List of top Questions asked in JEE Main

If the projections of a line segment on the \(x\), \(y\) and \(z\)-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :

If the curves \(\frac{x^2}{\alpha}+\frac{y^2}{4}=1\) and \(y^3=16x\) intersect at right angles, then a value of \(\alpha\) is :

If a curve passes through the point \(\left(2, \frac{7}{2}\right)\) and has slope \(\left(1-\frac{1}{x^2}\right)\) at any point \((x, y)\) on it, then the ordinate of the point on the curve whose abscissa is \(-2\) is :

Statement-1: The system of linear equations
\(x+(\sin\alpha)y+(\cos\alpha)z=0\)
\(x+(\cos\alpha)y+(\sin\alpha)z=0\)
\(x-(\sin\alpha)y-(\cos\alpha)z=0\)
has a non-trivial solution for only one value of \(\alpha\) lying in the interval \(\left(0, \frac{\pi}{2}\right)\).
Statement-2: The equation in \(\alpha\)
\(\begin{vmatrix}\cos\alpha & \sin\alpha & \cos\alpha\\ \sin\alpha & \cos\alpha & \sin\alpha\\ \cos\alpha & -\sin\alpha & -\cos\alpha\end{vmatrix}=0\)
has only one solution lying in the interval \(\left(0, \frac{\pi}{2}\right)\)

\(S=\tan^{-1}\left(\frac{1}{n^2+n+1}\right)+\tan^{-1}\left(\frac{1}{n^2+3n+3}\right)+\ldots+\tan^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)\), then \(\tan S\) is equal to :

Let \(f\) be a composite function of \(x\) defined by \(f(u)=\frac{1}{u^2+u-2},\ u(x)=\frac{1}{x-1}\). Then the number of points \(x\) where \(f\) is discontinuous is :

If the median and the range of four numbers \(\{x, y, 2x+y, x-y\}\), where \(0 \lt y \lt x \lt 2y\), are 10 and 28 respectively, then the mean of the numbers is :

Let \(R=\{(x, y) : x, y \in N \text{ and } x^2-4xy+3y^2=0\}\), where \(N\) is the set of all natural numbers. Then the relation \(R\) is :

Let A, other than I or \(-I\), be a \(2 \times 2\) real matrix such that \(A^2=I\), I being the unit matrix. Let \(\mathrm{Tr}(A)\) be the sum of diagonal elements of A.
Statement-1: \(\mathrm{Tr}(A)=0\)
Statement-2: \(\det(A)=-1\)

If the circle \(x^2 + y^2 - 6x - 8y + (25 - a^2) = 0\) touches the axis of \(x\), then \(a\) equals.

The point of intersection of the normals to the parabola \(y^2 = 4x\) at the ends of its latus rectum is :

For integers \(m\) and \(n\), both greater than 1, consider the following three statements : \(P : m\) divides \(n\) \(Q : m\) divides \(n^2\) \(R : m\) is prime, then

Let \(\theta_1\) be the angle between two lines \(2x + 3y + c_1 = 0\) and \(-x + 5y + c_2 = 0\) and \(\theta_2\) be the angle between two lines \(2x + 3y + c_1 = 0\) and \(-x + 5y + c_3 = 0\), where \(c_1, c_2, c_3\) are any real numbers :
Statement-1 : If \(c_2\) and \(c_3\) are proportional, then \(\theta_1 = \theta_2\).
Statement-2 : \(\theta_1 = \theta_2\) for all \(c_2\) and \(c_3\).

The number of solutions of the equation \(\sin 2x - 2\cos x + 4\sin x = 4\) in the interval \([0, 5\pi]\) is :

The sum of the rational terms in the binomial expansion of \(\left(2^{\frac{1}{2}} + 3^{\frac{1}{5}}\right)^{10}\) is :

If two lines \(L_1\) and \(L_2\) in space, are defined by
\(L_1 = \{x = \sqrt{\lambda}\,y + (\sqrt{\lambda} - 1),\ z = (\sqrt{\lambda} - 1)\,y + \sqrt{\lambda}\}\) and
\(L_2 = \{x = \sqrt{\mu}\,y + (1 - \sqrt{\mu}),\ z = (1 - \sqrt{\mu})\,y + \sqrt{\mu}\}\)
then \(L_1\) is perpendicular to \(L_2\), for all nonnegative reals \(\lambda\) and \(\mu\), such that :

If \(a_1, a_2, a_3, \ldots, a_n, \ldots\) are in A.P. such that \(a_4 - a_7 + a_{10} = m\), then the sum of first 13 terms of this A.P., is :

The sum of the series : \((2)^2 + 2(4)^2 + 3(6)^2 + \ldots\) upto 10 terms is :

The least integral value \(\alpha\) of \(x\) such that \(\dfrac{x-5}{x^2+5x-14} \gt 0\), satisfies :

The polymer used for optical lenses is :

Which of the following statements about aspirin is not true?

A molecule \(M\) associates in a given solvent according to the equation \(\mathrm{M} \rightleftharpoons (\mathrm{M})_n\). For a certain concentration of M, the van't Hoff factor was found to be 0.9 and the fraction of associated molecules was 0.2. The value of \(n\) is :

What is the structure of the major product when phenol is treated with bromine water?

Amongst the following alcohols which would react fastest with conc. \(\mathrm{HCl}\) and \(\mathrm{ZnCl_2}\) ?

Which of the following statements is incorrect?