NTA JEE Main 25th January 2023 Shift 2

Let $$z$$ be a complex number such that $$\left|\frac{z-2i}{z+i}\right| = 2$$, $$z \neq -i$$. Then $$z$$ lies on the circle of radius 2 and centre

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is

Let $$f(x) = 2x^n + \lambda$$, $$\lambda \in \mathbb{R}$$, $$n \in \mathbb{N}$$, and $$f(4) = 133$$, $$f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) - f(2))$$ is

$$\sum_{k=0}^{6} {}^{51-k}C_3$$ is equal to

The equations of two sides of a variable triangle are $$x = 0$$ and $$y = 3$$, and its third side is a tangent to the parabola $$y^2 = 6x$$. The locus of its circumcentre is:

Let $$\triangle, \nabla \in \{\wedge, \vee\}$$ be such that $$(p \to q) \triangle (p \nabla q)$$ is a tautology. Then

Let $$A, B, C$$ be $$3 \times 3$$ matrices such that $$A$$ is symmetric and $$B$$ and $$C$$ are skew-symmetric. Consider the statements
(S1) $$A^{13}B^{26} - B^{26}A^{13}$$ is symmetric
(S2) $$A^{26}C^{13} - C^{13}A^{26}$$ is symmetric
Then,

Let $$A = \begin{bmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & -i \\ 0 & 1 \end{bmatrix}$$, where $$i = \sqrt{-1}$$. If $$M = A^T BA$$, then the inverse of the matrix $$AM^{2023}A^T$$ is

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \log_{\sqrt{m}} \{\sqrt{2}(\sin x - \cos x) + m - 2\}$$, for some $$m$$, such that the range of $$f$$ is $$[0, 2]$$. Then the value of $$m$$ is _____.

The number of functions $$f : \{1, 2, 3, 4\} \to \{a \in \mathbb{Z} : |a| \leq 8\}$$ satisfying $$f(n) + \frac{1}{n}f(n+1) = 1$$, $$\forall n \in \{1, 2, 3\}$$ is