NTA JEE Main 10th April 2019 Shift 2

The tangent and normal to the ellipse $$3x^2 + 5y^2 = 32$$ at the point P(2, 2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is:

If $$5x + 9 = 0$$ is the directrix of the hyperbola $$16x^2 - 9y^2 = 144$$, then its corresponding focus is:

If $$\lim_{x \to 1} \frac{x^2 - ax + b}{x - 1} = 5$$, then a + b is equal to:

The negation of the Boolean expression $$\sim s \vee (\sim r \wedge s)$$ is equivalent to

If both the mean and the standard deviation of 50 observations $$x_1, x_2, \ldots, x_{50}$$ are equal to 16, then the mean of $$(x_1 - 4)^2, (x_2 - 4)^2, \ldots, (x_{50} - 4)^2$$ is

The angles A, B & C of a ΔABC are in A.P. and a:b = 1:$$\sqrt{3}$$. If c = 4 cm, then the area (in sq. cm) of this triangle is:

The sum of the real roots of the equation $$\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix} = 0$$, is equal to:

Let $$\lambda$$ be a real number for which the system of linear equations
$$x + y + z = 6$$,
$$4x + \lambda y - \lambda z = \lambda - 2$$ and
$$3x + 2y - 4z = -5$$
has infinitely many solutions. Then $$\lambda$$ is a root of the quadratic equation:

If $$\cos^{-1}x - \cos^{-1}\frac{y}{2} = \alpha$$, where $$-1 \leq x \leq 1$$, $$-2 \leq y \leq 2$$, $$x \leq \frac{y}{2}$$, then for all x, y, $$4x^2 - 4xy\cos\alpha + y^2$$ is equal to:

Let $$f(x) = \log_e \sin x$$, $$0 < x < \pi$$ and $$g(x) = \sin^{-1}(e^{-x})$$, $$(x \geq 0)$$. If $$\alpha$$ is a positive real number such that $$a = f \circ g(\alpha)$$ and $$b = f \circ g(\alpha)$$, then