Show that the discriminant of \(f(x)\) is \(100 - 4{p^2}\).

Find the values of \(p\) so that \(f(x) = 0\) has two equal roots.

correct substitution into \({b^2} - 4ac\)     A1

eg\(\;\;\;{(10 - p)^2} - 4(p)\left( {\frac{5}{4}p - 5} \right)\)

correct expansion of each term     A1A1

eg\(\;\;\;100 - 20p + {p^2} - 5{p^2} + 20p,{\text{ }}100 - 20p + {p^2} - (5{p^2} - 20p)\)

\(100 - 4{p^2}\)     AG     N0

[3 marks]

recognizing discriminant is zero for equal roots     (R1)

eg\(\;\;\;D = 0,{\text{ }}4{p^2} = 100\)

correct working     (A1)

eg\(\;\;\;{p^2} = 25\), \(1\) correct value of \(p\)

both correct values \(p =  \pm 5\)     A1     N2

[3 marks]

Total [6 marks]

Many candidates were able to identify the discriminant correctly and continued with good algebraic manipulation. A commonly seen mistake was identifying the constant as \(\frac{5}{4}p\) instead of \(\frac{5}{4}p - 5\). Mostly a correct approach to part b) was seen \((\Delta  = 0)\), with the common error being only one answer given for \(p\), even though the question said values (plural).

Many candidates were able to identify the discriminant correctly and continued with good algebraic manipulation. A commonly seen mistake was identifying the constant as \(\frac{5}{4}p\) instead of \(\frac{5}{4}p - 5\). Mostly a correct approach to part b) was seen \((\Delta  = 0)\), with the common error being only one answer given for \(p\), even though the question said values (plural).