1
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let

$$\alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)}$$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$g(x)=2^{\alpha x}+2^{\alpha(1-x)} .$$

Then, which of the following statements is/are TRUE ?
A
The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
B
The maximum value of $g(x)$ is $1+2^{\frac{1}{3}}$
C
The function $g(x)$ attains its maximum at more than one point
D
The function $g(x)$ attains its minimum at more than one point
2
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : R $$\to$$ R be given by

$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define

$$F(x) = \int\limits_0^x {f(t)dt}$$, x > 0

Then which of the following options is/are correct?
A
F(x) $$\ne$$ 0 for all x $$\in$$ (0, 5)
B
F has a local maximum at x = 2
C
F has two local maxima and one local minimum in (0, $$\infty$$)
D
F has a local minimum at x = 1
3
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0

Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.

Then which of the following options is/are correct?
A
$$|{x_n} - {y_n}|\, > 1$$
B
$${x_{n + 1}} - {x_n}\, > 2$$ for every n
C
x1 < y1
D
$${x_n} \in \left( {2n,\,2n + {1 \over 2}} \right)$$ for every n
4
JEE Advanced 2017 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
f : R $$\to$$ R is a differentiable function such that f'(x) > 2f(x) for all x$$\in$$R, and f(0) = 1 then
A
f(x) > e2x in (0, $$\infty$$)
B
f'(x) < e2x in (0, $$\infty$$)
C
f(x) is increasing in (0, $$\infty$$)
D
f(x) is decreasing in (0, $$\infty$$)
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