Let \(n\) be a fixed positive integer. A relation \(R\) is defined on the set \(\mathbb{Z}\) by \[ R=\{(x,y):(x-y)\text{ is divisible by }n,\; x,y\in\mathbb{Z}\}. \] Determine whether \(R\) is an equivalence relation.
Find the shortest distance between the lines \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k}. \]
<div>Derivative of\[\(\cos^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),\)\quad -\frac{\pi}{4}\(<x<\)\frac{\pi}{4}\]with respect to \(x\) is:</div>
The order and degree of the differential equation \[ 1+\left(\frac{d^3y}{dx^3}\right)^3 = \lambda \frac{d^2y}{dx^2} \] is:
If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is:
Assertion (A): A relation \(R\) on the set \(\{1,2,3\}\) defined as \[ R=\{(1,1),(1,2),(2,1),(2,2),(3,3)\} \] is an equivalence relation. Reason (R): A relation that is reflexive, symmetric and transitive is an equivalence relation.
Determine the values of \(x\) for which \[ f(x)=\frac{x-3}{x+1}, \quad x\neq -1 \] is an increasing function.
If \[ y\sqrt{x^2+1}=\log\left(\sqrt{x^2+1}-x\right), \] show that \[ (x^2+1)\frac{dy}{dx}+xy+1=0. \]
Sketch the graph defined by \[ \left\{(x,y): \frac{x^2}{25}+\frac{y^2}{25}=1\right\}. \] Find the area of the region of the minor segment cut off by the line \[ x=\frac{5}{2}, \] using integration.