List of practice Questions

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: 

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 \[ (x^2+1)\frac{dy}{dx}+xy+1=0. \] 

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A|B)=\frac{2}{3}, \] then \(P(A\cup B)\) is: