JEE Advance 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 $f \colon \mathbb{R} \to \mathbb{R}$  , $g \colon \mathbb{R} \to \mathbb{R}$  , and $h \colon \mathbb{R} \to \mathbb{R}$   by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$  , $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$  , and $h(x) = f(x+1) - g(x+2)$  . 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 $P(U) = \frac{1}{2}, P(V) = \frac{1}{10}$  , and $P(W) = \frac{1}{12}$  , then $P(T)$   is equal to

Let $\mathbb{R}$   denote the set of all real numbers. Define the function $f \colon \mathbb{R} \to \mathbb{R}$   by $f(x) = 2 - 2x^2 - x^2 \sin(1/x)$   if $x \neq 0$   and $f(0) = 2$  . 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 $Q^{-1} = Q^T$   and $PQ = QP$  , is

Let $L_1$   be the line of intersection of the planes $2x + 3y + z = 4$   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 $2x + y - 2z = 6$  . Suppose $L_2$   meets $M$   at $Q$  . Let $R$   be the foot of the perpendicular from $P$   to $M$  . Then which of the following is TRUE?

Consider functions $f \colon \mathbb{N} \to \mathbb{Z}$   where $f(n) = (n+1)/2$   if $n$   is odd and $(4-n)/2$   if $n$   is even, and $g \colon \mathbb{Z} \to \mathbb{N}$   where $g(n) = 3+2n$   if $n \ge 0$   and $-2n$   if $n < 0$  . Which statements are TRUE?

Let $z_1 = 1 + 2i$   and $z_2 = 3i$  . Let $S = \{(x, y) \in \mathbb{R} \times \mathbb{R} \colon |x + iy - z_1| = 2|x + iy - z_2| \}$  . Which statements are TRUE?

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

Let $P, Q, R$   be distinct points in a plane. Let $S$   be a point inside $\Delta PQR$   such that $\vec{SP} + 5\vec{SQ} + 6\vec{SR} = \vec{0}$  . Let $E$   and $F$   be mid-points of $PR$   and $QR$  . Find the value of $\frac{\text{length of } EQ}{\text{length of } SF}$  .

Find the number of 7-digit numbers formed using digits 0, 1, 2 (starting with 1 or 2) such that at least one of 0 and 1 appears exactly twice.

If $\lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1-t^2} dt + \beta x \cos x \right) = 2$  , find $\alpha + \beta$  .

Let $f(x+y)=f(x)f(y)$   with $f(x)>0$  . Let $a_1, \dots, a_{50}$   be in AP. If $f(a_{31}) = 64f(a_{25})$   and $\sum_{i=1}^{50} f(a_i) = 3(2^{25} + 1)$  , find $\sum_{i=6}^{30} f(a_i)$  .

Given $\frac{dy_1}{dx} - (\sin^2 x)y_1 = 0, y_1(1)=5$  ; $\frac{dy_2}{dx} - (\cos^2 x)y_2 = 0, y_2(1)=1/3$  ; $\frac{dy_3}{dx} - \frac{2-x^3}{x^3} y_3 = 0, y_3(1)=3/(5e)$  . Find $\lim_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x}$  .

Match List-I (Statistics values) to List-II (Numerical results) for the given frequency distribution of 19 elements with median 6.

Match List-I (Calculus properties) to List-II (Minimum natural number n or value).

Given $\vec{u} \times \vec{v} = \vec{w}$   and $\vec{v} \times \vec{w} = \vec{u}$   with $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$  . Match List-I to List-II.

A disk of radius $r$   and mass $m$   rolls without slipping inside a ring of radius $R$  . It is attached to a spring of constant $k$  . Find the angular frequency $\omega$   of small oscillations.

A particle of mass $2m$   collides perfectly elastically with a particle of mass $m$   at rest. Find the maximum angular deviation $\theta$   of the heavier particle in radians.

A square loop hinges on the X-axis and rotates at $\omega$   in a time-varying magnetic field $B(t) = B_0 \cos(\omega t) \hat{k}$   for $y \ge 0$  . Which plot represents the induced e.m.f.?

Figure 1 shows Vernier scale zero error and Fig 2 shows measurement of diameter $D$   of a tube. Calculate $D$  .

A square loop enters a magnetic field region with initial velocity $v_0$  . Given $K = \frac{B_0^2 L^2}{RM}$  , which statements are correct?

