NTA JEE Main 10th April 2019 Shift 2

The number of real roots of the equation $$5 + 2^x -1 = 2^x \cdot 2^x - 2$$ is:

If z and $$\omega$$ are two complex numbers such that $$z\omega = 1$$ and $$\arg(z) - \arg(\omega) = \frac{\pi}{2}$$, then:

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

The sum $$1 + \frac{1^3 + 2^3}{1 + 2} + \frac{1^3 + 2^3 + 3^3}{1 + 2 + 3} + \ldots + \frac{1^3 + 2^3 + 3^3 + \ldots + 15^3}{1 + 2 + 3 + \ldots + 15} - \frac{1}{2}(1 + 2 + 3 + \ldots + 15)$$ is equal to

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. with $$a_6 = 2$$. Then, the common difference of this A.P., which maximise the product $$a_1 \cdot a_4 \cdot a_5$$, is:

Let a, b and c be in G.P. with common ratio r, where $$a \neq 0$$ and $$0 \lt r \leq \frac{1}{2}$$. If 3a, 7b and 15c are the first three terms of an A.P., then the 4$$^{th}$$ term of this A.P. is:

The smallest natural number n, such that the coefficient of x in the expansion of $$\left(x^2 + \frac{1}{x^3}\right)^n$$ is $$^nC_{23}$$, is

Lines are drawn parallel to the line $$4x - 3y + 2 = 0$$, at a distance $$\frac{3}{5}$$ units from the origin. Then which one of the following points lies on any of these lines?

The locus of the centres of the circles, which touch the circle, $$x^2 + y^2 = 1$$ externally, also touch the y-axis and lie in the first quadrant, is:

If the line $$ax + y = c$$, touches both the curves $$x^2 + y^2 = 1$$ and $$y^2 = 4\sqrt{2}x$$, then c is equal to: