JEE Main 2026 — 4 April, Morning Shift

In a screw gauge when the circular scale is given five complete rotations it moves linearly by $2.5\mathrm{mm}$. If the circular scale has 100 divisions, the least count of screw gauge is mm.

The increase in the pressure required to decrease the volume ($\Delta V$) of water is $6.3\times10^{7}\mathrm{N/m}^2$. The percentage decrease in the volume is (Bulk modulus of water = $2.1\times10^{9}\mathrm{N/m}^2$)

The time taken by a block of mass m to slide down from the highest point to the lowest point on a rough inclined plane is 50% more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45° with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is

Two nuclei of mass number 3 combine with another nucleus of mass number 4 to yield a nucleus of mass number 10. If the binding energy per nucleon for mass numbers 3, 4 and 10 are 5.6 MeV, 7.4 MeV and 6.1 MeV respectively, then in the process, $\Delta Mc^2 =$ ______ MeV.

A solid sphere of mass M and radius R is divided into two unequal parts. The smaller part having mass M/8 is converted into a sphere of radius r and the larger part is converted into a circular disc of thickness t and radius 2R. If I₁ is moment of inertia of the sphere about its centre and I₂ is moment of inertia of the disc about its diameter, the ratio I₂/I₁ is

Two projectiles are projected with the same initial velocities at 15° and 30° with respect to the horizontal. The ratio of their ranges is 1 : x. The value of x is

The graph shows variation of stopping potential V₀ with frequency ν for three photosensitive metals X₁, X₂ and X₃. Which metal will give out electrons with greater kinetic energy for the same wavelength of incident radiation?

A string A of length 0.314 m, Young's modulus 2×10¹⁰ N/m² is connected to another string B of length and Young's modulus both twice of those of A. This series combination is suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg. The net change in length of the combination is ______ mm. (radius of both strings = 0.2 mm, g = 10 m/s², neglect string masses)

One gas of n₁ moles at T₁, V₁, P₁ and another gas of n₂ moles at T₂, V₂, P₂ are mixed resulting in pressure P and volume V of the mixture. The temperature of the mixture is

An ideal gas undergoes a process P = P₀[1 + (V₀/V)²]⁻¹. Two samples A and B (2 moles each) with initial volumes V₀ and 3V₀ undergo the process and attain same pressure. The difference T_B - T_A is

A voltmeter with internal resistance x Ω can measure up to 20 V. To increase its range to 30 V, the required modification is

Two 4‑bit binary numbers A = 1101 and B = 1010 are given as inputs to the logic circuit shown. The output Y is

A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm. The length of the image is ______ cm.

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed v. If x is the separation at any instant, the time rate of change of electrostatic energy is proportional to $x^\alpha$. The value of $\alpha$ is

An insulated wire is wound to form a flat coil with N = 200 turns. Innermost radius r₁ = 3 cm, outermost r₂ = 6 cm. If current 20 mA flows, the magnetic moment is $\alpha\times10^{-2}$ A·m². The value of $\alpha$ is

Consider a circuit with a capacitor (20 μF), resistor (100 Ω) and two identical diodes. Diode forward resistance = 10 Ω. The time constant of the circuit is $\alpha\times10^{-3}$ s. The value of $\alpha$ is

A telescope with objective diameter R is used to observe a distant star emitting light of wavelength 500 nm, at a resolution of 5×10⁻⁷ radian. The value of R is ______ cm.

An unpolarized light is incident on the plane interface of air‑dielectric medium at Brewster angle. Identify the expression representing the reflected wave.

A 1 kg block is subjected to two simultaneous forces $(2\hat{i}+3\hat{j}+4\hat{k})$ N and $(3\hat{i}-\hat{j}-2\hat{k})$ N and is moved a distance of 25 m along $(3\hat{i}-4\hat{j})$ direction. The work done in this process is ______ J.

The surface tension of a soap solution is 3.5×10⁻² N/m. The work required to increase the radius of a soap bubble from 1 cm to 2 cm is $\alpha\times10^{-6}$ J. The value of $\alpha$ is ______. ($\pi=22/7$)

The velocity of a particle executing SHM along x‑axis is v² = 50 – x², where x is displacement. If the time period of motion is $\frac{x}{7}$ s, the value of x is ______.

