The value of \(k\) for which the function \[ f(x)= \begin{cases} x^2\sin\left(\frac{1}{x}\right), & x\neq 0,\\ k(x+1), & x=0 \end{cases} \] is a continuous function, is:
\( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ? (i) \( F(x, y) = 3x + 2y \) (ii) \(F(x, y) = \sin \frac{y}{x} + \log y - \log x\)(iii) \( F(x, y) = e^{y/x} + 1 \) (iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)}
If \(E\) and \(F\) are two independent events such that \[ P(E)=\frac{3}{10}, \qquad P(E\cup F)=\frac{1}{2}, \] then \[ P(E|F)-P(F|E) \] is equal to:
Find: \[ \int \frac{\cos x}{(2+\sin x)(4+\sin x)}\,dx \]
Find a particular solution of the differential equation \[ (x+1)\frac{dy}{dx}=2e^{-y}-1, \] given that \[ y=0 \quad \text{when} \quad x=0. \]
If \[ (\sin x)^y=y^{\cos x}, \] then find \[ \frac{dy}{dx}. \]
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII.Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination.Based on the above information, answer the following questions. (i). Find the probability that a student selected at random is a regular student.
The principal value of \[ \sec^{-1}(\sqrt{2})+2\cosec^{-1}(-\sqrt{2}) \] is:
If \[ f(x)= \begin{cases} \dfrac{\sin x}{x}+\cos x, & x\neq 0,\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then the value of \(k\) is:
If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}, \] then the values of \(p\) and \(q\) are:
Assertion (A): A function \[ f:\mathbb{N}\to\mathbb{N} \] given by \[ f(x)=x^3+2,\quad \forall x\in\mathbb{N} \] is one-one but not onto. Reason (R): Since, for every \(y\in\mathbb{N}\) (codomain), there does not exist \[ x=(y-2)^{1/3}\in\mathbb{N} \] (domain) such that \[ f(x)=x^3+2=y. \]
Evaluate: \[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) +\cot^{-1}\left(\frac{1}{\sqrt{3}}\right) +\tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) +\tan^{-1}\left(\tan\frac{2\pi}{3}\right). \]
Show that the function \[ f(x)= \begin{cases} \dfrac{\cos x}{\frac{\pi}{2}-x}, & x\neq \frac{\pi}{2},\\ 1, & x=\frac{\pi}{2} \end{cases} \] is continuous at \[ x=\frac{\pi}{2}. \]
Find whether the function \[ f(x)= \begin{cases} x-1, & x<2,\\ 2x-3, & x\geq 2 \end{cases} \] is differentiable at \(x=2\) or not.
Let \[ A=\mathbb{R}-\{3\} \] and \[ B=\mathbb{R}-\{1\}. \] A function \(f:A\to B\) is defined by \[ f(x)=\frac{x-2}{x-3}. \] Find whether \(f\) is one-one and onto.
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:
