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95M-4
Sr. No. 7
EXAMINATION OF MARINE ENGINEER OFFICER
Function: Marine Engineering at Operational Level
MATHEMATICS
M.E.O. Class IV
(Time allowed - 3hours)
Morning Paper
India (2003)
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.
1.
a) If the roots of the equation
p(q – r)x2 + q(r – p)x + r(p – q) = 0 be equal, show that
1 1 2
 
p r q
b) Express 9 as a power of e.
2.
a) For what values of k are the roots ,  of the equation
(12 k + 3)x2 + k = (9 k + 1)x such that 3 +  = 1?
b) Find (235.7)–5.18.
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. Prove that the following hold for every natural numbers n:
a) nC0 + nC1 + … + nCn = 2n.
b) nC0 + 2nC1 + … + 2nnCn = 3n.
5.
6.
a) Find the equation of the circle which passes through the points A(–2, 3), B(5, 2) and C(6, –1).
b) A circle touches the line 5x – y = 3 at the point (2, 7) and the center is on the line x + 2y = 19.
Find the equation of the circle.
Find the derivatives of the following:
a) (x3 – 2x + 3)/ x
b)
 a2  x 2 
 2

2 
a x 
c)
x2  4
x 3 ( x  1)
d) (2x2 + 3) (x2 + 4)
7. Reduce 1 – cos + i sin to the modulus argument form.
8. Show that the function y =
2
( x  1) 2
has a maximum value
and a minimum value zero.
3
27
( x  3)
9. Differentiate
a) y =
10.
sin x x
b) cosec3 x5
Integrate using by parts:
a)  x ex dx
b) x2 sin x
c)
1  cos x
.
1  cos x
c) sin–1x
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