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1. 114.1

A line (with constant term $k$) perpendicular to $4x+3y+2=0$, is tangent to circle with integral radius (radius as integer). If the ratio of minimum to maximum possible distance from origin to such tangents is $1:3$, then possible value of $k$ such that the radius of circle is minimum

If the angle between the tangents drawn to the circle $x^2+y^2+2gx+2fy+c=0$ $(c>0)$ from the origin is $\frac{\pi }{2},$ then

Eight coins are tossed at a time, the probability of getting atleast 6 heads up, is

Arg z + Arg $\overline{z}(\overline{z}\notin R^{-})$

Which among the following equations represents a pair of straight lines?

Let $y=f\left(x\right)$ is a parabola of the form $y=x^2+ax+1$ and tangent to the parabola at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$.If it is known that no point of the parabola is below $x-$axis then the radius of circle when $a$ attains its maximum value in units is

If (sin A)/(sin B) = √3/2 , (cos A)/(cos B) = √5/2, then find the values of tan A and tan B.

An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value $-1$. The expected value of $X$, is

If a, b, c are sides of a triangle and $a^3,b^3,c^3$ are roots of $x^3-p{x}^2+qx-r=0$, then match the following list - I with list - II

$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \left(P\right)& si{n}^{3}A+si{n}^{3}B+si{n}^{3}C-3sinAsinBsinC=8D^3& 1.& p=14,q=5,r=8\\ \left(Q\right)& asi{n}^{2}A+bsi{n}^{2}B+csi{n}^{2}C=14D^2& 2.& p=36,q=7,r=2\\ \left(R\right)& sinAsinBsinC=2D^3& 3.& p=9,q=7,r=8\\ \left(S\right)& aco{s}^{2}A+bco{s}^{2}B+ccos2C=2(S-{\Delta }^2)& 4.& p=\frac{63}{2},q=5,r=27\\ & & 5.& p=1,q=8,r=8\end{array}$

(here $\Delta$ denotes area of triangle and S represents semi perimeter)

The value of $\sqrt{2}\int \frac{\sin xdx}{\sin (x-\frac{\pi }{4})}$ is

(where $C$ is constant of integration)

A number is chosen at random from among the first $50$ natural numbers. The probability that the number chosen is either a prime number or a mutiple of $5$ is

Let $f\left(x\right)$ be a real valued function, then the domain of $f\left(x\right)=\sqrt{x-4-2\sqrt{x-5}}-\sqrt{x-4+2\sqrt{x-5}}$ is

1. 7

If $x,y,z$ are the sides of pedal triangle of $△ABC,$ then $x+y+z$ is equal to

(For $△ABC,$ usual notations are used)

1. 2.614

If $g\left(x\right)=\underset{0}{\overset{x}{\int }}(|\sin t|+|\cos t|)dt$, then the value of $g(x+\frac{n\pi }{2})$,( where $n\in N$) is

If $5\cos \theta \sin \theta =1,$ then the value of ${\tan }^3\theta +{\cot }^3\theta$ is

The sum of all the solutions of the equation $|505x-1010|+|1515x+505|=2020$, is

Find the area enclosed by the parabola
4y = 3x² and the line 2y = 3x
+ 12

The value of $2{\sin }^2\frac{\pi }{6}+{\text{cosec}}^2\frac{7\pi }{6}\cdot {\cos }^2\frac{\pi }{3}$ is

The equation of the common tangent to the curves $y^2=8x$ and $xy=–1$ is

If $\underset{x\to \infty }{lim}{\left(\frac{(x-1)(x+3)}{x^2}\right)}^{4x}=e^P,$ find $P.$

The slope of the line which is inclined at an angle of ${75}^{\circ }$ with the $x-$axis in anticlockwise direction is

Choose the correct answer.
The probability of obtaining an even prime number on each die when a pair of dice is rolled is
(a) zero (b) $\frac{1}{3}$ (c) $\frac{1}{12}$ (d) $\frac{1}{36}$

Log6+ 2log5+log4-log3-log2=2

Prove it