JEE Main 2024 · 08 Apr 2024 · Shift 1

Q1.Three bodies A, B and C have equal kinetic energies and their masses are $400\mathrm{g},1.2\mathrm{kg}$ and 1.6 kg respectively. The ratio of their linear momenta is :

Q2.Average force exerted on a non-reflecting surface at normal incidence is $2.4\times 10^{-4}\mathrm{N}$. If $360\mathrm{W}/{\mathrm{cm}}^2$ is the light energy flux during span of 1 hour 30 minutes. Then the area of the surface is:

Q3.A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is:
(Assume $\mathrm{h}=6.63\times 10^{-34}\mathrm{J}\mathrm{s},{\mathrm{m}}_{\mathrm{e}}=9.0\times 10^{-31}\mathrm{kg}$ and ${\mathrm{m}}_{\mathrm{p}}=1836$ tim …

Q4.A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature $\left(27^{\circ }\mathrm{C}\right)$. The ratio of specific heat of gases at constant volume respectively is:

Q5.In an expression a × $10^{\mathrm{b}}$ :

Q6.In the given circuit, the terminal potential difference of the cell is :

Q7.Binding energy of a certain nucleus is $18\times 10^8\mathrm{J}$. How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus:

Q8.Paramagnetic substances:
A. align themselves along the directions of external magnetic field.
B. attract strongly towards external magnetic field.
C. has susceptibility little more than zero.
D. move from a region of strong magnetic field to weak magnetic field.

Choose the most …

Q9.A clock has $75\mathrm{cm},60\mathrm{cm}$ long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel x distance more than the tip of minute hand. The value of x in meter is nearly (Take $\pi =3.14$) :

Q10.Young's modulus is determined by the equation given by $Y=49000\frac{\mathrm{m}}{\ell }\frac{\text{ dyne }}{{\mathrm{cm}}^2}$ where $M$ is the mass and $\ell$ is the extension of wire used in the experiment. Now error in Young modules(Y) is estimated by taking data from …

Q11.Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio $\frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{d}}}$ and the ratio $\frac{V_b}{V_c}$ is:

Q12.Two planets A and B having masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ move around the sun in circular orbits of $r_1$ and $r_2$ radii respectively. If angular momentum of A is L and that of B is $3L$, the ratio of time period $\left(\frac{T_A}{T_B}\right)$ …

Q13.A LCR circuit is at resonance for a capacitor C, inductance L and resistance R. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

Q14.The output Y of following circuit for given inputs is :

Q15.Two charged conducting spheres of radii a and b are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

Q16.Correct Bernoulli's equation is (symbols have their usual meaning) :

Q17.A player caught a cricket ball of mass 150 g moving at a speed of $20\mathrm{m}/\mathrm{s}$. If the catching process is completed in 0.1 s, the magnitude of force exerted by the ball on the hand of the player is:

Q18.A stationary particle breaks into two parts of masses ${\mathrm{m}}_{\mathrm{A}}$ and ${\mathrm{m}}_{\mathrm{B}}$ which move with velocities ${\mathrm{v}}_{\mathrm{A}}$ and ${\mathrm{v}}_{\mathrm{B}}$ respectively. The ratio of their kinetic energies ( …

Q19.Critical angle of incidence for a pair of optical media is $45^{\circ }$. The refractive indices of first and second media are in the ratio:

Q20.The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to 1 mm. The main scale reading is 2 cm and second division of vernier scale coincides with a …

Q21.A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of $x$ and $y$ coordinates of center of mass of plate in $\frac{n}{9}$.The value of n is ____.

Q22.An electron with kinetic energy 5 eV enters a region of uniform magnetic field of $3\mu \mathrm{T}$ perpendicular to its direction. An electric field E is applied perpendicular to the direction of velocity and magnetic field. The value of E, so that electron moves along the sam …

Q23.A square loop PQRS having 10 turns, area $3.6\times 10^{-3}{\mathrm{m}}^2$ and resistance $100\Omega$ is slowly and uniformly being pulled out of a uniform magnetic field of magnitude $\mathrm{B}=0.5\mathrm{T}$ as shown. Work done in pulling the loop out of the field in 1.0 …

Q24.Resistance of a wire at $0^{\circ }\mathrm{C}$, $100^{\circ }\mathrm{C}$ and ${\mathrm{t}}^{\circ }\mathrm{C}$ is found to be $10\Omega$, $10.2\Omega$ and $10.95\Omega$ respectively. The temperature t in Kelvin scale is ____.

