How many positive roots does the function

\[
f(x) = x^4 – 8x^3 + 22x^2 – 24x
\]

have?

Compute the integral

\[
\int_{0}^{1} \frac{x – 4}{\sqrt{x}\,(\sqrt{x} + 2)} \, dx
\]

.

Given that
\[
\frac{dV}{dt} = (1 + t)^4
\]
and \( V(1) = 5 \), what is \( V(2) \)?

The sum to infinity of a geometric progression is 4. The sum to infinity of
the squares of each term in the progression is 10. What is the common
ratio of the geometric series?

How many solutions does the equation
\[
\cos(2x)\,\log x = \sin(2x)
\]
have in the range \( 0 < x < 3\pi \)?

Compute the shortest distance between the curves \( x^2 + 4x + y^2 + 6y + 10 = 0 \quad \text{and} \quad x^2 – 4x + y^2 – 8y + 12 = 0 \).

What is the probability of rolling the same number exactly three times
with five six-sided dice?

Given that, in the expansion of \( (3x + b)^7 \), the coefficient of \( x^4 \) is the same as the coefficient of \( x^2 \) in \( (3b + x)^4 \), find the positive constant \( b \).

Consider the tangent to the curve \( y = x^2 + bx \) at \( x = 2 \). For what values of \( b \) is the x-intercept greater than 4?

Given \( f(x) = \left( 9x^2 + 12 + \frac{4}{x^2} \right)^{1/2} \) and \( \frac{d^n f}{dx^n}(2) = -\frac{3}{4} \), find \( n \).

In which of the following ranges is \( (x^2 – 1)(x + 2)(x + 4) > 0 \)?

Suppose \( 5^{4 + 6 + \cdots + 2x} = 0.04^{-14} \). Given \( x \) is a positive integer, what is \( x \)?

An arithmetic-geometric series is defined by
\[
x_1 = 2
\]
\[
x_{n+1} = x_n + q
\]
Given \( x_{100} \) is 13, find the sum to infinity of a series with common ratio \( q \), and first term 5.

The roots of \( x^2 + 3x + c = 2 \) differ by 7. What is \( c \)?

Which of the following is a line of symmetry of the graph
\[
y = \frac{1}{\sin\left(4x + \frac{\pi}{3}\right)} \, ?
\]

Four unbiased coins are tossed. What is the probability of getting at most
two heads?

Define a recurrent sequence by
\[
x_{n+1} =
\begin{cases}
\frac{x_n}{2} & \text{for } x_n \text{ even} \\
3x_n + 1 & \text{for } x_n \text{ odd}
\end{cases}
\]
Given \( x_1 = 12 \), what is \( x_{100} \)?

What is the sum of roots of the equation
\[
2^y – 5\cdot \sqrt{2}^{y+2} + 24 = 0 \, ?
\]

For \( p > 0 \), find the area enclosed by the curves
\[
y = px^2 \quad \text{and} \quad x = py^2.
\]

What is the complete set of values for which
\[
\frac{x^2 + 2x}{\sqrt{x^3}}
\]
is increasing?