A-LevelMathematicsLogarithmic and exponential functions9709/22 · Oct/Nov 2023 · Q49 Marks

Sketch, on the same diagram, the graphs of y = |3 – x| and y = 9 – 2x.

(a)Sketch, on the same diagram, the graphs of y = |3 – x| and y = 9 – 2x.[2]
(b)Solve the inequality |3 − x| > 9 – 2x.[3]
(c)Use logarithms to solve the inequality 2^(3x-10) < 500. Give your answer in the form x < a, where the value of a is given correct to 3 significant figures.[3]
(d)List the integers that satisfy both of the inequalities |3 – x| > 9 – 2x and 2^(3x−10) < 500.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[B1] (a) Draw V-shaped graph with vertex on positive x-axis
[B1] (a) Draw (more or less) correct graph of y=9-2x with steeper negative gradient Dependent on first B mark, appropriately positioned with respect to first graph.
[M1] (b) Solve linear equation or inequality with signs of x and 2x different
[A1] (b) Obtain critical value 4
[A1] (b) Conclude x>4 only 6 must be discounted.
[B1] (b) State or imply non-modulus equation (or inequality) (3−x)² = (9−2x)²
[M1] (b) Attempt solution of three-term quadratic equation (or inequality) Dependent on previous B1.
[A1] (b) Conclude x >4 only 6 must be discounted.
[B1] (c) State or imply (3x-10)ln2<ln 500 Or equivalent perhaps involving different logarithm base.
[B1] (c) Obtain critical value 6.32
[B1] (c) Obtain x <6.32 Or greater accuracy.
[B1] (d) State 5 and 6 only

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.