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Find the shortest
distance between lines

and.

If the two lines $x+(a-1)y=1$ and $2x+a^{2}y=1$ $(a\in R-\{0,1\left\}\right)$ are perpendicular, then the distance of their point of intersection from the origin is :

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{|[|x|-5]|-5}}$ is

(where [.] denotes the greatest integer function)

If $f\left(x\right)=\{\begin{array}{cc}\frac{1-\sqrt{2}\sin x}{\pi -4x},& x\ne \frac{\pi }{4}\\ a,& x=\frac{\pi }{4}\end{array}$ is continuous at $x=\frac{\pi }{4},$ then value of $a$ is

If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial coefficients respectively in $(1+x{)}^{2020},$ then

Evaluate $\int \frac{(4x+1)dx}{x^2+3x+2}$.

Prove that,

1.3+3.5+5.7+.............+(2n-1)(2n+1)=(n(4n²+6n-1))/3.

If $n\left(A\right)=52$, $n(A\cup B)=80$, $n(A\cap B)=31,$ then $n(A\cap B^{\prime })=$

⁸C_r = ⁸C_p. So

If $\underset{0}{\overset{\infty }{\int }}\frac{dx}{x^{3/2}+1}=\frac{a\pi }{b^{3/2}},$ where $\text{gcd}(a,b)=1,$ then

Write the value of ${lim}_{x\to 0^{-}}\frac{\sin \left[x\right]}{\left[x\right]}.$

Given $\int$ sinx dx = -cosx and $\int$ cosx dx = sinx. If f(x) = 36 $\int$ [sin (2x) + cos (3x)] dx, then find the value of -$f\left(\pi \right)$ [ take constant of integration equal to zero]

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

$\begin{array}{cc}Monthlyconsumption\left(inunits\right)& Numberofconsumers\\ 65-85& 4\\ 85-105& 5\\ 105-125& 13\\ 125-145& 20\\ 145-165& 14\\ 165-185& 8\\ 185-205& 4\end{array}$

If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle 2 $\theta$ such that $|\overrightarrow{a}-\overrightarrow{b}|$ <1 and 0 $\le \theta \le \pi$, then $\theta$ lies in the interval

The set $\{6,36,216,1296\}$ can be written in Set builder form as:

The value of the angle ${\tan }^{–1}(\tan {65}^{\circ }-2\tan {40}^{\circ })$ in degrees is equal to

Let f:$[0,\frac{\pi }{2}]\to [0,1]$ be a differentiable function such that $f\left(0\right)=0,f\left(\frac{\pi }{2}\right)=1,$ then

The parametric coordinates of the point $(8,3\sqrt{3})$ on the hyperbola $9x^2-16y^2=144$ is

The range of ${\sin }^{-1}\left(2^x\right)$ is

If for the differential equation $y^1=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$ the general solution is $y=\frac{x}{log|Cx|}$ then $\varphi \left(\frac{x}{y}\right)$is given by

The maximum value of $y=6x-x^2-5$ is

The ratio of area of incircle and circumcircle of quadrilateral formed by lines $x=1,x=5,y=-1,y=3$ is

If $I=\underset{0}{\overset{x}{\int }}\left[\sin t\right]dt$, where $x\in (2n\pi ,(2n+1\left)\pi \right),n\in N$ and $[\cdot ]$ denotes the greatest integer function, then the value of $I$ is

The value of $\underset{n\to \infty }{lim}\frac{1\cdot 2+2\cdot 3+3\cdot 4+\cdots +n(n+1)}{n^3}$ is

A hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its conjugate hyperbola. The foci points of both hyperbolas are connected as shown. Then $S_1{S}_3{S}_2{S}_4$ always forms a

At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $25$ more workers, then the new level of production of items is :

The equations of the assymptotes of the hyperbola $3x^2+10\mathrm{xy}+8y^2+14x+22y+7=0$ are .

The principal value of $si{n}^{-1}\left[sin\left(\frac{2\pi }{3}\right)\right]$
[IIT 1986]

The Range of The function $f\left(x\right)={\sec }^{-1}x$ is .

The range of the function $f\left(x\right)=\frac{x+2}{|x+2|}$, $x\ne -2$ is