NTA JEE Main 9th January 2019 Shift 2

The number of all possible positive integral values of $$\alpha$$ for which the roots of the quadratic equation $$6x^2 - 11x + \alpha = 0$$ are rational numbers is:

If both the roots of the quadratic equation $$x^2 - mx + 4 = 0$$ are real and distinct and they lie in the interval (1, 5), then $$m$$ lies in the interval:
Note: In the actual JEE paper interval was $$[ 1, 5 ]$$

Let $$z_0$$ be a root of quadratic equation, $$x^2 + x + 1 = 0$$. If $$z = 3 + 6iz_0^{81} - 3iz_0^{93}$$, then $$\arg(z)$$ is equal to:

The number of natural numbers less than 7000 which can be formed by using the digits 0, 1, 3, 7, 9 (repetition of digits allowed) is equal to:

The sum of the following series $$1 + 6 + \frac{9(1^2 + 2^2 + 3^2)}{7} + \frac{12(1^2 + 2^2 + 3^2 + 4^2)}{9} + \frac{15(1^2 + 2^2 + \ldots + 5^2)}{11} + \ldots$$ up to 15 terms, is:

Let $$a$$, $$b$$ and $$c$$ be the 7$$^{th}$$, 11$$^{th}$$ and 13$$^{th}$$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $$\frac{a}{c}$$ is equal to:

The coefficient of $$t^4$$ in the expansion of $$\left(\frac{1 - t^6}{1 - t}\right)^3$$ is:

If $$0 \leq x < \frac{\pi}{2}$$, then the number of values of $$x$$ for which $$\sin x - \sin 2x + \sin 3x = 0$$, is:

Let $$S$$ be the set of all triangles in the $$xy$$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $$S$$ has area 50 sq. units, then the number of elements in the set $$S$$ is:

Let the equations of two sides of a triangle be $$3x - 2y + 6 = 0$$ and $$4x + 5y - 20 = 0$$. If the orthocenter of this triangle is at $$(1, 1)$$ then the equation of its third side is: