Surds

CHAPTER The Golden Ratio is an irrational number that is

approximately equal to 1,618. It is found by dividing

03
line segment into two parts such that the ratio of
the length of the longer part to the length of the shorter
par t equal to the ratio of the length of the entire line
segment to the length of the longer part.

CT
b

a + b

Golden ratio, 9=%=4ib- i = 1.618

We o f t e n u s e t h e G r e e k a l p h a b e t i p ( p r o n o u n c e d a s p h i )
to refer to this ratio. The golden ratio has bean observed
and employed in many diverse fields, such as architecture,
financial markets and ar t. Did you know that the exac t
1+15
value of the golden ratio is a surd, 2 See Extend,

question 2 on page 62.

I n t h i s c h a p t e r, w e w i l l l e a r n t o m a n i p u l a t e s u r d s a n d s o l v e
equations involving surds.

T h e h e i g h t a n d w i d t h a f t h e Po r t h e n o n i n At h e n a ,
G r e e c e , a r e a p p r ox i m a t e l y i n t h e r a t i o 1 : 1.6 1 8.
 Surds 51

3.1 Manipulating Surds

You Will Learn To
Ad d , s u b t r a c t a n d m u l t i p l y s u r d s
R a t i o n a l i s e t h e d e n o m i n a t o r o f a f r a c t i o n t h a t i nvo l ve s s u rd [ s )

rational number is a number that can be expressed in

Think
the form elia where a b and b are integers. For example, Deeper.O
A student thinks that x is a
1, 0.5 a n d - e a re r a t i o n a l n u m b e rs .
rational number since it can be

When a number cannot be expressed as a ratio of two w r i t te n a s 4 W h a t i s w ro n g ?

integers, it is an irrational number. For example, T, 2

and -45 are irrational numbers.

A surd is an irrational number that is written using a

Think
Deeper,O
radical (V or root sign. It has an infinite number of

non-recurring decimals. For example, V2 = 1.414... . Is J16 a surd?

We u s e t h e s e p ro p e r t i e s t o s i m p l i f y s u rd s .

For a, b > 0, Va x Jb = Vab

BIG IDEA
J E
Na Equivalence
Va x Ja - a We can write
Va x Vb fab as

We need to simplify surds when the numbers under the a* xbl =(abyt How do you
r a d i c a l s i g n h ave s q u a re n u m b e rs a s fa c to rs .
express E-A similarly?

For ex amp le, V27 = V9 x 3

V9 xV3 Na x Jb = Jah
=3xv3
- 343
3-3 cannot be further simplified. We say 3/3 is the simplest
form of V27.

Example 1 Try 1
Simplify each of the following. Simplify each of the following.

(a) (J7) (b) 432

(e) (v5) (b) Vr2
(c) V3 x J27 (d) 1500 (c) V12 x +3 WO AE
V20 Answers
(a) 7 (b) 4/2
(c) 6 (d) 5,2
52 Chapter C3

Solution M a ke
Connection
(a) (V5) = V5 x V5 Na x Va =u
= 5 N5 is a real root of the equation x* = 5.

(5) V12 = J4 x 3
- 4 xJ3 Va x Vb = Vab
BIG IDEA
- 2xV3 Equivalence
= 2v3
What is wrong with the statements below?

(c) V3 x J27 = J3 x 27 Ja x Jb - Vah

VEDX(D=VI
V8I
=9 But -DX(-T) = V-1 x J-1
-(F)
(d) (1500 - 1500 20 *EE Hence 1 = -1.

V75
V25 x 3
Tate
425 x J3 Va x Vb = Jab Note :
- 5 x V3 Always reduce surds to the
simplest form in the final answer.
= 5V3

We manipulate surds in a similar way as algebraic terms.

Think
Surds
Deeper,O
Algebraic terms

W hy c a n we
Like surds: 42, -342 manipulate surds
Note: Like surds are surds with the in similar way as
Like ter ms: x. -3.x
SW I 1 I e r a d i c a n d ( t h e o u m b e r u n d e r t h e algebraic terms?
radical sign).

Ad d a n d s u b t r a c t l i k e s u r d s . analogous
Add and subtrac t like ter ms.
to
242+3N2-42-(2+3-1N2 2x + 3x -x - (2 + 3 -1)r
= 4r
= 42
Multiply surds.
Multiply al gebraie terms.
(3JE)(445) = (3 x 4)(J5 xJ5) (3r)(4y) = (3 4)(x - y)
- 12410 = 12y

Example 2- Try 2-
Simplify each of the following. Simplify each of the following.

(a) 643 + 7J3 - 5V3 (b) V32 -V8 + 3V2 (a) 7V5 - 345 + 2/5
(b) ,50 - V8 -742
Solution A n swe rs

(a) 645
(a) 643 - 7J5 - 545 = (6 + 7 -5)V3
(b) 1042
= 8v3
 Surds 53

(b) V32 - V8 + 3V2 = J16 x2 - 4x2 +3N2 Think

- 42 - 2/2 +3V2 Deeper.C
- 542
$ J32 - 8 = 432 - 8 ? Why?

Example 3 Try 3
Simplify each of the following. Simplify each of the following.

(a) (4+ 2J5)(9 - 5v3) (a) (2 + 445 5-35)

(b) (3-445)
(b) (4 + 32)
(e) (45 +2)(J5 -2)
(c) (345 + 2JZ)(345 -242)
A n swe r s

Solution (a) -50 + 14V5

(b) 57 - 24v3
(c )
(a) (4 + 245)(o - 5V3)

= 4(9) - 4(545) + 245(9)-(25)(543) (a + b)(c + d) = ac + ad + bc + bd

=36 - 20V3 + 18J3 - 10(3)

= 6 -243

(b) (4+3JD) =4 +200(32) + (32) (a + b)2 = a° + 2ab + b

16 + 24v2 + 9(2)
- 34 + 24V2

te (A45 + 2JE)(845 2E)-(A5) (AVD) (a+hya-b)=w-b

- 9(5) - 4(2)
- 37

From Example 3, we see that the product of surds can be rational or irrational.
Under what condition is the product of two surds a rational number?

Activity 3A• [ I nve s t i g a l e t h e n a t u re o l t h e p ro d u c t o f t wo s u rd s ]

(i) Which of the following gives a rational number?

(a) V5 x V5 (b) v3 x2.2
What would you multiply the surd N k by to a g e t a r a t i o n a l n u m b e r ?

(ii) Which of the following give a rational number?

(a) (3+ J2)(3-J2) (b) (J2 +1)V2 -1)

(c) (1+33)(1+ 33) (d) (5+243)X5-243)
What would you multiply the surd a + hyk by to get a rational number?
(iii) Which of the following give a rational number?

(a) (V5 - V3)(J5 + J3) (b) (2J7 +J3)247 -J3)

(e) (245 + 4J3 )(245 -443) (d) (245 + 33)(2/5 + 3/3)
What would you multiply the surd ah + bJk by to get a rational number?
54 Chapter C3

afh + bJk and anfi - huR are called conjugate surds. In particular, a + bJR and a -bfk
a re a l s o c o n j u g a te s u rd s .

F ro m t h e a c t i v i t y, we h ave t h i s re s u l t .

T h e p ro d u c t o f c o n j u g a t e s u rd s i s a l ways a r a t i o n a l n u m b e r.

Rationalising the Denominator of a Fraction

When a surd exists in the denominator of a fraction, we usually simplify the fraction by
removing the surd from the denominator. Such a process of simplifying the fraction is called
rationalising the denominator.

We a p p l y t h e s e t e c h n i q u e s t o r a t i o n a l i s e t h e d e n o m i n a t o r o f a f r a c t i o n t h a t i nvo l ve s s u rd ( s ) .

de
If the denominator is in the form JR, multiply the numerator and denominator by R.

For example, 77 7

2
If the denominator is in the form a bek, multiply the numerator and denominator

by its conjugate a -buk.

2 2
For example,
21 V3 2113 3 2(24 D
- 3 -2(2-45).

If the denominator is in the form a fh + byk, multiply the numerator and denominator
by its conjugate anh - bJk.
10 10
For example,
2V2 -V3
245+43
242 -V3 22 115 - 10l245 +5 - 2(245 +45).
4(2) -3

Example 4 Try 4
Simplify each of the following by rationalising the Simplify each of the following
denominator. by rationalising the
d e n o m i n a t o r.
10
(a) D ta) +
(b)
5-245
(b)
3-15 (c)
15-32
2/5 + 242
43-32 A n swe rs
(c) 245 -32
(a)
27 (b) 10+ 4V5

11-4410
(c )
6
 S u rd s 5 5

Solution Take
Note :
W#+*$ Multiply /2 by .2 to
rationalise the denominator. A fraction that contains surd|s) in the

N2 denominator is not in the simplest form.

We a l w a y s r a t i o n a l i s e t h e d e n o m i n a t o r
to simplify the fraction.

(b) 0 E5 Multiply 3 - V5
3-15 3-V5 3+15
8(3+v5)
by 3 + V5 to
rationalise the BIG IDEA
denominator. Equivalence
32 (V5)
We have learned that multiplying the
8{3+J5)
9-5 numerator and denominator of a fraction
by the same whole number results in an
equivalent fraction. This concept can be
- 2(3+V5) extended to fraction containing surd(s).
- 6+ 2.5 When we rationalise the denominator
of such fraction, we will get another
fraction that has the same value as the
original one.

4N3 -342 _ 443 -342 x 245 +3E Multiply 2/5 - 3/2 by 243 + 3/2
(c)
2,3 -32 23-32 2V3 + 342 to rationalise the denominator.
8(3) + 12/6 - 6V6 - 902)

(23) -(32)
6 + 66
4(3) - 9(2)

6 + 66
-6
--1- VG

Example 5- -Try 5-
A triangle is such that its area is A rectangle is such that its

(9+4)3) em* and the base is a re a i s ( 8 6 + 1 1 4 7 ) c m ? a n d

the width is (2+ 3)7) cm.
(8 - 443) cm. Find its height in the form Find its length in the for m

(4+ b4)3) cm, where a and h are rational a + b u f 7 ) c m , w h e re G and h

numbers. are integers.

Answer
(8 -43) €c m (1+4/7) em
56 Chapter C3

Solution
Let h cm be the height of the triangle.

¼(8 -443)h = 9+443 Area

of triangle = 7 x base x height

(4- 243)h = 9 + 445

M= 9+ 4N5
4-2/3
9+4v5 x 4+23 Multiply 4 -2/3 by 4 + 2/3 to rationalise the denominator.
4 -243 4+ 243
36 + 18/3 + 1645 + 8(3)

4° -(245)
60 + 343
16 - 4(3)

2(30+1743)

15+ 1243
2 V-

The height of the triangle is {15 + 44,5

Let us now do the following activity to see how the concept of conjugate surds is involved
in solving quadratic equations.

Activity 3B [Investigate how conjugate aurda are involved in solving quadratic cquationa]

(i) Use the quadratic formula to find the exact solution(s) to each of the following equations.
(a) 5r. 3x- 2 =0
(b) 6x - 5x + 1 = 0
(c) 7-2x -1-0
(d) 3x +4r-7=0
(e) 4x - 4x - 1= 0
(f) x-243r-3-0 Make
(il) Which of the equations in part (i) have roots that are Connection
surds?

(iii) For each of the equations in part (ii), multiply the a Solving a quadratic equation

surd roots. ls the product a rational number or an may yield a pair of conjugate
surds.
irrational number? What is the relationship between
the two surd roots?
 Surds 57

Get the Master the

Exercise 3.1
Challenge
Basics Right C o n c e p ts fou ys e lf

Simplify each of the following. Simplify each of the following by

(a) 4/2 + 342 -2/2 r a t i o n a l i s i n g t h e d e n o m i n a t o r.
(b) 2/3 V3 +242 (a) 4V12 - 203 - 18

Le % e
Simplify each of the following.
(a) 2V3 x J6
(5) 7-1+ ⅝
(5) (2 +43)
(c) (242 -3)(342 -1)
Simplify each of the following by
Simplify each of the following by rationalising the denominator.
r a t i o n a l i s i n g t h e d e n o m i n a t o r.

76 (a)
(a) 443 -2 4V3 +2

(b)
2 (b)
(247-3) (247+3)
2.3
(c)
12 (c) 6(42+33)-&R
*N4 T 2V3) - 542-203
2V5 - 4

Explain why
Which number is larger, V5 or 7s? Va - Vb Va + b - 2Nab,
Explain how you obtained your answer where a > b> 0.
without using a calculator.

10 Projectile Motion. An object is

Simplify (7 + 32) - (5-205). released from rest from a building. Its
velocity, v m/s, is given by V205,
Simplify each of the following by where s m is the distance travelled
rationalising the denominator. by the object. Find the value of v in

(a)
3/2- 4 the simplest surd form when
4+ 3/2 (i) s = 40,
(ii) S = 90.
(b) 45+22 Hence find the change in velocity when
N3-242
the object travels from 40 m to 90 m.
(c)
5
11 (a) Express (3-J3) in the
2-13
(d)
50 -/48 form a + bJ3, where a and b are
integers.
(b) Given that s = 1 - V5, express
77+1
in the form e + d V5, where
5+2
c and d are fractions in the
simplest form.
58 Chapter C3

12
12 (a) An equilateral triangle of side x cm 15 Simplify
3 +V5 + 2,2
has an area of (3+ /15) cm".
Given that sin 61° 2
NE. find the Given that x = 3 | 22, find the value
exact value of x° in the form of -+
x(J3 + 45) cm*, where k is an k
integer. You may use the formula
1 (a) (i) Express (2 N3) in the form
2 ab sin C for the area of a+ b3, where a and b are b
triangle ABC, where a and b are integers.
the sides of the triangle and C is (ii) Hence find the two square
the included angle. roots of 28 - 1643.
(b) Find the square roots of
34 - 2442 in the form c + d2,
where and d are integers.

18 Physics. Two resistors with resistances

B
R ohms and R, ohms are connected in
paral le l in an el ec t ric circu it . Th e total
(b) The triangle ABC' is such that its

area is (3+ J15) em2, the height 11

resistance, R ohms, is given by
1
R R R The voltage, V volts,
AB is (V5 - V3) cm and angle
across the circuit is given by V = IR,
ABC is 90°. Find the length of BC where / amperes is the current flowing
t h ro u g h t h e c i rc u i t .
in the form (145 + 4N3) cm,
where p and 4 are integers. Given that R 16(642+ 75).
R, - V3 +342 and I - 5/6, find in
13 A right circular cone has a vertical
the simplest surd form, the value of
height of (2V3 -J2) cm and a slant a (i) V.
height of / cm. The volume of the cone (ii) R..
is VIS V48)r em*. Without using
(i) Show that
a c a l c u l a to r, ex p re ss / " i n t h e fo r m

(a - b6) em", where a and b are Va t Va1i Va+1 - Va.

integers. (ii) Hence find the value of
1
14 Given that p and g are two distinct 1+ V2 V2+3 V3+14
and real roots of the equation
7 1 3 = J20x, where p > g, express V8 + 19
L in the form mn + INTO where m
q 3
and F are integers.
 Surds 59

You Will LearnSTo 3.2 Selving Equations Involving Surds

olve equations involving surds
Find unknowns in equations involving surds

W T h e n we h ave a n e q u a t i o n t h a t i nvo l ve s a n u n k n ow n u n d e r t h e s q u a r e r o o t
sign, we could solve the equation by squaring both sides of the equation,

for example, E = 3 = (VE) -3°.

T h i s t e c h n i q u e m ay i n t ro d u c e s o l u t i o n s t h a t d o n o t s a t i s f y t h e o r i g i n a l e q u a t i o n . H e n c e
we must check the solutions to ensure that they satisfy the original equation.

Example 6 -Try 6
Solve each of the following equations. Solve each of the following
(a) V6-* =3 equations.
(a) V2x-1=5
(b) v2x-4-2Vx -3 = 0 (b) 24x-1-V6-X=0
(c) v5x+1 - VX =2 (c) V3x-2 = Vx-2 +2

Solution Answers
(a) 13
(a) (b)
(c) 2 or 6
Square both sides of the equation.
6-X=9
X=-3

Check: Substitute x - -3 into the original equation.

LHS = 6 - (-3) = 3
RHS = 3
Hence x = -3 is a solution.

(5) v2x-4 -2Vx-3 = 0 Think

V2x-4 = 24x-3 DeeperO
(V2x -4) = (2J-3) Square both s i d e s What if we s q u a re b o t h
of the equation.
2t-4 = 40t -3) sides of the equation

2x - 4 = 4x - 12 V2. - 4 2V -3 =0
2x = 8 directly?
X = 4

Check: Substitute x = 4 into the original equation. Take

LHS V2(4) - 4 - 2/4-7 0 Note:
RHS = 0
Hence x = 4 is a solution. Always check the solutions
after squaring both sides of
a surd equation.
60 Chapter C3

(c) V5t+1-Vi = 2
V5x+1 =2+VE
Think
Deeper.O
(N5c+1)' =(2+ JR) Square both sides
5x+ 1=4+44x+x of the equation. Given Va + Jb = Nc, does
it follow that a + b = c? Why?
4x-3 = 45
(4 -3)' = (4) Square both sides
167 - 24x ÷ 9 = 16r of the equation.
167 - 40x : 9 = 0
(4x - 1)(4x - 9) = 0 BIG IDEA
Equivalence
x=A ot

Substitute x = 2 into the (4x 3)*= (44) is derived

Check: Substitute x= 7 FI into the
original equation. original equation. from V5x+1 -VT =:
LHS (sx4+1-4-1 LHS
5x241-14-2
A re t h e t wo e q u a t i o n s
equivalent?
RUS - 2 RHS = 2
Hencex 1
4 is not a solution. Hence * = % is a solution.

Hence x ;.
When an equation with unknown constants has rational and
irrational terms, we can use the following property of surds
to find the unknown constants. Think
Deeper,O
Will the statement still hold
If a + bJk = c+ dek, where a, b, c, d are rational and
VR is irrational, then « = c and b = d. if JR is rational? Explain.

Example 7. Try 7
Given that
Given that (2 + aV5)(7-245) = -16 + bJ5, where a an d
b are integers, find the value of a and of b. (4- 33)(5 - 0N5)
= b- 7J3, where a and a b
are integers, find the value of
a and of b.

A n swe r s
a =-2 and b = 2
 Surds 61

Solution
(2 + 0N5)(7 -245) = -16 + b5
14 -4/5 | 7aV5 - 20(5) = -16 1 b45
(14 - 10€) (Ta -4)J5 = -16 + bJ5
Comparing rational terms, 14 - 10a = -16 (1)
Comparing irrational terms, 7a -4= b (2)
H ow d o 1 c h e c k
t h a t t h e a n swe r
Solving equations (1) and (2) simultaneously yields « = 3 and i corvect?
b - 17.

