1
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1

Let $$f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}$$. Then the value of $$\left| {\int_0^{\pi /2} {f(\theta )d\theta } } \right|$$ is _____________.

2
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1

Let $$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha$$ and $$\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta$$.

If $$\int\limits_{\beta - {8 \over 3}}^{2\alpha - 1} {Max\left\{ {{{9 - {x^2}} \over {5 - x}},x} \right\}dx = {\alpha _1} + {\alpha _2}{{\log }_e}\left( {{8 \over {15}}} \right)}$$ then $${\alpha _1} + {\alpha _2}$$ is equal to _____________.

3
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1

Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then $${{{R_2}} \over {{R_1}}}$$ is equal to ______________.

4
JEE Main 2021 (Online) 31st August Evening Shift
Numerical
+4
-1
If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = $${3 \over 2}$$ and the curve y = 1 + 4x $$-$$ x2, then 12 m is equal to _____________.
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