Simple deductive proofs (LHS = RHS): notes and practice questions

(a) Show that the sum of the squares of any two consecutive even integers is equal to $8n^2 + 8n + 4$, where $n \in \mathbb{Z}$.

(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive even integers is always a multiple of 4.

Show that the identity $p^4 + q^4 \equiv ((p+q)^2 - 2pq)^2 - 2(pq)^2$ is true for all $p, q \in \mathbb{R}$

The equation $x^2 - 4x + 1 = 0$ has two distinct real roots, $α$ and $β$.

Consider the equation $x^2 + mx + n = 0$, where $m, n \in \mathbb{Z}$, which has roots $α^4$ and $β^4$.

Without finding the values of $α$ and $β$, determine the values of $m$ and $n$.

(a) Show that the sum of the squares of any two consecutive even integers is equal to $8n^2 + 8n + 4$, where $n \in \mathbb{Z}$.

(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive even integers is always a multiple of 4.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

A mathematician is investigating properties of integers. They consider three consecutive integers.

(a) Show that the sum of the squares of three consecutive integers, where $n$ is the middle integer, can be expressed as $3n^2 + 2$.

(b) Hence, or otherwise, prove that the sum of the squares of three consecutive integers is even if the middle integer is even, and odd if the middle integer is odd.

The acceleration, $a \text{ ms}^{-2}$, of a particle moving in a straight line at time $t$ seconds, $t \ge 0$, is given by $a = -2v - 8$, where $v \text{ ms}^{-1}$ is the particle's velocity. At $t=0$, the particle is at the origin O and has an initial velocity $v_0 \text{ ms}^{-1}$, where $v_0 > 0$.

By solving an appropriate differential equation, show that the particle's velocity at time $t$ is given by $v(t) = (v_0 + 4)e^{-2t} - 4$.

The particle moves in the positive direction until it reaches its maximum displacement from O at time $T$. Show that $e^{2T} = \frac{v_0+4}{4}$.

Find an expression for the maximum displacement, $s_{\text{max}}$, in terms of $v_0$.

Let $v(T-k)$ represent the particle's velocity $k$ seconds before it reaches $s_{\text{max}}$, where $0 < k < T$. By using the result from part (b)(i), show that $v(T-k) = 4(e^{2k} - 1)$.

Similarly, let $v(T+k)$ represent the particle's velocity $k$ seconds after it reaches $s_{\text{max}}$. Deduce a similar expression for $v(T+k)$ in terms of $k$.

Hence, show that the speed of the particle $k$ seconds before it reaches $s_{\text{max}}$ is greater than or equal to its speed $k$ seconds after it reaches $s_{\text{max}}$.

(a) Consider any three consecutive positive integers. Show that their sum is a multiple of 3.

(b) Prove that the sum of the squares of any three consecutive positive integers can be written in the form $3m+2$, where $m$ is an integer.

The following diagram shows the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$, with asymptotes at $y = -\frac{3\pi}{4}$ and $y = \frac{\pi}{4}$.

Describe a sequence of transformations that transforms the graph of $y = \arctan x$ to the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$.

Show that $\arctan p + \arctan q = \arctan\left(\frac{p+q}{1-pq}\right)$ where $p, q > 0$ and $pq < 1$.

Using mathematical induction and the result from part (b), prove that

$\sum_{r=1}^{n} \arctan\left(\frac{1}{r^2+r+1}\right) = \arctan\left(\frac{n}{n+2}\right)$ for $n \in \mathbb{Z}^+$.

Let two consecutive positive odd integers be represented by $2n-1$ and $2n+1$, where $n \in \mathbb{Z}^+$ .

(a) Show that the sum of these two integers is a multiple of 4.

(b) Prove that the sum of the squares of these two integers is 2 more than a multiple of 8.

A curve is given by the equation $e^y = \cos(x)$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$.

(a) Use implicit differentiation to show that $\frac{dy}{dx} = -\tan(x)$.

(b) Show that $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 + 1 = 0$.

(c) Find an expression for $\frac{d^3y}{dx^3}$ in terms of $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.

(d) Hence, find the Maclaurin series for $y = \ln(\cos(x) )$ up to and including the term in $x^4$.

(a) Show that $2x+5 + \frac{3}{x-1} = \frac{2x^2 + 3x - 2}{x-1}$, for $x \in \mathbb{R}, x \neq 1$.

(b) Hence or otherwise, solve the equation $2\sin{\theta} + 5 + \frac{3}{\sin{\theta}-1} = 0$ for $0 \leq \theta \leq 2\pi$, $\theta \neq \frac{\pi}{2}$.

A company models the average cost per unit, $C(x)$, in thousands of dollars, for producing $x$ thousand units of a specialized component using the function $C(x) = \frac{kx-3}{x-k}$, where $x > 0$ represents the number of units in thousands, and $k$ is a positive constant.

(a) Write down the equations of the vertical and horizontal asymptotes of the graph of $C$.

(b) Show that $C'(x) = \frac{3-k^2}{(x-k)^2}$.

(c) Show that the graph of $C$ has no turning points.

(d) Find the equation, in terms of $k$, of the normal to the graph of $C$ at $x = 1$.

(e) The horizontal and vertical asymptotes of $C$ meet at the point $P$. The normal to the graph of $C$ at $x = 1$ passes through $P$ for certain values of $k$.

Show that these values of $k$ satisfy the equation $k^3-3k^2+k+2=0$.

(f) Hence, find the values of $k$ for which the normal to the graph of $C$ at $x = 1$ passes through $P$.

(a) A cubic equation with real coefficients has roots $z = -2$ and $z = 3 + 2i$.

(i) Write down the third root.

(ii) Verify that the real part of the complex roots is 3.

(b) Let $f(x) = (x+2)(x^2 - 6x + 13)$. Show that the line $y = 4x + 8$ is tangent to the curve $y = f(x)$ at the point $A(3, 20)$.

(c) Sketch the curve $y = f(x)$ and the tangent to the curve at point $A$, clearly showing where the tangent crosses the $x$-axis.

(d) Let a general cubic function be given by $g(x) = (x-r)((x-a)^2 + b^2)$, where $a, b, r \in \mathbb{R}$ and $b > 0$.

(i) Show that $g'(x) = (x-a)^2 + b^2 + 2(x-r)(x-a)$.

(ii) Hence, or otherwise, prove that the tangent to the curve $y = g(x)$ at the point $A(a, g(a) )$ intersects the $x$-axis at the point $R(r, 0)$.

(e) Deduce from part (d)(ii) that the complex roots of the equation $(z - r)((z-a)^2 + b^2) = 0$ can be expressed as $a \pm i\sqrt{g'(a)}$.

(f) Consider a cubic function of the form $g(z) = (z-r)((z-a)^2 + b^2) = 0$. Given that $r = -1$ and $b = 3$, and the $y$-coordinate of the point of tangency $A(a, g(a) )$ is $45$.

(i) Use this information to determine the roots of the corresponding equation for $z \in \mathbb{C}$.

(ii) State the coordinates of the complex conjugate root with negative imaginary part, $C_2$, in the Argand diagram.

(g) The point of inflection for the curve $y=g(x)$ is denoted by $P$.

(i) Show that the $x$-coordinate of $P$ is $\frac{1}{3}(2a+r)$. You are not required to demonstrate a change in concavity.

(ii) Hence describe numerically the horizontal position of point $P$ relative to the horizontal positions of the points $R$ and $A$.

(h) Consider the special case where $a=r$.

(i) Sketch the curve $y = (x-r)((x-a)^2 + b^2)$ for $a=r=1$ and $b=2$.

(ii) For $a=r$ and $b > 0$, state in terms of $r$, the coordinates of points $P$ and $A$.

A pharmaceutical company is formulating a new drug. They are testing two active ingredients, $x_1$ and $x_2$, such that their total concentration is $24$ mg/mL. The drug's efficacy is modelled by the product of the concentrations of the two ingredients.

Find the efficacy, $E$, as a function of $x_1$ only.

Determine the concentration of $x_1$ that maximizes the drug's efficacy.

Hence, show that the maximum efficacy for two ingredients with a total concentration of $24$ mg/mL is $144$.

Let $M_n(S)$ represent the maximum efficacy for a drug with $n$ active ingredients and a total concentration of $S$ mg/mL. For $n = 2$, the maximum efficacy can be expressed as $M_2(S) = \left(\frac{S}{2}\right)^2$.

Verify that $M_2(S) = \left(\frac{S}{2}\right)^2$ is true for $S = 24$.

The relationship between the geometric mean and arithmetic mean states that for $n$ positive real numbers $x_1, x_2, ..., x_n$, their geometric mean $(x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}$ is always less than or equal to their arithmetic mean $\frac{x_1 + x_2 + ... + x_n}{n}$.

Show that the geometric mean and arithmetic mean are equal when $x_1 = x_2 = ... = x_n$.

Use this result to prove that $M_n(S) = \left(\frac{S}{n}\right)^n$.

Using the formula for $M_n(S)$, determine the value of:

$M_3(24)$;

$M_4(24)$;

$M_5(24)$.

For a fixed total concentration of $S = 24$ mg/mL, the company wants to find the optimal number of active ingredients, $n$, to maximize the drug's efficacy. Let $P(S)$ denote this maximum efficacy.

Write down the value of $P(24)$ and the value of $n$ at which it occurs.

Determine the value of $P(30)$ and the value of $n$ at which it occurs.

Consider the continuous function $h$, defined by $\ln(h(x) ) = x\ln\left(\frac{S}{x}\right)$, where $x \in \mathbb{R}^+$. A sketch of the graph of $y = h(x)$ is shown in the following diagram. Point A is the maximum point on this graph.

Find, in terms of $S$, the $x$-coordinate of point A.

Verify that $h(x) = M_x(S)$, when $x \in \mathbb{Z}^+$.

The company has a total concentration of $S = 150$ mg/mL available. Use your answer to part (h) to find the largest possible efficacy. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}^+$.

Show that $3x + 2 - \frac{2}{x+1} = \frac{3x^2 + 5x}{x+1}$, for $x \in \mathbb{R}, x \neq -1$.

Hence or otherwise, solve the equation $3\cos(\theta) + 2 - \frac{2}{\cos(\theta)+1} = 0$ for $0 \le \theta \le 2\pi, \theta \neq \pi$.

A student claims that $2^n > n^2$ for all integers $n \ge 5$.

(a) Show that $2n^2 > (n+1)^2$ for all integers $n \ge 3$.

(b) Use mathematical induction and the result from part (a) to prove that the student's claim is valid for all integers $n \ge 5$.