NTA JEE Mains 21st Jan 2026 Shift 1

Let $$\overrightarrow{a}=-\widehat{i}+2\widehat{j}+2\widehat{k},\overrightarrow{b}=8\widehat{i}+7\widehat{j}-3\widehat{k} \text { and } \overrightarrow{c}$$ be a vector such that $$\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$$. If $$\overrightarrow{c}\cdot(\widehat{i}+\widehat{j}+\widehat{k})=4$$, then $$\mid\overrightarrow{a}+\overrightarrow{c}\mid^{2}$$ is equal to :

Let PQ and MN be two straight lines touching the circle $$x^{2}+y^{2}-4x-6y-3=0$$ at the points A and B respectively. Let O be the centre of the circle and $$\angle AOB=\pi/3$$. Then the locus of the point of intersection of the lines PQ and MN is:

The number of relations, defined on the set {a, b, c, d}, which are both reflexive and symmetric, is equal to:

If $$x^{2}+x+1=0$$, then the value of $$\left(x+\frac{1}{x}\right)^{4}+\left(x^{2}+\frac{1}{x^{2}}\right)^{4}+\left(x^{3}+\frac{1}{x^{3}}\right)^{4}+...+\left(x^{25}+\frac{1}{x^{25}}\right)^{4}$$ is:

Let $$f: R \rightarrow (0, \infty)$$ be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f" (3) = 4. Then $$\lim_{x \rightarrow 1}\left(\log_{a}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$$ ls equal to :

The value of $$\operatorname{cosec}10°-\sqrt{3}\sec10°$$ is equal to :

If the coefficient of x in the expansion of $$(ax^{2}+bx+c)(1-2x)^{26}$$. is - 56 and the coefficients of $$x^{2}\text{ and }x^{3}$$ are both zero, then a + b + c is equal to:

Let a point A lie between the parallel lines $$L_{1}\text{ and }L_{2}$$ such that its distances from $$L_{1}\text{ and }L_{2}$$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines $$L_{1}\text{ and }L_{2}$$, respectively, is:

Let O be the vertex of the parabola $$x^{2}=4y$$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2: 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:

Let $$\overrightarrow{c} \text{ and } \overrightarrow{d}$$ be vectors such that $$\mid\overrightarrow{c}+\overrightarrow{d}\mid=\sqrt{29}$$ and $$\overrightarrow{c}\times( 2\widehat{i}+3\widehat{j}+4\widehat{k})=(2\widehat{i}+3\widehat{j}+4\widehat{k})\times\overrightarrow{d}$$. If $$\lambda_{1}, \lambda_{2}( \lambda_{1}> \lambda_{2})$$ are the possible values of $$(\overrightarrow{c}+\overrightarrow{d})\cdot(-7\widehat{i}+2\widehat{j}+3\overrightarrow{k})$$, then the equation $$K^{2}x^{2}+(K^{2}-5K+\lambda_{1})xy+\left(3K+\frac{\lambda_{2}}{2} \right)y^{2}-8x+12y+\lambda_{2}=0$$ represents a circle, for K equal to :

Let $$a_{1},a_{2},a_{3},...$$ be a G.P. of increasing positive terms such that $$a_{2}.a_{3}.a_{4}=64\text{ and }a_{1}+a_{3}+a_{5}=\frac{813}{7}.\text{ Then }a_{3}+a_{5}+a_{7}$$ is equal to :

The sum of all the roots of the equation $$(x-1)^2-5\mid x-1\mid+\ 6=0$$ is:

Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x >  y, be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x - 4,y,5} one after an other without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is :

Let the foci of a hyperbola coincide with the foci of the ellipse $$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

If the domain of the function $$f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^{2}-3x+1)$$ is the interval $$[\alpha, \beta]$$, then $$\alpha+2\beta$$ is equal to:

The area of the region, inside the ellipse $$x^{2}+4y^{2}=4$$ and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:

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

Let $$(\alpha,\beta,\gamma)$$ be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line $$\overrightarrow{r}=(-\widehat{i}+3\widehat{j}+\widehat{k})+\lambda(2\widehat{i}+3\widehat{j}-\widehat{k}).$$ Then the length of the projection of the vector $$\alpha\widehat{i}+\beta\widehat{j}+\gamma\widehat{k}$$ on the vector $$6\widehat{i}+2\widehat{j}+3\widehat{k}$$ is:

