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, zaxis 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 xaxis.
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, zaxis 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 xaxis.
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, zaxis 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 xaxis.
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, zaxis 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 xaxis.
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, zaxis 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)= x3 if x=3
g(x) = 3x 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, zaxis 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 xaxis.
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, zaxis 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)= x3 if x=3
g(x) = 3x 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 xaxis.
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(ab).
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
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