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1

### IIT-JEE 1984

Subjective
If $$\alpha$$ be a repeated root of a quadratic equation $$f(x)=0$$ and $$A(x), B(x)$$ and $$C(x)$$ be polynomials of degree $$3$$, $$4$$ and $$5$$ respectively,
then show that $$\left| {\matrix{ {A\left( x \right)} & {B\left( x \right)} & {C\left( x \right)} \cr {A\left( \alpha \right)} & {B\left( \alpha \right)} & {C\left( \alpha \right)} \cr {A'\left( \alpha \right)} & {B'\left( \alpha \right)} & {C'\left( \alpha \right)} \cr } } \right|$$ is
divisible by $$f(x)$$, where prime denotes the derivatives.

Solve it.
2

### IIT-JEE 1982

Subjective
Let $$f$$ be a twice differentiable function such that

$$f''\left( x \right) = - f\left( x \right),$$ and $$f'\left( x \right) = g\left( x \right),h\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \right]^2}$$

Find $$h\left( {10} \right)$$ if $$h(5)=11$$

$$11$$
3

### IIT-JEE 1981

Subjective
Let $$y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}$$. Find $${{dy} \over {dx}}$$

$${e^{x\,\sin {x^3}}}\left[ {\sin {x^3} + 3{x^3}\cos {x^3}} \right] + {\left( {\tan x} \right)^x}\left[ {{{2x} \over {\sin x}} + \log \,\tan x} \right]$$
4

### IIT-JEE 1980

Subjective
Given $$y = {{5x} \over {3\sqrt {{{\left( {1 - x} \right)}^2}} }} + {\cos ^2}\left( {2x + 1} \right)$$; Find $${{dy} \over {dx}}$$.

$${{dy} \over {dx}} = \left\{ {\matrix{ {{5 \over 3}.{1 \over {{{\left( {1 - x} \right)}^2}}} - 2\,\sin \left( {4x + 2} \right),} & {x < 1} \cr { - {5 \over 3}.{1 \over {{{\left( {x - 1} \right)}^2}}} - 2\,\sin \left( {4x + 2} \right),} & {x > 1} \cr } } \right.$$

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