JEE Main 8 April 2023 Shift 1 Question Paper

Question 1

Let $I(x)=\int \frac{(x+1)}{x(1+xe^{x})^{2}}\,dx,\ x>0$. If $\lim_{x	o\infty} I(x)=0$, then $I(1)$ is equal to

Accepted Answer

(D) $\frac{e+2}{e+1}-\log_{e}(e+1)$

Answer

(A) $\frac{e+1}{e+2}-\log_{e}(e+1)$

Answer

(B) $\frac{e+1}{e+2}+\log_{e}(e+1)$

Answer

(C) $\frac{e+2}{e+1}+\log_{e}(e+1)$

Question 2

If the equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k})$ is $ax+by+cz=4$, then $(a-b+c)$ is equal to

Accepted Answer

(C) 22

Answer

(A) 20

Answer

(B) 24

Answer

(D) 18

Question 3

Let $R$ be the focus of the parabola $y^{2}=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10,10)$ be the centroid of the triangle $PQR$. If $c-m=6$, then $(PQ)^{2}$ is

Accepted Answer

(A) 325

Answer

(B) 317

Answer

(C) 296

Answer

(D) 346

Question 4

Let $C(\alpha,\beta)$ be the circumcenter of the triangle formed by the lines $4x+3y=69$, $4y-3x=17$ and $x+7y=61$. Then $(\alpha-\beta)^{2}+\alpha+\beta$ is equal to

Accepted Answer

(B) 17

Answer

(A) 18

Answer

(C) 16

Answer

(D) 15

Question 5

Let $P=\begin{bmatrix}\frac{\sqrt{3}}{2} & \frac{1}{2}\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{bmatrix}$, $A=\begin{bmatrix}1 & 1\ 0 & 1\end{bmatrix}$ and $Q=PAP^{T}$. If $P^{T}Q^{2007}P=\begin{bmatrix}a & b\ c & d\end{bmatrix}$, then $2a+b-3c-4d$ equal to

Accepted Answer

(B) 2005

Answer

(A) 2007

Answer

Answer

(D) 2004

Question 6

Let $\alpha,\beta,\gamma$ be the three roots of the equation $x^{3}+bx+c=0$. If $\beta\gamma=1=-\alpha$, then $b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3}$ is equal to

Accepted Answer

(C) 19

Answer

(A) 21

Answer

(B) $\frac{169}{8}$

Answer

(D) $\frac{155}{8}$

Question 7

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is

Accepted Answer

(A) $126(5!)^{2}$

Answer

(B) $7(360)^{2}$

Answer

(C) 720

Answer

(D) $7(720)^{2}$

Question 8

In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine C is

Accepted Answer

(C) $\frac{5}{14}$

Answer

(A) $\frac{2}{7}$

Answer

(B) $\frac{9}{28}$

Answer

(D) $\frac{3}{7}$

Question 9

The number of arrangements of the letter of the word "INDEPENDENCE" in which all the vowels always occur together is

Accepted Answer

(A) 16800

Answer

(B) 14800

Answer

(C) 18000

Answer

(D) 33600

Question 10

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x},\ x\in[0,\pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7\pi}{12}\right)f''\left(\frac{7\pi}{12}\right)$ is equal to

Accepted Answer

(B) $\frac{2}{9}$

Answer

(A) $\frac{-2}{3}$

Answer

(C) $-\frac{1}{3\sqrt{3}}$

Answer

(D) $\frac{-2}{3\sqrt{3}}$

Question 11

If the points with vectors $\alpha\hat{i}+10\hat{j}+13\hat{k}$, $6\hat{i}+11\hat{j}+11\hat{k}$, $\frac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k}$ are collinear, then $(19\alpha-6\beta)^{2}$ is equal to

Accepted Answer

(A) 36

Answer

(B) 16

Answer

(C) 25

Answer

(D) 49

Question 12

If the coefficients of the three consecutive terms in the expansion of $(1+x)^{n}$ are in the ratio $1:5:20$, then the coefficient of the fourth term is

Accepted Answer

(A) 3654

Answer

(B) 1827

Answer

(C) 5481

Answer

(D) 2436

Question 13

Let $S_{k}=\frac{1+2+\dots+K}{K}$ and $\sum_{j=1}^{n} S_{j}^{2}=\frac{n}{A}(Bn^{2}+Cn+D)$, where $A,B,C,D\in N$ and $A$ has least value. Then

Accepted Answer

(A) $A+B$ is divisible by $D$

Answer

(B) $A+B=5(D-C)$

Answer

(C) $A+C+D$ is not divisible by $B$

Answer

(D) $A+B+C+D$ is divisible by 5

Question 14

Let $A=\begin{bmatrix}2 & 1 & 0\ 1 & 2 & -1\ 0 & -1 & 2\end{bmatrix}$. If $|\,\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\,2A))\,|=(16)^{n}$, then $n$ is equal to

Accepted Answer

(A) 10

Answer

(B) 9

Answer

(C) 12

Answer

(D) 8

Question 15

Negation of $p\Rightarrow q\Rightarrow q\Rightarrow p$ is

Accepted Answer

(C) $q\wedge(\sim p)$

Answer

(A) $(\sim p)\vee q$

Answer

(B) $(\sim q)\wedge p$

Answer

(D) $p\vee(\sim q)$

Question 16

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is

Accepted Answer

(A) $3\sqrt{6}$

Answer

(B) $6\sqrt{3}$

Answer

(C) $6\sqrt{2}$

Answer

(D) $2\sqrt{6}$

Question 17

The area of the region $\{(x,y):x^{2}\le y\le 8-x^{2},\, y\le 7\}$ is

Accepted Answer

(D) 20

Answer

(A) 21

Answer

(B) 18

Answer

(C) 24

Question 18

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A	imes B$ each having at least 3 and at most 6 element is :

Accepted Answer

(A) 792

Answer

(B) 752

Answer

(C) 782

Answer

(D) 772

Question 19

$\lim_{x	o 0}\left(\left(\frac{1-\cos^{2}(3x)}{\cos^{3}(4x)}ight)\left(\frac{\sin^{3}(4x)}{(\log_{e}(2x+1))^{5}}ight)ight)$ is equal to

Accepted Answer

(B) 18

Answer

(A) 9

Answer

(C) 15

Answer

(D) 24

Question 20

If for $z=\alpha+i\beta$, $|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha\beta$ are the roots of the equation

Accepted Answer

(A) $x^{2}+7x+12=0$

Answer

(B) $x^{2}+3x-4=0$

Answer

(C) $x^{2}+2x-3=0$

Answer

(D) $x^{2}+x-12=0$

Question 21

Let $[t]$ denotes the greatest integer $\le t$. Then $\frac{2}{\pi}\int_{\pi/6}^{5\pi/6}\left(8[\operatorname{cosec} x]-5[\cot x]\right)dx$ is equal to

Accepted Answer

14

Question 22

Let $[t]$ denotes the greatest integer $\le t$. If the constant term in the expansion of $\left(3x^{2}-\frac{1}{2x^{5}}\right)^{7}$ is $\alpha$, then $[\alpha]$ is equal to

Accepted Answer

1275

Question 23

Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k}$, $\vec{b}=\alpha\hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}	imes\vec{c}=\vec{a}	imes\vec{b}$. If $\vec{a}\cdot\vec{c}=-12$, $\vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5$, then $\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})$ is equal to

Accepted Answer

11

Question 24

The largest natural number $n$ such that $3^{n}$ divides $66!$ is

Accepted Answer

31

Question 25

If $a_{n}$ is the greatest term in the sequence $a_{n}=\frac{n^{3}}{n^{4}+147}$, $n=1,2,3,\ldots$, then $\alpha$ is equal to

Accepted Answer

5

Question 26

Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x,y)\in A	imes A: x-y 	ext{ is odd positive integer or } x-y=2\}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to

Accepted Answer

19

Question 27

Consider a circle $C_{1}:x^{2}+y^{2}-4x-2y=\alpha-5$. Let its mirror image in the line $y=2x+1$ be another circle $C_{2}:5x^{2}+5y^{2}-10fx-10gy+36=0$. Let $r$ be the radius of $C_{2}$. Then $\alpha+r$ is equal to

Accepted Answer

2

Question 28

If the solution curve of the differential equation $(y-2\log_{e}x)dx+(x\log_{e}x^{2})dy=0$, $x>1$ passes through the points $\left(e,\frac{4}{3}\right)$ and $(e^{4},\alpha)$, then $\alpha$ is equal to

Accepted Answer

3

Question 29

Let $\lambda_{1},\lambda_{2}$ be the values of $\lambda$ for which the points $\left(\frac{5}{2},1,\lambda\right)$ and $(-2,0,1)$ are at equal distance from the plane $2x+3y-6z+7=0$. If $\lambda_{1}>\lambda_{2}$, then the distance of the point $(\lambda_{1}-\lambda_{2},\lambda_{2},\lambda_{1})$ from the line $\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}$ is

Accepted Answer

9

Question 30

Let the mean and variance of 8 numbers $x$, $y$, 10, 12, 6, 12, 4, 8, be 9 and 9.25 respectively. If $x>y$, then $3x-2y$ is equal to

Accepted Answer

25

Question 31

A charge particle moving in magnetic field $B$, has the components of velocity along $B$ as well as perpendicular to $B$. The path of the charge particle will be

Accepted Answer

(C) helical path with the axis along magnetic field $B$

Answer

(A) helical path with the axis perpendicular to the direction of magnetic field $B$

Answer

(B) straight along the direction of magnetic field $B$

Answer

(D) circular path

Question 32

Two projectiles A and B are thrown with initial velocities of $40$ m/s and $60$ m/s at angles $30^{\circ}$ and $60^{\circ}$ with the horizontal respectively. The ratio of their ranges respectively is $(g = 10$ m/s$^{2})$

Accepted Answer

(D) $4 : 9$

Answer

(A) $\sqrt{3} : 2$

Answer

(B) $2 : \sqrt{3}$

Answer

(C) $1 : 1$

Question 33

Certain galvanometers have a fixed core made of non magnetic metallic material. The function of this metallic material is

Accepted Answer

(B) to bring the coil to rest quickly

Answer

(A) to oscillate the coil in magnetic field for longer period of time

Answer

(C) to produce large deflecting torque on the coil

Answer

(D) to make the magnetic field radial

Question 34

A TV transmitting antenna is $98$ m high and the receiving antenna is at the ground level. If the radius of the earth is $6400$ km, the surface area covered by the transmitting antenna is approximately:

Accepted Answer

(B) $3942$ km$^{2}$

Answer

(A) $1240$ km$^{2}$

Answer

(C) $4868$ km$^{2}$

Answer

(D) $1549$ km$^{2}$

Question 35

In a reflecting telescope, a secondary mirror is used to:

Accepted Answer

(D) move the eyepiece outside the telescopic tube

Answer

(A) reduce the problem of mechanical support

Answer

(B) remove spherical aberration

Answer

(C) make chromatic aberration zero

Question 36

Given below are two statements:
Statement I: If heat is added to a system, its temperature must increase.
Statement II: If positive work is done by a system in a thermodynamic process, its volume must increase.
In the light of the above statements, choose the correct answer from the options given below

Accepted Answer

(D) Statement I is false but Statement II is true

Answer

(A) Statement I is true but Statement II is false

Answer

(B) Both Statement I and Statement II are true

Answer

(C) Both Statement I and Statement II are false

Question 37

The weight of a body on the earth is $400$ N. Then weight of the body when taken to a depth half of the radius of the earth will be:

Accepted Answer

(D) $200$ N

Answer

(A) Zero

Answer

(B) $300$ N

Answer

(C) $100$ N

Question 38

An aluminium rod with Young's modulus $Y = 7.0 	imes 10^{10}$ N/m$^{2}$ undergoes elastic strain of $0.04\%$. The energy per unit volume stored in the rod in SI unit is:

Accepted Answer

(A) $5600$

Answer

(B) $8400$

Answer

(C) $2800$

Answer

(D) $11200$

Question 39

At any instant the velocity of a particle of mass $500$ g is $(2t\,\hat{i} + 3t^{2}\,\hat{j})$ ms$^{-1}$. If the force acting on the particle at $t = 1$ s is $(\hat{i} + x\hat{j})$ N. Then the value of $x$ will be:

Accepted Answer

(A) $3$

Answer

(B) $4$

Answer

(C) $6$

Answer

(D) $2$

Question 40

For the logic circuit shown, the output waveform at Y is:

Accepted Answer

(D) (4)

Answer

(A) (1)

Answer

(B) (2)

Answer

(C) (3)

Question 41

For a nucleus $^{A}_{Z}X$ having mass number A and atomic number Z: A. The surface energy per nucleon $(b_{s}) = a_{1}A^{2/3}$. B. The Coulomb contribution to the binding energy $b_{c} = -a_{2}\dfrac{Z(Z-1)}{A^{4/3}}$. C. The volume energy $b_{v} = a_{3}A$. D. Decrease in the binding energy is proportional to surface area. E. While estimating the surface energy, it is assumed that each nucleon interacts with 12 nucleons, $(a_{1}, a_{2}$ and $a_{3}$ are constants). Choose the most appropriate answer from the options given below:

Accepted Answer

(A) C, D only

Answer

(B) B, C, E only

Answer

(C) A, B, C, D only

Answer

(D) B, C only

Question 42

Given below are two statements:
Statement I: If E be the total energy of a satellite moving around the earth, then its potential energy will be $\dfrac{E}{2}$.
Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy E.
In the light of the above statements, choose the most appropriate answer from the options given below

Accepted Answer

(B) Both Statement I and Statement II are incorrect

Answer

(A) Both Statement I and Statement II are correct

Answer

(C) Statement I is incorrect but Statement II is correct

Answer

(D) Statement I is correct but Statement II is incorrect

Question 43

Dimension of $\dfrac{1}{\mu_{0}\,\epsilon_{0}}$ should be equal to

Accepted Answer

(C) $L^{2}/T^{2}$

Answer

(A) $T^{2}/L^{2}$

Answer

(B) $L/T$

Answer

(D) $T/L$

Question 44

In this figure the resistance of the coil of galvanometer G is $2\,\Omega$. The emf of the cell is $4$ V. The ratio of potential difference across $C_{1}$ and $C_{2}$ is:

Accepted Answer

(B) $\dfrac{4}{5}$

Answer

(A) $1$

Answer

(C) $\dfrac{3}{4}$

Answer

(D) $\dfrac{5}{4}$

Question 45

Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius R, with distance r from the centre O is represented by:

Accepted Answer

(A) (1)

Answer

(B) (2)

Answer

(C) (3)

Answer

(D) (4)

Question 46

Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is

Accepted Answer

(D) $\frac{\sqrt{5}A}{2}$

Answer

(A) $\frac{\sqrt{5}A}{4}$

Answer

(B) $\frac{5A}{2}$

Answer

(C) $\frac{\sqrt{5}A^{2}}{2}$

Question 47

The engine of a train moving with speed $10\,	ext{ms}^{-1}$ towards a platform sounds a whistle at frequency $400\,	ext{Hz}$. The frequency heard by a passenger inside the train is (neglect air speed. Speed of sound in air $330\,	ext{ms}^{-1}$)

Accepted Answer

(B) $400\,	ext{Hz}$

Answer

(A) $200\,	ext{Hz}$

Answer

(C) $412\,	ext{Hz}$

Answer

(D) $388\,	ext{Hz}$

Question 48

An air bubble of volume $1\,	ext{cm}^{3}$ rises from the bottom of a lake $40\,	ext{m}$ deep to the surface at a temperature of $12^\circ	ext{C}$. The atmospheric pressure is $1	imes10^{5}\,	ext{Pa}$, the density of water is $1000\,	ext{kg/m}^{3}$ and $g=10\,	ext{m/s}^{2}$. There is no difference of the temperature of water at the depth of $40\,	ext{m}$ and on the surface. The volume of air bubble when it reaches the surface will be

Accepted Answer

(A) $5\,	ext{cm}^{3}$

Answer

(B) $2\,	ext{cm}^{3}$

Answer

(C) $4\,	ext{cm}^{3}$

Answer

(D) $3\,	ext{cm}^{3}$

Question 49

A cylindrical wire of mass $(0.4\pm0.01)\,	ext{g}$ has length $(8\pm0.04)\,	ext{cm}$ and radius $(6\pm0.03)\,	ext{mm}$. The maximum error in its density will be

Accepted Answer

(C) $4\%$

Answer

(A) $1\%$

Answer

(B) $3.5\%$

Answer

(D) $5\%$

Question 50

Proton (P) and electron (e) will have same de-Broglie wavelength when the ratio of their momentum is (assume, $m_{p}=1849\,m_{e}$)

Accepted Answer

(D) $1:1$

Answer

(A) $1:43$

Answer

(B) $43:1$

Answer

(C) $1:1849$

Question 51

An electric dipole of dipole moment is $6.0	imes10^{-6}\,	ext{Cm}$ placed in a uniform electric field of $1.5	imes10^{3}\,	ext{NC}^{-1}$ in such a way that dipole moment is along electric field. The work done in rotating dipole by $180^\circ$ in this field will be ______ $	ext{mJ}$

Accepted Answer

18

Question 52

Two vertical parallel mirrors A and B are separated by $10\,	ext{cm}$. A point object O is placed at a distance of $2\,	ext{cm}$ from mirror A. The distance of the second nearest image behind mirror A from the mirror A is ______ $	ext{cm}$

Accepted Answer

18

Question 53

The momentum of a body is increased by $50\%$. The percentage increase in the kinetic energy of the body is ______ $\%$

Accepted Answer

125

Question 54

The moment of inertia of semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is $\frac{1}{x}MR^{2}$, where $R$ is the radius and $M$ is the mass of semicircular ring. The value of $x$ will be

Accepted Answer

1

Question 55

An organ pipe $40\,	ext{cm}$ long is open at both ends. The speed of sound in air is $360\,	ext{ms}^{-1}$. The frequency of the second harmonic is ______ $	ext{Hz}$

Accepted Answer

900

Question 56

An air bubble of diameter $6\,	ext{mm}$ rises steadily through a solution of density $1750\,	ext{kg/m}^{3}$ at the rate of $0.35\,	ext{cm/s}$. The co-efficient of viscosity of the solution (neglect density of air) is ______ $	ext{Pas}$ (given, $g=10\,	ext{ms}^{-2}$)

Accepted Answer

10

Question 57

An oscillating LC circuit consists of a $75\,	ext{mH}$ inductor and a $1.2\,\mu	ext{F}$ capacitor. If the maximum charge to the capacitor is $2.7\,\mu	ext{C}$. The maximum current in the circuit will be ______ $	ext{mA}$.

Accepted Answer

9

Question 58

The magnetic intensity at the centre of a long current carrying solenoid is found to be $1.6	imes10^{3}\,	ext{Am}^{-1}$. If the number of turns is 8 per cm, then the current flowing through the solenoid is ______ $	ext{A}$.

Accepted Answer

2

Question 59

A current of $2\,	ext{A}$ flows through a wire of cross-sectional area $25.0\,	ext{mm}^{2}$. The number of free electrons in a cubic meter are $2.0	imes10^{28}$. The drift velocity of the electrons is ______ $	imes10^{-6}\,	ext{ms}^{-1}$ (given, charge on electron $=1.6	imes10^{-19}\,	ext{C}$)

Accepted Answer

25

Question 60

A nucleus with mass number 242 and binding energy per nucleon as $7.6\,	ext{MeV}$ breaks into fragment each with mass number 121. If each fragment nucleus has binding energy per nucleon as $8.1\,	ext{MeV}$, the total gain in binding energy is ______ $	ext{MeV}$

Accepted Answer

121

Question 61

$2IO_3^- + xI^- + 12H^+ \rightarrow 6I_2 + 6H_2O$
What is the value of x?

Accepted Answer

(D) 10

Answer

(A) 12

Answer

(B) 2

Answer

(C) 6

Question 62

Which of the following metals can be extracted through alkali leaching technique?

Accepted Answer

(B) Sn

Answer

(A) Cu

Answer

(C) Pb

Answer

(D) Au

Question 63

Match List I with List II. Choose the correct answer from the options given below:

Accepted Answer

(D) A-II, B-IV, C-I, D-III

Answer

(A) A-II, B-III, C-IV, D-I

Answer

(B) A-II, B-IV, C-III, D-I

Answer

(C) A-IV, B-III, C-I, D-II

Question 64

Which of the following represent the Freundlich adsorption isotherms? Choose the correct answer from the options given below:

Accepted Answer

(B) A, B, D only

Answer

(A) B, C, D only

Answer

(C) A, B only

Answer

(D) A, C, D only

List I	List II
A.Saccharin	I.High potency sweetener
B.Aspartame	II.First artificial sweetening agent
C.Alitame	III.Stable at cooking temperature
D.Sucralose	IV.Unstable at cooking temperature

List-I (Reagents used)	List-II (Compound with functional group detected)
A.Alkaline solution of copper sulphate and sodium citrate	I.see figure
B.Neutral $FeCl_3$ solution	II.see figure
C.Alkaline chloroform solution	III.see figure
D.Potassium iodide and sodium hypochlorite	IV.see figure

List-I (Species)	List-II (Maximum allowed concentration in ppm in drinking water)
A. $F^-$	I.< 50 ppm
B. $SO_4^{2-}$	II.< 5 ppm
C. $NO_3^-$	IV.< 500 ppm
D.Zn	III.< 2 ppm

JEE Main 8 April 2023 Shift 1 Question Paper with Answers

Physics — JEE Main 8 April 2023 Shift 1

Chemistry — JEE Main 8 April 2023 Shift 1

Mathematics — JEE Main 8 April 2023 Shift 1

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