Continuity of a Function

Q.

Find the value of $\sin {75}^{\circ }$.

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+x+5=0$, then $y=\sum {\alpha }^2+\alpha \beta \gamma$ satisfies the equation

1. $y^3+y+2=0$

2. $y^3–y^2–y–2=0$

3. $y^3+3y^2–y–3=0$

4. $y^3+4y^2+5y+20=0$

Q.

What is the integration of $cotx$?

Q.

Let $f\left(x\right)={\sin }^{-1}x$ and $g\left(x\right)=\frac{x^2-x-2}{2x^2-x-6}$. If $g\left(2\right)=\underset{x\to 2}{\lim }g\left(x\right)$, then then the domain of the function $f\circ g$is:

1. $(-\infty ,-2]\cup [\frac{-4}{3},\infty )$

2. $(-\infty ,-1]\cup [2,\infty )$

3. $(-\infty ,-2]\cup [-1,\infty )$

4. $(-\infty ,2]\cup [\frac{-3}{2},\infty )$

Q.

If $\mathrm{f}(\mathrm{x}+1)+\mathrm{f}(\mathrm{x}-1)=\sqrt{3}\mathrm{f}\left(\mathrm{x}\right)$ for all $x\in R$, then the period of$f\left(x\right)$ is

1. 3

2. 6

3. 12

4. 9

Q. $\underset{x\to a}{lim}\frac{{(a+2x)}^{\frac{1}{3}}-{\left(3x\right)}^{\frac{1}{3}}}{{(3a+x)}^{\frac{1}{3}}-{\left(4x\right)}^{\frac{1}{3}}}(a\ne 0)$ is equal to:

1. ${\left(\frac{2}{9}\right)}^{\frac{4}{3}}$
2. ${\left(\frac{2}{3}\right)}^{\frac{4}{3}}$
3. $\left(\frac{2}{3}\right){\left(\frac{2}{9}\right)}^{\frac{1}{3}}$
4. $\left(\frac{2}{9}\right){\left(\frac{2}{3}\right)}^{\frac{1}{3}}$

Q.

What is the integration of $\sin wt$?

Q. The function $f\left(x\right)=\left[x\right]$ where $\left[x\right]$ the greatest integer function is continuous at

1. $1.5$
2. $4$
3. $-2$
4. $1$

Q.

If $\int (e^{2x}+2e^x-e^{-x}-1)e^{\left(e^x+e^{-x}\right)}dx=g\left(x\right)e^{\left(e^x+e^{-x}\right)}+C$, where $C$ is a constant, then $g\left(0\right)$ is equal to

1. $2$

2. $e$

3. $1$

4. $e^2$

Q.

Find the value of $\sin {50}^{\circ }$.

Q.

If $a\times (b\times c)=(a\times b)\times c$, where $a,b,c$ are any three vectors such that $a\bullet b\ne 0,c\ne 0$, then $a,c$ are $?$

1. Inclined at an angle of $\frac{\pi }{6}$ between them

2. Perpendicular

3. Parallel

4. $\frac{\pi }{3}$Inclined at angle of between them

Q. Show that the function $f:R\to \{x\in R:-1<x<1\}$ defined by $f\left(x\right)=\frac{x}{1+|x|},x\in R$ is one one and onto function.

Q.

$2{\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{24}{25}\right)=$

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{2\mathrm{\pi }}{3}$

3. $\frac{5\mathrm{\pi }}{3}$

4. $Noneofthese$

Q.

If C is a point at which Rolle’s theorem holds for the function, $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathbf{log}}_{\mathbf{e}}\mathbf{\left(}\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{\alpha }}{\mathbf{7}\mathbf{x}}\mathbf{\right)}$ in the interval $\mathbf{[}\mathbf{3}\mathbf{,}\mathbf{}\mathbf{4}\mathbf{]}$, where$\mathbf{}\mathit{\alpha }\mathbf{\in }\mathbf{𝐑}$, then $\mathit{f}\mathbf{\prime }\mathbf{\prime }\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}$is equal to

1. $\frac{-1}{24}$

2. $\frac{\mathbf{-}\mathbf{1}}{\mathbf{12}}$

3. $\sqrt{\frac{3}{7}}$

4. $\frac{1}{12}$

Q. Which of the following is NOT a function?

1. $f:N\to N$ defined by $f\left(x\right)=2x+5$
2. $f:(-\infty ,0]\to [0,\infty )$ defined by $f\left(x\right)=\sqrt{-x}$
3. $f:Z\to N$ defined by $f\left(x\right)=x-100$
4. $f:N\to R$ defined by $f\left(x\right)=\frac{2x-5}{3}$

Q.

The domain of the function $f\left(x\right)=\sqrt{\cos x}$is

1. $\left[\frac{3\mathrm{\pi }}{2},2\mathrm{\pi }\right]$

2. $\left[0,\frac{\mathrm{\pi }}{2}\right]$

3. $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$

4. $\left[0,\frac{\mathrm{\pi }}{2}\right]\cup \left[\frac{3\mathrm{\pi }}{2},2\mathrm{\pi }\right]$

Q. Find $\underset{x\to 0}{lim}f\left(x\right)$ and $\underset{x\to 1}{lim}f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{cc}2x+3,& x\le 0\\ 3(x+1),& x>0\end{array}$

Q.

On the interval $\left(0,\frac{\pi }{2}\right)$ the function $\log \sin x$ is

1. Increasing

2. Decreasing

3. Neither increasing nor decreasing

4. None of the above

Q.

If $f:[1,\infty )\to [2,\infty )$is given by $f\left(x\right)=x+\frac{1}{x}$, then $f^{(-1)}\left(x\right)$ equals

Q.

If $\alpha$ and $\beta$are the roots of the equation $a{x}^2+bx+c=0,$ then the equation, whose roots are $2\alpha +3\beta$ and $3\alpha +2\beta$, is

1. $ac{x}^2+(a+c)bx-{(a+c)}^2=0$

2. $ac{x}^2–(a+c)bx-{(a+c)}^2=0$

3. $ab{x}^2–(a+b)cx-{(a+c)}^2=0$

4. None of these

Q. The value of $\underset{x\to 0}{lim}\frac{\sqrt{3+x}-\sqrt{3}}{x}$ is

1. $\frac{1}{\sqrt{3}}$
2. $\frac{1}{2\sqrt{3}}$
3. $\sqrt{3}$
4. $\frac{1}{3\sqrt{3}}$

Q.

The value of $\underset{x\to 0^{+}}{\lim }\frac{{\cos }^{-1}\left(x\right)·{\sin }^{-1}\left(x\right)}{x(1–x^2)}$ , where $\left[x\right]$ denotes the greatest integer $\le x$ is

1. $0$

2. $\frac{\mathrm{\pi }}{4}$

3. $\frac{\mathrm{\pi }}{2}$

4. $\mathrm{\pi }$

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$ is continuous at $x=0$, then the ordered pair $(p,q)$ is equal to:

1. $(-\frac{3}{2},\frac{1}{2})$
2. $(\frac{5}{2},\frac{1}{2})$
3. $(-\frac{1}{2},\frac{3}{2})$
4. $(-\frac{3}{2},-\frac{1}{2})$

Q.

For which, interval, the function$\frac{x^2-3x}{x-1}$ satisfies all the conditions of Rolle’s theorem

1. $[0,3]$

2. $[-3,0]$

3. $[1.5,3]$

4. For no interval

Q.

If $a,b,c$ are in $G.P$ then ${\log }_a\left(x\right),{\log }_b\left(x\right),{\log }_c\left(x\right)$ are in

Q.

If the value of the integral ${\int }_0^{\frac{1}{2}}\frac{x^2}{{(1-x^2)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}dx$ is $\frac{k}{6}$, then $k$ is equal to

1. $2\sqrt{3}+\mathrm{\pi }$

2. $3\sqrt{2}+\mathrm{\pi }$

3. $3\sqrt{2}-\mathrm{\pi }$

4. $2\sqrt{3}-\mathrm{\pi }$

Q. Let f(x) = $\{\begin{array}{cc}\frac{x^3+x^2-16x+20}{(x-2{)}^2},& ifx\ne 2\\ k,& ifx=2\end{array}.$ if f(x) be continuous for all x, then k =

1. 7
2. -7
3. $\pm 7$
4. None of these

Q.

The value of $\underset{x\to 0}{\lim }\frac{\left[{\sin }^2\left(x\right)+\cos \left(x\right)-1\right]}{x^2}$ is

1. $1$

2. $\frac{1}{2}$

3. $-\frac{1}{2}$

4. $0$

Q.

The domain of the real-valued function $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{5-4\mathrm{x}-{\mathrm{x}}^2}+{\mathrm{x}}^2\log \left(\mathrm{x}+4\right)$ is

1. $-5\le \mathrm{x}\le 1$

2. $-5\le \mathrm{x}\mathrm{and}\mathrm{x}\ge 1$

3. $-4<\mathrm{x}\le 1$

4. $\mathrm{\varphi }$

5. $0\le \mathrm{x}\le 1$

Q.

If $\tan \theta =\frac{\sin \left(\alpha \right)-\cos \left(\alpha \right)}{\sin \left(\alpha \right)+\cos \left(\alpha \right)}$, then $\sin \left(\alpha \right)+\cos \left(\alpha \right)$ and $\sin \left(\alpha \right)-\cos \left(\alpha \right)$ must be equal to

1. $\sqrt{2}\cos \left(\theta \right),\sqrt{2}\sin \left(\theta \right)$

2. $\sqrt{2}\sin \left(\theta \right),\sqrt{2}\cos \left(\theta \right)$

3. $\sqrt{2}\sin \left(\theta \right),\sqrt{2}\sin \left(\theta \right)$

4. $\sqrt{2}\cos \left(\theta \right),\sqrt{2}\cos \left(\theta \right)$