1
JEE Main 2020 (Online) 9th January Morning Slot
+4
-1
Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $$\in$$ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $$\in$$ (a, b), $${{f(c) - f(a)} \over {f(b) - f(c)}}$$ is greater than :
A
1
B
$${{b - c} \over {c - a}}$$
C
$${{b + a} \over {b - a}}$$
D
$${{c - a} \over {b - c}}$$
2
JEE Main 2020 (Online) 9th January Morning Slot
+4
-1
If $$f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr {{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \over 3}}}} \over {{x^{{4 \over 3}}}}};} & {x > 0} \cr } } \right.$$
is continuous at x = 0, then a + 2b is equal to :
A
0
B
-1
C
-2
D
1
3
JEE Main 2020 (Online) 8th January Evening Slot
+4
-1
Let S be the set of all functions ƒ : [0,1] $$\to$$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $$\in$$ (0,1), depending on ƒ, such that
A
$$\left| {f(c) - f(1)} \right| < \left| {f'(c)} \right|$$
B
$$\left| {f(c) + f(1)} \right| < \left( {1 + c} \right)\left| {f'(c)} \right|$$
C
$$\left| {f(c) - f(1)} \right| < \left( {1 - c} \right)\left| {f'(c)} \right|$$
D
None
4
JEE Main 2020 (Online) 8th January Evening Slot
+4
-1
$$\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}$$ is equal to
A
$$- {1 \over 5}$$
B
$$- {1 \over 10}$$
C
0
D
$${1 \over 10}$$
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