List of top Questions asked in JEE Main

The area of the region \( \{(x,y): x^2-8x \le y \le -x\} \) is :

Let \(f:\mathbb{R}\to\mathbb{R}\) be such that \(f(x+y)=f(x)f(y)\) for all \(x,y\in\mathbb{R}\) and \(f(0)\neq0\). Let \(g:[1,\infty)\to\mathbb{R}\) be a differentiable function such that \[ x^2g(x)=\int_1^x\big(t^2f(t)-tg(t)\big)\,dt \] Then \(g(2)\) is equal to :

If \( \sin\!\left(\tan^{-1}(x\sqrt2)\right)=\cot\!\left(\sin^{-1}\!\sqrt{1-x^2}\right),\; x\in(0,1) \), then the value of \(x\) is :

Let \(x=-9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \( \frac13 \). Let \(P(\alpha,0), \alpha>0\), be a focus of \(E\) and \(AB\) be a chord passing through \(P\). Then the locus of the mid point of \(AB\) is :

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \( \frac{\sqrt3}{2} \) and its directrices are \( x=\pm \frac{4\sqrt6}{3} \). Let \( H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :

Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x-y=3\) is:

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue ball and one green ball is:

Let the mean and the variance of seven observations \(2,4,\alpha,8,\beta,12,14\), \( \alpha < \beta \), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha+2\) and \(2\beta+1\) is :

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

The sum \( 1 + \frac{1}{2}(1^2+2^2) + \frac{1}{3}(1^2+2^2+3^2) + \ldots \) upto \(10\) terms is equal to:

Let \( S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0 \}. \) Then \( \displaystyle \sum_{z \in S} z^8 \) is equal to:

Consider the quadratic equation \( (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, \; n \in \mathbb{R}. \) Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is:

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3} \). Then \( f \) is:

Decomposition of a hydrocarbon follows the equation} \[ k = (5.5 \times 10^{11}\, s^{-1})\, e^{\frac{-28000}{T}} \] The activation energy of reaction is _____ kJ mol\(^{-1}\). (Nearest Integer)} Given: \(R = 8.3\, J\,K^{-1}\,mol^{-1}\)}

In a closed flask at 600 K, one mole of \(X_2Y_4(g)\) attains equilibrium as given below :} \[ X_2Y_4(g) \rightleftharpoons 2XY_2(g) \] At equilibrium, 75% \(X_2Y_4(g)\) was dissociated and the total pressure is 1 atm. The magnitude of \(\Delta G^\circ\) (in kJ mol\(^{-1}\)) at this temperature is _____. (Nearest Integer)}

500 mL of 0.2 M \(MnO_4^-\) solution in basic medium when mixed with 500 mL of 1.5 M KI solution, oxidises iodide ions to liberate molecular iodine. This liberated iodine is then titrated with a standard \(M\) thiosulphate solution in presence of starch till the end point. If 300 mL of thiosulphate was consumed, the value of \(x\) is _____.

An alkane (Y) requires 8 moles of oxygen for complete combustion and on chlorination with \(Cl_2/h\nu\), (Y) gives only one monochlorinated product (Z). The total number of primary carbon atoms in (Y) is _____.

An excess of \(AgNO_3\) is added to \(100\) mL of a \(0.05\) M solution of tetraaquodichloridochromium (III) chloride. The number of moles of \(AgCl\) precipitated will be ______\(\times 10^{-3}\). (Nearest integer)}

Treatment of a gas \(X\) with a freshly prepared ferrous sulphate solution gives a compound \(Y\) as a brown ring. The compounds \(X\) and \(Y\) are :

Consider the following reactions. Total number of electrons in the \(\pi\)-bonds and lone pair of electrons in the product (X) is :}

The number of compounds from the following which can undergo reaction with \(Br_2/KOH\) (alcoholic) to give respective products and these products can also be obtained separately by Gabriel phthalimide reaction is :}

Consider the following organic reaction sequence. Choose the final product (X) from the following (consider the major product in all intermediate reactions).

Consider the following reaction}

Statement I:} In the above reaction, product formed will be a mixture of benzyl alcohol and iodobenzene.
Statement II:} In the above reaction, the \(-O-CH_2-\) bond is cleaved to give the product. Choose the correct answer.

An optically active alkyl bromide \(C_4H_9Br\) reacts with ethanolic KOH to form major compound [A] which reacts with bromine to give compound [B]. Compound [B] reacts with ethanolic KOH and sodamide to give compound [C]. One molecule of water adds to compound [C] on warming with mercuric sulphate and dilute sulphuric acid at \(333\,K\) to form compound [D]. The functional group in compound D is confirmed by :

\(X\) is the product obtained by hydrolysis of prop-1-yne in the presence of mercuric sulphate under dilute acidic medium. \(Y\) is obtained by the reaction of ethanenitrile with methyl magnesium bromide in dry ether followed by hydrolysis. IUPAC name of product obtained from \(X\) and \(Y\) in presence of barium hydroxide followed by heating is :