NTA JEE Main 31st January 2023 Shift 2

A body is moving with constant speed, in a circle of radius 10 m. The body completes one revolution in 4 s. At the end of 3rd second, the displacement of body (in m) from its starting point is:

Match List I with List II
List I                                    List II
A. Angular momentum     I. [ML$$^2$$T$$^{-2}$$]
B. Torque                         II. [ML$$^{-2}$$T$$^{-2}$$]
C. Stress                          III. [ML$$^2$$T$$^{-1}$$]
D. Pressure gradient      IV. [ML$$^{-1}$$T$$^{-2}$$]
Choose the correct answer from the options given below:

A stone of mass 1 kg is tied to end of a massless string of length 1 m. If the breaking tension of the string is 400 N, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is:

A body of mass 10 kg is moving with an initial speed of 20 m s$$^{-1}$$. The body stops after 5 s due to friction between body and the floor. The value of the coefficient of friction is: (Take acceleration due to gravity $$g = 10$$ m s$$^{-2}$$)

A body weight $$W$$, is projected vertically upwards from earth's surface to reach a height above the earth which is equal to nine times the radius of earth. The weight of the body at that height will be:

Under the same load, wire A having length 5.0 m and cross section $$2.5 \times 10^{-5}$$ m$$^2$$ stretches uniformly by the same amount as another wire B of length 6.0 m and a cross section of $$3.0 \times 10^{-5}$$ m$$^2$$ stretches. The ratio of the Young's modulus of wire A to that of wire B will be:

Heat energy of 735 J is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but do not oscillate. The increase in the internal energy of the gas will be:

A hypothetical gas expands adiabatically such that its volume changes from 08 litres to 27 litres. If the ratio of final pressure of the gas to initial pressure of the gas is $$\dfrac{16}{81}$$. Then the ratio of $$\dfrac{C_p}{C_v}$$ will be.

For a solid rod, the Young's modulus of elasticity is $$3.2 \times 10^{11}$$ N m$$^{-2}$$ and density is $$8 \times 10^3$$ kg m$$^{-3}$$. The velocity of longitudinal wave in the rod will be

Considering a group of positive charges, which of the following statements is correct?

The number of turns of the coil of a moving coil galvanometer is increased in order to increase current sensitivity by 50%. The percentage change in voltage sensitivity of the galvanometer will be:

The $$H$$ amount of thermal energy is developed by a resistor in 10 s when a current of 4 A is passed through it. If the current is increased to 16 A, the thermal energy developed by the resistor in 10 s will be

A long conducting wire having a current $$I$$ flowing through it, is bent into a circular coil of $$N$$ turns. Then it is bent into a circular coil of $$n$$ turns. The magnetic field is calculated at the centre of coils in both the cases. The ratio of the magnetic field in first case to that of second case is:

An alternating voltage source $$V = 260 \sin(628t)$$ is connected across a pure inductor of 5 mH. Inductive reactance in the circuit is:

Match List I and List II
List I                    List II
A. Microwaves       I. Physiotherapy
B. UV rays               II. Treatment of cancer
C. Infra-red rays     III. Lasik eye surgery
D. X-rays                  IV. Aircraft navigation
Choose the correct answer from the option given below:

A microscope is focused on an object at the bottom of a bucket. If liquid with refractive index $$\dfrac{5}{3}$$ is poured inside the bucket, then microscope have to be raised by 30 cm to focus the object again. The height of the liquid in the bucket is:

If the two metals $$A$$ and $$B$$ are exposed to radiation of wavelength 350 nm. The work functions of metals $$A$$ and $$B$$ are 4.8 eV and 2.2 eV. Then choose the correct option

The radius of electron's second stationary orbit in Bohr's atom is $$R$$. The radius of 3$$^{rd}$$ orbit will be

Given below are two statements:
Statement I: In a typical transistor, all three regions emitter, base and collector have same doping level.
Statement II: In a transistor, collector is the thickest and base is the thinnest segment.
In the light of the above statements, choose the most appropriate answer from the options given below.

