Secondary Mathematics Review: Algebra, Geometry, and Number Skills

Practice flashcards covering algebra, simultaneous equations, geometry, sequences, standard form, and measurement definitions based on the lecture notes.

In isosceles triangle $LMN$ where $LN = MN$, the angle $MLN$ is given by $2x + 15^\circ$ and the angle $LMN$ is given by $3x - 10^\circ$. What is the value of $x$?

$x = 25$. Because sides $LN$ and $MN$ are equal in length, their base angles are equal: $2x + 15 = 3x - 10 \implies 25 = x$.

In a cricket match, the number of runs Peter scored was $10$ more than half of Marty's runs. If Peter scored a half-century ($50\,\text{runs}$), how many runs did Marty score?

$80\,\text{runs}$. Setting up the relationship gives $50 = \frac{1}{2}m + 10 \implies \frac{1}{2}m = 40 \implies m = 80$.

Naomi has $80\text{c}$ more than Aimee, and together they have \n$4.00. How much money does Aimee have?

$\$1.60$ (or $160\text{c}$). Letting Aimee's amount be $a$, Naomi has $a + 0.80$. Then $a + (a + 0.80) = 4.00 \implies 2a = 3.20 \implies a = 1.60$.

What are the coordinates of the intersection point $P$ of the two straight lines $2x + 3y = 12$ and $y - 2x = 4$?

$(0, 4)$. Rearranging the second equation yields $y = 2x + 4$. Substituting this into the first equation gives $2x + 3(2x + 4) = 12 \implies 8x + 12 = 12 \implies x = 0$, which then gives $y = 4$.

What is the solution to the linear inequality $2 - 5x < 12$?

$x > -2$. Subtracting $2$ from both sides yields $-5x < 10$. Dividing by $-5$ flips the inequality symbol, resulting in $x > -2$.

Simplify the algebraic fraction $\frac{x^2 - 16}{2x - 8}$.

$\frac{x + 4}{2}$. Factoring numerator and denominator gives $\frac{(x - 4)(x + 4)}{2(x - 4)}$, which simplifies to $\frac{x + 4}{2}$ after cancelling the common factor $(x - 4)$.

For the linear sequence $6, 11, 16, 21, 26, 31, 36$, what is the first-order difference and the general rule for term $t$ in the form $t = mx + c$?

The first-order difference is $+5$, and the general rule is $t = 5x + 1$.

Using the elimination method to solve the simultaneous equations $3x + y = 17$ and $x + y = 7$, what are the values of $x$ and $y$?

$x = 5$ and $y = 2$. Subtracting the second equation from the first gives $2x = 10 \implies x = 5$. Substituting $x = 5$ into $x + y = 7$ yields $y = 2$.

According to the measurement reference sheet, what is the formal definition of bearings?

Bearings are measured from North, in a clockwise direction.

According to the measurement reference sheet, how many square metres are in $1\,\text{hectare}$, and how many cubic centimetres are in $1\,\text{litre}$?

$1\,\text{hectare} = 10\,000\,\text{m}^2$ and $1\,\text{litre} = 1000\,\text{cm}^3$.

A triangle has interior angles measuring $60^\circ$, $2x$, and $x$. What is the value of $x$?

$x = 40$. The interior angles of a triangle sum to $180^\circ$: $60 + 2x + x = 180 \implies 3x = 120 \implies x = 40$.

What is the Lowest Common Multiple (LCM) of $3$ and $8$?

$24$. The multiples of $8$ are $8, 16, 24, \dots$, of which $24$ is the smallest multiple divisible by $3$.

Factorise the quadratic expression $x^2 + 13x + 36$.

$(x + 4)(x + 9)$. The two numbers that multiply to give $36$ and add to give $13$ are $4$ and $9$.

Convert the number $0.000\,000\,005$ into standard form.

$5 \times 10^{-9}$.

The number $24$ is divided into two parts such that their product is $128$. What are the two parts?

$16$ and $8$. Setting up the equation $x(24 - x) = 128 \implies x^2 - 24x + 128 = 0$, which factorises to $(x - 16)(x - 8) = 0$.