List of top Mathematics Questions asked in JEE Main

Let
\(A=[a_{ij}]\)
be a \(3\times3\) matrix, where
\(a_{ij}=\left\{\begin{matrix}1&,&\text{if } i=j\\-x&,&\text{if } |i-j|=1\\2x+1&,&\text{otherwise.}\end{matrix}\right.\)
Let a function \(f:\mathbb{R}\to\mathbb{R}\) be defined as \(f(x)=\det(A)\). Then the sum of maximum and minimum values of \(f\) on \(\mathbb{R}\) is equal to:

If \(\alpha\) and \(\beta\) are the distinct roots of the equation
\(x^2+(3)^{1/4}x+3^{1/2}=0\)
, then the value of \(\alpha^{96}\left(\alpha^{12}-1\right)+\beta^{96}\left(\beta^{12}-1\right)\) is equal to :

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are :

The value of the integral
\(\displaystyle\int_{-1}^{1}\log_e\left(\sqrt{1-x}+\sqrt{1+x}\right)dx\)
is equal to:

Let \(a\) be a positive real number such that
\(\displaystyle\int_0^a e^{x-[x]}\,dx = 10e-9\)
where \([x]\) is the greatest integer less than or equal to \(x\). Then \(a\) is equal to:

Let \(f : [-3, 1] \to R\) be given as
\(f(x) = \begin{cases} \min\{(x + 6), x^2\}, & -3 \le x \le 0 \\ \max\{\sqrt{x}, x^2\}, & 0 \le x \le 1. \end{cases}\)
If the area bounded by \(y = f(x)\) and x-axis is A, then the value of \(6A\) is equal to ___________.

Let \(f : [-1, 1] \to R\) be defined as \(f(x) = ax^2 + bx + c\) for all \(x \in [-1, 1]\), where \(a, b, c \in R\) such that \(f(-1) = 2\), \(f'(-1) = 1\) for \(x \in (-1, 1)\) the maximum value of \(f''(x)\) is \(\frac12\). If \(f(x) \le \alpha\), \(x \in [-1, 1]\), then the least value of \(\alpha\) is equal to _________.

Let the coefficients of third, fourth and fifth terms in the expansion of
\(\left(x + \frac{a}{x^2}\right)^n,\ x \ne 0\)
be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to ___________.

Let \(\vec{x}\) be a vector in the plane containing vectors
\(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\)
and
\(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\).
If the vector \(\vec{x}\) is perpendicular to \(\left(3\hat{i} + 2\hat{j} - \hat{k}\right)\) and its projection on \(\vec{a}\) is \(\frac{17\sqrt{6}}{2}\), then the value of \(|\vec{x}|^2\) is equal to __________.

Let
\(I_n = \int_1^e x^{19}(\log|x|)^n\,dx\)
where \(n \in N\). If \((20)I_{10} = \alpha I_9 + \beta I_8\), for natural numbers \(\alpha\) and \(\beta\), then \(\alpha - \beta\) equals to ___________.

Consider a set of \(3n\) numbers having variance 4. In this set, the mean of first \(2n\) numbers is 6 and the mean of the remaining \(n\) numbers is 3. A new set is constructed by adding 1 into each of first \(2n\) numbers, and subtracting 1 from each of the remaining \(n\) numbers. If the variance of the new set is \(k\), then \(9k\) is equal to __________.

Let \(\tan\alpha\), \(\tan\beta\) and \(\tan\gamma\); \(\alpha, \beta, \gamma \ne \frac{(2n-1)\pi}{2}\), \(n \in N\) be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of \(\Delta ABC\) coincides with origin and its orthocentre lies on y-axis, then the value of
\(\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos\alpha\cos\beta\cos\gamma}\right)^2\)
is equal to ____________.

If 1, \(\log_{10}(4^x - 2)\) and \(\log_{10}\left(4^x + \frac{18}{5}\right)\) are in arithmetic progression for a real number x, then the value of the determinant
\(\begin{vmatrix} 2\left(x - \frac12\right) & x - 1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}\)
is equal to ___________.

