CAIE A-Level Mathematics 2 Pure Mathematics 2 Question Bank

CAIE A-Level Mathematics 2 Pure Mathematics 2 Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise Pure Mathematics 2 through algebra, logarithms, exponentials, trigonometry, differentiation, integration and numerical solution methods with exact and iterative working.

Covered syllabus points

2.1 Algebra2.2 Logarithmic and exponential functions2.3 Trigonometry2.4 Differentiation2.5 Integration2.6 Numerical solution of equations

Question 1

Question 1(a)

(a)

（a）Sketch the graph of y=|3 x-6| ．
（a）Sketch the graph of y=|3 x-6| ．

Table for Question 1(a) — CAIE A-Level Mathematics

Question 1(b)

(b)

Solve the inequality 5x-3<|3x-6|.

Question 2(a)

[Maximum number: 3]

Show that the equation log⁡4(2x+1)=2log⁡4(3x−1)−2\log _{4}(2 x+1)=2 \log _{4}(3 x-1)-2 can be written as a quadratic equation in x.

Question 4

[Maximum number: 5]

Solve the equation cot⁡θtan⁡(θ+45∘)=7\cot \theta \tan \left(\theta+45^{\circ}\right)=7 for 0∘<θ<90∘0^{\circ}<\theta<90^{\circ}.

Question 5

[Maximum number: 8]

The polynomial p(x) is defined by p(x)=9x3+18x2+5x+4\mathrm{p}(x)=9 x^{3}+18 x^{2}+5 x+4.

Question 5(a)

(a)

Find the quotient when p(x) is divided by (3 x+2), and show that the remainder is 6 .

[ 3 ]

Question 5(b)

(b)

Find the value of ∫02p(x)3x+2 dx\int_{0}^{2} \frac{\mathrm{p}(x)}{3 x+2} \mathrm{~d} x, giving your answer in the form a+ln⁡ba+\ln b where a and b are integers.

[ 5 ]

Question 6

Question 6(a)

(a)

By sketching a suitable pair of graphs, show that the equation

∣x−2∣=2sin⁡12x|x-2|=2 \sin \frac{1}{2} x

has only one root in the interval 0<x<π0<x<\pi.

[ 2 ]

Question 6(b)

(b)

Show by calculation that this root lies between 1 and 1.5.

[ 2 ]

Question 6(c)

(c)

Use the iterative formula xn+1=2−2sin⁡12xnx_{n+1}=2-2 \sin \frac{1}{2} x_{n} with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

Question 4

[Maximum number: 3]
Figure for Question 4 — CAIE A-Level Mathematics

The diagram shows the curve with equation y=6e2x−e3xy=6 \mathrm{e}^{2 x}-\mathrm{e}^{3 x}. The shaded region is bounded by the axes and the curve.

Question 4(a)

(a)

Find the exact x-coordinate of the maximum point.

[ 3 ]

Question 4(b)

(b)

Find the area of the shaded region. Give your answer in the form pq\frac{p}{q}, where p and q are integers.

Question 6

[Maximum number: 6]

It is given that 3sin⁡2θ=cos⁡θ3 \sin 2 \theta=\cos \theta where θ\theta is an angle such that 0∘<θ<90∘0^{\circ}<\theta<90^{\circ}.

Question 6(a)

(a)

Find the exact value of sin⁡θ\sin \theta.

[ 2 ]

Question 6(b)

(b)

Find the exact value of sec⁡θ\sec \theta.

[ 2 ]

Question 6(c)

(c)

Find the exact value of cos⁡2θ\cos 2 \theta.

[ 2 ]

Question 6

[Maximum number: 9]
Figure for Question 6 — CAIE A-Level Mathematics

The diagram shows the curve with equation y=ln⁡(2x+1)x+3y=\frac{\ln (2 x+1)}{x+3}. The curve has a maximum point M.

Question 6(a)

(a)

Find an expression for dy dx\frac{\mathrm{d} y}{\mathrm{~d} x}.

[ 2 ]

Question 6(b)

(b)

Show that the x-coordinate of M satisfies the equation x=x+3ln⁡(2x+1)−0.5x=\frac{x+3}{\ln (2 x+1)}-0.5.

[ 2 ]

Question 6(c)

(c)

Show by calculation that the x-coordinate of M lies between 2.5 and 3.0 .

[ 2 ]

Question 6(d)

(d)

Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

[ 3 ]

Question 6

Question 6(a)

(a)

Sketch the graph of y=3sin⁡x+2y=3 \sin x+2 for 0⩽x⩽2π0 \leqslant x \leqslant 2 \pi.

Figure for Question 6(a) — CAIE A-Level Mathematics
[ 2 ]

Question 6(b)

(b)

Determine the number of solutions in the interval 0⩽x⩽2π0 \leqslant x \leqslant 2 \pi of each of the following equations.

[ 1 ]

Question 6(b)(ii)

(i)

3sin⁡x+2=5−x3 \sin x+2=5-x

[ 1 ]

Question 8

[Maximum number: 6]

The equation of a curve is y=4e1−2x3x−1y=4 \mathrm{e}^{1-2 x} \sqrt{3 x-1}.
Find the exact coordinates of the stationary point of the curve.