JEE Main 2026 — 6 April, Evening Shift

The percentage error in the calculated volume of a sphere, if there is $2\%$ error in its diameter measurement, is

A solid sphere (A) of mass 5m and spherical shell (B) of mass m, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of $a_A$ and $a_B$ respectively. The ratio of $a_A$ and $a_B$ is

A body of mass $1\mathrm{kg}$ moves along a straight line with a velocity $v = 2x^2$. The work done by the body during displacement from $x = 0$ to $5\mathrm{m}$ is J.

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both compartments at same P, V, T. Heating is started from left side until pressure changes to $\frac{27P}{8}$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is ______ litres. (for this ideal gas $C_p/C_v = 1.5$)

For an electromagnetic wave propagating through vacuum, $\vec{k}$, $\vec{E}$ and $\omega$ represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by:

Two identical bodies A and B of equal masses have initial velocities $\vec{v}_1 = 4\hat{i}\mathrm{m/s}$ and $\vec{v}_2 = 4\hat{j}\mathrm{m/s}$ respectively. The body A has acceleration $\vec{a}_1 = 6\hat{i}+6\hat{j}\mathrm{m/s}^2$ while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in ______ path.

A cylindrical vessel of $40\mathrm{cm}$ radius is completely filled with water and its capacity is $528\mathrm{dm}^3$ (dm: decimeter). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at $70\mathrm{cm}$ below the top of water level, then horizontal range of water falling on the ground in the beginning is ______ cm.

If 2 mole of an ideal monoatomic gas at temperature T, is mixed with 6 mole of another ideal monoatomic gas at temperature 2T then the temperature of mixture is:

A spring stretches by $2\mathrm{mm}$ when it is loaded with a mass of $200\mathrm{g}$. From equilibrium position the mass is further pulled down by $2\mathrm{mm}$ and released. The frequency associated with the system and maximum energy in the spring are ______ Hz and ______ J, respectively. (Take $g = 10\mathrm{m/s}^2$)

The electric potential as a function of x, y is given by $V = 5(x^2 - y^2)$ V. The electric field at a point (2, 3) m is ______ V/m.

A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as $B = 0.4 \sin(300t)$ Tesla. The normal to the plane of loop makes an angle of 60° with the field. The maximum induced emf produced in the loop is ______ mV.

A sphere of capacitance 100 pF is charged to a potential of 100 V. Another identical undercharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is $\alpha \times 10^{-7}$ J. The value of $\alpha$ is ______. (combined capacitance of spheres is 200 pF).

The energy released if hydrogen atoms are combined to form $^4_2\mathrm{He}$ is ______ MeV. (Take binding energies per nucleon of hydrogen as 1.1 MeV and of helium as 7.2 MeV).

Angle of minimum deviation is equal to the half of the angle of prism in an equilateral prism. The refractive index of the prism is

In a Young double slit experiment, the wavelength of incident light is $6000\mathrm{\AA}$, the separation between slits is $5\mathrm{cm}$, the distance between slits plane and screen is $50\mathrm{cm}$ as shown in the figure. If the resultant intensity at P is equal to the intensity due to individual slits, the path difference between interfering waves is ______ Å.

A block takes t time to slide down a plane inclined at $45^\circ$ to the horizontal. If the surface is made smooth (frictionless), the block takes time $t/2$ to slide down the plane. The coefficient of friction between the block and the inclined plane is $\left(\frac{\alpha}{100}\right)$. The value of $\alpha$ is ______.

The de Broglie wavelength for an electron accelerated through the potential difference $V_1$ volt is $\lambda_1$. When the potential difference is changed to $V_2$ volt, the associated de Broglie wavelength is increased by $50\%$. If $(V_1/V_2) = (9/\alpha)$, then the value of $\alpha$ is ______.

A moving coil of a galvanometer when shunted with $2\Omega$ resistance gives a full scale deflection for a current of $500\mathrm{mA}$. When a resistance of $470\Omega$ is connected in series it gives a full scale deflection for $10\mathrm{V}$ potential applied on it. The value of resistance of galvanometer coil is $\Omega$.

Two cells of emfs 1V and 2V and internal resistance 2Ω and 1Ω, respectively connected in parallel, gave current of 1A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be $\frac{\alpha}{5}$ A. The value of $\alpha$ is ______.

