18M.1.SL.TZ2.S_6

pestleMathematicsAISLPaper 118M· sl-2-5-modelling-functionssource ↗

Let $f\left( x \right) = p{x^2} + qx - 4p$ , where p ≠ 0. Find Find the number of roots for the equation $f\left( x \right) = 0$ .

Justify your answer.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

evidence of discriminant (M1)
eg ${b^2} - 4ac,\,\,\Delta$

correct substitution into discriminant (A1)
eg ${q^2} - 4p\left( { - 4p} \right)$

correct discriminant A1
eg ${q^2} + 16{p^2}$

$16{p^2} > 0\,\,\,\,\left( {{\text{accept}}\,\,{p^2} > 0} \right)$ A1

${q^2} \geqslant 0\,\,\,\,\left( {{\text{do not accept}}\,\,{q^2} > 0} \right)$ A1

${q^2} + 16{p^2} > 0$ A1

$f$ has 2 roots A1 N0

METHOD 2

y-intercept = −4p (seen anywhere) A1

if p is positive, then the y-intercept will be negative A1

an upward-opening parabola with a negative y-intercept R1
eg sketch that must indicate p > 0.

if p is negative, then the y-intercept will be positive A1

a downward-opening parabola with a positive y-intercept R1
eg sketch that must indicate p > 0.

$f$ has 2 roots A2 N0

[7 marks]

Examiners’ report

[N/A]