DP Mathematics: Analysis and Approaches Questionbank

Given that $L$ meets ${\Pi }_1$ at the point $\text{P}$, find the coordinates of $\text{P}$.

Particle $P_2$ also moves in a straight line. The position of $P_2$ is given by $\mathit{r}=\left(\begin{array}{c}-1\\ 6\end{array}\right)+t\left(\begin{array}{c}4\\ -3\end{array}\right)$.

The speed of $P_1$ is greater than the speed of $P_2$ when $t>q$.

Find the value of $q$.

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

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 $c$.

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 $n$.

Show that $f$ is an even function.

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

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

Determine the value of $m$.

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

Given that $\left|\mathit{a}+\mathit{b}\right|$ is a minimum, find $\mathit{p}$.

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

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

Determine the length of time between the first airplane arriving at $\text{P}$ and the second airplane arriving at $\text{P}$.

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.

Write down the amplitude of this graph

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

$C$.

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

Find the length of EB.

Find the area, in km², of triangle ABC.

Show that ${\left(\text{sin}x+\text{cos}x\right)}^2=1+\text{sin}2x$.

Calculate the curved surface area of the remaining solid.

Let $f\left(x\right)=\text{tan}\left(x+\pi \right)\text{cos}\left(x-\frac{\pi }{2}\right)$ where .

Express $f\left(x\right)$ in terms of sin $x$ and cos $x$.

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

Find $p_{av}$(0.007).

Find the area of the shaded region.

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.

Find the distance from A to C.

the length of BD;

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

Express $\sin 2x$ in terms of $u$.

Hence show that $f(x)=0$ can be expressed as $u^3-7u^2+15u-9=0$.

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

Express this volume in $\text{c}{\text{m}}^3$.

Find $\frac{\text{d}A}{\text{d}r}$.

Find the volume of water in the container.

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

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.

Find the slant height of the cone-shaped container.

Find the length of arc ABC.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

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

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 length of BC.

Write down your answer to part (a) correct to the nearest integer.

Find the area of the triangle OPQ.

Find, in terms of a and b $\overrightarrow{\text{AF}}$.

Find, in terms of $b$, the equation of the line L which passes through M and is perpendicular to the plane П.

Given that a $\times$ b $=$ b $\times$ c $\ne$ 0 prove that a $+$ c $=$ sb where s is a scalar.

The vector equation of line $L$ is given by r $=\left(\begin{array}{c}-1\\ 3\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)$.

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.

find the value of $a$;

Three planes have equations:

$2x-y+z=5$

$x+3y-z=4$     , where $a\text{, }b\in \mathbb{R}$.

$3x-5y+az=b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

A straight line, $L_{\theta }$, has vector equation r $=\left(\begin{array}{c}5\\ 0\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}5\\ \text{sin}\theta \\ \text{cos}\theta \end{array}\right)\text{, }\lambda \text{, }\theta \in \mathbb{R}$.

The plane ${\Pi }_p$, has equation $x=p\text{, }p\in \mathbb{R}$.

Show that the angle between $L_{\theta }$ and ${\Pi }_p$ is independent of both $\theta$ and $p$.

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

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

Find $\overrightarrow{\text{PQ}}$.

Find the area of triangle ABC.

Find $\left|\overrightarrow{\text{PQ}}\right|$.

Find the angle between PQ and PR.

Find DC.

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

Find the angle between $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Show that $p=\frac{q^2+q-1}{2q+1}$.

Find the value of $r$.

Find DB.

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 $q$.

Solve ${\log }_2(2\sin x)+{\log }_2(\cos x)=-1$, for .

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

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

Find an expression for the velocity of $P_1$ at time $t$.

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

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

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

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

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

Show that $l_1$ and $l_2$ are never perpendicular to each other.

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.

the condition on the value of $p$.

Show that $\sin 2x+\cos 2x-1=2 \sin x\left(\cos x-\sin x\right)$.

Show that the point $(-1, 0, 3)$ lies on $L_1$.

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

Find the value of $l$.

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

Given that point $\text{A}$ is $12$ metres from the base of the windmill, find the height of point $\text{C}$ above the ground.

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 the value of $d$.

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

The line $L$ is the intersection of ${\Pi }_1$ and ${\Pi }_2$. Verify that the vector equation of $L$ can be written as $\mathit{r}=\left(\begin{array}{c}0\\ -2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$.

Find the coordinates of $\text{P}$.

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

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

Find the distance from the camp to point C.

Find the value of $b$.

Find the bearing that Jacob must take to point C.

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

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

Find the vertical height of B above the ground.

Find the size, in degrees, of angle BÂC.

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

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

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

Determine an expression for $f^{\prime }(x)$ in terms of $x$.

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

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

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

Calculate the length of MN.

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 volume of one cone-shaped container.

Calculate the volume of the paperweight.

Find the slant height, $l$, of the cone.

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

Find the area of the park.

(i)     Find the value of $c$.

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

(iii)     Find the value of $a$.

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

Find a unit vector in the direction of $L_2$.

parallel.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Write down your answer to part (b) in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$.

Calculate the radius of the balloon following this increase.

Find the distance between the two submarines at this time.

Find how many sets of four points can be selected which can form the vertices of a quadrilateral.

Find how many sets of three points can be selected which can form the vertices of a triangle.

Write down the value of $\lambda$ corresponding to the point $\text{A}$.

Let $\text{C}$ be the point on $l_1$ with coordinates (1, 0, 1) and $\text{D}$ be the point on $l_2$ with parameter $\mu =-2$.

Find the area of the quadrilateral $\text{CDBA}$.

Find the Cartesian equation of the plane ${\mathrm{\Pi }}_1$, passing through the points A , B and D.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\mu$.

Find a vector equation of the line L passing through the points A and B.

Find the coordinates of the point of intersection of the line L with the plane Π.

Verify that 2j + k is perpendicular to the plane Π${}_2$.

Find the coordinates of the point of intersection of the planes defined by the equations $x+y+z=3,x-y+z=5$ and $x+y+2z=6$.

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

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 $\text{AD}$

local maximum points;

Find the value of $p$.

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 < $k$ < $n$.

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

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.

Find $\left|\overrightarrow{\text{AB}}\right|$.

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

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.

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

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Given that $\tan \theta >0$, find $\tan \theta$.

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

Find the volume, in cm³, of this calculator case.

Find the minimum height of the ball above the ground.

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

Find the coordinates of the point where ${\text{Π}}_1$, ${\text{Π}}_2$ and ${\text{Π}}_3$ intersect.

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

Show that the equation $2 {\cos }^2 x+5 \sin x=4$ may be written in the form $2 {\sin }^2 x-5 \sin x+2=0$.

Hence, solve the equation $2 {\cos }^2 x+5 \sin x=4, 0\le x\le 2\mathrm{\pi }$.

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

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.

Find the value of $a$.

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

Hence find the equation of ${\Pi }_1$, expressing your answer in the form $ax+by+cz=d$, where $a, b, c, d\in \mathrm{\mathbb{Z}}$.

Describe these two transformations.

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

Find the height of the pyramid, $\text{VX}$.

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

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

Find the area of triangle ABC.

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

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

Find the value of $C$.

$B$.

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

By writing $\frac{\pi }{12}$ as $\left(\frac{\pi }{4}-\frac{\pi }{6}\right)$, find the value of cos $\frac{\pi }{12}$ in the form $\frac{\sqrt{a}+\sqrt{b}}{c}$, where $a$, $b$ and $c$ are integers to be determined.

Hence, or otherwise, show that S = $\frac{1}{2}\left(1+\sqrt{2}\right)\left(1+\sqrt{3}\right)\left(1+\text{i}\right)$.

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

Show that the volume of a cone shaped glass is $160\text{ c}{\text{m}}^3$, correct to 3 significant figures.

Calculate the radius, $r$, of a hemisphere shaped glass.

Show that there is $20\text{ c}{\text{m}}^3$ of orange paste in each special dessert.

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.

Find the radius, $r$, of the circular base of the cone.

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 value of $q$.

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

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

Find the equation of $L$.

Calculate the volume of this pan.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

In the context of this model, state what the value of 19 represents.

Calculate the volume of the balloon.

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

Calculate Abdallah’s estimate for the area.

Show that submarine B travels in the same direction as originally planned.

Show that P is the midpoint of AD.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\lambda$;

Find the Cartesian equation of the plane containing P, Q and R.

Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}6\\ 8\\ -5\end{array}\right)$

Find a vector equation of the line that passes through P and Q.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$.

At $\text{X}$, find the value of $p$ and the value of $\theta$.

$A$ and $B$ are acute angles such that $\text{cos}A=\frac{2}{3}$ and $\text{sin}B=\frac{1}{3}$.

Show that $\text{cos}\left(2A+B\right)=-\frac{2\sqrt{2}}{27}-\frac{4\sqrt{5}}{27}$.

