List of top Questions asked in JEE Main- 2023

The shortest distance between the lines $\frac{x−1}2 =\frac{y−2}1 =\frac{z−6}{-3} $ and $\frac{x−1}2 =\frac{y+8}{-7} =\frac{z−4}{5} $ is

The value of the integral \(\int^2_1\bigg(\frac{t^4+1}{t^6+1}\bigg)\)dt is 

Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?

Let R be a relation defined on N as a ⁡R⁡b if 2a+3b is a multiple of 5 ,a , b∈N. Then R is

If the tangent at a point 𝑃 on the parabola \(𝑦^2=3𝑥\) is parallel to the line \(𝑥+2𝑦=1\) and the tangents at the points 𝑄 and 𝑅 on the ellipse \(\frac{𝑥^2}{ 4} +\frac{𝑦^2}{ 1} =1\) are perpendicular to the line 𝑥−𝑦=2, then the area of the triangle PQR is : 

If \(\overrightarrow a =\hat i+2\hat k,\overrightarrow b =\hat i+\hat j+\hat k,\overrightarrow c =7\hat i−3\hat j+4\hat k,r ×\hat b +\hat b ×\overrightarrow c =\overrightarrow0\) and \(\overrightarrow r .\overrightarrow a =0\). Then \(\overrightarrow r. \overrightarrow c\)  is equal to 

If the lines \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}\)  and \(\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}\) intersect at the point 𝑃, then the distance of the point 𝑃 from the plane 𝑧=𝑎 is : 

The value of the integral \(∫_{\frac 12}^2 \frac{tan^{-1}x}{x}dx\) is equal to

The plane \(2x−y+z=4\) intersects the line segment joining the points \(A(a,−2,4)\) and \(B(2,b,−3)\) at the point C in the ratio \(2:1\) and the distance of the point C from the origin is \(\sqrt5\). If \(ab<0 \) and \(P\) is the point \((a−b,b,2b−a)\). Then \(CP^2 \) is equal to

The area of the region A = \({(x,y):|cos⁡x−sin⁡x|≤y≤sin⁡x,0≤x≤\frac{π}{ 2} }\) is

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is

Let \(\overrightarrow 𝑎 =4\hat 𝑖+3\hat 𝑗ˆ\) and \(\overrightarrow 𝑏 =3\hat 𝑖−4\hat 𝑗\)+\(5\hat 𝑘\). If \(\overrightarrow 𝑐\)  is a vector such that \(\overrightarrow 𝑐 ⋅(\overrightarrow 𝑎 ×\overrightarrow𝑏⃗ )+25=0\), \(\overrightarrow𝑐 ⋅(\hat𝑖+\hat𝑗+ \hat𝑘)=4\), and projection of \(\overrightarrow𝑐\)  on \(\overrightarrow𝑎\)  is \(1\) , then the projection of \(\overrightarrow 𝑐\)  on \(\overrightarrow 𝑏\) equals 

The set of all values of \(λ\) for which the equation \(cos^2⁡2x−2sin^4⁡x−2cos^2⁡x=λ\) has a real solution x, is 

The set of all values of \(t∈R\), for which the matrix \(\begin{bmatrix} e^t & e^{−t}(sin⁡\ t−2cos\ ⁡t) & e^{−t}(−2sin⁡\ t−cos\ ⁡t) \\[0.3em] e^t & e^{−t}(2sin\ ⁡t+cos\ ⁡t) & e^{−t}(sin\ ⁡t−2cos\ ⁡t) \\[0.3em] e^t & e^{−tcos\ ⁡t}& e^{−tsin\ ⁡t} \end{bmatrix}\) is invertible, is

Let \(𝑦=𝑦(𝑥)\) be the solution of the differential equation \(𝑥\;𝑙𝑜𝑔_𝑒⁡𝑥\frac{𝑑𝑦}{ 𝑑𝑥} +𝑦=𝑥^2𝑙𝑜𝑔_𝑒⁡𝑥,(𝑥>1)\).  If \(𝑦(2)=2\), then \(𝑦(𝑒)\) is equal to 

In a part of a circuit shown: Find V_A – V_B in volts. It is given that current is decreasing at a rate of 1 ampere/s.

A particle undergoing SHM follows the position-time equation given as x = A sin \((\omega t+\frac{\pi}{3})\) . If the SHM motion has a time period of T, then velocity will be maximum at time \(t=\frac{T}{β}\) for first time after t = 0. Value of β is equal to

A block of mass 1 g is equilibrium with the help of a current carrying square loop which is partially lying in constant magnetic field (B) as shown. Resistance of the loop is 10 Ω. Find the voltage (V) (in volts) of the battery in the loop.

Initial volume of 1 mole of a monoatomic gas is 2 litres. It is expanded isothermally to a volume of 6 litres. Change in internal energy is xR. Find x.

An object is placed at a distance of 40 cm from the pole of a converging mirror. The image is formed at a distance of 120 cm from the mirror on the same side. If the focal length is measured with a scale where each 1 cm has 20 equal divisions. If the fractional error in the measurement of focal length is \(\frac{1}{10\ k}\).Find k.

In two circuit shown above the value of current I₁ (in amperes) is equal to \(-\frac{y}{5}A\) . Value of y is equal to

For given cell, at T K
Pt | H₂(g) | H⁺ || Fe³⁺ ; Fe²⁺ | Pt
   ( 1 bar ) ( 1 M )
E_cell = .712 V
E^°_cell = .770 V
if \(\frac{[Fe^{2+}]}{[Fe^{3+}]}\) is t \((\frac{2.303\ RT}{F}=.058)\)
then find \((\frac{t}{5})\)

600 mL of 0.04 M HCl is mixed with 400 mL of 0.02 M H₂SO₄. Find out the pH of resulting solution (Nearest integer).

\( \lim_{x \to 0} \frac{48 x^4}{x} \int_0^x \frac{t^3}{t^6 + 1} \, dt \) is equal to:

If \(a_n=\frac{-2}{4n^2-16n+15}\) and a₁ + a₂ + …… + a₂₅ = m/n where m and n are coprime, then the value of m + n is