2 2 1
               (2) 
 
4 
 
8 
Show that the function g() = | - 3|,  ? 
, is continuous but not differentiable at  = 3. 
Sol: g() = | - 3|    , 
g(x)= x-3 if x=3 
g(x) = 3-x if x< 3 
For continuity: L.H.L =0  
R.H.L =0. 
g(3)=0 so continuous at x=3.                         (2) 
4 
Page 5


 
 
     
 
     
Time: 3hrs    
General Instructions: 
1. All questions are compulsory.
2. The question paper consists of 26
comprises of 6 questions of 1 mar
Section C comprises of 7 questions of 6 marks each.
3. Use of calculators is not permitted
 
1 
Find the principal value of tan
 
2 
The 
-coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
Find its -coordinate. 
3 Let N N f ? : be defined by 
 
4 If ,
1 2
2 1
?
?
?
?
?
?
= A and ) (
2
= x x f
 
 
5 Find the value of the following 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
                                                                  
 
7 
Evaluate: dx x
?
- 2 1
) (sin 
 
 
8 
Show that the function g(
) = 
 
9 Write
?
?
?
+
?
?
?
?
?
?
+
- 2
1
2
1
1
2
sin
2
1
tan
x
x
10 Find the intervals in which the function f(x) = 2x
decreasing. Also find the points on which the tangents are para
 
11 A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability dist
How many times must a man toss a fair
head is more than 80% . 
 
 
12 If 


 
	

 




, prove that  
 
 
      
     
                                    
compulsory. 
he question paper consists of 26 questions divided into three sections A, B
questions of 1 mark each. Section B comprises of 13 questions of 4 marks e
questions of 6 marks each. 
se of calculators is not permitted. 
Section A 
.
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,
be defined by . 4 ) ( x x f = Is function f  an onto function?G
, 3 2
2
- - x then find ). (A f 
Find the value of the following determinant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
are unit vectors along x, y, z-axis respectively, find the value of 
). 
                                                                 Section B 
) = |
  3|, 
 ? , is continuous but not differentiable at 
?
?
?
?
?
?
?
?
?
?
?
+
- - 2
2
1
1
1
cos
2
1
y
y
in simplest form. 
Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
three times, find the probability distribution of number of tails. 
OR 
How many times must a man toss a fair coin, so that the probability of having at least 
prove that  
x
y
dx
dy 2
= . 
              Class: XII 
                                 M. M:100 
questions divided into three sections A, B and C. Section A 
questions of 4 marks each and 
 
1 
coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. 
1 
an onto function?Give reason. 1 
1 
 
 
1 
 
1 
 
4 
is continuous but not differentiable at 
 
 3. 
 
 
4 
 
4 
+ 36x + 1 is strictly increasing or 
axis. 
4 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed 
coin, so that the probability of having at least one 
4 
 
4 
 
XII A 2 of 3 
 
 
 
13 Evaluate:
?
+ + ) cos( ) cos( b x a x
dx
 
OR 
Evaluate:
?
+ + + 1
2 3
x x x
dx
 
 
 
 
4 
14 Form the differential equation of family of ellipses having foci on y axis and centre at 
origin. 
OR 
Solve the differential equation: x x y
dx
dy
x log 2
2
= + 
 
4 
 
15 Show that the lines 
7
5
5
3
3
1 +
=
+
=
+ z y x
and 
5
6
3
4
1
2 - =
- =
- z y x
intersect each other. 
Find the point of intersection also. 
 
 
4 
 
16 Solve  that  differential equation: , 0 ) ( ) 3 (
2 2
= + + + dy xy x dx y xy  
 
 1,  
 1 
4 
17 Let N be a set of all natural numbers and let R be a relation on N×N, defined by  
(a, b) R (c, d) ? ad = bc for all (a, b), (c, d) ?N×N. Show that R is an equivalence relation 
on N×N. 
 
4 
 
18 Using properties of determinants, show that: . 4
2 2 2
2
2
2
c b a
ca ab a
c bc ca
bc b ab
=
- - - 
 
OR 
Find the matrix X such that 
?
?
?
?
?
?
?
?
?
? - - - =
?
?
?
?
?
?
?
?
?
?
- - 10 20 10
0 4 3
10 8 1
4 2
1 0
1 2
X 
 
 
 
4 
 
19 
b a, and c
r
are three vectors such that . , a c b c b a
r r
r
r
r
r
= × = × Prove that , ,b a c
r
are mutually at 
right angles and . , 1 a c b
r r
r
= = 
 
 
4 
 
Section C 
 
20 Find the equation of the line passing through the point (2,-3,-5) and perpendicular to the 
plane 6
  3  5  2 
 0. Also, find the point of intersection of this line and the plane. 
6 
 
 
21 
 
 
In answering a question on a MCQ test with 4 choices per question, a student knows the 
answer, guesses or copies the answer. Let 1/2be the probability that he knows the answer, 
1/4 be the probability that he guesses and ¼ that he copies it. Assuming that a student, who 
copies the answer, will be correct with the probability3/4, what is the probability that 
student knows the answer, given that he answered it correctly? 
Ram does not know the answer to one of the questions in the test. The evaluation process 
has negative marking. Which value would Ram violate if he resorts to unfair means? How 
would an act like the above hamper his character development in the coming years?   
 
 
6 
XII A 3 of 3 
 
 
 
 
22 
A point on the hypotenuse of a right angled triangle is at distances a and b from the sides. 
Show that the minimum length of the hypotenuse is ( )
3 / 2 3 / 2
b a +
2 / 3
. 
OR 
If the sum of the lengths of the hypotenuse and a side of a triangle is given, show that the 
area of the triangle is maximum when the angle between them is 


. 
 
