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1

JEE Advanced 2013 Paper 2 Offline

MCQ (Single Correct Answer)
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

If the function $${e^{ - x}}f\left( x \right)$$ assumes its minimum in the interval $$\left[ {0,1} \right]$$ at $$x = {1 \over 4}$$, which of the following is true?

A
$$f'\left( x \right) < f\left( x \right),{1 \over 4} < x < {3 \over 4}$$
B
$$f'\left( x \right) > f\left( x \right),0 < x < {1 \over 4}$$
C
$$f'\left( x \right) < f\left( x \right),0 < x < {1 \over 4}$$
D
$$f'\left( x \right) < f\left( x \right),{3 \over 4} < x < 1$$
2

JEE Advanced 2013 Paper 2 Offline

MCQ (Single Correct Answer)
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

Which of the following is true for $$0 < x < 1?$$

A
$$0 < f\left( x \right) < \infty $$
B
$$ - {1 \over 2} < f\left( x \right) < {1 \over 2}$$
C
$$ - {1 \over 4} < f\left( x \right) < 1$$
D
$$ - \infty < f\left( x \right) < 0$$
3

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.

Consider the statements:
$$P:$$ There exists some $$x \in R$$ such that $$f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)$$
$$Q:\,\,$$ There exists some $$x \in R$$ such that $$2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)$$
Then

A
both $$P$$ and $$Q$$ are true
B
$$P$$ is true and $$Q$$ is false
C
$$P$$ is false and $$Q$$ is true
D
both $$P$$ and $$Q$$ are false
4

IIT-JEE 2012 Paper 2 Offline

MCQ (Single Correct Answer)
Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.

Which of the following is true?

A
$$g$$ is increasing on $$\left( {1,\infty } \right)$$
B
$$g$$ is decreasing on $$\left( {1,\infty } \right)$$
C
$$g$$ is increasing on $$(1, 2)$$ and decreasing on $$\left( {2,\infty } \right)$$
D
$$g$$ is decreasing on $$(1, 2)$$ and increasing on $$\left( {2,\infty } \right)$$

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