Figure 1 shows triangle $ABC$ where

(a) Show that $x$ satisfies the equation

(b) Hence find the exact area, in $\mathrm{cm}^2$, of triangle $ABC$

Give your answer in the form $9(p\sqrt{q} + \sqrt{r})$ where $p, q$ and $r$ are integers.

(a) Using a formula on page 2, show that

(b) Hence, or otherwise, solve in degrees to one decimal place

Figure 1 shows circle $C_1$ with radius 7 cm and circle $C_2$ with centre $O$ and radius 7 cm

The circles $C_1$ and $C_2$ intersect at the point $A$ and at the point $B$

The size of angle $AOB$ is $\theta$ radians

The perimeter of the region $R$ shown shaded in Figure 1 is $\frac{28\pi}{9}$ cm

(a) Find the exact value of $\theta$

(b) Find the area, in $\mathrm{cm}^2$ to 3 significant figures, of the region $R$

Figure 3 shows a right pyramid with vertex $V$ and square base, $ABCD$, of side 12 cm

The size of the angle between the plane $VAB$ and the base $ABCD$ is $30^\circ$

(a) Show that the height of the pyramid is $2\sqrt{3}$ cm

(b) Find, in cm, the exact length of $VA$

(c) Find, in $\mathrm{cm}^2$, the exact area of triangle $VAD$

(d) Find, in cm, the exact length of the perpendicular from $D$ to $VA$

(e) Find, in degrees to one decimal place, the size of the obtuse angle between the plane $VAB$ and the plane $VAD$

Triangle $ABC$ is such that

(a) Find in degrees, to one decimal place, the two possible values for angle $CAB$

(b) Hence, find in cm, to 3 significant figures, the shortest possible length of $AC$

(a) Using a formula from page 2, show that

Figure 2 shows part of the curve $C$ with equation $y = \mathrm{sin} x$ and part of the curve $D$ with equation $y = \mathrm{cos} 2x$

Curve $C$ and curve $D$ intersect at the point $A$

(b) Use algebra to find the exact coordinates of $A$

The shaded region $R$ is rotated through $360^\circ$ about the $x$-axis.

(c) Use algebraic integration to find the exact volume of the solid generated.

Give your answer in the form $\frac{a\sqrt{a}}{b}\pi$

where $a$ is a prime number and $b$ is an integer.

A circle has centre $O$ and radius 14 cm

A sector $AOB$ of the circle has a perimeter of 49 cm

Find, in $\mathrm{cm}^2$, the area of sector $AOB$

Figure 4 shows a right pyramid $ABCDE$ with

• vertex $E$

• horizontal square base $ABCD$ of side 6cm

• $EA = EB = EC = ED = 12$ cm

• vertical height of $h$ cm

(a) Show that $h = 3\sqrt{14}$

(b) Find, in degrees to one decimal place, the size of the angle between $EB$ and the plane $ABCD$

The midpoint of $AE$ is $P$ and the midpoint of $BE$ is $Q$

(c) Find, to the nearest degree, the size of the obtuse angle between the plane $EPQ$ and the plane $PQCD$

Solve the equation

for $0^\circ \leq A \leq 360^\circ$

In triangle $ABC$, $AC = 12$ cm, $BC = 14$ cm, $AB = x$ cm and angle $ABC = 30^\circ$

(a) Show that $x = P\sqrt{3} \pm \sqrt{95}$ where $P$ is a prime number.

(b) Hence or otherwise, find in $\mathrm{cm}^2$, the difference between the two possible areas of triangle $ABC$

Give your answer in the form $m\sqrt{n}$ where $m$ is a prime number and $n$ is an integer.

(a) Show that $\mathrm{cos}^4 \theta - \mathrm{sin}^4 \theta = \mathrm{cos} 2\theta$

(b) Hence, or otherwise, solve the equation

Give your solutions to 2 decimal places.

(c) Using calculus, find the exact value of $\int_{\frac{\pi}{16}}^{\frac{\pi}{8}} (\mathrm{cos}^4 2x - \mathrm{sin}^4 2x - 8 \mathrm{sin} 4x) \mathrm{d}x$

Give your answer in the form $a - b\sqrt{2}$ where $a$ and $b$ are rational numbers.

Figure 1 shows sector $OABC$ of a circle with centre $O$ and radius 6 cm

The size of angle $AOC$ is $\theta$ radians.

The perimeter of sector $OABC$ is $(12 + \pi)$ cm

The region shown shaded in Figure 1 is bounded by the arc $ABC$ and the line $AC$

The area of this region is $S \mathrm{cm}^2$

Find the exact value of $S$

Figure 2 shows a right pyramid $VABCD$ with vertex $V$ and rectangular base $ABCD$

Find in degrees, to one decimal place, the size of the angle between the plane $CVD$ and the base $ABCD$

Triangle $ABC$ is such that

Given that angle $ABC$ is obtuse,

find, in cm to one decimal place, the length of $AB$

Figure 1 shows a right prism $ABCDEF$

Triangle $CDE$ is a cross section of the prism.

• $\angle DCE = \angle ABF = 90^\circ$

• $\angle EDC = \angle FAB = 40^\circ$

• $DE = AF = 15$ cm

• $EF = DA = CB = 20$ cm

Find, in degrees to one decimal place, the size of the angle between the line $FD$ and the plane $DCBA$

(i) (a) Using a formula given on page 2, show that

(b) Hence, solve the equation

(ii) Using a formula given on page 2, solve, giving your solutions as exact values

In triangle $ABC$, $AB = 2x$ cm, $BC = 3x$ cm and $AC = 4x$ cm

The area of triangle $ABC$ is $50 \mathrm{cm}^2$

Find, to 2 decimal places, the value of $x$

Figure 1 shows triangle $ABC$ where

The area of triangle $ABC = 36 \mathrm{cm}^2$

(a) Show that $x = 9$

(b) Find, in cm to 3 significant figures, the length of $BC$

(c) Find, in degrees to one decimal place, the size of

(i) $\angle ABC$

(ii) $\angle ACB$

Figure 2 shows the sector $OPQ$ of a circle with centre $O$ and radius $r$ cm.

The area of the sector is $A \mathrm{cm}^2$

(a) Show that $A = \frac{r}{2}(21 - r)$

The area of the sector must be greater than or equal to $27 \mathrm{cm}^2$

(b) Find the set of possible values of $r$

(c) Hence write down the set of possible values of $\theta$

Figure 1 shows sector $ROS$ of a circle with centre $O$ and radius 2 cm
The size of angle $ROS$ is $\theta$ radians.

The area of sector $ROS$ is $\frac{\pi}{2} \mathrm{cm}^2$

(a) Find the exact value of $\theta$

The perimeter of sector $ROS$ is $P \mathrm{cm}$

(b) Find the exact value of $P$

Figure 1 shows the sector $AOB$ of a circle with centre $O$ and radius $3r$ cm

A circle with radius $r$ cm touches $OA$ and $OB$ and the arc $AB$

Angle $AOB$ is $\theta$ radians, where $0 < \theta < \frac{\pi}{2}$

(a) Find the exact value of $\theta$

The area of the region shown shaded in Figure 1 is $8\pi$$\mathrm{cm}^{2}$

(b) Find the value of $r$

Figure 5 shows a right triangular prism $ABCDEF$ where $ABCD$ is a rectangle.

Using a formula from page 2,

Without using a calculator,

The angle between the plane $AEB$ and the plane $ABCD$ is $65^\circ$

Using formulae from page 2, show that

Hence, or otherwise,

(c) solve, giving exact values in terms of $\pi$

(d) use algebraic integration to find the exact value of

Figure 4 shows a right pyramid $ABCDV$

The base of the pyramid is a rectangle where,

The edges $VA$, $VB$, $VC$ and $VD$ are all of equal length.