6 
23 Make a rough sketch of the region given below and find its area using methods of 
integration :
,  ; 0 =  = 


 3, 0 =  = 2
  3 , 0 = 
 = 3". 
OR 
Find the area of  region bounded by the curve y=
2
5 x - and y= . 1 - x 
 
6 
 
24 
Given two matrices
?
?
?
?
?
?
?
?
?
? - =
2 1 0
4 3 2
0 1 1
A and 
?
?
?
?
?
?
?
?
?
?
- - - - =
5 1 2
4 2 4
4 2 2
B verify that BA = 6I. Use the 
result to solve the system: , 3 = - y x , 17 4 3 2 = + + z y x . 7 2 = + z y 
 
6 
 
 
25 
Evaluate : #
$%$
&'()
*
$+,-'
*
$

.
 
 
 
6 
 
 
 
26 A factory owner purchases two types of machines A and B for his factory. The 
requirements and the limitations for the machines are as follows: 
 
Machine 
Area 
Occupied 
Labour 
Force 
Daily input    
(in units) 
A 1000 m² 12 men 60 
B 1200 m² 8 men 40 
 
He has maximum area of 9000 m² available, and 72 skilled labourers who can operate both 
the machines. How many machines of each type should he buy to maximize the daily 
output? 
 
6 
 
_______________________________________________________________________________________ 
 
 
                                                  Mathematics (Set - A) 
Date:                         Class: XII 
 
1 
Find the principal value of .
8
9
tan tan
1
?
?
?
?
?
?
?
?
?
?
?
?
- p
 


8
 
1 
2 
The -coordinate of a point P on the line joining the points Q(2,2,1) and R(5,1,-2) is 4. Find its -
coordinate. 
 
-1 
1 
3 
Let N N f ? : be defined by . 4 ) ( x x f = Is function f  an onto function. 
No , 1 has no preimage. 
1 
4 
If ,
1 2
2 1
?
?
?
?
?
?
= A show that , 3 2 ) (
2
- - = x x x f find ). (A f
 
,
0 0
0 0
?
?
?
?
?
?
= A 
1 
5 
Find the value of the following dererminant: 
b a c
a c b
c b a
+
+
+
2
2
2
. 
0 
 
1 
 
6 
If k j i
ˆ
,
ˆ
,
ˆ
are unit vectors along x, y, z-axis respectively, find the value of 
).
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
)
ˆ ˆ
.(
ˆ
j i k k i j k j i × + × + ×
 
1
 
 
1 
                                                                  Section B  
 
7 
Evaluate dx x
?
- 2 1
) (sin
 
Sol:  
dx x
?
- 2 1
) (sin
 
Put x=sin? 
. cos ) sin (sin
2 1
? ? ? d
?
- 
? ? ? d
?
cos ) (
2
                                                         (2)  
Apply Integration by parts.

 
c x x x x x + - - +
- - 2 sin 1 2 ) (sin
1 2 2 1
               (2) 
 
4 
 
8 
Show that the function g() = | - 3|,  ? 
, is continuous but not differentiable at  = 3. 
Sol: g() = | - 3|    , 
g(x)= x-3 if x=3 
g(x) = 3-x if x< 3 
For continuity: L.H.L =0  
R.H.L =0. 
g(3)=0 so continuous at x=3.                         (2) 
4 
For differentiability:  
L.H.L =1  
R.H.L =-1 so not differentiable at x=3.          (2) 
 
9 
Write
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
- +
?
?
?
?
?
?
+
- - 2
2
1
2
1
1
1
cos
2
1
1
2
sin
2
1
tan
y
y
x
x
in simplest form. 
Put x=tan? , y= tanø                                   (1) 
Simplest form=        .


                            (3) 
4 
10 Find the intervals in which the function f(x) = 2x
3
 – 15x
2
 + 36x + 1 is strictly increasing or 
decreasing. Also find the points on which the tangents are parallel to x-axis. 
Sol:  
f(x) = 2x
3
 – 15x
2
 + 36x + 1 


= 6

- 30 + 36. 


= 6

- 30 + 36=0.        (1) 
 = 2,3 
(-8,2) and (3, 8) st increasing. 
(2,3) st. decreasing.                   (2) 
Points: (2,29) (3,28)         (1) 
 
4 
 
11 
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, 
find the probability distribution of number of tails. 
Sol:  p=1/4 
q=3/4 
X= Number of tails. 
X=0,1,2,3. 
X 0 1 2 3 
P(X) 27/64 27/64 9/64 1/64 
 
OR 
How many times must a man toss a fair coin, so that the probability of having at least one head is 
more than 80% . 
Sol:  
n=? 
p=1/2, 
q=1/2. 
P(at least one head)=80/100. 
Apply binomial distribution to get n=3. 
4 
 
12 
If 

+ 

= 



, prove that  
x
y
dx
dy 2
= . 
Apply log on both sides                                                                (2) 
Then differentiate both the sides to get the required answer.       (2) 
4 
13 
Evaluate: dx
b x a x
?
+ + ) cos( ) cos(
1
 
Multi. And divide by sin(a-b). 
dx
b x a x
b x a x
b a
?
+ +
+ - +
- ) cos( ) cos(
)) ( sin(
) sin( / 1
       (2) 
dx b x a x
?
+ + + )} tan( ) {tan(
 
4