NTA JEE Main 26th August 2021 Shift 1

The equation $$\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}$$ represents a circle with:

The sum of the series $$\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \ldots + \frac{2^{100}}{x^{2^{100}}+1}$$ when $$x = 2$$ is:

If the sum of an infinite GP, $$a, ar, ar^2, ar^3, \ldots$$ is 15 and the sum of the squares of its each term is 150, then the sum of $$ar^2, ar^4, ar^6, \ldots$$ is:

If $$^{20}C_r$$ is the co-efficient of $$x^r$$ in the expansion of $$(1 + x)^{20}$$, then the value of $$\sum_{r=0}^{20} r^2(^{20}C_r) $$ is equal to:

The sum of solutions of the equation $$\frac{\cos x}{1+\sin x} = |\tan 2x|$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \left\{-\frac{\pi}{4}, \frac{\pi}{4}\right\}$$ is:

Let $$ABC$$ be a triangle with $$A(-3, 1)$$ and $$\angle ACB = \theta$$, $$0 < \theta < \frac{\pi}{2}$$. If the equation of the median through B is $$2x + y - 3 = 0$$ and the equation of angle bisector of C is $$7x - 4y - 1 = 0$$, then $$\tan \theta$$ is equal to:

If a line along a chord of the circle $$4x^2 + 4y^2 + 120x + 675 = 0$$, passes through the point $$(-30, 0)$$ and is tangent to the parabola $$y^2 = 30x$$, then the length of this chord is:

On the ellipse $$\frac{x^2}{8} + \frac{y^2}{4} = 1$$, let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line $$x + 2y = 0$$. Let S and S' be the foci of the ellipse and $$e$$ be its eccentricity. If A is the area of the triangle SPS', then the value of $$(5 - e^2) \cdot A$$ is

If the truth value of the Boolean expression $$((p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)) \rightarrow (p \wedge q)$$ is false, then the truth values of the statements $$p$$, $$q$$, $$r$$ respectively can be:

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If $$\alpha$$ and $$\sqrt{\beta}$$ are the mean and standard deviation respectively for correct data, then $$(\alpha, \beta)$$ is: