**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 1^{2} + 2^{2} + 3^{2} + ................. + n^{2} = 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) / n^{2}] = (n + 1)^{2} [TB 95/13] **[2]**

**Q. 10.** Solve the equation x^{2} - 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} = 2^{x} [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 x^{2} in the expansion of (1 + x )^{m} is 6. [TB 171/12] **[2]**

**Q. 16.** If the coefficient of a^{r - 1}, a^{r} and a^{r + 1} in the expansion of (1 + a)^{n} are in arithmetic progression, prove that n^{2} - n(4r + 1) + 4r^{2} - 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 S_{1}, S_{2}, S_{3} are the sum of thfirst n natural umbers, their squares and their cubes, respectively. Show that 9S_{2}^{2} = S_{3} (1 + 8S_{1}) [TB 200/24]

**Q. 22.** Find sin x/2, cos x/2 and tan x/2 when **[3]**

**Q. 23.** **[3]**