Q25.An electric field, $\overrightarrow{\mathrm{E}}=\frac{2\hat{\mathrm{i}}+6\hat{\mathrm{j}}+8\hat{\mathrm{k}}}{\sqrt{6}}$ passes through the surface of $4{\mathrm{m}}^2$ area having unit vector $\hat{n}=\left(\frac{2\hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}\right)$. The electr …

Q26.A liquid column of height 0.04 cm balances excess pressure of soap bubble of certain radius. If density of liquid is $8\times 10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and surface tension of soap solution is $0.28{\mathrm{Nm}}^{-1}$, then diameter of the soap bubble is ____ c …

Q27.A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$ then the value of $a$ is ____.

Q28.Three vectors $\overrightarrow{\mathrm{OP}}$, $\overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{OR}}$ each of magnitude A are acting as shown in figure. The resultant of the three vectors is $\mathrm{A}\sqrt{\mathrm{x}}$. The value of x is ____.

Q29.A parallel beam of monochromatic light of wavelength 600 nm passes through single slit of 0.4 mm width. Angular divergence corresponding to second order minima would be ____ $\times 10^{-3}\mathrm{rad}$.

Q30.In an alpha particle scattering experiment distance of closest approach for the $\alpha$ particle is $4.5\times 10^{-14}\mathrm{m}$. If target nucleus has atomic number 80, then maximum velocity of $\alpha$-particle is ____ $\times 10^5\mathrm{m}/\mathrm{s}$ a …

Q1.Given below are two statements:

Statement I:
IUPAC name of Compound A is 4-chloro-1, 3-dinitrobenzene:

Statement II:
IUPAC name of Compound B is 4-ethyl-2-methylaniline.

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

Q2.Which among the following compounds will undergo fastest ${\mathrm{S}}_{\mathrm{N}}2$ reaction.

Q3.Combustion of glucose $({\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6)$ produces ${\mathrm{CO}}_2$ and water. The amount of oxygen (in g) required for the complete combustion of 900 g of glucose is:
[Molar mass of glucose in $\mathrm{g}{\mathrm{mol}}^{-1}=180$]

Q4.Identify the major products A and B respectively in the following set of reactions.

Q5.Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R:

Assertion A : The stability order of +1 oxidation state of Ga , In and Tl is $\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}$.

Reason R : The inert pair effect stabilizes the lower oxida …

Q6.Match List I and with List II

Q7.For the given hypothetical reactions, the equilibrium constants are as follows:
$\mathrm{X}\rightleftharpoons \mathrm{Y};{\mathrm{K}}_1=1.0$
$\mathrm{Y}\rightleftharpoons \mathrm{Z};{\mathrm{K}}_2=2.0$
$\mathrm{Z}\rightleftharpoons \mathrm{W};{\mathrm{K}}_3=4.0$
The equi …

Q8.Thiosulphate reacts differently with iodine and bromine in the reaction given below :
$2{\mathrm{S}}_2{\mathrm{O}}_3^{2-}+{\mathrm{I}}_2\to {\mathrm{S}}_4{\mathrm{O}}_6^{2-}+2{\mathrm{I}}^{-}$ …

Q9.An octahedral complex with the formula ${\mathrm{CoCl}}_3{\mathrm{nNH}}_3$ upon reaction with excess of ${\mathrm{AgNO}}_3$ solution given 2 moles of AgCl . Consider the oxidation state of Co in the complex is ' x '. The value of " $\mathrm{x}+\mathrm{n}$ " is ______。

Q10.The incorrect statement regarding the given structure is

Q11.In the given compound, the number of $2^{\circ }$ carbon atom/s is ______ .

Q12.Which of the following are aromatic?
A.
B.
C.
D.

Q13.Among the following halogens
${\mathrm{F}}_2,{\mathrm{Cl}}_2,{\mathrm{Br}}_2$ and ${\mathrm{I}}_2$
Which can undergo disproportionation reaction?

Q14.Given below are two statements:

Statement I: $\mathrm{N}{\left({\mathrm{CH}}_3\right)}_3$ and $\mathrm{P}{\left({\mathrm{CH}}_3\right)}_3$ can act as ligands to form transition metal complexes.
Statement II: As N and P are from same group, the nature of bonding of …

Q15.Iron (III) catalyses the reaction between iodide and persulphate ions, in which
A. ${\mathrm{Fe}}^{3+}$ oxidises the iodide ion
B. ${\mathrm{Fe}}^{3+}$ oxidises the persulphate ion
C. ${\mathrm{Fe}}^{2+}$ reduces the iodide ion
D. ${\mathrm{Fe}}^{2+}$ reduces the persulphate ion …

Q16.Number of complexes with even number of electrons in ${\mathrm{t}}_{2\mathrm{g}}$ orbitals is - …

Q17.Identify the product $(\mathrm{P})$ in the following reaction:

i) ${\mathrm{Br}}_2/\mathrm{Red}\mathrm{P}$
ii) ${\mathrm{H}}_2\mathrm{O}$

Q18.A hypothetical electromagnetic wave is show below.

