JEE Main 2023 Question Paper PDF: Download PDF with Solutions for 10 April Shift 1

If I(x) = \(\int e^{sin^2x}\)\((cos \;x \;sin \;2x - sin \;x)dx\) and I (0) = 1, then \(I(π/3)\) is equal to

• (1) \(-e^{\frac{3}{4}}\)
• (2) \(e^{\frac{3}{4}}\)
• (3) \(-\frac{1}{2}e^{\frac{3}{4}}\)
• (4) \(\frac{1}{2}e^{\frac{3}{4}}\)

The shortest distance between the lines \(\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}\) and \(\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}\) is

• (1) 6
• (2) 7
• (3) 8
• (4) 9

Let the ellipse \(E : x^2 + 9y^2 = 9\) intersect the positive x- and y-axes at the points A and B respectively Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle which vertices A, P and the origin O is m/n , where m and n are coprime, then m – n is equal to 

• (1) 15
• (2) 16
• (3) 17
• (4) 18

Let O be the origin and the position vector of the point P be \(\hat i-2\hat j+3\hat k\). If the position vectors of the points A, B and C are \(-2\hat i+\hat j-3\hat k,2\hat i+4\hat j-2\hat k\) and \(-4\hat i+2\hat j-\hat k\) respectively then the projection of the vector \(\overrightarrow{OP}\) on a vector perpendicular to the vectors \(\overrightarrow{ AB}\) and \(\overrightarrow {AC}\) is

• (1) \(\frac{7}{3}\)
• (2) \(\frac{8}{3}\)
• (3) 3
• (4) \(\frac{10}{3}\)

if f(x)= \(\frac{(tan1\degree)x+\log_e(123)}{x\log_e(1234)-(tan1\degree)},x>0\) , then the least value of \(f(f(x))+f(f(\frac{4}{x}))\)is

• (1) 0
• (2) 2
• (3) 4
• (4) 8

If the coefficient of x⁷ in expansion of \((ax- \frac{1}{bx^2})^{13}\) is equal to the coefficient of x^-5 in expansion of \((ax + \frac{1}{bx^2})13\), then a⁴b⁴ is ______

• (1) 22
• (2) 44
• (3) 11
• (4) 33

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If OP = μ, OR=v and OQ=αv+βv, then α, β² are the roots of the equation

• (1) \(3x^2-2x-1=0\)
• (2) \(3x^2+2x-1=0\)
• (3) \(x^2-x-2=0\)
• (4) \(x^2+x-2=0\)

Two dice are rolled and sum of numbers of two dice is N then probability that 2N < N! is m/n , where m and n are co-prime, then 11m – 3n is?

From a square of side 30 cm the squares of side x cm is cut off to make a cuboid of maximum volume. The surface area of cuboid with open top is?

• (1) 400 cm²
• (2) 464 cm²
• (3) 800 cm²
• (4) 900 cm²

\(96\cos\frac{π}{33}\cos \frac{2π}{33}\cos\frac{4π}{33}\cos\frac{8π}{33}\cos\frac{16π}{33}\) is equal to

• (1) 1
• (2) 2
• (3) 3
• (4) 4

For the system of linear equations
\(2x-y+3z=5\)
\(3x+2y-z=7\)
\(4x+5y+\alpha z=\beta\),
which of the following is NOT correct?

• (1) The system has infinitely many solutions for \(\alpha=-6\) and \(\beta=9\)
• (2) The system has a unique solution for \(\alpha\neq-5\) and \(\beta=8\)
• (3) The system is inconsistent for \(\alpha=-5\) and \(\beta=8\)
• (4) The system has infinitely many solutions for \(\alpha=-5\) and \(\beta=9\)

A line segment AB of length λ moves such that the points A and B remain on the periphery of a circle of radius λ. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius

• (1) \(\frac{\sqrt{19}}{7}\lambda\)
• (2) \(\frac{\sqrt{19}}{5}\lambda\)
• (3) \(\frac{{2}}{3}\lambda\)
• (4) \(\frac{{3}}{5}\lambda\)

Let f be a differentiable function such that x² f(x) - x = \(4 \int^x_0tf(t)dt,f(1)=\frac{2}{3}. \;\text{Then 18f(3)is equal to}\)
 

• (1) 150
• (2) 160
• (3) 180
• (4) 210

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033, then the sum of these three terms is equal to

• (1) 231
• (2) 220
• (3) 210
• (4) 241

Let the complex number \( z = x + iy \) be such that \(\frac{2z - 3i}{2z + i}\) is purely imaginary. If \( x + y^2 = 0 \), then \( y^4 + y^2 - y \) is equal to

• (1) \(\frac{2}{3}\)
• (2) \(\frac{3}{2}\)
• (3) \(\frac{3}{4}\)
• (4) \(\frac{4}{3}\)

Let two vertices of triangle ABC be (2, 4, 6) and (0, –2, –5), and its centroid be (2, 1, –1). If the image of third vertex in the plane x + 2y + 4z = 11 is (α, β, γ), then αβ + βγ + γα is equal to

• (1) 70
• (2) 72
• (3) 74
• (4) 76

The negation of the statement
(p v q) ∧ (q v(∼r)) is

• (1) (p v r) ∧ (∼q)
• (2) ((∼p v r)) ∧ (∼q)
• (3) ((∼p ) v (∼q) v(∼r)
• (4) ((∼p ) v (∼q) ∧(∼r)

Let P be the point of intersection of the line \(\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}\) and the plane x + y + z = 2. If the distance of the point P from the plane 3x – 4y + 12z = 32 is q, then q and 2q are the roots of the equation 

• (1) \(x^2+18x+72=0\)
• (2) \(x^2-18x-72=0\)
• (3) \(x^2+18x-72=0\)
• (4) \(x^2-18x+72=0\)

The slope of tangent at any point (x, y) on a curve y = y(x) is \(\frac{x^2+y^2}{2xy},x>0\) If y(2) = 0, then a value of y(8) is
 

• (1) 4√3
• (2) -4√2
• (3) 2√3
• (4) -2√3

If the order of matrix A is \(3 \times 3\) and \(|A|=2\), then the value of \(|3adj (|3A|A2)|\) is?

• (1) \(3^{10}.2^{21}\)
• (2) \(2^{10}.3^{21}\)
• (3) \(2^{12}.3^{15}\)
• (4) \(3^{12}.2^{15}\)

Let f : (-2, 2) → IR be defined by \(f(x) =   \begin{cases}     x[x]       & \quad -2<x<0\\     (x-1)[x],  & \quad 0\leq x<2   \end{cases}\)
Where [x] denotes the greatest integer function. If m and n respectively are the number of points in (-2, 2) at which y = |f(x)| is not continuous and not differentiable, then m + n is equal to ______.

Let y = p(x) be the parabola passing through the points (–1, 0), (0, 1) and (1, 0). If the area of the region \(\{x,y):(x+1)^2+(y-1)^2≤1,y≤p(x)\}\) is A, then 12(π-4A) is equal to__.

The number of elements in the set \(\{n ∈ Z : n^2-10n+19| < 6\}\) is __.

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ___________.

The coefficient of \(x^7\) in \((1-x+2x^3)^{10}\) is ___________.

If the number of ways in which a mixed double badminton can be played such that no couples played into a same game is 840. Then find the number of players?

Let a, b, c be three distinct positive real numbers such that (2a)^log_ea = (bc)^log_eb and b^log_e2 = a^log_ec. Then 6a + 5bc is equal to _____.

Let a common tangent to the curves y² = 4x and (x – 4)² + y² = 16 touch the curves at the points P and Q. Then (PQ)² is equal to ________.

3, 8, 13, ......,373 are in arithmetic series. The sum of numbers not divisible by three is

• (1) 9310
• (2) 8340
• (3) 9525
• (4) 7325

Given below are two statement :  
Statement I : If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double.  
Statement II : Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio.
In the light of the above statement, choose the correct answer from the options given below : 

• (1) Statement I is false but Statement II is true
• (2) Both Statement I and Statement II are true
• (3) Both Statement I and Statement II are false
• (4) Statement I is true but Statement II is false

A physical quantity P is given as
\( P = \frac{a^2b^2 }{ c\sqrt{d}} .\)
The percentage error in the measurement of a,b, c and d are1%, 2%, 3% and 4% respectively. The percentage error in the measurement of quantity P  will be

• (1) 11%
• (2) 12%
• (3) 9%
• (4) 13%

Given below are two statements:  
Statement I : Pressure in a reservoir of water is same at all points at the same level of water.  
Statement II : The pressure applied to enclosed water is transmitted in all directions equally. 
In the light of the above statements, choose the correct answer from the options given below : 

• (1) Statement I is false but Statement II is true
• (2) Both Statement I and Statement II are true
• (3) Statement I is true but Statement II is false
• (4) Both Statement I and Statement II are false

The energy of an electromagnetic wave contained in a small volume oscillates with

• (1) zero frequency
• (2) half the frequency of the wave
• (3) double the frequency of the wave
• (4) the frequency of the wave

Find the equivalent capacitance across points A and B in the given electric circuit.

• (1) \(\frac{C}{2}\)
• (2) 2C
• (3) \(\frac{5C}{3}\)
• (4) \(\frac{3C}{4}\)

Consider two containers A and B containing monoatomic gases at the same Pressure (P), Volume (V) and Temperature (T). The gas in A is compressed isothermally to \(\frac{1}{8}\) of its original volume while the gas in B is com pressed adiabatically to \(\frac{1}{8}\) of its original volume. The ratio of final pressure of gas in B to that of gas in A is

• (1) \(\frac{1}{8}\)
• (2) 8 
• (3) 4
• (4) \(8^{\frac{2}{3}}\)

The range of the projectile projected at an angle of 15° with hor izontal is 50 m. If the projectile is projected with same velocity at an angle of 45° with horizontal, then its range will be

• (1) 50 m
• (2) 100 m
• (3) 80 m
• (4) 120 m

For a particle performing linear SHM, its position (x) as a function of time (t) is given by x = Asin(ωt + δ). Given that, at t = 0, particle is at +A/2 and is moving towards x = +A. Find δ

• (1) π/3 rad
• (2) π/6 rad
• (3) π/4 rad
• (4) 5π/6 rad

The de Broglie wavelength of a molecule in a gas at room tem perature (300 K) is λ₁. If the temperature of the gas is increased to 600 K, then the de Broglie wavelength of the same gas molecule becomes

• (1) \(\frac{\lambda}{2}\)
• (2) \({2\lambda}\)
• (3) \({\lambda}\)
• (4) \(\frac{5\lambda}{2}\)

Find the equivalent resistance across A and B for given circuit.

• (1) 6.4 Ω
• (2) 4Ω
• (3) 3.2Ω
• (4) 8Ω

An object is placed Infront of a plane mirror 12 cm away from it. The object is kept fixed while the plane mirror is shifted towards the object by a distance of 4 cm. The length of shift in the position of image is equal to ___________ cm.

Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r & 3r respectively. The ratio of orbital speeds of the satellites respectively is

• (1) √3 :1
• (2) 1:√3
• (3) √2 :1
• (4) 1:2

The position-time graphs for two students A and B returning from school to their homes are shown in the figure.

A. A lives closer to the school
B. B lives closer to the school
C. A takes lesser time to reach home
D. A travels faster than B
E. B travels faster than A
Choose the correct answer from the options given below :

• (1) (A), (C) and (E) only
• (2) (A) and (E) only
• (3) (B) and (E) only
• (4) (A), (C) and (D) only

The angular momentum for the electron in Bohr’s orbit is L. If the electron is assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be

• (1) L
• (2) 1L
• (3) 2L
• (4) 1.5L

Given below are two statements: one is labeled as Statement (1) and the other is labeled as Statement (2).
Statement (1): Maximum power is dissipated in a circuit containing an inductor, a ca pacitor and a resistor connected in series with an AC source, when resonance occurs
Statement (2): Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage.
In the light of the above statements, choose the correct option:

• (1) (1) and (2) both are correct
• (2) (1) is correct but (2) is incorrect
• (3) (1) is incorrect but (2) is correct
• (4) Both (1) and (2) are incorrect

• (1) (A)-(I), (B)-(IV), (C)-(III), (D)-(II)
• (2) (A)-(I), (B)-(III), (C)-(IV), (D)-(II)
• (3) (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
• (4) (A)-(IV), (B)-(II), (C)-(I), (D)-(III)

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth \(d=\frac{R}{2}\) from the surface of earth, if its weight on the surface of earth is 200 N, will be 

• (1) 100 N
• (2) 300 N
• (3) 50 N
• (4) 150 N

A zener diode of power rating 1.6 W is used as a voltage regula tor. If the zener diode has a breakdown voltage of 8 V and it has to regulate voltage fluctuating between 3 V and 10 V, what is the value of resistance Rs for safe operation of the diode?

• (1) 10 Ω
• (2) 12 Ω
• (3) 13 Ω
• (4) 13.3 Ω

A particle of mass m moving with a velocity v collides with a particle of mass 2m at rest and sticks to it. Velocity of the combined mass equal to? 

• (1) v
• (2) \(\frac{v}{2}\)
• (3) \(\frac{v}{3}\)
• (4) \(\frac{v}{4}\)

A carrier wave of amplitude 15 V modulated by a sinusoidal base band signal of amplitude 3 V. The ratio of maximum amplitude to minimum amplitude in an amplitude modulated wave is

• (1) \(\frac{3}{2}\)
• (2) 5
• (3) 1
• (4) 2

A transverse harmonic wave on a string is given by y(x,t) = 5 sin(6t + 0.003x) where x and y are in cm and t in sec. The wave velocity is _______ ms-1

10 resistors each of resistance 10Ω can be connected in such as to get maximum and minimum equivalent resistance. The ratio of maximum and minimum equivalent resistance will be_______ .

A 1 m long metal rod XY completes the circuit as shown in f igure. The plane of the circuit is perpendicular to the magnetic field of flux density 0.15 T. If the resistance of the circuit is 5ω, the force needed to move the rod in direction, as indicated, with a constant speed of 4 m/s will be ____10⁻³ N

A closed circular tube of average radius 15 cm, whose inner walls are rough, is kept in vertical plane. A block of mass 1 kg just fit inside the tube. The speed of block is 22 m/s, when it is introduced at the top of tube. After completing five oscillations, the block stops at the bottom region of tube. The work done by the tube on the block is J. (Given: g = 10 ms⁻²)

Unpolarised light of intensity 32Wm^-2 passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is 3Wm^-2 , then the angle between pass axis of first two polaroids is ______º .

Three concentric shells A, B and C having surface charge density σ, –σ and σ respectively. The radii of A and B are 2 cm and 3 cm respectively. Electric potential at surface A is V_A and at C is V_C. If V_A = V_C then find the radius of C in cm.

If the earth suddenly shrinks to \(\frac{1}{ 64}\) th of its original volume with its mass remaining the same, the period of rotation of earth becomes \(\frac{24}{ x}\) h. The value of x is .

The current required to be passed through a solenoid of 15 cm length and 60 turns in order to demagnetize a bar magnet of magnetic intensity \(2.4 × 10^3Am^{−1}\) is A.

Two wires each of radius 0.2 cm and negligible mass, one made of steel and other made of brass are loaded as shown in the figure. The elongation of the steel wire is _____ x 10-6 m .

[Young’s modulus for steel = 2 x 1011 Nm-2 and g = 10ms-2]

The decay constant for a radioactive nuclide is \(1.5 × 10^{−5}s^{−1}\). Atomic weight of the substance is 60 g mole⁻¹. (\(N_A = 6×10^{23}\)). The activity of 1.0 µg of the substance is _____\(×10^{10}\) Bq.

The compound which does not exist.

• (1) BeCl₂
• (2) NaO₂
• (3) PbEt₄
• (4) (NH₄)₂ BeF₄

Number of molecules and moles in 2.8375 litre of O₂ at STP.

One which does not stabilize secondary and tertiary protein?

• (1) H-H linkage
• (2) S-S linkage 
   
• (3) Vanderwoal’s force 
   
• (4) Hydrogen bonding 
   

The major product ‘P’ formed in the given reaction is

• (1)
• (2)
• (3)
• (4)

Prolonged heating of ferrous ammonium sulphate is avoided to prevent :

• (1) Oxidation
• (2) Reduction
• (3) Hydrolysis
• (4) Breaking

Which of the following statements are correct?
(A) The M³⁺/M²⁺ reduction potential for iron is greater than manganese 
(B) The higher oxidation states of first row dblock elements get stabilized by oxide ion. 
(C) Aqueous solution of Cr²⁺ can liberate hydrogen from dilute acid. 
(D) Magnetic moment of V²⁺ is observed between 4.4-5.2 BM Choose the correct answer from the options given below: 

• (1) (A), (B) only
• (2) (B), (C) only
• (3) (C), (D) only
• (4) (A), (B), (D) only

The octahedral diamagnetic low spin complex among the following is

• (1) [CO(NH₃)₆]³⁺
• (2) [COF₆]^3-
• (3) [COCl₆]^3- 
• (4) [Fe(H₂O)₆]³⁺

Using column chromatography, mixture of two compounds ‘A’ and ‘B’ was separated. ‘A’ eluted first, this indicates ‘B’ has

• (1) high Rf , weaker adsorption
• (2) high Rf , stronger adsorption
• (3) low Rf , weaker adsorption
• (4) low Rf , stronger adsorption

Select the correct option:
\(2CO (g) + O_2 (g) → 2CO_2 (g) ; ΔH = -x KJ/mol\)
\(C (graphite) + O_2(g) → CO_2 (g) ; AH = -y KJ/mol\)
Then AH for, \(C(graphite) + \frac{1}{2} O_2 (g) → CO (g):\)

• (1) \(x-\frac{y}{2}\)
• (2) \(\frac{x-2y}{2}\)
• (3) \(x+\frac{2y}{2}\)
• (4) \(\frac{x-y}{2}\)

Isomeric amines with molecular formula C₈ H₁₁N give the following tests
Isomer (P) ⇒ Can be prepared by Gabriel phthalimide synthesis
Isomer (Q) ⇒ Reacts with Hinsberg’s reagent to give solid insoluble in NaOH
Isomer (R) ⇒ Reacts with HONO followed by β-naphthol in NaOH to give red dye.
Isomers (P), (Q) and (R) respectively are

• (1)
• (2)
• (3)
• (4)

Read the following two statements:
Statement I: Potassium dichromate is used in volumetric analysis.
Statement II: \(K_2Cr_2O_7\) is more soluble in water than \(Na_2Cr_2O_7\)

• (1) Both statements I and II are correct.
• (2) Both statements I and II are incorrect.
• (3) Statement I is correct and II is incorrect
• (4) Statement I is incorrect and II is correct.

Suitable reaction condition for preparation of Methyl phenyl ether is

• (1) PhO- Na+ , MeBr
• (2) PhO- Na+ , MeOH
• (3) Ph - Br, MeO- Na+
• (4) Benzene, MeBr

Identify the correct order of reactivity for the following pairs towards the respective mechanism




Choose the correct answer from the options given below :

• (1) (B), (C) and (D) only
• (2) (A), (B) and (D) only
• (3) (A), (C) and (D) only
• (4) (A), (B), (C) and (D)

The pair of compounds from the following pairs having both the compounds with net non-zero dipole moment is

• (1) Cis-dichloroethene; trans-dichloroethene
• (2) \(CH_2Cl_2 ; CHCl_3\)
• (3) 1,4-dichlorobenzene;1,3,5-trichlorobenzene
• (4) Benzene; p-Anisidine

The enthalpy change for the adsorption process and micelle for mation respectively are

• (1) \(∆H_{ads} < 0\ and\ ∆H_{mic} < 0\)
• (2) \(∆H_{ads} > 0\ and\ ∆H_{mic} < 0\)
• (3) \(∆H_{ads} < 0\ and\ ∆H_{mic} > 0\)
• (4) \(∆H_{ads} > 0\ and\ ∆H_{mic} > 0\)

Lime reacts exothermally with water to give ‘A’ which has low solubility in water. Aqueous solution of ‘A’ is often used for the test of CO2 , a test in which insoluble B is formed. If B is further reacted with CO2 then soluble compound is formed ‘A’ is

• (1) Quick lime
• (2) Lime water
• (3) Slaked lime
• (4) White lime

Which of the following is used as a stabilizer during the concen tration of sulphide ores?

• (1) Fatty acids
• (2) Pine oil
• (3) Cresol 
• (4) Xanthates

Given below are two reactions, involved in the commercial production of dihydrogen (H₂). The two reactions are carried out at temperature “T₁ ” and “T₂ ” respectively
\(C(S)+H_2O(g)\underrightarrow {T_1}\;CO(g)+H_2(g)\)
\(CO(g)+H_2O(g)\underrightarrow{T_2}\;CO_2(g)+H_2(g)\)
The temperature \(T_1\) and \(T_2\) are correctly related as 

• (1) \(T_1>T_2\)
• (2) \(T_1<T_2\)
• (3) T1=T2
• (4) T1=100 K, T2=1270 K

• (1) (A)-(III), (B)-(II), (C)-(I), (D)-(IV)
• (2) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
• (3) (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
• (4) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

• (1) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
• (2) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
• (3) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
• (4) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

In potassium ferrocyanide, there are ____ pairs of electrons in the \(t_{2g}\) set of orbitals.

The number of bent-shaped molecule(s) from the following is_______? 
SO₂, O₃, I₃^-, N₃^-

The number of correct statement/s involving equilibria in physical process from the following is
(A) Equilibrium is possible only in a closed system at a given temperature 
(B) Both the opposing processes occur at the same rate. 
(C) When equilibrium is attained at a given temperature, the value of all its parameters became equal 
(D) For dissolution of solids in liquids, the solubility is constant at a given temperature

The Sum of number of lone pairs in central atom in IF₅ and IF₇ is:

The number of incorrect statement/s about the black body from the following is _______
(A) Emit or absorb energy in the form of electromagnetic radiation 
(B) Frequency distribution of the emitted radiation depends on temperature
(C) At a given temperature, intensity vs frequency curve passes through a maximum value 
(D) The maximum of the intensity vs frequency curve is at a higher frequency at higher temperature compared to that at lower temperature

A molecule undergoes two independent first order reactions whose respective half lives are 12 min and 3 min. If both the reactions are occurring then the time taken for the 50% consumption of the reactant is _____ min. (Nearest integer)

In the following reactions, the total number of oxygen atoms in X and Y is _______
Na₂O + H₂O → 2X
Cl₂O₇ + H₂O → 2Y

\(FeO^{2-}_4+\underrightarrow{2.2v}\;Fe^{3+}\underrightarrow{+0.70v}\;Fe^{2+}\underrightarrow{-0.45v}\;Fe^0\)
\(E^0_{FeO_4^{2+}}\; is \;x × 10^{-3} V\). The Value of \(x\) is __

The degree of dissociation of monobasic acid is 0 3 By what percent is the observed depression in freezing point greater than the calculated depression in freezing point ?

At constant temperature, a gas is at a pressure of 940.3 mm Hg. The pressure at which its volume decreases by 40% is mm Hg. (Nearest integer)