NTA JEE Main 13th April 2023 Shift 2

Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + 2 = 0$$. Then $$\alpha^{14} + \beta^{14}$$ is equal to

Let $$S = \{z \in \mathbb{C} : \bar{z} = i(z^2 + \text{Re}(\bar{z}))\}$$. Then $$\sum_{z \in S} |z|^2$$ is equal to

All words, with or without meaning, are made using all the letters of the word $$MONDAY$$. These words are written as in a dictionary with serial numbers. The serial number of the word $$MONDAY$$ is

Let $$a_1, a_2, a_3, \ldots$$ be a G.P. of increasing positive numbers. Let the sum of its 6$$^{th}$$ and 8$$^{th}$$ terms be 2 and the product of its 3$$^{rd}$$ and 5$$^{th}$$ terms be $$\frac{1}{9}$$. Then $$6a_2 + a_4a_4 + a_6$$ is equal to

The coefficient of $$x^5$$ in the expansion of $$\left(2x^3 - \frac{1}{3x^2}\right)^5$$ is

Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15x - y = 82$$, $$6x - 5y = -4$$ and $$9x + 4y = 17$$. Then $$\alpha + 2\beta$$ and $$2\alpha - \beta$$ are the roots of the equation

Let the centre of a circle $$C$$ be $$\alpha, \beta$$ and its radius $$r < 8$$. Let $$3x + 4y = 24$$ and $$3x - 4y = 32$$ be two tangents and $$4x + 3y = 1$$ be a normal to $$C$$. Then $$(\alpha - \beta + r)$$ is equal to

If $$\lim_{x \to 0} \frac{e^{ax} - \cos(bx) - \frac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17$$, then $$5a^2 + b^2$$ is equal to

The statement $$(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q))$$ is equivalent to ____

Let for $$A = \begin{bmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$, $$|A| = 2$$. If $$|2 \ \text{adj}(2 \ \text{adj}(2A))| = 32^n$$, then $$3n + \alpha$$ is equal to