JEE Advanced 2025 Paper 1 Online

MCQ (More than One Correct Answer)

-2

Let ℝ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let

$$S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2| \}.$$

Then which of the following statements is (are) TRUE?

S is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$

S is a circle with centre $\left(\frac{1}{3}, \frac{8}{3} \right)$

S is a circle with radius $\frac{\sqrt{2}}{3}$

S is a circle with radius $\frac{2\sqrt{2}}{3}$

JEE Advanced 2024 Paper 1 Online

MCQ (More than One Correct Answer)

-2

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?

$\mathbb{Z} \cup T_1 \cup T_2 \subset S$

$T_1 \cap\left(0, \frac{1}{2024}\right)=\phi$, where $\phi$ denotes the empty set.

$T_2 \cap(2024, \infty) \neq \phi$

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 $i=\sqrt{-1}$.

JEE Advanced 2022 Paper 2 Online

MCQ (More than One Correct Answer)

-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 $|z|$ ?

$\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$

$\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$

$\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$

$\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$

JEE Advanced 2021 Paper 1 Online

MCQ (More than One Correct Answer)

-2

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?

$$\alpha$$ = $$-$$1

$$\alpha$$$$\beta$$ = 4

$$\alpha$$$$\beta$$ = $$-$$4

$$\beta$$ = 4

JEE Advanced Subjects