NTA JEE Main 8th April 2023 Shift 2

Let $$m$$ and $$n$$ be the numbers of real roots of the quadratic equations $$x^2 - 12x + [x] + 31 = 0$$ and $$x^2 - 5|x + 2| - 4 = 0$$ respectively, where $$[x]$$ denotes the greatest integer $$\leq x$$. Then $$m^2 + mn + n^2$$ is equal to

Let $$A = \left\{\theta \in (0, 2\pi) : \frac{1 + 2i\sin\theta}{1 - i\sin\theta} \text{ is purely imaginary}\right\}$$. Then the sum of the elements in $$A$$ is

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $$(6!)k$$ then $$k$$ is equal to

Let $$a_n$$ be $$n^{th}$$ term of the series $$5 + 8 + 14 + 23 + 35 + 50 + \ldots$$ and $$S_n = \sum_{k=1}^{n} a_k$$. Then $$S_{30} - a_{40}$$ is equal to

The absolute difference of the coefficients of $$x^{10}$$ and $$x^7$$ in the expansion of $$\left(2x^2 + \frac{1}{2x}\right)^{11}$$ is equal to

$$25^{190} - 19^{190} - 8^{190} + 2^{190}$$ is divisible by

The value of $$36(4\cos^2 9^\circ - 1)(4\cos^2 27^\circ - 1)(4\cos^2 81^\circ - 1)(4\cos^2 243^\circ - 1)$$ is

Let $$A(0, 1)$$, $$B(1, 1)$$ and $$C(1, 0)$$ be the mid-points of the sides of a triangle with incentre at the point $$D$$. If the focus of the parabola $$y^2 = 4ax$$ passing through $$D$$ is $$\left(\alpha + \beta\sqrt{2}, 0\right)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha}{\beta^2}$$ is equal to

Let $$O$$ be the origin and $$OP$$ and $$OQ$$ be the tangents to the circle $$x^2 + y^2 - 6x + 4y + 8 = 0$$ at the points $$P$$ and $$Q$$ on it. If the circumcircle of the triangle $$OPQ$$ passes through the point $$\left(\alpha, \frac{1}{2}\right)$$, then a value of $$\alpha$$ is

If $$\alpha > \beta > 0$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, and
$$\lim_{x \to \frac{1}{\alpha}} \left(\frac{1 - \cos(x^2 + bx + a)}{2(1 - \alpha x)^2}\right)^{\frac{1}{2}} = \frac{1}{k}\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)$$, then $$k$$ is equal to