NTA JEE Mains 6th April 2024 Shift 1

Let $$\alpha, \beta$$ be the distinct roots of the equation $$x^2 - (t^2 - 5t + 6)x + 1 = 0, t \in \mathbb{R}$$ and $$a_n = \alpha^n + \beta^n$$. Then the minimum value of $$\frac{a_{2023} + a_{2025}}{a_{2024}}$$ is

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $$A = \{n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of 3 nor a multiple of 4}\}$$. Then the number of elements in $$A$$ is

Let a variable line of slope $$m > 0$$ passing through the point $$(4, -9)$$ intersect the coordinate axes at the points $$A$$ and $$B$$. The minimum value of the sum of the distances of $$A$$ and $$B$$ from the origin is

If $$A(3, 1, -1)$$, $$B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right)$$, $$C(2, 2, 1)$$ and $$D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$$ are the vertices of a quadrilateral $$ABCD$$, then its area is

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m + n^2$$ is equal to

Let $$C$$ be the circle of minimum area touching the parabola $$y = 6 - x^2$$ and the lines $$y = \sqrt{3}|x|$$. Then, which one of the following points lies on the circle $$C$$?

Let $$f : (-\infty, \infty) - \{0\} \rightarrow \mathbb{R}$$ be a differentiable function such that $$f'(1) = \lim_{a \to \infty} a^2 f\left(\frac{1}{a}\right)$$. Then $$\lim_{a \to \infty} \frac{a(a+1)}{2} \tan^{-1}\left(\frac{1}{a}\right) + a^2 - 2\log_e a$$ is equal to

The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Let the relations $$R_1$$ and $$R_2$$ on the set $$X = \{1, 2, 3, \ldots, 20\}$$ be given by $$R_1 = \{(x, y) : 2x - 3y = 2\}$$ and $$R_2 = \{(x, y) : -5x + 4y = 0\}$$. If $$M$$ and $$N$$ be the minimum number of elements required to be added in $$R_1$$ and $$R_2$$, respectively, in order to make the relations symmetric, then $$M + N$$ equals