The number of strictly increasing functions f from the set {1, 2, 3, 4, 5, 6} to the set {1, 2, 3, ... , 9} such that $$f(i)\neq i \text{ for }1\leq i\leq 6$$, is equal to:

The value of $$\int_{\frac{-\pi}{6}}^{\frac{\pi}{6}}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\frac{\pi}{6})}\right)dx$$ is equal to :

$$6\int_{0}^{\pi} |(\sin3x+\sin2x+\sin x)|dx$$ is equal to _________.

Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.

Let S= {(m, n) :m, n $$\epsilon$$ {1, 2, 3, .... , 50}}. lf the number of elements (m, n) in S such that $$6^m+9^n$$ is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p +q is equal to ________.

Let $$f:R\rightarrow R$$ be a twice differentiable function such that the quadratic equation $$f(x)m^{2}-2 f'(x)m+ f''(x)=0$$ in m, has two equal roots for every $$x \epsilon R$$. If $$ f(0)=1,f'(0)=2$$, and $$(\alpha,\beta)$$ is the largest interval in which the function $$f(\log_{e}{x-x})$$ is increasing, then $$\alpha+\beta$$ is equal to ________.

For some $$\alpha,\beta\epsilon R$$, let $$A=\begin{bmatrix}\alpha &  2 \\ 1 &  2 \end{bmatrix}\text{ and }B=\begin{bmatrix}1 &  1 \\1 &   \beta \end{bmatrix}$$ be such that $$A^{2}-4A+2I=B^2-3B+I=0$$. Then $$(det(adj(A^3-B^3)))^2$$ is equal to _______.

Two strings (A, B) having linear densities $$\mu_{A}=2\times10^{-4}kg/m\text{ and },\mu_{B}=4\times10^{-4}kg/m$$ and lengths $$L_{A}=2.5m$$ and $$L_{B}=1.5m$$ respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $$t_{1}\text{ and } t_{2}$$, respectively, to reach the joint. The ratio $$t_{1}/ t_{2}$$ is :

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from 27°C to 87°C. The rate of consumption of the gas is ___ g/s. (take heat of combustion of $$gas=5.0\times10^{4}J/g$$) specific heat capacity of water =4200 J/ kg.°C

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. lf the resistance of the total circuit is 2Ω then the force needed to move the rod towards right with constant speed (v) of 1.5 m/s is ___ N

Consider a modified Bernoulli equation.
$$\left( P + \frac{A}{B t^2} \right) + \rho g (h + B t) + \frac{1}{2}\,\rho V^2 = \text{constant}$$
If t has tile dimension of time then the dimensions of A and B are ___.___ respectively.

In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness t and refractive index n(=1.5), the central fringe shifts by 0.2 cm. The value of t is____ cm

A current carrying solenoid is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. lf g is acceleration due to gravity then the acceleration (n) of the charged particle will satisfy :

A uniform rod of mass m and length l is suspended by means of two identical inextensible light strings as shown in the figure. Tension in one of the strings, immediately after the other string is cut, is ____ . (g is the acceleration due to gravity)

A parallel plate capacitor has capacitance C, when there is vacuum within the parallel plates. A sheet having thickness $$\left(\frac{1}{3}\right)^{rd}$$ of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is:

The given circuit works as:

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at 30°C. The coefficient of linear expansion of aluminium and steel are $$24\times10^{-6}/^{\circ}C\text{ and }1.2\times10^{-5}/^{\circ}C$$, respectively. The length of this composite rod when its temperature is raised to $$100^\circ$$ C, is ___ cm.

Potential energy (V) versus distance (x) is given by the graph. Rank various regions as per the magnitudes of the force (F) acting on a particle from high lo low.

