NTA JEE Main 2nd September 2020 Shift 2

Let $$f(x)$$ be a quadratic polynomial such that $$f(-1) + f(2) = 0$$. If one of the roots of $$f(x) = 0$$ is 3, then its other root lies in:

The imaginary part of $$(3 + 2\sqrt{-54})^{\frac{1}{2}} - (3 - 2\sqrt{-54})^{\frac{1}{2}}$$ can be:

Let $$n > 2$$ be an integer. Suppose that there are $$n$$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of $$n$$ is:

If the sum of first 11 terms of an A.P. $$a_1, a_2, a_3, \ldots$$ is $$0$$ $$(a_1 \ne 0)$$ then the sum of the A.P. $$a_1, a_3, a_5, \ldots, a_{23}$$ is $$ka_1$$ where $$k$$ is equal to:

Let $$S$$ be the sum of the first 9 terms of the series: $$\{x + ka\} + \{x^2 + (k+2)a\} + \{x^3 + (k+4)a\} + \{x^4 + (k+6)a\} + \ldots$$ where $$a \ne 0$$ and $$x \ne 1$$. If $$S = \frac{x^{10} - x + 45a(x-1)}{x-1}$$, then $$k$$ is equal to:

If the equation $$\cos^4\theta + \sin^4\theta + \lambda = 0$$ has real solutions for $$\theta$$ then $$\lambda$$ lies in interval:

The set of all possible values of $$\theta$$ in the interval $$(0, \pi)$$ for which the points $$(1, 2)$$ and $$(\sin\theta, \cos\theta)$$ lie on the same side of the line $$x + y = 1$$ is:

The area (in sq. units) of an equilateral triangle inscribed in the parabola $$y^2 = 8x$$, with one of its vertices on the vertex of this parabola is:

For some $$\theta \in \left(0, \frac{\pi}{2}\right)$$, if the eccentricity of the hyperbola, $$x^2 - y^2\sec^2\theta = 10$$ is $$\sqrt{5}$$ times the eccentricity of the ellipse, $$x^2\sec^2\theta + y^2 = 5$$, then the length of the latus rectum of the ellipse, is:

$$\lim_{x \to 0} \left(\tan\left(\frac{\pi}{4} + x\right)\right)^{1/x}$$ is equal to: