DP Mathematics: Analysis and Approaches Questionbank

Find the height of point $\text{A}$ above the ground.

Calculate the number of seconds it takes for the water wheel to complete one rotation.

Hence find the number of rotations the water wheel makes in one hour.

Find $k$.

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

Determine the rate of change of $h$ when the top of the bucket is at $\text{B}$.

Find the height of the ball above the ground when it is released.

Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.

Find the minimum height of the ball above the ground.

For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8+0.2\sqrt{2}$ metres above the ground.

A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $\text{P}$ is a point on the wheel. The wheel starts moving with $\text{P}$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $\text{P}$ above the ground after $t$ minutes is given by $h\left(t\right)=a \cos \left(bt\right)+c$, where $a, b, c \in \mathrm{ℝ}$.

Find the values of $a$, $b$ and $c$.

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

If the observer stands directly under point $\text{C}$ for one minute, point $\text{D}$ will pass over his head $n$ times.

Find the value of $n$.

Find the $x$-coordinates of $\text{A}$ and $\text{B}$.

Find the smallest possible value of $c$.

Find the value of $a$.

Find the value of $d$.

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

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

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

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

Find the value of $a$.

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.

Find the smallest possible value of $c$.

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

Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, r)$.

Given that $g\left(x\right)\ge 7$, find the smallest value of $r$.

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

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

Find the period of $f$.

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

Find the value of $b$.

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

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

Write down the amplitude of this graph

Find $\text{arctan}\frac{3}{4}$, giving the answer to 3 significant figures.

$A$.

Use your answers from part (a) to write down the value of $A$, $B$ and $D$.

Show that $\frac{b}{\sqrt{a^2+b^2}}=\text{cos}\theta$.

Hence prove your conjectures in part (e).

$C$.

Sketch the graph $y=3\text{sin}x+4\text{cos}x$, for $-2\pi ⩽x⩽2\pi$

Write down the period of this graph

Show that $\frac{a}{b}=\text{tan}\theta$.

Comment on your answer to part (c)(i).

Find the value of $C$.

$D$.

By considering the graph of $y=5\text{sin}x+12\text{cos}x$, find the value of $A$, $B$, $C$ and $D$.

$B$.

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

Find the exact area of the shaded region, giving your answer in the form $p\pi +q\sqrt{3}$, where $p$, $q\in \mathbb{Q}$.

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$-coordinate of A on the graph of $g$.

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

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

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.

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

Find $p_{av}$(0.007).

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

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 $d$.

Show that $f(2\pi )=2\pi$.

Find the coordinates of ${\text{P}}_0$ and of ${\text{P}}_1$.

Find the equation of $L$.

Show that the distance between the $x$-coordinates of ${\text{P}}_k$ and ${\text{P}}_{k+1}$ is $2\pi$.

A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.

How much time is there between the first low tide and the next high tide?

Find the difference in height between low tide and high tide.

Find the value of $p$;

Find the value of $q$;

Find the value of $r$.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

Find the coordinates of A.

For the graph of $f$, write down the amplitude.

For the graph of $f$, write down the period.

Hence, write $f\left(x\right)$ in the form $p\text{cos}\left(x+r\right)$.

Find the maximum speed of the ball.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

Find the value of $p$.

Find the value of $q$.

Use the model to find the depth of the water 10 hours after high tide.

The range of $f$ is $k$ ≤ $f\left(x\right)$ ≤ $m$. Find $k$ and $m$.

Find the range of $g$.

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

Find the period of $g$.

The equation $g\left(x\right)=12$ has two solutions where $\pi$ ≤ $x$ ≤ $\frac{4\pi }{3}$. Find both solutions.

Consider the graph of $g\left(x\right)=2\text{sin}px$, 0 ≤ $x$ < $2\pi$, where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y=-1$.

(i)     Find the value of $c$.

(ii)     Show that $b=\frac{\pi }{6}$.

(iii)     Find the value of $a$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.