Further Pure Math (Trigonometry Ratio)

 



1. (2011/june/Paper02/q3)

In triangle $A B C, A B=5 \mathrm{~cm}, A C=3 \mathrm{~cm}$, angle $B=25^{\circ}$ and angle $C$ is obtuse.

(a) Find, to the nearest degree, the size of angle $C$. (3)

The point $D$ lies on $B C$ produced and $A D=3 \mathrm{~cm}$.

(b) Find, to 3 significant figures, the length of $C D$. (3)


2. (2012/jan/paper01/q9) 


Figure 1 shows a triangular pyramid $A B C D .$

$$\begin{aligned}&\angle B A C=\angle D A C=\angle B A D=90^{\circ} \\&A D=5 \mathrm{~cm}, A C=8 \mathrm{~cm} \text { and } A B=6 \mathrm{~cm}\end{aligned}$$

(a) Find, in degrees to the nearest $0.1^{\circ}$, the size of $\angle B D C$.

(b) Find, to 3 significant figures, the area of triangle $B D C$.

(c) Find the area of triangle $D A C$. (1)

The point $E$ lies on $C D$ so that $A E$ is perpendicular to $C D$.

(d) Find the exact length of $A E$.

(e) Hence, or otherwise, find in degrees to the nearest $0.1^{\circ}$, the size of the angle between the planes $D A C$ and $B D C$. (4)


3. (2012/jan/paper02/q2)


Figure 1 shows a right pyramid with vertex $V$ and base $A B C D E F$ which is a regular hexagon. The diagonal $A D$ of the base is $10 \mathrm{~cm}$ and $X$ is the mid-point of $A D .$ The height $V X$ of the pyramid is $12 \mathrm{~cm}$

(a) Find the length of $V A$. (2)

(b) Find, in degrees to 1 decimal place, the size of the angle between the plane $V A B$ and the base. (4)


4. $\left(2012 / \mathrm{june} /\right.$ paper01/q $\left.^{2}\right)$

In triangle $A B C, A B=8 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and $C A=7 \mathrm{~cm} .$

(a) Find, to the nearest $0.1^{\circ}$, the size of angle $B A C$. (3)

(b) Find, to 3 significant figures, the area of triangle $A B C .$


5. (2012/june/paper02/q11)


Figure 1 shows a truncated right pyramid. The base $A B C D$ is a square with sides of length $10 \mathrm{~cm}$. The top $E F G H$ is a square with sides of length $4 \mathrm{~cm} .$ The base is parallel to the top and $A E=B F=C G=D H$

The point $P$ is on the line $A C$ such that angle $A P E$ is a right-angle and $E P=12 \mathrm{~cm} .$

(a) Find, in centimetres, the exact length of

(i) $A C$

(ii) $E G$

(iii) $A P$

(b) Find, in centimetres to 3 significant figures, the length of $A E$.

(c) Find, in degrees to 1 decimal place, the angle between the line $A E$ and the plane $A B C D .$ (2)

The point $Q$ is on the line $A B .$ Angle $A Q P$ is a right-angle.

(d) (i) Show that $P Q=3 \mathrm{~cm}$.

(ii) Write down, in centimetres, the length of $A Q$.

(e) Find, in degrees to 1 decimal place, the angle between the line $A E$ and the line $A B$.

(f) Find, in degrees to 1 decimal place, the angle between the plane $A B F E$ and the plane $A B C D$


6. (2013/jan/Paper01/q6)


Figure 1 shows triangle $A B C$ with $A B=10 \mathrm{~cm}, B C=6 \mathrm{~cm}$ and $\angle B A C=28^{\circ} .$ The point $D$ lies on $A C$ such that $B D=6 \mathrm{~cm}$

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

(b) Find, to 3 significant figures, the length of $A D .$

(c) Find, to 3 significant figures, the area of triangle $A B C$.