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

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

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Show that $\frac{1-\cos 2x}{2\sin x}\equiv \sin x,x\ne k\pi$ where $k\in \mathbb{Z}$.

Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.

Hence or otherwise, find the distance between $P_1$ and $P_2$ two seconds after they leave A.

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

Find the angle of elevation from B to C.

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Find $\overrightarrow{\text{AB}}$;

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

Find the equation of ${\Pi }_2$, giving your answer in the form $\mathit{r}\mathbf{.}\mathit{n}=d$.

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

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

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

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

Find the vectors $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Write down the radius of the planet.

Hence or otherwise, solve $\sin 2x+\cos 2x-1+\cos x-\sin x=0$ for $0<x<2\mathrm{\pi }$.

Given any two non-zero vectors, $\mathit{a}$ and $\mathit{b}$, show that ${\left|\mathit{a}\times \mathit{b}\right|}^2={\left|\mathit{a}\right|}^2{\left|\mathit{b}\right|}^2-{\left(\mathit{a}·\mathit{b}\right)}^2$.

Find the value of $r$.

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

Show that $l_1$ and $l_2$ do not intersect.

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 $p, q\in {\mathrm{\mathbb{Z}}}^{+}$.

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.

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 $2x-3-\frac{6}{x-1}=\frac{2x^2-5x-3}{x-1}, x\in \mathrm{ℝ}, x\ne 1$.

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

By using a suitable substitution for $a$, show that $1-3 \cos 2x+3 {\cos }^2 2x- {\cos }^3 2x=8 {\sin }^6 x$.

Find $\text{AV}$.

Find the distance between $L$ and $\prod _3$.

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

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

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

Find the volume of the sphere expressing your answer in the form $a\times {10}^k$, and $k\in \mathbb{Z}$.

Find the period of $f$.

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

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

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

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

Find $\cos 2\theta$.

Given that $\text{sin}x=\frac{1}{3}$, where , find the value of $\text{cos}4x$.

Hence or otherwise find ${\int }_0^{\frac{\pi }{6}}\left(\text{sec}2x+\text{tan}2x\right)\text{d}x$ in the form $\text{ln}\left(a+\sqrt{b}\right)$ where $a$, $b\in \mathbb{Z}$.

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

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

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

The length of path AD is 287 m.

Find the area of the region ABCD.

the length of the fence, BP.

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

Find $\frac{\text{d}A}{\text{d}r}$.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Calculate the length of AE.

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

Find the percentage error on Giovanni’s diagram.

The total external surface area of the pencil case rounded to 3 significant figures is 570 cm².

Using your graphic display calculator, calculate the value of r.

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

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

Find the length of the perimeter of ABCDE.

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

A second line $L_2$, has equation r = $\left(\begin{array}{c}1\\ 13\\ -14\end{array}\right)+s\left(\begin{array}{c}p\\ 0\\ 1\end{array}\right)$.

Given that $L_1$ and $L_2$ are perpendicular, show that $p=2$.

The lines $L_1$ and $L_1$ intersect at $C(9,13,z)$. Find $z$.

Hence or otherwise, find one point on $L_2$ which is $\sqrt{5}$ units from C.

Write down an equation for the area, $A$, of the curved surface in terms of $r$.

perpendicular.

The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.

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

Find the maximum speed of the ball.

Find the percentage error in Abdallah’s estimate.

Find the value of t when submarine B passes through P.

Find an expression for the distance between the two submarines in terms of t.

Deduce an expression for $\overrightarrow{\text{CD}}$ in terms of a and b only.

Given that П₃ is parallel to the line $L$, show that $a+2b-5c=0$.

Find, in terms of $b$, a Cartesian equation of the plane Π containing this triangle.

Point C (k , 12 , −k) is on L. Show that k = 14.

Write down the value of angle OBA.

Using vector algebra, show that $\overrightarrow{\text{AD}}=\overrightarrow{\text{BC}}$.

Find the vector equation of the straight line passing through M and normal to the plane $\mathrm{\Pi }$ containing ABCD.

Find the coordinates of X, Y and Z.

The vector $\left(\begin{array}{c}2\\ p\\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\text{AB}}$. Find the value of p.

Find the vector equation of the line (BC).

Find the point of intersection of $L_1$ and $L_2$.

Find a vector equation for $L$.

The point D has coordinates $(q^2,0,q)$. Given that $\overrightarrow{\text{DC}}$ is perpendicular to $L$, find the possible values of $q$.

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 $f_{n+1}(x)=\text{cos}\left(n\text{arccos}x\right)\text{cos}\left(\text{arccos}x\right)-\text{sin}\left(n\text{arccos}x\right)\text{sin}\left(\text{arccos}x\right)$.

Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

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.

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

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

Find the gradient of $L$.

Find the shortest distance from the point $\text{O}(0, 0, 0)$ to ${\Pi }_1$.

Express $\overrightarrow{\text{AB}}$ in terms of $m$.

Find the value of $m$.

Consider a unit vector $\mathit{u}$, such that $\mathit{u}=p\mathit{i}-\frac{2}{3}\mathit{j}+\frac{1}{3}\mathit{k}$, where $p>0$.

Point $\text{C}$ is such that $\overrightarrow{\text{BC}}=9\mathit{u}$.

Find the coordinates of $\text{C}$.

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

Find $k$.

Find the length of new fence required.

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

the value of $m$.

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

Find a Cartesian equation of the plane ${\text{Π}}_3$ which is perpendicular to ${\text{Π}}_1$ and ${\text{Π}}_2$ and passes through the origin $(0, 0, 0)$.

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

Find the angle of elevation of point $\text{C}$ from point $\text{B}$.

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

Hence or otherwise, solve the equation  $2 \sin 2\theta -3-\frac{6}{\sin 2\theta -1}=0$  for  $0\le \theta \le \mathrm{\pi }, \theta \ne \frac{\mathrm{\pi }}{4}$.

Find the value of $a$.

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

Find the vector $\overrightarrow{\text{AB}}$ and the vector $\overrightarrow{\text{AC}}$.

Hence find the coordinates of $\text{P}$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane ${\Pi }_3$.

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

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.

Find the three-figure bearing on which airplane $B$ is travelling.

Hence or otherwise solve $\text{lo}{\text{g}}_3\left(\text{2}\text{sin}x\right)=\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)$ for .

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

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

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 $a$, $b\in \mathbb{Z}$.

Write down the period of this graph

$D$.

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

Find the roots of $z^{24}=1$ which satisfy the condition , expressing your answers in the form $r{e}^{\text{i}\theta }$, where $r$, $\theta \in {\mathbb{R}}^{+}$.

Show that Re S = Im S.

Let .

Find, in terms of b, the solutions of $\text{sin}2x=-a,0⩽x⩽\pi$.

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

the size of angle DAB;

the length of AP;

The following diagram shows triangle PQR.

Find PR.

Sketch a graph of $y=f^{\prime }(x)$ for .

Express $\sin x$ in terms of $u$.

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.

Calculate the area of the park.

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

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Find the total cost of the ingredients of one special dessert.

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

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

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

Find the value of $p$.

Find the length of the path CE.

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

Let a = $\left(\begin{array}{c}2\\ k\\ -1\end{array}\right)$ and b = $\left(\begin{array}{c}-3\\ k+2\\ k\end{array}\right)$, $k\in \mathbb{R}$.

Given that a and b are perpendicular, find the possible values of $k$.

Find the vector $\overrightarrow{\text{AB}}$.

Let $\overrightarrow{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 0\end{array}\right)$. Find $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

Find a $\times$ b.

Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point $(1,0,-1)$.

Find the area of the postage stamp.

Write down $\overrightarrow{\text{PA}}$ and $\overrightarrow{\text{PB}}$.

Find, in terms of a and b $\overrightarrow{\text{OF}}$.

Show that $\mu =\frac{1}{13}$, and find the value of $\lambda$.

Show that $5a-7c=4$.

Show that p = 1, q = 1 and r = 4.

Find YZ.

Find the area of the parallelogram ABCD.

Find a vector equation for L₁.

Find a vector perpendicular to the plane Π${}_3$.

determine the coordinates of the point of intersection P.

Hence, show that, if the angle between the faces $\text{ABV}$ and $\text{ACV}$ is $\theta$, then $\text{cos}\theta =\frac{p\left(3p-20\right)}{6p^2-40p+100}$.

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

the area.

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

$L_3$ passes through the intersection of $L_1$ and $L_2$.

Write down a vector equation for $L_3$.

Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by

r_A = $\left(\begin{array}{c}4\\ 3\end{array}\right)+t\left(\begin{array}{c}5\\ 8\end{array}\right)$

r_B = $\left(\begin{array}{c}7\\ -3\end{array}\right)+t\left(\begin{array}{c}0\\ 12\end{array}\right)$

where distances are measured in kilometres.

