JEE Main 2023 · 30 Jan 2023 · Shift 2

Q1.A block of $\sqrt{3}\mathrm{kg}$ is attached to a string whose other end is attached to the wall. An unknown force F is applied so that the string makes an angle of $30^{\circ }$ with the wall. The tension T is :
(Given $\mathrm{g}=10{\mathrm{ms}}^{-2}$)

Q2.A flask contains hydrogen and oxygen in the ratio of $2:1$ by mass at temperature $27^{\circ }\mathrm{C}$. The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is :

Q3.The equivalent resistance between A and B is

Q4.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The nuclear density of nuclides ${}_5^{10}\mathrm{B},{}_3^6\mathrm{Li},{}_{26}^{56}\mathrm{Fe},{}_{10}^{20}\mathrm{Ne}$ and …

Q5.A thin prism ${\mathrm{P}}_1$ with an angle $6^{\circ }$ and made of glass of refractive index 1.54 is combined with another prism ${\mathrm{P}}_2$ made from glass of refractive index 1.72 to produce dispersion without average deviation. The angle of prism ${\mathrm{P}}_2$ …

Q6.The output Y for the inputs A and B of circuit is given by

Truth table of the shown circuit is :

Q7.A vehicle travels 4 km with speed of $3\mathrm{km}/\mathrm{h}$ and another 4 km with speed of $5\mathrm{km}/\mathrm{h}$, then its average speed is :

Q8.As shown in the figure, a point charge Q is placed at the centre of conducting spherical shell of inner radius $a$ and outer radius $b$. The electric field due to charge Q in three different regions I, II and III is given by : (I : $r<a$, II : $a<r<b$, III : $r>b$)

Q9.As shown in the figure, a current of 2A flowing in an equilateral triangle of side $4\sqrt{3}\mathrm{cm}$. The magnetic field at the centroid O of the triangle is :
(Neglect the effect of earth's magnetic field.)

Q10.In the given circuit, rms value of current (${\mathrm{I}}_{\mathrm{rms}}$) through the resistor R is :

Q11.A machine gun of mass 10 kg fires 20 g bullets at the rate of 180 bullets per minute with a speed of $100\mathrm{m}{\mathrm{s}}^{-1}$ each. The recoil velocity of the gun is :

Q12.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Efficiency of a reversible heat engine will be highest at $-273^{\circ }\mathrm{C}$ temperature of cold reservoir.
Reason R : The efficiency of Carnot's engine d …

Q13.For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is 1 kg, the angular frequency is ${\omega }_1$. When the mass block is 2 kg the angular frequency is ${\omega }_2$. The ratio ${\omega }_2/{\omega }_1$ is :

Q14.An electron accelerated through a potential difference $V_1$ has a de-Broglie wavelength of $\lambda$. When the potential is changed to ${\mathrm{V}}_2$, its de-Broglie wavelength increases by $50\%$. The value of $\left(\frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}\right)$ is equal …

Q15.A current carrying rectangular loop PQRS is made of uniform wire. The length $\mathrm{PR}=\mathrm{QS}=5\mathrm{cm}$ and $\mathrm{PQ}=\mathrm{RS}=100\mathrm{cm}$. If ammeter current reading changes from I to 2 I, the ratio of magnetic forces per unit length on the wire PQ due …

Q16.A force is applied to a steel wire 'A', rigidly clamped at one end. As a result elongation in the wire is 0.2 mm. If same force is applied to another steel wire 'B' of double the length and a diameter 2.4 times that of the wire 'A', the elongation in the wire 'B' will be (wires h …

Q17.An object is allowed to fall from a height R above the earth, where R is the radius of earth. Its velocity when it strikes the earth's surface, ignoring air resistance, will be :

Q18.A point source of 100 W emits light with $5\%$ efficiency. At a distance of 5 m from the source, the intensity produced by the electric field component is :

Q19.A faulty thermometer reads $5^{\circ }\mathrm{C}$ in melting ice and $95^{\circ }\mathrm{C}$ in steam. The correct temperature on absolute scale will be ____ K when the faulty thermometer reads $41^{\circ }\mathrm{C}$.

Q20.If the potential difference between B and D is zero, the value of $x$ is $\frac{1}{n}\Omega$. The value of $n$ is ____

Q21.The velocity of a particle executing SHM varies with displacement $(\mathrm{x})$ as $4{\mathrm{v}}^2=50-{\mathrm{x}}^2$. The time period of oscillations is $\frac{x}{7}s$. The value of $x$ is \ldots\ldots.. $($ Take $\pi =\frac{22}{7})$

Q22.In a Young's double slit experiment, the intensities at two points, for the path difference $\frac{\lambda }{4}$ and $\frac{\lambda }{3}$ ( $\lambda$ being the wavelength of light used) are ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$ respectively. If ${\mathrm{I}}_0$ denotes the intensit …

Q23.A radioactive nucleus decays by two different process. The half life of the first process is 5 minutes and that of the second process is 30 s. The effective half-life of the nucleus is calculated to be $\frac{\alpha }{11}$ s. The value of $\alpha$ is ______.

Q24.A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 s is $\frac{1}{3}{\alpha }^2\sqrt{P}$ m. The value of $\alpha$ will be ......

Q25.As shown in figure, a cuboid lies in a region with electric field $E=2x^2\hat{i}-4y\hat{j}+6\hat{k}$ N/C. The magnitude of charge within the cuboid is $n{ϵ}_0$ C. The value of n is ______ (if dimension of cuboid is $1\times 2\times 3$ m³)

Q26.In an ac generator, a rectangular coil of 100 turns each having area $14\times 10^{-2}$ m² is rotated at 360 rev/min about an axis perpendicular to a uniform magnetic field of magnitude 3.0 T. The maximum value of the emf produced will be ______ V. (Take $\pi =\frac{22}{7}$)

Q27.A stone tied to 180 cm long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is $\frac{1936}{x}$ ms⁻². The value of x ______. (Take $\pi =\frac{22}{7}$)

Q28.A uniform disc of mass 0.5 kg and radius r is projected with velocity 18 m/s at t=0 s on a rough horizontal surface. It starts off with a purely sliding motion at t=0 s. After 2 s it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after 2 s wil …

Q1.The most stable carbocation for the following is:

Q2.The correct order of ${\mathrm{pK}}_{\mathrm{a}}$ values for the following compounds is:
(a) (see structure)
(b) (see structure)
(c) (see structure)
(d) (see structure)

Q3.Decreasing order towards ${\mathrm{S}}_{\mathrm{N}}1$ reaction for the following compounds is:
(a) (see structure)
(b) (see structure)
(c) (see structure)
(d) (see structure)

Q4.In the above conversion of compound ( X ) to product (Y), the sequence of reagents to be used will be:

Q5.Maximum number of electrons that can be accommodated in shell with $\mathrm{n}=4$ are:

Q6.The $\mathrm{Cl}-\mathrm{Co}-\mathrm{Cl}$ bond angle values in a fac$[\mathrm{Co}({\mathrm{NH}}_3{)}_3{\mathrm{Cl}}_3]$ complex is/are:

Q7.Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: (see structure) can be easily reduced using $\mathrm{Zn}-\mathrm{Hg}/\mathrm{HCl}$ to (see structure)
Reason R: $\mathrm{Zn}-\mathrm{Hg}/\mathrm{HCl}$ is used to …

Q8.Chlorides of which metal are soluble in organic solvents:

Q9.Given below are two statements : One is labelled as Assertion A and the other labelled as Reason R.
Assertion A: Antihistamines do not affect the secretion of acid in stomach.
Reason R : Antiallergic and antacid drugs work on different receptors.
In the light of the above stateme …

Q10.The wave function $(\Psi )$ of 2 s is given by
${\Psi }_{2\mathrm{s}}=\frac{1}{2\sqrt{2\pi }}{\left(\frac{1}{{\mathrm{a}}_0}\right)}^{1/2}\left(2-\frac{\mathrm{r}}{{\mathrm{a}}_0}\right){\mathrm{e}}^{-\mathrm{r}/2{\mathrm{a}}_0}$
At $r=r_0$, radial node is formed. Thus, …

Q11.${\mathrm{KMnO}}_4$ oxidises ${\mathrm{I}}^{-}$ in acidic and neutral/faintly alkaline solution, respectively to

Q12.Bond dissociation energy of E-H bond of the "${\mathrm{H}}_2\mathrm{E}$" hydrides of group 16 elements (given below), follows order.
(A) O
(B) S
(C) Se
(D) Te

Q13.The water quality of a pond was analysed and its BOD was found to be 4. The pond has

Q14.Boric acid is solid, whereas ${\mathrm{BF}}_3$ is gas at room temperature because of

Q15.Given below are two statements:

Statement I: During Electrolytic refining, the pure metal is made to act as anode and its impure metallic form is used as cathode.
Statement II: During the Hall-Heroult electrolysis process, purified ${\mathrm{Al}}_2{\mathrm{O}}_3$ is mixed wit …

Q16.Formulae for Nessler's reagent is:

Q17.$1\mathrm{L}$, $0.02\mathrm{M}$ solution of $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5{\mathrm{SO}}_4\right]\mathrm{Br}$ is mixed with $1\mathrm{L}$, $0.02\mathrm{M}$ solution of …

Q18.1 mole of ideal gas is allowed to expand reversibly and adiabatically from a temperature of $27^{\circ }\mathrm{C}$. The work done is $3\mathrm{kJ}{\mathrm{mol}}^{-1}$. The final temperature of the gas is ____ K (Nearest integer). Given …

Q19.Iron oxide FeO, crystallises in a cubic lattice with a unit cell edge length of $5.0\mathring{A}$. If density of the FeO in the crystal is $4.0\mathrm{g}{\mathrm{cm}}^{-3}$, then the number of FeO units present per unit cell is ____ (Nearest integer)

Given : Molar mass of Fe a …

Q20.An organic compound undergoes first order decomposition. If the time taken for the 60% decomposition is 540 s, then the time required for $90\%$ decomposition will be is ____ s. (Nearest integer).

Given : $\ln 10=2.3;\log 2=0.3$

Q21.Lead storage battery contains $38\%$ by weight solution of ${\mathrm{H}}_2{\mathrm{SO}}_4$. The van't Hoff factor is 2.67 at this concentration. The temperature in Kelvin at which the solution in the battery will freeze is ____ (Nearest integer).
Given …

Q22.Consider the following equation :
$2{\mathrm{SO}}_2(\mathrm{g})+{\mathrm{O}}_2(\mathrm{g})\rightleftharpoons 2{\mathrm{SO}}_3(\mathrm{g}),\Delta \mathrm{H}=-190\mathrm{kJ}$
The number of factors which will increase the yield of ${\mathrm{SO}}_3$ at equilibrium from t …

Q23.The graph of $\log \frac{\mathrm{X}}{\mathrm{m}}$ vs $\log \mathrm{p}$ for an adsorption process is a straight line inclined at an angle of $45^{\circ }$ with intercept equal to 0.6020. The mass of gas adsorbed per unit mass of adsorbent at the pressure of 0.4 atm is ______ …

Q24.Number of compounds from the following which will not dissolve in cold ${\mathrm{NaHCO}}_3$ and NaOH solutions but will dissolve in hot NaOH solution is ______.

Q25.A short peptide on complete hydrolysis produces 3 moles of glycine (G), two moles of leucine (L) and two moles of valine (V) per mole of peptide. The number of peptide linkages in it are ______.

Q26.The strength of 50 volume solution of hydrogen peroxide is ______ $\mathrm{g}/\mathrm{L}$ (Nearest integer).

Given:
Molar mass of ${\mathrm{H}}_2{\mathrm{O}}_2$ is $34{\mathrm{g}\mathrm{mol}}^{-1}$
Molar volume of gas at STP = $22.7\mathrm{L}$.

Q27.The electrode potential of the following half cell at 298 K.
$\mathrm{X}|{\mathrm{X}}^{2+}(0.001\mathrm{M})||{\mathrm{Y}}^{2+}(0.01\mathrm{M})|\mathrm{Y}$ is ______ $\times 10^{-2}\mathrm{V}$ (Nearest integer).

Given: …

Q1.Consider the following statements:

P: I have fever
Q : I will not take medicine
R : I will take rest

The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

Q2.Let A be a point on the x-axis. Common tangents are drawn from A to the curves ${\mathrm{x}}^2+{\mathrm{y}}^2=8$ and ${\mathrm{y}}^2=16x$. If one of these tangents touches the two curves at Q and R, then $(\mathrm{QR}{)}^2$ is equal to

Q3.Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation ${\mathrm{x}}^2-\mathrm{px}+\frac{5}{4}\mathrm{p}=0$ are rational. Then the area of the region …

Q4.If the functions $f(x)=\frac{x^3}{3}+2bx+\frac{a{x}^2}{2}$ and $g(x)=\frac{x^3}{3}+ax+b{x}^2,a\ne 2b$ have a common extreme point, then $\mathrm{a}+2\mathrm{b}+7$ is equal to

Q5.The range of the function $\mathrm{f}(\mathrm{x})=\sqrt{3-\mathrm{x}}+\sqrt{2+\mathrm{x}}$ is

Q6.The solution of the differential equation
$\frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\frac{{\mathrm{x}}^2+3{\mathrm{y}}^2}{3{\mathrm{x}}^2+{\mathrm{y}}^2}\right),\mathrm{y}(1)=0$ is

Q7.Let $x=(8\sqrt{3}+13{)}^{13}$ and $y=(7\sqrt{2}+9{)}^9$. If [t] denotes the greatest integer $\le \mathrm{t}$, then

Q8.A vector v in the first octant is inclined to the x-axis at $60^{\circ }$, to the y-axis at $45^{\circ }$ and to the z-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(\mathrm{a},\mathrm{b},\mathrm{c})$, is normal to …

Q9.Let $\mathrm{f},\mathrm{g}$ and h be the real valued functions defined on $\mathbb{R}$ as $f(x)=\{\begin{array}{cc}\frac{x}{|x|},& x\ne 0\\ 1,& x=0\end{array}$, …

Q10.The number of ways of selecting two numbers $a$ and $b$, $a\in \{2,4,6,\dots ,100\}$ and $b\in \{1,3,5,\dots ,99\}$ such that 2 is the remainder when $\mathrm{a}+\mathrm{b}$ is divided by 23 is

Q11.If P is a $3\times 3$ real matrix such that ${\mathrm{P}}^{\mathrm{T}}=\mathrm{aP}+(\mathrm{a}-1)\mathrm{I}$, where $\mathrm{a}>1$, then

Q12.Let $\lambda \in \mathbb{R},\vec{a}=\lambda \hat{i}+2\hat{j}-3\hat{k},\vec{b}=\hat{i}-\lambda \hat{j}+2\hat{k}$.

