NTA JEE Mains 5th April Shift 2 2026

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - x + p = 0$$ and $$\gamma, \delta$$ be the roots of the equation $$x^2 - 4x + q = 0$$, where $$p, q \in \mathbb{Z}$$. If $$\alpha, \beta, \gamma, \delta$$ are in G.P., then $$|p + q|$$  equals :

Let $$z_1, z_2 \in \mathbb{C}$$ be the distinct solutions of the equation $$z^2 + 4z - (1 + 12i) = 0$$. Then $$|z_1|^2 + |z_2|^2$$ is equal to :

Let $$f : \mathbb{N} \to \mathbb{Z}$$ be defined by $$f(n) = \det\begin{bmatrix} n  & -1 & -5\\-2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{bmatrix}$$, $$k \in \mathbb{N}$$ and  $$\displaystyle\sum_{n=1}^{k} f(n) = 98$$, then $$k$$ is equal to :

Let $$M$$ be a $$3 \times 3$$ matrix such that $$M\begin{bmatrix}1\\0\\0\end{bmatrix} = \begin{bmatrix}1\\2\\3\end{bmatrix}$$, $$M\begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}0\\1\\0\end{bmatrix}$$, $$M\begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}-1\\1\\1\end{bmatrix}$$. If $$M\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}1\\7\\11\end{bmatrix}$$, then $$x + y + z$$ is equal to :

The sum of the first 10 terms of the series $$\frac{1}{1 + 1^4 \cdot 4} + \frac{2}{1 + 2^4 \cdot 4} + \frac{3}{1 + 3^4 \cdot 4} + \cdots$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$. Then $$m + n$$ is equal to :

Let $$A_1, A_2, \ldots, A_{39}$$ be 39 arithmetic means between the numbers 59 and 159. The mean of $$A_{25}, A_{28}, A_{31} and  A_{36}$$ is equal to :

The coefficient of $$x^2$$ in the expansion of $$\left(2x^2 + \frac{1}{x}\right)^{10}$$, $$x \neq 0$$, is :

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :

A box contains 5 blue, 6 yellow, and 4 red balls. The number of ways, of drawing 8 balls containing atleast two balls of each colour, is :

A  variable $$X$$ takes values $$0, 0, 2, 6, 12, 20, \ldots, n(n-1)$$ with frequencies $${^{n} C_{0}}, {^{n} C_{1}}, {^{n} C_{2}}, \ldots, {^{n} C_{n}}$$ respectively. If the mean of the data is 60, then the median is :

Let the point $$P$$ be the vertex of the parabola $$y = x^2 - 6x + 12$$. If a line passing through the point $$P$$ intersects the circle $$x^2 + y^2 - 2x - 4y + 3 = 0$$ at the points $$R$$ and $$S$$.then the maximum value of $$(PR + PS)^2$$ is :

Let the directrix of the parabola $$P: y^2 = 8x$$ cuts the x-axis at the point $$A$$.Let $$B(\alpha, \beta)$$, $$\alpha > 1$$, be a point on $$P$$ such that the  slope of $$AB$$ is $$3/5$$. If  $$BC$$ is a focal chord of chord of $$P$$. then six times the area off $$(\triangle ABC)$$ is :

Let the eccentricity $$e$$ of a hyperbola satisfy the equation $$6e^2 - 11e + 3 = 0$$. Its foci of the hyperbola are $$(3, 5)$$ and $$(3, -4)$$.then  the length of its latus rectum is :

Let a $$\triangle PQR$$,be such that $$P$$ and $$Q$$ lie on the line $$\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$$ and are  at a distance of 6 units  from $$R(1, 2, 3)$$. If $$(\alpha, \beta, \gamma)$$ is the centroid of $$\triangle PQR$$, then $$\alpha + \beta + \gamma$$ is equal to :

Let the distance of the point  $$(a, 2, 5)$$ from the image of the point $$(1, 2, 7)$$ in the line $$\frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2}$$ is 4,then the sum of all possible values of $$a$$ is equal to:

Let $$O$$ be the origin, $$\overrightarrow{OP} = \vec{a}$$ and $$\overrightarrow{OQ} = \vec{b}$$.If $$R$$ is the point on $$\overrightarrow{OP}$$ such that $$\overrightarrow{OP} = 5\overrightarrow{OR}$$, and $$M$$ is the point such that $$\overrightarrow{OQ} = 5\overrightarrow{RM}$$. Then $$\overrightarrow{PM}$$ is equal to :

Let $$f(x) = \displaystyle\lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^3}$$. Then the  number of solutions of the equation $$f(x) = \sin x$$, $$x \in \mathbb{R}$$, is :

