1
JEE Main 2025 (Online) 3rd April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $3 x+4 y+5=0$ is

A
15
B
12
C
$5 \sqrt{3}$
D
$15 \sqrt{5}$
2
JEE Main 2025 (Online) 3rd April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3} x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$. If $P_n=$ $\alpha^n+\beta^n$ and $Q_n=\gamma^n+\hat{o}^n$, then $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

A
4
B
3
C
5
D
7
3
JEE Main 2025 (Online) 2nd April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}$. If $\mathrm{P}_{10}=123, \mathrm{P}_9=76, \mathrm{P}_8=47$ and $\mathrm{P}_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

A
$x^2+x-1=0$
B
$x^2-x+1=0$
C
$x^2+x+1=0$
D
$x^2-x-1=0$
4
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If the set of all $a \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3 a$ has no real root, is the interval ( $\alpha, \beta$ ), and $X=|x \in Z ; \alpha < x < \beta|$, then $\sum\limits_{x \in X} x^2$ is equal to:
A

2139

B

2119

C

2109

D

2129

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