1
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?
A
$\mathbb{Z} \cup T_1 \cup T_2 \subset S$
B
$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set.
C
$T_2 \cap(2024, \infty) \neq \phi$
D
For any given $a, b \in \mathbb{Z}, \cos (\pi(a+b \sqrt{2}))+i \sin (\pi(a+b \sqrt{2})) \in \mathbb{Z}$ if and only if $b=0$, where $i=\sqrt{-1}$.
2
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
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 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1} $$. Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$$, the ordered pair (x, y) lies on the circle $${x^2} + {y^2} + 5x - 3y + 4 = 0$$, Then which of the following statements is (are) TRUE?
A
$$\alpha$$ = $$-$$1
B
$$\alpha$$$$\beta$$ = 4
C
$$\alpha$$$$\beta$$ = $$-$$4
D
$$\beta$$ = 4
4
JEE Advanced 2020 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let S be the set of all complex numbers z
satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE?
A
$$\left| {z + {1 \over 2}} \right|$$ $$ \le $$ $${{1 \over 2}}$$ for all z$$ \in $$S
B
|z| $$ \le $$ 2 for all z$$ \in $$S
C
$$\left| {z + {1 \over 2}} \right|\, \ge {1 \over 2}$$ for all z$$ \in $$S
D
The set S has exactly four elements
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