Question 11:
--- 11(a) ---
11(a) | Gives or or as an asymptote.
Condone other incorrect asymptotes.
𝑥𝑥 = √𝑟𝑟 𝑥𝑥 = −√𝑟𝑟 𝑦𝑦 = 1 | 1.1b | B1 | 2
𝑥𝑥 −𝑟𝑟 = 0
2
𝑥𝑥 = 𝑟𝑟
𝑥𝑥 = ±√𝑟𝑟
Gives and as asymptotes, with no incorrect asymptotes given. | 1.1b | B1
--- 11(b) ---
11(b) | Rearran𝑥𝑥g=es± √𝑟𝑟 𝑦𝑦 i=nto1 a non-fractional form.
2
𝑥𝑥 +𝑥𝑥−6
Accept any sensible2 alternative for , e.g. or
𝑘𝑘 = 𝑥𝑥 −1 | 3.1a | M1 | Let 𝑦𝑦 = 1
2
𝑥𝑥 +𝑥𝑥−6
2
𝑘𝑘 = 𝑥𝑥 −1
2 2
𝑘𝑘(𝑥𝑥 −1)= 𝑥𝑥 +𝑥𝑥−6
2
(𝑘𝑘−1)𝑥𝑥 −𝑥𝑥+6−𝑘𝑘 = 0
1−4(𝑘𝑘−1)(6−𝑘𝑘)≥ 0
2
1−4(6𝑘𝑘−𝑘𝑘 −6+𝑘𝑘) ≥ 0
2
4𝑘𝑘 −28𝑘𝑘+25≥ 0
,
7−2√6 7+2√6
𝑦𝑦 ≤ 2 𝑦𝑦 ≥ 2
Rearranges their equation into a correct three-term quadratic equation in
𝑘𝑘 𝑦𝑦 f
Condone missing
𝑥𝑥
Possibly implied by a correct discriminant. | 1.1b | A1
= 0
Correctly substitutes their coefficients into to obtain an expression in k only.
Accept any sensible alternative for , e.g. 2or
𝑏𝑏 −4𝑎𝑎𝑐𝑐 | 3.1a | M1
Obtains a correct quadratic equation/inequality in – may be unsimplified.
𝑘𝑘 𝑦𝑦 f
Or obtains the correct critical values of .
𝑘𝑘
Accept any sensible alternative for , e.g. or | 1.1b | A1
𝑘𝑘
Obtains the correct critical values.
𝑘𝑘 𝑦𝑦 f
Accept non-exact values to at least 3 sig figs, e.g. 1.05 and 5.95 | 1.1b | A1
Gives a correct range in terms of y using exact values.
Condone ‘and’.
Follow through their critical values if M2 scored and quadratic inequality seen.
Do not accept an alternative for y
Accept any equivalent expressions for and
7−2√6 7+2√6
NMS scores 0/6
2 2 | 2.2a | A1F
Total | 8
Q | Marking instructions | AO | Marks | Typical solution