Question
The quadratic equation \(k{x^2} + (k – 3)x + 1 = 0\) has two equal real roots.
Find the possible values of k.[5]
Write down the values of k for which \({x^2} + (k – 3)x + k = 0\) has two equal real roots.[2]
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]
Question
Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) .
(i) Find \({u_4}\) .
(ii) Find the sum of the infinite sequence.
Consider an arithmetic sequence with n terms, with first term (\( – 36\)) and eighth term (\( – 8\)) .
(i) Find the common difference.
(ii) Show that \({S_n} = 2{n^2} – 38n\) .
The sum of the infinite geometric sequence is equal to twice the sum of the arithmetic sequence. Find n .
Answer/Explanation
Markscheme
(i) correct approach (A1)
e.g. \({u_4} = (40){\frac{1}{2}^{(4 – 1)}}\) , listing terms
\({u_4} = 5\) A1 N2
(ii) correct substitution into formula for infinite sum (A1)
e.g. \({S_\infty } = \frac{{40}}{{1 – 0.5}}\) , \({S_\infty } = \frac{{40}}{{0.5}}\)
\({S_\infty } = 80\) A1 N2
[4 marks]
(i) attempt to set up expression for \({u_8}\) (M1)
e.g. \( – 36 + (8 – 1)d\)
correct working A1
e.g. \( – 8 = – 36 + (8 – 1)d\) , \(\frac{{ – 8 – ( – 36)}}{7}\)
\(d = 4\) A1 N2
(ii) correct substitution into formula for sum (A1)
e.g. \({S_n} = \frac{n}{2}(2( – 36) + (n – 1)4)\)
correct working A1
e.g. \({S_n} = \frac{n}{2}(4n – 76)\) , \( – 36n + 2{n^2} – 2n\)
\({S_n} = 2{n^2} – 38n\) AG N0
[5 marks]
multiplying \({S_n}\) (AP) by 2 or dividing S (infinite GP) by 2 (M1)
e.g. \(2{S_n}\) , \(\frac{{{S_\infty }}}{2}\) , 40
evidence of substituting into \(2{S_n} = {S_\infty }\) A1
e.g. \(2{n^2} – 38n = 40\) , \(4{n^2} – 76n – 80\) (\( = 0\))
attempt to solve their quadratic (equation) (M1)
e.g. intersection of graphs, formula
\(n = 20\) A2 N3
[5 marks]
Question
[Maximum mark: 6] [without GDC]
Consider the function \(f(x)=2x^{2}-8x+5\).
(a) Express \(f (x)\) in the form \(a(x-p)^{2}+q\), where \(a,p,q \in \mathbb{Z}\) .
(b) Find the minimum value of \(f (x)\).
Answer/Explanation
Ans.
(a) \(2x^{2}-8x+5=2(x^{2}-4x+4)+5-8=2(x-2)^{2}-3\)
OR vertex at (2,-3)\(\Rightarrow y=2(x-2)^{2}-3\)
=> \(a\) = 2, \(p\) = 2, \(q\) = –3
(b) Minimum value of \(f (x) = –3\)