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Question
The quadratic equation \(k{x^2} + (k – 3)x + 1 = 0\) has two equal real roots.
Find the possible values of k.
Write down the values of k for which \({x^2} + (k – 3)x + k = 0\) has two equal real roots.
Answer/Explanation
Markscheme
attempt to use discriminant (M1)
correct substitution, \({(k – 3)^2} – 4 \times k \times 1\) (A1)
setting their discriminant equal to zero M1
e.g. \({(k – 3)^2} – 4 \times k \times 1 = 0\) , \({k^2} – 10k + 9 = 0\)
\(k = 1\) , \(k = 9\) A1A1 N3
[5 marks]
\(k = 1\) , \(k = 9\) A2 N2
[2 marks]
Question
Let \(f(x) = k{x^2} + kx\) and \(g(x) = x – 0.8\). The graphs of \(f\) and \(g\) intersect at two distinct points.
Find the possible values of \(k\).
Answer/Explanation
Markscheme
attempt to set up equation (M1)
eg\(\;\;\;f = g,{\text{ }}k{x^2} + kx = x – 0.8\)
rearranging their equation to equal zero M1
eg\(\;\;\;k{x^2} + kx – x + 0.8 = 0,{\text{ }}k{x^2} + x(k – 1) + 0.8 = 0\)
evidence of discriminant (if seen explicitly, not just in quadratic formula) (M1)
eg\(\;\;\;{b^2} – 4ac,{\text{ }}\Delta = {(k – 1)^2} – 4k \times 0.8,{\text{ }}D = 0\)
correct discriminant (A1)
eg\(\;\;\;{(k – 1)^2} – 4k \times 0.8,{\text{ }}{k^2} – 5.2k + 1\)
evidence of correct discriminant greater than zero R1
eg\(\;\;\;{k^2} – 5.2k + 1 > 0,{\text{ }}{(k – 1)^2} – 4k \times 0.8 > 0\), correct answer
both correct values (A1)
eg\(\;\;\;0.2,{\text{ }}5\)
correct answer A2 N3
eg\(\;\;\;k < 0.2,{\text{ }}k \ne 0,{\text{ }}k > 5\)
[8 marks]