1
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
$$- \infty < f\left( x \right) < 0$$
2
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
$$f'\left( x \right) < f\left( x \right),{3 \over 4} < x < 1$$
3
JEE Advanced 2013 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ white balls, $$3$$ red balls and $$4$$ black balls. A third box $${B_3}$$ contains $$3$$ white balls, $$4$$ red balls and $$5$$ black balls.

If $$2$$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $$2$$ balls are drawn from box $${B_2}$$ is

A
$${{116} \over {181}}$$
B
$${{126} \over {181}}$$
C
$${{65} \over {181}}$$
D
$${{55} \over {181}}$$
4
JEE Advanced 2013 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ white balls, $$3$$ red balls and $$4$$ black balls. A third box $${B_3}$$ contains $$3$$ white balls, $$4$$ red balls and $$5$$ black balls.

If $$1$$ ball is drawn from each of the boxex $${B_1},$$ $${B_2}$$ and $${B_3},$$ the probability that all $$3$$ drawn balls are of the same colour is

A
$${{82} \over {648}}$$
B
$${{90} \over {648}}$$
C
$${{558} \over {648}}$$
D
$${{566} \over {648}}$$
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