NTA JEE Main 10th April 2023 Shift 1

Let the complex number $$z = x + iy$$ be such that $$\frac{2z - 3i}{2z + i}$$ is purely imaginary. If $$x + y^2 = 0$$, then $$y^4 + y^2 - y$$ is equal to

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

If the coefficient of $$x^7$$ in $$\left(ax - \frac{1}{bx^2}\right)^{13}$$ and the coefficient of $$x^{-5}$$ in $$\left(ax + \frac{1}{bx^2}\right)^{13}$$ are equal, then $$a^4 b^4$$ is equal to:

$$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$$ is equal to

A line segment $$AB$$ of length $$\lambda$$ moves such that the points $$A$$ and $$B$$ remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment $$AB$$ in the ratio 2:3, is a circle of radius

Let the ellipse $$E: x^2 + 9y^2 = 9$$ intersect the positive $$x$$- and $$y$$-axes at the points $$A$$ and $$B$$ respectively. Let the major axis of $$E$$ be a diameter of the circle $$C$$. Let the line passing through $$A$$ and $$B$$ meet the circle $$C$$ at the point $$P$$. If the area of the triangle with vertices $$A$$, $$P$$ and the origin $$O$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m - n$$ is equal to

The negation of the statement $$p \vee q \wedge q \vee \sim r$$ is

If $$A$$ is a $$3 \times 3$$ matrix and $$|A| = 2$$, then $$|3 \text{ adj}(|3A| \cdot A^2)|$$ is equal to

For the system of linear equations
$$2x - y + 3z = 5$$
$$3x + 2y - z = 7$$
$$4x + 5y + \alpha z = \beta$$,
which of the following is NOT correct?

If $$f(x) = \frac{\tan^{-1} x + \log_e 123}{x \log_e 1234 - \tan^{-1} x}$$, $$x > 0$$, then the least value of $$f(f(x)) + f\left(f\left(\frac{4}{x}\right)\right)$$ is