NTA JEE Main 8th April 2019 Shift 2

If three distinct numbers $$a$$, $$b$$, $$c$$ are in G.P. and the equations $$ax^{2} + 2bx + c = 0$$ and $$dx^{2} + 2ex + f = 0$$ have a common root, then which one of the following statements is correct?

The number of integral values of $$m$$ for which the equation, $$1 + m^{2}x^{2} - 21 + 3mx + 1 + 8m = 0$$ has no real root, is:

If $$z = \frac{\sqrt{3}}{2} + \frac{i}{2} (i = \sqrt{-1})$$, then $$(1 + iz + z^{5} + iz^{8})^{9}$$ is equal to:

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is:

The sum $$\sum_{k=1}^{20} k \cdot \frac{1}{2^k}$$ is equal to:

If the fourth term in the binomial expansion of $$\left(\sqrt{x^{\frac{1}{1+\log_{10}x}}} + x^{\frac{1}{12}}\right)^{6}$$ is equal to 200, and $$x > 1$$, then the value of $$x$$ is:

Suppose that the points $$h, k$$, $$(1, 2)$$ and $$(-3, 4)$$ lie on the line $$L_1$$. If a line $$L_2$$ passing through the points $$h, k$$ and $$(4, 3)$$ is perpendicular to $$L_1$$, then $$\frac{k}{h}$$ equals:

The tangent and the normal lines at the point $$(\sqrt{3}, 1)$$ to the circle $$x^{2} + y^{2} = 4$$ and the x-axis form a triangle. The area of this triangle (in square units) is:

The tangent to the parabola $$y^{2} = 4x$$ at the point where it intersects the circle $$x^{2} + y^{2} = 5$$ in the first quadrant, passes through the point:

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $$(0, 5\sqrt{3})$$, then the length of its latus rectum is: