A-LevelMathematicsQuadratics9709/12 · Feb/Mar 2023 · Q14 Marks
A line has equation y = 3x – 2k and a curve has equation y = x² – kx + 2, where k is a constant. Show that the line and the curve meet for all values of k.
- (a)Show that the line and the curve meet for all values of k.[4]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.
- [M1] x² -kx+2=3x-2k leading to x²-x(k+3)+(2+2k) [=0]
- [DM1] b²-4ac = (k+3)²-8(1+k) (ignore ‘= 0' at this stage)
- [A1] =(k-1)² accept (k-1)(k-1) Or use of calculus to show minimum of zero at k = 1 or sketch of f(k)=k² -2k+1.
- [A1] ≥ 0 Hence will meet for all values of k Clear conclusion.
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