NTA JEE Mains 9th April 2024 Shift 2

Let $$\alpha, \beta; \alpha > \beta$$, be the roots of the equation $$x^2 - \sqrt{2}x - \sqrt{3} = 0$$. Let $$P_n = \alpha^n - \beta^n, n \in \mathbb{N}$$. Then $$(11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12}$$ is equal to

Let $$z$$ be a complex number such that the real part of $$\frac{z-2i}{z+2i}$$ is zero. Then, the maximum value of $$|z - (6 + 8i)|$$ is equal to

Let $$a, ar, ar^2, \ldots$$ be an infinite G.P. If $$\sum_{n=0}^{\infty} ar^n = 57$$ and $$\sum_{n=0}^{\infty} a^3 r^{3n} = 9747$$, then $$a + 18r$$ is equal to

The sum of the coefficient of $$x^{2/3}$$ and $$x^{-2/5}$$ in the binomial expansion of $$\left(x^{2/3} + \frac{1}{2}x^{-2/5}\right)^9$$ is

Two vertices of a triangle $$ABC$$ are $$A(3, -1)$$ and $$B(-2, 3)$$, and its orthocentre is $$P(1, 1)$$. If the coordinates of the point $$C$$ are $$(\alpha, \beta)$$ and the centre of the circle circumscribing the triangle $$PAB$$ is $$(h, k)$$, then the value of $$(\alpha + \beta) + 2(h + k)$$ equals

Let the foci of a hyperbola $$H$$ coincide with the foci of the ellipse $$E : \frac{(x-1)^2}{100} + \frac{(y-1)^2}{75} = 1$$ and the eccentricity of the hyperbola $$H$$ be the reciprocal of the eccentricity of the ellipse $$E$$. If the length of the transverse axis of $$H$$ is $$\alpha$$ and the length of its conjugate axis is $$\beta$$, then $$3\alpha^2 + 2\beta^2$$ is equal to

$$\lim_{x \to \frac{\pi}{2}} \left(\frac{\int_{x^3}^{(\pi/2)^3} \left(\sin(2t^{1/3}) + \cos(t^{1/3})\right) dt}{\left(x - \frac{\pi}{2}\right)^2}\right)$$ is equal to

$$\lim_{x \to 0} \frac{e - (1+2x)^{\frac{1}{2x}}}{x}$$ is equal to

If the variance of the frequency distribution

is 160, then the value of $$c \in \mathbb{N}$$ is

Let $$B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$AB^{-1} = A^{-1}$$. If $$BCB^{-1} = A$$ and $$C^4 + \alpha C^2 + \beta I = O$$, then $$2\beta - \alpha$$ is equal to