A 4 kg mass moves under the influence of a force $$\overrightarrow{F}=\left(4t^{3}\widehat{i}-3t\widehat{j}\right)N$$ where t is the time in second. If mass starts from origin at t= 0, the velocity and position after t= 2 s will be:

The electric field in a plan a electromagnetic wave is given by :
$$E_{y}=69\sin[0.6\times10^{3}x-1.8\times10^{11}t]V/m$$
The expression for magneticfield associated with this electromagnetic wave is ___ T.

A light wave described by $$E=60[\sin(3\times10^{15})t+\sin(12\times10^{15})t]$$ (in SI units) falls on a metal surface of work function 2.8 eV. The maximum kinetic energy of ejected photoelectron is (approximately) ___ eV. $$(h=6.6\times10^{-34}J.s\text{ and }e=1.6\times10^{19}C)$$

Water flows through a horizontal tube as shown in the figure. The difference in height between the water colunms in vertical tubes is 5 cm and the area of cross-sections at A and B are $$6cm^{2}$$ and $$3cm^{2}$$ respectively. The rate of flow will be ____ $$cm^{3/s}$$. $$(take g=10m/s^{2})$$

A point charge of $$10^{-8}$$ is placed at origin. The work done in moving a point charge $$2 \mu C$$ from point A(4, 4, 2) m to B(2, 2, 1) m is _____J $$\left(\frac{1}{4\pi\epsilon _{0}}=9\times10^{9} \text{in SI units}\right)$$

A conducting circular loop of area $$1.0m^{2}$$ is placed perpendicular to a magnetic field which varies as $$B = \sin(100 t)$$ Tesla. If the resistance of the loop is $$100 \Omega$$, then the average thermal energy dissipated in the loop in one period is _______J.

In an experiment the values of two spring constants were measured as $$k_{1}=(10\pm0.2)N/m\text{ and }k_{2}=(20\pm0.3)N/m$$. lf these springs are connected in parallel, then the percentage error in equivalent spring constant is :

If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is ___ m. (Atomic number of gold = 79 and $$\frac{1}{4\pi\epsilon _{0}}=9\times10^{9} \text{in SI units} )$$

Initially a satellite of 100 kg is in a circular orbit of radius $$1.5R_{E}$$ This satellite can be moved to a circular orbit of radius $$3R_{E}$$ by supplying $$\alpha\times10^{6}J$$ of energy The value of $$\alpha$$ is ____. (Take Radius of Earth $$R_{E}=6\times10^{6}m\text{ and }g=10m/s^{2}$$)

Two identical thin rods of mass M kg and length L m are connected as shown in the figure below. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $$\frac{x}{12}ML^{2}\text{kg m}^{2}$$. The value of x is ____ .

10 mole of oxygen is heated at constant volume from $$30^{\circ}C  \text{to}  40^{\circ}C$$. The change in the internal energy of the gas is ____ cal (the molecular specific heat of oxygen at constant pressure, $$C_{p}= 7 \text{cal}/\text{mol}.^{\circ}C \text{and} R = 2 \text{cal}./\text{mol}.^{\circ}C).$$

A collimated beam of light of diameter 2 mm is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. lf first lens has focal length 40 mm, then the focal length of second lens is ____ mm.

The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1Ω is connected across these points is ____ J

In a microscope the objective is having focal length $$f_{0}=2 \text{cm}$$ and eye-piece is having focal length $$f_{e} = 4 \text{cm}$$ The tube length is 32 cm. the magnification produced by this microscope for normal adjustment is _______.

From the following, the least stable structure is :

Which of the following represents the correct trend for the mentioned property ?
A.$$F > P > S > B$$  - First Ionization Energy
B.$$Cl > F > S > P$$ - Electron Affinity
C. $$K > Al > Mg > B$$ - Metallic character
D. $$K_{2}O > Na_{2}O > MgO > AL_{2}O_{3}$$ - Basic character
Choose the correct answer from the options given below :

80 mL of a hydrocarbon on mixing with 264 mL of oxygen in a closed U-tube undergoes complete combustion. The residual gases after cooling to 273 K occupy 224 mL. When the system is treated with KOH solution, the volume decreases to 64 ml. The formula of the hydrocarbon is:

Given below are two statements:
Statement I: Among $$[Cu(NH_{3})_{4}]^{2+},[Ni(en)_{3})]^{2+},[Ni(NH_{3})_{6}]^{2+}$$ and $$[Mn(H_{2}O)_{6}]^{2+},[Mn(H_{2}O)_{6}]^{2+}$$ has the maximum number of unpaired electrons.
Statement II : The number of pairs among $$\left\{[Ni(Cl_{4}]^{2-},[Ni(CO)_{4}]\right\}$$, $$\left\{[NiCl_{4}]^{2-},[Ni(CN)_{4}]^{2-}\right\}$$ and $$\left\{[Ni(CO)_{4}],[Ni(CN)_{4}]^{2-}\right\}$$ that contain only diamagnetic species is two.
ln the light of the above statements, choose the correct answer from the options given below:

Identify the correct statements.
A. Arginine and Tryptophan are essential amino acids.
B. Histidine does not contain heterocyclic ring in its structure.
C. Proline is a six membered cyclic ring amino acid.
D. Glycine does not have chiral centre.
E. Cysteine has characteristic feature of side chain as $$MeS-CH_{2}-CH_{2}-.$$
CHoose the correct answer from the options given below :

Consider the following reactions.
$$PbCl_{2}+K_{2}CrO_{4}\rightarrow A+2KCI$$
(Hot solution)
$$A+NaOH\rightleftharpoons B+Na_{2}CrO_{4}$$
$$PbSO_{4}+4CH_{3}COONH_{4}\rightarrow (NH_{4})_{2}SO_{4}+X$$
In the above reactions, A, Band X are respectively.

$$\text{MnO}_4^{2-}$$, in acidic medium, disproportionates to :

Given below are two statements:
Statement I : When an electric discharge is passed through gaseous hydrogen, the hydrogen molecules dissociate and the energetically excited hydrogen atoms produce electromagnetic radiation of discrete frequencies.
Statement II: The frequency of second line of Balmer series obtained from He+ is equal to that of first line of Lyman series obtained from hydrogen a tom.

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

An organic compound "P" of molecular formula $$C_6H_{12}O_3$$ gives positive Iodoform test but negative Tollen's test. When "P" is treated with dilute acid, it produces "Q". "Q" gives positive Tollen's test and also iodoform test. The structure of "P" is :

14.0 g of calcium metal is allowed to react with excess HCI at 1.0 atm pressure and 273 K
Which of the following statements is incorrect?
[Given : Molar mass in g $$\text{mol}^{-1}$$ of Ca-40, Cl-35.5, H-1]

Given below are two statements:
Statement I: The number of pairs among $$[SiO_{2},CO_{2}],[SnO,SnO_{2}],[PbO,PbO_{2]}$$ and $$[GeO,GeO_{2}]$$, which contain oxides that are both amphoteric is 2.
Statement ll: $$BF_{3}$$ is an electron deficient molecule, can act as a Lewis add, forms adduct with $$NH_{3}$$ and has a trigonal planar geometry.
In the light of the above statements, choose the correct answer from the options given below:

Which of the following graphs between pressure 'p' versus volume 'V' represents the maximum work done?

Identify A in the following reaction.

A hydrocarbon 'P' $$(C_{4}H_{8})$$ on reaction with HCl gives an optically active compound 'Cl' $$(C_{4}H_{9}Cl)$$ which on reaction with one mole of ammonia gives compound 'R' $$(C_{4}H_{11}N)$$ on diazolization followed by hydrolysis gives 'S'. Identify P, Q, Rand S.

Elements P and Q form two types of non-volatile, non-ionizable compounds PQ and $$PQ_{2}$$. When 1g of PQ is dissolved in 50 g of solvent ''A', $$\Delta T_{b}$$was 1.176 K while when 1 g of $$PQ_{2}$$ is dissolved in 50g of solvent 'A'.$$\Delta T_{b}$$ was 0.689 K ($$K_{b}$$ of 'A' =5K kg $$mol^{-1}$$) The molar masses of elements P and Q (in g  $$mol^{-1}$$ )  respectively, are:

Identify correct statements from the following :
A Propanal and propanone are functional isomers.
B. Ethoxyethane and methoxypropane are metamers.
C. But-2-ene shows optical isomerism.
D. But-1-ene and but-2-ene are functional isomers.
E. Pentane and 2, 2-dimethyl propane are chain isomers.
Choose the correct answer from the options given below:

An organic compound (P) on treatment with aqueous ammonia under hot condition forms compound (Q) which on heating with $$Br_{2}$$ and KOH forms compound (R) having molecular formula $$C_{6}H_{7}N$$ Names of P, Q and R respectively are.