Let
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)
and
\(B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \ne \begin{bmatrix} 0 \\ 0 \end{bmatrix}\)
such that \(AB = B\) and \(a + d = 2021\), then the value of \(ad - bc\) is equal to ___________.

The value of

\(\sum_{r=0}^{6}\left({}^{6}C_r\cdot{}^{6}C_{6-r}\right)\)

is equal to :

If the Boolean expression \((p \wedge q) \odot (p \otimes q)\) is a tautology, then \(\odot\) and \(\otimes\) are respectively given by :

Let the tangent to the circle \(x^2 + y^2 = 25\) at the point \(R(3, 4)\) meet x-axis and y-axis at points \(P\) and \(Q\), respectively. If \(r\) is the radius of the circle passing through the origin \(O\) and having centre at the incentre of the triangle \(OPQ\), then \(r^2\) is equal to :

If the equation of plane passing through the mirror image of a point \((2, 3, 1)\) with respect to line
\(\frac{x+1}{2} = \frac{y-3}{1} = \frac{z+2}{-1}\)
and containing the line
\(\frac{x-2}{3} = \frac{1-y}{2} = \frac{z+1}{1}\)
is \(\alpha x + \beta y + \gamma z = 24\), then \(\alpha + \beta + \gamma\) is equal to :

If \(x,y,z\) are in arithmetic progression with common difference \(d\), \(x\ne 3d\), and the determinant of the matrix

\(\begin{bmatrix}3&4\sqrt{2}&x\\4&5\sqrt{2}&y\\5&k&z\end{bmatrix}\)

is zero, then the value of \(k^2\) is :

Let \(y=y(x)\) be the solution of the differential equation

\(\cos x\,(3\sin x+\cos x+3)\,dy=\left(1+y\sin x\,(3\sin x+\cos x+3)\right)dx,\ 0\le x\le\frac{\pi}{2},\ y(0)=0\)

. Then,

\(y\left(\frac{\pi}{3}\right)\)

is equal to :

Let \(L\) be a tangent line to the parabola \(y^2=4x-20\) at \((6,2)\). If \(L\) is also a tangent to the ellipse

\(\frac{x^2}{2}+\frac{y^2}{b}=1\)

, then the value of \(b\) is equal to :

Two tangents are drawn from a point \(P\) to the circle \(x^2+y^2-2x-4y+4=0\), such that the angle between these tangents is

\(\tan^{-1}\left(\frac{12}{5}\right)\)

, where

\(\tan^{-1}\left(\frac{12}{5}\right)\in(0,\pi)\)

. If the centre of the circle is denoted by \(C\) and these tangents touch the circle at points \(A\) and \(B\), then the ratio of the areas of \(\Delta PAB\) and \(\Delta CAB\) is :

The value of

\(\lim_{n\to\infty}\frac{[r]+[2r]+\dots+[nr]}{n^2}\)

, where \(r\) is a non-zero real number and \([r]\) denotes the greatest integer less than or equal to \(r\), is equal to :

Consider the function \(f:\mathbb{R}\to\mathbb{R}\) defined by

\(f(x)=\begin{cases}\left(2-\sin\left(\frac{1}{x}\right)\right)|x|, & x\ne 0\\ 0, & x=0\end{cases}\)

. Then \(f\) is :

Let O be the origin. Let

\(\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}\) and \(\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k}\), \(x,y\in\mathbb{R}\), \(x\gt 0\), be such that \(\left|\overrightarrow{PQ}\right|=\sqrt{20}\) and the vector \(\overrightarrow{OP}\) is perpendicular to \(\overrightarrow{OQ}\). If \(\overrightarrow{OR}=3\hat{i}+z\hat{j}-7\hat{k}\), \(z\in\mathbb{R}\), is coplanar with \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\), then the value of \(x^2+y^2+z^2\) is equal to :