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1

### IIT-JEE 1982

Subjective
For any real $$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$ is a point on the
hyperbola $${x^2} - {y^2} = 1$$. Show that the area bounded by this hyperbola and the lines joining its centre to the points corresponding to $${t_1}$$ and $$-{t_1}$$ is $${t_1}$$.

Solve it.
2

### IIT-JEE 1982

Subjective
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx}$$

$${3 \over \pi } + {1 \over {{\pi ^2}}}$$
3

### IIT-JEE 1982

Subjective
Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.}$$

Solve it.
4

### IIT-JEE 1981

Subjective
Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$

Solve it.

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NEET

Class 12