JEE Main 2023 01 Feb Shift 2

79 Questions • 180 Minutes • 2023

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For the system of linear equations $ax+y+z=1, x+ay+z=1, x+y+az=\beta$, which one of the following statements is NOT correct?

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Let $S=\left\{x \in R: 0<x<1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right)=\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$. If $n(S)$ denotes the number of elements in $S$ then :

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Let $f: R-\{0,1\} \rightarrow R$ be a function such that $f(x)+f\left(\frac{1}{1-x}\right)=1+x$. Then $f(2)$ is equal to :

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If $y(x)=x^x, x>0$, then $y''(2)-2y'(2)$ is equal to :

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The sum of the abosolute maximum and minimum values of the function $f(x)=\left|x^2-5x+6\right|-3x+2$ in the interval $[-1,3]$ is equal to :

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The value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x+\frac{\pi}{4}}{2-\cos 2x}dx$ is :

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The area of the region given by $\{(x,y): xy \leq 8, 1 \leq y \leq x^2\}$ is :

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Let $\alpha x=\exp\left(x^{\beta}y^{\gamma}\right)$ be the solution of the differential equation $2x^2ydy-(1-xy^2)dx=0$, $x>0, y(2)=\sqrt{\log_e 2}$. Then $\alpha+\beta-\gamma$ equals :

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Let $\overrightarrow{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\overrightarrow{b}=\hat{i}+3\hat{j}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

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Let $\overrightarrow{\mathrm{a}}=2\hat{\mathrm{i}}-7\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ , $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\widehat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\widehat{\mathrm{k}}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that $\overrightarrow{\mathrm{r}}\times\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}}\times\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{r}}\cdot\overrightarrow{\mathrm{b}}=0$, then $\left|\overrightarrow{\mathrm{r}}\right|$ is equal to:

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Let the plane $P$ pass through the intersection of the planes $2x+3y-z=2$ and $x+2y+3z=6$, and be perpendicular to the plane $2x+y-z+1=0$. If $d$ is the distance of $P$ from the point $(-7,1,1)$, then $d^2$ is equal to :

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Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{\mathrm{st}}$ die is less than the number appeared on the $2^{\mathrm{nd}}$ die, $B$ be the event that the number appeared on the $1^{\mathrm{st}}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{\mathrm{st}}$ die is odd and that on the $2^{\mathrm{nd}}$ is even. Then

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For a train engine moving with speed of $20\ \mathrm{ms^{-1}}$, the driver must apply brakes at a distance of $500\ \mathrm{m}$ before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed $\sqrt{x}\ \mathrm{ms^{-1}}$. The value of $x$ is ______. (Assuming same retardation is produced by brakes)

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A block is fastened to a horizontal spring. The block is pulled to a distance $x = 10\ \mathrm{cm}$ from its equilibrium position (at $x=0$) on a frictionless surface from rest. The energy of the block at $x=5\ \mathrm{cm}$ is $0.25\ \mathrm{J}$. The spring constant of the spring is ______ $\mathrm{N\ m^{-1}}$.

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A force $F=\left(5+3y^2\right)$ acts on a particle in the $y$-direction, where $F$ is newton and $y$ is in meter. The work done by the force during a displacement from $y=2\ \mathrm{m}$ to $y=5\ \mathrm{m}$ is ______ J.

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Moment of inertia of a disc of mass $M$ and radius '$R$' about any of its diameter is $\frac{MR^2}{4}$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac{x}{2}MR^2$. The value of $x$ is ______.

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The surface of water in a water tank of cross section area $750\ \mathrm{cm^2}$ on the top of a house is $h\ \mathrm{m}$. above the tap level. The speed of water coming out through the tap of cross section area $500\ \mathrm{mm^2}$ is $30\ \mathrm{cm\ s^{-1}}$. At that instant, $\frac{dh}{dt}$ is $x\times10^{-3}\ \mathrm{m\ s^{-1}}$. The value of $x$ will be ______.

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A cubical volume is bounded by the surfaces $x=0$, $x=a$, $y=0$, $y=a$, $z=0$, $z=a$. The electric field in the region is given by $\overrightarrow{E}=E_0x\hat{i}$. Where $E_0=4\times10^4\ \mathrm{NC^{-1}\ m^{-1}}$. If $a=2\ \mathrm{cm}$, the charge contained in the cubical volume is $Q\times10^{-14}\ \mathrm{C}$. The value of $Q$ is ______. (Take $\epsilon_0=9\times10^{-12}\ \mathrm{C^2\ N^{-1}m^{-2}}$)

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A square shaped coil of area $70\ \mathrm{cm^2}$ having $600$ turns rotates in a magnetic field of $0.4\ \mathrm{Wb\ m^{-2}}$, about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes $500$ revolution in a minute, the instantaneous emf when the plane of the coil is inclined at $60^{\circ}$ with the field, will be ______ $\mathrm{V}$. (Take $\pi=\frac{22}{7}$)

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Nucleus a having $Z=17$ and equal number of protons and neutrons has $1.2\ \mathrm{MeV}$ binding energy per nucleon. Another nucleus $B$ of $Z=12$ has total $26$ nucleons and $1.8\ \mathrm{MeV}$ binding energy per nucleons. The difference of binding energy of $B$ and $A$ will be ______ MeV.

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Paper Info

Exam: JEE
Year: 2023
Questions: 79
Duration: 180 mins

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