List of top Questions asked in JEE Main

If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.

There is a weak base 'B' having $\text{pK}_b = 5.691$ of molarity 0.02M. When 0.02M HCl solution has been added, then pH of resultant buffer solution has been found to be 9. Take total volume of resultant buffer solution to be 100 ml. Find the value of 'x' & 'y', where 'x' is volume of HCl solution in ml & 'y' is volume of 'B' solution in ml. Given $\log(5) = 0.691$

Which of the following graph is correct between log P$_{CO_2}$ vs log X$_{CO_2}$? [given P$_{CO_2}$ = Partial Pressure of CO$_2$, X$_{CO_2}$ = Mole fraction of CO$_2$ in solution]

If in the expansion of \( (1 + x^2)^2(1 + x)^n \), the coefficients of \( x \), \( x^2 \), and \( x^3 \) are in arithmetic progression, then the sum of all possible values of \( n \) (where \( n \geq 3 \)) is:

The coefficient of \( x^{48} \) in the expansion of \[ 1 + (1+x) + 2(1+x)^2 + 3(1+x)^3 + \dots + 100(1+x)^{100} \] is

In the binomial expansion of \( (ax^2 + bx + c)(1 - 2x)^{26} \), the coefficients of \( x, x^2 \), and \( x^3 \) are -56, 0, and 0 respectively. Then, the value of \( (a + b + c) \) is

For given vectors \( \mathbf{a} = -\hat{i} + \hat{j} + 2\hat{k} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \), where \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \), then the value of \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} \) is:

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

If three vectors are given as shown. If the angle between vectors \( \mathbf{p} \) and \( \mathbf{q} \) is \( \theta \) where \( \cos \theta = \frac{1}{\sqrt{3}} \), \( |\mathbf{p}| = 2 \), and \( |\mathbf{q}| = 2 \), then the value of \( |\mathbf{p} \times (\mathbf{q} - 3\mathbf{r})|^2 - 3|\mathbf{r}|^2 \) is:

Let the lines \[ L_1:\ \vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k),\ \lambda\in\mathbb R \] \[ L_2:\ \vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k),\ \mu\in\mathbb R \] intersect at the point $R$. Let $P$ and $Q$ be the points lying on the lines $L_1$ and $L_2$ respectively, such that \[ |PR|=\sqrt{29}\quad \text{and}\quad |PQ|=\sqrt{\frac{47}{3}}. \] If the point $P$ lies in the first octant, then find $27(QR)^2$.

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

\[ \left(\frac{1}{^{15}C_0}+\frac{1}{^{15}C_1}\right) \left(\frac{1}{^{15}C_1}+\frac{1}{^{15}C_2}\right) \cdots \left(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}\right) = \frac{\alpha^{13}}{^{14}C_0\cdot {}^{14}C_1\cdot {}^{14}C_2\cdots {}^{14}C_{12}} \] If so, then find the value of \(30\alpha\).

Let \( S \) be the number of 4-digit numbers \( abcd \), where \[ a>b>c>d \] and let \( P \) be the number of 5-digit numbers \( abcde \), where the product of digits is 20. Find \( S + P \):

Number of 4 letter words with or without meaning formed from the letters of the word PQRSTTUVV is:

If \( A = \{ 1, 2, 3, 4, 5, 6 \}, B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \), then the number of strictly increasing functions from \( A \to B \) such that \( f(i) \neq i \) for \( i = 1, 2, 3, 4, 5, 6 \) is

Let \( y^2 = 16x \), from point \( (16, 16) \) a focal chord is passing. Point \( (\alpha, \beta) \) divides the focal chord in the ratio 2:3, then the minimum value of \( \alpha + \beta \) is:

Consider an ellipse \[ E_1:\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \ (a>b) \quad \text{and} \quad E_2:\ \frac{x^2}{A^2}+\frac{y^2}{B^2}=1 \ (B>A), \] where $e=\dfrac{4}{5}$ for both the curves and $\ell_1$ is the length of latus rectum of $E_1$ and $\ell_2$ is the length of latus rectum of $E_2$. Let the distance between the foci of the first curve be $8$. Find the distance between the foci of the second curve. (Given $2\ell_1^2=9\ell_2$).

Ellipse \( E: \frac{x^2}{36} + \frac{y^2}{25} = 1 \), A hyperbola confocal with ellipse \( E \) and eccentricity of hyperbola is equal to 5. The length of latus rectum of hyperbola is, if principle axis of hyperbola is x-axis?