Ellipses, Hyperbolas, Circles and Radius Quiz

What is the value of 'x' in the given equation xI?

In terms of escape velocity ve, the initial velocity 'vi' required to project a body vertically upward is described as vi = x * ve. What is the value of 'x'?

What is the r.m.s. value of the alternating current i given by i = 42sin(2πT) + 10?

If a compass needle oscillates 20 times per minute at a place with a dip of 30° and 30 times per minute where the dip is 60°, what is the ratio of the total magnetic field due to the earth at these two places?

The image of an object formed by a convex refracting surface is real and located at 2/3 of the distance of 3 times the object's distance from the surface. What is the wavelength of light inside the surface compared to air?

A particle slides down a frictionless track from rest at a height of 2m. If we are asked to find the variance of its final position, what could be a plausible way to calculate it based on the given information?

If the eccentricities of the ellipse and hyperbola are $e_1$ and $e_2$ respectively, and a point $(x, y)$ is on the ellipse, then the value of $k$ in the equation $15x^2 - 3y^2 = k$ is equal to:

A circle C passes through the points A(2, 1) and B(3, 4), and the line segment AB is not a diameter of C. If the center of C lies on the circle $(x - 5)^2 + (y - 1)^2 = 13$, then the value of $r^2$, where $r$ is the radius of C, is:

The sum of the first 10 terms of the series $1 + \tan^{-1}(1/3) + \tan^{-1}(1/7) + \tan^{-1}(1/13) + \tan^{-1}(1/21) + ...$ is S. The value of $\tan(S)$ is:

If the matrix $A = \begin{bmatrix} \sin^{-1}(1/5) & 1/4 \ \sin^{-1}(2) & 1/3 \end{bmatrix}$, then the determinant of $A$ lies in the interval:

What is the area (in square units) of the region ${(x, y) : 0 \leq y \leq x \leq 1, 0 \leq y \leq x \leq 2}$?

If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression and $a_1 + a_4 + a_7 + \dots + a_{16} = 114$, then the value of $a_1 + a_6 + a_{11} + a_{16}$ is:

Let $A = {z \in \mathbb{C} : 1 \leq |z| \leq 2}$ and $B = {z \in A : |z - 1| \leq 1}$. Then B is:

If the function $f(x) = ax^2 + b$ is differentiable at every point of its domain, then the values of $a$ and $b$ are respectively:

The value of the integral $\int_{0}^{\frac{\pi}{2}} \frac{1 - e^{-\sin x}\cos x}{1 + e^{-\sin x}} dx$ is equal to: