DP Mathematics: Analysis and Approaches Questionbank

Hence, given that $f^{-1}$ does not exist, show that $b^2-3ac>0$.

Find $\left(f\circ g\right)\left(x\right)$.

Find an expression for $f^{-1}\left(x\right)$, stating its domain.

Arc $\text{APB}$ is twice the length of chord $\left[\text{AB}\right]$.

Find the value of $\theta$.

Find the temperature of the water when $t=15$.

Find the $x$-coordinates of the points of intersection of the two curves.

State the cross-sectional radius of the bowl at this point.

Use your graphic display calculator to explore the graph of $y=f_n\left(x\right)$ for

•   the odd values $n=3$ and $n=5$;

•   the even values $n=2$ and $n=4$.

Hence, copy and complete the following table.

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_n$.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

Show that $f$ is an even function.

Find the possible values for $p$.

Show that $b=\frac{\mathrm{\pi }}{6}$.

The equation $2x^2-5x+1=0$ has two real roots, $\alpha$ and $\beta$.

Consider the equation $x^2+mx+n=0$, where $m, n\in \mathrm{\mathbb{Z}}$ and which has roots $\frac{1}{{\alpha }^3}$ and $\frac{1}{{\beta }^3}$.
Without solving $2x^2-5x+1=0$, determine the values of $m$ and $n$.

Determine the value of $m$.

Show that $b=\frac{\mathrm{\pi }}{6}$.

$y$-axis.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

Point $\text{Q}$ has coordinates $(5, 12)$. Find the value of $a$.

Write down the equation of the horizontal asymptote.

Solve the inequality $0<\frac{2\left|x\right|-1}{\left|x\right|+1}<2$.

Show that the inverse function of $f$ is given by $f^{-1}\left(x\right)=\sqrt{x^2+1}$.

Find the equation of $L$.

Find the values of $d$ for which $g$ is an increasing function.

Find the coordinates of $\text{B}$.

Write down the equation of the vertical asymptote.

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

Determine the length of time between the first airplane arriving at $\text{P}$ and the second airplane arriving at $\text{P}$.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.

Write down the equation of the vertical asymptote.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Find $p_{av}$(0.007).

Sketch the graph of $y=f\left(x\right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.

Find $f^{\prime }(x)$.

Write down the $x$-coordinates of these two points;

Find the $x$-coordinate of point A.

Factorize $f(x)$ into a product of linear factors.

Find the largest possible domain $D$ for $f$ to be a function.

Given that $\left(x-5\right)$ is a factor of $P\left(x\right)$, find a relationship between $a$, $b$ and $c$.

Find $f^{\prime }(x)$.

Find $\left(f\circ g\right)\left(\left(x\text{,}y\right)\right)$.

State with a reason whether or not $f$ and $g$ commute.

Find the difference between the greatest possible expected value and the least possible expected value.

By sketching the graph of $p$ as a function of $q$, determine the range of values of $p$ for which there are possible values of $q$.

Show that $p=\frac{q^2+q-1}{2q+1}$.

Write down the values of x for which $f\left(x\right)>g\left(x\right)$.

Given that $p=\frac{4}{3}$, $q=\frac{4}{9}$ and A has coordinates $\left(-\frac{1}{2},0\right)$, determine the possible sets of values for $a$, $b$ and $c$.

Sketch the graphs $y=\text{si}{\text{n}}^{3}x+\text{ln}x$ and $y=1+\text{cos}x$ on the following axes for 0 < $x$ ≤ 9.

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

The following diagram shows part of the graph of $f$.

The region enclosed by the graph of $f$, the $x$-axis and the lines $x=-p$ and $x=p$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Find the area of triangle OBC in terms of $r$ and θ.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

Sketch the graph of the function g (x) = 10x + 40 on the same axes.

On the grid, sketch the graph of $f^{-1}$.

Find $q$.

Express this volume in $\text{c}{\text{m}}^3$.

Find $\frac{\text{d}A}{\text{d}r}$.

Write down an expression for $f\text{'}\left(x\right)$.

Show that $g^{-1}$ exists.

State each of the transformations in the order in which they are applied.

Complete the table below placing a tick (✔) to show whether the unknown parameters $a$ and $b$ are positive, zero or negative. The row for $c$ has been completed as an example.

State the values of $x$ for which $f\left(x\right)$ is decreasing.

Find the value of $K$.

In the context of the model, state what the horizontal asymptote represents.

Write down the minimum number of pizzas that can be ordered.

Sketch the graph of $T$ versus $t$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

Given that $q(x)$ has a factor $(x-4)$, find the value of $k$.

Find the value of $k$, giving your answer correct to four significant figures.

Sketch the graph of $f$ on the axes below.

find the $x$-coordinates of the $x$-intercepts.

Write down the value of $f\prime \left(4\right)$.

Show that the distance of $P$ from $\mathrm{O}$ at this time is $\frac{88}{27}$ metres.

find the coordinates of the vertex.

the vertical asymptote of the graph of $f$.

Write down the minimum total cost for this journey.

Find the new value of $x$ so that the total cost $C$ to travel from $\text{A}$ to $\text{B}$ via $\text{D}$ is a minimum.

Find the value of $d$.

Find the height of the water at $12:00$.

$f\left(\text{i}u\right)$.

$g\left(\text{i}u\right)$.

The oblique asymptote of the graph of $f$ can be written as $y=ax+b$ where $a, b\in \mathrm{\mathbb{Q}}$.

Find the value of $a$ and the value of $b$.

Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

By solving the differential equation $\frac{dy}{dt}=y$, show that $y=A{\text{e}}^t$ where $A$ is a constant.

Hence show that $x=-\frac{B}{2}{\text{e}}^{2t}+C$, where $C$ is a constant.

Find the two values for $\lambda$ that satisfy $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$.

the horizontal asymptote of the graph of $f$.

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

Write down the values of $h$ and $k$.

Find the value of $k$, giving a reason for your answer.

Find the times when the particle’s acceleration is $-1.9 {\text{m\hspace{0.05em}s}}^{-2}$.

Assuming Kaia’s annual salary can be approximately modelled by the equation $S=ax+b$, show that Kaia had a higher salary than Gemma in the year 2021, according to the model.

Plant $B$.

For $t>6$, prove that Plant $A$ was always taller than Plant $B$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

Show that $p=\frac{2}{3}$.

The range of $f$ is $a\le y\le b$, where $a, b\in \mathrm{ℝ}$.

Find the value of $a$ and the value of $b$.

Solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Find the coordinates of $\text{P}$.

Find $u$, $v$ and $w$ expressing your answers in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

Find the largest value of $a$ such that $f$ has an inverse function.

Find the value of $b$.

Hence, or otherwise, show that the exact value of $\text{tan}\frac{\pi }{12}=2-\sqrt{3}$.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Given that $x^2-1$ is a factor of $f(x)$ find the value of $a$ and the value of $b$.

Given that $\tan \theta >0$, find $\tan \theta$.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Write down the range of $f(x)$.

Explain why the inverse function $f^{-1}$ does not exist.

Find the function, $P\left(x\right)$ , that would define this footpath on the map.

Find the coordinates of the bridges relative to the centre of Orangeton.

Let f(x) = x⁴ + px³ + qx + 5 where p, q are constants.

The remainder when f(x) is divided by (x + 1) is 7, and the remainder when f(x) is divided by (x − 2) is 1. Find the value of p and the value of q.

Find the inverse of $f$.

(i)     Sketch the graph of $y=f(x)$ and hence justify whether or not $f$ is a bijection.

(ii)     Show that $A$ is a group under the binary operation of multiplication.

(iii)     Give a reason why $B$ is not a group under the binary operation of multiplication.

(iv)     Find an example to show that $f(a\times b)=f(a)\times f(b)$ is not satisfied for all $a,b\in A$.

(i)     Sketch the graph of $y=g(x)$ and hence justify whether or not $g$ is a bijection.

(ii)     Show that $g(a+b)=g(a)\times g(b)$ for all $a,b\in \mathbb{R}$.

(iii)     Given that $\{\mathbb{R},+\}$ and $\{D,\times \}$ are both groups, explain whether or not they are isomorphic.

Find the value of $\left(f\circ f\right)\left(1\right)$.

Write down the coordinates of $\text{B}$.

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

Find an expression for ${\theta }_p$ in terms of $p$.

Show that $\alpha$ + β < −2.

The triangle ABC is shown in the following diagram. Given that , find the range of possible values for AB.

Find the range of f.

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=f(x)$;

Find the coordinates of P.

Write down the domain of $g$.

Write down the range of $f$.

Explain why the probability of drawing three white marbles is $\frac{1}{6}$.

Sketch the graphs of $y=g\left(x\right)$ and $y=g^{-1}\left(x\right)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.

The graph of $f$ has a local minimum at point $\text{A}$.

Find the coordinates of $\text{A}$.

Write down the other solution of $f\left(x\right)=k$.

Find Jean-Pierre’s vertical speed after $10$ seconds. Give your answer in $\text{km} {\text{h}}^{-1}$ .

State the range of $f$.

Show that $T_0=100$.

$c=2$.

The cubic equation $x^3-k{x}^2+3k=0$ where $k>0$ has roots $\alpha , \beta$ and $\alpha +\beta$.

Given that $\alpha \beta =-\frac{k^2}{4}$, find the value of $k$.

Sketch the graph of $y=g^{-1}\left(x\right)$, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

Find, correct to the nearest $10$ years, the time taken after the plant’s death for $25\%$ of the carbon-14 to decay.

Find the range of $f$.

Show that $k=\frac{\ln 2}{5730}$.

Given that $\left(g\circ f\right)\left(x\right)\le 0$ for all $x\in \mathrm{ℝ}$, determine the set of possible values for $c$.

the horizontal asymptote of the graph of $f$.

Show that $a=8$.

Find $f\left(4\right)$.

Sketch a graph of $v$ against $t$, clearly showing any points of intersection with the axes.

Consider the case when $p=4$. The roots of the equation can be expressed in the form $x=\frac{a\pm \sqrt{13}}{6}$, where $a\in \mathrm{\mathbb{Z}}$. Find the value of $a$.

Sketch the graph of $f$ on the axes below.

Find an expression, in terms of $x$ for the travel time $T$, from $\text{A}$ to $\text{B}$, passing through $\text{D}$.

Find the value of $a$.

Given that $P\left(\left|X-m\right|\le a\right)=0.3$, determine the value of $a$.

Hence find $y$ as a function of $t$.

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

By varying the value of $b$, suggest two key features common to these curves.

The point $\text{S}(-1 , 1)$ also lies on $C$. The line $\text{[QS]}$ intersects $C$ at a further point. Determine the coordinates of this point.

$y^2=x^3, x\ge 0$

Write down the integer root of this equation.

Find $(f\circ g)\left(0\right)$.

The third term in the expansion is the mean of the second term and the fourth term in the expansion.

Find the possible values of $x$.

Write down the equation of the vertical asymptote.

Find $f^{-1}\left(x\right)$.

Show that $f$ is an odd function.

find the equation of the horizontal asymptote.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

Use the result from part (c)(ii) to show that $a^2-3b=0$.

Consider the equation $z^2+az+12=0$, where $z\in \mathrm{ℂ}$ and $a\in \mathrm{ℝ}$.

Given that $0<\alpha -\theta <\pi$, deduce that only one equilateral triangle ${\text{Z}}_1{\text{OZ}}_2$ can be formed from the point $\text{O}$ and the roots of this equation.

Hence, find the value of $f$(6).

Find the value of $p$ and the value of $q$.

Show that $f\left(x\right)$ has no vertical asymptotes.

Find the value of $a$.

Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.

Write down the $y$-intercept of the graph.

Write down the number of possible solutions to the equation $f(x)=5$.

Find the equation of $L$ in the form $y=mx+c$.

Hence, find the area of the region enclosed by the graphs of $f$ and $g$.

Write down the equation of the axis of symmetry for this graph.

Explain why $f$ is an even function.

Find the inverse function $g^{-1}$ and state its domain.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

Find the value of $f\left(x\right)$ when $x=\frac{1}{2}$.

Find $\left(g\circ f\right)\left(\left(x\text{,}y\right)\right)$.

find the coordinates of the $y$-intercept.

sketch the graph, showing clearly any asymptotic behaviour.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

Find the value of $b$.

The graph of $f\left(x\right)=x^2$ is transformed to obtain the graph of $g$.

Describe this transformation.

$\overrightarrow{\text{OA}}\bullet \overrightarrow{\text{OC}}$.

Show that $b=0.3-a$.

Find $\alpha$ and β in terms of $k$.

Explain why $f$ has no inverse on the given domain.

Explain why $f$ is not a function for $-\frac{3\pi }{4}⩽x⩽\frac{\pi }{4}$.

Sketch the graphs of $y=\frac{x}{2}+1$ and $y=\left|x-2\right|$ on the following axes.

Find $p$.

Consider the equation $z^4+a{z}^3+b{z}^2+cz+d=0$, where $a$, $b$, $c$, $d\in \mathbb{R}$ and $z\in \mathbb{C}$.

Two of the roots of the equation are log₂6 and $i\sqrt{3}$ and the sum of all the roots is 3 + log₂3.

Show that 6$a$ + $d$ + 12 = 0.

Write down the value of $k$.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Given that $q=8d^2$, find the other two real roots.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

Let $f\left(x\right)=-x^2+4x+5$ and $g\left(x\right)=-f\left(x\right)+k$.

Find the values of $k$ so that $g\left(x\right)=0$ has no real roots.

Solve $f\left(x\right)=0$.

Solve the equation $\left(f\circ g\right)\left(x\right)=x$.

Write down the coordinates of the local minimum point.

Consider the function $g\left(x\right)=3-x$.

Solve $f\left(x\right)=g\left(x\right)$.

The following table shows the probability distribution of a discrete random variable $X$ where $x=1, 2, 3, 4$.

Find the value of $k$, justifying your answer.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=x-2$ and $g\left(x\right)=ax+b$, where $a, b\in \mathrm{ℝ}$.

Given that $\left(f\circ g\right)\left(2\right)=-3$ and $\left(g\circ f\right)\left(1\right)=5$, find the value of $a$ and the value of $b$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

The quadratic equation $\left(k-1\right)x^2+2x+\left(2k-3\right)=0$, where $k\in \mathrm{ℝ}$, has real distinct roots.

Find the range of possible values for $k$.

Show that $m=4$.

Find the value of $h$ and the value of $k$.

Find the value of $p$ and the value of $q$.

Find the $x$-coordinate of the point of inflexion.

By considering the graph of $y=g\left(x\right)-f\left(x\right)$, or otherwise, solve $f\left(x\right)<g\left(x\right)$ for $x\in \mathrm{ℝ}$.

Find the value of $p$ and the value of $q$.

Hence find the equation of the tangent to the graph of $h$ at $x=4$.

Given that the areas of the two shaded regions are equal, show that $\theta =2 \sin \theta$.

Write down the equation of the vertical asymptote of the graph of $f$.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

Show that $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

By considering each curve from part (a), identify two key features that would distinguish one curve from the other.

Find the value of the gradient of the graph of $f\prime$ at $x=0$.

Write down the equation of the axis of symmetry.

Find $g\left(0\right)$.

Show that $g^{-1}\left(x\right)=1+\frac{\sqrt{4x^2+x}}{x}$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

After $10$ years, the population of marsupials is $3P_0$. It is known that $N=4P_0$.

Find the value of $k$ for this population model.

Write down the equation of the horizontal asymptote.

Plant $A$ correct to three significant figures.

By considering the sum of the roots $u$, $v$ and $w$, show that

$\text{cos}\frac{5\pi }{18}+\text{cos}\frac{7\pi }{18}+\text{cos}\frac{17\pi }{18}=0$.

Find the period of $f$.

Find the value of $p$.

Solve the inequality $\left|3x\arccos (x)\right|>1$.

Write down the maximum and minimum value of $v$.

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Consider the function $f\left(x\right)=x^4-6x^2-2x+4$, $x\in \mathbb{R}$.

The graph of $f$ is translated two units to the left to form the function $g\left(x\right)$.

Express $g\left(x\right)$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ where $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

The polynomial $x^4+p{x}^3+q{x}^2+rx+6$ is exactly divisible by each of $\left(x-1\right)$, $\left(x-2\right)$ and $\left(x-3\right)$.

Find the values of $p$, $q$ and $r$.

Find the value of $p$ and of $q$.

Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.

Given that $g\left(x\right)=2f\left(x-1\right)$, find the domain and range of $g$.

Find the point of intersection of $L_1$ and $L_2$.

Calculate the probability a flight is not late.

Show that $g\left(t\right)={\left(\frac{1+t}{1-t}\right)}^2$.

Hence solve  in the range 0 < $x$ ≤ 9.

Find the set of values of $k$ that satisfy the inequality .

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Write down the value of $q$;

Sketch the graph of $y=\frac{x-4}{2x-5}$, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.

Write down the x-coordinate of P and the x-coordinate of Q.

Show that $(f\circ g)(x)=x^4-4x^2+3$.

Hence find the value of n such that $\sum _{k=1}^n{x}_k=861$.

Show that $abc=8$.

State, in the context of the question, what the value of $34.50$ represents.

State, in the context of the question, what the value of $8.50$ represents.

Write down the zero of $f\left(x\right)$.

Hence find the exact value of ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

The model for the temperature of the water can also be expressed in the form $T=T_0{a}^{\frac{t}{10}}$ for $t\ge 0$ and $a$ is a positive constant.

Find the exact value of $a$.

Write down the value of $f\left(2\right)$.

The line $L$ is tangent to the graphs of both $f$ and the inverse function $f^{-1}$.

Find the shaded area enclosed by the graphs of $f$ and $f^{-1}$ and the line $L$.

Let $g\left(x\right)=\frac{1}{2}f\left(x\right)+1$ for $-4\le x\le 6$. On the axes above, sketch the graph of $g$.

Sketch the graph of $y=f\left(x\right)$ for $-3\le x\le 3$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

Show that $A_0=100$.

Find the value of $x$ for which $f\prime \left(x\right)=0$.

The function $f$ can be written in the form $f\left(x\right)=-2{\left(x-h\right)}^2+k$.

Write down the value of $h$ and the value of $k$.

The function $g$ is defined by $g\left(x\right)=\frac{ax+4}{3-x}$, where $x\in \mathrm{ℝ}, x\ne 3$ and $a\in \mathrm{ℝ}$.

Given that $g\left(x\right)=g^{-1}\left(x\right)$, determine the value of $a$.

Sketch the graph of $f$ on the following grid.

the boat travels directly to $\text{B}$.

Find the value of $x$ so that $T$ is a minimum.

Find the value of $a$.

Find the height of the water at $12:00$.

$x$-axis.

Show that $\frac{dx}{dt}-x=-A{\text{e}}^t$.

The number $351$ is a triangular number. Determine which one it is.

The line $L$ passes through the point $(k, 2)$.

Find the value of $k$.

Find $(f\circ g)\left(x\right)$.

State the domain of $g^{-1}$.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

Express $-3+\sqrt{3}\text{i}$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and .

Verify that $x=\text{tan}\theta$ and $x=-\text{cot}\theta$ satisfy the equation $x^2+\left(2\text{cot}2\theta \right)x-1=0$.

Given that ${\left(g\circ f\right)}^{-1}\left(a\right)=4$, find the value of $a$.

Find the value of $x$ for which the functions have the greatest difference.

Using the results from parts (b) and (c) find the exact value of $\text{tan}\frac{\pi }{24}-\text{cot}\frac{\pi }{24}$.

Give your answer in the form $a+b\sqrt{3}$ where $a$, $b\in \mathbb{Z}$.

Find the co-ordinates of all stationary points.

Find the exact value of $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$, giving the answer in the form $\text{ln}q\text{,}q\in \mathbb{Q}$.

Find an expression for $f^{-1}(x)$.

(i)     Write down the value of $h$.

(ii)     Find the value of $k$.

Show that $a=8$.

Show that $\cos \theta =\frac{3}{4}$.

The quadratic equation $x^2-2kx+(k-1)=0$ has roots $\alpha$ and $\beta$ such that ${\alpha }^2+{\beta }^2=4$. Without solving the equation, find the possible values of the real number $k$.

Find the value of $b$.

Given that ${\left(x-5\right)}^2$ is a factor of $P\left(x\right)$, write down the value of $P^{\prime }\left(5\right)$.

Write down the coordinates of the point of intersection.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

Given that $f^{-1}\left(a\right)=3$, determine the value of $a$.

Given that $\text{A}\hat{\text{O}}\text{C}=\text{B}\hat{\text{O}}\text{C}$, show that $k=7$.

$L_3$ passes through the intersection of $L_1$ and $L_2$.

Write down a vector equation for $L_3$.

Given that at least $7$ of these flights are on time, find the probability that exactly $10$ flights are on time.

Find the expected number of weeks in the year in which Lucca eats no bananas.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

State the range of $g$.

Show that one of the real roots is equal to 1.

Find the value of $p$.

$g\left(x\right)$ can be written in the form $p(x-2{)}^3+q$ , where $p, q \in \mathrm{ℝ}$.

Find the value of $p$ and the value of $q$.

Find the area of triangle $\text{AOB}$ in terms of $k$.

Kaelani has $450 \text{PGK}$.

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.

The $x$-intercept of the graph of $f$ is $\left(5, 0\right)$.

On the following grid, sketch the graph of $f$.

Find the coordinates of the point(s) where the graphs of $y=f\left(x\right)$ and $y=f^{-1}\left(x\right)$ intersect.

Show that $k=\frac{1}{10}\ln \frac{10}{7}$.

Find the $x$-coordinate of $\text{P}$.

Find the value of $k$.

Find an expression for $f\text{'}\left(x\right)$.

Solve $f\left(x\right)>0$ for $x>0$.

Consider the function $g\left(x\right)=x^3-3cx+d$ for $x\in \mathrm{ℝ}$ and where $c , d\in \mathrm{ℝ}$.

Find all conditions on $c$ and $d$ such that the graph of $y=g\left(x\right)$ has exactly one $x$-axis intercept, explaining your reasoning.

Describe a sequence of transformations that transforms the graph of $y=\text{arctan\hspace{0.05em}}x$ to the graph of $y=\text{arctan}\left(2x+1\right)+\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}$.

Sketch the graph of $f$ on the grid below.

State the domain of $g^{-1}$.

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Find the total distance travelled by $P$.

the $y$-axis.

the $x$-axis.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

Let the two values found in part (c)(ii) be ${\lambda }_1$ and ${\lambda }_2$.

Verify that $y=F{\text{e}}^{{\lambda }_{1}t}+G{\text{e}}^{{\lambda }_{2}t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

By differentiating $\frac{dy}{dt}=-x+y$ with respect to $t$, show that $\frac{d^{2}y}{d{t}^2}=2\frac{{\displaystyle dy}}{{\displaystyle dt}}$.

$y^2=x^3+1, x\ge -1$

Find an expression for ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$ in terms of $p$ and $q$.

Find the value of $q$.

Hence or otherwise, find the value of $p$.

write down the equation of the vertical asymptote.

Find the value of $p$ and the value of $q$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of $f$ in the $y$-axis followed by a reflection in the $x$-axis.

Plant $A$ correct to three significant figures.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

The sum of the first $n$ terms of the series is $-3 \ln x$.

Find the value of $n$.

The sum of the first $n$ terms of the series is $\ln \left(\frac{1}{x^3}\right)$.

Find the value of $n$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

State the domain and range of $f^{-1}$.

Plant $B$.

Find the area of triangle UVW.

Show that $\left(g\circ f\right)\left(x\right)=2x+11$.

The following diagram shows the graph of $y=f\left(x\right)$. The graph has a horizontal asymptote at $y=-1$. The graph crosses the $x$-axis at $x=-1$ and $x=1$, and the $y$-axis at $y=2$.

