1
JEE Advanced 2019 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Change Language
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
2
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
$$f''\left( {{\pi \over 2}} \right) + f\left( {{\pi \over 2}} \right) = 0$$
3
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
For every twice differentiable function $$f:R \to [ - 2,2]$$ with $${(f(0))^2} + {(f'(0))^2} = 85$$, which of the following statement(s) is(are) TRUE?
A
There exist r, s $$ \in $$ R, where r < s, such that f is one-one on the open interval (r, s)
B
There exists x0 $$ \in $$ ($$-$$4, 0) such that |f'(x0)| $$ \le $$ 1
C
$$\mathop {\lim }\limits_{x \to \infty } f(x) = 1$$
D
There exists $$\alpha $$$$ \in $$($$-$$4, 4) such that f($$\alpha $$) + f"($$\alpha $$) = 0 and f'($$\alpha $$) $$ \ne $$ 0
4
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : R $$ \to $$ R and g : R $$ \to $$ R be two non-constant differentiable functions. If f'(x) = (e(f(x) $$-$$ g(x))) g'(x) for all x $$ \in $$ R and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?
A
f(2) < 1 $$-$$ loge 2
B
f(2) > 1 $$-$$ loge 2
C
g(1) > 1 $$-$$ loge 2
D
g(1) < 1 $$-$$ loge 2
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