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PLUS TWO
MATHEMATICS REVISION QUESTIONS
CONTENTS
1
...
1
2
...
3
3
...
DETERMINANTS……………………………………………………
...
DIFFERENTIATION…………………………………………………10
6
...
INTEGRALS…………………………………………………………
...
DEFINITE INTEGRALS…………………………………………
...
AREA……………………………………………………………
...
DIFFERENTIAL EQUATION…………………………………
...
VECTOR………………………………………………………
...
THREE DIMENSIONAL GEOMETRY……………………
...
LINEAR PROGRAMMING………………………………
...
PROBABILITY……………………………………………
...
MODEL EXAMINATION………………………………
...
RELATIONS AND FUNCTIONS
1
...
Let Q be the set of rational numbers & * be the binary operation on Q
defined by a*b =

for all a, b in Q
...

b) Find the inverse element of a under * on Q
...

3
...

4
...

5
...
Find the value of 2*3
...
Find f f (x) and f f (1)
...
Let f(x) =

x 3, g(x) =

x 1

a) Find f g(x)
b) Find f -1(x) and g -1(x)
c) Find (g f)-1(x)
...
f: {1, 2, 3, 4}{5} defined by f = {(1, 5) (2, 5) (3, 5) (4, 5)}
...
f(x) =

g(y) =

Find g f(x)
...
Let A=N x N (N-set of natural numbers and * be a binary operation on A
defined by (ab) * (cd)=(ac-bd, ad+bc)
...

10
...
Find a function g: YN such that f g and g f are identity
functions in their respective domains
...

11
...

12
...
Consider the function f: NN
defined by f(x) = x+1, x N

a) Prove that f is not an on to function
b) If g(x)=

then find g f

c) Check whether g f is an on to function
...
Give an example of a relation on a set
a) A= {1, 2, 3, 4} which is reflexive & symmetric but not transitive
...

c) Let * be a binary operation on Q defined by a*b =

Find the

inverse of 9 with respect to *
...
Consider f: RR given by f(x) = 5x+2
a) Show that f is one-one
...
Let * be a binary operation on N defined a*b = HCF of a and b
...
Consider the set A={1, 2, 3, 4, 5} B={1, 4, 9, 16, 25} and a function
f:AB defined by f(1)=1, f(2)=4, f(3)=9, f(4)=16 and f(5)=25
...

b) Show that f is on to
...
A function f:[-1, 1]R defined by f(x)=
a) Show that f is one-one
...

c) Find
exist? Explain?
18
...

Find the identity element for * if exists
...
a) Find f g and g f if f(x)= and g(x)=
b) Let A=N x N and * be binary operation on A defined by (a, b)*(c, d) =
(a + c, b + d)
(i)
Show that * is commutative & associative
...

20
...

(ii) Find
(iii) Show that is invertible
...
INVERSE TRIGONOMETRIC FUNCTIONS
1
...
Prove that
3
...
Given an expression for tan(x + y)
Prove that
Using the above result
5
...
If sin (


...


7
...
Show that
9
...
Given that
Show that
11
...
Find the value of x if
13
...
Prove that

15
...
Evaluate
17
...
If

3
...
Let A=
a) Is A singular
b) Find adj A
c) Find
using adj A and
2
...
If the matrices A and B are of orders mxn and nxm respectively
...

4
...

5
...

II
...


Check whether

is a skew symmetric matrix
...
Consider the matrix A=
I
...


Is A non-singular, give reasons
...
Find x, y, z and t
2
If

+3

=3

express it as the sum of a symmetric and a skew symmetric

matrix
...

8
...


9
...


Find A + B

and B =

II
...
Let A=
I
...


Find
If

be a 3x3 matrix
...

, where I is the 3x3 identity matrix
...
Given A =

B=

12
...
Find the inverse of P by elementary row operation
...
Let A=


...


and C=

I
...

II
...
Consider a 2x2 matrix A=
where
I
...
Find A+
...
Consider the matrix A =
I
...


Find A2
Find K so that A2 = KA – 7 I
...
Find the value of a and b if the matrix
17
...

II
...
Consider the matrices A =
I
...

III
...


and A + 3B =

Find B
...

Find the transpose of AB
...
If a matrix A=

is a solution of the matrix equation
Find any one value of x
...
DETERMINANTS
1
...
Prove that

=

(3y+k)

3
...
Solve the linear equations
3x - 2y + 3z = 8
2x + y – z = 1
4x - 3y + 2z = 4
5
...
Without expanding prove that

=0

7
...


8
...
60
...
90
...
70
...

9
...
find the value of
(b) Show that

= abc + bc + ca + ab

(c) Solve the system of equation by matrix method

10
...

(b) Find the value of x, y and z satisfying the above system of equations
...
Evaluate

=

12
...
Verify A(adj A) = (adj A)A =

I

If A =

14
...
If A =
(a) Find
(b) Show that
(c) Find
16
...
2x+5y=1, 3x+2y=7

(b) Find the value of

17
...
(i) If A =
What is the value of

?

(ii) Find the equation of the line joining the points (1, 2) and (-3, -2)
using determinant
...
(i) Let B be a square matrix of order 5
...
(a) Find the value of x in which

=

(b) Using the property of determinants, show that the points
A (a, b + c), B (b, c + a), C (c, a + b) are collinear
...
Consider a system of equation which is given below:
+ +

=4

- + = 1 and
+ -

=2

(a) Express the above system in the matrix form AX = B
(b) Find
, the inverse of A
...


5
...

a) Find
b)
2
...


if

3
...


Find

5
...


Find

7
...

8
...
If

Find

10
...


Find

12
...


Show that (1

14
...


Find

16
...
find
+x

–m2 y = 0

17
...
is function continues at x=0?
18
...
Consider the function f(x) {

-5

, x= -2

a) what is the value of f(-2) ?
b) check whether the function f(x) is continuous at x = -2
20
...
if its continuous at x= 2 and a+b =2
21
...
Determine the value of K so that the function f(x) = { k (x2 +x +1: x<0
Cos x : x ≥ 0
Is continuous
23
...

24
...
sin

,x 0
1

,x=0

25
...
Show that the function f(x) defined by f(x) = sin (cos x) is a continuous
function
...
APPLICATION OF DIFFERENTIATION
1
...

a) Taking the sides of the square cut off as x
...

2
...

find the following
a) The rate of change of the perimeter of the rectangle
...

3
...
a) State Rolle’s Theorem
...
Find the intervals on which f(x) =
is strictly
increasing/decreasing
...
An edge of a variable cube is increasing at the rate of 3cm/s
...
I) if f(x) =

...
Find the slope of the tangent to the curve
at (xy)
...
A wire of length 28 cm is cut into two pieces & the pieces is to be made
into a square & the other into a circle
...

b) What should be the length of the two pieces so that the combined area
of the square & the circle is a minimum?
10
...
Show that the function
is strictly increasing
...
Find the equation of the tangent & normal at the point (1, 2) of the
parabola

...
Find the maximum & minimum values of
on [12]
14
...

13

15
...
Find slope of the
curve at (0, 5)
...

16
...

17
...


14

7
...

2
...


dx

4
...

6
...

8
...

10
...

12
...

14
...

x2=y

16
...

18
...

20
...

22
...
Show that
24
...

26
...

28
...

30
...

32
...

34
...

36
...

38
...
DEFINITE INTEGRALS
1
...
Evaluate
3
...


Using limit as a sum
...
Prove that
6
...
Evaluate
8
...


Evaluate
...
Evaluate
11
...


12
...


13
...
Evaluate
15
...
Evaluate
17
...
Find

as limit of a sum
...
Evaluate
20
...

22
...

as a limit of a sum
...
Evaluate
24
...


25
...


Evaluate
...


Evaluate
...
Evaluate
29
...


As the limit of a sum
...


9
...
Find the area of the region bounded by the curves y2=x and the lines x=1,
x=4 and the x-axis?
2
...
Draw a rough sketch of the curve

+

=1& line + =1
...

4
...

B) Find the area bounded by the parabola y=x2 & y=x in the first quadrant
as shown in the figure
...
A) Find the area enclosed by the ellipse

+

=1

B) Find the area of the region in the first quadrant enclosed by the line
y=x & the circle x2+y2=32
...
Using the figure (i) Define the equation of the ellipse & circle
...


-2

-1

1

-1

7
...

B) Find the x-co-ordinate of the point of intersection of the above two
curves?
C) Find the area of the shaded region
...
Find the area of the shaded region in the first quadrant of the following
figure
...
Using integration find the common area to the curves
and

...
Consider the curves
and

...

11
...

=1

=1
12
...

1) Find the vertices of the triangle
...

3) Using integration find its area
...
Make a rough sketch of the curves y= and y=
1) Find the points of intersection of the two curves
...

14
...

Find the area enclosed between the line & the parabola
...
Find the area of the smaller part of the circle
cut off by the
line

...
Find the area of the region bounded by the curve
and the x-axis
between x = - 4 and x = 2
...
Find the area bounded by the curves(x-1)2+y2=1and x2+y2=1
...
Consider the circle x2+y2=16 and the straight line y=
figure
...

B

A

x as shown in the

10
...
Find the family of curves for which the slope of the tangent at any point
(x, y) on it is
2
...
E -

+3

+4=0

3
...
Consider the D
...

b) Find its general solution
...
Consider the D
...


Check whether it is homogeneous in x & y & solve
...
Find the order & the degree of the D
...
Consider the D
...

b) Hence solve the D
...

8
...
E
a) Find the integrating factor
...

9
...
E
a) Prove that this is a homogeneous D
...
E
...
Consider the D
...

b) Find its general solution
11
...
E corresponding to the function
a) State the order & degree of D
...



...
E
12
...
E of the family of circles touching the x-axis at origin
...
Find the particular solution of the D
...
Write the equation of circle with center on y-axis & passing through the
origin
...
E of all such circles
b) Find the order & degree of the above D
...


15
...
Find the D
...
Consider the D
...

b) Express the given D
...

c) Hence solve the D
...

18
...
E
Solve

=

11
...
If = 2 - - 3 and
I
...


= -3 -2
...


Find the area of a parallelogram whose adjacent sides are and
...


Find the unit vector perpendicular to both and
...
Let =
I
...


= - +

-

Find the unit vector along

-

III
...

3
...
If = -2 + 3 and
I
...


Find

=3 -2 +2
...

and

III
...

IV
...

5
...

I
...

III
...
A)
I
...


Find

...

What is the radius of the circle passing through the points A, B and
C
...


B) If a= - 7 + 7 and b= 3 - 2 + 2
I
...


Find a vector perpendicular to each of the vector and
...


7
...
Find the direction cosines of the vector 2 - 2 -

II
...
(3 - 6 +2 )=11
8
...

b) Find the equation of the plane passing through the points
(2, 5, -3),(-2, -3, 5) and (5, 3, -3)
...
a) Write two non-zero vectors and
b) If is a unit of vector such that [
magnitude of vector x
c) If =

+ + ,

=

of the vectors +
10
...


Find +

+2 +3

such that x =
]
...


+2 -3

and

= 3 - +2

and -
...
Find a unit vector perpendicular to both + and -
...
Consider the points A (2, -1, 1), B (1, -3, -5), C(3, -4, -4)
I
...
Prove that the above points form a right angled triangle
...
a) Find the projection of the vector = 2 +3 +2 on the vector
=
b) If a and b are such that
c) =

+ + ,

= 2 + +3

=2,

+2 +

=3 and
...
find a unit vector

parallel to the vector 2 - +3
13
...

I
...

II
...

III
...

14
...

I
...

II
...

15
...
Consider the

product (
...
( x ) and x ( x )
...
Out of the above three products, which is not possible to find out
...
Find the volume of the parallelepiped whose co-terminal edges are
,

and
...


Show that x( x ) = (
...

16
...
Show that for any point
O
+
+
=
+
+
...

17
...


I
...


Find a x b
If the area of the parallelogram is
square units
...

18
...
Find the projection of BC on AB
II
...

19
...
Consider the vectors = 2 + -2
I
...


Find


...
b)2

21
...

I
...

III
...


IV
...

A
22
...

23
...

24
...

25
...

26
...

27
...
THREE DIMENSIONAL GEOMETRY
1
...

I
...

II
...

2
...
Fill in the blanks:
I
...

II
...

III
...
The foot of the perpendicular from (
) on the y-axis is ______
4
...

II
...
(A)
...
Find the direction
cosines
...

(B)
...

6
...

and
and
through the point (1, 1, 1)
7
...
Express them in vector form as

and

II
...
Find (
)=(
)
IV
...

8
...
Write the equation of family of planes through the intersection of
given planes
...
Find the member of the above family of planes which is perpendicular
to x + y + z = 0
III
...

9
...
Express them in vector form as
II
...
Find (
)–(
)
IV
...

10
...
a) Find the equation of the plane with intercept 4 on the Y-axis and
parallel to ZOX plane
...
a) Find the equation of the line passing through the intersection of the
planes x + y + 4z + 5 = 0 and 2x– y + 3z + 6 = 0 and through the point
(1, 0, 0)
b) Find the equation of the plane passing through the points (2, 5, -3),
(-2, -3, 5) and (5, 3, -3)
13
...

14
...
Find the equation of the plane through the intersection of the planes
and
and passes through the point
(2, 2, 1)
...
a) Find the vector equation of the line if its Cartesian equation is

b) Find p so that the lines

and

are at

right angles
...
Consider the equations of lines

and the plane
and

I
...

II
...

III
...

18
...
Consider the Cartesian equation of a line
I
...

II
...
Consider the vector equation of two planes

and

I
...

II
...

21
...

b) Find the perpendicular distance from the point (6, 5, 9) to this plane
...
The foot of the perpendicular from the origin to a plane is P (4, -2, 5)
I
...
Find the equation of the above plane in vector form and Cartesian
form
...
a) Find the angle between the lines having direction ratios <1, 1, 2> and
<
>
b) If the lines
Find the value of
...


24
...

Find the distance of the point (-1, -2, 3) from the plane
25
...

I
...

II
...

26
...
Find the foot of the perpendicular drawn from the origin to the plane
...
Write the vector equation and Cartesian equation of this perpendicular
...
a) A line makes angles 90 , 60 and 30 with the positive direction of x, y
and z axis respectively
...

b) Find the angle between the pair of lines,
and

...
Consider the planes
and
I
...

II
...

29
...
Prove

13
...
Consider the linear programming problem:
Maximize Z = x + y
Subject to 2x + y - 3 ≥ 0
x - 2y + 1 ≤ 0
y≤3
x ≥ 0, y ≥ 0
a) Draw its feasible region
...

c) Find the corner at which Z attains its maximum
2
...

3
...
The vitamin contents of one kg food
is given below:
Food
Vit A
Vit B
Vit C
X
Y

1
2

2
2

3
1

One kg of food X costs Rs
...
20
...

4
...
The profit from crops X and Y per hectare are estimated as Rs
...
9000 respectively
...

Further no more than 800 litres of herbicide should be used in order to
protect fish and wild life using a pond which collects drainage from this
land
...

a) By suitably defining the variables write the objective function of
the problem
...

c) Solve the problem by graphical method and find the allocation of
the land to each crop
...
A furniture dealer sells only tables and chairs
...
12, 000 to invest
and a space to store 90 pieces
...
400 and a chair
Rs
...
He can sell a table at a profit of Rs
...
25
...
The dealer wants to get
maximum profit
...

b) Write the constraints
...

6
...
The production
time of one ball of type B is double the type A (time in units)
...
The
supply of raw material is sufficient for the production of 1500 balls (both
A and B) per day
...
3 from a ball of type A and Rs
...
Then
a) By defining suitable variables, write the objective function
...

c) How many balls should be produced in each type per day in order
to get maximum profit?
7
...
The shaded
region is the feasible region
...
Z = px + qy
Y

10

2x
+y
C =1
5
0

B
X+3y=15

0

A
5

10

15

X

a) What are the co-ordinates of the corners of feasible region?
b) Write the constraints
...
Z occurs at A and B, what is the relation between p
and q?
d) If q=1, write the objective function
...
Z
...
A manufacturer produces nuts and bolts
...
It
takes 3 hours on machine, A and 1 hour on machine B to produce a
package of bolts
...
17
...
7
...
How many packages of each should be
produced each day so as to maximize his profit, if he operates his
machines for at the most 12 hours a day?
a) By suitably defining the variables write the objective function of
the problem
...

c) Solve the problem by graphical method and find the number of
packages of nuts and bolts to be manufactured
...
Consider the following LPP: Minimize Z = 200 x + 500 y
...

b) Find the co-ordinates of the corner points of the feasible region
...

10
...

b) Find the corner points of the feasible region
...

11
...
Three machines are
needed for this purpose
...

Type of cakes
Machine
I
II
III

A

B

12
18
6

6
0
9

Each machine is available for at most 6 hours per day
...
The bakery owner wants to make
maximum profit per day by making Rs
...
5 from type A and Rs
...

a) Write the objective function by defining suitable variables
...

c) Find the maximum profit graphically
...
A company makes 2 products X and Y
...
The second requires 4 hours for
assembling and 2 hours for packing
...
The profit margin for X is Rs
...
21/a) Convert this into a linear programming problem
...


14
...
A card is drawn from a well-shuffled pack of cards
...
A fair die is thrown twice
...
Let B denote the event that the sum of the out comes in the
two trails is 6
...

b) Write down the set of favorable outcomes for event B
...


3
...
Getting a 1 or 6 is considered a ‘success’
...

b) Obtain the probability distribution of the number of successes
...
(a) What is meant by mutually exclusive events?
(b) Find the probability of drawing a one-rupee coin from a purse with
two compartments, one of which contains 3 fifty-paisa and 2 one-rupee
coins and the other 2 fifty-paisa and 3 one-rupee coins
...
(a) A and B independently try to solve a problem
...
Find the
probability that
I
...

II
...

(b) Find the probability distribution of the number of heads in three tosses
of a fair coin
...
(a) P(A) =

, P(B) =

P (A B) =
...
Probability that A
solves is and that B solves is
...


7
...
Two balls are drawn
from the urn one after the other without replacement
...
(a) Three fair coins are tossed and let X be the number of heads turning
up
...

(b) There are 5% defective items in a large bulk of items
...
(a) State and prove the theorem of total probability
...
The probability distribution of a random variable X is given below
...

11
...

a) If A is the event that the number on the card is even, then write A
...

12
...
One student is selected such that each has
the same chance of being selected; the age X of the selected student is
recorded
...

b) Find E(X)
c) Find var (X)
13
...
Let A be event ‘odd number on the
first throw’ and B be the event ‘odd number on the second throw’
...

(b) If P(A) = 0
...
5 and P(B/A) = 0
...
P(A B)
II
...
P(A B)
14
...
xn occur
with probabilities P1, P2…
...
The probability of husband’s selection is and that of wife’s
selection is
...
(a) For any two events A and B write an expression for P(A/B)
...
1%, 2% and 3% of the bulbs produced by A, B
and C respectively are defective
...
Find the probability that this
bulb has been produced by machine A
...
(a) Write the probability function of the Binomial distribution
...
It is
not possible to just look at a bulb and tell whether or not it is defective
...

17
...
Find the probability that one of them
is black and other is red
...

18
...
Find the probability of getting an odd number at
least once
...
One ball is drawn at random from one of the bags
& it is found to be red
...

19
...

(ii) A box contains 30 defective and 30 non-defective bulbs
...
The events A and B are defined as follows:
A: ‘the first bulb is defective’
B: ‘the second bulb is non-defective’
...
Prove that A and B are independent events
...
Of their
outputs, 1, 2 and 3 percent are respectively defective bulbs
...
What is the probability that it
is manufactured by the machine Y?
20
...
The probability that A
solves the problem is and that B solves the problem is Find the
probability that
a) Both of them solve the problem
b) The problem is solved
...
If A and B are two independent events, then
a) Prove that A and B’ are independent events
...


MODEL EXAMINATION
HSE II

MATHEMATICS

Time: 2 hours
Marks: 80

1
...

2
...
8, P(B) = 0
...
4, then find P(A/B)
b) Find the probability distribution of number of heads X in two tosses of a
coin
...


3
...
a) u =
Find

,v=
and

b) Find

if y =

c) If y=
, Show that
5
...
Let A =
a) Find
b) Hence solve the system of equations

7
...
a) Find the slope of the curve

at (1, 2)

b) Find the equation of tangent to
which is parallel to the line

...


9
...
What is the maximum
volume of such a box obtained?
OR
10
...
A soldier is placed at
the point (3, 2)
...
Let S be
the distance between these two points
I
...

II
...

11
...

II
...


12
...

OR

13
...


Prove that
I=

II
...
a) Draw the rough sketch of the curves y= x2 and x=y2
b) Find the point of intersection of the two curves
...


15
...
What is the degree of the differential equation
II
...

III
...

16
...


Express these lines in vector form as
and

II
...

IV
...

OR
17
...

II
...


18
...


Find

and
+

and

-

II
...
Consider the points A (2, -1, 1), B (1, -3, -5), C (3, -4, -4)
I
...
Prove that the above points form a right angled triangle
...
Five cards are drawn successively with replacement from a well-shuffled deck
of 52 cards
...
All the five cards are spades?
II
...

OR
21
...
One of the two bags is selected at random and a ball is selected
from one bag which is found to be red
...

22
...
Show that ‘*’ is commutative and associative
...
Find the identity element for ‘*’ if any
...
A company makes 2 products X and Y
...
The second requires 4 hours for
assembling and 2 hours for packing
...
The profit margin for X is Rs
...
21/c) Convert this into a linear programming problem
...




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