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1

### IIT-JEE 1994

Subjective
If $$x$$ is not an integral multiple of $$2\pi$$ use mathematical induction to prove that : $$\cos x + \cos 2x + .......... + \cos nx = \cos {{n + 1} \over 2}x\sin {{nx} \over 2}\cos ec{x \over 2}$$\$

Solve it.
2

### IIT-JEE 1993

Subjective
Using mathematical induction, prove that
$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right)} \right\}$$

Solve it.
3

### IIT-JEE 1993

Subjective
Prove that $$\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,}$$ where $$k = \left( {3n} \right)/2$$ and $$n$$ is an even positive integer.

Solve it.
4

### IIT-JEE 1992

Subjective
Let $$p \ge 3$$ be an integer and $$\alpha$$, $$\beta$$ be the roots of $${x^2} - \left( {p + 1} \right)x + 1 = 0$$ using mathematical induction show that $${\alpha ^n} + {\beta ^n}.$$
(i) is an integer and (ii) is not divisible by $$p$$

Solve it.

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