NTA JEE Mains 21st Jan 2026 Shift 2

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of $$a$$ is :

Let A= {2, 3, 5, 7, 9}. Let R be the relation on A defined by x R y if and only if $$2x\leq3y$$. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:

If the system of equations
3x + y + 4z = 3
$$2x+\alpha y-z = -3$$
x+ 2y + z = 4
has no solution, then the value of $$\alpha$$ is equal to :

If the area of the region $$\left\{\left(x,y\right): 1-2x \leq y \leq4-x^{2}, x\geq 0, y\geq0 \right\}$$ is $$\frac{\alpha}{\beta} , \alpha,\beta \epsilon N$$, gcd $$\left(\alpha,\beta\right)=1$$, then the value of $$\left(\alpha+\beta\right)$$ is

Let $$f: R\rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$ for all $$x\in R$$ and f'(a-1)=0, where a is a real number. Let g(x)= $$f(\tan^{2}x- 2\tan x+a)$$, $$0 < x < \frac{\pi}{2}$$.
Consider the following two statements :
(I) $$\text{g is increasing in } \left(0, \frac{\pi}{4} \right)$$
(II) $$\text{g is deceasing in } \left( \frac{\pi}{4} , \frac{\pi}{2} \right)$$
Then,

For the matrices $$A=\begin{bmatrix}3  -4 \\1  -1 \end {bmatrix}$$ and $$B=\begin{bmatrix}-29  49 \\-13  18 \end{bmatrix}$$, if  $$\left(A^{15} + B \right) \begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}0 \\0 \end{bmatrix}$$, then among the following which one is true ?

Let one end of a focal chord of the parabola $$y^{2}=16x$$ be (16,16). If $$P\left(\alpha,\beta\right)$$ divides this focal chord internally in the ratio 5 : 2, then the minimum value of $$\alpha+\beta$$ is equal to :

Let $$A =\left\{x: |x^{2}-10|\leq6 \right\}$$  and $$B= \left\{x:|x-2|>1 \right\}$$. Then 

If the line $$\alpha x+4y=\sqrt{7}$$, where $$\alpha \epsilon R$$, touch the ellipse $$3x^{2}+4y^{2}=1$$ at the point P in the first quadrant, then one of the focal distances of P is:

Let y = y(x) be the solution of the differential equation $$\sec x \frac{dy}{dx}-2y=2+3\sin x, x\epsilon \left(-\frac{\pi}{2}, \frac{\pi}{2} \right), y(0)=-\frac{7}{4}$$. Then $$y\left(\frac{\pi}{6}\right)$$ is equal to:

The largest $$n\epsilon N$$, for which $$7^{n}$$ divides 101!, is :

Let the line $$L_{1}$$ be parallel to the vector $$-3\widehat{i} +2\widehat{j} + 4\widehat{k}$$ and pass through the point (2, 6, 7), and the line $$L_{2}$$ be parallel to the vector $$2\widehat{i} +\widehat{j} + 3\widehat{k}$$ and pass through the point (4, 3, 5). If the line $$L_{3}$$ is parallel to the vector $$-3\widehat{i} +5\widehat{j} + 16\widehat{k}$$ and intersects the lines $$L_{1}$$ and $$L_{2}$$ at the points C and D, respectively, then $$|\overrightarrow{CD}|^2$$ is equal to :

For a triangle ABC, let $$\overrightarrow{p} = \overrightarrow{BC}, \overrightarrow{q}= \overrightarrow{CA}$$ and $$\overrightarrow{r} = \overrightarrow{BA}$$. If $$|\overrightarrow{p}| = 2\sqrt{3}, |\overrightarrow{q}|=2$$ and $$\cos\theta = \frac{1}{\sqrt{3}}$$ where $$\theta$$ is the angle between $$\overrightarrow{p}$$ and $$\overrightarrow{q}$$, then $$|\overrightarrow{p} \times \left(\overrightarrow{q}-\overrightarrow{3r}\right)|^2 +3|\overrightarrow{r}|^2$$ is equal to :

The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + .... + (x + 2n - 2)(x + 2n) = $$\dfrac{8n}{3}$$ are two consecutive even integers, is:

Let $$y^{2}=12x$$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $$\angle OPA =90^\circ$$. Then the locus of the centroid of such triangles OPA is:

Let $$f(x) = x^{3}+ x^{2}f'(1)+2xf''(2)+f'''(3)$$, $$x\epsilon R$$. Then the value of f'(5) is :

A random varaible X takes values 0,1,2,3 with probabilities $$\frac{2a+1}{30},\frac{8a-1}{30},\frac{4a+1}{30}$$, b respectively, where $$a,b \epsilon R$$. let $$\mu$$ and $$\sigma$$ respectively be the mean and standard deviation of X such that $$\sigma^{2}+\mu^{2}=2$$. Then $$\frac{a}{b}$$ is equal to :

Let z be the complex number satisfying $$|z-5|\leq 3$$ and having maximum positive principal argument. Then $$34|\frac{5z-12}{5iz+16}|^2$$ is equal to :

Let the line L pass through the point ( - 3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point ( - 2, r, 1) is $$\sqrt{\frac{14}{3}}$$, then the sum of all possible values of r is:

Let $$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$. If $$a_{1}+a_{2}+....+a_{10}=62$$, then $$a_{1}$$ is equal to: 

If $$\left(\frac{1}{{}^{15}C_0} + \frac{1}{{}^{15}C_1}\right) \left(\frac{1}{{}^{15}C_1} + \frac{1}{{}^{15}C_2}\right) \cdots \left(\frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}}\right) = \frac{\alpha^{13}}{{}^{14}C_0 {}^{14}C_1 \cdots {}^{14}C_{12}}$$ then $$30\alpha$$ is equal to __________

If $$\displaystyle \int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$$, where $$a,b\epsilon N$$ then (2a + b} is equal to _________

Let [·] denote the greatest integer function and $$f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^{n} \left[\frac{k^2}{3^x}\right]$$. Then $$12 \sum_{j=1}^{\infty} f(j)$$ is equal to _______

If P is a point on the circle $$x^{2}+y^{2}=4$$, Q is a point on the straight line 5x + y + 2 = 0 and x- y + 1 = 0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________

Let the maximum value of $$\left(\sin^{-1}x\right)^2+\left(\cos^{-1}x\right)^2$$ for $$x\epsilon \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$$ be $$\frac{m}{n}\pi^{2}$$, where gcd
(m, n) = l. Then m + n is equal to ____________

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/ s. The frequency of this simple harmonic oscillator is _____Hz. [take $$\pi = \frac{22}{7}$$]

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is :
(Take $$k= \frac{1}{4\pi \epsilon_{0}}$$ and assume $$kQ^{2}$$ > $$Gm^{2}$$ where G is the Gravitational constant)

Given below are two statements :
Statement I : In a Young's double slit experiment, the angular separation of fringes will increase as the screen is moved away from the plane of the slits
Statement II: In a Young's double slit experiment, the angular separation of fringes will increase when monochromatic source is replaced by another monochromatic source of higher wavelength
In the light of the above statements, choose the correct answer from the options given below :

A battery with EMF E and internal resistance r is connected across a resistance R. The power consumption in R will be maximum when :

The charge stored by the capacitor C in the given circuit in the steady state is ______ $$\mu C$$.

The energy of an electron in an orbit of the Bohr's atom is $$-0.04E_{0} eV$$ where $$E_{0}$$ is the ground state energy. lf L is the angular momentum of the electron in this orbit and h is the Planck's constant, then $$\frac{2\pi L}{h}$$ is __________:

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are _______ and ____________ respectively.

Keeping tl1e significant figures in view, the sum of the physical quantities 52.01 m, 153.2 m and 0.123 m is:

An infinitely long straight wire carrying current I is bent in a planer shape as shown in the diagram. The radius of the circular part is r. The magnetic field at the centre O of the circular loop is :

The r.m.s. speed of oxygen molecules at $$47^\circ$$ is equal to that of the hydrogen molecules kept at ________ $$C^\circ$$. (Mass of oxygen molecule/mass of hydrogen molecule = 32/2)

A body of mass 2 kg is moving along x-direction such that its displacement as function of time is given by x(t) = $$\alpha t^{2} +\beta t +ym$$, where $$\alpha=1m/s^{2}, \beta=1m/s$$ and y=1m. The work done on the body during the time interval t = 2 s to t = 3 s, is _________ J.

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.

If refractive index of the material of prism is $$\sqrt{2}$$, the angle $$\theta$$ of prism is.

A large drum having radius R is spinning around its axis with angular velocity $$\omega$$, as shown in figure. The minimum value of $$\omega$$ so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M as $$\mu$$, is :

Surface tension of two liquids (having same densities), $$T_{1}$$ and $$T_{2}$$, are measured using capillary rise method utilizing two tubes with inner radii of $$r_{1}$$ and $$r_{2}$$
where $$r_{1} > r_{2}$$. The measured liquid heights in these tubes are $$h_{1}$$ and $$h_{2}$$ respectively. [Ignore the weight of the liquid about the lowest point of miniscus]. The heights $$h_{1}$$ and $$h_{2}$$ and surfaces tensions $$T_{1}$$ and $$T_{2}$$ satisfy the relation :

The pulley shown in the figure is made using a thin rim and two rods of length equal to the diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is ___________
(assume no slipping of string on pulley).

