NTA JEE Mains 30th Jan 2024 Shift 1

If $$z = x + iy$$, $$xy \neq 0$$, satisfies the equation $$z^2 + i\bar{z} = 0$$, then $$|z^2|$$ is equal to :

Let $$S_a$$ denote the sum of first $$n$$ terms an arithmetic progression. If $$S_{20} = 790$$ and $$S_{10} = 145$$, then $$S_{15} - S_5$$ is :

If $$2\sin^3 x + \sin 2x \cos x + 4\sin x - 4 = 0$$ has exactly $$3$$ solutions in the interval $$\left[0, \frac{n\pi}{2}\right]$$, $$n \in \mathbb{N}$$, then the roots of the equation $$x^2 + nx + (n - 3) = 0$$ belong to :

A line passing through the point $$A(9, 0)$$ makes an angle of $$30°$$ with the positive direction of $$x$$-axis. If this line is rotated about $$A$$ through an angle of $$15°$$ in the clockwise direction, then its equation in the new position is

If the circles $$(x + 1)^2 + (y + 2)^2 = r^2$$ and $$x^2 + y^2 - 4x - 4y + 4 = 0$$ intersect at exactly two distinct points, then

The maximum area of a triangle whose one vertex is at $$(0, 0)$$ and the other two vertices lie on the curve $$y = -2x^2 + 54$$ at points $$(x, y)$$ and $$(-x, y)$$ where $$y > 0$$ is :

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let $$f : \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be a differentiable function such that $$f(0) = \frac{1}{2}$$. If $$\lim_{x \to 0} \frac{x \int_0^x f(t) dt}{e^{x^2} - 1} = \alpha$$, then $$8\alpha^2$$ is equal to :

Let $$M$$ denote the median of the following frequency distribution.

Then $$20M$$ is equal to :

If $$f(x) = \begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3 + \sin^2 2x \\ 3 + 2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3 + 2\sin^4 x & \sin^2 2x \end{vmatrix}$$ then $$\frac{1}{5}f'(0)$$ is equal to ________.