Model Question Paper 1
II P.U.C
MATHEMATICS (35)
Time : 3 hours 15 minute
Max. Marks : 100
Instructions :
(i)
The question paper has five parts namely A, B, C, D and E. Answer all the parts.
(ii)
Use the graph sheet for the question on Linear programming in PART E.
PART-A
Answer All the questions:
10 1 = 10
1.
Operation * is defined by a*b=a. Is * is a binary operation on
2.
Write the principal value branch of ( )
3.
Define a diagonal matrix.
4.
If
5.
Write the points of discontinuity for the function ( )
6.
Evaluate
7.
Find the direction ratios of the vector, joining the points
(
),
from P to Q.
8.
Find the equation of the plane with the intercept 2, 3 and 4 on x, y and z axes
respectively.
9.
Define optimal solution in the linear programming.
10.
If A and b are independent events with ( )
/ find |
|.
, -
cos ecx cos ecx cot x dx
and ( )
) and
find (
PART-B
Answer any Ten questions:
10 2 = 20
11.
Find the gof and fog if ( )
12.
Write the function
13.
Prove that 2
14.
Find the area of a triangle whose vertices are (
determinant.
15.
Find
16.
If
17.
Find the approximate change in the volume V of a cube of a side x meters
caused by
by 2 %.
if
and ( )
. /
in the simplest form.
. /
)(
) using
.
show that
18.
Evaluate
19.
Evaluate
20.
Find the order and degree of differential equation . /
21.
Find a vector in the direction of the vector
units.
22.
If
are
23.
Find the vector equation of the line, passing through the points (
24.
Two coins are tossed once, find (
appears.
and
.
.
that has magnitude 7
, then show that the vectors
and
) and
) where E: no tail appears, F: no head
PART-C
Answer any Ten questions:
10 3=30
25.
Show that the relation R in the set of real numbers R defined as
*(
+ is neither
)
nor symmetric nor transitive
26.
Prove that
27.
Find the values of x, y and z in the following matrices
(
/ | |
/
(
)(
)(
28.
Differentiate
29.
If
30.
If ( )
31.
Evaluate
32.
Evaluate
33.
Find the area between the curves
34.
Form the differential equation representing the family of curves
(
) where a and b arbitrary constants.
35.
Find the area of a triangle having the points
its vertices using vector method.
36.
Prove that [
37.
Find the distance between parallel lines
with respect to x.
/find
is (a) Strictly increasing
[ ]
n 2i 3j 6k
r 3i 3j 5k
38.
(b) Strictly decreasing.
and
) as
[ ].
) and
Two cards are drawn successively with replacement from a well shuffled
deck of 52 cards. Find the probability distribution of the number of aces.
PART-D
Answer any Six questions:
6 5 = 30
39.
Let
be a function defined by ( )
Show that
where S is the range of function f, is invertible. Find the inverse of f.
40.
Verify (
if
,
),
) and
41.
Solve the following system of equations by matrix method
and
.
42.
If
43.
The length of a rectangle is decreasing at the rate of 3 cm/minute and the
width y is increasing at the
of 2 cm/minute. When x = 10 cm and y = 6
cm, find the rate of change of (i) the perimeter and (ii) the area of the
rectangle
44.
Find the integral of
45.
Using integration find the area bounded by the triangle whose vertices are
( ) ( )
( )
46.
Solve the differential equation
47.
Derive equation of plane perpendicular to a given vector and passing through
a given point both in the vector and Cartesian form.
48.
There are 5 % defective items in a large bulk of items. What is the probability
that sample of
items will include not more than one defective item?
prove that
. /
and hence prove that
w.r.t x and hence evaluate
PART-E
Answer any One question:
49.
(a) Prove that
|
evaluate |
(b)
50
Prove that |
1
( )
( )
when
10 = 10
)
( ) and hence
6
(a) Solve the following linear programming problem graphically:
Maximize,
subjected to the constraints:
6
(b) Find the relationship between a and b so that the function f defined by
ax 1,
f x
bx 3,
if x 3
is continuous at x = 3.
if x 3
