List of top Mathematics Questions asked in JEE Main

If the function \( f(x) \) is defined as \( f(x)=\begin{vmatrix}\frac{1}{1+e^{\sin x}} & \sin x \\ \sqrt{1-x^{2}} & 2\end{vmatrix} \). Then the value of \( \frac{1}{\pi}\int_{-\pi/2}^{\pi/2} f(x)\,dx \) is equal to ________.

The number of pairs \( (x, y) \) satisfying the equations \( \sin x+\sin y=\sin (x+y) \) and \( |x|+|y|=1 \) is ________.

The equations \(ax^{2}+bx+a=0\), \((a, b \in \mathbb{R})\) and \(x^{3}-2x^{2}+2x-1=0\) have two common roots. Then the value of \(a+b\) is ________.

Let \(A\) be a matrix satisfying \(A^{3}=3A+2I\) where \(A \neq nI\), \(n \in \mathbb{Z}\). If \(A^{2}+\alpha A+\beta I=O\), then maximum value of \(|\alpha+\beta|\) is ________.

An equilateral triangle is inscribed in the ellipse \(\frac{x^{2}}{3}+y^{2}=1\) such that one of the vertex of the triangle is \((0, 1)\) and one of the altitude of the triangle is along the \(y\)-axis. If A is the area of the equilateral triangle. Then \(625A^{2}\) is equal to ________.

Suppose \(\left|\begin{array}{cc}f'(x) & f(x) \\ f''(x) & f'(x)\end{array}\right|=0\), where \(f(x)\) is continuously differentiable function with \(f'(x) \neq 0\ \forall x \in \mathbb{R}\) and satisfies \(f(0)=1\) and \(f'(0)=4\). If \(f(x)=e^{\lambda x}+\mu\), where \(\lambda, \mu \in \mathbb{R}\) then the value of \(\lambda^{2}+\mu^{2}\) is

The volume of the greatest cylinder that can be inscribed in a cone of height \(\frac{3}{\sqrt[3]{\pi}}\) and semi-vertical angle \(45^{\circ}\) is ________.

The intercept on the line \(x=y\) by the circle \(x^{2}+y^{2}-2y=0\) is AB. The equation of the circle with AB as a diameter is

Let the area of the region enclosed between the two circles \(x^{2}+y^{2}=a^{2}\) and \((x-a)^{2}+y^{2}=a^{2}\) be \(\frac{m}{3}(4\pi-3\sqrt{3})\) sq. units for some \(a \geq 2\). If \(a\) is a root of \(x^{3}-3x^{2}-6x+8=0\), then

If \(\vec{a}=6 \hat{i}+2 \hat{j}-\lambda_{1} \hat{k}\), \(\vec{b}=3 \hat{i}+\left(3-\lambda_{2}\right) \hat{j}+2 \hat{k}\) and \(\vec{c}=2 \hat{i}+\left(1+\lambda_{3}\right) \hat{j}-3 \hat{k}\) are three vectors such that \(\vec{a}=3 \vec{c}\) and \(\vec{b}\) is perpendicular to \(\vec{c}\) then \(\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)\) is

If the foot of the perpendicular from the point \((0, 0, 0)\) on the line \(\frac{x+2}{5}=\frac{y+2}{2}=\frac{z+8}{3}\) is \((a, b, c)\), then \(a^{2}+b^{2}+c^{2}\) is

If the vectors \(\overrightarrow{AB}=3 \hat{i}+8 \hat{k}\) and \(\overrightarrow{AC}=5 \hat{i}-2 \hat{j}+6 \hat{k}\) are the sides of a triangle \(ABC\), then the length of the median through \(A\) is

Let \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) be observations with mean \(m\) and standard deviation \(s\). Then, which of the following statements are correct:
A. Mean of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(m+5k\)
B. Mean of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(km\).
C. The standard deviation of the observations \(x_{1}+k, x_{2}+k, x_{3}+k, x_{4}+k, x_{5}+k\) is \(s\).
D. The standard deviation of the observations \(kx_{1}, kx_{2}, kx_{3}, kx_{4}, kx_{5}\) is \(k^{5}s\).
Choose the correct answer from the options given below :

If \((1+\cos x) \frac{dy}{dx}-(y-3) \sin x=0\) and \(y(0)=0\), then the value of \(y\left(\frac{\pi}{2}\right)\) is

Suppose L is the line joining the points \((0, 0)\) and \((1, 2)\). If a line parallel to L meets the curve \(y=2x^{2}+3x+1\) only at a single point P, then the point P is

The value of \(\int_{0}^{1} \log _{e}\left(\frac{1}{x}-1\right) dx\) is

A manufacturer can sell \(x\) items at a price of ₹ \(\left(5-\frac{x}{100}\right)\) each. The cost price of \(x\) items is ₹ \(\left(\frac{x}{5}+500\right)\). The number of items he should sell to reach maximum profit is

If a straight line passing through point P \((3, -5)\) is such that its intercepted portion between the co-ordinate axes is bisected at P, then the equation of the line is

The first two terms of a G.P. are \( x^{-3} \) and \( x^{n} \), respectively. If \( x^{42} \) is the sixth term of the same progression, then \( n^{2} \) is equal to

The sum of all those terms of the arithmetic progression 2, 6, 10, 14, ....... , 598 which are not divisible by 3, is equal to

In how many ways can we select four cards from an ordinary pack of playing cards so that two of them are of the same denomination and the remaining two are of some other same denomination?

If the coefficient of \( x^{7} \) and \( x^{8} \) in the expansion of \( \left(2+\frac{x}{3}\right)^{n} \) are equal, then \( n \) is

A college awarded 38 medals for cricket, 15 for tennis and 20 for football. If these medals were bagged by a total of 58 students and only 3 students got medals for all the three games, then how many students received medals for exactly two of the three games?

If \( \alpha \) and \( \beta \) are the roots of \( ax^{2}+bx+c=0 \), then the value of \( \frac{1}{(a\alpha+b)^{2}}+\frac{1}{(a\beta+b)^{2}} \) is

Let \( A=\left[\begin{array}{cccc}1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 9 & 9 & 9 \\ 8 & 8 & 8 & 8\end{array}\right] \), then determinant of \( A= \)