JEE Advanced 2018 Paper 1 Offline | Limits, Continuity and Differentiability Question 16 | Mathematics

JEE Advanced 2018 Paper 1 Offline

MCQ (More than One Correct Answer)

-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?

There exist r, s $$ \in $$ R, where r < s, such that f is one-one on the open interval (r, s)

There exists x₀ $$ \in $$ ($$-$$4, 0) such that |f'(x₀)| $$ \le $$ 1

$$\mathop {\lim }\limits_{x \to \infty } f(x) = 1$$

There exists $$\alpha $$$$ \in $$($$-$$4, 4) such that f($$\alpha $$) + f"($$\alpha $$) = 0 and f'($$\alpha $$) $$ \ne $$ 0

JEE Advanced 2018 Paper 1 Offline

MCQ (More than One Correct Answer)

-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?

f(2) < 1 $$-$$ log_e 2

f(2) > 1 $$-$$ log_e 2

g(1) > 1 $$-$$ log_e 2

g(1) < 1 $$-$$ log_e 2

JEE Advanced 2017 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

Let $$f(x) = {{1 - x(1 + |1 - x|)} \over {|1 - x|}}\cos \left( {{1 \over {1 - x}}} \right)$$

for x $$ \ne $$ 1. Then

$$\mathop {\lim }\limits_{x \to {1^ + }} f(x)$$ = 0

$$\mathop {\lim }\limits_{x \to {1^ - }} f(x)$$ does not exist

$$\mathop {\lim }\limits_{x \to {1^ - }} f(x)$$ = 0

$$\mathop {\lim }\limits_{x \to {1^ + }} f(x)$$ does not exist

JEE Advanced 2017 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

Let f : R $$ \to $$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?

$${e^x} - \int_0^x {f(t)\sin t\,dt} $$

$$f(x) + \int_0^{{\pi \over 2}} {f(t)\sin t\,dt} $$

$$f(x) - \int_0^{{\pi \over 2} - x} {f(t)\cos t\,dt} $$

x⁹ $$-$$ f(x)

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