The angle between $VA$ and $ABCD$ is $45^\circ$

(a) Show that $VA = \frac{\sqrt{10}}{2}x$ cm

(b) Find in cm, the exact height of the pyramid in terms of $x$

Find, in degrees to one decimal place,

(c) the size of angle $VBA$

(d) the size of the obtuse angle between the plane $AVC$ and the plane $BVD$

Given that the volume of the pyramid is $9\sqrt{5}$$\mathrm{cm}^{3}$

(e) find the value of $x$

(a) Using a formula on page 2 show that $\cos 2A = 2\cos^2 A - 1$

(b) Hence show that $(2\cos^2 A - 1)^2 = \frac{\cos 4A + 1}{2}$

The curve with equation $y = \frac{\sin 2x}{2} + \frac{(2\cos^2 x - 1)^2}{2} + \frac{1}{8}$ has a stationary point $P$ in the range $0 \leq x \leq \frac{\pi}{6}$

(c) Find the exact coordinates of $P$

Figure 3 shows a right pyramid with a horizontal square base.

$O$ is the point of intersection of the diagonals of the base.

The vertex $V$ of the pyramid is vertically above $O$

(a) Show that $VO = \frac{\sqrt{2}}{2}x \mathrm{cm}$

(b) Find, in degrees, the size of the angle $AVC$

(c) Find, in degrees to one decimal place, the size of the angle between the plane $VAB$ and the plane $VDC$

The volume of the pyramid is $200 \mathrm{cm}^3$

Given that the volume of a pyramid $= \frac{1}{3} \times \text{base area} \times \mathrm{height}$

(d) Find to 3 significant figures, the value of $x$

(a) Using formulae on page 2, show that

(b) Show that $f(\theta) = \mathrm{sin} 2\theta$

(c) Solve, in radians to 3 significant figures, for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$, the equation

(d) Using calculus, find the exact value of

(a) Using the formulae on page 2, show that

(b) Show that

(c) Solve, for $0^\circ \leq x \leq 180^\circ$ the equation, $\frac{\mathrm{d}y}{\mathrm{d}x} = 0$

Give your answers to the nearest whole number.

Figure 2 shows triangle $ABC$ where

(a) Find, to 3 significant figures, the value of $x$

(b) Find, in degrees to one decimal place, the size of

(i) angle $ABC$

(ii) angle $ACB$

The bisector of angle $ABC$ meets $AC$ at the point $M$

(c) Find the area, in $\mathrm{cm}^2$ to 3 significant figures, of triangle $BMC$.

In triangle $ABC$, $AB = 3x$ cm, $BC = 5x$ cm and $\angle ABC = 110^\circ$

(a) Find, in degrees to one decimal place, the size of $\angle BCA$

The area of triangle $ABC$ is $24 \mathrm{cm}^2$

(b) Find, to 3 significant figures, the value of $x$

A logo, $AEBCD$, is shown shaded in Figure 2.

The straight line $ABC$ is the diameter of the semicircle $ADC$

$AEB$ is an arc of a circle with centre $O$

All angles are measured in radians.

• $BC = 2x$ cm

• $OA = OB = x$ cm

• length of arc $AEB = 1.8x$ cm

The perimeter of the logo is $P$

(a) Show that $P = ax(\pi + \pi \mathrm{sin} 0.9 + b)$ where $a$ and $b$ are constants to be found.

Given that $x = 10$ cm,

(b) find, in $\mathrm{cm}^2$ to 3 significant figures, the area of the logo.

Figure 1 shows a circle, centre $O$, with radius $r$ cm.

The points $A$, $P$ and $B$ lie on the circle.

The obtuse angle $AOB = \theta$ radians.

The area of the sector $APBO$, shown shaded, is $372.4 \mathrm{cm}^2$ and the length of the arc $APB$ is $53.2 \mathrm{cm}$.

Find, to 3 significant figures where appropriate, the value of

(i) $r$

(ii) $\theta$

Figure 4 shows a right pyramid with vertex $V$ and base $ABCDE$ which is a regular pentagon.

Find, in degrees to one decimal place, the size of the angle between the plane $VBC$ and the base $ABCDE$

(a) Use a formula on page 2 to show that $\mathrm{sin}^2 A = \frac{1}{2}(1 - \mathrm{cos} 2A)$

(b) Show that $\mathrm{sin}^4 x + \mathrm{cos}^4 x = \frac{3 + \mathrm{cos} 4x}{4}$

(c) Hence solve, in degrees to one decimal place, the equation

Figure 2 shows a rectangle $ABCD$ with $AB=10\,\mathrm{cm}$ and $AD=15\,\mathrm{cm}$. $E$ is the point inside the rectangle such that $DE=13\,\mathrm{cm}$ and $\angle BAE=35^\circ$.

