1
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$ with $$f(2)=\frac{1}{2}$$ and $$f(4)=\frac{1}{4}$$.

Consider the following two statements :

(A) : $$f(x) \leq 1$$, for all $$x \in[2,4]$$

(B) : $$f(x) \geq \frac{1}{8}$$, for all $$x \in[2,4]$$

Then,

A
Neither statement (A) nor statement (B) is true
B
Only statement (A) is true
C
Only statement (B) is true
D
Both the statements $$(\mathrm{A})$$ and (B) are true
2
JEE Main 2023 (Online) 10th April Evening Shift
MCQ (Single Correct Answer)
+4
-1

Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime}(x) > 0, x \in(0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0, a)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to :

A
$$\frac{3 \pi}{4}$$
B
$$\pi$$
C
$$\frac{5 \pi}{4}$$
D
$$\frac{3 \pi}{2}$$
3
JEE Main 2023 (Online) 10th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

The slope of tangent at any point (x, y) on a curve $$y=y(x)$$ is $${{{x^2} + {y^2}} \over {2xy}},x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ is :

A
$$- 4\sqrt 2$$
B
$$2\sqrt 3$$
C
$$4\sqrt 3$$
D
$$- 2\sqrt 3$$
4
JEE Main 2023 (Online) 10th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$$^2$$) is equal to :

A
1025
B
900
C
800
D
675
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