DP Mathematics: Analysis and Approaches Questionbank

Consider a triangle $\text{ABC}$, where $\mathrm{AC}=12, \mathrm{CB}=7$ and $\text{B}\hat{\text{A}}\text{C}=25°$.

Find the smallest possible perimeter of triangle $\mathrm{ABC}$.

Solve the equation $(f\circ g)\left(x\right)=2 \cos 2x$ where $0\le x\le \pi$.

Hence, given that $\text{arg}\left(z_1{z}_2\right)=\frac{\mathrm{\pi }}{4}$, find the value of $b$.

Find the least positive value of $x$ for which $\cos \left(\frac{x}{2}+\frac{\mathrm{\pi }}{3}\right)=\frac{1}{\sqrt{2}}$.

Show that ${\int }_0^{m}f\left(x\right)dx=\frac{32}{7}{\sin }^7 m$, where $m$ is a positive real constant.

It is given that ${\int }_m^{\frac{\mathrm{\pi }}{2}}f\left(x\right)dx=\frac{127}{28}$, where $0\le m\le \frac{\mathrm{\pi }}{2}$. Find the value of $m$.

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.

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 $k=\frac{\mathrm{\pi }}{4}$.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Use the cosine rule to find the two possible values for AC.

Find the difference between the areas of the two possible triangles ABC.

Show that $\text{sin}\theta =\frac{\sqrt{15}}{4}$.

Find the two possible values for the length of the third side.

The following diagram shows triangle PQR.

Find PR.

Find the value of $\text{tan}\theta$.

Show that $\text{AC}=7\text{ cm}$.

The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.

Find the exact perimeter of this shape.

Triangle ABC has a = 8.1 cm, b = 12.3 cm and area 15 cm². Find the largest possible perimeter of triangle ABC.

Let SR = $x$. Use the cosine rule to show that $x^2-\left(76\text{cos}{43}^{\circ }\right)x+420=0$.

Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.

Find BD.

The area of triangle ABD is 18.5 cm². Find the possible values of θ.

Given that $\text{cos}{75}^{\circ }=q$, show that $\text{cos}{105}^{\circ }=-q$.

Show that $\text{B}\stackrel{\wedge }{\text{A}}\text{D}={75}^{\circ }$.

By considering triangle $\text{ABD}$, show that $\text{B}{\text{D}}^2=5r^2-2r^{2}q\sqrt{6}$.

By considering triangle $\text{CBD}$, find another expression for $\text{B}{\text{D}}^2$ in terms of $r$ and $q$.

Use your answers to part (c) to show that $\text{cos}{75}^{\circ }=\frac{1}{\sqrt{6}+\sqrt{2}}$.