The highest power of 7 that divides 100! is

Find the value of ${\lim }_{n\to \infty }\left(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{3n}\right)$.

The number of 6-digit positive integers whose sum of the digits is at least 52 is

Solve the differential equation $\frac{dy}{dx}=x-xy$.

The solution of the differential equation

$$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}$$

is

Evaluate $\iint \sqrt{\frac{a^2{b}^2-b^2{x}^2-a^2{y}^2}{a^2{b}^2+b^2{x}^2+a^2{y}^2}}dxdy$ over the positive quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Show that every square matrix is uniquely expressible as the sum of a symmetric and a skew symmetric matrix.

How many triplets of real numbers $(x,y,z)$ are simultaneous solutions of the equations $x+y=2$ and $xy-z^2=1$?

The graph of $2x^2+xy+3y^2-11x-20y+40=0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $y/x$ over all the points $(x,y)$ on the ellipse. Find the value of $a+b$.

The value of $\frac{1}{2\sin 10^{\circ }}-2\sin 70^{\circ }$ is

Let $S_1=\{(x,y)|{\log }_{10}(1+x^2+y^2)\le 1+{\log }_{10}(x+y)\}$ and $S_2=\{(x,y)|{\log }_{10}(2+x^2+y^2)\le 2+{\log }_{10}(x+y)\}$. What is the ratio of the area of $S_2$ to the area of $S_1$?

Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $[{\log }_{10}4x]-[{\log }_{10}x]=0$? Here $[x]$ denotes the greatest integer that is less than or equal to $x$.

The chance of a student getting admitted to colleges A and B are 60% and 40%, respectively. Assume that the colleges admit students independently. If the student is told that he has been admitted to at least one of these colleges, what is the probability that he has got admitted to college A?

A function $f:{\mathbb{R}}^2\to \mathbb{R}$ is called degenerate on $x_i$ if $f(x_1,x_2)$ remains constant when $x_i$ varies $(i=1,2)$. Define

$$f(x_1,x_2)={\left|e^{2\pi i/x_1}\right|}^{x_2}$$

for $x_1\ne 0$, where $i=\sqrt{-1}$. Then which of the following statements is true?