A concave mirror of focal length $10\mathrm{cm}$ forms an image which is double the size of object when the object is placed at two different positions. The distance between the two positions of the object is ______ cm.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3}$. Then $f$ is:

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0$ , $n \in \mathbb{R}$ . Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$ , is:

Let $S = \left\{z \in \mathbb{C}: z^2 + \sqrt{6} iz - 3 = 0\right\}$ . Then $\sum_{z \in S} z^8$ is equal to:

The sum of all possible values of $\theta \in [0, 2\pi ]$, for which the system of equations: $x cos3θ - 8y - 12z = 0, x cos2θ + 3y + 3z = 0, x + y + 3z = 0$ has a non-trivial solution, is equal to:

Let $A = [[1,0,0],[3,1,0],[9,3,1]]$ and $B = [b_ij], 1≤i,j≤3.$ If $B = A^99 - I,$ then the value of $\frac{(b_{31} - b_{21})}{b_{32}}$ is:

The value of $\sum_{n=1}^{2008} \frac{\sqrt{n^{4}+2 n^{3}+3 n^{2}+2 n+1}}{n(n+1)}$ is equal to

A building has ground floor and $10$ more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10$th floor. The number of ways, in which any $4$ persons exit at a floor and the remaining $5$ persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

Let C be a circle having centre in the first quadrant and touching the x-axis at a distance $3$ units from the origin. If the circle C has an intercept of length $6\sqrt3$ on y-axis, then the length of the chord of the circle C on the line $x-y=3$ is:

The eccentricity of an ellipse $E$ with centre at the origin O is $\frac{\sqrt3}{2}$ and its directrices are $x = ±4\sqrt6/3$. Let $H: \frac{x²}{a²} - \frac{y²}{b²} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of $E.$ Then the distance between the foci of $H$ is :

Let $\mathrm{x}=9$ be a directrix of an ellipse E , whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $\mathrm{P}(\alpha, 0), \alpha>0$, be a focus of E and AB be a chord passing through P . Then the locus of the mid point of AB is :

If $\sin \left(\tan^{-1}(x\sqrt{2})\right) = \cot \left(\sin^{-1}\sqrt{1-x^2}\right)$, $x\in (0,1)$, then the value of x is :

The shortest distance between the lines $\frac{x-4}{1} = \frac{y-3}{2} = \frac{z-2}{-3}$ and $\frac{x+2}{2} = \frac{y-6}{4} = \frac{z-5}{-5}$ is:

Let $\vec{a} = 2\hat{i} +3\hat{j} +3\hat{k}$ and $\vec{b} = 6\hat{i} +3\hat{j} +3\hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is :

Let $\lim_{x\to 2}\frac{(\tan(x-2))(rx^{2} + (p-2)x - 2p)}{(x-2)^{2}} = 5$ for some $r,p∈R.$ If the set of all possible values of $q$ such that the roots of the equation $rx^{2} - px + q = 0$ lie in $(0,2),$ be the interval $(α,β],$ then $4(α+β)$ equals:

Let $A = \begin{vmatrix}1 & 3 & -1\\ 2 & 1 & \alpha \\ 0 & 1 & -1\end{vmatrix}$ be a singular matrix. Let $f(x) = \int_{0}^{x}(t^{2} + 2t + 3)dt$, $x∈[1,α].$ If M and m are respectively the maximum and the minimum values of f in $[1,α]$ then $3(M-m)$ is equal to:

Let $f:R→R$ be such that $f(xy)=f(x)f(y)$ for all $x,y∈R$ and $f(0)≠0. Let g:[1,∞)→R$ be a differentiable function such that $x^{2}g(x) = \int_{1}^{x}(t^{2}f(t) - tg(t))dt$. Then $g(2)$ is equal to :

The area of the region $\{(x,y): x^{2} - 8x \leq y \leq -x\}$ is:

The value of the integral $\int_{-1}^{1}\frac{x^{3}+|x|+1}{x^{2}+2|x|+1}dx$ is equal to :

Let $R = {(x,y) ∈ N×N : log_e(x+y) ≤ 2}.$ Then the minimum number of elements required to be added in $R$ to make it a transitive relation, is _____.

If $(1-x^3)^{10} = \sum_{r=0}^{10} a_r x^{30-2r}(1-x)^r$, then $\frac{9a_9}{a_{10}}$ is equal to ______.

Let the line $x-y=4$ intersect the circle $C: (x-4)²+(y+3)²=9$ at the points $Q$ and $R.$ If $P(α,β)$ is a point on $C$ such that $PQ=PR,$ then $(6α+8β)²$ is equal to ______.

Let the image of the point P(0,-5,0) in the line $\frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-2}$ be the point R and the image of the point Q(0,-1/2,0) in the line $\frac{x-1}{-1} = \frac{y+9}{4} = \frac{z+1}{1}$ be the point S. Then the square of the area of the parallelogram PQRS is ______.

Let $f(x) = \begin{cases} x^3+8, & x<0 \\ x^2-4, & x\ge 0 \end{cases}$ and $g(x) = \begin{cases} (x-8)^{1/3}, & x<0 \\ (x+4)^{1/2}, & x\ge 0 \end{cases}$. Then the number of points where the function $g(f(x))$ is discontinuous, is ______.

Which of the following contain the same number of atoms? (Given : Molar mass in $\mathrm{g} \mathrm{mol}^{-1}$ of $\mathrm{H}, \mathrm{He}, \mathrm{O}$ and S are $1,4,16$ and 32 respectively) A. 2 g of $\mathrm{O}_{2}$ gas B. 4 g fo $\mathrm{SO}_{2}$ gas C. 1400 mL of $\mathrm{O}_{2}$ at STP D. 0.05 L of He at STP

E, $0.0625 \mathrm{~mol}^{\circ}$ of $\mathrm{H}_{2}$ gas Choose the correct answer from the options given below :

The Bohr radius of a hydrogen like species is 70.53 pm . The species and the stationary state(n) are respectively (Given: Hydrogen atom Bohr radius is 52.9 pm )

Given below are two statements:

Statement I: The number of compounds among $\mathrm{SO}_{2}, \mathrm{SO}_{3}, \mathrm{SF}_{4}, \mathrm{SF}_{6}$ and $\mathrm{H}_{2} \mathrm{~S}$ in which sulphur does not obey the Octet rule is 3 .

Statement II : Among $\left[\mathrm{H}_{2} \mathrm{O}, \mathrm{ClF}_{3}, \mathrm{SF}_{4}\right],\left[\mathrm{NH}_{3}\right.$, $\left.\mathrm{BrF}_{5}, \mathrm{SF}_{4}\right], \quad\left[\mathrm{BrF}_{5}, \quad \mathrm{ClF}_{3}, \quad \mathrm{XeF}_{4}\right] \quad$ and $\quad\left[\mathrm{XeF}_{4}\right.$, $\left.\mathrm{ClF}_{3}, \mathrm{H}_{2} \mathrm{O}\right]$, the number of sets in which all the molecules have one lone pair of electrons on the central atom is 1 .

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

Match List-I with List-II Given $\mathrm{V}_{1}$ and $\mathrm{V}_{2}$ are initial and final volumes respectively:

Choose the correct answer from the options given below :

Given below are two statements :

Statement I: $\mathrm{H}_{2} \mathrm{O}$ molecules move from the chamber 1 to chamber 2.

Statement II: The osmotic pressure of a solution prepared by dissolving 50 mg of potassium sulphate (molar mass $=174 \mathrm{~g} / \mathrm{mol}$ ) in 2 L of water (at $27^{\circ} \mathrm{C}$ ) is 0.0107 bar. (Given : $\mathrm{R}=0.083 \mathrm{dm}^{3}$ bar $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ and assume complete dissociation of electrolyte)

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

Given is a concentrated solution of a weak electrolyte $\mathrm{A}_{x} \mathrm{~B}_{y}$ of concentration ' c ' and dissociation constant ' K '. the degree of dissociation is given by:

For a general redox reaction

Which of the following statement is incorrect?

In a period, the first ionisation enthalpy of the element at extreme left and the negative electron gain enthalpy of the extreme right element, except noble gases, are respectively.

Given below are two statements :

Statement I: $\mathrm{F}_{2} \mathrm{O}<\mathrm{H}_{2} \mathrm{O}<\mathrm{Cl}_{2} \mathrm{O}$ is the correct trend in terms of bond angle.

Statement II : $\mathrm{SiF}_{4}, \mathrm{SnF}_{4}$ and $\mathrm{PbF}_{4}$ are ionic in nature.

