NTA JEE Main 2025 April 3rd Shift 1

Let A be a matrix of order $$3 \times 3$$ and $$|A| = 5$$. If $$|2\text{adj}(3A \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$$, $$\alpha, \beta, \gamma \in \mathbb{N}$$ then $$\alpha + \beta + \gamma$$ is equal to

Let a line passing through the point $$(4, 1, 0)$$ intersect the line $$L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$ at the point $$A(\alpha, \beta, \gamma)$$ and the line $$L_2 : x - 6 = y = -z + 4$$ at the point $$B(a, b, c)$$. Then $$\begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix}$$ is equal to

Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 + \sqrt{3}x - 16 = 0$$, and $$\gamma$$ and $$\delta$$ be the roots of $$x^2 + 3x - 1 = 0$$. If $$P_n = \alpha^n + \beta^n$$ and $$Q_n = \gamma^n + \delta^n$$, then $$\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$$ is equal to

The sum of all rational terms in the expansion of $$(2 + \sqrt{3})^8$$ is

Let $$A = \{-3, -2, -1, 0, 1, 2, 3\}$$. Let R be a relation on A defined by $$xRy$$ if and only if $$0 \leq x^2 + 2y \leq 4$$. Let $$l$$ be the number of elements in R and $$m$$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $$l + m$$ is equal to

A line passing through the point $$P(\sqrt{5}, \sqrt{5})$$ intersects the ellipse $$\frac{x^2}{36} + \frac{y^2}{25} = 1$$ at A and B such that $$(PA) \cdot (PB)$$ is maximum. Then $$5(PA^2 + PB^2)$$ is equal to :

The sum $$1 + 3 + 11 + 25 + 45 + 71 + \ldots$$ upto 20 terms, is equal to

If the domain of the function $$f(x) = \log_e\left(\frac{2x - 3}{5 + 4x}\right) + \sin^{-1}\left(\frac{4 + 3x}{2 - x}\right)$$ is $$[\alpha, \beta)$$, then $$\alpha^2 + 4\beta$$ is equal to

If $$\sum_{r=1}^{9} \left(\frac{r+3}{2^r}\right) \cdot \,^{9}C_r = \alpha\left(\frac{3}{2}\right)^9 - \beta$$, $$\alpha, \beta \in \mathbb{N}$$, then $$(\alpha + \beta)^2$$ is equal to

The number of solutions of the equation $$2x + 3\tan x = \pi$$, $$x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$$ is