JEE (Advanced) 2025 Paper-1

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

$$f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4,$$

$$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4,$$

$$h(x) = f(x+1) - g(x+2).$$

If $$f(x) \neq g(x)$$ for every $$x \in \mathbb{R}$$, then the coefficient of $$x^3$$ in $$h(x)$$ is

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 the problem,

$$W$$: $$S_2$$ can solve the problem and $$S_3$$ cannot solve the problem,

$$T$$: $$S_3$$ can solve the problem.

For any event $$E$$, let $$P(E)$$ denote the probability of $$E$$. If

$$P(U) = \frac{1}{2}, \quad P(V) = \frac{1}{10}, \quad$$ and $$\quad P(W) = \frac{1}{12},$$

then $$P(T)$$ is equal to

Let $$\mathbb{R}$$ denote the set of all real numbers. Define the function $$f: \mathbb{R} \to \mathbb{R}$$ by

$$f(x) = \begin{cases} 2 - 2x^2 - x^2 \sin \frac{1}{x} & \text{if } x \neq 0, \\ 2 & \text{if } x = 0. \end{cases}$$

Then which one of the following statements is TRUE?

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

$$Q^{-1} = Q^T \quad \text{and} \quad PQ = QP,$$

is

Let $$L_1$$ be the line of intersection of the planes given by the equations

$$2x + 3y + z = 4 \quad \text{and} \quad x + 2y + z = 5.$$

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 equation

$$2x + y - 2z = 6.$$

Suppose that the line $$L_2$$ meets the plane $$M$$ at the point $$Q$$. Let $$R$$ be the foot of the perpendicular drawn from $$P$$ to the plane $$M$$.

Then which of the following statements is (are) TRUE?

Let $$\mathbb{N}$$ denote the set of all natural numbers, and $$\mathbb{Z}$$ denote the set of all integers. Consider the functions $$f: \mathbb{N} \to \mathbb{Z}$$ and $$g: \mathbb{Z} \to \mathbb{N}$$ defined by

$$f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd}, \\ (4-n)/2 & \text{if } n \text{ is even}, \end{cases}$$

and

$$g(n) = \begin{cases} 3 + 2n & \text{if } n \geq 0, \\ -2n & \text{if } n < 0. \end{cases}$$

Define $$(g \circ f)(n) = g(f(n))$$ for all $$n \in \mathbb{N}$$, and $$(f \circ g)(n) = f(g(n))$$ for all $$n \in \mathbb{Z}$$.

Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}$$ 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 following statements is (are) TRUE?

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 ________.

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 the triangle $$\triangle PQR$$ such that

$$\overrightarrow{SP} + 5\,\overrightarrow{SQ} + 6\,\overrightarrow{SR} = \vec{0}.$$

Let $$E$$ and $$F$$ be the mid-points of the sides $$PR$$ and $$QR$$, respectively. Then the value of

$$\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$$

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 and 1 appears exactly twice in $$x$$, is equal to ________.