List of top Questions asked in JEE Main- 2013

The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$. The function $f(x) = x log x$ is an increasing function in $[1,2]$ and $g (x)=2-x$ is a decreasing function in $[1, 2]$ and the graphs represented by these functions intersect at a point in $[1,2]$

How many grams of methyl alcohol should be added to $10$ litre tank of water to prevent its freezing at $268\, K$ ? $\left(K_{f} \, for\, water\, is \,1.86 K\, kg \, mol^{-1} \right)$

The integral $\int \frac{xdx}{2-x^{2}+\sqrt{2-x^{2}}}$ equals

A diode detector is used to detect an amplitude modulated wave of $60\%$ modulation by using a condenser of capacity $250$ picofarad in parallel with a load resistance $100$ $\Omega$. Find the maximum modulated frequency which could be detected by it.

A beam of unpolarized light of intensity $I_0$ is passed through a polaroid $A$ and then through another polaroid $B$ which is oriented so that its principal plane makes an angle of 45$^{\circ}$ relative to that of $A$. The intensity of the emergent light is

Experimentally it was found that a metal oxide has formula $ M_{0.98}O. $ Metal $M$, present as $ M^{2+} $ and $ M^{3+} $ in its oxide.Fraction of the metal which exists as $ M^{3+} $ would be

$XeO_4$ molecule is tetrahedral having :

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude is $5\, s$. In another $10\, s$ it will decrease to a times its original magnitude, where $\alpha$ equals

A uniform wire (Young?s modulus $2 \times 10^{11}\,Nm^{-2}$) is subjected to longitudinal tensile stress of $5 \times 10^{7}\,Nm^{-2}$. If the overall volume change in the wire is $0.02\%$, the fractional decrease in the radius of the wire is close to:

To establish an instantaneous current of $2 A$ through a $1 \mu F$ capacitor; the potential difference across the capacitor plates should be changed at the rate of :

If the integral $\int \frac{cos 8x+1}{cot 2x-tan 2x} dx=A cos 8x+k,$ where $k$ is an arbitrary constant, then $A$ is equal to:

$6$ litres of an alkene require $27$ litres of oxygen at constant temperature and pressure for complete combustion. The alkene is:

Consider the following reaction, $xMnO_4^- + yC_2O_4^{2-}+zH^+ \rightarrow xMn^{2+}+2yCO_2 + \frac{z}{2}H_2O$ The values of x, y and z in the reaction are, respectively

Let $a_1, a_2, a_3,...$ be an $A.P$, such that $\frac{a_{1}+a_{2}+\ldots+a_{p}}{a_{1}+a_{2}+a_{3}+\ldots+a_{q}} = \frac{p^{3}}{q^{3}} ; p\ne q$ Then $\frac{a_{6}}{a_{21}}$ is equal to:

Potassium dichromate when heated with concentrated sulphuric acid and a soluble chloride, gives brown-red vapours of :

The cost of running a bus from $A$ to $B$ , is $Rs.\left(av+\frac{b}{v}\right)$ where $v$ km/h is the average speed of the bus. When the bus travels at $30\, km/h$, the cost comes out to be $Rs.\, 75$ while at $40\, km/h$, it is $Rs. \,65$. Then the most economical speed (in $km/ h$) of the bus is :

The coagulating power of electrolytes having ions $Na^+, Al^{3+}$ and $Ba^{2+}$ for arsenic sulphide sol increases in the order

Two point dipoles of dipole moment $\overrightarrow{p_{1}}$ and $\overrightarrow{p_{2}}$ are at a distance $x$ from each other and $\overrightarrow{p_{1}} \left|\right| \overrightarrow{p_{2}}.$ The force between the dipoles is :

Two balls of same mass and carrying equal charge are hung from a fixed support of length $l$. At electrostatic equilibrium, assuming that angles made by each thread is small, the separation, $x$ between the balls is proportional to :

Let $x \in \left(0,1\right).$ The set of all x such that $sin^{-1} x >\, cos^{-1} x, $ is the interval:

Number of atoms in the following samples of substances is largest in :

The rate of a reaction doubles when its temperature changes from $300\, K$ to $310\,K$. Activation energy of such a reaction will be $(R = 8.314 JK^{-1} \,mol^{-1}$ and $log \,2 = 0.301)$

In nucleophilic substitution reaction, order of halogens as incoming (attacking) nucleophile is: $I^- > \, Br^{-} > \, Cl^-$ The order of halogens as departing nucleophile should be:

Which one of the following statements about packing in solids is ?

Rate of dehydration of alcohols follows the order: