1
JEE Advanced 2017 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only CORRECT combination?
A
(III) (iii) (R)
B
(IV) (iv) (S)
C
(II) (ii) (Q)
D
(I0 (i) (P)
2
JEE Advanced 2016 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1
The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is
A
$${1 \over {64}}$$
B
$${1 \over {32}}$$
C
$${1 \over {27}}$$
D
$${1 \over {25}}$$
3
JEE Advanced 2013 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
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
$$ - \infty < f\left( x \right) < 0$$
4
JEE Advanced 2013 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
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
$$f'\left( x \right) < f\left( x \right),{3 \over 4} < x < 1$$
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