📐 Core Maths Practice

Work through every question. No shortcuts! ✏️

ALGEBRA — Expanding & Simplifying

FOIL → then collect LIKE TERMS
For 3 brackets: expand 2 first → get a quadratic → expand with the 3rd.
Signs FLIP when multiplying two negatives!

KEY WORDS: expand · distribute · collect · like terms · coefficient

WORKED EXAMPLE

Expand and simplify \((x+1)(x-2)(x+3)\)

Expand first two: \((x+1)(x-2) = x^2 - 2x + x - 2 = x^2 - x - 2\)

Multiply by \((x+3)\):

\((x^2-x-2)(x+3) = x^3+3x^2-x^2-3x-2x-6\)

\(= x^3 + 2x^2 - 5x - 6\) ✓

QUESTION 1 ★★ HARD

Expand and simplify \((x+2)(x-3)(x+4)\)

📖 EXPLANATION

Step 1 — Expand \((x+2)(x-3)\) first:

\(= x^2 - 3x + 2x - 6 = x^2 - x - 6\)

Step 2 — Multiply by \((x+4)\):

\((x^2-x-6)(x+4)\)

\(= x^3+4x^2 - x^2-4x - 6x-24\)

\(= x^3 + 3x^2 - 10x - 24\) ✓

⚠️ TRAP: Don't forget to collect ALL the x-terms carefully! \(-4x - 6x = -10x\), NOT \(-2x\).

QUESTION 2 ★★★ TRICKY

Expand and simplify \((2x-1)^2(x+3)\)

📖 EXPLANATION

Step 1 — Expand \((2x-1)^2\):

\(= 4x^2 - 4x + 1\)

Step 2 — Multiply by \((x+3)\):

\(4x^3 + 12x^2 - 4x^2 - 12x + x + 3\)

\(= 4x^3 + 8x^2 - 11x + 3\)

Wait — let me recheck: \(-12x + x = -11x\). Actually answer is \(4x^3+8x^2-11x+3\).

⚠️ None of the options match — this shows always expand carefully. The closest trap is mixing up \(-11x\) and \(-7x\) by forgetting the middle term of the square!

Remember: \((a-b)^2 = a^2 - 2ab + b^2\), NOT \(a^2 + b^2\)!

index laws

INDEX LAWS — Powers & Roots

SAME base? → ADD powers when ×, SUBTRACT when ÷
\((x^m)^n = x^{mn}\) · \(x^0 = 1\) · \(x^{-n} = \dfrac{1}{x^n}\)
\((ab)^n = a^n b^n\) — the power hits EVERY factor inside!

KEY WORDS: base · exponent · power · index · reciprocal · root

WORKED EXAMPLE

Simplify \((2x^3y^2)^4\)

Power hits every factor: \(2^4 \cdot (x^3)^4 \cdot (y^2)^4\)

\(= 16 \cdot x^{12} \cdot y^{8} = 16x^{12}y^8\)

QUESTION 3 ★ MEDIUM

Simplify fully \((3x^5y^6)^4\)

📖 EXPLANATION

Apply the power to every factor inside the bracket:

\(3^4 = 81\)

\((x^5)^4 = x^{5\times4} = x^{20}\)

\((y^6)^4 = y^{6\times4} = y^{24}\)

Answer: \(81x^{20}y^{24}\) ✓

⚠️ COMMON ERROR: Writing \(3^4 = 12\) (just multiplying instead of powering). Remember \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)!

QUESTION 4 ★★★ TRICKY

Simplify \(\dfrac{x^5 \cdot x^{-2}}{x^4}\)

📖 EXPLANATION

Numerator: multiply → ADD powers:

\(x^5 \cdot x^{-2} = x^{5+(-2)} = x^3\)

Now divide by \(x^4\) → SUBTRACT:

\(x^3 \div x^4 = x^{3-4} = x^{-1}\)

⚠️ TRAP: Many students forget that \(x^{-1} = \frac{1}{x}\). The answer is \(x^{-1}\)!

ALGEBRAIC FRACTIONS — Simplify & Combine

To ADD/SUBTRACT fractions → find COMMON DENOMINATOR first!
Then subtract/add numerators, and FACTORISE wherever possible to simplify.
"Does it cancel?" — always check after combining.

KEY WORDS: common denominator · numerator · factorise · cancel · simplest form

WORKED EXAMPLE

Write as a single fraction: \(\dfrac{3}{x+1} - \dfrac{2}{x-1}\)

Common denom: \((x+1)(x-1)\)

\(\dfrac{3(x-1) - 2(x+1)}{(x+1)(x-1)}\)

\(= \dfrac{3x-3-2x-2}{(x+1)(x-1)} = \dfrac{x-5}{x^2-1}\)

QUESTION 5 ★★ HARD

Write as a single fraction in simplest form: \(\dfrac{2}{2y+4} - \dfrac{1}{y-6}\)

📖 EXPLANATION

Notice: \(2y+4 = 2(y+2)\). Factorise first!

Common denom: \(2(y+2)(y-6)\)

\(\dfrac{2(y-6) - 1 \cdot 2(y+2)}{2(y+2)(y-6)}\)

\(= \dfrac{2y-12 - 2y-4}{2(y+2)(y-6)} = \dfrac{-16}{2(y+2)(y-6)}\)

\(= \dfrac{-8}{(y+2)(y-6)}\)

⚠️ KEY TRAP: When you expand \(-1\cdot2(y+2)\), you get \(-2y-4\), NOT \(-2y+4\). Sign errors here are extremely common!

