1
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let

$$\alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)}$$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$g(x)=2^{\alpha x}+2^{\alpha(1-x)} .$$

Then, which of the following statements is/are TRUE ?
A
The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
B
The maximum value of $g(x)$ is $1+2^{\frac{1}{3}}$
C
The function $g(x)$ attains its maximum at more than one point
D
The function $g(x)$ attains its minimum at more than one point
2
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $$(\bar{z})^{2}+\frac{1}{z^{2}}$$ are integers, then which of the following is/are possible value(s) of $|z|$ ?
A
$\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$
B
$\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$
C
$\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$
D
$\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$
3
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, the circle $G_{i}$ touches $G_{i+1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?
A
If $n=4$, then $(\sqrt{2}-1) r < R$
B
If $n=5$, then $r < R$
C
If $n=8$, then $(\sqrt{2}-1) r < R$
D
If $n=12$, then $\sqrt{2}(\sqrt{3}+1) r > R$
4
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let

\begin{aligned} & \vec{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k} \text {, } \\ & \vec{b}=\hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}, \quad b_{2}, b_{3} \in \mathbb{R} \text {, } \\ & \vec{c}=c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}, \quad c_{1}, c_{2}, c_{3} \in \mathbb{R} \end{aligned}

be three vectors such that $b_{2} b_{3}>0, \vec{a} \cdot \vec{b}=0$ and

$$\left(\begin{array}{ccc} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{array}\right)\left(\begin{array}{l} 1 \\ b_{2} \\ b_{3} \end{array}\right)=\left(\begin{array}{r} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{array}\right) .$$

Then, which of the following is/are TRUE?
A
$\vec{a} \cdot \vec{c}=0$
B
$\vec{b} \cdot \vec{c}=0$
C
$|\vec{b}|>\sqrt{10}$
D
$|\vec{c}| \leq \sqrt{11}$
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