If $((\vec{a}+\vec{b})\times (\vec{a}\times \vec{b}))\times (\vec{a}-\vec{b})=8\hat{i}-40\hat{j}-24\hat{k}$, then …

Q13.Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ be two vectors. Let $|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=4$ and $\vec{a}\cdot \vec{b}=2$. If $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$, then the value of …

Q14.Let ${\mathrm{a}}_1=1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4,\dots$. be consecutive natural numbers. Then …

Q15.The parabolas : $a{x}^2+2bx+cy=0$ and $d{x}^2+2ex+fy=0$ intersect on the line $\mathrm{y}=1$. If $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}$ are positive real numbers and $\mathrm{a},\mathrm{b},\mathrm{c}$ are in G.P., then

Q16.If a plane passes through the points $(-1,\mathrm{k},0),(2,\mathrm{k},-1)$, $(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2y+1}{2}=\frac{\mathrm{z}+1}{-1}$, then the value of $\frac{{\mathrm{k}}^2+1}{(\mathrm{k}-1)(\mathrm{k}-2)}$ is

Q17.Let $\mathrm{a},\mathrm{b},\mathrm{c}>1,{\mathrm{a}}^3,{\mathrm{b}}^3$ and ${\mathrm{c}}^3$ be in A.P., and ${\log }_{\mathrm{a}}\mathrm{b}$, ${\log }_{c}a$ and ${\log }_{b}c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is …

Q18.Let S be the set of all values of ${\mathrm{a}}_1$ for which the mean deviation about the mean of 100 consecutive positive integers ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots \dots ,{\mathrm{a}}_{100}$ is 25 . Then S is

Q19.${\lim }_{n\to \infty }\frac{3}{n}\left\{4+{\left(2+\frac{1}{n}\right)}^2+{\left(2+\frac{2}{n}\right)}^2+\dots +{\left(3-\frac{1}{n}\right)}^2\right\}$ is equal to

Q20.For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations
$\mathbf{x}-y+z=5$
$2\mathrm{x}+2\mathrm{y}+\alpha \mathrm{z}=8$
$3x-y+4z=\beta$
has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

Q21.$50^{\text{th }}$ root of a number x is 12 and $50^{\text{th }}$ root of another number y is 18 . Then the remainder obtained on dividing ( $\mathrm{x}+\mathrm{y}$ ) by 25 is ____ .

Q22.Let $\mathrm{A}=\{1,2,3,5,8,9\}$. Then the number of possible functions $\mathrm{f}:\mathrm{A}\to \mathrm{A}$ such that $\mathrm{f}(\mathrm{m}\cdot \mathrm{n})=\mathrm{f}(\mathrm{m})\cdot \mathrm{f}(\mathrm{n})$ for every …

Q23.Let $\mathrm{P}\left({\mathrm{a}}_1,{\mathrm{b}}_1\right)$ and $\mathrm{Q}\left({\mathrm{a}}_2,{\mathrm{b}}_2\right)$ be two distinct points on a circle with center $\mathrm{C}(\sqrt{2},\sqrt{3})$. Let O be the origin and OC be perpendicular to both CP and CQ . If the …

Q24.The $8^{\text{th }}$ common term of the series
$S_1=3+7+11+15+19+\dots .$,
$S_2=1+6+11+16+21+\dots$.
is ____ .

Q25.Let a line L pass through the point P(2,3,1) and be parallel to the line $x+3y-2z-2=0=x-y+2z$. If the distance of L from the point $(5,3,8)$ is $\alpha$, then $3{\alpha }^2$ is equal to ____.

Q26.If $\int \sqrt{\sec 2x-1}dx=\alpha {\log }_e\left|\cos 2x+\beta +\sqrt{\cos 2x\left(1+\cos 2x\right)}\right|$ + constant, then $\beta -\alpha$ is equal to ____.

Q27.If the value of real number $a>0$ for which $x^2-5ax+1=0$ and $x^2-ax-5=0$ have a common real root is $\frac{3}{\sqrt{2\beta }}$ then $\beta$ is equal to ____.

Q28.The number of seven digits odd numbers, that can be formed using all the seven digits 1,2,2,2,3,3,5 is ____.

Q29.A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the sam …

Q30.Let A be the area of the region

Then 540 A is equal to ____.