Find the minimum distance between the two ships.

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

Find $c$.

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

Find the value of $k$.

Find the value of $y$.

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 $y=-1$.

Find the area of triangle PQR.

Find AB.

Show that $\cos \theta =\frac{3}{4}$.

Calculate the area of triangle ABC.

Find the length of AC.

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

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

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

Find the value of $p$;

Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.
“Seaview” $(S)$ is at an angle of depression of 25°.
“Nauti Buoy” $(N)$ is at an angle of depression of 35°.
The following three dimensional diagram shows Barry and the two yachts at S and N.
X lies at the foot of the cliff and angle $\text{SXN}=$ 70°.

Find, to 3 significant figures, the distance between the two yachts.

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.

Determine the acute angle between ${\Pi }_1$ and ${\Pi }_2$.

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

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

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

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

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

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

Use a vector method to show that $\text{B}\hat{\text{A}}\text{C}=60°$.

Show that the Cartesian equation of the plane $\Pi$ that contains the triangle $\text{ABC}$ is $-x+y+z=-2$.

Find a vector equation of the line $L$.

Hence determine the minimum distance, $d_{\text{min}}$, from $\text{D}$ to $\Pi$.

Find the volume of right-pyramid $\text{ABCD}$.

Find the length of each blade of the windmill.

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

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

Show that at the point $\text{P}, \lambda =\frac{3}{4}$.

Find the reflection of the point $\text{B}$ in the plane ${\Pi }_3$.

A second ornament is in the shape of a cuboid with a rectangular base of length $2x \text{cm}$, width $x \text{cm}$ and height $y \text{cm}$. The cuboid has the same volume as the pyramid.

The cuboid has a minimum surface area of $S {\text{cm}}^2$. Find the value of $S$.

Find $\text{CÂB}$.

Let $D\left(t\right)$ represent the distance between airplane $A$ and airplane $B$ for $0\le t\le 2.5$.

Find the minimum value of $D\left(t\right)$.

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 $m$.

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

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

Find the area of the shaded region.

Find the distance from point A to point B.

Hence prove your conjectures in part (e).

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

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

Find $\overrightarrow{AB}$.

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

Find BD.

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

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

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.

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

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

Write down a formula for $A$, the surface area to be coated.

Hence find the cube roots of $z$ in modulus-argument form.

Find an expression for the slant height of the cone in terms of r.

Find the height, $h$, of this cylinder-shaped container.

Calculate the length of CF.

Find the area of sector OABC.

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

Calculate the total length of the track.

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

Hence, write down a vector equation for $L_1$.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

Find the angle between the faces ABD and BCD.

Find the Cartesian equation of ${\mathrm{\Pi }}_3$.

Show that L does not intersect the $y$-axis for any negative value of $b$.

Find $\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{AB}}$.

Explain why ABCD is a parallelogram.

By using a suitable scalar product of two vectors, determine whether $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is acute or obtuse.

Show that the line (BC) lies in the plane Π${}_1$.

Find the acute angle between the planes Π${}_2$ and Π${}_3$.

Comment on the positions of $\text{V}$ in relation to the plane $\text{ABC}$.

Find the equation of the horizontal asymptote of the graph.

the perimeter.

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

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

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

Write down a direction vector for $L_3$.

local minimum points.

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

local maximum points;

local minimum points;

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

A second line, $L_2$, is parallel to $L_1$ and passes through (1, 2, 3).

Write down a vector equation for $L_2$.

Find the value of $r$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Find the area of triangle ABC.

Find the value of $q$;

Find $\overrightarrow{\text{AB}}$.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

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.

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

The volume of the planet can be expressed in the form $\mathrm{\pi }\left(a\times {10}^k\right) {\text{km}}^3$ where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

Find the value of $a$ and the value of $k$.

Show that $f\text{'}\text{'}\left(x\right)=-\frac{1}{4\sqrt{{\left(1+x\right)}^3}}$.

It is given that the lines $L_1$ and $L_2$ have a unique point of intersection, $\text{A}$, when $a\ne k$.

Find the value of $k$, and find the coordinates of the point $\text{A}$ in terms of $a$.

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

Hence, find the volume of the cone.

Solve the equation $2 {\cos }^2 x+5 \sin x=4, 0\le x\le 2\mathrm{\pi }$.

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

Hence determine the value of $\theta$.

Find the smallest possible value of $c$.

Find the smallest possible value of $c$.

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 $\text{cot} \theta$.

