(Paper) Maths: Trig Function, Math Induction & Complex Numbers, Binomial and Sequence & Series
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PAPER : Trig
Function, Math Induction & Complex Numbers, Binomial and Sequence &
Series
Q. 1. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length 15 cm. [TB 5 /7] [1]
Q. 2. Prove that cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1 [TB 74/22] [1]
Q. 3. Prove
that if x and y are not odd multiple of 
where
[1]
Q. 4. Prove that cos ( A + B ) = cos A cos B - sin A sin B. [ TB 64/3] [2]
Q. 5. Solve sin 2x - sin 4x + sin 6x = 0 [ TB 77/23 ] [2]
Q. 6. 
[2]
Q. 7. Prove
by mathematical induction that for all 
12
+ 22 + 32 + ................. + n2 = n(n + 1)
(2n + 1) / 6 [TB 89/1] [2]
Q. 8. Prove by mathematical induction that (2n + 7) < (n + 3)2. [TB 95/24] [2]
Q. 9. Prove by mathematical induction that: (1 + 3/1) (1 + 5/4) (1 + 7/9)................... [1 + (2n + 1) / n2] = (n + 1)2 [TB 95/13] [2]
Q. 10. Solve the equation x2 - x + 2 = 0 [TB 109/6] [2]
Q. 11. Find the conjugate of (3 - 2t) (2 + 3t) / (1 + 2t ) (2 - t) [ TB 109/12] [2]
Q. 12. Find the number of non - zero integral solutions of the equation | 1 - t | x = 2x [TB 113/18] [2]
Q. 13. Find the middle term in the expansion of (1 + x)2n, where n is a positive integer. [TB 168/6] [2]
Q. 14.
Write the general term in the expansion of 
[2]
Q. 15. Find a positive value of m for which the coefficient of x2 in the expansion of (1 + x )m is 6. [TB 171/12] [2]
Q. 16. If the coefficient of ar - 1, ar and ar + 1 in the expansion of (1 + a)n are in arithmetic progression, prove that n2 - n(4r + 1) + 4r2 - 2 = 0. [TB 172/11] [2]
Q. 17. The sum of n terms of two arithmetic progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms. [TB 182/6] [2]
Q. 18. Find the sum of the series 2 + 11 + 101 + 1001 + ............. To n terms. [2]
Q. 19. Find the least n for which the sum of the series 1 + 2 + 4 + 8 + ........... is greater than 10000. [2]
Q. 20. The
sum of two numbers is 6 times their geometric means, show that the numbers
arein the ratio 
[2]
Q. 21. If S1, S2, S3 are the sum of thfirst n natural umbers, their squares and their cubes, respectively. Show that 9S22 = S3 (1 + 8S1) [TB 200/24]
Q. 22. Find
sin x/2, cos x/2 and tan x/2 when 
[3]
Q. 23. 
[3]