DP Mathematics: Analysis and Approaches Questionbank

Find the times when $P$ comes to instantaneous rest.

Find an expression for $s$ in terms of $t$.

Find the maximum displacement of $P$, in metres, from its initial position.

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

Show that, at these times, $\tan 6t=2$.

Hence show that $\frac{v_2}{v_1}=\frac{v_3}{v_2}=-{\text{e}}^{-\frac{\mathrm{\pi }}{2}}$.

Show that $\text{AC}=\tan \theta$.

Find the value of $\theta$ when the area of the shaded region is equal to the area of sector $\text{OADB}$.

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

Find the length of new fence required.

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 $\frac{392}{{\theta }^2}\left(\theta +\sin \theta \right)$.

Find the value of $r$.

Hence, find the exact area of the non-shaded region.

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

Hence determine the value of $\theta$.

Find the area of one of the shaded segments in terms of $\theta$.

Given that the area of the logo is $13.4 {\text{cm}}^2$, find the value of $\theta$.

Find the area of the shaded sector.

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

Find the area of the shaded region.

Find the value of θ, giving your answer in radians.

Find AM.

Find $\text{A}\stackrel{\wedge }{\text{M}}\text{P}$ in radians.

Find the area of the shaded region.

This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius $\text{OB}=9\text{ cm}$ and four equal sectors of a smaller circle of radius $\text{OA}=3\text{ cm}$.
The angle $\text{BOC}=$ 20°.

Find the area of the pendant.

Find an expression for the shaded area in terms of $\alpha$, $\theta$ and $r$.

Show that $\alpha =4\arcsin \frac{1}{4}$.

Hence find the value of $r$ given that the shaded area is equal to 4.

A sector of a circle with radius $r$ cm , where $r$ > 0, is shown on the following diagram.
The sector has an angle of 1 radian at the centre.

Let the area of the sector be $A$ cm² and the perimeter be $P$ cm. Given that $A=P$, find the value of $r$.

Find an expression for the volume of water $V({\text{m}}^3)$ in the trough in terms of $\theta$.

Calculate $\frac{\text{d}\theta }{\text{d}t}$ when $\theta =\frac{\pi }{3}$.

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.

The following diagram shows the chord [AB] in a circle of radius 8 cm, where $\text{AB}=12\text{ cm}$.

Find the area of the shaded segment.

Find the length of arc ABC.

Find the perimeter of sector OABC.

Find the area of sector OABC.

Write down the exact value of $\theta$ in radians.

Find the value of $r$.

Find AB.

Find the area of the shaded region, in terms of θ.

The area of the shaded region is 12 cm². Find the value of θ.

The following diagram shows a circle with centre O and radius r cm.

The points A and B lie on the circumference of the circle, and $\text{A}\stackrel{\wedge }{\text{O}}\text{B}$ = θ. The area of the shaded sector AOB is 12 cm² and the length of arc AB is 6 cm.

Find the value of r.

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

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

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

the perimeter.

the area.

Find $\text{B}\hat{\text{A}}\text{C}$.

Find the area of $R$.

Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.