1
JEE Main 2024 (Online) 31st January Morning Shift
Numerical
+4
-1

If the integral $$525 \int_\limits0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\operatorname{Cos}^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$$ is equal to $$(n \sqrt{2}-64)$$, then $$n$$ is equal to _________.

2
JEE Main 2024 (Online) 31st January Morning Shift
Numerical
+4
-1

Let $$S=(-1, \infty)$$ and $$f: S \rightarrow \mathbb{R}$$ be defined as

$$f(x)=\int_\limits{-1}^x\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61} d t \text {, }$$

Let $$\mathrm{p}=$$ Sum of squares of the values of $$x$$, where $$f(x)$$ attains local maxima on $$S$$, and $$\mathrm{q}=$$ Sum of the values of $$\mathrm{x}$$, where $$f(x)$$ attains local minima on $$S$$. Then, the value of $$p^2+2 q$$ is _________.

3
JEE Main 2024 (Online) 31st January Morning Shift
Numerical
+4
-1

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$f(x)=\frac{4^x}{4^x+2}$$ and $$M=\int_\limits{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x, N=\int_\limits{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2}$$. If $$\alpha M=\beta N, \alpha, \beta \in \mathbb{N}$$, then the least value of $$\alpha^2+\beta^2$$ is equal to __________.

4
JEE Main 2024 (Online) 30th January Morning Shift
Numerical
+4
-1

The value of $$9 \int_\limits0^9\left[\sqrt{\frac{10 x}{x+1}}\right] \mathrm{d} x$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is