JEE Advanced 2025 Paper 1 Online

Let $L_1$ be the line of intersection of the planes given by the equations$2x + 3y + z = 4$ and $x + 2y + z = 5$.Let $L_2$ be the line passing through the point $P(2, -1, 3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equa...

Let ℝ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let\(S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2| \}.\)Then which of the followin...

For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying
$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $ $ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $ $ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3}...

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by$$ f(n) = \begin{cases} \frac{(n + 1)}{2} & \text{if } n \text{ is odd,} \\ \frac{(4-n)}{2} & \text{if...

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be a function such that f(x) > 0 for all x ∈ ℝ, and f(x+y) = f(x)f(y) for all x, y ∈ ℝ.
Let the real numbers a₁, a₂, ..., a₅₀ be in an arithmetic progression. If f(a₃₁) = 64f(a₂₅), and
...

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by$f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$Then ...

Let α and β be the real numbers such that
$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $
Then the value of α + β is ___________.

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and cho...

Consider the matrix$$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that$$ ...

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly $10$ elements, be denoted by $\mathcal{S}$.Then the number of elements in $\mathcal{S}$ is ______________...

Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example, $2210222$ is in $S$, but $0210222$ is NOT in $S$.Then the number of elements $x$ in $S$ such that at least one of the digits $0$ ...

Three students $S_1, S_2,$ and $S_3$ are given a problem to solve. Consider the following events:
U: At least one of $S_1, S_2,$ and $S_3$ can solve the problem,
V: $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve t...

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$.
Define the functions $f: \mathbb{R} \to \mathbb{R}$, $g: \mathbb{R} \to \mathbb{R}$, and $h: \mathbb{R} \to \mathbb{R}$ by
$f(x) = a...

Consider the following frequency distribution:table {width: 100%;border-collapse: collapse;margin: 20px 0;} th {background-color: blue;color: white;border: 1px solid lightgrey;padding: 10px;} td {border: 1px solid lightgrey;padding: 10px;te...

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside th...

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that
$...