1
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points $P, Q$ and $R$, then the area of $\triangle P Q R$ is :

A
4
B
6
C
5
D
7
2
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left(\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2\right)$ is :

A
$24 \pi^2$
B
$18 \pi^2$
C
$22 \pi^2$
D
$31 \pi^2$
3
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ satisfies $f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0, \mathrm{a}>0$, then the area of the region $\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(a x), 0 \leq x \leq 2\}$ is :

A
$\mathrm{e}^2-1$
B
$e^4+1$
C
$\mathrm{e}^2+1$
D
$e^4-1$
4
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $\sum_\limits{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}$, then $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=1}^n\left(\frac{1}{T_r}\right)$ is equal to :

A
$\frac{2}{3}$
B
$\frac{1}{3}$
C
1
D
0
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