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95M-4
Sr. No. 2
EXAMINATION OF MARINE ENGINEER OFFICER
Function: Marine Engineering at Operational Level
MATHEMATICS
M.E.O. Class IV
(Time allowed - 3hours)
Morning Paper
India (2004)
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
(4) Illustration of an Answer with clear sketches / diagrams carries weightage.
2
1.
1
1



a) Solve  x    2 x   4  – 11 = 0.
x
x



b) If a + b = c + d where a and c are rational and
b,
d are irrational, then a = c, b = d.
ax  ax a
 .
ax  ax x
log c a
b) Show that logba = .
log c b
2.
a) Solve
3.
a) If a, b, c are in A.P., prove that b + c, c + a, a + b are in A.P.
b) Solve the equation 1 + 6 + 11 + 16 + … + x = 148.
4.
a) If A + B + C = , show that
cos 2A + cos 2B + cos 2C = –1 – 4 cosA cosB cosC.
b) Evaluate tan 105°.
5.
a) Find the equation of the locus of a point which is equidistant from the points (–4, 0), (3, 5).
b) Show that the line joining the points (5, 6), (2, 2) is perpendicular to the line joining the
points (3, –1) and (–1, 2).
6.
a) Find the point of intersection of the lines 3x – 4y = 7, and 3x + 4y = 17.
b) Find the value of ‘m’ so that the line x – 6y + m = 0 may pass through the intersection of the
lines 2x + 3y + 4 – 0, x + 4y + 1 = 0.
7.
a) Find the condition that the line y = mx + c becomes a tangent to the circle
x2 + y2 = a2.
b) The equation of a chord of the circle x2 + y2 + 4x – 6y = 0 is given by
x + 2y = 0. Find the equation of the circle described on this chord as diameter.
8.
Integrate
3

1 
a)  x 

x

1
c)
ax  b  ax  c
9.
x3
b)
2x  1
2x
d)
2x
Find the root mean square value of the expression
a sin pt + b cos qt over the interval (0, 2).
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