NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a. Sum of the series is equal to
(A) 348551 (B) -1000
(C) 5151 (D) None of the above
b. The value of tan is
(A)
(B)
(C) (D)
c. In a triangle
ABC, let a = BC, b = CA and c = AB. If , then
(A)
(B)
(C) (D)
None of the above
d. The circles and
cut orthogonally
if the value of p is
(A) 3 (B) -2
(C) -3 (D) 1
e. The eccentricity
of the ellipse is
(A)
3
(B)
(C) 5 (D)
f. The derivative of – cos (log x) is
(A)
sin
(log x) (B)
(C) – sin (log x) (D)
g. The value of the is
(A) 0 (B) 1
(C) e (D) Does not exist
h. The integral is equal to
(A)
(B)
(C) 0 (D) 1
i. The area under
the curve between
x = 0 and x = 1 is
(A)
1 (B)
(C) (D)
j. The solution of is
(A) (B)
(C) (D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Show
that the sum to n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + … is . (8)
b. If
are the
roots of the quadratic equation
and
are the roots of the quadratic
equation
then
show that
. (8)
Q.3 a. If
A + B + C = ,
prove that
. (8)
b. Show that sin is a root of the
equation
. (8)
Q.4 a. Find
the value of such
that the circles
and
touch each other. (8)
b. For what values of k the
points and
are
collinear? (8)
Q.5 a. Find the equation
of the circle for which is a tangent and
are normals. (8)
b. Find
the values of a, b such that the line ax + by + 1 = 0 is tangent to the
hyperbola and
is parallel to the line y = 2x + 4. (8)
Q.6 a. Evaluate the limit. (8)
b. Consider the function . Find
using first
principle. Is
continuous
at x = 0? (8)
Q.7 a. Find
the local maximum and minimum values of in
. (8)
b. Find the area of the
region bounded by ,
and x = 1. (8)
Q.8 a. Evaluate
the following integral . (8)
b. Evaluate
the following definite integral . (8)
Q.9 a. Solve the differential equation
. (8)
b. Solve the differential equation
. (8)