Solving roots of quadratics (factorising, CTS, quadratic formula) + discriminant: notes and practice questions

Consider the functions $f(x) = x^2 - 4$ and $g(x) = \frac{p}{x}$, where $p$ is a non-zero real constant.

(a) Write down an expression for $(f \circ g)(x)$.

(b) Given that $(f \circ g)(2) = 5$, find the possible values of $p$.

Consider the series $\sin\theta + k\sin\theta + \frac{1}{2}\sin\theta + \dots$, where $0 < \theta < \frac{\pi}{2}$ and $k \in \mathbb{R}, k \neq 0$.

Consider the case where the series is geometric.

(a) (i) Show that $k = \pm \frac{1}{\sqrt{2}}$.

(a) (ii) Given that $k > 0$ and the sum to infinity is $\sqrt{2}$, find the value of $\sin\theta$.

Now consider the case where the series is arithmetic with common difference $d$.

(b) (i) Show that $k = \frac{3}{4}$.

(b) (ii) Write down $d$ in the form $c\sin\theta$, where $c \in \mathbb{Q}$.

(b) (iii) The sum of the first $n$ terms of the series is $-14\sin\theta$.

Find the value of $n$.

Consider the function $g(x) = \frac{ax^2+x+k}{x-4}$.

The graph of $y=g(x)$ passes through the point $(1, -2)$ and has an oblique asymptote with equation $y = -3x-11$.

(a) Write down the equation of the vertical asymptote.

(b) Find the value of:

(i) $a$

(ii) $k$

(c) Hence, find the exact coordinates of any points where the graph of $y=g(x)$ intersects the x-axis.

Let $f(x) = \frac{1}{\sqrt{1-2x}}$ for $x < \frac{1}{2}$.

(a) Show that $f''(x) = 3(1-2x)^{-\frac{5}{2}}$.

(b) Use mathematical induction to prove that $f^{(n)}(x) = \frac{(2n)!}{2^n n!} (1-2x)^{-\frac{2n+1}{2}}$ for $n \in \mathbb{Z}, n \ge 2$.

Let $g(x) = \ln(1+kx)$, where $k$ is a real constant.

Consider the function $h$ defined by $h(x) = f(x) \times g(x)$ for $x < \frac{1}{2}$.

It is given that the coefficient of the $x^2$ term in the Maclaurin series for $h(x)$ is $-4$.

(c) Find the possible values of $k$.

Two spacecraft, S1 and S2, travel along straight paths, represented by the lines $L_1$ and $L_2$ respectively. The paths of the spacecraft intersect at a docking station D. A probe is located at a point P on the path of $L_2$. This is shown in the following diagram.

The direction vector of $L_1$ is $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$. The vector $\vec{\text{DP}}$ is given by $\begin{pmatrix} k \\ 4 \\ 0 \end{pmatrix}$, where $k \ge 0$.

The acute angle between the paths $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(a) Show that $2k+4 = \sqrt{k^2+16}$.

(b) Find the value of $k$.

(c) Hence, find the shortest distance from the probe at P to the path $L_1$.

The paths $L_1$ and $L_2$ lie on a plane, $\Pi$.

(d) Find a vector normal to the plane $\Pi$.

A satellite dish is modelled as a right circular cone with its vertex at V. The base of the cone lies in the plane $\Pi$ and is centred at P. The path $L_1$ is tangent to the circular base of the cone. The volume of the cone is $\frac{128\pi\sqrt{2}}{9}$ cubic units. The position vector of P is $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

(e) Find the two possible position vectors for V.

(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 $0 \leq \theta \leq 2\pi$, $\theta \neq \frac{\pi}{2}$.

The graphs of the functions $f(x) = kx^2 - 3x$ and $g(x) = x - k$ intersect at two distinct points.

(a) Find the set of possible values for $k$.

(b) Consider the case when $k=1$. The x-coordinates of the intersection points can be written in the form $x = m \pm \sqrt{n}$, where $m, n \in \mathbb{Z}$. Find the values of $m$ and $n$.

A rectangle is inscribed in an ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The sides of the rectangle are parallel to the coordinate axes. The vertices of the rectangle are located at $(\pm x, \pm y)$, where $x > 0$ and $y > 0$.

(a) Show that the area of the rectangle, $A$, can be expressed as $A = \frac{12x}{5}\sqrt{25-x^2}$.

(b) Show that $\frac{dA}{dx} = \frac{12(25-2x^2)}{5\sqrt{25-x^2}}$.

(c) Hence, find the exact dimensions of the rectangle with the maximum possible area.

Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$.

Hence, show that $\theta = \frac{\pi}{8}$ and $\theta = \frac{3\pi}{8}$ are solutions of the equation $8\cos^4\theta - 8\cos^2\theta + 1 = 0$.

Hence, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

The functions $f$ and $g$ are defined by

$f(x) = 2\ln x$, where $x > 0$

$g(x) = \ln(k(x-2) )$, where $x > 2, k \in \mathbb{R}^+$.

The graphs of $y = f(x)$ and $y = g(x)$ intersect at two distinct points.

(a) State the equation of the vertical asymptote to the graph of $y = g(x)$.

(b) (i) Show that, at the points of intersection, $x^2 - kx + 2k = 0$.

(b) (ii) Hence show that $k^2 - 8k > 0$.

(b) (iii) Find the range of possible values of $k$.

The graphs intersect at $x=p$ and $x=q$, where $p<q$.

(c) In the case where $k=10$, find the value of $q-p$. Express your answer in the form $a\sqrt{b}$, where $a, b \in \mathbb{Z}^+$.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

Consider the function $f$ defined by $f(x) = x^3 - 6x^2 + 8x$.

Find the $x$-intercepts of the graph of $y=f(x)$.

The graph of $y=f(x)$ for $0 \le x \le 4$ is shown below. The graph encloses two regions with the $x$-axis, shaded in the diagram.

Find the total area of the shaded regions.

The total surface area of a closed right cylinder is 8, equal to the total shaded area found in part (b). The cylinder has a height of $\frac{4-\pi}{\pi}$.

Find the radius, $r$, of the cylinder.

Hence, find the volume of the cylinder.

A function $f$ is defined by $f(x) = \frac{4x-1}{2x+3}$, where $x \in \mathbb{R}, x \neq -\frac{3}{2}$.

The graph of $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider the function $g(x) = mx - \frac{1}{3}$, where $m \in \mathbb{R}, m \neq 0$.

(i) Write down the number of solutions to $f(x) = g(x)$ for $m < 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$ for which $f(x) = g(x)$ has two distinct solutions for $x \le 0$.

The functions $f$ and $g$ are defined by $f(x) = \sin x$ and $g(x) = \cot x$, for $0 < x < \frac{\pi}{2}$.

The curves $y = f(x)$ and $y = g(x)$ intersect at a point P whose x-coordinate is $k$.

Show that $\sin^2 k = \cos k$.

Hence, show that the tangent to the curve $y = f(x)$ at P and the tangent to the curve $y = g(x)$ at P are perpendicular.

Find the value of $\cos k$. Give your answer in the form $\frac{a+\sqrt{b}}{c}$, where $a, c \in \mathbb{Z}$ and $b \in \mathbb{Z}^+$.

A project manager has $n$ employees available for a new project. The employees are to be divided into two teams, Team Alpha and Team Beta. For the project to be successful, Team Alpha must have exactly four members and Team Beta must have at least four members.

The manager will randomly assign four employees to Team Alpha, with the rest forming Team Beta.

Write down an expression for the number of ways that the teams could be formed.

Two of the employees, Chloe and David, have a history of conflict and cannot be in the same team. The manager agrees to this condition, and finds that this restriction reduces the number of possible team formations to two-fifths of the original number.

Determine the value of $n$.