List of top Questions asked in JEE Main

Let \( f(x) = \begin{cases} x^3 + 8 & x < 0 \\ x^2 - 4 & x \ge 0 \end{cases} \) and \( g(x) = \begin{cases} (x-8)^{1/3} & x < 0 \\ (x+4)^{1/2} & x \ge 0 \end{cases} \) then find number of points of discontinuity of \( g(f(x)) \).

Let \(O\) be the vertex of the parabola \(y^2 = 4x\). Let \(P\) and \(Q\) be two points on parabola such that chords \(OP\) and \(OQ\) are perpendicular to each other. If the locus of mid-point of segment \(PQ\) is a conic \(C\), then latus rectum of \(C\) is

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are _________ and _________} respectively.

A point light source emits E.M. waves in free space. A detector, placed at a distance of $L$ m, measures the intensity as $I_0$. The detector is now shifted to another location on the same spherical surface ensuring the angle between original location and new location as $45^{\circ}$. The measured intensity at new location will be ————.

The energy released when $\frac{7}{17.13} \text{ kg}$ of ${}^7_3\text{Li}$ is converted into ${}^4_2\text{He}$ by proton bombardment is $\alpha \times 10^{32} \text{ eV}$. The value of $\alpha$ is ____. (Nearest integer)
(Mass of ${}^7_3\text{Li} = 7.0183 \text{ u}$, mass of ${}^4_2\text{He} = 4.004 \text{ u}$, mass of proton $= 1.008 \text{ u}$ and $1 \text{ u} = 931 \text{ MeV}/c^2$ and Avogadro number $= 6.0 \times 10^{23}$)

If \( 26 \left( \dfrac{2^3 \cdot \binom{12}{2}}{3} + \dfrac{2^5 \cdot \binom{12}{4}}{5} + \dots + \dfrac{2^{13} \cdot \binom{12}{12}}{13} \right) = 3^{13 - \alpha} \), then find the value of \( \alpha \).

If \(\int_0^{\pi/4} \left[ \cot(x - \frac{\pi}{3}) - \cot(x + \frac{\pi}{3}) + 1 \right] dx = \alpha \log(\sqrt{3} - 1)\) then \(9\alpha\) is

The mean & variance of \(x_1, x_2, x_3, x_4\) is 1 and 13 respectively and the mean and variance of \(y_1, y_2, \dots, y_6\) be 2 and 1 respectively, the variance of \(x_1, x_2, \dots, x_4, y_1, y_2, \dots, y_6\) will be

The number of values of \(z \in \mathbb{C}\) satisfying \(|z-4-8i| = \sqrt{10}\) and \(|z-3-5i| + |z-5-11i| = 4\sqrt{5}\) is

Locus of the mid-point of chord of circle \(x^2 + y^2 - 6x - 8y - 11 = 0\), subtending a right angle at the center is

Let \( \frac{x^2}{f(a^2 + 2a + 7)} + \frac{y^2}{f(3a + 14)} = 1 \) represent an ellipse. The major axis of the given ellipse is the y-axis and \( f \) is a decreasing function. If the range of \( a \) is \( R - [\alpha, \beta] \), then \( \alpha + \beta \) is:

A person has 3 different bags & 4 different books. The number of ways in which he can put these books in the bags so that no bag is empty, is

If \( \alpha = 3 + 4 + 8 + 9 + 13 + \dots \) up to 40 terms and \( (\tan \beta)^{\frac{\alpha}{1020}} \) is the root of the equation \( x^2 - x - 2 = 0 \), then the value of \( \sin^2 \beta + 3\cos^2 \beta \) is:

The line passing through point of intersection of \(3x + 4y = 1\) and \(4x + 3y = 1\) intersects axes at P and Q, then locus of midpoint of PQ is

Let \(\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}\), \(\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = 0\). If \(\vec{a} \cdot \vec{c} = 15\), then the value of \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\) is

A person goes to college either by bus, scooter or car. The probability that he goes by bus is \(\frac{2}{5}\), by scooter is \(\frac{1}{5}\) and by car is \(\frac{3}{5}\). The probability that he entered late in college if he goes by bus is \(\frac{1}{7}\), by scooter is \(\frac{3}{7}\) and by car is \(\frac{1}{7}\). If it is given that he entered late in college, then the probability that he goes to college by car is

Statement 1: \( f(x) = e^{|\sin x| - |x|} \) is differentiable for all \( x \in \mathbb{R} \).
Statement 2: \( f(x) \) is increasing in \( x \in \left( -\pi, -\frac{\pi}{2} \right) \).

If \((x\sqrt{1-x^2}) \, dy - (y\sqrt{1-x^2} - x^2 \cos^{-1} x) \, dx = 0\) and \(\lim_{x \to 1^-} y(x) = 1\), then \(y\left(\frac{1}{2}\right)\) is

If \(\alpha = 3\sin^{-1}\left(\frac{6}{11}\right)\) and \(\beta = 3\cos^{-1}\left(\frac{4}{9}\right)\), consider statements:
Statement 1: \(\cos(\alpha + \beta) > 0\)
Statement 2: \(\cos\alpha < 0\)
Then which of the following is true?

Which of the following statement is correct about Hoffmann’s bromamide degradation reaction?
(a) Alkyl amide do not react.
(b) Secondary amide do not form secondary amide.
(c) Ratio of NaOH and Br₂ is 4 : 2.
(d) Na₂CO₃, NaBr and H₂O also formed along with amine.

Product is:

Find total number of aromatic compounds from the given organic molecules.

Which of the following statement(s) are incorrect?

Statement-1: During fractional distillation, higher boiling point will condense first.
Statement-2: In fractionating column, one which has higher boiling point value will be more in concentration above the fractionating column.