Get the Master the

Exercise 3.2
Challenge
Basies Right Concepts Yourself

Solve each of the following equations. Find the values of the real numbers a
and b.
(a) xV3 -x - 2
(b) V2x - 3 = 5 (a) 3 + J5 =b(3-v5) -2(3+ J5)
(c) 2I -x = V3 (b) a-b7=(2+J72-+5
(d) VR-X=VR-2 (e) u+ J2 = (3-2JZ) + bJR
Find the values of the integers a and b. Without using calculator, find the
(a) a + hV3 = 2-V5 values of the integers a and b such
(b) a + bJ7 =4J7 12 7 -4-7
that ya + by7 = _18
(c) a + bJ2 =3(1-J2) : 442
Find the possible values of the
Solve each of the following equations. real numbers a and b such that
(a) v2+x -x =0 (4+V3(10-b/27) = 28 + 16/5.
(b) (6 - 5x - x =-2x
(c) x+5+x= 1 Find the values of the integers a and b.

(a) («+32)(3-442) = -18 + b42

Without using a calculator, find the
values of the integers a and b far (b) (3-243)(4 + aJ3) = b - 2343
which the solution of the equation
(c) (a+ b45) 5 -3VB) = 6-445
(a) xV/24 + J96 - VIO8 + 1VI2
2 + 4/7
4-4-J7
is va + b, (d)
0+b/71
(b) aJ40 = 145 + Vio is a+ Nb
10 Solve the equation
Explain why the equation * - 2r = V27- 4x+3.
V3x- 7 + V2x-1 0
has no solutions.
Explain why the equation
VP +3t+7 V.x + 3r - 9 = 2
has no real roots.
62 Chapter C3

Extend

1 In geometry, the spiral of Theodorus, also known as the square root spiral or the
P ythagorean spiral, is a spiral constructed a by a adding a sequence of contiguous
right triangles.

A 13

(i) Find the exact length of the hypotenuse in the 14th triangle.
(ï) Suggest a formula that determines the exact length of the hypotenuse in the nth
triangle. Then, find the exact length of the hypotenuse in the 105th triangle.
(iii) Jim says that at least 17 triangles must be drawn for the spiral to have one complete
revolution. For the spiral to have two complete revolutions, at least 17 x 2 = 34
triangles are needed. Comment on his statements. Do you agree? Why or why not?

Regardless of how the spirai of Theodorus

i s c o n t i n u e d , n o t wo hy p e t e n u s e s w i l l eve r
coincide. You may use the Internet to explore
more of its other interesting properties.

2 I n a l g e b r a , a n e s te d s u rd i s a s u rd t h a t c o n t a i n s a n o t h e r s u rd i n i ts ex p re ss i o n .

For example, y1+ v1+Vi+... and +2+/2+ . are nested surds.

Let us evaluate the following nested surds to find out their value.

(i) Let VI =X.

Use the technique of squaring to show that x? - x - 1 = 0.
Solve the equation and use the result to show that
1+V5 1.618,
2
which is the Golden Ratio (p).

Is 1+ 41+ V1+ ... a rational number or an irrational number ?

(ii) Use a similar approach to evaluate

Is 12 + J2 + 2+ . a rational or an irrational number?

(iii) Show that if n > 0, then Vn =}(1+ JT+4n).
 Surde 63

Summary

Surds

Manipulation of surds Solving equations involving surds

U s i n g p ro p e r t i e s : We can square b o t h s i d e s o f a n

-E
For a, b > 0, Va x Vb = Vab equation involving surds to solve for
the unknown, e.g.

VR-1 =3 = (Vx-1) =3°.

Va x Va = a
Always check the solutions by substituting
Rationalising the denominator: the solutions into the original equation.
If the denominator is in the form VR,
multiply the numerator and denominator
by JR.

I f t h e d e n o m i n a to r i s i n t h e fo r m a + b u k , Finding unknowns in equations

multiply the numerator and denominator involving surds
by its conjugate a bek. We can use the following property of
surds to find unknowns in an equation.
If the denominator is in the form

anh -bJk, multiply the numerator Ifa+byk=c+dek, where a, b, c, d

and denominator by its conjugate are rational and VR is irrational, then
avh -byk. a = c and b= d.

Journal Writing

(a) What is a surd? Why is V2 a surd, but not /4?

(b) What does 'rationalising the denominator' of a fraction mean?
(c) Why is it necessary to check for extraneous solutions when solving equations involving
s u r d s ? H ow d o we c h e c k fo r t h e s e ex t r a n e o u s s o l u t i o n s ?
64 Chapter C3

Revision Exercise 3

Set A

A1 () Given that (V5- 2)r = V5 + 2, The lengths of the parallel sides, AB

express x in the form a + bV5, and DC, are (3J7 - 2) cm and
w h e re Q a n d b a re i n t e g e r s .
(247 + 3) respectively. Given
(ii) Hence evaluate x *
1 without
that the area of the trapezium is
using a calculator.
¿(13+ 747) cm*, express the
height of the trapezium in the form
42 (a) Simplty 4 V2 - 450 - 7a * 7m
(b) Without using a calculator, find the (p7 + q) cm, where p and q are
value of the integer k such that

T
integers.

o
1J2xNE 2,1216
3 2 Set B

- kJ3. B1 Solve each of the following equations.

(a) Vx-8-VT =3
A3 Solve each of the following equations.
(b) V18 - Vx-1 =4
(a) V7 - 1=3
(b) 2x + V3 - 4x = 0 (c ) r
V1 +2
Vn-2
(c)
V1- 8x Im

B2 Find the possible values of the real

numbers a and b such that
A4 Solve the equation
VOx 2 - V4x 3+VR+1. (a -65X2 + bN5) = -82.
B3 The diagram shows a triangle ABC in
A5 Find the value of 3 + 242
3- 242 • giving which AB AC and the length of its

your answer in the form a + b42, base BC is (83 - 242) cm.

where a and b are integers. b

A6
Given that 7 7E - p. express V6
in terms of p.
B
A7 The diagram shows a trapezium. (8V5 -242) cm

(37 - 2) em Given that the area of the triangle is

46 cm*, find in the simplest surd form,
(i) the height of the triangle,
(iF) the perimeter of the triangle.

DF IC
(2V7 + 3) cm
 S u rd e 6 5

B4 (i) The length of each side of an (ii) Given that the volume of the prism
cm.
equilateral triangle is 6(V5 - 1) c1 is ¾(17/3+ 30) cm*, find the
W i t h o u t u s i n g a c a l c u l a t o r, ex p re s s
height of the prism in the form
the area of the triangle in the form
(a5 + h) cm, where u and b are h
(a + b5) em*, where u and h are
i n te g e rs .
integers.
(i) The triangle in part (i) is the base
B6 (i) Without using a calculator, evaluate
of a prism. Given that the volume
(2-45X4+ 245 + J25).
of the prism is 9(V3 -1) cm*,
(ii) Hence find the values of the
show that the height of the prism
can be expressed in the form integers m and n such that
$25 + 235
( m + " f 3 ) e m , w h e re m a n d n " 4/5 + 4
4 - 245 + /25
a re f r a c t i o n s i n t h e s i m p l e st fo r m .

B7 The diagram shows a triangle.

B5 The diagram shows prism in which its
c ro ss - s e c t i o n i S a re g u l a r h ex a g o n o f
side (2 + V3) cm.

(2+45) czo

The height BC (7-3

(i) Given that sin 60°
2
V3 find the
the base AB is (9 -248) cm and
cross-sec tional area of the prism in angle ABC is 90°. Find
(i) the area of the triangle in the
the form ; p/3 + 4) em*, where
form (a+ b2) cm?, where
p and q are integers. You may use
a and b are fractions in the
the formula 15ab sin C for the area simplest form,
of triangle ABC, where a and b are (ii) an expression for AC in the
the sides of the triangle and C is form (e+ d42) cm', where
the included angle. and d are real numbers.