1
JEE Advanced 2020 Paper 1 Offline
+3
-1
Consider the rectangles lying the region

$$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$$

and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
A
$${{3\pi \over 2}}$$
B
$$\pi$$
C
$${\pi \over {2\sqrt 3 }}$$
D
$${{\pi \sqrt 3 } \over 2}$$
2
JEE Advanced 2017 Paper 1 Offline
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$\in$$(0, $$\infty$$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty$$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty$$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only INCORRECT combination?
A
(I) (iii) (P)
B
(II) (iv) (Q)
C
(II) (ii) (P)
D
(III) (i) (R)
3
JEE Advanced 2017 Paper 1 Offline
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$\in$$(0, $$\infty$$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty$$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty$$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only CORRECT combination?
A
(I) (ii) (R)
B
(III) (iv) (P)
C
(II) (iii) (S)
D
(IV) (i) (S)
4
JEE Advanced 2017 Paper 1 Offline
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$\in$$(0, $$\infty$$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty$$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty$$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only CORRECT combination?
A
(III) (iii) (R)
B
(IV) (iv) (S)
C
(II) (ii) (Q)
D
(I0 (i) (P)
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