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

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

Show that the three planes do not intersect.

The sphere is to be melted down and remoulded into the shape of a cone with a height of 14.8 cm.

Find the radius of the base of the cone, correct to 2 significant figures.

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

Find $\cos \theta$.

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

Calculate the curved surface area of the cone which has been removed.

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 $g$.

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 $d$.

Find AM.

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

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

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

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 the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

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

Calculate the length of side AC in km.

Find the value of $r$ when the area of the curved surface is maximized.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the cost of $100\text{ c}{\text{m}}^3$ of chocolate mousse.

Find the value of $x$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Calculate the area of triangle EAD.

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 perimeter of sector OABC.

Find the length of the path BD.

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

Find the radius of the sphere in cm, correct to one decimal place.

Find the area of ABCD.

Show that the two submarines would collide at a point P and write down the coordinates of P.

Verify that $\text{P}$ is the point of intersection of the two lines.

Given that area $\mathrm{\Delta }\text{OAB}=k(\text{area }\mathrm{\Delta }\text{CAD})$, find the value of $k$.

Given that П₁ and П₂ meet in a line $L$, verify that the vector equation of $L$ can be given by r $=\left(\begin{array}{c}\frac{5}{4}\\ 0\\ -\frac{7}{4}\end{array}\right)+\lambda \left(\begin{array}{c}\frac{1}{2}\\ 1\\ -\frac{5}{2}\end{array}\right)$.

Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.

Find and simplify an expression for a • b in terms of $t$.

Find the two possible coordinates of $\text{V}$.

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

Let $f\left(x\right)=4\text{cos}\left(\frac{x}{2}\right)+1$, for $0⩽x⩽6\pi$. Find the values of $x$ for which $f\left(x\right)>2\sqrt{2}+1$.

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

Consider the vectors a = $\left(\begin{array}{c}3\\ 2p\end{array}\right)$ and b = $\left(\begin{array}{c}p+1\\ 8\end{array}\right)$.

Find the possible values of p for which a and b are parallel.

Given that c = a + 2b, find c.

Calculate the probability that Iqbal passes at least two of the papers he attempts.

Find the coordinates of A.

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

Hence or otherwise find the shortest distance from R to the line through P and Q.

Draw and label the angle of depression on the diagram.

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

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

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

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

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

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

Show that $\text{AC}=\tan \theta$.

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

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.

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

Find a vector equation of $L_1$.

Find the possible values of $a$ when the acute angle between $L_1$ and $L_2$ is $45°$.

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

Find the minimum distance between $l_1$ and $l_2$.

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

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

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 $\mathrm{ABC}$.

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.

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 $r$.

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

Find the possible range of values for $\left|\mathit{a}+\mathit{b}\right|$.

Find $\mathit{q}$ such that $\left|\mathit{q}\right|=\left|\mathit{b}\right|$ and $\mathit{q}$ is perpendicular to $\mathit{a}$.

Find the area of this piece of land.

Show that airplane $A$ travels at a greater speed than airplane $B$.

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

Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

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

Show that $\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)=\text{lo}{\text{g}}_3\sqrt{\text{cos}2x+2}$.

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

$A$.

State the range of $f$.

Find the size of angle APB.

Show that $\text{sec}2x+\text{tan}2x=\frac{\text{cos}x+\text{sin}x}{\text{cos}x-\text{sin}x}$.

Calculate the radius of the base of the cone which has been removed.

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

the area of triangle ABD;

the area of quadrilateral ABCD;

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

Find the range of $g$.

Find the period of $g$.

Find the value of this minimum area.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

Calculate the total volume of the barn.

Show that Farmer Brown is incorrect.

Calculate the total length of metal required for one support.

Find the value of $h$.

Find the percentage error in her estimate.

Write down the value of x.

Find the curved surface area of the cone.

(i)     Find $w$.

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

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 the vector $\overrightarrow{\text{AC}}$.

Hence or otherwise, find the area of the triangle ABC.

Find the coordinates of A.

Calculate the volume of this hemisphere in cm³.

Give your answer correct to one decimal place.

Find the value of $a$.

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

Find the value of t when the two submarines are closest together.

Find a vector equation for L.

Find the area of the parallelogram ABCD.

Find the Cartesian equation of $\mathrm{\Pi }$.

Find the Cartesian equation of plane Π containing the points $\text{A}\left(6,2,1\right)$ and $\text{B}\left(3,-1,1\right)$ and perpendicular to the plane $x+2y-z-6=0$.