1

### 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.

Solve it.
2

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 $$x = 1,\,y = 0,\,z = 0,\,w = 0$$ 3 ### IIT-JEE 1980 Subjective For what values of $$m,$$ does the system of equations $$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr }$$$

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

$$m \in \left( { - \alpha ,\, - {{15} \over 2}} \right) \cup \left( {30,\,\alpha } \right)$$
4

### IIT-JEE 1980

Subjective
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.

$$\left[ { - 1,\left. 1 \right)\, \cup \left[ {3,\left. \alpha \right)} \right.} \right.$$

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