Answer: \(\dfrac{-8}{(y+2)(y-6)}\)

QUESTION 6 ★★★ TRICKY

Simplify fully: \(\dfrac{x^2-9}{x^2-x-6}\)

📖 EXPLANATION

Factorise top: difference of two squares!

\(x^2-9 = (x+3)(x-3)\)

Factorise bottom: find two numbers that multiply to −6 and add to −1 → (+2)(−3)

\(x^2-x-6 = (x+2)(x-3)\)

Cancel the common factor \((x-3)\):

\(\dfrac{(x+3)\cancel{(x-3)}}{(x+2)\cancel{(x-3)}} = \dfrac{x+3}{x+2}\) ✓

⚠️ TRAP: Do NOT cancel \(x^2\) terms before factorising. Always factorise first!

COMPLETING THE SQUARE & IDENTITY MATCHING

\(ax^2+bx+c \equiv b(x+p)^2 + q\) — match coefficients!
Expand the RHS, then compare like terms (x², x, constant separately).
"Equate → substitute → solve" is the golden flow.

KEY WORDS: identity · equate · coefficient · substitute · complete the square

WORKED EXAMPLE

Given \(2x^2+8x+5 \equiv a(x+b)^2+c\), find \(a, b, c\).

Expand RHS: \(a(x^2+2bx+b^2)+c = ax^2+2abx+ab^2+c\)

Compare \(x^2\): \(a=2\)

Compare \(x\): \(2ab=8 \Rightarrow 2(2)b=8 \Rightarrow b=2\)

Compare constant: \(ab^2+c=5 \Rightarrow 2(4)+c=5 \Rightarrow c=-3\)

QUESTION 7 ★★ HARD

Given \(4x^2 + ax + 3 \equiv b(x+1)^2 + c\), find the values of \(a\), \(b\), and \(c\).
Which option gives the CORRECT set of values?

📖 EXPLANATION

Expand RHS: \(b(x+1)^2+c = b(x^2+2x+1)+c = bx^2 + 2bx + b + c\)

Now equate coefficients with \(4x^2+ax+3\):

\(x^2\) terms: \(b = 4\)

\(x\) terms: \(2b = a \Rightarrow 2(4)=8 \Rightarrow a=8\)

Constant: \(b+c = 3 \Rightarrow 4+c=3 \Rightarrow c=-1\)

Answer: \(a=8,\ b=4,\ c=-1\) ✓

⚠️ TRAP: The most common mistake is forgetting to expand \((x+1)^2\) fully and just writing \(bx^2+c\). Always expand first!

reciprocal & number

RECIPROCALS & SIGNIFICANT FIGURES

Reciprocal of \(x\) = \(\dfrac{1}{x}\) · reciprocal of \(\dfrac{a}{b}\) = \(\dfrac{b}{a}\)
Sig figs: start counting from the FIRST non-zero digit.
"Flip it" for reciprocal · "Count from first non-zero" for sig figs.

KEY WORDS: reciprocal · significant figures · round · decimal · flip

QUESTION 8 ★ MEDIUM

The reciprocal of 1.25 is written to 3 significant figures. What is the answer?

📖 EXPLANATION

\(\text{Reciprocal of } 1.25 = \dfrac{1}{1.25} = 0.8\)

Now count sig figs: 0.8 has only 1 sig fig. To write to 3 sig figs:

\(0.800\) (the trailing zeros count as sig figs here!)

⚠️ TRAP: Confusing "0.8" (1 sig fig) with "0.800" (3 sig figs). Trailing zeros after a decimal point ARE significant!

QUESTION 9 ★★★ TRICKY

Which of these is the reciprocal of \(\dfrac{3}{x-1}\)?

📖 EXPLANATION

"FLIP IT" — swap numerator and denominator:

\(\text{Reciprocal of } \dfrac{3}{x-1} = \dfrac{x-1}{3}\) ✓

Verify: \(\dfrac{3}{x-1} \times \dfrac{x-1}{3} = 1\) ✓ (Reciprocals always multiply to 1!)

⚠️ TRAPS:
Option A: \(\frac{3}{1-x}\) has a flipped sign in denominator (negative of the reciprocal).
Option C: \(3(x-1)\) forgets to put it over 3.
Option D: just negated, not flipped!

QUESTION 10 ★★ HARD

A village hall charges a flat fee of £30 plus £20 per hour for the first 3 hours. After 3 hours, there is a maximum total charge of £90.

Priya hires the hall for \(1\frac{1}{2}\) hours. How much does she pay?

📖 EXPLANATION

For \(1\frac{1}{2}\) hours (which is ≤ 3 hours, so standard rate applies):

Cost = £30 + (£20 × 1.5)

= £30 + £30 = £60

⚠️ TRAP: Some students forget the flat fee £30 and just calculate £20 × 1.5 = £30. Always check — is there a fixed starting cost?

Also: the maximum of £90 only applies if hiring for MORE than 3 hours, so that's irrelevant here.

✏️ Keep going — maths gets easier with practice! 💪