DP Mathematics: Analysis and Approaches Questionbank

Given that $\text{A}\hat{\text{B}}\text{C}$ is acute, find $\sin \theta$.

Find $\cos \left(2\times \text{C}\hat{\text{A}}\text{B}\right)$.

Calculate the length of $\text{BD}$.

Show that angle $\text{EDC}=48.0°$, correct to three significant figures.

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

Pedro draws a circle, with centre at point $\text{E}$, passing through point $\text{C}$. Part of the circle is shown in the diagram.

Show that point $\text{A}$ lies outside this circle. Justify your reasoning.

Find the size of $\text{A}\hat{\text{B}}\text{C}$.

Find the length of new fence required.

Consider quadrilateral $\mathrm{PQRS}$ where $\left[\mathrm{PQ}\right]$ is parallel to $\left[\mathrm{SR}\right]$.

In $\mathrm{PQRS}$, $\mathrm{PQ}=x$, $\mathrm{SR}=y$, $\mathrm{R}\hat{\mathrm{S}}\mathrm{P}=\alpha$ and $\mathrm{Q}\hat{\mathrm{R}}\mathrm{S}=\beta$.

Find an expression for $\mathrm{PS}$ in terms of $x, y, \sin \beta$ and $\sin \left(\alpha +\beta \right)$.

Show that arc $\text{APB}$ has length $6\mathrm{\pi }-3\theta$.

Show that $L=\sqrt{18-18 \cos \theta }$.

Write down an expression for $r$ in terms of $\theta$.

The area of field that the horse can reach is $460 {\text{m}}^2$. Find the value of $\theta$.

Hence, find the size of $\text{D}\hat{\text{A}}\text{E}$.

Show that the area of the field that the horse can reach is $\frac{392}{{\theta }^2}\left(\theta +\sin \theta \right)$.

The following diagram shows triangle $\text{ABC}$, with $\text{AB}=10$, $\text{BC}=x$ and $\text{AC}=2x$.

Given that $\cos \hat{\text{C}}=\frac{3}{4}$, find the area of the triangle.

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

Show that $y=2x \sin \frac{\mathrm{\pi }}{n}$.

Use the results from parts (b) and (c) to show that $A=P=4n \tan \frac{\mathrm{\pi }}{n}$.

Given that $\text{sin}\theta =\frac{3}{5}$, find the value of $\text{cos}\theta$.

Find the value of $\text{cos}2\theta$.

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 $\mathrm{ABC}$.

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

Hence determine the value of $\theta$.

Given that $\text{A}\hat{\text{V}}\text{B}=40°$, find $\text{AB}$.

Find the area of this piece of land.

Find the distance from point $\text{B}$ to point $\text{D}$.

Find the distance from point $\text{A}$ to point $\text{C}$.

Find $\text{CÂB}$.

Find the length of the chord $\text{[AB]}$.

Find the area of the shaded sector.

Given that $\text{cos}\hat{A}=\frac{5}{6}$ find the value of $\text{sin}\hat{A}$.

Find the distance from the camp to point C.

Find the area of triangle ABC.

Find the distance from point A to point B.

Show that $\text{A}\stackrel{\wedge }{\text{B}}\text{C}$ is 101°.

Find the bearing that Jacob must take to point C.

Jacob hikes at an average speed of 3.9 km/h.

Find, to the nearest minute, the time it takes for Jacob to reach point C.

Find $\text{B}\stackrel{\wedge }{\text{C}}\text{A}$.

In triangle ABC, AB = 5, BC = 14 and AC = 11.

Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.

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

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

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

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

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.

Find DB.

Find DC.

The following diagram shows triangle PQR.

Find PR.

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.

Find BC.

Find the area of triangle ABC.

Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.

Use Giovanni's diagram to calculate the length of AX.

Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.

Find the percentage error on Giovanni’s diagram.

