DP Mathematics: Analysis and Approaches Questionbank

Deduce a quadratic equation with integer coefficients, having roots ${\mathrm{cosec}}^2 \frac{\mathrm{\pi }}{8}$ and ${\mathrm{cosec}}^2 \frac{3\mathrm{\pi }}{8}$.

Show that $y=x+50 \text{cot} \theta$ .

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

Verify that $\text{arctan\hspace{0.05em}}\left(2x+1\right)=\text{arctan\hspace{0.05em}}\left(\frac{x}{x+1}\right)\mathrm{+}\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}, x>0$.

By considering limits, show that the graph of $y=f\left(x\right)$ has a horizontal asymptote and state its equation.

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.

By using the expression for $f\text{'}\left(x\right)$ and the result $\sqrt{x^2}=\left|x\right|$, show that $f$ is decreasing for $x<0$.

Show that $f\text{'}\left(x\right)=\frac{2x}{\sqrt{x^2}\left(x^2+1\right)}$ for $x\in \mathrm{ℝ}, x\ne 0$.

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

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

Show that $f$ is an even function.

Determine the value of $m$.

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

It is given that $\text{cosec} \theta =\frac{3}{2}$, where $\frac{\mathrm{\pi }}{2}<\theta <\frac{{\displaystyle 3\mathrm{\pi }}}{{\displaystyle 2}}$. Find the exact value of $\text{cot} \theta$.

By using the substitution $u=\text{sec} x$ or otherwise, find an expression for $\underset{0}{\overset{\frac{\pi }{3}}{\int }}{\text{sec}}^n x \tan x dx$ in terms of $n$, where $n$ is a non-zero real number.

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 $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Using l’Hôpital’s rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

Show that a finite limit only exists for $k=\frac{\mathrm{\pi }}{4}$.

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

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

Use an appropriate trigonometric identity to show that $f_{n+1}(x)=\text{cos}\left(n\text{arccos}x\right)\text{cos}\left(\text{arccos}x\right)-\text{sin}\left(n\text{arccos}x\right)\text{sin}\left(\text{arccos}x\right)$.

On a new set of axes, sketch the graphs of $y=f_2(x)$ and $y=f_4(x)$ for −1 ≤ $x$ ≤ 1.

Hence show that $f_{n+1}(x)+f_{n-1}(x)=2x{f}_n\left(x\right)$, $n\in {\mathbb{Z}}^{+}$.

local maximum points;

Hence express $f_3(x)$ as a cubic polynomial.

local maximum points;

local minimum points;

On the same set of axes, sketch the graphs of $y=f_1(x)$ and $y=f_3(x)$ for −1 ≤ $x$ ≤ 1.

Use an appropriate trigonometric identity to show that $f_2(x)=2x^2-1$.

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

local minimum points.

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 $a$, $b\in \mathbb{Z}$.

Solve the equation ${f_n}^{\mathrm{\prime }}(x)=0$ and hence show that the stationary points on the graph of $y=f_n(x)$ occur at $x=\text{cos}\frac{k\pi }{n}$ where $k\in {\mathbb{Z}}^{+}$ and 0 < $k$ < $n$.

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 least value of $a$ such that $g$ has an inverse.

For the value of $a$ found in part (b), write down the domain of $g^{-1}$.

For the value of $a$ found in part (b), find an expression for $g^{-1}\left(x\right)$.

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

State the range of $f$.

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

Find the $x$-coordinates of A and of C , giving your answers in the form $a+\text{arctan}b$, where $a$, $b\in \mathbb{R}$.

Hence, show that $\int {\text{e}}^x\text{cos}{}^{2}x\text{d}x=\frac{{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{10}\text{cos}2x+\frac{{\text{e}}^x}{2}+c$,  $c\in \mathbb{R}$.

Find the area enclosed by the curve and the $x$-axis between B and D, as shaded on the diagram.

Use integration by parts to show that $\int {\text{e}}^x\text{cos}2x\text{d}x=\frac{2{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{5}\text{cos}2x+c$,  $c\in \mathbb{R}$.

Solve the equation ${\sec }^{2}x+2\tan x=0,0⩽x⩽2\pi$.