1
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
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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