List of top Questions asked in JEE Main

A \(25\times 10^{-3}\,\mathrm{m^3}\) volume cylinder is filled with 1 mol of \(\mathrm{O_2}\) gas at room temperature (300 K). The molecular diameter of \(\mathrm{O_2}\), and its root mean square speed, are found to be 0.3 nm, and 200 m/s, respectively. What is the average collision rate (per second) for an \(\mathrm{O_2}\) molecule ?

In a meter bridge experiment, the circuit diagram and the corresponding observation table are shown in figure

Sl. No.	\(R\,(\Omega)\)	\(l\,(\mathrm{cm})\)
1.	1000	60
2.	100	13
3.	10	1.5
4.	1	1.0

Which of the readings is inconsistent?

The sum
\(\frac{3\times 1^3}{1^3} + \frac{5\times(1^3 + 2^3)}{1^2 + 2^2} + \frac{7\times(1^3 + 2^3 + 3^3)}{1^2 + 2^2 + 3^2} + \dots\)
upto 10 terms is:

If the line \(x - 2y = 12\) is tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the point \(\left(3, -\frac{9}{2}\right)\), then the length of the latus rectum of the ellipse is :

The line \(x = y\) touches a circle at the point \((1, 1)\). If the circle also passes through the point \((1, -3)\), then its radius is :

If the circles \(x^2 + y^2 + 5Kx + 2y + K = 0\) and \(2(x^2 + y^2) + 2Kx + 3y - 1 = 0\), \((K \in \mathbb{R})\), intersect at the points \(P\) and \(Q\), then the line \(4x + 5y - K = 0\) passes through \(P\) and \(Q\), for :

Which one of the following Boolean expressions is a tautology?

If the system of linear equations
\(x + y + z = 5\)
\(x + 2y + 2z = 6\)
\(x + 3y + \lambda z = \mu\), \((\lambda, \mu \in \mathbb{R})\), has infinitely many solutions, then the value of \(\lambda + \mu\) is :

If \(\int \frac{dx}{(x^2 - 2x + 10)^2} = A\left(\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right) + C\)
where \(C\) is a constant of integration then :

If the coefficients of \(x^2\) and \(x^3\) are both zero, in the expansion of the expression \((1 + ax + bx^2)(1 - 3x)^{15}\) in powers of \(x\), then the ordered pair \((a, b)\) is equal to :

Let \(A(3, 0, -1)\), \(B(2, 10, 6)\) and \(C(1, 2, 1)\) be the vertices of a triangle and \(M\) be the midpoint of \(AC\). If \(G\) divides \(BM\) in the ratio \(2 : 1\), then \(\cos(\angle GOA)\) (\(O\) being the origin) is equal to :

If \(y = y(x)\) is the solution of the differential equation
\(\frac{dy}{dx} = (\tan x - y)\sec^2 x\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\),
such that \(y(0) = 0\), then \(y\left(-\frac{\pi}{4}\right)\) is equal to :

All the pairs \((x, y)\) that satisfy the inequality
\(2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot \frac{1}{4^{\sin^2 y}} \le 1\)
also satisfy the equation

Let \(f(x) = e^x - x\) and \(g(x) = x^2 - x\), \(\forall\, x \in R\). Then the set of all \(x \in R\), where the function \(h(x) = (f \circ g)(x)\) is increasing, is :

If \(\Delta_1 = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}\) and \(\Delta_2 = \begin{vmatrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & 1 \\ \cos 2\theta & 1 & x \end{vmatrix}\), \(x \ne 0\); then for all \(\theta \in \left(0, \frac{\pi}{2}\right)\) :

Let \(f : R \to R\) be differentiable at \(c \in R\) and \(f(c) = 0\). If \(g(x) = |f(x)|\), then at \(x = c\), \(g\) is :

Let \(f(x) = x^2\), \(x \in R\). For any \(A \subseteq R\), define \(g(A) = \{x \in R : f(x) \in A\}\). If \(S = [0, 4]\), then which one of the following statements is not true ?

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is :

The region represented by \(|x - y| \le 2\) and \(|x + y| \le 2\) is bounded by a :

If the length of the perpendicular from the point \((\beta, 0, \beta)\) \((\beta \ne 0)\) to the line, \(\frac{x}{1} = \frac{y-1}{0} = \frac{z+1}{-1}\) is \(\sqrt{\frac{3}{2}}\), then \(\beta\) is equal to :

If \(a_1, a_2, a_3, \ldots, a_n\) are in A.P. and \(a_1 + a_4 + a_7 + \ldots + a_{16} = 114\), then \(a_1 + a_6 + a_{11} + a_{16}\) is equal to :

The value of \(\displaystyle\int_0^{2\pi} \left[\sin 2x\,(1 + \cos 3x)\right] dx\), where \([t]\) denotes the greatest integer function is :

If for some \(x \in R\), the frequency distribution of the marks obtained by 20 students in a test is :

Marks	2	3	5	7
Frequency	\((x+1)^2\)	\(2x - 5\)	\(x^2 - 3x\)	\(x\)

then the mean of the marks is

If \(a > 0\) and \(z = \dfrac{(1+i)^2}{a - i}\), has magnitude \(\sqrt{\frac{2}{5}}\), then \(\bar{z}\) is equal to :

If \(f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x < 0 \\ q, & x = 0 \\ \dfrac{\sqrt{x + x^2} - \sqrt{x}}{x^{3/2}}, & x > 0 \end{cases}\) is continuous at \(x = 0\), then the ordered pair \((p, q)\) is equal to