NTA JEE Main 25th February 2021 Shift 2

Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 - 6x - 2 = 0$$. If $$a_n = \alpha^n - \beta^n$$ for $$n \geq 1$$, then the value of $$\dfrac{a_{10} - 2a_8}{3a_9}$$ is:

If $$\alpha, \beta \in R$$ are such that $$1 - 2i$$ (here $$i^2 = -1$$) is a root of $$z^2 + \alpha z + \beta = 0$$, then $$(\alpha - \beta)$$ is equal to:

The minimum value of $$f(x) = a^{a^x} + a^{1 - a^x}$$, where $$a, x \in R$$ and $$a > 0$$, is equal to:

If $$0 < x, y < \pi$$ and $$\cos x + \cos y - \cos(x + y) = \frac{3}{2}$$, then $$\sin x + \cos y$$ is equal to:

If the curve $$x^2 + 2y^2 = 2$$ intersects the line $$x + y = 1$$ at two points $$P$$ and $$Q$$, then the angle subtended by the line segment $$PQ$$ at the origin is

A hyperbola passes through the foci of the ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

The contrapositive of the statement "If you will work, you will earn money" is:

If for the matrix, $$A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$$, $$AA^T = I_2$$, then the value of $$\alpha^4 + \beta^4$$ is:

Let $$A$$ be a $$3 \times 3$$ matrix with det$$(A) = 4$$. Let $$R_i$$ denote the $$i^{th}$$ row of $$A$$. If a matrix $$B$$ is obtained by performing the operation $$R_2 \to 2R_2 + 5R_3$$ on $$2A$$, then det$$(B)$$ is equal to:

The following system of linear equations
$$2x + 3y + 2z = 9$$
$$3x + 2y + 2z = 9$$
$$x - y + 4z = 8$$