95M-4
Sr. No. 5
EXAMINATION OF MARINE ENGINEER OFFICER
Function: Marine Engineering at Operational Level
MATHEMATICS
M.E.O. Class IV
(Time allowed - 3hours)
Morning Paper
India (2002)
Total Marks 100
NB : (1)All Questions are Compulsory
(2)All Questions carry equal marks
(3)Neatness in handwriting and clarity in expression carries weightage
1. a) Find the sum to ‘n’ terms of the sequence {an} when an = 5 – 6n, n  N.
b) Insert three A.Ms between 3 and 19.
sin A  cos A  1
= sec A + tan A.
sin A  cos A  1

9
5
b) Prove that cos 2 cos – cos 3 cos
= sin 5 sin
.
2
2
2
2. a) Prove that
3. a) If Cr denotes the binomial coefficient nCr, prove that
(2 n)!
C20 + C21 + … + C24 =
.
(n! ) 2
b) Expand (a + b)6 – (a – b)6. Hence find the value of

 
6
2 1 

6
2 1 .
4. Differentiate with respect to x:
1
a) (sin x)
tan1 x
c) sin–1(3x – 4x3)
 3  x 3
b) (x – 4) 

 1 x 
 1 x 2  1 x 2
d) tan–1 
 1  x 2  1  x 2



5. At the end of t seconds, the distances of a moving point from two rectangular axes are
given by x = a + c cos t, y = b + c sin t. Show that the resultant velocity and acceleration
are constant in magnitude. Show also the equation to the locus of the moving point.

between 0 and  for x, choosing the same
2
section and with the same axis. Obtain from the graphs an approximate value for x which

satisfies the equation x = sin x + .
2
6. Draw the graphs of y = sin x and y = x –
7. Find the area of the loop of the curve ay2 = (a – x)x2.
8. The center of a regular hexagon is at the origin and one vertex is given by
the diagram. Determine the other vertices.
Y B
C
A
X
o
D
F
E
9. Find the area between the parabolas, y2 = 20x, x2 = 16y.
10. Insert four G.Ms between 2 and 64.
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3 + i on