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

The correct order of first $\left(\Delta_{\mathrm{i}} \mathrm{H}_{1}\right)$ and second $\left(\Delta_{\mathrm{i}} \mathrm{H}_{2}\right)$ ionisation enthalpy values of cr and Mn are : A. $\Delta_{\mathrm{i}} \mathrm{H}_{1}: \mathrm{Cr}>\mathrm{Mn}$ B. $\Delta_{\mathrm{i}} \mathrm{H}_{2}: \mathrm{Cr}>\mathrm{Mn}$ C. $\Delta_{\mathrm{i}} \mathrm{H}_{1}: \mathrm{Mn}>\mathrm{Cr}$ D. $\Delta_{\mathrm{i}} \mathrm{H}_{2}: \mathrm{Mn}>\mathrm{Cr}$

Choose the correct answer from the options given below :

Which of the following sequence of hybridisation, geometry and magnetic nature are correct for the given coordination compounds ? A. $\left[\mathrm{NiCl}_{4}\right]^{2-}-\mathrm{sp}^{3}$, tetrahedral, paramagnetic B. $\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}-\mathrm{sp}^{3} \mathrm{~d}^{2}$, octahedral, paramagnetic C. $\left[\mathrm{Ni}(\mathrm{CO})_{4}\right]-\mathrm{sp}^{3}$, tetrahedral, paramagnetic D. $\left[\mathrm{Ni}(\mathrm{CN})_{4}\right]^{2-}-\mathrm{dsp}^{2}$, square planar diamagnetic

Choose the correct answer from the options given below :

Given below are two statements :

Statement I: A mixture of $\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}$ (sugar) and NaCl can be separated by dissolving sugar in alcohol, due to differential solubility

Statement II : Rose essence from rose petals is separated by steam distillation due to its high volatility and insolubility in $\mathrm{H}_{2} \mathrm{O}$.

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

Shown below is the structure of methyl acetate with three different $\alpha, \beta$ and $\gamma$ carbon-oxygen bonds.

The correct order of bond length of these bonds is :

An optically active alkyl bromide $\mathrm{C}_{4} \mathrm{H}_{9} \mathrm{Br}$, reacts with ethanolic KOH to form major compound [A] which reacts with bromine to give compound [B]. Compound [B] reacts with ethanolic KOH and sodamide to give compound [C]. One molecule of water adds to compound [C] on warming with mercuric sulphate and dilute sulphuric acid at 333 K to form compound [D]. The functional group in compound D will be confirmed by:

Treatment of a gas ' X ' with a freshly prepared ferrous sulphate solution gives a compound ' Y ' as a brown ring. The compounds X and Y are.

An excess of $\mathrm{AgNO}_{3}$ is added to 100 mL of a 0.05 M solution of tetraaquadichloridochromium (III) chloride. The number of moles of AgCl precipitated will be $\_\_\_\_$ $\times 10^{-3}$. (Nearest integer)

An alkane (Y) requires 8 moles of oxygen for complete combustion and on chlorination with $\mathrm{Cl}_{2} / \mathrm{hv}$, (Y) gives only one monochlorinated product (Z). The total number of primary carbon atoms in (Y) is $\_\_\_\_$ .

500 mL of $0.2 \mathrm{M} \mathrm{MnO}_{4}^{-}$solution in basic medium when mixed with 500 mL of 1.5 M KI solution, oxidises iodide ions to liberate molecular iodine. This liberated iodine is then titrated with a standard x M thiosulphate solution in presence of starch till the end point. If 300 mL of thiosulphate was consumed, then the value of x is $\_\_\_\_$ .

In a closed flask at 600 K , one mole of $\mathrm{X}_{2} \mathrm{Y}_{4}(\mathrm{~g})$ attains equilibrium as given below: $\mathrm{X}_{2} \mathrm{Y}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{XY}_{2}(\mathrm{~g})$ At equilibrium, $75 \% \mathrm{X}_{2} \mathrm{Y}_{4}(\mathrm{~g})$ was dissociated and the total pressure is 1 atm . The magnitude of $\Delta_{\mathrm{r}} \mathrm{G}^{\ominus}$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) at this temperature is $\_\_\_\_$ . (Nearest Integer) (Given: $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} ; \ln 10=2.3$, $\log 2=0.3, \log 3=0.48, \log 5=0.69, \log 7=0.84)$

Decomposition of a hydrocarbon follows the equation $\mathrm{k}=\left(5.5 \times 10^{11} \mathrm{~s}^{-1}\right) \mathrm{e}^{\frac{-28000 \mathrm{~K}}{\mathrm{~T}}}$. The activation energy of reaction is $\_\_\_\_$ $\mathrm{kJ} \mathrm{mol}^{-1}$. (Nearest Integer) Given : $\mathrm{R}=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$