Junior Cycle Mathematics Revision Guide

Revision Test Overview

• Subject: Junior Cycle Maths Revision Test.

• Time Allotted: $90\,minutes$.

• Instructions:

  • Answer all questions.

  • Show all workings clearly.

  • Calculators are permitted unless instructions specify otherwise.

Section A: Algebra

• Evaluating Algebraic Expressions:

  • Process: Substitute given numerical values into the variables of the expression and solve.

  • Example (Question 1): Evaluate $3x^2 - 2y$ where $x = 4$ and $y = 5$.

    • Substitute: $3(4)^2 - 2(5)$

    • Square the value: $3(16) - 10$

    • Multiply: $48 - 10$

    • Final Result: $38$

• Expanding and Simplifying:

  • Process: Use the distributive law to multiply terms outside parentheses by terms inside, then combine like terms.

  • Example (Question 2): Expand and simplify $2(3x - 5) + 4(x + 2)$.

    • Distribute: $(6x - 10) + (4x + 8)$

    • Group like terms: $6x + 4x - 10 + 8$

    • Simplify: $10x - 2$

• Factorisation:

  • Highest Common Factor (HCF): Identify the largest factor that divides into all terms.

  • Example (Question 3): Factorise completely $6x^2 + 9x$.

    • Identify HCF: The HCF of $6$ and $9$ is $3$; the HCF of $x^2$ and $x$ is $x$. Common factor is $3x$.

    • Divide terms by HCF: $3x(2x + 3)$

• Solving Linear Equations:

  • Process: Isolate the variable using inverse operations.

  • Example (Question 4): Solve $5x - 7 = 18$.

    • Add $7$ to both sides: $5x = 18 + 7$

    • Simplify: $5x = 25$

    • Divide by $5$: $x = 5$

• Solving Simultaneous Equations:

  • Method (Elimination): Align equations to eliminate one variable by addition or subtraction.

  • Example (Question 5): Solve $x + y = 10$ and $2x - y = 5$.

    • Add the equations to eliminate $y$: $(x + 2x) + (y - y) = 10 + 5$

    • Solve for $x$: $3x = 15 \rightarrow x = 5$

    • Substitute $x$ back into the first equation: $5 + y = 10$

    • Solve for $y$: $y = 5$

    • Solution: $(5, 5)$

• Solving Quadratic Equations:

  • Method (Factorisation): Find two numbers that multiply to the constant term and add to the coefficient of the middle term.

  • Example (Question 6): Solve $x^2 - 7x + 10 = 0$.

    • Factorise: $(x - 2)(x - 5) = 0$

    • Set each factor to zero: $x - 2 = 0$ or $x - 5 = 0$

    • Solutions: $x = 2, x = 5$

• Rearranging Formulas (Change of Subject):

  • Process: Use algebraic operations to isolate a specific variable.

  • Example (Question 7): Rearrange the formula $V = \pi r^2 h$ to make $h$ the subject.

    • Divide both sides by $\pi r^2$: $\frac{V}{\pi r^2} = h$

    • Final Formula: $h = \frac{V}{\pi r^2}$

Section B: Coordinate Geometry

• Distance Formula:

  • The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

  • Example (Question 8): Distance between $A(2, 3)$ and $B(8, 11)$.

    • Substitute: $d = \sqrt{(8 - 2)^2 + (11 - 3)^2}$

    • Calculate: $d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100}$

    • Result: $10\,units$

• Midpoint Formula:

  • The midpoint is given by: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

  • Example (Question 9): Midpoint of $(-4, 6)$ and $(2, 10)$.

    • Substitute: $M = \left(\frac{-4 + 2}{2}, \frac{6 + 10}{2}\right)$

    • Calculate: $M = \left(\frac{-2}{2}, \frac{16}{2}\right)$

    • Result: $(-1, 8)$

• Slope (Gradient) Formula:

  • The slope $m$ is calculated by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • Example (Question 10): Slope of the line joining $(1, 2)$ and $(5, 14)$.

    • Substitute: $m = \frac{14 - 2}{5 - 1} = \frac{12}{4}$

    • Result: $m = 3$

• Equation of a Line:

  • Formula: $y - y_1 = m(x - x_1)$

  • Example (Question 11): Find the equation of the line with slope $3$ passing through $(2, -1)$.

    • Substitute: $y - (-1) = 3(x - 2)$

    • Simplify: $y + 1 = 3x - 6$

    • Standard form: $3x - y - 7 = 0$ or $y = 3x - 7$

• Parallel and Perpendicular Lines:

  • Parallel Lines: Have identical slopes ($m_1 = m_2$).

  • Perpendicular Lines: Slopes are negative reciprocals ($m_1 \times m_2 = -1$).

