JEE Main 2024 · 04 Apr 2024 · Shift 1

Q1.An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

Q2.The electric field in an electromagnetic wave is given by $\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}}40\cos \omega \left(\mathrm{t}-\frac{\mathrm{z}}{\mathrm{c}}\right){\mathrm{NC}}^{-1}$. The magnetic field induction of this wave is (in SI unit):

Q3.Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left({}_0^1\mathrm{n}\right)$ and uranium isotope $\left({}_{92}^{235}\mathrm{U}\right)$ is correct:

Q4.In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object ( u ) and for the position of image ( v ) are $\Delta \mathrm{u}$ and $\Delta \mathrm{v}$, respectively. The error in the measurement of the focal …

Q5.Given below are two statements :

Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $P_1-P_2=\rho \mathrm{g}\left({\mathrm{h}}_2-{\mathrm{h}}_1\right)$
Statement II : In ventury tube shown …

Q6.The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8\Omega$ and $10\Omega$ respectively. After inserting in a hot bath of temperature $400^{\circ }\mathrm{C}$, the resistance of platinum wire is :

Q7.A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is:

Q8.On celcius scale the temperature of body increases by $40^{\circ }\mathrm{C}$. The increase in temperature on Fahrenheit scale is:

Q9.An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D . Focal length of each of the convex lens is :

Q10.Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light ( ${\mathrm{I}}_1<{\mathrm{I}}_2$ ) of same wavelengths :

Q11.A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t . Which of the following curve best describes acceleration of the block with time :

Q12.If a rubber ball falls from a height h and rebounds upto the height of $\mathrm{h}/2$. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

Q13.The equation of stationary wave is :
$y=2a\sin \left(\frac{2\pi nt}{\lambda }\right)\cos \left(\frac{2\pi x}{\lambda }\right)$
Which of the following is NOT correct

Q14.A body travels 102.5 m in ${\mathrm{n}}^{\text{th }}$ second and 115.0 m in $(\mathrm{n}+2{)}^{\text{th }}$ second. The acceleration is :

Q15.To measure the internal resistance of a battery, potentiometer is used. For $\mathrm{R}=10\Omega$, the balance point is observed at $\ell =500\mathrm{cm}$ and for $\mathrm{R}=1\Omega$ the balance point is observed at $\ell =400\mathrm{cm}$. The internal resistance of …

Q16.An infinitely long positively charged straight thread has a linear charge density $\lambda {\mathrm{Cm}}^{-1}$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron …

Q17.The value of net resistance of the network as shown in the given figure is :

Q18.P-T diagram of an ideal gas having three different densities ${\rho }_1,{\rho }_2,{\rho }_3$ (in three different cases) is shown in the figure. Which of the following is correct :

Q19.The co-ordinates of a particle moving in $x-y$ plane are given by :
$x=2+4t,y=3t+8t^2$.
The motion of the particle is :

Q20.In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to :
A. pure inductor.
B. pure capacitor.
C. pure resistor.
D. combination of an inductor and capacitor.

Choose the correct answer from the o …

Q21.An infinite plane sheet of charge having uniform surface charge density $+{\sigma }_{\mathrm{s}}\mathrm{C}/{\mathrm{m}}^2$ is placed on $\mathrm{x}-\mathrm{y}$ plane. Another infinitely long line charge having uniform linear charge density …

Q22.A hydrogen atom changes its state from $\mathrm{n}=3$ to $\mathrm{n}=2$. Due to recoil, the percentage change in the wave length of emitted light is approximately $1\times 10^{-n}$. The value of $n$ is ____ .
[Given Rhc $=13.6\mathrm{eV}$, hc …

Q23.The magnetic field existing in a region is given by $\overrightarrow{\mathrm{B}}=0.2(1+2\mathrm{x})\hat{\mathrm{k}}\mathrm{T}$. A square loop of edge 50 cm carrying 0.5 A current is placed in $\mathrm{x}-\mathrm{y}$ plane with its edges parallel to the $x-y$ axes, as sho …

Q24.A alternating current at any instant is given by $\mathrm{i}=\left[6+\sqrt{56}\sin \left(100\pi \mathrm{t}+\frac{\pi }{3}\right)\right]$ A. The rms value of the current is ____ A.

Q25.Twelve wires each having resistance $2\Omega$ are joined to form a cube. A battery of 6 V emf is joined across point a and c. The voltage difference between e and f is ______ V.

Q26.A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne/cm then the new radius is ______ cm. Take: $\left(\pi =\frac{22}{7}\right)$.

Q27.Two wavelengths ${\lambda }_1$ and ${\lambda }_2$ are used in Young's double slit experiment ${\lambda }_1=450\mathrm{nm}$ and ${\lambda }_2=650\mathrm{nm}$. The minimum order of fringe produced by ${\lambda }_2$ which overlaps with the fringe produced by ${\lambda }_1$ …

Q28.An elastic spring under tension of 3 N has a length a. Its length is b under tension 2 N. For its length ($3a-2b$), the value of tension will be ______ N.

Q29.Two forces ${\overrightarrow{\mathrm{F}}}_1$ and ${\overrightarrow{\mathrm{F}}}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between …

Q30.A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights ${\mathrm{h}}_1$ and ${\mathrm{h}}_2$, respectively, above the initial level. The ratio …

Q1.What pressure (bar) of ${\mathrm{H}}_2$ would be required to make emf of hydrogen electrode zero in pure water at $25^{\circ }\mathrm{C}$?

Q2.The correct sequence of ligands in the order of decreasing field strength is :

Q3.Match List -I with List II:

Q4.What will be the decreasing order of basic strength of the following conjugate bases?
${}^{-}\mathrm{OH},\mathrm{R}\overline{\mathrm{O}},{\mathrm{CH}}_3\mathrm{CO}\overline{\mathrm{O}},\mathrm{C}\overline{\mathrm{l}}$

Q5.In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to :

Q6.Identify (B) and (C) and how are (A) and (C) related?

Q7.One of the commonly used electrode is calomel electrode. Under which of the following categories calomel electrode comes?

Q8.Number of complexes from the following with even number of unpaired "d" electrons is ____. …

Q9.Which one of the following molecules has maximum dipole moment?

Q10.Number of molecules/ions from the following in which the central atom is involved in ${\mathrm{sp}}^3$ hybridization is ____.
${\mathrm{NO}}_3{}^{-},{\mathrm{BCl}}_3,{\mathrm{ClO}}_2{}^{-},{\mathrm{ClO}}_3$

Q11.Which among the following is incorrect statement?

Q12.Given below are two statements:

Statement I: Acidity of $\alpha$-hydrogens of aldehydes and ketones is responsible for Aldol reaction.

Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product. In the light of above statements, choose the mo …

Q13.Which of the following nitrogen containing compound does not give Lassaigne's test?

Q14.Which of the following is the correct structure of L-Glucose ?

Q15.The element which shows only one oxidation state other than its elemental form is :

Q16.Identify the product in the following reaction :

Q17.Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is ______.
B, C, N, S, O, F, P, Al, Si

Q18.The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is :
(Given : Molar Mass Na : 23 and Cl: 35.5 gmol⁻¹)

Q19.Identify the correct set of reagents or reaction conditions 'X' and 'Y' in the following set of transformation.

Q20.The correct order of first ionization enthalpy values of the following elements is :
(A) O
(B) N
(C) Be
(D) F
(E) B

Choose the correct answer from the options given below :

Q21.The enthalpy of formation of ethane (C₂H₆) from ethylene by addition of hydrogen where the bond energies of C-H, C-C, H-H are 414 kJ, 347 kJ, 615 kJ and 435 kJ respectively is - ____ kJ.

Q22.The number of correct reaction(s) among the following is ____.
(A)
(B)
(C)
(D)

Q23.X g of ethylamine is subjected to reaction with NaNO₂/HCl followed by water; evolved dinitrogen gas which occupied 2.24 L volume at STP.

X is ____ × 10⁻¹ g.

Q24.The de-Broglie's wavelength of an electron in the 4th orbit is ____ πa₀. (a₀ = Bohr's radius)

Q25.Only 2 mL of KMnO₄ solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid (2 M) in acidic medium. The molarity of KMnO₄ solution should be ____ M.

Q26.Consider the following reaction
${\mathrm{MnO}}_2+\mathrm{KOH}+{\mathrm{O}}_2\to \mathrm{A}+{\mathrm{H}}_2\mathrm{O}$.
Product 'A' in neutral or acidic medium disproportionate to give products ' B ' and ' C ' along with water. The sum of spin-only magnetic moment value …

Q27.Consider the following transformation involving first order elementary reaction in each step at constant temperature as shown below. …

Q28.2.5 g of a non-volatile, non-electrolyte is dissolved in 100 g of water at $25^{\circ }\mathrm{C}$. The solution showed a boiling point elevation by $2^{\circ }\mathrm{C}$. Assuming the solute concentration in negligible with respect to the solvent concentration, the vapour press …

Q29.The number of different chain isomers for ${\mathrm{C}}_7{\mathrm{H}}_{16}$ is _____.

Q30.Number of molecules/species from the following having one unpaired electron is _____.
${\mathrm{O}}_2,{\mathrm{O}}_2^{-1},\mathrm{NO},{\mathrm{CN}}^{-1},{\mathrm{O}}_2^{2-}$

Q1.Let $f(x)=\{\begin{array}{ll}-2,& -2\le x\le 0\\ x-2,& 0<x\le 2\end{array}$ and $h(x)=f(|x|)+|f(x)|$. Then ${\int }_{-2}^{2}h(x)dx$ is equal to

Q2.Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

Q3.If 2 and 6 are the roots of the equation $a{x}^2+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is :

Q4.The vertices of a triangle are $A(-1,3)$, $B(-2,2)$ and $C(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Q5.Let $f(x)=x^5+2e^{x/4}$ for all $x\in R$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in R$. Then the value of $8g^{\prime }(2)$ is