Let $$(2^{1-a} + 2^{1+a})$$, $$f(a)$$, $$(3^a + 3^{-a})$$ be in A.P. and $$\alpha$$ be the minimum value of  $$f(a)$$, Then the value of the integral $$\displaystyle\int_{\log_e(\alpha - 1)}^{\log_e(\alpha)} \frac{dx}{e^{2x} - e^{-2x}}$$ is equal to :

Let $$f : [1, \infty) \to \mathbb{R}$$ be a differentiable defined as $$f(x) = \displaystyle\int_1^x f(t)\,dt + (1 - x)(\log_e x - 1) + e$$. Then the value of  $$f(f(1))$$ is :

Let $$f(x)$$ and $$g(x)$$ be twice differentiable functions satisfying  $$f''(x) = g''(x)$$ for all $$x$$, $$f'(1) = 2g'(1) = 4$$, and $$g(2) = 3f(2) = 9$$. Then $$f(25) - g(25)$$ is equal to :

Let $$A = \{1, 4, 7\}$$ and $$B = \{2, 3, 8\}$$.  Then the number of elements, in the relation $$R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B) \times (A \times B) : a_1 + b_2 \text{ divides } a_2 + b_1\}$$. is :

From the point $$(-1, -1)$$, two rays are sent making angle  of $$45°$$ with the line $$x + y = 0$$. The rays get reflected from the mirror $$x + 2y = 1$$. If the equations of the reflected rays are $$ax + by = 9$$ and $$cx + dy = 7$$,$$a,b,c,d \in \mathbb{Z}$$ thenthe value of  $$ad + bc$$ is :

Let $$S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\left(\frac{5\theta}{2}\right) = \cos 7\theta \cos\left(\frac{7\theta}{2}\right)\right\}$$. Then $$n(S)$$ is equal to :

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function such that $$f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x, x \in R$$.Let  the maximum value of $$f$$ on $$\mathbb{R}$$ be  $$\alpha$$. The area of the region bounded by the curves $$g(x) = x^2$$ and $$h(x) = \beta x^3$$, $$\beta > 0$$, is $$\alpha^2$$. Then $$30\beta^3$$ is equal to :

Let $$y = y(x)$$ be the solution of $$(\tan x)^{1/2}\,dy = (\sec^3 x - (\tan x)^{3/2}\,y)\,dx$$, $$0 < x < \frac{\pi}{2}$$. If $$y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}$$, and $$y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha$$, then $$\alpha^4$$ is equal to :

Match the List I With List II:

Where (Plank's constant), G(gravitational constant) and c (speed of light in vacuum) as fundamental units.
Choose the  correct answer from the options given below:

In an experiment to determine the resistance of a given wire using Ohm’s law, the voltmeter and ammeter readings are noted as $$10 V$$ and $$5 A$$, respectively. The least counts of voltmeter and ammeter are $$500 mV$$ and $$200 mA$$, respectively. The estimated error in the resistance measurement  is_______Ω.

A mass of 1 kg is kept on a inclined plane with 30° inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of 4 m/s. The work done by the frictional force in time 2 s is ________ J. (Take g = 10 m/s²)

The velocity (v) versus time (t) plot of a particle is shown in the figure, for a time interval of 40 s. The total distance travelled by the particle and the and the average velocity during this period are, respectively___________, 

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s, it rotates through an angle $$\theta_1$$. In the next 2 s, it rotates through an angle $$\theta_2$$. Then the ratio $$\frac{\theta_2}{\theta_1}$$ is :

An object of uniform density rolls up a curved path with the initial velocity $$v_{\circ}$$ as shown in the figure. If the maximum height attained by an object is  $$\frac{7v_0^2}{10g}$$. (g = acceleration due to gravity), The object is a :

A body of mass $$m$$ is taken from the surface of the earth to a height equal to twice the radius of earth$$(R_e)$$. The increase in potential energy will be  :
(g is acceleration due to gravity at the surface of earth) 

Eight mercury drops, each of radius $$r$$ coalesce to form a bigger  drop. The surface energy released in this process is_________ .
( S is the surface tension of mercury).

An ideal gas at pressure $$P$$ and temperature $$T$$ is expanding such that $$PT^3 = \text{constant}$$. The coefficient of volume expansion of the gas is :

Match the List I with List II:

Choose the correct answer from the options given below: 

A metal rod of length $$L$$ rotates about one end at origin with a uniform angular velocity $$\omega$$. The magnetic field radially falls off as $$B(r) = B_0 e^{-\lambda r}$$. $$\lambda$$ being a positive constant. The EMF induced (neglecting the centripetal force on electrons in the rod) is :

Under steady state condition the potential difference across the capacitor in the circuit is________  V.

