List of top Questions asked in JEE Main

If for \(x \ge 0\), \(y = y(x)\) is the solution of the differential equation \((x+1)\,dy = \left((x+1)^2 + y - 3\right)dx\), \(y(2) = 0\), then \(y(3)\) is equal to ______.

The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdots\) to \(\infty\) is equal to :

Let the observations \(x_i\ (1\le i\le10)\) satisfy the equations,
\(\sum_{i=1}^{10}(x_1-5)=10\) and \(\sum_{i=1}^{10}(x_1-5)^2=40\).
If \(\mu\) and \(\lambda\) are the mean and the variance of the observations, \(x_1-3,x_2-3,\dots,x_{10}-3\), then the ordered pair \((\mu,\lambda)\) is equal to :

If for some \(\alpha\) and \(\beta\) in R, the intersection of the following three places
\(x + 4y - 2z = 1\)
\(x + 7y - 5z = b\)
\(x + 5y + \alpha z = 5\)
is a line in \(\mathbb{R}^3\), then \(\alpha + \beta\) is equal to :

If the matrices \(A = \begin{bmatrix}1&1&2\\1&3&4\\1&-1&3\end{bmatrix}\), \(B = \mathrm{adj}A\) and \(C = 3A\), then \(\frac{|\mathrm{adj}B|}{|C|}\) is equal to :

If \(f'(x) = \tan^{-1}(\sec x + \tan x)\), \(-\frac{\pi}{2} \lt x \lt \frac{\pi}{2}\), and \(f(0) = 0\), then \(f(1)\) is equal to :

A circle touches the y-axis at the point \((0, 4)\) and passes through the point \((2, 0)\). Which of the following lines is not a tangent to this circle?

The number of real roots of the equation,
\(e^{4x}+e^{3x}-4e^{2x}+e^x+1=0\) is :

In a box, there are 20 cards, out of which 10 are lebelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :

Let \(f\) be any function continuous on [a, b] and twice differentiable on (a, b). If for all \(x\in(a,b)\), \(f'(x)>0\) and \(f''(x)<0\), then for any \(c\in(a,b)\),
\(\frac{f(c)-f(a)}{f(b)-f(c)}\)
is greater than :

Let C be the centroid of the triangle with vertices (3, \(-1\)), (1, 3) and (2, 4). Let P be the point of intersection of the lines \(x+3y-1=0\) and \(3x-y+1=0\). Then the line passing through the points C and P also passes through the point :

The value of
\(\int_0^{2\pi}\frac{x\sin^8x}{\sin^8x+\cos^8x}\,dx\)
is equal to :

The integral
\(\int\frac{dx}{(x+4)^{\frac87}(x-3)^{\frac67}}\)
is equal to :
(where C is a constant of integration)

If
\(f(x)=\begin{cases}\dfrac{\sin(a+2)x+\sin x}{x}; & x<0\\[2mm] b\ ; & x=0\\[2mm] \dfrac{(x+3x^2)^{\frac13}-x^{\frac13}}{x^{\frac43}}; & x>0\end{cases}\)
is continuous at x = 0, then a + 2b is equal to :

The value of
\(\cos^3\left(\frac{\pi}{8}\right)\cos\left(\frac{3\pi}{8}\right)+\sin^3\left(\frac{\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\)
is :

If for all real triplets (a, b, c), \(f(x)=a+bx+cx^2\); then
\(\int_0^1 f(x)\,dx\)
is equal to :

Let z be complex number such that
\(\left|\frac{z-i}{z+2i}\right|=1\)
and \(|z|=\frac{5}{2}\).
Then the value of \(|z+3i|\) is :

If the number of five digit numbers with distinct digits and 2 at the 10^th place is 336 k, then k is equal to :

\(\mathrm{NaClO_3}\) is used, even in spacecrafts, to produce \(\mathrm{O_2}\). The daily consumption of pure \(\mathrm{O_2}\) by a person is 492 L at 1 atm, 300 K. How much amount of \(\mathrm{NaClO_3}\), in grams, is required to produce \(\mathrm{O_2}\) for the daily consumption of a person at 1 atm, 300 K?
\(\mathrm{NaClO_3(s) + Fe(s) \to O_2(g) + NaCl(s) + FeO(s)}\)
\(R = 0.082\ \mathrm{L\ atm\ mol^{-1}\ K^{-1}}\)

Complexes (\(\mathrm{ML_5}\)) of metals Ni and Fe have ideal square pyramidal and trigonal bipyramidal grometries, respectively. The sum of the \(90^\circ\), \(120^\circ\) and \(180^\circ\) L-M-L angles in the two complexes is _________.

Preparation of Bakelite proceeds via reactions.

For an electrochemical cell
\(\mathrm{Sn(s)\,|\,Sn^{2+}(aq,1M)\,\|\,Pb^{2+}(aq,1M)\,|\,Pb(s)}\)
the ratio \(\frac{[\mathrm{Sn^{2+}}]}{[\mathrm{Pb^{2+}}]}\) when this cell attains equilibrium is _________.
(Given \(E^0_{\mathrm{Sn^{2+}|Sn}} = -0.14\,\mathrm{V}\), \(E^0_{\mathrm{Pb^{2+}|Pb}} = -0.13\,\mathrm{V}\), \(\frac{2.303RT}{F} = 0.06\))

White Phosphorus on reaction with concentrated NaOH solution in an inert atmosphere of \(\mathrm{CO_2}\) gives phosphine and compound (X). (X) on acidification with HCl gives compound (Y). The basicity of compound (Y) is :

Which of the following compounds is likely to show both Frenkel and Schottky defects in its crystalline form?

A metal (A) on heating in nitrogen gas gives compound B. B on treatment with \(\mathrm{H_2O}\) gives a colourless gas which when passed through \(\mathrm{CuSO_4}\) solution gives a dark blue-violet coloured solution. A and B respectively, are :