NTA JEE Main 6th April 2023 Shift 2

For $$\alpha, \beta, z \in C$$ and $$\lambda > 1$$, if $$\sqrt{\lambda - 1}$$ is the radius of the circle $$|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$$, then $$|\alpha - \beta|$$ is equal to ______.

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ______.

If $$(20)^{19} + 2(21)(20)^{18} + 3(21)^2(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$$, then $$k$$ is equal to ______.

The value of $$\tan 9° - \tan 27° - \tan 63° + \tan 81°$$ is ______.

Let the eccentricity of an ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ is reciprocal to that of the hyperbola $$2x^2 - 2y^2 = 1$$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ______.

If the mean and variance of the frequency distribution
$$x_i$$         2,    4,    6,    8,   10,   12,   14,   16
$$f_i$$         4,    4,    $$\alpha$$,  15,    8,    $$\beta$$,     4,     5
are 9 and 15.08 respectively, then the value of $$\alpha^2 + \beta^2 - \alpha\beta$$ is ______.

Let a curve $$y = f(x)$$, $$x \in (0, \infty)$$ pass through the points $$P\left(1, \dfrac{3}{2}\right)$$ and $$Q\left(a, \dfrac{1}{2}\right)$$. If the tangent at any point $$R(b, f(b))$$ to the given curve cuts the y-axis at the point $$S(0, c)$$ such that $$bc = 3$$, then $$(PQ)^2$$ is equal to

The number of points, where the curve $$y = x^5 - 20x^3 + 50x + 2$$ crosses the x-axis, is ______.

Let $$f(x) = \dfrac{x}{(1+x^n)^{1/n}}$$, $$x \in \mathbb{R} - \{-1\}$$, $$n \in \mathbb{N}$$, $$n > 2$$. If $$f^n(x) = (f \circ f \circ f \ldots$$ upto n times$$(x)$$, then $$\lim_{n \to \infty} \int_0^1 x^{n-2}(f^n(x))dx$$ is equal to ______.

If the lines $$\dfrac{x-1}{2} = \dfrac{2-y}{3} = \dfrac{z-3}{\alpha}$$ and $$\dfrac{x-4}{5} = \dfrac{y-1}{2} = \dfrac{z}{\beta}$$ intersect, then the magnitude of the minimum value of $$8\alpha\beta$$ is ______.