Two cars A and B each of mass $$10^{3}$$ kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is __________ J.s.

A capacitor C is first charged fully with potential difference of $$V_{0}$$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s 25% of the initial energy in the capacitor is transferred to the inductor. The value of t is ____________s.

Two known resistances of $$R \Omega$$ and $$2R \Omega$$ and one unknown resistance $$X \Omega$$ are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is $$X \Omega$$, then the value of X is __________ $$\Omega$$.

The total length of potentiometer wire AB is 50 cm in the arrangement as shown in figure. If P is the point where the galvanometer shows zero reading then the length AP is ____ cm.

A spherical body of radius r and density $$\sigma$$ falls freely through a viscous liquid having density $$\rho$$ and viscosity $$\eta$$ and attains a terminal velocity $$\upsilon_{0}$$. Estimated maximum error in the quantity $$\eta$$ is: (Ignore errors associated with $$\sigma,\rho$$ and g, gravitational acceleration)

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, $$\sigma$$ = 10mho/m. The ratio of maximum conduction current density to maximum displacement current
density is ___________.
[Take $$\frac{1}{4\pi \epsilon_{0}} = 9 \times 10^{9} Nm^{2}/C^{2}$$]

A diatomic gas ( y= 1.4) does 100 J of work when it is expanded isobarically. Then the heat given to the gas ____ J.

In a Young's double slit experiment set up, the two slits are kept 0.4. mm apart and screen is placed at 1 m from slits. If a thin transparent sheet of thickness $$20\mu m$$ is introduced in front of one of the slits then center bright fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by $$\frac{\alpha}{10}$$, where $$\alpha$$ is __________.

A particle having electric charge $$3\times 10^{-19}$$ C and mass $$6 \times 10^{-27}$$ kg is accelerated by applying an electric Potential of 1.21 V. Wavelength of the matter wave
associated with the particle is $$\alpha \times 10^{-12}m$$. The value of $$\alpha$$ is __________. (Take Planck's constant = $$6.6 \times 10^{-34}$$ J.s. )

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be ________ cm/ s.

The correct order of reactivity of the following benzyl halides towards reaction with KCN is:

Given below are four compounds :
(a) n-propyl choride
(b) iso-propyl chloride
(c) sec-butyl chloride
(d) neo-pentyl chloride
Percentage of carbon in the one which exhibits optical isomerism is:

By usual analysis, 1.00 g of compound (X) gave 1.79 g of magnesium pyrophosphate. The percentage of phosphorus in compound (X) is: (nearest integer)
(Given, molar mass in $$gmol^{-1}$$ : 0 = 16, Mg = 24, P = 31 )

Given below are two statements:
Statement I : Crystal Field Stabilization Energy (CFSE) of $$\left[Cr\left( H_{2}O \right)_{6} \right]^{2+}$$ is greater than that of $$\left[Mn\left( H_{2}O \right)_{6} \right]^{2+}$$.
Statement II: Potassium ferricyanide has a greater spin-only magnetic moment than sodium ferrocyanide.
In the light of the above statements, choose the correct answer from the options given below :

Aqueous HCI reacts with $$MnO_{2} \left(s\right)$$ to form $$MnCl_{2}\left(aq\right)$$, $$Cl_{2}\left(g\right)$$ and $$H_{2}O\left(l\right)$$. What is the weight (in g) of $$Cl_{2}$$ liberated when 8.7 g of $$MnO_{2} \left(s\right)$$ is reacted with excess aqueous HCI solution ?
(Given Molar mass in g $$mol^{-1}$$ Mn = 55, Cl = 35.5, 0 = 16, H = l )

For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature?

Match List - I with List - II.

Choose the correct answer from the options given below :

Consider the following data :
$$\Delta_f H^\ominus$$ (methane, g) = - X kJ $$mol^{-1}$$
Enthalpy of sublimation of graphite = Y kJ $$mol^{-1}$$
Dissociation enthalpy of $$H_{2}$$ = Zkj $$mol^{-1}$$
The bond enthalpy of C - H bond is given by :

The correct order of the rate of the reaction for the following reaction with respect to nucleophiles is:
$$CH_{3}Br + Nu^{\ominus} \rightarrow CH_{3}Nu+Br^{\ominus}$$

Given below are two statements:
Statement I: Compound (X), shown below, dissolves in $$NaHCO_{3}$$ solution and has two chiral carbon atoms

Statement II: Compound (Y), shown below, has two carbons with $$sp^{3}$$ hybridization, one carbon with $$sp^{2}$$ and one carbon with sp hybridization

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

Given below are two statements:
Statement I: The correct order in terms of bond dissociation enthalpy is $$Cl_{2} > Br_{2} > F_{2} > I_{2}$$.
Statement II : The correct trend in the covalent character of the metal halides is $$[SnCl_{4} > SnCl_{2}]$$, $$[PbCl_{4}> PbCl_{2}]$$, and $$[UF_{4} > UF_{6}]$$.
In The light oh the above statements, choose the correct answer from the options given below:

Consider the following spectral lines for atomic hydrogen :
A. First line of Paschen series
B. Second line of Balmer series
C. Third line of Paschen series
D. Fourth line of Bracket series
The correct arrangement of the above lines in ascending order of energy is :

Match List - I with List - II.

