Mr Roche's Second Year HL — Christmas Revision

Tap a topic below to jump to it. Try each sample question, then reveal the worked solution.

1.1 Simplifying expressions

Combine like terms

Identify like terms in an expression.
Add and subtract coefficients correctly.

Sample question

Simplify \( 3x + 7x - 4 + 2 - 5x \).

Solution

Combine the \(x\)-terms: \[ 3x + 7x - 5x = (3 + 7 - 5)x = 5x. \] Combine the constants: \[ -4 + 2 = -2. \] So the simplified expression is \[ 5x - 2. \]

1.2 Removing brackets

Expand brackets

Use the distributive law: \(a(b + c) = ab + ac\).
Expand a product of two binomials.

Sample question

Expand and simplify \( (x + 4)(x - 7) \).

Solution

Multiply term by term: \[ (x + 4)(x - 7) = x\cdot x + x(-7) + 4x + 4(-7). \] \[ = x^{2} - 7x + 4x - 28 = x^{2} - 3x - 28. \]

1.3 Evaluating expressions

Substitution

Substitute given values into an expression.
Follow order of operations correctly.

Sample question

Evaluate \( 3x^{2} - 2x + 1 \) for \( x = -2 \).

Solution

Substitute \(x=-2\): \[ 3(-2)^{2} - 2(-2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17. \]

1.4 Solving linear equations

One variable equations

Collect like terms on each side.
Isolate the variable using inverse operations.

Sample question

Solve the equation \( 3(x - 2) + 5 = 20 \).

Solution

Expand: \[ 3(x - 2) + 5 = 3x - 6 + 5 = 3x - 1. \] So \[ 3x - 1 = 20 \Rightarrow 3x = 21 \Rightarrow x = 7. \]

1.5 Problems using linear equations

Translating word problems

Represent the unknown with a letter.
Form and solve an equation from the information.

Sample question

A number is tripled and then decreased by \(5\). The result is \(19\). Find the number.

Solution

Let the number be \(x\). “Tripled and decreased by \(5\)” gives \(3x - 5\). \[ 3x - 5 = 19 \Rightarrow 3x = 24 \Rightarrow x = 8. \]

1.6 Algebraic division

Dividing monomials

Divide coefficients.
Subtract powers using \(x^{m}/x^{n} = x^{m-n}\).

Sample question

Simplify \( \dfrac{18a^{3}b^{2}}{6ab} \).

Solution

Divide coefficients: \(18/6 = 3\). Subtract powers: \(a^{3}/a = a^{2}\), \(b^{2}/b = b\). So \[ \dfrac{18a^{3}b^{2}}{6ab} = 3a^{2}b. \]

1.7 Number line

Plotting numbers and inequalities

Plot integers and fractions.
Show solutions of inequalities with open/closed dots.

Sample question

Solve the inequality \( x > -2 \) and describe how it is shown on a number line.

Solution

The solution set is all real numbers greater than \(-2\). On a number line we draw an open dot at \(-2\) and shade (or draw an arrow) to the right.

1.8 Solving inequalities

Linear inequalities

Solve inequalities like equations.
Reverse the inequality sign when multiplying or dividing by a negative number.

Sample question

Solve \( 3 - 2x \ge 7 \).

Solution

\[ 3 - 2x \ge 7 \Rightarrow -2x \ge 4. \] Divide by \(-2\) and reverse the sign: \[ x \le -2. \]

2.1 Common factors

Factorising using the HCF

Find the highest common factor of terms.
Write the expression as HCF × bracket.

Sample question

Factorise \( 12x - 18 \).

Solution

HCF of \(12x\) and \(18\) is \(6\): \[ 12x - 18 = 6(2x - 3). \]

2.2 Grouping

Factorising by grouping terms

Group terms in pairs.
Factorise each pair, then factor again.

Sample question

Factorise \( 3x + 3y + 2x + 2y \).

Solution

Group: \[ (3x + 3y) + (2x + 2y) = 3(x + y) + 2(x + y). \] Factorise again: \[ = (3 + 2)(x + y) = 5(x + y). \]

2.3 Difference of two squares

Special factor pattern

Recognise \(a^{2} - b^{2}\).
Write \(a^{2} - b^{2} = (a-b)(a+b)\).

Sample question

Factorise \( 9a^{2} - 25 \).

Solution

Recognise \( (3a)^{2} - 5^{2} \): \[ 9a^{2} - 25 = (3a - 5)(3a + 5). \]

2.4 Quadratic expressions

Factorising quadratics

Factorise quadratics with \(a = 1\).
Use splitting-the-middle-term for \(a \ne 1\).

Sample question

Factorise \( 2x^{2} - x - 6 \).

Solution

Look for two numbers with product \(2 \times (-6) = -12\) and sum \(-1\): \( -4\) and \(3\). Rewrite: \[ 2x^{2} - 4x + 3x - 6 = 2x(x - 2) + 3(x - 2). \] \[ = (2x + 3)(x - 2). \]

2.5 Algebraic fractions

Simplifying using factors

Factorise numerator and denominator fully.
Cancel common factors.

Sample question

Simplify \[ \dfrac{x^{2} - 9}{x^{2} + 4x + 3}. \]

Solution

Factorise: \[ x^{2} - 9 = (x - 3)(x + 3),\quad x^{2} + 4x + 3 = (x + 1)(x + 3). \] Cancel \((x + 3)\): \[ \dfrac{x^{2} - 9}{x^{2} + 4x + 3} = \dfrac{x - 3}{x + 1},\quad x \ne -3,-1. \]

6.1 Perimeter & area

Rectangles and triangles

Use \(P = 2(l + w)\) for rectangles.
Use \(A = \tfrac{1}{2}bh\) for triangles.