A particle of charge $$q$$ and mass $$m$$ is projected  from origin with an initial velocity has $$\vec{v} = \frac{v_0}{\sqrt{2}}\hat{x} + \frac{v_0}{\sqrt{2}}\hat{y}$$. There exists a uniform magnetic field $$\vec{B} = B_0\hat{z}$$ and a space varying electric field $$\vec{E} = E_0 e^{-\lambda x}\hat{x}$$ within the region $$0 \leq x \leq L$$. After travelling a distance such that x-coordinate has changed from $$x=0$$ to $$x=L$$, the change in the kinetic energy  is_________.

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): The Electromagnetic waves exert pressure on the surface on which they ar allowed to fall.

Reason (R): There is no mass associated with the electromagnetic waves.
In the light of the above statements,  Choose the correct answer from the options given below :

A thin convex lens and a thin concave lens are kept in  contact and are co-axial. Which of the following statements is correct for this combination of two lenses ?

An object AB is placed 15 cm to the left of a convex lens P of focal length $$10 cm$$. Another convex lens Q is now placed $$15 cm$$ right of lens P . If the focal length of lens Q is 15 cm,  final image is______.

The maximum intensity in a Young's double slit experiment is $$I_0$$, distance between the slits (d) is  $$5\lambda$$, where $$\lambda$$ is the wavelength of light used. The intensity of the fringe, exactly opposite to one of the slits on the screen, placed at $$D = 10d$$ is_________.

An electron is travelling with a velocity $$v$$ in free space and when it enters a medium, its velocity is reduced by 20%. The de Broglie wavelength of electron in the medium is $$\alpha \lambda_0$$.where $$\lambda_0$$ is its de Broglie wavelength in free space. The value of $$\alpha$$ is :

Assuming the experimental mass of $$^{12}_{6}C$$ as 12 u, The mass defect of $${}^{12}\text{C}$$ atom is (in MeV/$$c^2$$) :
(Mass of proton  = 1.00727 u, mass of neutron = 1.00866 u, 1 u = 931.5$$ $$MeV/$$c^2$$ and c is the speed of the light in vacuum).

In a semiconductor p-n  diode, the doping concentrations on p-side has $$10^{15}$$ atoms/cm³ and the n-side has $$10^{18}$$ atoms/cm³respectively.  Which one of the following statements is true?

A copper wire of length 3 m is stretched by 3 mm by applying an external force. The volume of the wire is $$600 \times 10^{-6}$$ m³. The elastic potential energy stored in the wire in stretched condition would be _______  J :
(Given Young's modulus of copper $$ = 1.1 \times 10^{11}$$ N/m²)

The heat extracted out of x gram of water initialy at  $$50°$$C to  cool it down to $$0°$$C is sufficient to evaporate  $$(1000 - x)$$ gram  of water also initialy at $$50°$$C. The value of $$x$$ (closest integer) is_______.  (Take Latent heat of water  $$L = 2256$$ kJ/kg K, specific heat capacity of water $$c = 4200$$ J/kg·K)

In a series LCR circuit with , $$R = 20\,\Omega$$, $$L = 1.6$$ H, $$C = 40\,\mu$$F is connected to a variable frequency a.c. source.  The inductive reactance at resonant frequency is ____________ $$\Omega$$.

When an external resistance of $$5\,\Omega$$ is connected across terminals of a cell,  a current of $$0.25$$ A flow through it. When the $$5\,\Omega$$ resistor is replaced by a  $$2\,\Omega$$ resistor, a current of $$0.5$$ A flow through it. The internal resistance of the cell is________ $$\Omega$$.

A circular loop of radius $$r = 20$$ cm and resistance $$R = 2\,\Omega$$ is placed in a time varying  magnetic field $$B = (2t^2 + 2t + 3)$$ T. At $$t = 0$$, for the plane of the loop being perpendicular to the magnetic field and, The induced current in the loop  at $$t = 3$$ s is $$\frac{\alpha}{50}$$ A. The value of $$\alpha$$ is__________.
(Take $$\pi = 22/7$$)

What volume of hydrogen gas at STP would be liberated by action of 50 mL of $$H_2SO_4$$ of 50% purity (density = 1.3 g mL$$^{-1}$$) on 20 g of zinc?

Given : Molar mass of H, O, S, Zn are 1, 16, 32, 65 g mol$$^{-1}$$ respectively.

Which of the following statement(s) is/are true?

$$\textbf{A.}$$ If two orbitals have the same value of $$(n + l)$$, the orbital with lower value of $$n$$ will have lower energy.

$$\textbf{B.}$$ Energies of the orbitals in the same subshell increase with increase in atomic number.

$$\textbf{C.}$$ The size of $$2p_x$$ orbital is less than the size of $$3p_x$$ orbital.

$$\textbf{D.}$$ Among $$5f$$, $$6s$$, $$4d$$, $$5p$$ and $$5d$$ orbitals, none of the orbitals have 2 radial nodes.

Choose the correct answer from the options given below:

The covalent radii of atoms A and B are $$r_A$$ and $$r_B$$ respectively. The covalent bond length and total length of $$AB$$ molecule are respectively  :

Consider the following data for the reaction 
$$\text{X}_2(g) + \text{Y}_2(g) \rightleftharpoons 2\text{XY}(g)$$
at 600 K.  The $$\Delta_r G^\circ$$ (in $$kJ/mol^{-1}$$) for the reaction is :

The correct order of molar heat capacities measured at 298 K and 1 bar is :

The reaction $$A(g) \rightleftharpoons B(g) + C(g)$$ was initiated with the amount `$$a$$` of $$A(g)$$. At equilibrium it is found that the amount of $$A(g)$$ remaining is $$(a - x)$$ at a total pressure of $$p$$.
The equilibrium constant $$K_p$$ of the reaction can be calculated from the expression:

One half cell in a voltaic cell is constructed by dipping silver rod in $$\text{AgNO}_3$$ solution of unknown concentration, other half cell is Zn rod dipped in 1 molar solution of $$\text{ZnSO}_4$$. A voltage of $$1.60\,\text{V}$$ is measured at $$298\,\text{K}$$ for this cell. What is the concentration of $$\text{Ag}^+$$ ions used in terms of $$\log x$$ $$(x = [\text{Ag}^+])$$?

$$E^\ominus_{\text{Zn}^{2+}/\text{Zn}} = -0.76\,\text{V}, \quad$$ $$E^\ominus_{\text{Ag}^{+}/\text{Ag}} = +0.80\,\text{V}, \quad$$ $$\frac{2.303RT}{F} = 0.059\,\text{V}$$

Given below are two statements:

$$\textbf{Statement I :}$$ The number of pairs among $$[\text{Al}_2\text{O}_3, \text{Cr}_2\text{O}_3]$$, $$[\text{Cl}_2\text{O}_7, \text{Mn}_2\text{O}_7]$$, $$[\text{Na}_2\text{O}, \text{V}_2\text{O}_3]$$ and $$[\text{CO}, \text{N}_2\text{O}]$$ that contain oxides of same nature (acidic, basic, neutral or amphoteric) is 4.

$$\textbf{Statement II :}$$ Among $$\text{Na}_2\text{O}$$, $$\text{Al}_2\text{O}_3$$, $$\text{CO}$$ and $$\text{Cl}_2\text{O}_7$$, the most basic and acidic oxides are $$\text{Na}_2\text{O}$$ and $$\text{Cl}_2\text{O}_7$$, respectively.

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

Given below are two statements:

$$\textbf{Statement I :}$$ Aluminium upon reaction with $$\text{NaOH}$$ forms $$[\text{Al(OH)}_6]^{3-}$$ ion.

$$\textbf{Statement II :}$$ The geometry of $$\text{ICl}_4^{-}$$, $$\text{ClO}_3^{-}$$ and $$\text{IBr}_2^{-}$$ is square planar, pyramidal and linear respectively.

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

Given below are two statements:

$$\textbf{Statement I :}$$ Presence of large number of unpaired electrons in transition metal atoms results in higher enthalpies of their atomisation.

$$\textbf{Statement II :}$$ $$d_{xy} = d_{xz} = d_{yz} < d_{x^2-y^2} = d_{z^2}$$ and $$d_{x^2-y^2} = d_{z^2} < d_{xy} = d_{xz} = d_{yz}$$ are the d-orbital splittings in $$[\text{Fe}(\text{H}_2\text{O})_6]^{3+}$$ and $$[\text{Ni}(\text{Cl})_4]^{2-}$$ complex ions respectively.