Choose the correct answer from the options given below:

Given below are some of the statements about Mn and $$Mn_{2}O_{7}$$. Identify the correct statements.
A. Mn forms the oxide $$Mn_{2}O_{7}$$, in which Mn is in its highest oxidation state.
B. Oxygen stabilizes the Mn in higher oxidation states by forming multiple bonds with Mn.
C. $$Mn_{2}O_{7}$$ is an ionic oxide.
D. The structure of $$Mn_{2}O_{7}$$ consists of one bridged oxygen.
Choose the correct answer from the options given below :

Decompasition of A is a first order reaction at T(K) and is given by $$A(g) \rightarrow B(g)+C(g)$$.
In a closed 1 L vessel, 1 bar A(g) is allowed to decompose at T(K). After 100 minutes, the total pressure was 1.5bar. What is the rate constant $$(in min^{-1})$$ of the reaction ? (log 2 = 0.3)

The correct increasing order of C - H(A), C - 0 (B), C = O(C) and C = N (D) bonds in terms of covalent bond length is :

On heating a mixture of common salt and $$K_{2}Cr_{2}O_{7}$$ in equal amount along with concentrated $$H_{2}SO_{4}$$ in a test tube, a gas is evolved. Formula of the gas evolved and oxidation State of the central metal atom in the gas respectively are:

Given below are two statements:
Statement I: The correct order in terms of atomic/ionic radii is $$Al >Mg > Mg^{2+} >Al^{3+}$$
Statement II: The correct order in terms of the magnitude of electron gain enthalpy is Cl > Br >S >O.
In the light of the above statements, choose the correct answer from the options given below:

Consider the above sequence of reactions. The number of bromine atom(s) in the final product (P) will be:

The correct statements are :
A. Activation energy for enzyme catalysed hydrolysis of sucrose is lower than that of acid catalysed hydrolysis.
B. During denaturation, secondary and tertiary structures of a protein are destroyed but primary structure remains intact.
C. Nucleotides are joined together by glycosidic linkage between $$C_{1}$$ and $$C_{4}$$ carbons of the pentose sugar.
D. Quaternary structure of proteins represents overall folding of the polypeptide chain.
Choose the correct answer from the options given below :

The first and second ionization constants of $$H_{2}X$$ are $$2.5 \times 10^{-8}$$ and $$1.0 \times 10^{-13}$$ respectively. The concentration of $$X^{2-}$$ in $$0.1 MH_{2} X$$ solution is _________ $$\times 10^{-15}M$$. (Nearest Integer)

A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y'(molar mass=300 g $$mol^{-1}$$ ) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is __________ $$\times 10^{-2}$$. (nearest integer)
[Given : $$K_{b}$$ of the solvent =5.0 K kg $$mol^{-1}$$]
Assume the solution to be dilute and no association or dissociation of X takes place in solution.

The osmotic pressure of a living cell is 12 atm at 300 K. The strength of sodium chloride solution that is isotonic with the living cell at tltis temperature is __________ $$g L^{-1}$$.
(Nearest integer)
Given: R = 0.08 L atm $$K^{-1} mol^{-1}$$
Assume complete dissociation of NaCl
(Given : Molar mass of Na and Cl are 23 and 35.5 g $$mol^{-1}$$ respectively.)

MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K.
$$MX(s) \rightleftharpoons M^{+} (aq)+X^{-}(aq); K_{sp}=10^{-10}$$
If the standard reduction potential for $$M^{+}(aq)\xrightarrow {+e^{-}}M(s)$$ and $$\left(E_{M^{+}/M}^\ominus\right)=0.79V$$, then the value of the standard reduction potential for the metal/metal insoluble salt electrode $$E^{\ominus}_{X^{-}/MX (s)/M }$$ is __________mV.(nearest integer)
[Given : $$\frac{2.303RT}{F} = 0.059V$$]

Identify the metal ions among $$Co^{2+}, Ni^{2+}, Fe^{2+}, V^{3+} \text{ and } Ti^{2+}$$ having a spin-only magnetic moment value more than 3.0 BM. The sum of unpaired electrons present in the high spin octahedral complexes formed by those metal ions is __________.