1

### IIT-JEE 1980

Subjective
Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$

Solve it.
2

### 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.

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

### IIT-JEE 1979

Subjective
If $$\alpha ,\,\beta$$ are the roots of $${x^2} + px + q = 0$$ and $$\gamma ,\,\delta$$ are the roots of $${x^2} + rx + s = 0,$$ evaluate $$\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \right)$$ $$\left( {\beta - \delta } \right)$$ in terms of $$p,\,q,\,r$$ and $$s$$.

deduce the condition that the equations have a common root.

$$q{\left( {r - p} \right)^2} - p\left( {r - p} \right)\left( {s - q} \right) + {\left( {s - q} \right)^2};\,\,{\left( {q - s} \right)^2} = \left( {r - p} \right)\left( {ps - qr} \right)$$
4

### IIT-JEE 1978

Subjective
Sketch the solution set of the following system of inequalities: $${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$\$

Solve it.

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