7. (2013/june/paper01/q9)


Figure 3 shows a triangular prism $A B C D E F$

$A C D E$ is a rectangle. In triangle $A B C, A C=12 \mathrm{~cm}, \angle B A C=60^{\circ}$ and $\angle B C A=30^{\circ}$

(a) Find the exact length of $B C$. (3)

The point $P$ lies on the line $A C$ and $\angle B P C=90^{\circ}$

(b) Show that $B P=3 \sqrt{3} \mathrm{~cm}$ (2)

The angle between the plane $A F C$ and the plane $A C D E$ is $25^{\circ}$

(c) Find, to 3 significant figures, the length of $B F$

(d) Find the size of the angle between the line $B D$ and the plane $A C D E$, giving your answer in degrees to 1 decimal place. (4)

(e) Find, to 3 significant figures, the volume of the prism $A B C D E F$.


8. (2013/june/paper02/q1)


In triangle $A B C, A B=10 \mathrm{~cm}, A C=16 \mathrm{~cm}$ and $\angle B A C=35^{\circ}$, as shown in Figure $1 .$

(a) Find, to 3 significant figures, the area of the triangle $A B C$. (2)

(b) Find, in degrees to the nearest $0.1^{\circ}$, the size of the angle $A B C .$


9. (2014/jan/рарег01/q6)

In triangle $A B C, A B=x \mathrm{~cm}, B C=7 \mathrm{~cm}, A C=(5 x-6) \mathrm{cm}$ and $\angle B A C=60^{\circ}$

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

Using your value of $x$

(b) find, in degrees to 1 decimal place, the size of $\angle A C B .$


10. (2014/jan/paper02/q7)


Figure 3 shows a prism $A B C D E F G H I J$ which consists of a triangular prism $A B E F G H$ on top of a cuboid BCDEFHIJ.

$$ A B=A E=5 \mathrm{~cm}, \quad E B=8 \mathrm{~cm}, \quad E D=10 \mathrm{~cm}, \quad C I=15 \mathrm{~cm}$$

$P$ is the midpoint of $D C$.

Calculate, in $\mathrm{cm}$ to 3 significant figures,

(a) the length of $P G$ (3)

(b) the length of $A C$.

Find, in degrees to the nearest $0.1^{\circ}$,

(c) the size of the angle between $P G$ and the plane $C D J I$,

(d) the size of the angle between the plane $A G I C$ and the plane $C D J I$. $(3)$


11. (2014/june/paper01/q10)


A paperweight $A B C D E F G H I$ consists of a cuboid $B C D E F G H I$ and a right pyramid $A B C D E$ as shown in Figure $1 .$

$$E F=3 \mathrm{~cm}, \quad F I=4 \mathrm{~cm}, \quad I H=5 \mathrm{~cm}$$

The volume of the pyramid is equal to the volume of the cuboid.

(a) Show that the height of the pyramid is $9 \mathrm{~cm}$.

Find, in $\mathrm{cm}$ to 3 significant figures, the length of

(b) $A E$

(c) $E H$.

Find, in degrees to the nearest $0.1^{\circ}$, the size of

(d) the angle between $A E$ and the plane $E B C D$,

(e) the obtuse angle between the plane $A B E$ and the plane $B E I H$.


12. (2014/june/paper02/q11)

In triangle $A B C, \angle B A C=60^{\circ}, A B=(3 x-1) \mathrm{cm}, A C=(3 x+1) \mathrm{cm}$ and $B C=2 \sqrt{7 x} \mathrm{~cm}$.

(a) Show that $(9 x-1)(x-3)=0$

(b) Hence find the value of $x$, justifying your answer.

(c) Find, to the nearest $0.1^{\circ}$, the size of angle $A B C$.

(d) Find the exact value, in $\mathrm{cm}^{2}$, of the area of triangle $A B C$. (2)


13. (2015/jan/paper01/q1)

An equilateral triangle has sides of length $x \mathrm{~cm}$.

(a) Show that the area of the triangle is $\frac{\sqrt{3}}{4} x^{2} \mathrm{~cm}^{2}$

The length of each side of the equilateral triangle is increasing at a rate of $0.1 \mathrm{~cm} / \mathrm{s}$.

(b) Find the length of each side of the triangle when the area of the triangle is increasing at a rate of $\frac{\sqrt{3}}{10} \mathrm{~cm}^{2} / \mathrm{s}$


14. (2015/jan/paper02/q1)


In triangle $A B C, A B=x \mathrm{~cm}, A C=2 x \mathrm{~cm}$ and $\angle A B C=100^{\circ}$, as shown in Figure 1 .

(a) Find, in degrees to the nearest $0.1^{\circ}$, the size of $\angle B A C$.

Given that the area of triangle $A B C$ is $16 \mathrm{~cm}^{2}$,

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


15. (2015/jan/paper02/q9)


Figure 2 shows a triangular pyramid $A B C D$. $A B=B C=C A=10 \mathrm{~cm}$ and $D A=D B=D C=13 \mathrm{~cm}$. The point $E$ is the midpoint of $A C$.

(a) Find the exact length of

(i) $D E$

(ii) $B E$

(b) Find, in degrees to 1 decimal place, the size of the angle between the line $B D$ and the line $D E$.

(c) Find, in degrees to 1 decimal place, the size of the angle between the line $B D$ and the plane $A B C$.

(d) Find, in degrees to 1 decimal place, the size of the angle between the plane $A D C$ and the plane $A B C$.

(e) Find, to 3 significant figures, the volume of the pyramid $A B C D$.


16. (2015/june/paper01/q6)


Figure 1 shows $\triangle A B C$ with $A B=22 \mathrm{~cm}, A C=14 \mathrm{~cm}$ and $B C=20 \mathrm{~cm}$.

(a) Find, to 3 decimal places, the size of each of the three angles of $\triangle A B C$.

The bisector of angle $B A C$ meets $B C$ at $P$.

(b) Find, in $\mathrm{cm}$ to 3 significant figures, the length of $A P$.

(c) Find, to the nearest $\mathrm{cm}^{2}$, the area of $\triangle A B C$.


17. $(2015 /$ june $/$ paper02/q 7 text $)$


Figure 2 shows a solid $V A B C D E F G H$ which is formed by joining a cuboid $A B C D E F G H$ to a right pyramid $V A B C D .$ The height of the cuboid and the height of the pyramid are both $h \mathrm{~cm}$ and $F G=8 \mathrm{~cm}$ and $G H=6 \mathrm{~cm}$. The total volume of the solid is $256 \mathrm{~cm}^{3}$.

(a) Show that $h=4$

(b) Find, in $\mathrm{cm}$ to 3 significant figures, the length of $V F$.

Find, to the nearest $0.1^{\circ}$

(c) the angle between $V A$ and the plane $A B C D$, (3)

(d) the acute angle between the plane $V A B$ and the plane $A B H E$. (4)


18. $(2016 / \mathrm{jan} /$ paper01/q8)

A particle $P$ is moving along the positive $x$-axis. At time $t$ seconds $(t \geqslant 0)$, the acceleration $a \mathrm{~m} / \mathrm{s}^{2}$ of $P$ is given by $a=6-4 t$

When $t=0, P$ is at rest and the displacement of $P$ from the origin $O$ is 5 metres.

At time $t$ seconds, the velocity of $P$ is $v \mathrm{~m} / \mathrm{s}$ and the displacement of $P$ from $O$ is $s$ metres.

(a) Find, in terms of $t$, an expression for

(i) $v$

(ii) $S$

For $t>0, P$ comes to instantaneous rest at the point $A .$

(b) Find

(i) the value of $t$ when $P$ reaches $A$,

(ii) the distance $O A$


19. (2016/june/paper01/q3)

A right pyramid $A B C D E$ has a square base $A B C D$ of side $10 \mathrm{~cm}$.

The height of the pyramid is $8 \mathrm{~cm}$.

(a) Find, to 3 significant figures, the length of $A E$.

(b) Find, in degrees to the nearest degree, the size of the angle between the plane $A B E$ and the base $A B C D$ (3)


20. (2016/june/paper02/q1)

A triangle has sides of length $10 \mathrm{~cm}, 8 \mathrm{~cm}$ and $9 \mathrm{~cm}$.