Giovanni adds a point D to his diagram, such that BD = 45 m, and another triangle is formed.

Find the angle of elevation of A from D.

Calculate the area of triangle EAD.

Calculate the total volume of the barn.

Calculate the length of MN.

Calculate the length of AE.

Show that Farmer Brown is incorrect.

Calculate the total length of metal required for one support.

Find the length of EB.

Write down the angle of elevation of B from E.

Find the vertical height of B above the ground.

Write down an equation for the area of ABCDE using the above information.

Show that the equation in part (a)(i) simplifies to $3x^2+19x-414=0$.

Find the length of CD.

Show that angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{E}={67.4}^{\circ }$, correct to one decimal place.

Find the length of the perimeter of ABCDE.

Calculate the length of CF.

Find the angle of elevation from B to C.

A second truck arrives whose ladder, when fully extended, is 30 metres long. The base of this ladder is also 4 metres above the ground. For safety reasons, the maximum angle of elevation that the ladder can make is 70º.

Find the maximum height on the wall that can be reached by the ladder on the second truck.

Find the length of one of the straight pieces in the wireframe.

Find the total length of wire needed to construct this wireframe. Give your answer in centimetres correct to the nearest millimetre.

Show that the size of angle BÂC is 20.2°, correct to 3 significant figures.

Calculate the area of triangle ABC.

Find the length of AC.

A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.

Draw and label the angle of depression on the diagram.

Calculate CD, the height of the observation deck above the ground.

Calculate the angle of depression from A to B.

Show that $\text{BD}=93\text{ m}$ correct to the nearest metre.

Calculate angle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}$.

Find the area of ABCD.

Calculate Abdallah’s estimate for the area.

Find the percentage error in Abdallah’s estimate.

On the diagram, draw and label with an x the angle of depression of B from P.

Find the size of angle APB.

Find the size of the angle of depression of B from P.

Calculate the length of side AC in km.

Calculate the area of the park.

Find the length $\text{AB}$, giving your answer to four significant figures.

Find the total area of the two rectangles that make up the roof.

Find the maximum number of complete panels that can be fitted to the whole roof.

Find the percentage error in her estimate.

Olivia investigates arranging the panels, such that the longer edge of each panel is parallel to $\text{AB}$ or $\text{AC}$.

State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.

Find the length of the path BD.

Show that angle DBC is 48.7°, correct to three significant figures.

Find the area of the park.

Find the length of the path CE.

Calculate the total length of the track.

Find the size, in degrees, of angle BÂC.

Find the area, in km², of triangle ABC.

the length of BD;

the size of angle DAB;

the area of triangle ABD;

the area of quadrilateral ABCD;

the length of AP;

the length of the fence, BP.

Find the distance from A to C.

Find the size of angle $\text{B}\stackrel{\wedge }{\text{A}}\text{C}$.

Find the size of angle $\text{C}\stackrel{\wedge }{\text{A}}\text{D}$.

Find the size of angle $\text{A}\stackrel{\wedge }{\text{C}}\text{D}$.

The length of path AD is 287 m.

Find the area of the region ABCD.

Find the length of BC.

Find the area of the postage stamp.

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 $\text{cos}{75}^{\circ }=\frac{1}{\sqrt{6}+\sqrt{2}}$.

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

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

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

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

The diagram below shows a triangular-based pyramid with base $\text{ADC}$.
Edge $\text{BD}$ is perpendicular to the edges $\text{AD}$ and $\text{CD}$.

$\text{AC}=28.4\text{cm}$,  $\text{AB}=x\text{cm}$,  $\text{BC}=x+2\text{cm}$,  $\text{A}\hat{\text{B}}\text{C}=0.667$,  $\text{B}\hat{\text{A}}\text{D}=0.611$

Calculate $\text{AD}$

Hence or otherwise, find $\text{BC}$.

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

Show that $p=\frac{q^2+q-1}{2q+1}$.

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 $q$.