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1

IIT-JEE 2006

Subjective
Match the following :

Column $$I$$
(A) $$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot x - \log {{\left( {\sin x} \right)}^{\sin x}}} \right)dx}$$
(B) Area bounded by $$- 4{y^2} = x$$ and $$x - 1 = - 5{y^2}$$
(C) Cosine of the angle of intersection of curves $$y = {3^{x - 1}}\log x$$ and $$y = {x^x} - 1$$ is
(D) Let $${{dy} \over {dx}} = {6 \over {x + y}}$$ where $$y(0)=0$$ then value of $$y$$ when $$x+y=6$$ is

Column $$II$$
(p) $$1$$
(q) $$0$$
(r) $$6\ln 2$$
(s) $${4 \over 3}$$

$$\left( A \right) - \left( p \right),\left( B \right) - \left( s \right),\left( C \right) - \left( p \right),\left( D \right) - \left( r \right)$$
2

IIT-JEE 2006

Subjective
The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx}}$$ is.

$$5051.$$
3

IIT-JEE 2005

Subjective
$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable
function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f'(x)=g(x).$$
If $${f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.$$ Prove that there exists some $$c \in \left( { - 3,3} \right)$$
such that $$g(c).g''(c)<0.$$

Solve it.
4

IIT-JEE 2005

Subjective
If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\left[ {\matrix{ {f\left( { - 1} \right)} \cr {f\left( 1 \right)} \cr {f\left( 2 \right)} \cr } } \right] = \left[ {\matrix{ {3{a^2} + 3a} \cr {3{b^2} + 3b} \cr {3{c^2} + 3c} \cr } } \right],\,\,f\left( x \right)$$ is a quadratic
function and its maximum value occurs at a point $$V$$. $$A$$ is a point of intersection of $$y=f(x)$$ with $$x$$-axis and point $$B$$ is such that chord $$AB$$ subtends a right angle at $$V$$. Find the area enclosed by $$f(x)$$ and chord $$AB$$.

$${{125} \over 3}$$ sq. units

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