Exam-Style Problems

Stage 9 - Q1 - 2 marks

19087

Simplify.

\[\frac{3x+27}{3}\]

Hint

Factor the entire numerator first.

Stage 9 - Q2 - 2 marks

19088

Simplify.

\[\frac{10x}{5}\]

Hint

The remaining coefficient is the quotient.

Stage 9 - Q3 - 3 marks

19089

Simplify for nonzero \(x\).

\[\frac{7x^2}{21x}\]

Hint

Cancel factors, not terms in a sum.

Stage 9 - Q4 - 3 marks

19090

Simplify.

\[\frac{9x-9}{36}\]

Hint

The common numerical factor divides every numerator term.

Stage 9 - Q5 - 4 marks

19091

Simplify and state the restriction.

\[\frac{3x^2+6x}{3x}\]

Hint

The simplified expression retains the original restriction.

Stage 9 - Q6 - 4 marks

19092

Simplify and state the restriction.

\[\frac{5x^2+10x}{5x}\]

Hint

The simplified expression retains the original restriction.

Stage 9 - Q7 - 5 marks

19093

A learner cancels \(x\) in \(\frac{x+3}{x}\) and gets \(4\). Use \(x=7\) to test the claim.

Hint

Evaluate the original expression and the proposed answer separately.

Stage 9 - Q8 - 5 marks

19094

Find \(k\) so the identity holds for every \(x\).

\[\frac{9x+k}{9}=x+3\]

Hint

Equality for every \(x\) requires matching constant terms.

Stage 9 - Q9 - 5 marks

19095

Simplify the expression, state its domain, then evaluate at \(x=2\).

\[\frac{3x^2+12x}{3x}\]

Hint

Do not restore the excluded value after cancellation.

Stage 9 - Q10 - 3 marks

19096

Solve the equation, keeping its original restriction.

\[\frac{x^2-9}{x-3}=6\]

Hint

Record the forbidden value before cancelling.