Solutions

Work out 1¹⁄₅ + 2¹⁄₃.

1. Convert both mixed numbers into improper fractions:

1¹⁄₅ = ^{(1 × 5) + 1}⁄₅ = ⁶⁄₅

2¹⁄₃ = ^{(2 × 3) + 1}⁄₃ = ⁷⁄₃

2. Find a lowest common denominator for 5 and 3, which is 15:

⁶⁄₅ = ^{6 × 3}⁄_{5 × 3} = ¹⁸⁄₁₅

⁷⁄₃ = ^{7 × 5}⁄_{3 × 5} = ³⁵⁄₁₅

3. Add the numerators together:

¹⁸⁄₁₅ + ³⁵⁄₁₅ = ⁵³⁄₁₅

4. Convert back to a mixed number: 53 ÷ 15 = 3 remainder 8.

3⁸⁄₁₅

Given m = 2 × 3^a × 7^b and n = 3 × 7² × 11.

(a) Given that m > 126, find the smallest values of a and b which make m a multiple of 126:

Find the prime factorisation of 126:

126 = 2 × 63 = 2 × 3² × 7¹

For m to be a multiple of 126, its prime factor powers for 3 and 7 must be greater than or equal to those of 126:

Power of 3 must be at least 2 ⟹ a ≥ 2

Power of 7 must be at least 1 ⟹ b ≥ 1

If a = 2 and b = 1, then m = 2 × 3² × 7¹ = 126, which is NOT greater than 126.

So increase one exponent to make m the next smallest multiple:

a = 3, b = 1: m = 2 × 3³ × 7¹ = 2 × 27 × 7 = 378

Check: 378 ÷ 126 = 3. ✓ And 378 > 126. ✓

a = 2, b = 2: m = 2 × 3² × 7² = 2 × 9 × 49 = 882 (larger)

So the smallest values are a = 3, b = 1.

(b) Explain why 33n is a perfect square:

Substitute the value of n into the expression:

33n = 33 × (3 × 7² × 11)

Break down 33 into prime factors (33 = 3 × 11):

33n = (3 × 11) × 3 × 7² × 11 = 3² × 7² × 11² = (3 × 7 × 11)²

Every prime factor in the prime factorisation of 33n has an even exponent, which makes it a perfect square.

Arrange the fractions starting with the largest: ⁶⁄₅, ³⁄₂₅, ¹²¹⁄₁₀₀, ⁵¹⁄₅₀

Convert all fractions to a common denominator of 100:

⁶⁄₅ = ^{6 × 20}⁄_{5 × 20} = ¹²⁰⁄₁₀₀

³⁄₂₅ = ^{3 × 4}⁄_{25 × 4} = ¹²⁄₁₀₀

¹²¹⁄₁₀₀

⁵¹⁄₅₀ = ^{51 × 2}⁄_{50 × 2} = ¹⁰²⁄₁₀₀

Comparing the values: ¹²¹⁄₁₀₀ > ¹²⁰⁄₁₀₀ > ¹⁰²⁄₁₀₀ > ¹²⁄₁₀₀

¹²¹⁄₁₀₀ > ⁶⁄₅ > ⁵¹⁄₅₀ > ³⁄₂₅

Given M = ( −2   1 ; 3   4 ) and N = ( 1   0 ; 1   −4 ), find M + N:

Add corresponding matrix entries:

M + N = ( −2 + 1   1 + 0 ; 3 + 1   4 + (−4) ) = ( −1   1 ; 4   0 )

Express the total area of the shape in terms of π in its simplest form:

1. Area of the rectangle (ABCD): Length × Width = 5 cm × 3 cm = 15 cm².

2. Area of the sector (CDE): The radius is CD = AB = 5 cm. The sector angle is 30°.

Area of sector = ^θ⁄₃₆₀ × πr² = ³⁰⁄₃₆₀ × π × 5² = ¹⁄₁₂ × 25π = ^25π⁄₁₂

3. Total Area: 15 + ^25π⁄₁₂

(a) Evaluate ¹⁄_9⁻²:

A negative power in the denominator moves to the numerator as a positive power:

¹⁄_9⁻² = 9² = 81

(b) Evaluate √(2¹⁄₄):

Convert the mixed number to an improper fraction:

2¹⁄₄ = ^{(2 × 4) + 1}⁄₄ = ⁹⁄₄

Take the square root of the numerator and denominator:

√(⁹⁄₄) = ^√9⁄_√4 = ³⁄₂

Calculate the radius of the circle given PQ = 5 cm and QR = 12 cm:

1. Line segment PR passes through the centre O, making it a diameter. The angle subtended by a diameter at the circumference is always a right angle (∠PQR = 90°).

2. Apply Pythagoras' theorem to right-angled triangle △PQR to find diameter PR:

PR = √(PQ² + QR²) = √(5² + 12²) = √(25 + 144) = √169 = 13 cm

3. Calculate the radius: Radius = ^Diameter⁄₂ = ¹³⁄₂ = 6.5 cm

M400 is increased by 15%. Find the new amount:

Increase = 15% of 400 = ¹⁵⁄₁₀₀ × 400 = 15 × 4 = 60

New Amount = 400 + 60 = 460

Given the shoe sizes 2, 7, 3, 4, 5, 3.

(a) Find the range:

Range = Maximum Value − Minimum Value = 7 − 2 = 5

(b) Find the mean shoe size:

Sum all values and divide by the total count (6):

Mean = ^{2 + 7 + 3 + 4 + 5 + 3}⁄₆ = ²⁴⁄₆ = 4

The ratio of interior angle to exterior angle of a regular polygon is 5 : 1.