Q6.If the system of equations $x+(\sqrt{2}\sin \alpha )y+(\sqrt{2}\cos \alpha )z=0$, $x+(\cos \alpha )y+(\sin \alpha )z=0$, $x+(\sin \alpha )y-(\cos \alpha )z=0$ has a non-trivial solution, then $\alpha \in \left(0,\frac{\pi }{2}\right)$ is equal to

Q7.The sum of all rational terms in the expansion of ${\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)}^{15}$ is equal to

Q8.Let the first three terms 2, p and $q\ne 2$, of a G.P. be respectively the $7^{th}$, $8^{th}$ and $13^{th}$ terms of an A.P. If the $5^{th}$ term of the G.P. is the $n^{th}$ term of the A.P., then $n$ is equal to

Q9.Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2x^2-3x+8}{2x^2+3x+8}$ be $\frac{m}{n}$, where $\gcd (m,n)=1$. Then $m+n$ is equal to

Q10.There are 5 points $P_1,P_2,P_3,P_4,P_5$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $P_6,P_7,\dots ,P_{11}$ on the side BC and 7 points $P_{12},P_{13},\dots ,P_{18}$ on the side CA of the triangle. The number o …

Q11.One of the points of intersection of the curves $y=1+3x-2x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2},2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(\ell \sqrt{5}+m)-n{\log }_e(1+\sqrt{5})$, where $\ell ,m,n\in N$. Then …

Q12.A square is inscribed in the circle $x^2+y^2-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_i,y_i)$ are the vertices of the square, then $\sum (x_i^2+y_i^2)$ is equal to

Q13.Let $\alpha \in (0,\infty )$ and $A=\left[\begin{array}{ccc}1& 2& \alpha \\ 1& 0& 1\\ 0& 1& 2\end{array}\right]$. If $\det (\text{adj}(2A-A^T))=2^8$, then $(\det (A){)}^2$ is equal to

Q14.If the domain of the function ${\sin }^{-1}\left(\frac{3x-22}{2x-19}\right)+{\log }_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha ,\beta ]$, then $3\alpha +10\beta$ is equal to

Q15.If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)dy-(2x^2+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi }{4}$, then $y(0)$ is equal to

Q16.Let a unit vector which makes an angle of $60^{\circ }$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45^{\circ }$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}+\left(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is

Q17.Let the point, on the line passing through $P(1,-2,3)$ and $Q(5,-4,7)$, farther from the origin and at a distance of 9 units from the point $P$, be $(\alpha ,\beta ,\gamma )$. Then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to

Q18.Let $f:R\to R$ be a function given by $f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2},& x<0\\ \alpha ,& x=0\\ \frac{\beta \sqrt{1-\cos x}}{x},& x>0\end{array}$ where $\alpha ,\beta \in R$. If f is continuous at $x=0$, then ${\alpha }^2+{\beta }^2$ is equal to

Q19.Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z}{)}^2+|z|=0$, $z\in C$. Then $4({\alpha }^2+{\beta }^2)$ is equal to

Q20.Let $\alpha ,\beta \in R$. Let the mean and the variance of 6 observations $-3,4,7,-6,\alpha ,\beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is

Q21.If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and ${\int }_0^k[x^2]dx=\alpha -\sqrt{\alpha }$, where $[x]$ denotes the greatest integer function, then …

Q22.Let ABC be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\overrightarrow{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of …

Q23.Let $a=1+\frac{{}^2{C}_2}{3!}+\frac{{}^3{C}_2}{4!}+\frac{{}^4{C}_2}{5!}+\dots$, $b=1+\frac{{}^1{C}_0+{}^1{C}_1}{1!}+\frac{{}^2{C}_0+{}^2{C}_1+{}^2{C}_2}{2!}+\frac{{}^3{C}_0+{}^3{C}_1+{}^3{C}_2+{}^3{C}_3}{3!}+\dots$. Then $\frac{2b}{a^2}$ is equal …

Q24.If ${\int }_0^{\frac{\pi }{4}}\frac{{\sin }^{2}x}{1+\sin x\cos x}dx=\frac{1}{a}{\log }_e\left(\frac{a}{3}\right)+\frac{\pi }{b\sqrt{3}}$, where $a,b\in N$, then $a+b$ is equal to ____.

Q25.Let A be a $3\times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=3\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$. Then the maximum value of $\det (A)$ is ____.

Q26.Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying $A^2+xA+yI=O$ lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is $e$ and the length of the latus rec …

Q27.Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi )=1$. Then $y\left(\frac{\pi }{2}\right)+10$ is equal to ____.

Q28.If ${\lim }_{x\to 1}\frac{(5x+1{)}^{1/3}-(x+5{)}^{1/3}}{(2x+3{)}^{1/2}-(x+4{)}^{1/2}}=\frac{m\sqrt{5}}{n(2n{)}^{2/3}}$, where $\gcd (m,n)=1$, then $8m+12n$ is equal to ____.

Q29.Let the length of the focal chord PQ of the parabola $y^2=12x$ be 15 units. If the distance of PQ from the origin is $p$, then $10p^2$ is equal to ____.