The frequency of the wave is $\mathrm{x}\times 10^{19}\mathrm{Hz}$.
$\mathrm{x}=$ ______ (nearest integer)

Q19.Consider the figure provided.
1 mol of an ideal gas is kept in a cylinder, fitted with a piston, at the position A , at $18^{\circ }\mathrm{C}$. If the piston is moved to position B , keeping the temperature unchanged, then ' x ' L atm work is done in this reversible process. …

Q20.Number of amine compounds from the following giving solids which are soluble in NaOH upon reaction with Hinsberg's reagent is ______.

Q21.The number of optical isomers in following compound is : ______.

Q22.The 'spin only' magnetic moment value of ${\mathrm{MO}}_4{}^{2-}$ is
______ BM. (Where M is a metal having least metallic radii. among $\mathrm{Sc},\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Mn}$ and Zn ).
(Given atomic number : …

Q23.Number of molecules from the following which are exceptions to octet rule is ______.
${\mathrm{CO}}_2,{\mathrm{NO}}_2,{\mathrm{H}}_2{\mathrm{SO}}_4,{\mathrm{BF}}_3,{\mathrm{CH}}_4,{\mathrm{SiF}}_4,{\mathrm{ClO}}_2,{\mathrm{PCl}}_5$, …

Q24.If 279 g of aniline is reacted with one equivalent of benzenediazonium chloride, the maximum amount of aniline yellow formed will be ______ g. (nearest integer)
(consider complete conversion)

Q25.Consider the following reaction
$\mathrm{A}+\mathrm{B}\to \mathrm{C}$
The time taken for A to become $1/4^{\text{th}}$ of its initial concentration is twice the time taken to become 1/2 of the same. Also, when the change of concentration of B is plotted against time, the …

Q26.Major product B of the following reaction has ______ $\pi$-bond.

Q27.A solution containing 10g of an electrolyte ${\mathrm{AB}}_2$ in 100 g of water boils at $100.52^{\circ }\mathrm{C}$. The degree of ionization of the electrolyte ($\alpha$) is ______ $\times 10^{-1}$.
(nearest integer)
[Given : Molar mass of …

Q28.Match List I with List II:
List-I (Compound): A. Fe₄[Fe(CN)₆]₃·xH₂O B. [Fe(CN)₅NOS]⁴⁻ C. [Fe(SCN)]²⁺ D. (NH₄)₃PO₄·12MoO₃
List-II (Colour): I. Violet II. Blood Red III. Prussian Blue IV. Yellow
Choose the correct answer from the options given below:

Q1.The value of $\mathrm{k}\in \mathbb{N}$ for which the integral $I_n={\int }_0^1{\left(1-x^k\right)}^{n}dx,n\in \mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is :

Q2.The sum of all the solutions of the equation $(8{)}^{2\mathrm{x}}-16\cdot (8{)}^{\mathrm{x}}+48=0$ is :

Q3.Let the circles $C_1:(x-\alpha {)}^2+(y-\beta {)}^2=r_1^2$ and ${\mathrm{C}}_2:(\mathrm{x}-8{)}^2+{\left(\mathrm{y}-\frac{15}{2}\right)}^2={\mathrm{r}}_2^2$ touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres …

Q4.Let $\mathrm{P}(\mathrm{x},\mathrm{y},\mathrm{z})$ be a point in the first octant, whose projection in the xy -plane is the point Q . Let $\mathrm{OP}=\gamma$; the angle between OQ and the positive x -axis be $\theta$; and the angle between OP and the positive z -axis be …

Q5.The number of critical points of the function $\mathrm{f}(\mathrm{x})=(\mathrm{x}-2{)}^{2/3}(2\mathrm{x}+1)$ is :

Q6.Let $\mathrm{f}(\mathrm{x})$ be a positive function such that the area bounded by $y=f(x),y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation, whose general solution is $\mathrm{y}={\mathrm{c}}_1\mathrm{f}(\mathrm{x})+{\mathrm{c}}_2$, …

Q7.Let $f(x)=4{\cos }^{3}x+3\sqrt{3}{\cos }^{2}x-10$. The number of points of local maxima of $f$ in interval ( $0,2\pi$ ) is:

Q8.Let $\mathrm{A}=\left[\begin{array}{lll}2& \mathrm{a}& 0\\ 1& 3& 1\\ 0& 5& \mathrm{b}\end{array}\right]$. If ${\mathrm{A}}^3=4{\mathrm{A}}^2-\mathrm{A}-21\mathrm{I}$, where I is the identity matrix of order $3\times 3$, then $2\mathrm{a}+3\mathrm{b}$ is eq …

Q9.If the shortest distance between the lines
${\mathrm{L}}_1:\overrightarrow{\mathrm{r}}=(2+\lambda )\hat{\mathrm{i}}+(1-3\lambda )\hat{\mathrm{j}}+(3+4\lambda )\hat{\mathrm{k}},\lambda \in \mathbb{R}$ …

Q10.Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest positive product, is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $n-m$ equals :

Q11.If $\sin \mathrm{x}=-\frac{3}{5}$, where $\pi <\mathrm{x}<\frac{3\pi }{2}$,
then $80\left({\tan }^{2}x-\cos x\right)$ is equal to :

Q12.Let $\mathrm{I}(\mathrm{x})=\int \frac{6}{{\sin }^2\mathrm{x}(1-\cot \mathrm{x}{)}^2}\mathrm{dx}$. If $\mathrm{I}(0)=3$, then $\mathrm{I}\left(\frac{\pi }{12}\right)$ is equal to :

Q13.The equations of two sides AB and AC of a triangle ABC are $4x+y=14$ and $3x-2y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side BC internally in the ratio $2:1$. The equation of the side BC is :

Q14.Let $[t]$ be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f:A\to \mathbb{Z}$ be the function $f(x)=\left[{\log }_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from A to …

Q15.Let z be a complex number such that $|z+2|=1$ and $\mathrm{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\mathrm{Re}(\overline{z+2})|$ is :

Q16.If the set $R=\{(a,b);a+5b=42,a,b\in \mathbb{N}\}$ has $m$ elements and $\sum _{n=1}^m\left(1-i^{n!}\right)=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is :

Q17.For the function $f(x)=(\cos x)-x+1,x\in \mathbb{R}$, between the following two statements
(S1) $f(x)=0$ for only one value of $x$ is $[0,\pi ]$.
(S2) $f(x)$ is decreasing in $\left[0,\frac{\pi }{2}\right]$ and increasing in $\left[\frac{\pi }{2},\pi \right]$.

Q18.The set of all $\alpha$, for which the vector $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in \mathbb{R}$ is :

Q19.Let $y=y(x)$ be the solution of the differential equation $(1+y^2)e^{\tan x}dx+{\cos }^{2}x(1+e^{2\tan x})dy=0$, $y(0)=1$. Then $y\left(\frac{\pi }{4}\right)$ is equal to :

Q20.Let $H:\frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha ,6)$, $\alpha >0$ lies on H. If $\beta$ is the product of the focal distances of the point …

Q21.Let $A=\left[\begin{array}{cc}2& -1\\ 1& 1\end{array}\right]$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then n is equal to ______.

Q22.If the orthocentre of the triangle formed by the lines $2x+3y-1=0$, $x+2y-1=0$ and $ax+by-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$, then the value of $|a-b|$ is ______.

Q23.Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of X and Y respectively, then $7\bar{X}+4\bar{Y}$ is equal to …

Q24.The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Q25.Let the positive integers be written in the form :

If the ${\mathrm{k}}^{\text{th }}$ row contains exactly k numbers for every natural number k , then the row in which the number 5310 will be, is ______ .

Q26.If the range of $f(\theta )=\frac{{\sin }^4\theta +3{\cos }^2\theta }{{\sin }^4\theta +{\cos }^2\theta },\theta \in \mathbb{R}$ is $[\alpha ,\beta ]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha }{\beta }$, is equal to _____ …

Q27.Let $\alpha ={\sum }_{\mathrm{r}=0}^{\mathrm{n}}{\left(4{\mathrm{r}}^2+2\mathrm{r}+1\right)}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}$
and $\beta =\left({\sum }_{\mathrm{r}=0}^{\mathrm{n}}\frac{{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If …

Q28.Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k},\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\overrightarrow{\mathrm{c}}=17\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that …

Q29.Let the area of the region enclosed by the curve $\mathrm{y}=\min \{\sin \mathrm{x},\cos \mathrm{x}\}$ and the x -axis between $\mathrm{x}=-\pi$ to $\mathrm{x}=\pi$ be A . Then ${\mathrm{A}}^2$ is equal to ______ .

Q30.The value of
${\lim }_{x\to 0}2\left(\frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}\dots \dots \sqrt[10]{\cos 10x}}{x^2}\right)$ is ______ .