Given that $\angle AED$ is obtuse, find the area, in $\mathrm{cm}^2$ to one decimal place, of triangle $BCE$.

Figure 1 shows sector $ORS$ of a circle with centre $O$ and radius $4\,\mathrm{cm}$. The size of angle $ROS$ is $\theta$ radians. The area of sector $ORS$ is $2\pi\,\mathrm{cm}^2$.

(a) Find the exact value of $\theta$.

(b) Find the perimeter, in $\mathrm{cm}$ to 3 significant figures, of the sector $ORS$.

The point $T$ lies on $OR$ such that $OT:TR=1:3$. The shaded region is bounded by $TR$, $TS$ and the arc $RS$. The area of this region is $A\,\mathrm{cm}^2$.

(c) Find the exact value of $A$.

Given that

(a) without using a calculator, find the value of $a$.

Show your working clearly.

Triangle $PQR$ is such that

(b) Show that

Given that

and that the area of triangle $PQR$ is $A\,\mathrm{cm}^2$,

(c) find the exact value of $A$ in the form

where $p$, $q$, $r$ and $s$ are integers.

Using suitable results for $\mathrm{sin}\,(A+B)$ and $\mathrm{sin}\,(A-B)$,

(a) show that

Figure 4 shows a sketch of part of the curve with equation

(b) Use calculus to find the total area of the regions bounded by the curve and the $x$-axis between $x=0$ and $x=\frac{\pi}{2}$.

Give your answer to 3 significant figures.

Figure 1 shows two circles, $C_1$ and $C_2$, each with a radius of $6\,\mathrm{cm}$.

The centre of $C_1$ is $O_1$ such that $O_1$ lies on $C_2$. The centre of $C_2$ is $O_2$ such that $O_2$ lies on $C_1$.

The circles intersect at the points $A$ and $B$ and enclose the region $R$, shown shaded in Figure 1.

The area of region $R$ is $P\,\mathrm{cm}^2$.

Find the exact value of $P$, giving your answer in the form

where $a$, $b$ and $c$ are integers.

(i) (a) Using a formula from page 2, show that

Given that $\mathrm{tan}\,2\alpha=1$,

(b) show that

where $a$ and $b$ are integers whose values need to be found.

(ii) (a) Using formulae from page 2, show that

can be written as $\mathrm{tan}\,x^\circ=1$.

(b) Hence, or otherwise, solve

for $-90<y\leq90$.

Figure 1 shows a right pyramid $VABCD$ with vertex $V$ and square base $ABCD$. Each of the edges of the pyramid has the same length.

Find the size, in degrees to one decimal place, of the angle between the plane $CVD$ and the base $ABCD$.

Solve

(a) $$\mathrm{cos}\,(3x-15)^\circ=\frac{\sqrt{3}}{2}, \qquad 0\leq x<180,$$

(b) $$3\mathrm{tan}\,y^\circ+4\mathrm{sin}\,y^\circ=0, \qquad -180\leq y<180,$$

giving your answers to one decimal place where appropriate,

(c) $$\mathrm{cos}\,\theta^\circ=3\mathrm{sin}^2\theta^\circ-1, \qquad -180\leq\theta<180,$$

giving your answers to one decimal place where appropriate.

Angle $\alpha$ is acute such that $\mathrm{cos}\,\alpha=\frac35$.

Angle $\beta$ is obtuse such that $\mathrm{sin}\,\beta=\frac12$.

(a) Find the exact value of

(i) $\mathrm{tan}\,\alpha$,

(ii) $\mathrm{tan}\,\beta$.

(b) Hence show that

where $m$ and $n$ are positive integers whose values are to be found.

Figure 1 shows a sector $OPQ$ of a circle with centre $O$. The radius of the circle is $18\,\mathrm{cm}$ and the angle $POQ$ is $\frac{2\pi}{3}$ radians.

(a) Find the length of the arc $PQ$, giving your answer as a multiple of $\pi$.

Figure 2 shows the sector $OPQ$ and the kite $OPTQ$. $PT$ is the tangent to the circle at $P$ and $QT$ is the tangent at $Q$, such that angle $PTQ=\alpha$ radians.

