1
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
For $$a \in R,\,|a|\, > 1$$, let

$$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( {{1 \over {{{(an + 1)}^2}}} + {1 \over {{{(an + 2)}^2}}} + ... + {1 \over {{{(an + n)}^2}}}} \right)}}} \right) = 54$$
A
$$-$$6
B
$$-$$7
C
8
D
$$-$$9
2
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : R be a function. We say that f has

PROPERTY 1 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}$$ exists and is finite, and

PROPERTY 2 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}$$ exists and is finite. Then which of the following options is/are correct?
A
f(x) = sin x has PROPERTY 2
B
f(x) = x2/3 has PROPERTY 1
C
f(x) = |x| has PROPERTY 1
D
f(x) = x|x| has PROPERTY 2
3
JEE Advanced 2019 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : R $$\to$$ R be given by

$$f(x) = \left\{ {\matrix{ {{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x < 0;} \cr {{x^2} - x + 1,} & {0 \le x < 1;} \cr {{2 \over 3}{x^3} - 4{x^2} + 7x - {8 \over 3},} & {1 \le x < 3;} \cr {(x - 2){{\log }_e}(x - 2) - x + {{10} \over 3},} & {x \ge 3;} \cr } } \right\}$$

Then which of the following options is/are correct?
A
f is increasing on ($$-$$$$\infty$$, 0)
B
f' is not differentiable at x = 1
C
f is onto
D
f' has a local maximum at x = 1
4
JEE Advanced 2018 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : (0, $$\pi$$) $$\to$$ R be a twice differentiable function such that $$\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x$$ for all x$$\in$$ (0, $$\pi$$).

If $$f\left( {{\pi \over 6}} \right) = - {\pi \over {12}}$$, then which of the following statement(s) is (are) TRUE?
A
$$f\left( {{\pi \over 4}} \right) = {\pi \over {4\sqrt 2 }}$$
B
$$f(x) < {{{x^4}} \over 6} - {x^2}$$ for all x$$\in$$(0, $$\pi$$)
C
There exists $$\alpha$$$$\in$$(0, $$\pi$$) such that f'($$\alpha$$) = 0
D
$$f''\left( {{\pi \over 2}} \right) + f\left( {{\pi \over 2}} \right) = 0$$
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