(a) Calculate, in degrees to the nearest $0.1^{\circ}$, the size of the largest angle of this triangle.

(b) Find, to 3 significant figures, the area of this triangle.


21. (2017/jan/paper01/q5)


Figure 2 shows the quadrilateral $A B C D$ in which $A B=B C$.

$D C=8 \mathrm{~cm} \quad A C=12 \mathrm{~cm} \quad \angle A B C=120^{\circ} \quad \angle C A D=35^{\circ}$

Find

(a) the exact length, in $\mathrm{cm}$, of $A B .$

Given that angle $A D C$ is obtuse, find

(b) the size, in degrees to 1 decimal place, of angle $A D C$,

(c) the area, in $\mathrm{cm}^{2}$ to 3 significant figures, of the quadrilateral $A B C D .$


22. (2017/jan/paper02/q10)


Figure 1 shows a right prism $A B C D E F G H I J$. The base, $D E F G$, is horizontal and is a rectangle with $D G=E F=10 \mathrm{~cm}$. The midpoint of $E D$ is $M$.

The planes $A B C D E$ and $J I H G F$ are vertical.

$$\begin{aligned}&A E=C D=G H=F J=8 \mathrm{~cm} \\&A B=B C=H I=I J=6 \mathrm{~cm} \\&\text { Angle } B A C=30^{\circ}\end{aligned}$$

(a) Show that the length of $M D$ is $3 \sqrt{3} \mathrm{~cm}$. (2)

(b) Show that the length of $B M$, the height of the prism, is $11 \mathrm{~cm}$. (2)

(c) Find, in $\mathrm{cm}$ to 3 significant figures, the length $B G$. (3)

Find, in degrees to 1 decimal place

(d) the size of the angle between the planes $B C H I$ and $C H F E$,

(e) the size of the angle between the planes $A B I J$ and $B E F I$. (5)


23. (2017/june/paper01/q5)

In triangle $A B C, A B=10 \mathrm{~cm}, B C=7 \mathrm{~cm}$ and angle $B A C=40^{\circ}$

(a) Find, in degrees to the nearest $0.1^{\circ}$, the two possible sizes of angle $A C B$.

(b) Find, in $\mathrm{cm}$ to 3 significant figures, the difference between the two possible lengths of $A C$. (4)


24. (2017/june/paper02/q10)


Figure 2 shows a solid cuboid $A B C D E F G H$ with $E F=8 \mathrm{~cm}$ and $E H=3 \mathrm{~cm}$.

The angle between the diagonal $A H$ of the cuboid and the plane $A B C D$ is $45^{\circ}$.

The midpoint of $\mathrm{CH}$ is $\mathrm{N}$.

Find, in $\mathrm{cm}$ to 3 significant figures,

(a) the length of $\mathrm{CH}$,

(b) the length of $A H$,

(c) the length of $F N$. (3)

Find, in degrees to 1 decimal place, the size of

(d) the angle between the plane $B C E F$ and the plane $F G H E$, (3)

(e) angle $F N G$. (3)


25. (2017/june/paper02/q5)

In triangle $A B C, A B=x \mathrm{~cm}, B C=(4 x-5) \mathrm{cm}, A C=(2 x+3) \mathrm{cm}$ and angle $A B C=60^{\circ}$.

Find, to 3 significant figures,

(a) the value of $x$,

(b) the area of triangle $A B C$.


26. (2018/jan/paper01/q6)


Figure 1 shows the triangle $A B C$ with $A B=x \mathrm{~cm}, B C=(2 x-2) \mathrm{cm}, A C=(x+4) \mathrm{cm}$ and $\angle B A C=\theta^{\circ}$

Given that $\tan \theta^{\circ}=\sqrt{255}$ and without finding the value of $\theta$

(a) show that $\cos \theta^{\circ}=\frac{1}{16}$(2)

Hence find

(b) the value of $x$,

(c) the size, in degrees to 1 decimal place, of $\angle A B C$,

(d) the area, in $\mathrm{cm}^{2}$ to 3 significant figures, of triangle $A B C$. (2)


27. (2018/jan/paper02/q11)


A pyramid with a rectangular base $A B C D$ and vertex $E$ is shown in Figure $6 .$

The rectangular base is horizontal with $A B=12 \mathrm{~cm}$ and $B C=8 \mathrm{~cm}$.

The diagonals of the base intersect at the point $O$.

The vertex $E$ of the pyramid is vertically above $O$.

The height of the pyramid is $h \mathrm{~cm}$ and $A E=B E=C E=D E=10 \mathrm{~cm}$.

(a) Show that $h=4 \sqrt{3}$

(b) Find, in degrees to 1 decimal place, the size of angle $O C E$.

The angle between $O E$ and the plane $C B E$ is $\theta^{\circ}$

(c) Show that $\cos \theta^{\circ}=\frac{2 \sqrt{7}}{7}$ (3)

The point $P$ is the midpoint of $B E$ and the point $Q$ is the midpoint of $C E$.

(d) Find, in degrees to 1 decimal place, the size of the angle between the plane $O P Q$ and the plane $E P Q .$ (4)


28. (2018/june/paper01/q11)


Figure 3 shows the right pyramid $A B C D E$. The base of the pyramid, $A B C D$, is a rectangle with $C D=16 x \mathrm{~cm}$ and $A D=12 x \mathrm{~cm}$. The diagonals of the base intersect at the point $X$. The edges $E A, E B, E C$ and $E D$ are all of equal length. The size of the angle between $E A$ and the base $A B C D$ is $45^{\circ}$

Find, in terms of $x$

(a) the height, $E X$, of the pyramid,

(b) the length of $E A$ (2)

Find, in degrees to the nearest $0.1^{\circ}$, the size of

(c) the acute angle between the planes $A E B$ and $A B C D$,

(d) the acute angle between the planes $B E D$ and $A E C$.

The area of triangle $A E D$ is $250 \mathrm{~cm}^{2}$

(e) Find, to 4 significant figures, the value of $x$


29. (2018/june/paper01/q3)

In triangle $A B C, A B=12 \mathrm{~cm}, B C=9 \mathrm{~cm}$ and angle $B A C=42^{\circ}$

(a) Find, in degrees to the nearest $0.1^{\circ}$, each of the two possible sizes of angle $A B C .$

(b) Find, to 2 significant figures, the smaller of the two possible areas of triangle $A B C$.


30. (2018/june/рaper02/q1)

In triangle $A B C, A B=9 \mathrm{~cm}, B C=6 \mathrm{~cm}$ and $C A=8 \mathrm{~cm}$

Find, in degrees to the nearest $0.1^{\circ}$, the size of angle $B A C .$


31. (2019/june/paper01/q3)

In triangle $A B C, A C=7 \mathrm{~cm}, B C=10 \mathrm{~cm}$ and angle $B A C=65^{\circ}$

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

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


32. (2019/june/paper02/q11)


Figure 1 shows a right pyramid with vertex $V$ and square base, $A B C D$, of side $16 \mathrm{~cm}$.

The size of angle $A V C$ is $90^{\circ}$

(a) Show that the height of the pyramid is $8 \sqrt{2} \mathrm{~cm}$.

(b) Find, in $\mathrm{cm}$, the length of $V A$.

(c) Find, in $\mathrm{cm}$, the exact length of the perpendicular from $D$ onto $V A$.

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

(d) the angle between the plane $V A B$ and the base $A B C D$,

(e) the obtuse angle between the plane $V A B$ and the plane $V A D$. (3)


33. (2019/june/paper02/q4)

In triangle $A B C, A B=5 x \mathrm{~cm}, B C=(3 x-1) \mathrm{cm}, A C=(2 x+5) \mathrm{cm}$ and angle $A B C=60^{\circ}$

Find, to 3 significant figures, the value of $x$. (5)


34. (2019/juneR/paper01/q2)


