NTA JEE Main 8th April 2019 Shift 2

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:

Let $$f: R \rightarrow R$$ be a differentiable function satisfying $$f'(3) + f'(2) = 0$$. Then $$\lim_{x \to 0} \frac{1 + f(3 + x) - f(3)}{1 + f(2 - x) - f(2)}^{\frac{1}{x}}$$ is equal to:

Which one of the following statements is not a tautology?

A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:

Two vertical poles of height, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:

If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:

Let the numbers 2, b, c be in an A.P. and $$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{pmatrix}$$. If $$\det(A) \in [2, 16]$$, then $$c$$ lies in the interval:

If the system of linear equations
$$x - 2y + kz = 1$$
$$2x + y + z = 2$$
$$3x - y - kz = 3$$
has a solution $$(x, y, z), z \neq 0$$, then $$(x, y)$$ lies on the straight line whose equation is:

Let $$f(x) = a^x$$ ($$a > 0$$) be written as $$f(x) = f_1(x) + f_2(x)$$, where $$f_1(x)$$ is an even function and $$f_2(x)$$ is an odd function. Then $$f_1(x + y) + f_1(x - y)$$ equals:

Let $$f: [-1, 3] \rightarrow R$$ be defined as
$$f(x) = \begin{cases} x + x, & -1 \le x < 1 \\ x + x, & 1 \le x < 2 \\ x + x, & 2 \le x \le 3 \end{cases}$$
where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at: