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1

IIT-JEE 1985

Subjective
Find the sum of the series : $$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$$

Answer

$${{{2^{mn}} - 1} \over {{2^{mn}}\left( {{2^n} - 1} \right)}}$$
2

IIT-JEE 1984

Subjective
If $$n$$ is a natural number such that
$$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}$$ and $${p_1},{p_2},\,\,......,\,{p_k}$$ are distinct primes, then show that $$In$$ $$n \ge k$$ $$in$$ 2

Answer

Solve it.
3

IIT-JEE 1984

Subjective
If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$$

Answer

Solve it.
4

IIT-JEE 1983

Subjective
Find three numbers $$a,b,c$$ between $$2$$ and $$18$$ such that
(i) their sum is $$25$$
(ii) the numbers $$2,$$ $$a, b$$ are consecutive terms of an A.P. and
(iii) the numbers $$b,c,18$$ are consecutive terms of a G.P.

Answer

$$5, 8, 12$$

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