Figure 2 shows triangle $A B C$ in which

$$A B=2 x \mathrm{~cm} \quad A C=3 x \mathrm{~cm} \quad B C=4 x \mathrm{~cm}$$

(a) Show that $\sin A B C=\frac{3 \sqrt{15}}{16}$

Given that the area of triangle $A B C$ is $\frac{75 \sqrt{15}}{64} \mathrm{~cm}^{2}$

(b) find the value of $x$. (2)


35. (2019/juneR/paper02/q8)


Figure 3 shows a right prism $A B C D E F .$ The cross section $B C F$ of the prism is a triangle.

$$A B=D C=12 \mathrm{~cm} \quad B C=A D=8 \mathrm{~cm} \quad B F=A E=10 \mathrm{~cm} \quad \angle F B C=\angle E A D=60^{\circ}$$

The point $N$ lies on $B C$ such that $F N$ is perpendicular to $B C$.

(a) Show that $B N=5 \mathrm{~cm}$(2)

(b) Find, in $\mathrm{cm}$ to 3 significant figures, the length of $E N$.

The midpoint of $B F$ is $X$ and the midpoint of $F C$ is $Y$.

(c) Find, in degrees to one decimal place, the size of the angle between the plane $A B C D$ and the plane $A X Y D$

(d) Find, in degrees to one decimal place, the size of the angle $A Y E .$


36. (2011/june/paper01/q10)


Figure 2 shows the pyramid $V A B C D$. The base $A B C D$ is a rectangle with $C D=6 x \mathrm{~cm}$ and $A D=8 x \mathrm{~cm}$. The diagonals of the base intersect at the point $N$. The edges $V A, V B$, $V C$ and $V D$ are all of equal length. The angle between $V A$ and the base $A B C D$ is $60^{\circ} .$

Find, in terms of $x$

(a) the height, $V N$, of the pyramid,

(b) the length of $V A$. (3)

Find, in degrees to the nearest $0.1^{\circ}$,

(c) the size of the angle between the planes $A V B$ and $A B C D$,

(d) the size of the angle between the planes $B V D$ and $A V C$. (3)

The volume of the pyramid is $1110 \mathrm{~cm}^{3}$.

(e) Find, to the nearest whole number, the value of $x$. (3)


37. (2016/jan/paper02/q12)


Figure 3 shows a right prism $A B C D E F G H$. The cross section $A B C D$ of the prism is a trapezium with $A B=D C$. The point $M$ lies on $A D$ and $B M$ is perpendicular to $A D$.

$$A B=8 \mathrm{~cm} \quad C D=8 \mathrm{~cm} \quad B C=8 \mathrm{~cm} \quad A D=16 \mathrm{~cm} \quad D E=20 \mathrm{~cm}$$

Given that $B M=p \sqrt{q} \mathrm{~cm}$ where $q$ is a prime number,

(a) find the value of $p$ and the value of $q$.

(b) Find the size of angle $B A M$ in degrees.

Find, in degrees to the nearest $0.1^{\circ}$

(c) the size of the angle between $E B$ and the plane $A D E H$,

(d) the size of the angle between the plane $B C E H$ and the plane $A D E H$.(3)


38. (2016/jan/paper01/q7)


Figure 1 shows the triangle $A B C$ with $A B=4 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and angle $B C A=30^{\circ}$

The point $D$ lies on $A C$ such that $B D=4 \mathrm{~cm}$ and angle $B D C$ is obtuse.

Find

(a) the size of angle $B D C$, giving your answer in degrees correct to 1 decimal place,

(b) the length, in $\mathrm{cm}$, of $A D$, giving your answer correct to 3 significant figures,

(c) the area, in $\mathrm{cm}^{2}$, of triangle $A B D$, giving your answer correct to 3 significant figures.


