1
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let a differentiable function $$f$$ satisfy $$f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3$$. Then $$12 f(8)$$ is equal to :

A
19
B
34
C
17
D
1
2
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\alpha \in (0,1)$$ and $$\beta = {\log _e}(1 - \alpha )$$. Let $${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$$. Then the integral $$\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} $$ is equal to

A
$$ - \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$$
B
$$\beta - {P_{50}}(\alpha )$$
C
$${P_{50}}(\alpha ) - \beta $$
D
$$\beta + {P_{50}} - (\alpha )$$
3
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of $$\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$$ is equal to

A
$$\frac{10}{3}-\sqrt{3}+\log _{e} \sqrt{3}$$
B
$$\frac{7}{2}-\sqrt{3}-\log _{e} \sqrt{3}$$
C
$$\frac{10}{3}-\sqrt{3}-\log _{e} \sqrt{3}$$
D
$$-2+3\sqrt{3}+\log _{e} \sqrt{3}$$
4
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2-p x+\frac{5}{4} p=0$ are rational. Then the area of the region $\left\{(x, y): 0 \leq y \leq(x-q)^2, 0 \leq x \leq q\right\}$ is :
A
$\frac{125}{3}$
B
243
C
164
D
25
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