1
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.
2
IIT-JEE 1980
Subjective
+4
-0
Let $$y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}}$$

Find all the real values of $$x,$$ for which $$y$$ takes real values.

3
IIT-JEE 1980
Subjective
+4
-0
Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$
4
IIT-JEE 1980
Subjective
+4
-0
For what values of $$m,$$ does the system of equations $$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr }$$\$

has solution satisfying the conditions $$x > 0,\,y > 0.$$

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