(a) Find the size of an exterior angle:

Interior and exterior angles on any vertex lie on a straight line and sum to 180°.

Total parts = 5 + 1 = 6 parts

Size of 1 part (exterior angle) = ¹⁄₆ × 180° = 30°

(b) Calculate the number of sides of the polygon:

The sum of all exterior angles of a polygon is always 360°.

Number of sides = ^360°⁄_{Exterior Angle} = ^360°⁄_30° = 12

A box contains 12 white, 3 red, and 5 black balls (Total = 20).

(a) Find the probability that a chosen ball is white:

^{White Balls}⁄_{Total Balls} = ¹²⁄₂₀ = ³⁄₅ (or 0.6)

(b) Find the probability that it is either black or red:

^{Black + Red}⁄_{Total Balls} = ^{5 + 3}⁄₂₀ = ⁸⁄₂₀ = ²⁄₅ (or 0.4)

Calculate the height of the cuboid not in contact with water:

1. Find the total height (H) of the cuboid using its total volume formula (V = length × width × height):

300 = 10 × 4 × H ⟹ 300 = 40H ⟹ H = ³⁰⁰⁄₄₀ = 7.5 cm

2. The depth of the water is 2 cm.

3. Height not in contact with water = Total Height − Water Depth = 7.5 cm − 2 cm = 5.5 cm

Construct the locus of all points that are 2 cm from line AB:

The locus of points at a fixed distance from a straight line consists of two parallel lines on either side of line AB, along with semi-circular caps of radius 2 cm around endpoints A and B.

Use a metric ruler to mark parallel reference boundaries exactly 2 cm above and below line AB.

(a) Factorise completely: (a − 1)² − b²

Use the difference of two squares identity, A² − B² = (A − B)(A + B), where A = (a − 1) and B = b:

[(a − 1) − b][(a − 1) + b] = (a − 1 − b)(a − 1 + b)

(b) Make R the subject of the formula ^a⁄₂ = ^R⁄₆ − ^b⁄₃:

Clear the denominators by multiplying the entire equation by the lowest common multiple, which is 6:

6 × (^a⁄₂) = 6 × (^R⁄₆) − 6 × (^b⁄₃)

3a = R − 2b

Isolate R by adding 2b to both sides:

R = 3a + 2b

Given f(x) = 2x + 1.

(a) Find f(x) − 3:

Subtract 3 from the complete function definition:

(2x + 1) − 3 = 2x − 2

(b) Find the value of x when f(x) = 7:

2x + 1 = 7 ⟹ 2x = 6 ⟹ x = 3

Calculate angles marked a and b:

1. Triangle △AFE is equilateral, so all its interior angles are 60°. Thus, ∠AEF = 60°.

2. Since ABCD is a rectangle, line AD is parallel to BC.

3. Angles on straight line FEG sum to 180°. Thus, ∠a = 180° − ∠AEF = 180° − 60° = 120°.

4. By alternate interior angles across parallel horizontal lines AD ∥ BC, the angle ∠DEG = ∠b. Since ∠DEG and ∠a are supplementary angles on a straight line: ∠DEG = 180° − 120° = 60° ⟹ ∠b = 60°.

Expand and simplify (x − 1)² − (x² − 5):

1. Expand the first binomial squared term: (x − 1)² = x² − 2x + 1.

2. Distribute the negative sign across the second parenthesis block: −(x² − 5) = −x² + 5.

3. Combine expressions and collect like terms:

(x² − 2x + 1) − x² + 5 = (x² − x²) − 2x + (1 + 5) = −2x + 6

Estimate ^{√8.7 × 0.25}⁄_0.87 by rounding each number to 1 significant figure:

8.7 → 9

0.25 → 0.3

0.87 → 0.9

Substitute values: ^{√9 × 0.3}⁄_0.9 = ^{3 × 0.3}⁄_0.9 = ^0.9⁄_0.9 = 1

Mpoetsi and Jesi shared profit in the ratio 12 : 8. Jesi received M300.

(a) Write the ratio 12 : 8 in its simplest form:

Divide both parts by their highest common factor, 4: ¹²⁄₄ : ⁸⁄₄ = 3 : 2

(b) Calculate the total profit they shared:

Jesi represents 2 parts of the simplified ratio, which equals M300.

Value of 1 part = ³⁰⁰⁄₂ = M150

Total parts = 3 + 2 = 5 parts

Total profit = 5 × 150 = 750

M750

Given AB = ( −8 ; 6 ), find |AB|:

Apply the magnitude formula via Pythagoras' theorem:

|AB| = √((−8)² + 6²) = √(64 + 36) = √100 = 10

10 units

(a) Given 3x + y = 180, show that x + y = 80:

1. Angles inside triangle △BCD must add up to 180°:

∠DBC + ∠BCD + ∠BDC = 180°

y + (y + 10) + (2x + 10) = 180

2. Collect and group like terms:

2x + 2y + 20 = 180

3. Subtract 20 from both sides:

2x + 2y = 160

4. Divide the entire equation by 2:

x + y = 80 (Proven)

(b) Find the value of x:

Set up the system of two linear equations:

1. 3x + y = 180
2. x + y = 80

Subtract equation (2) from equation (1) to eliminate variable y:

(3x − x) = 180 − 80 ⟹ 2x = 100 ⟹ x = 50