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Indices Exercises

Question 1

a) 243

b) 343

c) 512

Question 2

a) 1⁄512

b) 1⁄243

c) 1⁄256

Question 3

a) 5

b) 3

c) 4

Logarithms Exercises

Question 1

4 × 4 × 4 × 4 = 256

So log₄(256) = 4

Question 2

0.0016 = 16/10,000 = 1/625 = 1/5⁴ = 5^-4

So log₅(0.0016) = -4

Question 3

729 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶

So log₃(729) = 6

Inequalities Exercises

Question 1

5x > 10

Therefore
x > 10/5
= x > 2
 

Question 2

3x + 2 > 8

= 3x + 2 - 2 > 8 -2
= 3x  > 6
= 3x/3 > 6/3
= x > 2

Question 3

3x - 7 < 5

= 3x - 7 + 7 < 5 + 7
= 3x < 12
= 3x/3 < 12/3
= x < 4

Permutation and Combination Exercises 

Question 1

= 10!/(10 - 3)!
= 10 × 9 × 8
= 720
 

Question 2

The number of permutations of 4 letters chosen from 26 is
²⁶P₄ = 26 × 25 × 24 × 23 = 358,800

Question 3

= ¹⁰C₅
= 10!/(5!)(5!)
= (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2 × 1)
= 30,240/120
= 252
 

Arithmetic and Geometric Progression Exercises 

Question 1

x_n = a + d(n - 1)

The first term a = 4 (1)
The tenth term = 67  

x₁₀ = a + d(10 - 1) = 67  
a + 9d = 67 (2)
4 + 9d = 67
9d = 63
d = 63 ÷ 9 = 7

The common difference is 7 
 

Question 2

This sequence has a difference of 5 between each pair of numbers.

The values of a and d are:
a = -12
d = 5

x_n = a + d(n − 1)
= -12 + 5(n − 1)
= -12 + 5n − 5
= 5n − 17

So, the 32nd term is:
x₃₂ = 5 × 32 − 17 = 160 − 17 = 143

Question 3

This sequence has a difference of 4 between each pair of numbers.

The values of a and d are:
a = 3
d = 4

x_n = a + d(n-1)
= 3 + 4(n-1)
= 3 + 4n - 4
= 4n - 1

So, the 50th term is:
x₅₀ = 4 × 50 - 1 = 200 - 1 = 199

Data Exercises 

Question 1

D is the correct answer.
The number of teeth has to be a whole number, so is discrete.

Question 2

B is the answer.
B is discrete because the numbers of brothers and sisters can only be values like 0, 1, 2 etc...
The other three are all continuous because they can take any value within a range, such as 160.3 cm or 75.35 kg.
 

Question 3

C is the answer.
The weight of a cat is continuous because it can take any value within certain limits.
 

Mean, Median, Mode Exercises 

Question 1

Add the numbers: 8 + 9 + 13 + 18 = 48

Divide by how many numbers (i.e. we added 4 numbers).
Then 48 ÷ 4 = 12.

Question 2

Put the numbers in order first: 3, 4, 5, 5, 5, 6, 7, 7, 8, 9

5 occurs most often, so the mode is 5

Question 3

Put the numbers in order first: 2, 2, 4, 6, 11

The median is the middle number = 4

Probability Exercises 

Question 1

The factors of six are 1, 2, 3 and 6, so the Number of ways it can happen = 4
There are six possible scores when a die is thrown, so the Total number of outcomes = 6

So the probability that the score is a factor of six = 4/6 = 2/3

Question 2

Here are 4 Queens and 4 Kings, so the Number of ways it can happen = 8
There are 52 cards altogether, so the Total number of outcomes = 52

So the probability either a king or a queen = 8/52 = 2/13

Question 3

Represent 'Heads up' by H and 'Tails up' by T.
There are 8 possible ways the coin can land: 
(H, H, H), (H, H, T), (H, T, H), (H, T, T),
(T, H, H), (T, H, T), (T, T, H) and (T, T, T)
 
Of these, 3 have one Head and two Tails: (H, T, T), (T, H, T) and (T, T, H)

So:
The Number of ways it can happen = 3
The Total number of outcomes = 8


Therefore, the probability of obtaining one head and two tails = 3/8
 

Standard Deviation and Variance Exercises 

Question 1

Firstly find the mean:
Mean = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100
 
Next find the variance. To calculate the Variance, take each difference, square it, and then average the result:
(75 - 100)² + (83 - 100)² + (96 - 100)² + (100 - 100)² + (121 - 100)² + (125 - 100)²
= (-25)² +  (-17)² +  (-4)² +  (0)² +  (21)² +  (25)²
= 625 + 289 + 16 + 0 + 441 + 625
= 1,996 

So the Variance = 1,996 ÷ 6 = 332.66...
 
The Standard Deviation is just the square root of the Variance
= √(332.66...)
= 18.2 correct to 1 decimal places

Question 2

Firstly find the mean number of words per page:
Mean = (271 + 354 + 296 + 301 + 333 + 326 + 285 + 298 + 327 + 316 + 287 + 314) ÷ 12
= 3,708 ÷ 12
= 309
 
Next find the variance. To calculate the Variance, take each difference, square it, and then average the result:
(271 - 309)² + (354 - 309)² + (296 - 309)² + (301 - 309)² + (333 - 309)² + (326 - 309)² + (285 - 309)² + (298 - 309)² + (327 - 309)² + (316 - 309)² + (287 - 309)² + (314 - 309)²
= (-38)² +  (45)² +  (-13)² +  (-8)² +  (24)² +  (17)² +  (-24)² +  (-11)² +  (18)² +  (7)² +  (-22)² +  (5)²
= 1,444 + 2,025 + 169 + 64 + 576 + 289 + 576 + 121 + 324 + 49 + 484 + 25
= 6,146 

So the Variance = 6,146 ÷ 12 = 512.166...
 
The Standard Deviation is just the square root of the Variance
= √(512.166...)
= 22.6 correct to 1 decimal place

Question 3

If each number is multiplied by 3, then the mean is also multiplied by 3.

The values of the differences, therefore, are also multiplied by 3
= The values of the squares of the differences are multiplied by 9 (3²)
= The value of the variance is multiplied by 9
= The value of the standard deviation is multiplied by √9 = 3

Linear Equations Exercises 

Question 1

First find the slope:
m = Rise/Run = 5/1 = 5

Next use the formula y - y₁ = m(x - x₁)
Substitute x₁ = -1, y₁ = -3 and m = 5
Therefore y - (-3) = 5(x - (-1))

Question 2

First find the slope:
m = Rise/Run = 5/4 = 1.25

Next use the formula y - y₁ = m(x - x₁)
Substitute x₁ = 2, y₁ = -2 and m = 1.25
= y - (-2) = 1.25(x - 2)
= y + 2 = 1.25x - 2.5
= y = 1.25x - 2.5 - 2
= y = 1.25x - 4.5

Question 3

First find the slope:
m = Rise/Run = -8/6 = -4/3

Next use the formula y - y₁ = m(x - x₁)
Substiute x₁ = -4, y₁ = 6 and m = -4/3
= y - 6 = -4/3(x - (-4))
= y - 6 = -4/3(x + 4)
= y - 6 = -4/3x -16/3
= y = -4/3x -16/3 + 6
= y = -4/3x - 16/3 + 18/3
= y = -4/3x + 2/3