(b) (i) Find $\alpha$ in terms of $\pi$.

(ii) Calculate, to 3 significant figures, the area of the region, shown shaded in Figure 2, which is bounded by the arc $PQ$ and the tangents $PT$ and $QT$.

(a) Solve the equation

Give your solutions to the nearest whole number.

Given that

(b) show that

(c) Hence solve the equation

Give your solutions to the nearest whole number.

Figure 1 shows the right pyramid $ABCDE$. Its base $ABCD$ is a horizontal rectangle with $AD=12\,\mathrm{cm}$ and $CD=16\,\mathrm{cm}$. The height $ME$ is $14\,\mathrm{cm}$, where $M$ is the point of intersection of the diagonals of the base. The sloping edges are all of equal length.

(a) Calculate, to 3 significant figures, the length of a sloping edge.

Calculate, in degrees to one decimal place, the size of

(b) the angle between $AE$ and the base,

(c) the angle between the plane $AED$ and the base.

Figure 3 shows triangle $ABC$ with $AB=12\,\mathrm{cm}$, $BC=6\,\mathrm{cm}$ and $AC=2x\,\mathrm{cm}$. The point $D$ is the midpoint of $AC$ and $BD=6\,\mathrm{cm}$.

$\angle ABD=\theta^\circ$ and $\angle DBC=\phi^\circ$, where $\theta\neq0$ and $\phi\neq0$.

(a) Show that

(b) Hence, or otherwise, show that

(c) Show that

(d) Hence show that

Here is a formula

(a) Find the exact value of $P$ when $t=\frac{10}{3}$.

(b) Find

(i) the largest value of $P$,

(ii) the smallest value of $P$.

(c) Find the least value of $t$ for which $P=4$.

(a) Show that

(b) Hence express $2\mathrm{sin}\,7x\,\mathrm{cos}\,x$ in the form

where $m$ and $n$ are integers, giving the value of $m$ and the value of $n$.

(c) Use calculus to evaluate

Solve

(a) $$\mathrm{sin}\left(x+\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}, \qquad 0\leq x\leq2\pi,$$

giving your answers in terms of $\pi$,

(b) $$3\mathrm{sin}\,\theta+5\mathrm{cos}\,\theta=0, \qquad -360^\circ\leq\theta\leq360^\circ,$$

giving your answers to the nearest degree,

(c) $$1+\mathrm{sin}\,2y=2\mathrm{cos}^2 2y, \qquad -180^\circ\leq y\leq0^\circ.$$

(a) Show that

(b) Hence show that

(c) Solve, for $0\leq\theta\leq\frac{\pi}{2}$, the equation

Give each solution in terms of $\pi$.

(d) Show that

(e) Use calculus to find the exact value of

Figure 1 shows triangle $ABC$ in which $AB=10\,\mathrm{cm}$ and $AC=12\,\mathrm{cm}$. The point $D$ lies on $BC$ such that $BD=6\,\mathrm{cm}$, $DC=2\,\mathrm{cm}$ and $AD=x\,\mathrm{cm}$.

(a) Show that $x=11$.

(b) Find the area, in $\mathrm{cm}^2$ to 3 significant figures, of triangle $ADB$.

In Figure 2, $AB$ and $AC$ are tangents to a circle with centre $O$ and radius $r\,\mathrm{cm}$.

The points $B$ and $C$ lie on the circle so that $OBC$ is a sector of this circle and

Given that the area of the shaded region is $10\,\mathrm{cm}^2$, find, to 3 significant figures, the value of $r$.

In triangle $ABC$, $AC=7\,\mathrm{cm}$, $BC=10\,\mathrm{cm}$ and angle $BAC=65^{\circ}$.

(a) Find, to the nearest $0.1^{\circ}$, the size of angle $ABC$.

(b) Find, in $\mathrm{cm}^{2}$ to 3 significant figures, the area of triangle $ABC$.

Figure 1 shows a sector $OAB$ of a circle where angle $AOB=\theta$ radians. The circle has centre $O$ and radius $15\,\mathrm{cm}$. The point $C$ divides $OA$ in the ratio $2:1$ and the point $D$ divides $OB$ in the ratio $2:1$.