A body of mass 2 kg begins to move under a time‑dependent force $\vec{F}=(2t\hat{i}+6t^2\hat{j})$ N. The power produced by the force at t=2 s is ______ W.

An inductor of 10 mH, capacitor 0.1 μF and resistor 100 Ω are connected in series across an AC supply 220 V, 70 Hz. The power factor is 0.5. The difference between inductive reactance and capacitive reactance is $\sqrt{3}\alpha$ Ω. The value of $\alpha$ is ______.

Let [·] denote the greatest integer function. If the domain of the function $\mathrm{f(x) = cos^{- 1}\left(\frac{4x + 2[x]}{3}\right)}$ is $[\alpha ,\beta ]$ , then $12(\alpha +\beta)$ is equal to :

If the set of all solutions of $\left|\mathbf{x}^2 +\mathbf{x} - \mathbf{g}\right| = \left|\mathbf{x}\right| + \left|\mathbf{x}^2 -\mathbf{g}\right|$ is $[\alpha ,\beta ]\cup [\gamma ,\infty)$ , then $(\alpha^2 +\beta^2 +\gamma^2)$ is equal to:

Let $\mathbf{z}$ be a complex number such that $|z + 2| = |z - 2|$ and $\arg \left(\frac{z + 3}{z - i}\right) = \frac{\pi}{4}$ . Then $|z|^2$ is equal to:

The number of functions $\mathrm{f}:\{1,2,3,4\} \to \{\mathrm{a},\mathrm{b},\mathrm{c}\}$ which are not onto, is :

Let $\mathrm{S} = \left\{\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] : a,b,c,d \in \{0,1,2,3,4\} \text{ and } (ad-bc=3)\right\}$. Then the number of matrices in S for which the sum of the diagonal elements is equal to 4, is:

Let $\mathrm{A} = \left[ \begin{array}{ccc}1 & 1 & 2\\ -2 & 0 & 1\\ 1 & 3 & 5 \end{array} \right]$ . Then the sum of all elements of the matrix $\operatorname{adj}(\operatorname{adj}(2(\operatorname{adj}A)^{-1}))$ is equal to :

The first term of an A.P. of $30$ non-negative terms is $\frac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is :

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is :

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + (1+x)^5 + \ldots + (1+x)^{99} + (1+kx)^{100}$ is $\left(\frac{101}{4}+43n\right)\binom{100}{3}$ for some $n\in\mathbb{N}$, be $p$. Then the value of $p+n$ is:

Suppose that the mean and median of the non-negative numbers $21, 8, 17, a, 51, 103, b, 13, 67, (a > b),$ are $40$ and $21,$ respectively. If the mean deviation about the median is $26$, then $2a$ is equal to:

Let the line L1 : x + 3 = 0 intersect the lines L2 : x – y = 0 and L3 : 3x + y = 0 at the points A and B, respectively. Let the bisector of the obtuse angle between the lines L2 and L3 intersect the line L1 at the point C. Then BC² : AC² is equal to :

Let the vertex A of a triangle ABC be (1, 2), and the mid-point of the side AB be (5, –1). If the centroid of this triangle is (3, 4) and its circumcenter is (α, β), then 21(α+β) is equal to :

Suppose the two chords, drawn from the point (1, 2) on the circle $\mathbf{x}^2 +\mathbf{y}^2 +\mathbf{x} - 3\mathbf{y} = 0$ are bisected by the y-axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is $(\alpha ,\beta)$ , then $6(\alpha +\beta)$ is equal to:

A line with direction ratios 1, -1, 2 intersects the lines $\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}$ and $\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}$ at the points P and Q, respectively. If the length of the line segment PQ is α, then 225α² is equal to:

The square of the distance of the point (-2, -8, 6) from the line $\frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z}{-1}$ along the line $\frac{x + 5}{1} = \frac{y + 5}{-1} = \frac{z}{2}$ is equal to:

