CAIE
Mathematics
(9709)
2023
Oct/Nov
Questions & Answers
Q1a.
Short Answer
1 mark
Sketch the graph of y = 4x − 2.
Model Answer:
A straight line with gradient 4 and y-intercept −2.
Explanation: The equation is linear in x, so its graph is a straight line. The coefficient of x gives the gradient and the constant term gives the y-intercept.
Q1b.
Short Answer
4 marks
Solve the inequality 1 + 3x < 4x − 2.
Model Answer:
x > 3
Explanation: Rearranging gives 1 + 3x < 4x − 2, so 3 < x.
Q2.
Short Answer
4 marks
The parametric equations of a curve are x = ln(t^2), y = e^(2−t^2), for t > 0. Find the gradient of the curve at the point where t = e, simplifying your answer.
Model Answer:
The gradient is −e^(2−e^2)/e.
Explanation: Using dy/dx = (dy/dt)/(dx/dt), with x = ln(t^2) and y = e^(2−t^2), gives dx/dt = 2/t and dy/dt = −2t e^(2−t^2). Hence dy/dx = −t^2 e^(2−t^2). At t = e, this is −e^2 e^(2−e^2) = −e^(4−e^2).
Q3.
Short Answer
5 marks
The polynomial 2x^3 + ax^2 − 11x + b is denoted by p(x). It is given that p(x) is divisible by 2x − 1 and that when p(x) is divided by x + 1 the remainder is 12. Find the values of a and b.
Model Answer:
a = 1, b = 9
Explanation: Since 2x−1 is a factor, p(1/2)=0. Since the remainder on division by x+1 is 12, p(−1)=12. Solving the two equations gives a = 1 and b = 9.
Q4a.
Structured
4 marks
On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − 4 − 3i| ≤ 2 and Re z ≤ 3.
Model Answer:
The region inside and on the circle centred at (4, 3) with radius 2, intersected with the half-plane Re z ≤ 3.
Explanation: The modulus inequality represents a closed disc. The condition Re z ≤ 3 restricts to the part of the disc on or to the left of the vertical line x = 3.
Q4b.
Short Answer
2 marks
Find the greatest value of arg z for points in this region.
Model Answer:
The greatest value of arg z is arctan(5/2).
Explanation: The maximum argument occurs at the point in the region with the steepest line from the origin, which is the upper-left boundary point of the circle. This gives arg z = arctan(5/2).
Q5.
Short Answer
6 marks
Find the exact value of ∫ from 0 to 6 of x/(x^2 + 4) dx.
Model Answer:
1/2 ln 10
Explanation: Let u = x^2 + 4, so du = 2x dx. Then the integral becomes 1/2 ∫ from 4 to 40 of du/u = 1/2 ln(40/4) = 1/2 ln 10.