If the system of linear equations

$ \begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4 \end{aligned} $

has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is

1

If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}$, $\hat{i} + 2…

Topic: JEE Main 2025 (2 April Morning Shift)

2

Let $P_n = \alpha^n + \beta^n$, $n \in \mathbb{N}$. If $P_{10} = 123$, $P_9 = 76$, $P_8 = 47$ and $P_1 = 1$, then the q…

Topic: JEE Main 2025 (2 April Morning Shift)

3

The largest $n \in \mathbb{N}$ such that $3^n$ divides $50!$ is:

Topic: JEE Main 2025 (2 April Morning Shift)

4

Let one focus of the hyperbola $\textbf{H}: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be at $(\sqrt{10}, 0)$ and the cor…

Topic: JEE Main 2025 (2 April Morning Shift)

5

Let $z$ be a complex number such that $|z| = 1$. If $\dfrac{2 + k\bar{z}}{k + z} = kz$, $k \in \mathbb{R}$, then the ma…

Topic: JEE Main 2025 (2 April Morning Shift)

6

Let $ABCD$ be a tetrahedron such that the edges $AB$, $AC$ and $AD$ are mutually perpendicular. Let the areas of the tr…

Topic: JEE Main 2025 (2 April Morning Shift)

7

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$, where $a > 0$, attains its local maximum and local minimum values a…

Topic: JEE Main 2025 (2 April Morning Shift)

8

Let the focal chord $PQ$ of the parabola $y^2 = 4x$ make an angle of $60^\circ$ with the positive $x$-axis, where $P$ l…

Topic: JEE Main 2025 (2 April Morning Shift)

9

If $S$ and $S'$ are the foci of the ellipse $\dfrac{x^2}{18} + \dfrac{y^2}{9} = 1$ and $P$ be a point on the ellipse, t…

Topic: JEE Main 2025 (2 April Morning Shift)

10

If $\theta \in[-2 \pi, 2 \pi]$, then the number of solutions of $2 \sqrt{2} \cos ^2 \theta+(2-\sqrt{6}) \cos \theta-\sq…

Topic: JEE Main 2025 (2 April Morning Shift)

1

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2…

Topic: JEE Main 2025 (2 April Morning Shift)

2

For $\alpha, \beta, \gamma \in \mathbf{R}$, if $\lim _\limits{x \rightarrow 0} \frac{x^2 \sin \alpha x+(\gamma-1) \math…

Topic: JEE Main 2025 (2 April Morning Shift)

3

The number of sequences of ten terms, whose terms are either $0$, $1$ or $2$, that contain exactly five $1$’s and exact…

Topic: JEE Main 2025 (2 April Morning Shift)

4

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x-1)}{\l…

Topic: JEE Main 2025 (2 April Morning Shift)

5

Let $a_1, a_2, a_3, \ldots$ be in an A.P. such that $ \displaystyle \sum_{k=1}^{12} 2a_{2k-1} = -\dfrac{72}{5}a_1, \qua…

Topic: JEE Main 2025 (2 April Morning Shift)

6

Let $A = {\theta \in [0,2\pi] : 1 + 10,\mathrm{Re}\left(\dfrac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta}\ri…

Topic: JEE Main 2025 (2 April Morning Shift)