1
IIT-JEE 1983
Subjective
+2
-0
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 $$${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$$
2
IIT-JEE 1983
Subjective
+2
-0
Find all real values of $$x$$ which satisfy $${x^2} - 3x + 2 > 0$$ and $${x^2} - 2x - 4 \le 0$$
3
IIT-JEE 1982
Subjective
+2
-0
Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.
4
IIT-JEE 1982
Subjective
+5
-0
$$mn$$ 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.
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