1
JEE Main 2024 (Online) 5th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

For the function

$$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$

consider the following two statements :

(I) $$f$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$.

(II) $$f^{\prime}$$ is decreasing in $$\left(0, \frac{\pi}{2}\right)$$.

Between the above two statements,

A
only (I) is true.
B
both (I) and (II) are true.
C
only (II) is true.
D
neither (I) nor (II) is true.
2
JEE Main 2024 (Online) 4th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x)=3 \sqrt{x-2}+\sqrt{4-x}$$ be a real valued function. If $$\alpha$$ and $$\beta$$ are respectively the minimum and the maximum values of $$f$$, then $$\alpha^2+2 \beta^2$$ is equal to

A
42
B
38
C
24
D
44
3
JEE Main 2024 (Online) 4th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the sum of the maximum and the minimum values of the function $$f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$$ be $$\frac{m}{n}$$, where $$\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$$. Then $$\mathrm{m}+\mathrm{n}$$ is equal to :

A
217
B
182
C
201
D
195
4
JEE Main 2024 (Online) 1st February Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :
A
$\left(0, \frac{1}{\sqrt{5}}\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$
B
$\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(\frac{1}{\sqrt{5}}, \infty\right)$
C
$\left(-\frac{1}{\sqrt{5}}, 0\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$
D
$\left(-\infty, \frac{1}{\sqrt{5}}\right) \cup\left(0, \frac{1}{\sqrt{5}}\right)$
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