Answer

1. (a) 135 (b) $4.26$

2. (a) $70.2$ (b) $34.7$(c) 20(d) $\frac{40}{\sqrt{89}}$ (e) $54.8^{\circ}$

3. (a) $V A=13 \mathrm{~cm}(b) \theta=70 \cdot 2^{\circ}$

4. (a) $38 \cdot 2^{\circ}$ (b) 17.3

5. (a) (i) $AC = 10\sqrt{2}$ (ii) $EG = 4 \sqrt{2}$ (iii) $AP = 3\sqrt{2}$ (b) $AE = 12.7$ cm (3sf) (c) $EAP =70.5^{\circ}$ (d) (i) $PQ =  3$ (ii) $AQ = 3$ (e) $EAB =76.4^{\circ}$ (f) $76.0^{\circ}$

6. $\left(\right.$ a) $77.0^{\circ}$ (b) $5.09 \quad$ (c) $29.5$ 

7. (a) $\quad B C=6 \sqrt{3}$ (b) Show (c) $B F=11.1$ (d) $19.9^{\circ}$ (e) 347

8. (a) $A=45.9$ (b) $108.7^{\circ}$

9. (a) $x=2.79$ (b) $C=20 \cdot 2$

10. $(a) P G=19.8$ (b) $A C=13.6$ (c) $\theta=40.9^{\circ}$ (d) $\phi=72 \cdot 9$

11. (a) show (b) $A E=9.55$ (c) EH= $7.07$ (d) $70.4$ (e) $114.3^{\circ}$

12. (a) Show (b) $x=3$ (c) $70.9^{\circ}$ (d) $20 \sqrt{3}$

13. (a) Show (b) $x=2$

14. (a) $50.5^{\circ}$ (b) $x=4.55$

15. (a)(i) 12 (ii) $5 \sqrt{3}$ (b) $40.3^{\circ}$ (c) $63.6^{\circ}$ (d) $76.1$ (e) $V=168$

16. (a) $A=62.964$, $B=38.573^{\circ}, C=78.4637$ (b) $14.6$ (c) 13

17. (a) Show (b) $V F=9.43$ (c) $A=38.7$ (d) $\theta=33.69$

18. (a)(i) $ v=6 t-2 t^{2}$ (ii) $s=3 t^{2}-\frac{2 t^{3}}{3}+5$ (b) (i) $t=3$ (ii) 14

19. (a) $A E=10.7$ (b) $\theta=58^{\circ}$

20. (a) $\theta=71.8^{\circ}$ (b) $A=34.2$

21. (a) $4\sqrt{3}$ (b) 120.6 (c) 40.6

22. (a) Show (c) $B G=15.7$ (d) $67 \cdot 6^{\circ}$ (e) $34.7^{\circ}$

23. $\left(\right.$ a) $66.7^{\circ}, 113.3^{\circ}$ (b) $5.54$

24. (a) CH=8.54 (b) $A H=12.1$. (c) $F N=9.55$ (d) $70.7^{\circ}$ (e) $18.3^{\circ}$

25. (a) $x=4.86$ (b) $30.3$

26. (a) Show (b) $x=8$ (c) $58.8^{\circ}$ (d) 47.9

27. (a) Show (b) $43.9^{\circ}$ (c) Show (d) $98.2^{\circ}$

28. (a) $10x$ (b) $EA=10\sqrt{2}x$ (c) $59.0^{\circ}$ (d) $73.7^\circ$ (e) $x=1.804$

29. (a) $\angle B=74.9^{\circ}, 21.1^{\circ}$ (b) $A=19$

30. $\angle A=40.8^{\circ}$

31. $(a) 39.4^{\circ}$ (b) $33.9$

32. (a) Show (b) $VA=16$ (c) $DX=8 \sqrt{3}$ (d) $54.7^{\circ}$ (e) $109.5$

33. $x=2 \cdot 15$

34. (a) Show (b) $x=\frac 54$

35. (a) Show (b) $EN=14.8$ (c) $19.8^{\circ}$ (d) $42.8^{\circ}$

36. (a) $\quad V N=5 x \sqrt{3}$ (b) $V A=10 x$ (c) $65.2^{\circ}$ (d) $73.7^{\circ}$ (e) $x=2$

37. (a) $p=4, q=3$ (b) $60^{\circ}$(c) $16.5^{\circ}$(d) $19.1^{\circ}$

38. (a) $ 141.3^{\circ} $ (b) $ A D  =6.24$ (c) $A=7.81$


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