Length, breadth, and thickness of a strip are 10.5 cm, 0.05 mm, and $6.0 \mu m$  . Find volume in $cm^3$   with correct significant figures.

Three connected strings $S_1, S_2, S_3$   have linear densities $\mu, 4\mu, 16\mu$  . A wave $y = y_0 \cos(\omega t - kx)$   is sent. Which statements are correct?

Elevator height $y = 8[1 + \sin(2\pi t/T)]$  , $T = 40\pi$   s. Object mass 50 kg. Max variation of weight (in N) is?

A cube contains $35 \times 10^7$   photons of $10^{15}$   Hz. Amplitude of magnetic field is $\alpha \times 10^{-9}$   T. Find $\alpha$  .

Two black body plates P and Q transfer power $W_0$  . Two identical plates are inserted. Find ratio $W_0 / W_S$  .

Glass sphere $n = \sqrt{3}$   with air cavity. Ray reflected from point O is fully polarized. Find $\sin \theta$   (incidence at inner surface).

Single slit diffraction $bd/D = m\lambda$  . $D = 1$   m (err 1 cm), $d = 5$   mm (err 1 mm), $m = 3, \lambda = 600$   nm. Find absolute error in slit width $b$   in $\mu m$  .

Electron in $n = 3$   orbit has same de Broglie wavelength as a thermal neutron at $T$  . $T = \frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}$  . Find $\alpha$  .

Match dipole configurations (List-I) to resultant electric field at midpoint X (List-II).

Match AC circuit loads (List-I) to current $i(t)$   functions (List-II) for $V(t) = 300 \sin(400t)$  .

Match energy dependencies on Z (List-I) to physical phenomena (List-II).

Heating $\text{NH}_4\text{NO}_2$   at 60-70 $^\circ$  C and $\text{NH}_4\text{NO}_3$   at 200-250 $^\circ$  C produces nitrogen compounds X and Y. Find X and Y.

Reaction of permanganate ion with iodide ion in neutral aqueous medium produces:

Identify INCORRECT statement(s) regarding MO energy levels for homonuclear diatomic molecules.

Which pair(s) of ions is(are) diamagnetic?

Compare C-X bond properties in structures P, Q, and R.

Current (in A) flowing for 48.25 min to produce 1 mole of $Cr^{3+}$   from dichromate ions.

$[H^+]$   in $1.00 \times 10^{-3}$   M weak monobasic acid ($K_a = 4.00 \times 10^{-11}$  ) is $X \times 10^{-7}$   M. Find $X$  .

Ratio of coeff of $V_m^2$   to coeff of $V_m$   in cubic van der Waals equation for $a=6, b=0.06$   at 300K, 300atm.

Expansion work (kJ) when 144g water is electrolyzed at 300K.

Synthesis of Nylon 6,6 involves monomer X (positive carbylamine). Find grams of N2 from 10 moles of X via Dumas method.

Reaction sequence starting with 16 moles of X. Yields provided. Find grams of S.

Match group reagents (List-I) with metal ions (List-II).

Match organic reaction names (List-I) with List-II reactants/products.

Match compounds (List-I) with observations/tests (List-II).

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 $f(x) = a_{1} + 10x + a_{2}x^{2} + a_{3}x^{3} + x^{4}$  , $g(x) = b_{1} + 3x + b_{2}x^{2} + b_{3}x^{3} + x^{4}$  , and $h(x) = f(x + 1) - g(x + 2)$  . 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. If $P(U) = \frac{1}{2}, P(V) = \frac{1}{10}$  , and $P(W) = \frac{1}{12}$  , 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 $Q^{-1} = Q^{T}$   and $PQ = QP$  , is

Let $L_{1}$   be the line of intersection of the planes $2x + 3y + z = 4$   and $x + 2y + z = 5$  . Let $L_{2}$   be the line passing through $P(2, -1, 3)$   and parallel to $L_{1}$  . Let $M$   be the plane $2x + y - 2z = 6$  . Suppose $L_{2}$   meets $M$   at $Q$  . Let $R$   be the foot of the perpendicular from $P$   to $M$  . Which statement(s) is (are) TRUE?

Let $\mathbb{N}$   be natural numbers and $\mathbb{Z}$   be integers. $f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd} \\ (4-n)/2 & \text{if } n \text{ is even} \end{cases}$   and $g(n) = \begin{cases} 3+2n & n \geq 0 \\ -2n & n < 0 \end{cases}$  . Which statement(s) is (are) TRUE?

