NTA JEE Main 12th April 2019 Shift 1

Let P be the point of intersection of the common tangents to the parabola $$y^2 = 12x$$ and the hyperbola $$8x^2 - y^2 = 8$$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:

If the truth value of the statement $$p \to (\sim q \vee r)$$ is false F, then the truth values of the statements p, q, r are respectively

If the data $$x_1, x_2, \ldots, x_{10}$$ is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2000, then the standard deviation of this data is:

If $$B = \begin{pmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{pmatrix}$$ is the inverse of a 3$$\times$$3 matrix A, then the sum of all values of $$\alpha$$ for which det(A) + 1 = 0, is:

If A is a symmetric matrix and B is skew-symmetric matrix such that $$A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix}$$, then AB is equal to:

The value of $$\sin^{-1}\frac{12}{13} - \sin^{-1}\frac{3}{5}$$ is equal to:

For $$x \in R$$, Let [x] denotes the greatest integer $$\leq x$$, then the sum of the series $$\left[-\frac{1}{3}\right] + \left[-\frac{1}{3} - \frac{1}{100}\right] + \left[-\frac{1}{3} - \frac{2}{100}\right] + \ldots + \left[-\frac{1}{3} - \frac{99}{100}\right]$$ is

For $$x \in \left(0, \frac{3}{2}\right)$$, let $$f(x) = \sqrt{x}$$, $$g(x) = \tan x$$ and $$h(x) = \frac{1 - x^2}{1 + x^2}$$. If $$\phi(x) = ((h \circ f) \circ g)(x)$$, then $$\phi\left(\frac{\pi}{3}\right)$$ is equal to:

If $$e^y + xy = e$$, the ordered pair $$\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)$$ at x = 0 is equal to

A 2m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is: