NTA JEE Mains 8th April 2024 Shift 1

The sum of all the solutions of the equation $$(8)^{2x} - 16 \cdot (8)^x + 48 = 0$$ is :

Let $$z$$ be a complex number such that $$|z + 2| = 1$$ and $$\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}$$. Then the value of $$|\text{Re}(z + 2)|$$ is

If the set $$R = \{(a, b) : a + 5b = 42, a, b \in \mathbb{N}\}$$ has $$m$$ elements and $$\sum_{n=1}^{m}(1 - i^{n!}) = x + iy$$, where $$i = \sqrt{-1}$$, then the value of $$m + x + y$$ is

If $$\sin x = -\frac{3}{5}$$, where $$\pi < x < \frac{3\pi}{2}$$, then $$80(\tan^2 x - \cos x)$$ is equal to

The equations of two sides AB and AC of a triangle ABC are $$4x + y = 14$$ and $$3x - 2y = 5$$, respectively. The point $$\left(2, -\frac{4}{3}\right)$$ divides the third side BC internally in the ratio $$2 : 1$$. The equation of the side BC is

Let the circles $$C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$$ and $$C_2 : (x - 8)^2 + \left(y - \frac{15}{2}\right)^2 = r_2^2$$ touch each other externally at the point $$(6, 6)$$. If the point $$(6, 6)$$ divides the line segment joining the centres of the circles $$C_1$$ and $$C_2$$ internally in the ratio $$2 : 1$$, then $$(\alpha + \beta) + 4(r_1^2 + r_2^2)$$ equals

Let $$H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be the hyperbola, whose eccentricity is $$\sqrt{3}$$ and the length of the latus rectum is $$4\sqrt{3}$$. Suppose the point $$(\alpha, 6), \alpha > 0$$ lies on $$H$$. If $$\beta$$ is the product of the focal distances of the point $$(\alpha, 6)$$, then $$\alpha^2 + \beta$$ is equal to

Let $$A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$$. If $$A^3 = 4A^2 - A - 21I$$, where $$I$$ is the identity matrix of order $$3 \times 3$$, then $$2a + 3b$$ is equal to

Let $$[t]$$ be the greatest integer less than or equal to $$t$$. Let $$A$$ be the set of all prime factors of 2310 and $$f : A \rightarrow \mathbb{Z}$$ be the function $$f(x) = \left[\log_2\left(x^2 + \left[\frac{x^3}{5}\right]\right)\right]$$. The number of one-to-one functions from $$A$$ to the range of $$f$$ is

For the function $$f(x) = (\cos x) - x + 1, x \in \mathbb{R}$$, between the following two statements (S1) $$f(x) = 0$$ for only one value of $$x$$ in $$[0, \pi]$$. (S2) $$f(x)$$ is decreasing in $$\left[0, \frac{\pi}{2}\right]$$ and increasing in $$\left[\frac{\pi}{2}, \pi\right]$$.