TIME ALLOWED: THREE HOURS
MAXIMUM MARKS = 100
NOTE:(i) Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
(ii) All the parts (if any) of each Question must be attempted at one place instead of at different
places.
(iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
(iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(v) Extra attempt of any question or any part of the attempted question will not be considered.
(vi) Use of Calculator is allowed.
Q. No. 1.
(a) Find the directional derivative of
2
2
)
,
,
(
yz
y
x
z
y
x
f
+
=
at the point (2,-1, 1 ) in
the direction of the vector
?
2
2
k
j
i
+
+
(10)
(b) Evaluate
dy
x
dx
y
xy
c
2
2)
(
+
+
∫
where c is bounded by the line y = x and the
curve y = x2
(10)
Q. No. 2.
(a) Find the constants a, b, and c so that
F=(x+2y+az) i + (bx – 3y – z ) j + (4x + cy + 2 z) k
is irrotational and hence find the function ψ such that
ψ
∇
=
F
(10)
(b) The forces F1,F2,F3,F4,F5 and F6 act along the sides of a regular hexagone taken
in order. Verify that all the forces will be in equilibrium if,
∑ F= 0, and F1 – F4 = F3 –F6 =F5 –F2 .
(10)
Q. No. 3.
(a) A system of forces acts on a plate in the form of an equilateral triangle of side 2a.
The moment of the forces about the three vertices are M1, M2 and M3
respectively. Find the magnitudes of the resultant.
(10)
(b)
If a particle P move with a velocity V given by V2 = n2 (ax2 + 2bx + c). Show that
P executes a simple harmonic motion. Find the centre, the amplitude and the time
period of the motion?
(10)
Q. No. 4.
(a) What is the difference between linear differential equation and Bernoulli’s
equation? Also find the solution of the following differential equation.
y
y
dx
dy
x
−
=
+
1
(10)
(b) Use the method of undetermined coefficient