Given below are two statements:
Statement I :
The number of species among $$SF_{4},NH_4^+,[NiCl_{4}]^{2-},XeF_{4},[PtCl_{4}]^{2-},SeF_{4}$$ and $$[Ni(CN)_{4}]^{2-}$$, tha t have tetrahedral geometry is 3.
Statement II :
In the set $$[NO_{2},BeH_{2},BF_{3},AlCl_{3}]$$ all the molecules have incomplete octet around central atom.
In the light of the above statements, choose the correct answer from the options given below :

In Carius method, 0.75 g of an organic compound gave 1.2 g of barium sulphate, find percentage of sulphur (molar mass 32 g $$mol^{-1}$$ ) Molar mass of barium sulphate is 233 g $$mol^{-1}$$

For the reaction $$N_{2}O_{4}\rightleftharpoons2NO_{2}$$ , graph is plotted as shown below. Identify correct statements.
A. Standard free energy change for the reaction is $$-5.40kJmol^{-1}$$.
B. As $$\triangle G^{\ominus}$$ in graph is positive, $$N_{2}O_{4}$$ will not dissociate into $$NO_{2}$$ at all.
C. Reverse reaction will go to completion.
D. When 1 mole of $$N_{2}O_{4}$$ changes into equilibrium mixture, value of $$\triangle G^{\ominus}$$ = -0.84kJ $$mol^{-1}$$.
E. When 2 mole of $$NO_{2}$$ changes into equilibrium mixture, $$\triangle G^{\ominus}$$ for equilibrium mixture is -6.24kJ $$mol^{-1}$$.

Choose the correct answer from the options given below:

Consider the following reaction sequence

The percentage of nitrogen in product 'T' formed is ____ %. (Nearest integer)
(Given molar mass in g $$mol^{-1}$$ H : 1, C: 12, N : 14, 0: 16)

Pre-exponential factors of two different reactions of same order are identical. Let activation energy of first reaction exceeds the activation energy of second reaction by 20 kJ $$mol^{-1}$$. If $$k_{1}\text{ and }k_{2}$$ are the rate constants of first and second reaction respectively at 300 K, then In $$\frac{k_{2}}{k_{1}}$$ will be ___.
(nearest integer) $$[R=8.3JK^{-1}mol^{-1}]$$

Use the following data :

One mole each of $$A_{2}(g)$$ and $$B_{2}(g)$$ are taken in a 1 L closed flask and allowed to establish the equilibrium at 500K
$$A_{2}(g)+B_{2}(g)\rightleftharpoons2AB(g)$$
The value of x in $$( kJ mol^{-1})$$ is ____ . (Nearest integer)
(Given: log K=2.2 R= 8.3 kJ $$K^{-1} mol^{-1}$$)

Consider the following reactions:
$$NaCl+K_{2}Cr_{2}O_{7}+H_{2}SO_{4}\rightarrow A+KHSO_{4}+NaHSO_{4}+H_{2}O$$
$$A+NaOH\rightarrow B+NaCl+H_{2}O$$
$$B+H_{2}SO_{4}+H_{2}O_{2}\rightarrow C+Na_{2}SO_{4}+H_{2}O$$
In the product 'C, 'X' is the number of $$O_{2}^{2-}$$ units, 'Y' is the total number oxygen atoms present and 'Z' is the oxidation state of Cr·. The value of X + Y + Z is ______

The pH and conductance of a weak acid (HX) was found to be 5 and $$4\times10^{-5}S$$. respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 $$cm^{2}$$ were at a distance of 15 cm apart. The value of the limiting molar conductivity is ______ S $$m^{2}mol^{-1}$$ (nearest integer)
(Given : degree of dissociation of the weak acid ($$\alpha$$) < < 1)