1
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha \gamma + \beta \delta$ is equal to:

A

2

B

-6

C

6

D

-2

2
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :
A

10

B

4

C

8

D

2

3
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :
A
$4\left(\log _{\mathrm{e}} 2-2\right)$
B
$\log _{e^2} 2+2$
C
$2-\log \mathrm{e}^2$
D
$4\left(\log _e 2+2\right)$
4
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $\mathrm{B}=\mathrm{PAP}{ }^{\top}, \mathrm{C}=\mathrm{P}^{\top} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is :
A

127

B

2049

C

258

D

65

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