Solutions to Year 13 revision sheet 11
1.
a) Draw the diagram always.
x
55  62
= -0.583333

12
Using the normal dist. Tables gives probability = 0.281 (3sf)
b)
Using:
gives:
Find <70 then find<60 and take away
70  62
60  62
 0.666.. and
from each other.
12
12
From the tables we get; 0.7486 – 0.4364 = 0.312 (3sf)
c)
First we must find the probability of
58  62
 0.333 From the tables: probability of less than 58%
failing: so,
12
is 0.3707 and so the probability of passing is 1 – 0.3707 = 0.6293.
The number of people passing is: 16000 x 0.6293 = 10069.
2.
3.
Use 3D Pythagoras;
(3  1) 2  (7  1) 2  (6  1) 2 =
93  9.64
b)
4 3 12 2
 =

10 9 90 15
c)
 4 6   6 4  48 8

     
 10 9   10 9  90 15
4.
f ( x)  3x 2  12 x  15 Make equal to zero and solve:
3( x  5)( x  1)  0 and so the x coordinate is 5 and -1.
Substitute x = 5 and -1 into original to find the y coordinates:
(5 , -92) , (-1 , 16)
Double differentiate to find the nature (max or min)
f ( x)  6 x  12 . When x = 5 it gives 18 (positive) and so (5 , -92) is a
Minimum, when x = -1 it gives -18 (negative) and so (-1, 16) is a max.
5.
sin 2 (2 x)  14 and so, sin(2 x)   12
Therefore: 2x = 30 or -30. Using ASTC to find the second angles:
Between 0 and 360: we get: 2x = 30, 2x = 150 , 2x = 390 , 2x = 510
And we get: 2x = 330 , 2x = 210 , 2x = 690 , 2x = 570
So x = 15 , 75 , 195 , 255 , 165 , 105 , 345 , 285
6.
a) ar 3  16 and ar 6  76 and so,
7.
If AX = B
ar 6 76
and so, r 3  4.75

ar 3 16
Therefore r = 1.680987703… and a = 3.368421053…
3.368... 1.6809...n  1000000
b)
Divide both sides by 3.368… and then take logs on both sides.
log(296875)
= 24.263 and so the first term is the 25th
n
log(1.6809...)
A-1 =
A-1B =
then X = A-1B
1  0 5 

 and so we get;
0  30  6 2 
1  0 5   1 4 


 =
0  30  6 2   5 6 
1  25 30 


30  4 36 
8.
From 1st equation, x = 3y + 2 Sub this into the 2nd equation, gives:
(3 y  2)2  8 y  17 Expand and make equal to zero.
9 y 2  4 y  13  0 Factorise or use the formula: (9 y  13)( y  1) = 0
Then sub back in to find x, Gives: y = -13/4 , x = -7.75 and y = 1, x = 5
9.
y intercept when x = 0 is at -6.
x intercept when y = 0, (2x – 3)(x + 2) = 0 and so when x = -2, 3/2
Vertex at 2 BA and so is at; 41 for the x coordinate, sub back in to find
y gives: -6.125
10.a)
f 1 ( x) 
x 3
2
g f ( x) 
b)
1
1 =
2x  3 1
1
2 x  2 2 x  1


2x  2 2x  2 2x  2
x  3 2 x  1
2 x 2  4 x  6  4 x  2

Then cross multiply:
2
2x  2
2 x 2  4  0 and so
And so,
x 2
c)
n
11.
Use:
E ( x)   pi xi
i 1
Money
Prob
Prob x
money
4
3/6
2
5
2/6
10/6
6
1/6
1
And so expected return after 1 roll is 2 + 10/6 + 1 = $4.67.
After 10 rolls the expected return is $46.67