List of top Mathematics Questions on Miscellaneous asked in JEE Main

Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:

Let \[ f(x)=\int \frac{dx}{2\left(\frac{3}{2}\right)^x+2x\left(\frac12\right)^x} \] such that \(f(0)=-26+24\log_e(2)\). If \(f(1)=a+b\log_e(3)\), where \(a,b\in\mathbb{Z}\), then \(a+b\) is equal to:

Let \([\,\cdot\,]\) denote the greatest integer function. Then \[ \int_{-\pi/2}^{\pi/2} \frac{12(3+[x])}{3+[\sin x]+[\cos x]}\,dx \] is equal to:

The sum of all the elements in the range of
\[ f(x)=\operatorname{sgn}(\sin x)+\operatorname{sgn}(\cos x) +\operatorname{sgn}(\tan x)+\operatorname{sgn}(\cot x), \]
where
\[ x\neq \frac{n\pi}{2},\ n\in\mathbb{Z}, \]
and
\[ \operatorname{sgn}(t)= \begin{cases} 1, & t>0 \\ -1, & t<0 \end{cases} \]
is:

Given below are two statements: Statement I: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one-one. Statement II: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many-one. In the light of the above statements, choose the correct answer.

Let \[ A=\{z\in\mathbb{C}:|z-2|\le 4\} \quad\text{and}\quad B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}. \] Then the maximum value of \(|z_1-z_2|\), where \(z_1\in A\) and \(z_2\in B\), is:

Considering the principal values of inverse trigonometric functions, the value of \[ \tan\!\left(2\sin^{-1}\!\frac{2}{\sqrt{13}}-2\cos^{-1}\!\frac{3}{\sqrt{10}}\right) \] is equal to:

Three persons enter a lift at the ground floor. The lift will go up to the 10th floor. The number of ways in which the three persons can exit the lift at three different floors, if the lift does not stop at the 1st, 2nd and 3rd floors, is equal to _______.

If \[ \sum_{r=1}^{25}\left(\frac{r}{r^4+r^2+1}\right)=\frac{p}{q}, \] where \(p\) and \(q\) are positive integers such that \(\gcd(p,q)=1\), then \(p+q\) is equal to _______.

If \( A \) and \( B \) are binomial coefficients of the 30\(^\text{th}\) and 12\(^\text{th}\) terms of the binomial expansion \( (1 + x)^{2n-1} \), and \( 2A = 5B \), then the value of \( n \) is

The system of equations is given as: \[ (\lambda + 1)x + (\lambda + 2)y + (\lambda - 1)z = 0 \] \[ \lambda x + (\lambda - 1)y + (\lambda + 1)z = 0 \] \[ (\lambda - 1)x + (\lambda + 1)y + (\lambda + 2)z = 0 \] If the above system of equations has infinite solutions, then \( \lambda^2 + \lambda \) is:

Find the length of the chord whose midpoint is \( \left( \frac{3}{2}, 0 \right) \) of the ellipse \[ \frac{x^2}{2} + \frac{y^2}{4} = 1. \]