Let $z_{1} = 1 + 2i$   and $z_{2} = 3i$  . Let $S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_{1}| = 2|x + iy - z_{2}| \}$  . Then which of the following 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

Let $P, Q, R$   be distinct points in the xy-plane. Let $S$   be a point inside $\Delta PQR$   such that $\vec{SP} + 5\vec{SQ} + 6\vec{SR} = \vec{0}$  . Let $E$   and $F$   be the mid-points of $PR$   and $QR$  . Find the ratio of the length of segment $QS$   to the length of segment $EF$  .

Let $X$   be the set of all seven-digit numbers formed using digits 0, 1, 2 (first digit non-zero). Number of elements $x \in X$   such that at least one of the digits 0 and 1 appears exactly twice is:

Let $\alpha, \beta$   be real numbers such that $\lim_{x \to 0} \frac{1}{x^{3}} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1-t^{2}} dt + \beta x \cos x \right) = 2$  . Find $\alpha + \beta$  .

Let $f(x) > 0$   and $f(x+y) = f(x)f(y)$  . Let $a_{1}, ..., a_{50}$   be in AP. If $f(a_{31}) = 64f(a_{25})$   and $\sum_{i=1}^{50} f(a_{i}) = 3(2^{25} + 1)$  , find $\sum_{i=6}^{30} f(a_{i})$  .

For $x > 0$  , let $y_{1}, y_{2}, y_{3}$   satisfy $\frac{dy_{1}}{dx} - y_{1}\sin^{2}x = 0, y_{1}(1)=5$  ; $\frac{dy_{2}}{dx} - y_{2}\cos^{2}x = 0, y_{2}(1)=1/3$  ; $\frac{dy_{3}}{dx} - y_{3}(\frac{2-x^{3}}{x^{3}}) = 0, y_{3}(1)=\frac{3}{5e}$  . Find $\lim_{x \to 0^{+}} \frac{y_{1}y_{2}y_{3} + 2x}{e^{3x}\sin x}$  .

Match the frequency distribution statistics. Sum of frequencies = 19, median = 6. (P) $7f_{1} + 9f_{2}$   (Q) $19\alpha$   (Mean dev. from mean) (R) $19\beta$   (Mean dev. from median) (S) $19\sigma^{2}$   (Variance).

Match the functions of $n$  . (P) Min $n$   for $f(x) = [\frac{10x^3-45x^2+60x+35}{n}]$   continuous on [1,2]. (Q) Min $n$   for $g(x) = (2n^2-13n-15)(x^3+3x)$   increasing. (R) Smallest $n > 5$   such that $x=3$   is local minima of $h(x)=(x^2-9)^n(x^2+2x+3)$  . (S) Count of points where $l(x) = \sum_{k=0}^4 (\sin|x-k| + \cos|x-k+0.5|)$   is NOT differentiable.

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$  , $\vec{u} \times \vec{v} = \vec{w}$  , $\vec{v} \times \vec{w} = \vec{u}$  . Match the quantities: (P) $|\vec{v}|^2$   (Q) If $\alpha = \sqrt{3}$  , $\gamma^2$   (R) If $\alpha = \sqrt{3}$  , $(\beta + \gamma)^2$   (S) If $\alpha = \sqrt{2}$  , $t+3$  .

The center of a disk of radius $r$   and mass $m$   is attached to a spring of constant $k$  , inside a ring of radius $R > r$  . The disk rolls without slipping. In equilibrium, the disk is at the bottom. Assuming small displacement, find the angular frequency $\omega$   where $T = 2\pi/\omega$  .

In a scattering experiment, a particle of mass $2m$   collides with another of mass $m$   initially at rest. Assuming perfectly elastic collision, the maximum angular deviation $\theta$   of the heavier particle in radians is:

A conducting square loop rotates about the X-axis with angular speed $\omega$   in a time dependent magnetic field $\vec{B}(t) = B_{0}(\cos \omega t)\hat{k}$   for $y \ge 0$  . Which plot represents the induced e.m.f. in the loop as a function of time?

Measurement of diameter D of a tube using Vernier scale. Fig 1 shows zero error configuration, Fig 2 shows measurement. The value of D is:

A square loop enters a magnetic field $B_{0}$   for $y \ge 0$   with $v_{0}$  . $K = B_{0}^{2}L^{2}/RM$  . Which statements are correct?

