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1

### IIT-JEE 1999

Subjective
Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4

then show that the roots of the equation $$\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$$
$$+ [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$$ and $$20{x^2} + 10{(a - d)^2}x - 9 = 0$$ are reciprocals of each other.

solve it
2

### IIT-JEE 1996

Subjective
The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma$$ lie.

$$\beta \,\, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
3

### IIT-JEE 1991

Subjective
If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are $${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$,....................$$\,{1 \over {n + 1}}$$ respectively, then find the values of $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

$${{{}^n(2n + 1)\,(4n + 1) - 3} \over 3}$$
4

### IIT-JEE 1991

Subjective
Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

solve it

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