NTA JEE Main 11th April 2023 Shift 1

The number of integral solution $$x$$ of $$\log_{x + \frac{7}{2}}\left(\frac{x-7}{2x-3}\right)^2 \geq 0$$ is

Let $$w_1$$ be the point obtained by the rotation of $$z_1 = 5 + 4i$$ about the origin through a right angle in the anticlockwise direction, and $$w_2$$ be the point obtained by the rotation of $$z_2 = 3 + 5i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_1 - w_2$$ is equal to

The number of triplets $$(x, y, z)$$ where $$x, y, z$$ are distinct non negative integers satisfying $$x + y + z = 15$$, is

Let $$x_1, x_2, \ldots, x_{100}$$ be in an arithmetic progression, with $$x_1 = 2$$ and their mean equal to 200. If $$y_i = ix_i - i$$, $$1 \leq i \leq 100$$, then the mean of $$y_1, y_2, \ldots, y_{100}$$ is

The number of elements in the set $$S = \{\theta \in [0, 2\pi]: 3\cos^4\theta - 5\cos^2\theta - 2\sin^6\theta + 2 = 0\}$$ is

Consider ellipses $$E_k: kx^2 + k^2y^2 = 1$$, $$k = 1, 2, \ldots, 20$$. Let $$C_k$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$E_k$$. If $$r_k$$ is the radius of the circle $$C_k$$, then the value of $$\sum_{k=1}^{20} \frac{1}{r_k^2}$$ is

Let R be a rectangle given by the lines $$x = 0$$, $$x = 2$$, $$y = 0$$ and $$y = 5$$. Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$, $$\alpha \in (0, 2)$$ and $$\beta \in (0, 5)$$, be such that the line segment AB divides the area of the rectangle R in the ratio 4:1. Then, the mid-point of AB lies on a

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then how many received medals in exactly two of three events?

Let A be a 2 $$\times$$ 2 matrix with real entries such that $$A' = \alpha A + 1$$, where $$\alpha \in \mathbb{R} - \{-1, 1\}$$. If det$$(A^2 - A) = 4$$, the sum of all possible values of $$\alpha$$ is equal to