If $y = tan⁻¹((3cosx-4sinx)/(4cosx+3sinx)) + 2 \tan^{-1}(\frac{x}{1+\sqrt{1-x²}})$, then $dy/dx$ at $x = \sqrt3/2$ is equal to :

Let $f$ be a real polynomial of degree n such that $f(x) = f'(x) f''(x),$ for all $x ∈ ℝ. If f(0)=0,$ then $36[∫_0^2 f(x)dx + f''(2) + f'(2)]$ is equal to :

The area of the region ${(x,y): y ≤ π – |x|, y ≤ |x sin x|, y ≥ 0}$ is :

Let y = y(x) be the solution of the differential equation $\frac{dy}{dx} = (1+x+x^2)(1-y+y^2)$, $y(0)=1/2.$ Then $(2y(1)-1)$ is equal to :

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is p, then 96p is equal to ______.

Consider the parabola $\mathrm{P:y^2 = 4kx}$ and the ellipse $\mathrm{E:}\frac{\mathrm{x}^2}{\mathrm{a}^2} +\frac{\mathrm{y}^2}{\mathrm{b}^2} = 1$. Let the line segment joining the points of intersection of P and E be their latus rectum. Then $\mathrm{e}^2 + 2\sqrt{2}$ is equal to ______.

If $\mathrm{A} = \frac{\sin 3^{\circ}}{\cos 9^{\circ}} +\frac{\sin 9^{\circ}}{\cos 27^{\circ}} +\frac{\sin 27^{\circ}}{\cos 81^{\circ}}$ and $\mathrm{B} = \tan 81^{\circ} - \tan 3^{\circ}$, then $\frac{\mathrm{B}}{\mathrm{A}}$ is equal to ____.

Let $\bar{\mathrm{a}}_{\mathrm{k}} = (\tan \theta_{\mathrm{k}})\bar{\mathrm{i}} +\bar{\mathrm{j}}$ and $\bar{\mathrm{b}}_{\mathrm{k}} = \bar{\mathrm{i}} - (\cos \theta_{\mathrm{k}})\bar{\mathrm{j}}$, where $\theta_{\mathrm{k}} = \frac{2^{\mathrm{k} - 1}\pi}{2^{\mathrm{n} + 1}}$, for some $n∈ℕ, n>5.$ Then the value of $\frac{\sum_{k=1}^{n}|\bar{\mathrm{a}}_{\mathrm{k}}|^{2}}{\sum_{k=1}^{n}|\bar{\mathrm{b}}_{\mathrm{k}}|^{2}}$ is ______.

The number of points, at which the function $\mathrm{f(x)} = \max \{6x, 2 + 3x^2\} +|x - 1| \cos \left|x^2 - \frac{1}{4}\right|, x\in (-\pi ,\pi)$, is not differentiable, is ______.

Number of moles and number of molecules in 1.4187 L of $\mathrm{SO}_{2}$ at STP respectively are

What is the ratio of wave number of first line (lowest energy line) of Balmer series of H atomic spectrum to first line of its Brackett series?

Which of the following is correct set of 4 quantum number of $19{ }^{\text {th }}$ electron in Chromium (Atomic number $=24$ ) in accordance with Aufbau principle?

Given below are two statements:

Statement I: For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure.

Statement II : In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.

In the light of the above statements, choose the correct answer from the options given below:

At $\mathrm{T}(\mathrm{K})$, the equilibrium constant of $\mathrm{A}_{2}(\mathrm{~g})+\mathrm{B}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{C}(\mathrm{g})$ is $2.7 \times 10^{-5}$.

What is the equilibrium constant for $\frac{1}{3} \mathrm{~A}_{2}(\mathrm{~g})+\frac{1}{3} \mathrm{~B}_{2}(\mathrm{~g}) \rightleftharpoons \frac{1}{3} \mathrm{C}(\mathrm{g})$ at the same temperature?

In order to oxidise a mixture of 1 mole each of $\mathrm{FeC}_{2} \mathrm{O}_{4}, \mathrm{Fe}_{2}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}, \mathrm{FeSO}_{4}$ and $\mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}$ in acidic medium, the number of moles of $\mathrm{KMnO}_{4}$ required is :

Consider the first order reaction $\mathrm{R} \rightarrow \mathrm{P}$.

