1
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let $$f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$$ be a function which

satisfies $$f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy}$$. then $$(a+b)$$ is equal to

A
$$- 2\pi (\pi + 2)$$
B
$$- \pi (\pi - 2)$$
C
$$- \pi (\pi + 2)$$
D
$$- 2\pi (\pi - 2)$$
2
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1

The integral $$16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}$$ is equal to

A
$${{11} \over {12}} + {\log _e}4$$
B
$${{11} \over 6} + {\log _e}4$$
C
$${{11} \over {12}} - {\log _e}4$$
D
$${{11} \over 6} - {\log _e}4$$
3
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

The minimum value of the function $$f(x) = \int\limits_0^2 {{e^{|x - t|}}dt}$$ is :

A
2
B
$$2(e-1)$$
C
$$e(e-1)$$
D
$$2e-1$$
4
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

$$\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx}$$ is equal to :

A
$${\pi \over 2}$$
B
$${\pi \over 3}$$
C
$${\pi \over 6}$$
D
$$2\pi$$
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