List of top Mathematics Questions asked in JEE Main

Let $\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{q} = \hat{i} + 2\hat{j} + \hat{k}$ be two vectors. If a vector $\vec{r} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ is perpendicular to each of the vectors $(\vec{p} + \vec{q})$ and $(\vec{p} - \vec{q})$, and $|\vec{r}| = \sqrt{3}$, then $|\alpha| + |\beta| + |\gamma|$ is equal to __________.

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is __________.

Let $M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{ \pm 3, \pm 2, \pm 1, 0 \} \right\}$. Define $f: M \to \mathbb{Z}$ as $f(A) = \det(A)$, for all $A \in M$, where $\mathbb{Z}$ is the set of all integers. Then the number of $A \in M$ such that $f(A) = 15$ is equal to __________.

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is $100k$, then k is equal to __________.

Consider the following frequency distribution:

If the sum of all frequencies is 584 and median is 45, then $|\alpha - \beta|$ is equal to __________

Let $S = \left\{ n \in \mathbb{N} : \begin{pmatrix} 0 & i \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \forall a, b, c, d \in \mathbb{R} \right\}$, where $i = \sqrt{-1}$. Then the number of 2-digit numbers in the set S is __________.

The values of a and b, for which the system of equations $2x + 3y + 6z = 8$ $x + 2y + az = 5$ $3x + 5y + 9z = b$ has no solution, are :

Let the vectors $(2 + a + b)\hat{i} + (a + 2b + c)\hat{j} - (b + c)\hat{k}$, $(1 + b)\hat{i} + 2\hat{j} - b\hat{k}$ and $(2 + b)\hat{i} + 2\hat{j} + (1 - b)\hat{k}$, $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true ?

If $\alpha, \beta$ are roots of the equation $x^2 + 5(\sqrt{2})x + 10 = 0, \alpha>\beta$ and $P_n = \alpha^n - \beta^n$ for each positive integer n, then the value of $\frac{P_{17}P_{20} + 5\sqrt{2} P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2} P_{18}^2}$ is equal to __________.

Let $y=y(x)$ be solution of the following differential equation $e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0, y(\pi/2) = 0$. If $y(0) = \log_e(\alpha + \beta e^{-2})$, then $4(\alpha + \beta)$ is equal to __________.

If the value of $\left( 1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + ....... \text{upto } \infty \right)^{\log_{0.25} \left( \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + ....... \text{upto } \infty \right)}$ is $l$, then $l^2$ is equal to __________.

The term independent of '$x$' in the expansion of $\left( \frac{x+1}{x^{2/3}-x^{1/3}+1} - \frac{x-1}{x-x^{1/2}} \right)^{10}$, where $x \neq 0, 1$ is equal to __________.

The sum of all values of $x$ in $[0, 2\pi]$, for which $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$, is equal to :

The Boolean expression $(p \implies q) \wedge (q \implies \sim p)$ is equivalent to :

The value of the definite integral $\int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt[3]{\tan 2x}}$ is :

The number of real roots of the equation $e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$ is :

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + ....... + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$, then the value of $\gamma$ is :

Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability that $B_3$ contains exactly 3 balls is $k \left(\frac{3}{4}\right)^9$ then $k$ lies in the set :

The locus of the centroid of the triangle formed by any point P on the hyperbola $16x^2 - 9y^2 + 32x + 36y - 164 = 0$, and its foci is :

Let $f(x) = 3 \sin^4 x + 10 \sin^3 x + 6 \sin^2 x - 3, x \in [-\frac{\pi}{6}, \frac{\pi}{2}]$. Then, $f$ is :

Let a parabola $P$ be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $O(0, 0)$ to the parabola $P$ which meet $P$ at $S$ and $R$, then the area (in sq. units) of $\Delta SOR$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 1 + xe^{y-x}, -\sqrt{2}<x<\sqrt{2}, y(0) = 0$. Then, the minimum value of $y(x), x \in (-\sqrt{2}, \sqrt{2})$ is equal to :

Let an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a^2>b^2$, passes through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha, 0), \alpha>0$, of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q$, then $PQ^2$ is equal to :

Let $\vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k}$, $\vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k}$ and $\vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b} = -1$ and $\vec{b} \cdot \vec{c} = 10$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to ________.

If $ A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} $ and $ M = A + A^2 + A^3 + \cdots + A^{20} $, then the sum of all the elements of the matrix $ M $ is equal to ________.