List of top Questions asked in JEE Main

Let a plane P contain two lines
\(\vec{r} = \hat{i} + \lambda(\hat{i} + \hat{j})\), \(\lambda \in R\) and
\(\vec{r} = -\hat{j} + \mu(\hat{j} - \hat{k})\), \(\mu \in R\).
If \(Q(\alpha, \beta, \gamma)\) is the foot of the perpendicular drawn from the point M(1, 0, 1) to P, then \(3(\alpha + \beta + \gamma)\) equals ______ .

Let p, q, r be three statements such that the truth value of
\((p \wedge q) \rightarrow (\sim q \vee r)\) is F. Then the truth values of p, q, r are respectively :

Let S be the set of all integer solutions, (x, y, z), of the system of equations
\(x - 2y + 5z = 0\)
\(-2x + 4y + z = 0\)
\(-7x + 14y + 9z = 0\)
such that \(15 \le x^2 + y^2 + z^2 \le 150\). Then, the number of elements in the set S is equal to ______ .

Let \(e_1\) and \(e_2\) be the eccentricities of the ellipse,
\(\frac{x^2}{25}+\frac{y^2}{b^2}=1\)
(\(b\lt5\)) and the hyperbola,
\(\frac{x^2}{16}-\frac{y^2}{b^2}=1\)
respectively satisfying \(e_1e_2=1\). If \(\alpha\) and \(\beta\) are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair \((\alpha,\beta)\) is equal to :

Let \(A\) be a \(3\times3\) matrix such that
\(\mathrm{adj}\,A=\begin{bmatrix}2&-1&1\\-1&0&2\\1&-2&-1\end{bmatrix}\)
and \(B=\mathrm{adj}(\mathrm{adj}\,A)\).
If \(|A|=\lambda\) and \(|(B^{-1})^T|=\mu\), then the ordered pair \((|\lambda|,\mu)\) is equal to :

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to ______ .

If a \(\Delta ABC\) has vertices \(A(-1,7)\), \(B(-7,1)\) and \(C(5,-5)\), then its orthocentre has coordinates :

\(\displaystyle\lim_{x\to a}\frac{(a+2x)^{1/3}-(3x)^{1/3}}{(3a+x)^{1/3}-(4x)^{1/3}}\ (a\ne0)\) is equal to :

If \(x^3\,dy+xy\,dx=x^2\,dy+2y\,dx\); \(y(2)=e\) and \(x\gt1\), then \(y(4)\) is equal to :

If the term independent of \(x\) in the expansion of
\(\left(\frac32x^2-\frac{1}{3x}\right)^9\)
is \(k\), then \(18k\) is equal to :

Suppose \(f(x)\) is a polynomial of degree four, having critical points at \(-1, 0, 1\). If \(T=\{x\in\mathbb{R}\mid f(x)=f(0)\}\), then the sum of squares of all the elements of \(T\) is :

If \(z_1,\ z_2\) are complex numbers such that
\(\mathrm{Re}(z_1)=|z_1-1|\), \(\mathrm{Re}(z_2)=|z_2-1|\), and
\(\arg(z_1-z_2)=\frac{\pi}{6}\), then \(\mathrm{Im}(z_1+z_2)\) is equal to :

Let \(R_1\) and \(R_2\) be two relation defined as follows :
\(R_1=\{(a,b)\in\mathbb{R}^2 : a^2+b^2\in\mathbb{Q}\}\) and
\(R_2=\{(a,b)\in\mathbb{R}^2 : a^2+b^2\notin\mathbb{Q}\}\),
where \(\mathbb{Q}\) is the set of all rational numbers. Then :

Let \(a, b, c\in\mathbb{R}\) be such that \(a^2+b^2+c^2=1\). If
\(a\cos\theta=b\cos\left(\theta+\frac{2\pi}{3}\right)=c\cos\left(\theta+\frac{4\pi}{3}\right)\),
where \(\theta=\frac{\pi}{9}\), then the angle between the vectors \(a\hat{i}+b\hat{j}+c\hat{k}\) and \(b\hat{i}+c\hat{j}+a\hat{k}\) is :

If the surface area of a cube is increasing at a rate of \(3.6\ \mathrm{cm^2/sec}\), retaining its shape; then the rate of change of its volume (in \(\mathrm{cm^3/sec}\)), when the length of a side of the cube is \(10\ \mathrm{cm}\), is :

Let \(x_i\ (1 \le i \le 10)\) be ten observations of a random variable \(X\). If
\(\sum_{i=1}^{10}(x_i-p)=3\)
and
\(\sum_{i=1}^{10}(x_i-p)^2=9\)
where \(0 \ne p \in \mathbb{R}\), then the standard deviation of these observations is :

The volume strength of 8.9 M \(\mathrm{H_2O_2}\) solution calculated at 273 K and 1 atm is ______. (R = 0.0821 L atm \(\mathrm{K^{-1}\,mol^{-1}}\)) (rounded off ot the nearest integer)

The photoelectric current from Na (Work function, \(w_0\) = 2.3 eV) is stopped by the output voltage of the cell
Pt(s) | \(\mathrm{H_2}\) (g, 1 Bar) | HCl (aq., pH =1) | AgCl(s) | Ag(s).
The pH of aq. HCl required to stop the photoelectric current form K(\(w_0\) = 2.25 eV), all other conditions remaining the same, is _______ \(\times\) \(10^{-2}\) (to the nearest integer).
Given, \(2.303\frac{RT}{F}\) = 0.06 V;
\(E^0_{\mathrm{AgCl|Ag|Cl^-}}\) = 0.22 V

The antifertility drug “Novestrol” can react with :

If the boiling point of H₂O is 373 K, the boiling point of H₂S will be :

Thermal power plants can lead to :

Aqua regia is used for dissolving noble metals (Au, Pt, etc.). The gas evolved in this process is :

In a molecule of pyrophosphoric acid, the number of P - OH, P = O and P - O - P bonds/moiety(ies) respectively are :

Tyndall effect is observed when:

Glycerol is separated in soap industries by :