1

IIT-JEE 1983

Subjective
If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show that $$${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$$

Answer

Solve it.
2

IIT-JEE 1982

Subjective
$$mn$$ squares of equal size are arranged to from a rectangle of dimension $$m$$ by $$n$$, where $$m$$ and $$n$$ are natural numbers. Two squares will be called ' neighbours ' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares.Show that this is possible only if all the numbers used are equal.

Answer

Solve it.
3

IIT-JEE 1982

Subjective
Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.

Answer

Solve it.
4

IIT-JEE 1980

Subjective
Find the solution set of the system $$$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \cr } $$$

Answer

$$x = 1,\,y = 0,\,z = 0,\,w = 0$$

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