Property 3

Q. 15. Solve : 2[x]=4{x}+x where [] is gif and {} is fractional part function

Q. Find the integral $\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx$.

1. $0$
2. $1$
3. $2$
4. $\infty$

Q. Show that the function $f$ defined by $f\left(x\right)=\{\begin{array}{cc}2x+3& \text{if}-3\le x<-2\\ x+1& \text{if}-2\le x<0\\ -x+2& \text{if}0\le x\le 1\end{array}$ is not differentiable at $x=-2$ and $x=0.$

Q. If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ

Q. The value of ${\int }_0^2\left(x\right\}dx$, {x} denotes fractional part of x is

Q. Find the value of the following:
${\sin }^2\frac{\pi }{4}+{\cos }^2\frac{\pi }{4}-{\tan }^2\frac{\pi }{3}$

Q. The value of $\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx$ is :

1. $0$
2. $\frac{2}{3}$
3. $\frac{4}{3}$
4. $-\frac{4}{3}$

Q. ${\int }_0^a$ $\ln (\cot a+\tan x)dx$ = , where $aϵ(0,\frac{\pi }{2})$ is

1. $a\ln \left(\sin a\right)$
2. $–a\ln \left(\sin a\right)$
3. $–a\ln \left(\cos a\right)$
4. none of these

Q.

arg(z1*z2) = arg(z1) + arg(z2)

Q. find the derivative of log(log(x+1)) with respect to x

Q. If $\underset{}{\int }(x^4+3x+2)dx=\frac{x^5}{a}+\frac{3x^2}{b}+cx+K$, then the value of $a+b+c$ is (where $K$ is constant of integration and $a,b$ and $c$ are fixed constants)

Q. Prove that:
$\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{7\pi }{15}=\frac{1}{16}$

Q. The value of ${\int }_{-1}^{2}f\left(x\right)\mathrm{dx},\mathrm{where}f\left(x\right)=\left(x+1\right|+\left(x\right|+\left(x-1\right|$ is

1. $\frac{9}{2}$
2. $\frac{19}{2}$
3. $\frac{15}{2}$
4. $\frac{21}{2}$

Q. ${\int }_1^{\infty }\frac{1}{x^2}dx$ does not have a finite value

1. False
2. True

Q. f(x)=(x+3)/(x+2) prove that x=[2f(x)-3]/[1-f(x)]