Sample question

A rectangle measures \(12\text{ cm} \times 7\text{ cm}\). Find (a) its perimeter, (b) its area.

Solution

(a) \(P = 2(l + w) = 2(12 + 7) = 2 \times 19 = 38\text{ cm}.\) (b) \(A = lw = 12 \times 7 = 84\text{ cm}^{2}.\)

6.2 Parallelogram

Area of a parallelogram

Use \(A = bh\) where \(h\) is perpendicular height.

Sample question

A parallelogram has base \(15\text{ cm}\) and perpendicular height \(9\text{ cm}\). Find its area.

Solution

\[ A = bh = 15 \times 9 = 135\text{ cm}^{2}. \]

6.3 Circles

Area & circumference

Use \(C = 2\pi r\) or \(C = \pi d\).
Use \(A = \pi r^{2}\).

Sample question

A circle has radius \(7\text{ cm}\). Find (a) its circumference, (b) its area, correct to one decimal place.

Solution

(a) \(C = 2\pi r = 2\pi \times 7 = 14\pi \approx 43.98\text{ cm}\approx 44.0\text{ cm}.\) (b) \(A = \pi r^{2} = \pi \times 7^{2} = 49\pi \approx 153.94\text{ cm}^{2}\approx 153.9\text{ cm}^{2}.\)

6.4 Rectangular solids

Volume & surface area

Use \(V = lwh\).
Add the areas of all faces for surface area.

Sample question

A cuboid measures \(8\text{ cm} \times 5\text{ cm} \times 3\text{ cm}\). Find its volume and total surface area.

Solution

Volume: \[ V = 8 \times 5 \times 3 = 120\text{ cm}^{3}. \] Surface area: \[ A = 2(lw + lh + wh) = 2(8\cdot5 + 8\cdot3 + 5\cdot3) = 2(40 + 24 + 15) = 2 \times 79 = 158\text{ cm}^{2}. \]

6.5 Prisms

Volume of a prism

Find area of cross-section.
Multiply by the length of the prism.

Sample question

A triangular prism has a triangular cross-section with base \(10\text{ cm}\) and height \(6\text{ cm}\). The length of the prism is \(12\text{ cm}\). Find its volume.

Solution

Cross-section area: \[ A = \tfrac{1}{2}bh = \tfrac{1}{2}\times 10 \times 6 = 30\text{ cm}^{2}. \] Volume: \[ V = A \times \text{length} = 30 \times 12 = 360\text{ cm}^{3}. \]

15.1 Quadratic equations

Solving by factorising

Write the equation in the form \(ax^{2} + bx + c = 0\).
Factorise and use the zero product property.

Sample question

Solve \( x^{2} + 5x + 6 = 0 \).

Solution

Factorise: \[ x^{2} + 5x + 6 = (x + 2)(x + 3) = 0. \] So \[ x + 2 = 0 \Rightarrow x = -2,\quad x + 3 = 0 \Rightarrow x = -3. \]

15.2 Quadratic formula

Using the quadratic formula

Identify \(a\), \(b\), \(c\) correctly.
Substitute into \(x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

Sample question

Solve \( 3x^{2} - 2x - 1 = 0 \) using the quadratic formula.

Solution

Here \(a = 3\), \(b = -2\), \(c = -1\).
Then \[ x = \dfrac{-(-2) \pm \sqrt{(-2)^{2} - 4(3)(-1)}}{2\cdot3} = \dfrac{2 \pm 4}{6}. \] So \[ x = 1,\quad x = -\dfrac{1}{3}. \]

17.1 Cylinder

Volume & surface area of a cylinder

Use \(V = \pi r^{2}h\).
Use \(A_{\text{total}} = 2\pi r(r + h)\).

Sample question

A cylinder has radius \(5\text{ cm}\) and height \(12\text{ cm}\). Find (a) its volume, (b) its total surface area, correct to one decimal place.

Solution

(a) \[ V = \pi r^{2}h = \pi \times 5^{2} \times 12 = 300\pi \approx 942.5\text{ cm}^{3}. \] (b) \[ A = 2\pi r(r + h) = 2\pi \times 5(5 + 12) = 10\pi \times 17 = 170\pi \approx 534.1\text{ cm}^{2}. \]

17.2 Sphere & hemisphere

Area & volume

Use \(A_{\text{sphere}} = 4\pi r^{2}\), \(V_{\text{sphere}} = \frac{4}{3}\pi r^{3}\).
For a hemisphere: half the volume, half the surface area plus a circular base if required.

Sample question

A sphere has radius \(6\text{ cm}\). Find its volume, correct to one decimal place.

Solution

\[ V = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi \times 6^{3} = \frac{4}{3}\pi \times 216 = 288\pi \approx 904.8\text{ cm}^{3}. \]

17.3 Rates of flow

Volume, time & rate

Convert between \(\text{cm}^{3}\) and litres.
Use \(\text{volume} = \text{rate} \times \text{time}\).

Sample question

A cylindrical tank has radius \(10\text{ cm}\) and height \(60\text{ cm}\). Water flows in at a rate of \(1.2\text{ L/min}\). How long will it take to fill the tank? \(\big(1\text{ L} = 1000\text{ cm}^{3}\big)\)

Solution

Tank volume: \[ V = \pi r^{2}h = \pi \times 10^{2} \times 60 = 6000\pi\text{ cm}^{3} \approx 18\,850\text{ cm}^{3}. \] In litres: \[ \approx 18.85\text{ L}. \] Time: \[ t = \dfrac{18.85}{1.2} \approx 15.7\text{ minutes}. \] So it takes about \(16\) minutes to fill the tank.