List of top Questions asked in JEE Main

Calculate the time interval between 33% decay and 67% decay if half-life of a substance is 20 minutes.

Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight? (Assume radius of earth is 6400 km)

The half-life of \(\mathrm{Au^{198}}\) is 2.7 days. The activity of 1.50 mg of \(\mathrm{Au^{198}}\) if its atomic weight is \(198\ \mathrm{g\,mol^{-1}}\) is, (\(N_a = 6\times 10^{23}\)/mol).

A bimetallic strip consists of metals A and B. It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that of metal B. When the bimetallic strip is placed in a cold bath, it will :

Find out the surface charge density at the intersection of point \(x = 3\ \mathrm{m}\) plane and x-axis, in the region of uniform line charge of 8 nC/m lying along the z-axis in free space.

The magnetic field in a region is given by \(\vec{B} = B_o\left(\frac{x}{a}\right)\hat{k}\). A square loop of side d is placed with its edges along the x and y axes. The loop is moved with a constant velocity \(\vec{v} = v_0\hat{i}\). The emf induced in the loop is :

For the given circuit, comment on the type of transformer used.

Red light differs from blue light as they have :

The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of 100 V. What should nearly be the ratio of their wavelengths? (\(m_p = 1.00727\ \mathrm{u}\), \(m_e = 0.00055\ \mathrm{u}\))

Calculate the value of mean free path (\(\lambda\)) for oxygen molecules at temperature \(27^\circ\mathrm{C}\) and pressure \(1.01\times 10^5\) Pa. Assume the molecular diameter 0.3 nm and the gas is ideal. (\(k = 1.38\times 10^{-23}\ \mathrm{JK^{-1}}\))

In order to determine the Young's Modulus of a wire of radius 0.2 cm (measured using a scale of least count = 0.001 cm) and length 1 m (measured using a scale of least count = 1 mm), a weight of mass 1 kg (measured using a scale of least count = 1 g) was hanged to get the elongation of 0.5 cm (measured using a scale of least count 0.001 cm). What will be the fractional error in the value of Young's Modulus determined by this experiment?

A charge Q is moving \(\overrightarrow{dl}\) distance in the magnetic field \(\overrightarrow{B}\). Find the value of work done by \(\overrightarrow{B}\).

The refractive index of a converging lens is 1.4. What will be the focal length of this lens if it is placed in a medium of same refractive index? Assume the radii of curvature of the faces of lens are R₁ and R₂ respectively.

A large block of wood of mass \(M = 5.99\ \mathrm{kg}\) is hanging from two long massless cords. A bullet of mass \(m = 10\ \mathrm{g}\) is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance \(h = 9.8\ \mathrm{cm}\) before the (block + bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is :
(take \(g = 9.8\ \mathrm{ms^{-2}}\))

\(S_n(x)=\log_{a^{1/2}}x+\log_{a^{1/3}}x+\log_{a^{1/6}}x+\log_{a^{1/11}}x+\log_{a^{1/18}}x+\log_{a^{1/27}}x+\ldots\) up to n-terms, where \(a>1\). If \(S_{24}(x)=1093\) and \(S_{12}(2x)=265\), then value of a is equal to ____________.

What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min? The radius of the tap and viscosity of water are 0.5 cm and \(10^{-3}\) Pa s, respectively. (Density of water : \(10^3\) kg/m³)

Statement I : A cyclist is moving on an unbanked road with a speed of 7 km h^-1 and takes a sharp circular turn along a path of radius of 2 m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve. (\(g = 9.8\ \mathrm{m/s^2}\))

Statement II : If the road is banked at an angle of \(45^\circ\), cyclist can cross the curve of 2 m radius with the speed of 18.5 km h^-1 without slipping.

In the light of the above statements, choose the correct answer from the options given below.

A mosquito is moving with a velocity \(\vec{v} = 0.5t^2\hat{i} + 3t\hat{j} + 9\hat{k}\) m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2 s?

In \(\Delta ABC\), the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of \(\Delta ABC\) is 30 cm² and R and r are respectively the radii of circumcircle and incircle of \(\Delta ABC\), then the value of 2R + r (in cm) is equal to ___________.

Consider the statistics of two sets of observations as follows :

	Size	Mean	Variance
Observation I	10	2	2
Observation II	n	3	1

If the variance of the combined set of these two observations is \(\frac{17}{9}\), then the value of n is equal to ___________.

Let \(A=\begin{bmatrix}a_1\\a_2\end{bmatrix}\) and \(B=\begin{bmatrix}b_1\\b_2\end{bmatrix}\) be two \(2\times1\) matrices with real entries such that \(A=XB\), where
\(X=\frac{1}{\sqrt3}\begin{bmatrix}1&-1\\1&k\end{bmatrix}\), and \(k\in R\).
If \(a_1^2+a_2^2=\frac23(b_1^2+b_2^2)\) and \((k^2+1)b_2^2\ne-2b_1b_2\), then the value of k is __________.

Let \(\vec c\) be a vector perpendicular to the vectors, \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i+2\hat j+\hat k\). If \(\vec c\cdot(\hat i+\hat j+3\hat k)=8\) then the value of \(\vec c\cdot(\vec a\times\vec b)\) is equal to __________.

For real numbers \(\alpha, \beta, \gamma\) and \(\delta\), if
\(\int \frac{(x^2-1)+\tan^{-1}\left(\frac{x^2+1}{x}\right)}{(x^4+3x^2+1)\tan^{-1}\left(\frac{x^2+1}{x}\right)}dx\)
\(=\alpha\log_e\left(\tan^{-1}\left(\frac{x^2+1}{x}\right)\right)+\beta\tan^{-1}\left(\frac{\gamma(x^2+1)}{x}\right)+\delta\tan^{-1}\left(\frac{x^2+1}{x}\right)+C\)
where C is an arbitrary constant, then the value of \(10(\alpha+\beta\gamma+\delta)\) is equal to ______________.

Let n be a positive integer. Let
\(A=\sum_{k=0}^{n}(-1)^k\,{}^nC_k\left[\left(\frac12\right)^k+\left(\frac34\right)^k+\left(\frac78\right)^k+\left(\frac{15}{16}\right)^k+\left(\frac{31}{32}\right)^k\right]\).
If \(63A=1-\frac{1}{2^{30}}\), then n is equal to _____________.

Let \(f:R\to R\) and \(g:R\to R\) be defined as
\(f(x)=\begin{cases}x+a, & x<0\\ |x-1|, & x\ge 0\end{cases}\) and \(g(x)=\begin{cases}x+1, & x<0\\ (x-1)^2+b, & x\ge 0\end{cases}\),
where a, b are non-negative real numbers. If \((g\circ f)(x)\) is continuous for all \(x\in R\), then \(a+b\) is equal to ____________.