On the following set of axes, sketch the graph of $y={\left[f\left(x\right)\right]}^2+1$, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.

Write down the equation of the vertical asymptote.

Find all the intercepts of the graph of $f\left(x\right)$ with both the $x$ and $y$ axes.

The line $y=kx-5$ is a tangent to the curve of $f$. Find the values of $k$.

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

Determine the range of $f\left(x\right)$ for $p$ ≤ $x$ ≤ $q$.

Write down the domain and range of $f^{-1}$.

Given that ${\left(x-5\right)}^2$ is a factor of $P\left(x\right)$, and that $a=2$, find the values of $b$ and $c$.

Hence or otherwise, factorize $q(x)$ as a product of linear factors.

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

Solve f '(x) = f "(x).

Show that the discriminant of $f^{\prime }\left(x\right)$ is $k^2-16$.

$\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{OC}}$.

Calculate the area of triangle $\text{AOC}$.

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

Write down a direction vector for $L_3$.

Let $\text{C}$ be a point on the graph of $h$. The tangent to the graph of $h$ at $\text{C}$ is parallel to the graph of $f$.

Find the $x$-coordinate of $\text{C}$.

SpeedWay increases the number of flights from city $\text{A}$ to city $\text{B}$ to $20$ flights each week, and improves their efficiency so that more flights are on time. The probability that at least $19$ flights are on time is $0.788$.

A flight is chosen at random. Calculate the probability that it is on time.

Find $(g\circ f)(x)$.

Find the probability that Lucca eats at least one banana in a particular day.

Determine whether $f_n$ is an odd or even function, justifying your answer.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

Write down the range of $f$.

Show that $\text{OC}=r\text{cos}\theta$.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

Express $g\left(x\right)$ in the form $A+\frac{B}{x-2}$ where A, B are constants.

Given that the area of triangle OBC is $\frac{3}{5}$ of the area of sector OAB, find θ.

Given that x_{k + 1} = x_k + a, find a.

State the equation of the vertical asymptote on the graph of $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

Find $f\left(1\right)$.

Hence or otherwise, given that $g\left(2a\right)=f^{-1}\left(2a\right)$, find the value of $a$.

Find the value of $a$.

Describe a sequence of transformations that transforms the graph of $y=\sqrt{x}$ for $x\ge 0$ to the graph of $y=-1-\sqrt{x+3}$ for $x\ge -3$.

Find the time taken for the water to have a temperature of $50 °\text{C}$. Give your answer correct to the nearest second.

Hence find the values of $x$ where the graph of $f$ is both negative and increasing.

Sketch the graph of $f$, showing clearly the value of the $x$-intercept and the approximate position of point $\text{P}$.

Find the value of $p$ and the value of $q$.

Find the exact value of $a$.

the vertical asymptote of the graph of $f$.

Find the value of $t$ when $P$ reaches its maximum velocity.

Write down an expression for $f^{-1}\left(x\right)$.

Find the values of $x$ for which $f\left(x\right)=0$.

Hence determine the value of $\theta$.

Write down the minimum value of $T$.

Find the smallest possible value of $c$.

Prove the identity ${\left(p+q\right)}^3-3pq\left(p+q\right)\equiv p^3+q^3$.

Find the smallest possible value of $c$.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.

Hence, find the coordinates of the rational point $\text{Q}$ where this tangent intersects $C$, expressing each coordinate as a fraction.

For $a=r$ and $b>0$, state in terms of $r$, the coordinates of points $\text{P}$ and $\text{A}$.

Hence, find $\text{E}\left(X\right)$.

Hence or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Sketch the graph of $f″$, the second derivative of $f$. Indicate clearly the $x$-intercept and the $y$-intercept.

Consider the graphs of $y=\frac{x^2}{x-3}$ and $y=m\left(x+3\right)$, $m\in \mathbb{R}$.

Find the set of values for $m$ such that the two graphs have no intersection points.

Hence, solve the inequality $0<\frac{2x-1}{x+1}<2$.

Find the equation of the horizontal asymptote. 

As $x\to \pm \mathrm{\infty }$ the graph of $f\left(x\right)$ approaches an oblique straight line asymptote.

Divide $2x^2-5x-12$ by $x+2$ to find the equation of this asymptote.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

Consider the vectors a = $\left(\begin{array}{c}3\\ 2p\end{array}\right)$ and b = $\left(\begin{array}{c}p+1\\ 8\end{array}\right)$.

Find the possible values of p for which a and b are parallel.

Find $f(2)$.

Using your graph state the range of values of $c$ for which $f(x)=c$ has exactly two distinct real roots.

Write down the equation of the horizontal asymptote to the graph of f.

State the domain of $P$.

For the graph of $f$, find the $y$-intercept.

Hence or otherwise, write down $\underset{x\to \mathrm{\infty }}{\text{lim}}\left(\frac{6x-1}{2x+3}\right)$.

show that there are no $x$-intercepts.

Given that $f$ is an increasing function, find all possible values of $k$.

Find $h\left(x\right)$.

With reference to your graph, explain why $f$ is a function on the given domain.

Sketch the graph of $y=g\left(t\right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

Write down the number of solutions to $f(x)=-6$.

Find the range of values of $k$ for which $f(x)=k$ has no solution.

On the following axes, sketch the two possible graphs of $y=f\left(x\right)$ giving the equations of any asymptotes in terms of $p$ and $q$.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

Find $f^{-1}\left(x\right)$, stating its domain.

Hence find $g^{-1}\left(x\right)$.

Use an algebraic method to determine whether $f$ is a self-inverse function.

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

$c=1$.

Write down the value of $\left(f\circ f\right)\left(2\right)$.

Find the equations of all the asymptotes on the graph of $y=g\left(x\right)$.

Sketch the graph of $y=f_1\left(x\right)$, stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

Find an expression for $g^{-1}\left(x\right)$, justifying your answer.

the $x$-axis.

the $y$-axis.

Find $h\left(4\right)$.

Solve the equation ${\log }_3 \sqrt{x}=\frac{1}{2 {\log }_2 3}+{\log }_3\left(4x^3\right)$, where $x>0$.

the boat is taken from $\text{A}$ to $\text{P}$, and the bus from $\text{P}$ to $\text{B}$.

Find the value of $d$.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

By substituting $Y=\frac{dy}{dt}$, show that $Y=B{\text{e}}^{2t}$ where $B$ is a constant.

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

Find the equation of $L$.

Write down the value of the discriminant of $f\prime$.

Find the value of $x$ such that $f\left(x\right)=0$.

Show that $f$ is an odd function.

Show that $2k^2-k+0.12=0$.

Write down an expression for the product of the roots, in terms of $k$.

Find the particle’s acceleration when its speed is at its greatest.

Hence, solve the inequality $0<\frac{2x-1}{x+1}<2$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

Sketch the curve $y=f\left(x\right)$, clearly indicating the coordinates of the endpoints.

Show that $\text{cot}2\theta =\frac{1-\text{ta}{\text{n}}^2\theta }{2\text{tan}\theta }$.

For this value of $a$, find an expression for $f^{-1}\left(x\right)$, stating its domain.

State the range of $f$.

Write down the $y$-intercept of the graph of $y=f\left(x\right)$.

Solve .

Write down the range of $f$.

Find the area of the region enclosed by the graph of $f$ and the line $L$.

After $m$ complete months Tommaso will, for the first time, have enough money in his account to buy the bicycle.

Find the value of $m$.

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Find $g^{\prime }(x)$.

Expand the expression for $f(x)$.

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

Find an expression for the velocity, $v$ m s⁻¹, of the rocket during the first stage.

Find the distance that the rocket travels during the first stage.

Find the value of $m$.

Calculate the probability that at least $7$ of these flights are on time.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Draw the line $y=-6$ on the axes.

Write down the probability of drawing three blue marbles.

The bag contains a total of ten marbles of which $w$ are white. Find $w$.

Jill plays the game nine times. Find the probability that she wins exactly two prizes.