Applications Practice 01

Given that $x + \dfrac{1}{x} = 3$, the value of $x^2 + \dfrac{1}{x^2}$ is

What is the value of $1^2 - 2^2 + 3^2 - 4^2 + \cdots + 19^2 - 20^2 \, ?$

The point $(3, k)$ lies on the circle with centre $(1, 2)$ and radius $\sqrt{20}$. The sum of the possible values of $k$ is

Given that $\sin\theta - \cos\theta = \dfrac{1}{2}$, the value of $\sin\theta \cos\theta$ is

The functions $f$ and $g$ are defined for all real $x$ by $f(x) = x + 3$ and $g(x) = x^2$. Find all real values of $x$ for which $f(g(x)) = g(f(x))$.

Given that $\log_a b = 2$ and $\log_b c = 3$, the value of $\log_c (a b^2)$ is

The 2nd, 4th and 8th terms of an arithmetic progression form a non-constant geometric progression. What is the common ratio of this geometric progression?

The curve $y = x^3 + ax^2 + bx$ has a stationary point at $x = 1$, and the tangent to the curve at $x = 0$ has gradient $-9$. What is the $y$-coordinate of the other stationary point of the curve?

The curve $y = f(x)$ has a single vertical asymptote at $x = 2$ and a single horizontal asymptote at $y = -1$. The curve $y = g(x)$ is obtained by the following sequence of transformations applied to $y = f(x)$: a translation by $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$, then a reflection in the $x$-axis, then a stretch parallel to the $y$-axis with scale factor $2$. What are the asymptotes of $y = g(x)$?

The polynomial $p(x) = x^3 + ax^2 + bx + 12$ has $(x-2)$ as a factor, and leaves a remainder of $-18$ when divided by $(x+1)$. What is the value of $a + b$?

The total area of the finite region(s) enclosed between the curve $y = x^3 - 4x$ and the $x$-axis is

The function $f(\theta) = 4\sin\theta + 3\cos\theta$ attains its maximum value at $\theta = \alpha$, where $0 < \alpha < \dfrac{\pi}{2}$. The value of $\tan(2\alpha)$ is

Let $a$ and $b$ be positive real numbers. Consider the following three statements.

I. If $a + b \geq 2\sqrt{ab}$, then $a = b$.

II. $a^2 + b^2 \geq 2ab$.

III. If $\dfrac{1}{a} + \dfrac{1}{b} \leq \dfrac{4}{a+b}$, then $a = b$.

Which of the statements are true for all positive real $a$ and $b$?

In the expansion of $(1 + ax)^n$, where $a$ is a non-zero real constant and $n$ is a positive integer, the coefficient of $x^2$ is $60$ and the coefficient of $x^3$ is $160$. What is the value of $a + n$?

The function $f$ is defined for all real $x$ by $f(x) = |x^2 - 6x + 5|.$ The equation $f(x) = k$ has exactly four distinct real solutions. Find the complete set of values of $k$.

The line $\ell$ is tangent to the curve $y = x^4 - 8x^2 + 5x$ at two distinct points. The $y$-intercept of $\ell$ is

The equation $\log_{2}(x^2 + a) \;=\; 1 + \log_{2}(x + 1)$ in the unknown $x$ has exactly one real solution. Find the complete set of values of the real constant $a$ for which this holds.

Let $f(x) = x^2 - x - 1$. How many distinct real values of $x$ satisfy $f(f(x)) = x$?

Evaluate $\int_0^2 \frac{x^3}{x^3 + (2-x)^3}\, dx.$

Find the number of ordered pairs of integers $(a, b)$ with $2 \le a < b \le 100$ for which $\log_a b$ is rational.