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95M-4
Sr. No. 6
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) If Cos  = n,
Sin  = m, Show that (m² - n²)sin² = 1-n²
Cos 
Sin 
8
(b) Prove that Sin  - Cos8 = -Cos2(1-2Sin²Cos²)
2. (a) Solve 6x½ - 5x¼ + 1 = 0
(b) If the sum of the roots of the equation
Prove that the product of the root is
1
+ 1
x+a
x+b
_ 1 (a² + b²)
= 1 is zero,
2
3. Find the equation of the lines through the origin making an angle of 60º with the line x + y3
+ 33 = 0. Also find the coordinates of the points where they meet the line.
4. (a) If Sn denotes the sum of n terms of an A.P. Show that its mean difference d is given by
d = Sn – 2Sn – 1 + Sn – 2
(b) The mth term of an A.P. is n and nth term is m. Find the (m + n)th term and the common
difference.
5. (a) Using Binomial Theorem Prove that,
C1 + C3 + C5 + ------- = 2n – 1, where Cr means nCr
(b) Find the 4th term of
(3x _ x³6)
7
6. (a) Evaluate (4 + 3-20)½ + (4 – 3-20)½
(b) Find the smallest positive integer n such that
1+i n =1
1-i
( )
7. (a) Find dy/dx of the following by using definition (any 2)
(i) ex (ii) ax (iii) sinx (iv) logx
(b) Find dy/dx of (any 3)
x
(i) x4 e³x tanx,
(ii) x² + y² - 3x + 8y + 10 = 0, (iii) 1 + Sinx (iv) x
1 – Cosx
x

8. (a) If t = ax² + 2bx + c, show that acceleration is proportional to the cube of velocity.
(b) The horse power P transmitted by a belt is given by
Find the speed V for P to be the greatest and find the
maximum value of P in terms of K, T, W and g.
P = K [TV _ WV³ ]
g
9. Integrate any three

(i)
1 Cos 1 dx
(ii)
x²
x
x

x5 dx
1 + x12
(c)
 Cos 6x Cos 4x dx
(d)
(x² + 1)(x² + 2)
x
10. (a) Find the area bounded by the parabola y² = 2x and the straight line x – y = 4
(b) A population grown at the rate of 5%. How long will it take to double?
--------------------------X--------------------------
dx