NTA JEE Mains 29th Jan 2024 Shift 2

Let the set $$C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\}$$. Then $$\sum_{(x,y) \in C}(x + y)$$ is equal to ______.

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - \sqrt{6}x + 3 = 0$$ such that $$\text{Im}(\alpha) > \text{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by $$3$$ and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta} + \alpha^{98} = 3^n(a + ib), i = \sqrt{-1}$$. Then $$n + a + b$$ is equal to ______.

Remainder when $$64^{32^{32}}$$ is divided by $$9$$ is equal to ______.

Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2 = 4x$$. If $$P$$ also lies on the chord of the parabola $$x^2 = 8y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha - 28)(\beta - 8)$$ is equal to ______.

Let the slope of the line $$45x + 5y + 3 = 0$$ be $$27r_1 + \frac{9r_2}{2}$$ for some $$r_1, r_2 \in R$$. Then $$\lim_{x \to 3}\left(\int_3^x \frac{8t^2}{\frac{3r_2 x}{2} - r_2 x^2 - r_1 x^3 - 3x} dt\right)$$ is equal to ______.

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$ax + by + c = 0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations $$x + y + z = 6, 2x + 5y + \alpha z = \beta$$ and $$x + 2y + 3z = 4$$, has infinitely many solutions. Then $$(PQ)^2$$ is equal to ______.

Let $$f(x) = \sqrt{\lim_{r \to x}\left\{\frac{2r^2[(f(r))^2 - f(x)f(r)]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}}\right\}}$$ be differentiable in $$(-\infty, 0) \cup (0, \infty)$$ and $$f(1) = 1$$. Then the value of $$ae$$, such that $$f(a) = 0$$, is equal to ______.

If $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1 - \sin 2x} \, dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3}$$, where $$\alpha, \beta$$ and $$\gamma$$ are rational numbers, then $$3\alpha + 4\beta - \gamma$$ is equal to ______.

Let the area of the region $$\{(x, y) : 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2 + 2, 2x + 2\}\}$$ be $$A$$. Then $$12A$$ is equal to ______.

Let $$O$$ be the origin, and $$M$$ and $$N$$ be the points on the lines $$\frac{x-5}{4} = \frac{y-4}{1} = \frac{z-5}{3}$$ and $$\frac{x+8}{12} = \frac{y+2}{5} = \frac{z+11}{9}$$ respectively such that $$MN$$ is the shortest distance between the given lines. Then $$\vec{OM} \cdot \vec{ON}$$ is equal to ______.