Dimensions of a strip: 10.5 cm, 0.05 mm, 6.0 $\mu m$  . Volume in $cm^{3}$   with correct significant figures:

Three strings with densities $\mu, 4\mu, 16\mu$  . Wave $y = y_{0}\cos(\omega t - kx)$  . Which statements about reflection and transmission at P and Q are correct?

Elevator height $y = 8[1 + \sin(2\pi t/T)]$   where $T = 40\pi$   s. Max variation of weight of 50 kg object in N is:

Cube with $35 \times 10^{7}$   photons of $10^{15}$   Hz. Amplitude of magnetic field is $\alpha \times 10^{-9}$   T. Find $\alpha$  .

Two black body plates P and Q at temperatures $T_{P}$   and $T_{Q}$  . Power transfer $W_{0}$  . Two identical plates introduced between them. Steady state power $W_{S}$  . Ratio $W_{0}/W_{S}$   is:

Glass sphere $n = \sqrt{3}$   with air cavity. Light reflected from O is fully polarized. Angle of incidence $\theta$  . Find $\sin\theta$  .

Single slit diffraction. $bd/D = m\lambda$  . $D = 1$   m (least count 1 cm), $d = 5$   mm (least count 1 mm). Absolute error in $b$   in $\mu m$   for $m=3, \lambda=600$   nm is:

Electron in $n=3$   orbit of H-like atom. Neutron with same de Broglie wavelength has thermal energy $k_{B}T$  . If $T = \frac{Z^{2}h^{2}}{\alpha\pi^{2}a_{0}^{2}m_{N}k_{B}}$  , find $\alpha$  .

Match configuration of dipole pairs to resultant electric field at midpoint X. (P) Dipoles parallel (Q) Dipoles antiparallel (R) and (S) configurations as shown.

Circuit with load impedance Z and source $V(t) = 300\sin(400t)$  . Match load components in List-I to currents in List-II.

Match functional dependence of energy E on atomic number Z to phenomena in List-II. (P) $E \propto Z^{2}$   (Q) $E \propto (Z-1)^{2}$   (R) $E \propto Z(Z-1)$   (S) E independent of Z.

Heating $\text{NH}_{4}\text{NO}_{2}$   at $60-70^{\circ}\text{C}$   and $\text{NH}_{4}\text{NO}_{3}$   at $200-250^{\circ}\text{C}$   forms X and Y. X and Y are:

Correct order of wavelength maxima of absorption band for the complexes:

Product formed from reaction of permanganate ion with iodide ion in neutral aqueous medium:

Which hydrocarbon has the most acidic hydrogen (H)?

Incorrect statements regarding MO energy levels for homonuclear diatomic molecules:

The pair(s) of diamagnetic ions is(are):

For compounds P, Q, R as shown, which statement is correct?

Current (in amperes) flowing for 48.25 min to produce 1 mole of $Cr^{3+}$   from dichromate ions is:

Concentration of $H^{+}$   in $10^{-3}$   M solution of weak acid ($K_{a} = 4 \times 10^{-11}$  ) is $X \times 10^{-7}$   M. Value of X is:

For van der Waals gas ($a=6, b=0.06$  ) at 300K, 300 atm. Ratio of coefficient of $V_{m}^{2}$   to $V_{m}$   in cubic equation is:

Expansion work done (in kJ) when 144g water is electrolyzed completely at 300K, constant pressure:

Monomer X for Nylon 6,6 gives positive carbylamine test. Nitrogen gas evolved (in g) from Dumas analysis of 10 moles of X is:

Reaction sequence with 16 moles of X. Yields specified. Mass (in grams) of S produced is:

Match group reagents in List-I for precipitating metal ions in List-II. (P) H2S in NH4OH (Q) (NH4)2CO3 in NH4OH (R) NH4OH in NH4Cl (S) H2S in dil HCl.

Match named reactions in List-I with major products in List-II which act as reactants. (P) Stephen (Q) Sandmeyer (R) Hoffmann bromamide (S) Cannizzaro.

Match compounds in List-I with observations in List-II. (P) Aniline/Phenol deriv. (Q) Amino acid deriv. (R) Nitrogen deriv. (S) Hydrazine deriv.