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

Identify the correct statements from the following

$$\textbf{A.}$$ $$[\text{Fe}(\text{C}_2\text{O}_4)_3]^{3-}$$ is the most stable complex among $$[\text{Fe}(\text{OH})_6]^{3-}$$, $$[\text{Fe}(\text{C}_2\text{O}_4)_3]^{3-}$$ and $$[\text{Fe}(\text{SCN})_6]^{3-}$$

$$\textbf{B.}$$ The stability of $$[\text{Cu}(\text{NH}_3)_4]^{2+}$$ is greater than that of $$[\text{Cu}(\text{en})_2]^{2+}$$

$$\textbf{C.}$$ The hybridization of Fe in $$\text{K}_4[\text{Fe}(\text{CN})_6]$$ is $$d^2sp^3$$

$$\textbf{D.}$$ $$[\text{Fe}(\text{NO}_2)_3\text{Cl}_3]^{3-}$$ exhibits linkage isomerism

$$\textbf{E.}$$ $$\text{NO}_2^{-}$$ and $$\text{SCN}^{-}$$ ligands are NOT ambidentate ligands

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:

IUPAC name of the some alkenes are given below.
Find out the correct stability order.

$$\textbf{A.}$$ 2-Methylbut-2-ene

$$\textbf{B.}$$ $$\textit{cis}$$-But-2-ene

$$\textbf{C.}$$ 2,3-Dimethylbut-2-ene

$$\textbf{D.}$$ Prop-1-ene

Choose the correct answer from the options given below:

Identify the correct IUPAC name of  hydrocarbon (x) containing three  primary carbons atoms and with molar mass $$72$$ g mol$$^{-1}$$ :

Complete the following reaction sequence and give the name of major product 'P'.

Given below are two statements:

$$\textbf{Statement I :}$$ The condensation reaction between $$\text{CH}_3-\text{CH}=\text{O}$$ and 

under optimum pH will produce 

$$\textbf{Statement II :}$$ The molecule 

will generate $$\text{Ph}-\text{CH}=\text{O}$$ in the presence of dilute acid.

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

Given below are two statements:

$$\textbf{Statement I :}$$ Heating benzamide with bromine in an ethanolic solution of sodium hydroxide will give benzylamine.

$$\textbf{Statement II :}$$ Nitration of aniline with $$\text{HNO}_3/\text{H}_2\text{SO}_4$$ at $$288\,\text{K}$$ produces $$m$$-nitroaniline in higher amount than $$o$$-nitroaniline (pH adjusted).

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

Identify the incorrect  statements about  tertiary structure of proteins?

Given below are two statements : 
Statement I:

 are two anomers of D-(+)-glucose.
Statement II: the Open chain forms of D-glucose and D-fructose contain three similar chiral carbons at C₃, C₄, and C₅. 
In the light of the above statements, choose the correct answer from the options given below:

A paper is dipped in a dil. $$\text{H}_2\text{SO}_4$$ solution of  'X' upon treatment with $$\text{SO}_2$$ gas turns into green.The compound  'X' is :

The total number of unpaired electrons present in the $$d^3$$, $$d^4$$ (low spin), $$d^5$$ (high spin), $$d^6$$ (high spin), and $$d^7$$ (low spin) octahedral complexes systems is :

$$\text{RMgI}$$ when treated with ice cold water liberated a gas which occupied $$1.4\,\text{dm}^3/\text{g}$$ at STP. The gas produced is further reacted with iodine in presence of $$\text{HIO}_3$$ to give compound (X). Compound (X) in presence of Na and dry ether produced compound (Y). Molar mass of compound (Y) is $$\underline{\hspace{2cm}}$$ $$\text{g mol}^{-1}$$. (Nearest integer)

20 g hemoglobin in a 1 L aqueous solution (A) at 300 K is separated from pure water by a semi-permeable membrane. At equilibrium, the height of solution in a tube dipped in solution (A) is found to be $$80.0 mm$$ higher than the tube dipped in water.
The molar mass of hemoglobin is ______ $$kg mol^{-1}$$. (Nearest integer)
(Given: $$g = 10 \, m \, s^{-2}$$, $$R = 8.3 \, kPa \, dm^{3} \, K^{-1} \, mol^{-1}$$, density of solution = $$1000 \, kg \, m^{-3}$$)

At 298 K, the molar conductivity of an $$x\%$$ (w/w) MX solution is 123.5 S cm$$^{2} mol^{-1}$$. The conductance of same solution is $$1.9 \times 10^{-3} S$$. The value of x is ______ $$\times 10^{-2}$$.
(Given: Cell constant = 1.3 cm$$^{-1}$$, molar mass of MX = 75 $$g mol^{-1}$$, density of aqueous solution of $$MX$$ at 298 K is 1.0 $$g mL^{-1}$$)

For the reaction $$A \to p$$ at  $$T K$$, the half  life ($$t_{1/2}$$) is plotted as a function of initial concentration $$[A]_o$$ of  $$A$$ as give below.

The value of $$x$$ in the given figure is ______ s (Nearest integer)