The fraction of molecules decomposed in the given first order reaction can be expressed as

A monoatomic anion $\left(\mathrm{A}^{-}\right)$has 45 neutrons and 36 electrons. Atomic mass, group in the periodic table and physical state at room temperature of the element (A) respectively are :

Given below are two statements :

Statement I : The covalency of oxygen is generally two but it can exceed upto four. The oxidation state of oxygen in $\mathrm{SO}_{2}$ is -2 and in $\mathrm{OF}_{2}$ it is +2 .

Statement II: The anomalous behaviour of oxygen when compared to the other elements of group 16 is due to its small size and high electronegativity .

In the light of the above statements, choose the correct answer from the options given below:

The correct statements among the following are,

A. $\mathrm{Mo}(\mathrm{VI})$ and $\mathrm{W}(\mathrm{VI})$ are less stable than $\mathrm{Cr}(\mathrm{VI})$.

B. $\mathrm{Ce}^{4+}$ and $\mathrm{Tb}^{4+}$ are oxidant while $\mathrm{Eu}^{2+}$ and $\mathrm{Yb}^{2+}$ are reductant.

C. Cm and Am have seven unpaired electrons.

D. Actinoid contraction is greater from element to element than lanthanoid contraction.

Choose the correct answer from the options given below :

Correct statements from the following are

A. Potassium dichromate is an oxidising agent and it oxidises $\mathrm{FeSO}_{4}$ to $\mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}$ in acidic medium.

B. Sodium dichromate can be used as primary standard in volumetric estimation.

C. $\mathrm{CrO}_{4}^{2-}$ and $\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}$ are interconvertible in aqueous solution by varying the pH of the solution.

D. $\mathrm{Cr}-\mathrm{O}-\mathrm{Cr}$ bond angle in $\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}$ is $126^{\circ}$.

Choose the correct answer from the options given below :

Match the List-I with List-II Choose the correct answer from the options given below:

Increasing order of electron withdrawing power of following functional groups is :

a. -CN

b. -COOH

c. $-\mathrm{NO}_{2}$

d. -I

Match the List-I with List-II

Choose the correct answer from the options given below :

Match the LIST-I with LIST-II :

Choose the correct answer from the options given below:

Identify the colour of compound ' X ' in the sequence of the reaction.

Consider the following sequence of reactions to give the major product (X) P g of the major product ( X ) formed is reacted with $\mathrm{NaHCO}_{3}$ soluion to liberate a gas which occupied $11.2 \mathrm{dm}^{3}$ at STP. $\mathrm{P}=$ $\_\_\_\_$ . g. (Given molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{H}: 1, \mathrm{C}: 12$, $\mathrm{O}: 16, \mathrm{Cl}: 35.5)$

2.0 g of a bromo hydrocarbon ( X ) was subjected to Carius analysis, gave 3.36 g of AgBr . The percentage of carbon in the compound ( X ) is $26.7 \%$. Total number of carbon atoms in the empirical formula for compound ( X ) is $\_\_\_\_$ . (Given molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{H}: 1, \mathrm{C}: 12$, Br : 80, Ag : 108)

The pH a solution obtained by mixing 5 mL of $0.1 \mathrm{M} \mathrm{NH}_{4} \mathrm{OH}$ solution with 250 mL of 0.1 M $\mathrm{NH}_{4} \mathrm{Cl}$ solution is $\_\_\_\_$ $\times 10^{-2}$. (Nearest integer)

Given : $\mathrm{pK}_{\mathrm{b}}\left(\mathrm{NH}_{4} \mathrm{OH}\right)=4.74$

A non-volatile, non-electrolyte solid solute when dissolved in 40 g of a solvent, the vapour pressure of the solvent decreased from 760 mm Hg to 750 mm Hg . If the same solution boils at 320 K , then the number of moles of the solvent present in the solution is $\_\_\_\_$ . (Nearest integer)[0pt] [Given : boiling point of the pure solvent $=319.5 \mathrm{~K}, \mathrm{~K}_{\mathrm{b}}$ of the solvent $=0.3 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ ]