List of top Questions asked in JEE Main

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x + y - 2}{x - y}$ passes through the points $(2, 1)$ and (k + 1, 2), k > 0, then

The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _________.

Total number of isomers (including stereoisomers) obtained on monochlorination of methylcyclohexane is____.

Given below are two statements:
Statement I: The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement II: The rms speed of oxygen molecules in a gas is $v$ If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2 v$ In the light of the above statements, choose the correct answer from the options given below:

Consider a cylindrical tank of radius $1 m$ is filled with water. The top surface of water is at $15 m$ from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of $5 m$ from the bottom. A force of $5 × 10^5 N$ is applied on the top surface of water using a piston. The speed of efflux from the hole will be:

(Given atmosphere pressure $P_A = 1.01 × 10^5 Pa$, density of water $ρ_w 1000 Kg/m^3$ and gravitational acceleration $g = 10 m/s^2$)

Let $X = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix}$,
Y = αI + βX + γX² and
Z = α²l - αβX + (β² - αϒ)X² ,α,β,ϒ ∈ R.
If $Y^{-1} = \begin{bmatrix} \frac{1}{5} & -\frac{2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & -\frac{2}{5} \\ 0 & 0 & \frac{1}{5} \\ \end{bmatrix}$,
then ( α - β + ϒ )² is equal to ________.

If $\vec a= 2\hat i +\hat j + 3\hat k$, $\vec b = 3\hat i + 3\hat j + \hat k$ and $\vec c = c_1\hat i + c_2\hat j + c_3\hat k$ are coplanar vectors and $\vec a.\vec c = 5$, $\vec b⊥\vec c$ then $122( c_1 + c_2 + c_3 )$ is equal to _____ .

The area of the polygon, whose vertices are the non-real roots of the equation $\bar z = iz^2$ is

Match List-I with List-II.

List-I (Metal)	List-II (Emitted light wavelength (nm))
(A) Li	(I) 670.8
(B) Na	(II) 589.2
(C) Rb	(III) 780.0
(D) Cs	(IV) 455.5

Choose the most appropriate answer from the options given below:

20 mL of 0.1 M NH₄OH is mixed with 40 mL of 0.05 M HCl.The pH of the mixture is nearest to:(Given: K_b(NH₄OH) = 1 × 10^–5,log 2 = 0.30,log 3 = 0.48, log 5 = 0.69, log 7 = 0.84, log 11 =1.04)

The series of positive multiples of 3 is divided into sets: {3}, {6, 9, 12}, {15, 18, 21, 24, 27},…… Then the sum of the elements in the 11^th set is equal to ________.

Let $y = y(x)$ be the solution of the differential equation $(1 - x^2) \, dy = (xy + (x^3 + 2) \sqrt{1 - x^2}) \, dx \quad -1 < x < 1$, and $y(0) = 0.$ If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1 - x^2} \, y(x) \, dx = k$ then $k^{−1}$ is equal to_______.

A copper block of mass 5.0 kg is heated to a temperature of 500°C and is placed on a large ice block. What is the maximum amount of ice that can melt?

[Specific heat of copper 0.39 J g^–1 °C^–1 and latent heat of fusion of water : 335 J g^–1]

$\begin{array}{l}\displaystyle \lim_{ x\to 0}\left(\frac{(x+2\cos x)^3 + 2(x+2\cos x)^2 + 3 \sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}{x}} \end{array}$

is equal to _________.

A parallel beam of light of wavelength $900 \,nm$ and intensity $100 \,Wm ^{-2}$ is incident on a surface perpendicular to the beam Tire number of photons crossing $1 \,cm ^2$ area perpendicular to the beam in one second is :

A ball is thrown vertically upwards with a velocity of $19.6 ms^{–1}$ from the top of a tower. The ball strikes the ground after $6 s$. The height from the ground up to which the ball can rise will be $(k/5) m$. The value of k is ______ (use $g = 9.8 m/s^2$)

In 1^st case, Carnot engine operates between temperatures 300 K and 100 K. In 2^nd case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in 2^nd case) will be:

Let

$S=\left\{x∈[-6,3]-\left\{-2,2 \right\} :\frac{|x+3|-1}{|x|-2}>=0 \right\} \space and \space \left\{T={x∈Z:x^2-7|x|+9<=0}\right\}$

Then the number of elements in S ⋂ T is

$\int_{0}^{2} \left( |2x^2 - 3x| + \left[x - \frac{1}{2}\right] \right) \, dx,$
where [t] is the greatest integer function, is equal to

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^th term is a_n and the common difference is 10ar², is equal to

If $\lim_{n\rightarrow \infty}$($\sqrt{n^2-n-1}$ + nα+β)=0 then 8(α + β) is equal to

Let f: ℝ → ℝ be defined as
$f(x) = \left\{ \begin{array}{ll} [e^x] & x < 0 \\ [a e^x + [x-1]] & 0 \leq x < 1 \\ [b + [\sin(\pi x)]] & 1 \leq x < 2 \\ [[e^{-x}] - c] & x \geq 2 \\ \end{array} \right.$
Where a, b, c ∈ ℝ and [t] denotes greatest integer less than or equal to t.
Then, which of the following statements is true?

A fish swimming in water body when taken out from the water body is covered with a film of water of weight 36 g. When it is subjected to cooking at 100 °C, then the internal energy for vaporization in kJ mol^–1 is ________. [nearest integer]
[Assume steam to be an ideal gas. Given Δ_vapH^Θ for water at 373 K and 1 bar is 41.1 kJ mol^–1; R = 8.31 J K^–1 mol^–1]

The sum of all real value of $x$ for which $\frac{3x^2-9x+17}{x^2+3x+10}=\frac {5x^2-7x+19}{3x^2+5x+12}$ is equal to________.

If
$\sum_{k=1}^{10} \frac{k}{k^4 + k^2 + 1} = \frac{m}{n}$
where m and n are co-prime, then m + n is equal to