Given below are two statements
Statement I: For transmitting a signal, size of antenna ($$l$$) should be comparable to wavelength of signal (at least $$l = \dfrac{\lambda}{4}$$ in dimension).
Statement II: In amplitude modulation, amplitude of carrier wave remains constant (unchanged).
In the light of the above statements, choose the most appropriate answer from the options given below

Two bodies are projected from ground with same speeds 40 m s$$^{-1}$$ at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of 60°, with horizontal then sum of the maximum heights, attained by the two projectiles, is ______ m. (Given $$g = 10$$ m s$$^{-2}$$)

A ball is dropped from a height of 20 m. If the coefficient of restitution for the collision between ball and floor is 0.5, after hitting the floor, the ball rebounds to a height of ______ m.

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3 : 5. The moment of inertia of these discs respectively about their diameters will be in the ratio of $$\dfrac{x}{6}$$. The value of $$x$$ is ______.

A water heater of power 2000 W is used to heat water. The specific heat capacity of water is 4200 J kg$$^{-1}$$ K$$^{-1}$$. The efficiency of heater is 70%. Time required to heat 2 kg of water from 10°C to 60°C is ______ s. (Assume that the specific heat capacity of water remains constant over the temperature range of the water).

The displacement equations of two interfering waves are given by $$y_1 = 10 \sin\left(\omega t + \dfrac{\pi}{3}\right)$$ cm, $$y_2 = 5\left[\sin(\omega t) + \sqrt{3}\cos(\omega t)\right]$$ cm respectively. The amplitude of the resultant wave is ______ cm.

Two parallel plate capacitors $$C_1$$ and $$C_2$$ each having capacitance of 10 $$\mu$$F are individually charged by a 100 V D.C. source. Capacitor $$C_1$$ is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor $$C_2$$ is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor $$C_1$$ is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be ______ V.
(Assuming Dielectric constant = 10)

For the given circuit, in the steady state, $$|V_B - V_D|$$ = ______ V.

A series LCR circuit consists of $$R = 80$$ $$\Omega$$. $$X_L = 100$$ $$\Omega$$, and $$X_C = 40$$ $$\Omega$$. The input voltage is 2500 $$\cos(100\pi t)$$ V. The amplitude of current, in the circuit, is ______ A.

Two light waves of wavelengths 800 and 600 nm are used in Young's double slit experiment to obtain interference fringes on a screen placed 7 m away from plane of slits. If the two slits are separated by 0.35 mm, then shortest distance from the central bright maximum to the point where the bright fringes of the two wavelength coincide will be ______ mm.

If the binding energy of ground state electron in a hydrogen atom is 13.6 eV, then, the energy required to remove the electron from the second excited state of Li$$^{2+}$$ will be: $$x \times 10^{-1}$$ eV. The value of $$x$$ is ______.

When a hydrocarbon A undergoes complete combustion it requires 11 equivalents of oxygen and produces 4 equivalents of water. What is the molecular formula of A?

Arrange the following orbitals in decreasing order of energy.
A. n = 3, l = 0, m = 0
B. n = 4, l = 0, m = 0
C. n = 3, l = 1, m = 0
D. n = 3, l = 2, m = 1
The correct option for the order is:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The first ionization enthalpy of 3d series elements is more than that of group 2 metals
Reason (R): In 3d series of elements successive filling of d-orbitals takes place.
In the light of the above statements, choose the correct answer from the options given below:

Incorrect statement for the use of indicator in acid-base titration is:

Given below are two statements:
Statement I: H$$_2$$O$$_2$$ is used in the synthesis of Cephalosporin
Statement II: H$$_2$$O$$_2$$ is used for the restoration of aerobic conditions to sewage wastes.
In the light of the above statements, choose the most appropriate answer from the options given below:

The element playing significant role in neuromuscular function and interneuronal transmission is:

The Lewis acid character of boron tri halides follows the order:

In Dumas method for the estimation of N$$_2$$, the sample is heated with copper oxide and the gas evolved is passed over:

A hydrocarbon 'X' with formula C$$_6$$H$$_8$$ uses two moles of H$$_2$$ on catalytic hydrogenation of its one mole. On ozonolysis, 'X' yields two moles of methane dicarbaldehyde. The hydrocarbon 'X' is:

The normal rain water is slightly acidic and its pH value is 5.6 because of which one of the following?

Evaluate the following statements for their correctness.
A. The elevation in boiling point temperature of water will be same for 0.1 M NaCl and 0.1 M urea.
B. Azeotropic mixture boil without change in their composition.
C. Osmosis always takes place from hypertonic to hypotonic solution.
D. The density of 32% H$$_2$$SO$$_4$$ solution having molarity 4.09 M is approximately 1.26 g mL$$^{-1}$$.
E. A negatively charged sol is obtained when KI solution is added to silver nitrate solution.
Choose the correct answer from the options given below:

Match List I with List II
List I                                                                List II
A. Physisorption                                       I. Single Layer Adsorption
B. Chemisorption                                      II. 20-40 kJ mol$$^{-1}$$
C. N$$_2$$(g) + 3H$$_2$$(g) $$\xrightarrow{Fe(s)}$$ 2NH$$_3$$(g)        III. Chromatography
D. Analytical Application or Adsorption   IV. Heterogeneous catalysis
Choose the correct answer from the options given below:

Which one of the following statements is incorrect?

Which of the following elements have half-filled f-orbitals in their ground state?
A. Sm, B. Eu, C. Tb, D. Gd, E. Pm
(Given: atomic number Sm = 62; Eu = 63; Tb = 65; Gd = 64, Pm = 61)
Choose the correct answer from the options given below:

In the following halogenated organic compounds the one with maximum number of chlorine atoms in its structure is:

Cyclohexylamine when treated with nitrous acid yields (P). On treating (P) with PCC results in (Q). When (Q) is heated with dil. NaOH we get (R). The final product (R) is:
(1)
(2)
(3)
(4)

An organic compound [A] (C$$_4$$H$$_{11}$$N), shows optical activity and gives N$$_2$$ gas on treatment with HNO$$_2$$. The compound [A] reacts with PhSO$$_2$$Cl producing a compound which is soluble in KOH. The structure of A is:

Which of the following compounds are not used as disinfectants?
A. Chloroxylenol
B. Bithional
C. Veronal
D. Prontosil
E. Terpineol
Choose the correct answer from the options given below:

Given below are two statements:
Statement I: Upon heating a borax bead dipped in cupric sulphate in a luminous flame, the colour of the bead becomes green.
Statement II: The green colour observed is due to the formation of copper(I) metaborate.
In the light of the above statements, choose the most appropriate answer from the options given below:

Compound A, C$$_5$$H$$_{10}$$O$$_5$$, given a tetraacetate with AC$$_2$$O and oxidation of A with Br$$_2$$ - H$$_2$$O gives an acid, C$$_5$$H$$_{10}$$O$$_6$$. Reduction of A with HI gives isopentane. The possible structure of A is:

Assume carbon burns according to following equation:
$$2C(s) + O_2(g) \rightarrow 2CO(s)$$
when 12 g carbon is burnt in 48 g of oxygen, the volume of carbon monoxide produced is ______ $$\times 10^{-1}$$ L at STP [nearest integer]
[Given: Assume CO as ideal gas, Mass of C is 12 g mol$$^{-1}$$, mass of O is 16 g mol$$^{-1}$$ and molar volume of an ideal gas at STP is 22.7 L mol$$^{-1}$$]

Amongst the following, the number of species having the linear shape is
XeF$$_2$$, I$$_3^+$$, C$$_3$$O$$_2$$, I$$_3^-$$, CO$$_2$$, SO$$_2$$, BeCl$$_2$$ and BCl$$_2^\ominus$$

Enthalpies of formation of CCl$$_4$$(g), H$$_2$$O(g), CO$$_2$$(g) and HCl are -105, -242, -394 and -92 kJ mol$$^{-1}$$ respectively. The magnitude of enthalpy of the reaction given below is kJmol$$^{-1}$$. (nearest integer)
$$CCl_4(g) + 2H_2O(g) \rightarrow CO_2(g) + 4HCl(g)$$

At 298 K, the solubility of silver chloride in water is $$1.434 \times 10^{-3}$$ g L$$^{-1}$$. The value of $$-\log K_{sp}$$ for silver chloride is
(Given mass of Ag is 107.9 g mol$$^{-1}$$, and mass of Cl is 35.5 g mol$$^{-1}$$)

The number of alkali metal(s), from Li, K, Cs, Rb having ionization enthalpy greater than 400 kJ mol$$^{-1}$$ and forming stable super oxide is

A sample of a metal oxide has formula M$$_{0.83}$$O$$_{1.00}$$. The metal M can exist in two oxidation states +2 and +3. In the sample of M$$_{0.83}$$O$$_{1.00}$$, the percentage of metal ions existing in +2 oxidation state is %. (nearest integer)

The resistivity of a 0.8M solution of an electrolyte is $$5 \times 10^{-3}$$ $$\Omega$$cm. Its molar conductivity is $$ 10^4$$ $$\Omega^{-1}$$ cm$$^2$$ mol$$^{-1}$$. (Nearest integer)

The rate constant for a first order reaction is 20 min$$^{-1}$$. The time required for the initial concentration of the reactant to reduce to its $$\dfrac{1}{32}$$ level is ______ $$\times 10^{-2}$$ min. (Nearest integer)

If the CFSE of [Ti(H$$_2$$O)$$_6$$]$$^{3+}$$ is -96.0 kJ/mol, this complex will absorb maximum at wavelength ______ nm. (nearest integer)
Assume Planck's constant (h) = $$6.4 \times 10^{-34}$$ Js, Speed of light (c) = $$3.0 \times 10^8$$ m/s and Avogadro's constant (N$$_A$$) = $$6 \times 10^{23}$$/mol.

The number of molecules which gives haloform test among the following molecules is:

The equation $$e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0$$, $$x \in R$$ has:

The complex number $$z = \dfrac{i-1}{\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}}$$ is equal to:

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$a_7 = 3$$, the product $$(a_1 a_4)$$ is minimum and the sum of its first $$n$$ terms is zero then $$n! - 4a_{n(n+2)}$$ is equal to

The coefficient of $$x^{-6}$$, in the expansion of $$\left(\dfrac{4x}{5} + \dfrac{5}{2x^2}\right)^9$$, is

If $$^{2n+1}P_{n-1} : ^{2n-1}P_n = 11 : 21$$, then $$n^2 + n + 15$$ is equal to:

The set of all values of $$a^2$$ for which the line $$x + y = 0$$ bisects two distinct chords drawn from a point $$P\left(\dfrac{1+a}{2}, \dfrac{1-a}{2}\right)$$ on the circle $$2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$$, is equal to:

Let H be the hyperbola, whose foci are $$(1 \pm \sqrt{2}, 0)$$ and eccentricity is $$\sqrt{2}$$. Then the length of its latus rectum is:

$$\lim_{x \to \infty} \dfrac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3$$

The number of values of $$r \in \{p, q, \sim p, \sim q\}$$ for which $$((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$$ is a tautology, is:

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $$\alpha(> 0)$$, and the mean and standard deviation of marks of class B of $$n$$ students be respectively 55 and $$30 - \alpha$$. If the mean and variance of the marks of the combined class of $$100 + n$$ students are respectively 50 and 350, then the sum of variances of classes A and B is

Among the relations
$$S = \{(a,b): a, b \in R - \{0\}, 2 + \dfrac{a}{b} > 0\}$$ and $$T = \{(a,b): a, b \in R, a^2 - b^2 \in Z\}$$,

If a point $$P(\alpha, \beta, \gamma)$$ satisfying $$(\alpha \; \beta \; \gamma) \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = (0 \; 0 \; 0)$$ lies on the plane $$2x + 4y + 3z = 5$$, then $$6\alpha + 9\beta + 7\gamma$$ is equal to

Let $$(a, b) \subset (0, 2\pi)$$ be the largest interval for which $$\sin^{-1}(\sin\theta) - \cos^{-1}(\sin\theta) > 0$$, $$\theta \in (0, 2\pi)$$, holds. If $$\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$$ and $$\alpha - \beta = b - a$$, then $$\alpha$$ is equal to:

Let $$f: R - \{2, 6\} \to R$$ be real valued function defined as $$f(x) = \dfrac{x^2+2x+1}{x^2-8x+12}$$. Then range of $$f$$ is

The absolute minimum value, of the function $$f(x) = |x^2 - x + 1| + [x^2 - x + 1]$$, where $$[t]$$ denotes the greatest integer function, in the interval $$[-1, 2]$$, is

Let $$y = y(x)$$ be the solution of the differential equation $$(3y^2 - 5x^2)y\,dx + 2x(x^2 - y^2)\,dy = 0$$ such that $$y(1) = 1$$. Then $$|(y(2))^3 - 12y(2)|$$ is equal to:

Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{c} = 5\hat{i} - 3\hat{j} + 3\hat{k}$$, be three vectors. If $$\vec{r}$$ is a vector such that, $$\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{r} \cdot \vec{a} = 0$$, then $$25|\vec{r}|^2$$ is equal to

Let the plane $$P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0$$ be parallel to the line $$L: \dfrac{x+2}{2} = \dfrac{y-3}{3} = \dfrac{z+4}{5}$$. If the intercept of $$P$$ on the y-axis is 1, then the distance between $$P$$ and $$L$$ is

Let $$P$$ be the plane, passing through the point $$(1, -1, -5)$$ and perpendicular to the line joining the points $$(4, 1, -3)$$ and $$(2, 4, 3)$$. Then the distance of $$P$$ from the point $$(3, -2, 2)$$ is

The foot of perpendicular from the origin $$O$$ to a plane $$P$$ which meets the co-ordinate axes at the point $$A, B, C$$ is $$(2, a, 4)$$, $$a \in N$$. If the volume of the tetrahedron $$OABC$$ is 144 unit$$^3$$, then which of the following points is NOT on $$P$$?

The sum $$1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$$ is ______.

If the constant term in the binomial expansion of $$\left(\dfrac{x^{5/2}}{2} - \dfrac{4}{x^l}\right)^9$$ is -84 and the coefficient of $$x^{-3l}$$ is $$2^\alpha \beta$$ where $$\beta < 0$$ is an odd number, then $$|\alpha l - \beta|$$ is equal to ______.

Let $$S$$ be the set of all $$a \in N$$ such that the area of the triangle formed by the tangent at the point $$P(b, c)$$, $$b, c \in N$$, on the parabola $$y^2 = 2ax$$ and the lines $$x = b$$, $$y = 0$$ is 16 unit$$^2$$, then $$\sum_{a \in S} a$$ is equal to

Let $$A = [a_{ij}]$$, $$a_{ij} \in Z \cap [0, 4]$$, $$1 \le i, j \le 2$$. The number of matrices $$A$$ such that the sum of all entries is a prime number $$p \in (2, 13)$$ is ______.

Let $$A$$ be a $$n \times n$$ matrix such that $$|A| = 2$$. If the determinant of the matrix $$\text{Adj}\left(2 \cdot \text{Adj}(2A^{-1})\right)$$ is $$2^{84}$$, then $$n$$ is equal to ______.

Let $$\alpha > 0$$. If $$\int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$$ then $$\alpha$$ is equal to :

If $$\phi(x) = \dfrac{1}{\sqrt{x}} \int_{\pi/4}^{x} \left(4\sqrt{2}\sin t - 3\phi'(t)\right) dt$$, $$x > 0$$ then $$\phi'\left(\dfrac{\pi}{4}\right)$$ is equal to ______.

Let the area of the region $$\{(x,y): |2x-1| \le y \le |x^2-x|, 0 \le x \le 1\}$$ be $$A$$. Then $$(6A + 11)^2$$ is equal to ______.

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three vectors such that $$|\vec{a}| = \sqrt{31}$$, $$4|\vec{b}| = |\vec{c}| = 2$$ and $$2(\vec{a} \times \vec{b}) = 3(\vec{c} \times \vec{a})$$. If the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\dfrac{2\pi}{3}$$, then $$\left(\dfrac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$$ is equal to ______.

Let $$A$$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $$[0, 60]$$ is less than or equal to $$a$$. If $$P(A) = \dfrac{11}{36}$$, then $$a$$ is equal to ______.