(3a)
3√3/2 – 4√2/3 – √24
=3√3/2 – 4√2/√3 -√6*4
=(3√3/√2 × √2/√2) – (4√2/√3 × √3/√3) – 2√6
=3√6/2 – 4√6/3 – 2√6
=(3/2 – 4/3 – 2)√6
=(9 – 8 – 12)√6
= -11/6√6

(3bi)
prob(only one passes) = m passes and N fails or M fails and N passes
Prob (M passes) = 2/3; prob(M fails) = 1 – 2/3 = 1/3
Prob(N passes) = 4/5; prob(N fails) = 1 – 4/5 = 1/5
Prob(only one passes) = (2/3 × 1/5) + (1/3 × 4/5)
= 2/15 + 4/15
= 6/15
= 2/15

(3bii)
prob(at least one passes) = prob(only one passes) + prob(both parties)
= 2/5 + (2/3 × 4/5)
= 2/5 + 8/15
= 6/15 + 8/15
= 14/15

(4a)
17²=X²+15²
17²=X²+225
289-225=X²
64=X²
C=√64=8
Tanθ=18/15
Tanθ/1+2tanθ=18/15/1+2(8/15)=8/15/1+16/15
=8/15÷31/15=8/15*15/31=8/31

(4b)
Log^10/Log^64=½
Log10^10=Log10^64^½
y=√64=±8

(5a)
Obtuse Reflex
<MOP + <MOP = 360(angle at a point)
Obtuse <MOP + 196° = 360°
Obtuse <MOP = 360 – 196
= 164°

OMP + OPM + Obtuse MOP = 180°(<s in a triangle)
But OMP = OPM = x(radius of the circle and hence base angles are the same)

2x + 164 = 180
2x = 180 – 164
2x = 16°
X = 16/2 = 8°

Also MNP = 1/2×obtuseMOP
= 1/2 × 164°
= 82°

Now; MNP + NMP + NPM = 180° (<s in a triangle)
82 + (52+8) + (m+8) = 180°
m + 150 = 180°
M = 180 – 150
M = 30°

(5b)

(6a)
2^m*(1/8)^n = 128
2^m *2^-n = 2^7
m-3^n=7 ———(eq1)
4^m÷2^-4n=1/6
2^2m÷2^-4n=2-4
2m+4n=-4
m+2n=2 —–(eq2)
Subtract from (2)
m+2n-(n-3n)=-2-7
m+2n-m+3n=-9
5n/5=-9/5
n=-9/5
Sub for n in eq2
m+2(-9/5)=-2
m-18/5=-2
m=-2+18/5=8/5
;(m-n)=5/5-(-9/5)=17/5

(6b)
(-2,½) and (1-⅔)
y2-y1/x1-x2 = y-y1/x-x1
-⅔-½/1-(-2) = y-½/x-(-2)
-7/6/3=y-½/x+2
-7/18*y-½/x+2
(y-½)18=-7(x+2)
18y-9=-7x-14
18y=-7x-14+9
18y/18=-7x/18-5/18
y=-7x/18-5/18
18y=-7x-5

(8)
5th –> 11
8th –> 20
a + (n -1)d = 11
a + (5ˉ¹)d = 11
a + 4d = 11 –> (i)
a + 7d = 20 –> )ii)
a + 7d – a – 4d = 20 – 11
7d – 4d = 9
3d = 9
D = 9/3 = 3
D = 3
From eqn (i)
a + 4d = 11
a + 4 x 3 = 11
a + 12 = 11
a = 11 – 12
a = 1

(9i) DRAW THE DIAGRAM

n (U) = 110
n (P) = 25
n (B) = 45
n (M) = 48
n (Pnm) = 10
n (Bnm) = 8
n (pnBnm) = 5

(9ii) for biology = 45 = 36

(9iii)
x + 14 + 1 + 5 + 5 + 36 + 3 + 35 = 110
x + 25 + 36 + 38 = 110
x = 110 – 99
x = 11

(9b)
15 = 22 + 18 + 2 x 1 + 10 +20/5
15 x 5 = 40 + 2x + 31
45 = 71 + 2x
45 – 71 + 2x
2x/2 = -26/2
X = 13
22, 18, 2x + 1, 10 , 20
For 2x (-B) +1 = -26 +1 = -25
22, 18, -25, 10, 20
-25, 10, 18, 20, 22
The media is 18

(10a)
Loan borrowed = 80/100 × 350,000
=280,000
Amount paid to the bank after 8 years = P + I
= P + PRT/100
=280000 + 280000×7×8/100
=280,000 + 156,800
Amount = #436,800

Total cost of house to the man = #350,000 + Interest
= #350,000 + 156,800
= #506,800

(10b)
Percentage increase in cost of house = 506,800 – 350,000 × 100%
= 156,800/350,000 × 100%
= 44.8%

(c) Percentage loss = loss/cost price × 100%
=(506,800 + 10,000) – 460,000/(506,800 + 10,000) × 100%
= 516,800 – 460,000/516,800 × 100%
56,800/516,800 × 100%
= 0.1099 × 100%
= 10.99%

(12)
Draw the diagram
x² = 20² + 8² – 2x 20 x 8 x Cos 50º
x² = 400 + 64 – 320 x 0.6427
x² = 464 – 205.7
√x² = √258.3
x = 16.1
x = 16km

(12ai)
TJ = 16km

(12aii)
sinØ/20 = sin50/16
sinØ = 20 x sin 50/16
sinØ 0.9576
Ø = sinˉ¹ 0.9576 = 73.3º
The bearing 90 -73.3 = 16.7º

(12b)
Draw the diagram
y = √12² + 5²
y = √144 + 25
y = √169 = 13
tan Ø = 5/12 = (0.04167)
Ø = tanˉ¹ (0.4167) = 22.6º
Sin Ø = 5/6 + x
Sin 22.6º = 5/6 + x
(0.3843) (6 + x) = 5
2.3058 + 0.3843x = 5
0.3843x = 5-2.3058
0.3843x/0.3843 = 2.6942/0.3943
X = 7.01
X ≈ 7

(13a)
Draw the diagram

(i) Extend DO to touch AB at M
AOD + AOM = 180°(<s on a straight line)
130 + AOM = 180°
AOM = 180 – 130 = 50°

Also;
BMO = BAO + AOM(Ext < = sum of two opposite interior <s of a triangle)
BMO = 26 + 50 = 76°
Also; ABD = 1/2AOD(angle at centre = twice angle at circum)
ABD = 1/2 × 130
=65°
Hence ODB + ABD + BMO = 180°(sum of <s in a triangle)
ODB + 65° + 76° = 180°
ODB = 180 – 141
= 39°

(ii) BOD + ODB + DBO = 180°(sum of <s in a triangle)
But ODB = DBO(base angles of an isosceles triangle)
BOD + 39° + 39° = 180°
BOD + 78° = 180°
BOD = 180 – 78 = 102°

(13b)
|1 2 6
3|1,3 2,3 (6,3)
4|1,4 2,4 (6,4)
5|1,5 2,5 (6,5)

prob (greater than 7) = 3/9 = 1/3