  • Example (Question 12): Compare $y = 2x + 1$ and $y = 2x - 5$.

    • Both lines have a slope of $m = 2$.

    • Conclusion: The lines are parallel.

Section C: Speed, Distance & Time

• Average Speed Formula:

  • $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

  • Example (Question 13): A car travels $180\,km$ in $3\,hours$.

    • Calculation: $\frac{180}{3} = 60\,km/h$

• Unit Conversion (Speed):

  • km/h to m/s: Since $1\,km = 1000\,m$ and $1\,hour = 3600\,seconds$, multiply by $\frac{1000}{3600}$ (or divide by $3.6$).

  • Example (Question 14): Convert $72\,km/h$ to $m/s$.

    • Calculation: $72 \times \frac{1000}{3600} = 20\,m/s$

• Time Conversion for Speed:

  • When time is given in minutes, convert to hours to find $km/h$.

  • Example (Question 15): A cyclist travels $15\,km$ in $45\,minutes$.

    • Convert time: $45\,minutes = 0.75\,hours$

    • Calculation: $\frac{15}{0.75} = 20\,km/h$

• Multi-stage Journeys:

  • Example (Question 16):

    • Stage 1: $120\,km$ at $80\,km/h$. Time = $\frac{120}{80} = 1.5\,hours$.

    • Stage 2: $90\,km$ at $60\,km/h$. Time = $\frac{90}{60} = 1.5\,hours$.

    • Total Journey Time: $1.5 + 1.5 = 3\,hours$.

    • Average Speed: $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{120 + 90}{3} = \frac{210}{3} = 70\,km/h$.

Section D: Financial Maths

• Value Added Tax (VAT):

  • Example (Question 17): Jacket costs $€80$ before $23\%$ VAT.

    • VAT Amount: $80 \times 0.23 = €18.40$

    • Total Price: $€80 + €18.40 = €98.40$

• Profit Percentage:

  • Formula: $\text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100$

  • Example (Question 18): Phone bought for $€250$, sold for $€320$.

    • Profit: $320 - 250 = €70$

    • Calculation: $\frac{70}{250} \times 100 = 28\%$

• Discounts (Sale Price):

  • Example (Question 19): Runners cost $€150$, reduced by $20\%$.

    • Discount: $150 \times 0.20 = €30$

    • Sale Price: $150 - 30 = €120$

• Compound Interest:

  • Formula: $A = P(1 + i)^t$, where $P$ is principal, $i$ is interest rate (decimal), and $t$ is time.

  • Example (Question 20): $€2,000$ invested at $5\%$ for $3\,years$.

    • Calculation: $2000(1 + 0.05)^3 = 2000(1.157625)$

    • Final Amount: $€2,315.25$

Section E: Sets

• Set Notations and Operations:

  • Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$.

  • (a) Union (A ∡ B): All elements in either set.

    • Answer: $\{1, 2, 3, 4, 5, 6\}$

  • (b) Intersection (A ∩ B): Elements found in both sets.

    • Answer: $\{3, 4\}$

  • (c) Difference (A \ B): Elements in set A that are not in set B.

    • Answer: $\{1, 2\}$

• Venn Diagrams and Surveys:

  • Problem (Question 22): Total playing football ($F$) = $18$, basketball ($B$) = $12$, playing both ($F \cap B$) = $5$.

  • Drawing Steps:

    • Place the overlap ($5$) in the intersection of circles F and B.

    • Determine "Only Football": $18 - 5 = 13$.

    • Determine "Only Basketball": $12 - 5 = 7$.

  • Solution: $13$ students play only football.

Section F: Probability and Counting

• Basic Probability:

  • Formula: $P(E) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}$

  • Question 23 (Fair Die):

    • (a) Probability of getting a 6: $\frac{1}{6}$

    • (b) Probability of an even number ($2, 4, 6$): $\frac{3}{6} = \frac{1}{2}$

  • Question 24 (Counters): Bag contains $4\,red$, $3\,blue$, $5\,green$ (Total = $12$).

    • Probability of picking green: $\frac{5}{12}$

• Sample Space and Multi-stage Events:

  • Example (Question 25): Two coins are tossed.

    • Sample Space: $\{HH, HT, TH, TT\}$

    • Probability of exactly one head: Outcomes are $HT$ and $TH$.

    • Result: $\frac{2}{4} = \frac{1}{2}$

• Fundamental Principle of Counting:

  • Multiply the number of choices for each stage of the event.

  • Example (Question 26): Password with $2\,letters$ followed by $3\,digits$.

    • Assuming standard English alphabet ($26\,letters$) and digits $0$-$9$ ($10\,digits$):

    • Calculation: $26 \times 26 \times 10 \times 10 \times 10$

    • Result: $676,000\,passwords$