Given : A circle, $2x^2+2y^2=5$and a parabola, $y^2=4\sqrt{5}x$.Statement - I : An equation of a common tangent to these curves is $y=\text{x+}\sqrt{5}$Statement - II : If the line, $y=mx+\frac{\sqrt{5}}{m}\left(m\ne 0\right)$is their common tangent, then m satisfies $m^4-3m^2+2=0.$

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1), (1, 1) and (1, 0) is

The area (in square units) bounded by the curves $y=\sqrt{x},2y-x+3=0$, x-axis, and lying in the first quadrant is

The intercepts on x-axis made by tangents to the curve, $y={\int }_0^x\left|t\right|dt,x\in R,$which are parallel to the line $y=2x$, are equal to

If x, y, z are in A.P. and $ta{n}^{-1}x,ta{n}^{-1}yandta{n}^{-1}z$are also in A.P., then

If $\int f\left(x\right)dx=\psi \left(x\right)$, then $\int x^{5}f(x^3\text{)}dx$is equal to

The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$, and having centre at (0, 3) is

A ray of light along $x+\sqrt{3}y=\sqrt{3}$gets reflected upon reaching x-axis, the equation of the reflected rays is