1
IIT-JEE 2011 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that
$$f\left( x \right) = f\left( {1 - x} \right)$$ for all $$x \in \left[ { - 1,2} \right]$$

Let $${R_1} = \int\limits_{ - 1}^2 {xf\left( x \right)dx,}$$ and $${R_2}$$ be the area of the region bounded by $$y=f(x),$$ $$x=-1,$$ $$x=2,$$ and the $$x$$-axis. Then

A
$${R_1} = 2{R_2}$$
B
$${R_1} = 3{R_2}$$
C
$${2R_1} = {R_2}$$
D
$${3R_1} = {R_2}$$
2
IIT-JEE 2010 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The area bounded by the curve $$y=f(x)$$ and the lines $$x=0,$$ $$y=0$$ and $$x=t,$$ lies in the interval

A
$$\left( {{3 \over 4},3} \right)$$
B
$$\left( {{{21} \over {64}},{{11} \over {16}}} \right)$$
C
$$\left( {9,10} \right)$$
D
$$\left( {0,{{21} \over {64}}} \right)$$
3
IIT-JEE 2009 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1

Let $$f$$ be a non-negative function defined on the interval $$[0,1]$$.

If $$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} }$$, and $$f(0) = 0$$, then

A
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) > {1 \over 3}$$
B
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) > {1 \over 3}$$
C
$$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
D
$$f\left( {{1 \over 2}} \right) > {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
4
IIT-JEE 2008 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
The area of the region between the curves $$y = \sqrt {{{1 + \sin x} \over {\cos x}}}$$
and $$y = \sqrt {{{1 - \sin x} \over {\cos x}}}$$ bounded by the lines $$x=0$$ and $$x = {\pi \over 4}$$ is
A
$$\int\limits_0^{\sqrt 2 - 1} {{t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt}$$
B
$$\int\limits_0^{\sqrt 2 - 1} {{4t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt}$$
C
$$\int\limits_0^{\sqrt 2 + 1} {{4t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt}$$
D
$$\int\limits_0^{\sqrt 2 + 1} {{t \over {\left( {1 + {t^2}} \right)\sqrt {1 - {t^2}} }}dt}$$
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