NTA JEE Main 10th April 2019 Shift 1

If a directrix of a hyperbola centered at the origin and passing through the point $$(4, -2\sqrt{3})$$ is $$5x = 4\sqrt{5}$$ and its eccentricity is e, then:

If $$\lim_{x \to 1}\frac{x^4 - 1}{x - 1} = \lim_{x \to k}\frac{x^3 - k^3}{x^2 - k^2}$$, then k is

Which one of the following Boolean expression is a tautology?

If for some $$x \in$$ R, the frequency distribution of the marks obtained by 20 students in a test is:

Then the mean of the marks is:

ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are $$\cot^{-1}(3\sqrt{2})$$ and $$\text{cosec}^{-1}(2\sqrt{2})$$ respectively, then the height of the tower (in metres) is

If the system of linear equations $$x + y + z = 5$$, $$x + 2y + 2z = 6$$, $$x + 3y + \lambda z = \mu$$, ($$\lambda, \mu \in$$ R), has infinitely many solutions, then the value of $$\lambda + \mu$$ is:

If $$\Delta_1 = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$$ and $$\Delta_2 = \begin{vmatrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & 1 \\ \cos 2\theta & 1 & x \end{vmatrix}$$, $$x \neq 0$$; then for all $$\theta \in \left(0, \frac{\pi}{2}\right)$$:

Let $$f(x) = x^2$$, $$x \in R$$. For any $$A \subseteq R$$, define $$g(A) = \{x \in R : f(x) \in A\}$$. If $$S = [0, 4]$$, then which one of the following statements is not true?

Let $$f : R \to R$$ be differentiable at $$c \in R$$ and $$f(c) = 0$$. If $$g(x) = |f(x)|$$, then at $$x = c$$, g is:

If $$f(x) = \begin{cases} \frac{\sin(p+1)x + \sin x}{x}, & x < 0 \\ q, & x = 0 \\ \frac{\sqrt{x + x^2} - \sqrt{x}}{x^{3/2}}, & x > 0 \end{cases}$$ is continuous at $$x = 0$$, then the ordered pair (p, q) is equal to: