CAIE A-Level Mathematics 3. Pure Mathematics 3 Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise Pure Mathematics 3 techniques for algebra, functions, calculus, vectors and equations through exact and numerical work.

Covered syllabus points

3.1 Algebra 3.2 Logarithmic and exponential functions 3.3 Trigonometry 3.4 Differentiation 3.5 Integration 3.6 Numerical solution of equations 3.7 Vectors 3.8 Differential equations

[Maximum number: 4]

Solve the inequality $|3 x-4| \leqslant|2 x+5|$ .

(a)

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

[ 3 ]

(b)

Hence solve the equation $\log _{3}(4 y+1)=1+2 \log _{3}(2 y-1)$ , giving your answer correct to 2 decimal places.

[ 2 ]

[Maximum number: 4]

The complex number u is given by $u=-1-\mathrm{i} \sqrt{3}$ .

(a)

Express u in the form $r(\cos \theta+\mathrm{i} \sin \theta)$ , where r>0 and $-\pi<\theta \leqslant \pi$ . Give the exact values of r and $\theta$ .
The complex number v is given by $v=5\left(\cos \frac{1}{6} \pi+\mathrm{i} \sin \frac{1}{6} \pi\right)$ .

[ 2 ]

(b)

Express the complex number $\frac{v}{u}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ where r>0 and $-\pi<\theta \leqslant \pi$ .

[ 2 ]

[Maximum number: 3]

The polynomial p(x) is defined by

\mathrm{p}(x)=x^{4}-10 x^{3}+20 x^{2}-30 x+40 .

Find the quotient when p(x) is divided by $\left(x^{2}+3\right)$ and show that the remainder is -11.

[Maximum number: 7]

Figure for Question 5 — CAIE A-Level Mathematics

The diagram shows the curve with equation $y=8 \mathrm{e}^{-\frac{1}{2} x}-1$ . The curve meets the axes at the points A and B. The shaded region is bounded by the curve and the line segment AB.

(a)

Show that the x-coordinate of B is $6 \ln 2$ .

[ 2 ]

(b)

Find the area of the shaded region. Give your answer in the form $p \ln 2-q$ , where p and q are positive integers.

[ 5 ]

[Maximum number: 5]

Given that $\int_{a}^{a+14} \frac{1}{3 x} \mathrm{~d} x=\ln 2$ , find the value of the positive constant a.

[Maximum number: 8]

The equation of a curve is $y=\tan ^{-1}(4 x)$ .

(a)

Find the exact values of x when the gradient of the curve is $\frac{1}{4}$ .

[ 3 ]

(b)

Find the exact value of $\int_0^{0.25}y\,dx$ .

[ 5 ]

[Maximum number: 11]

The polynomial p(x) is defined by

\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

(a)

Find the value of k.

It is given that the equation p(x)=0 has exactly two real roots, denoted by $\alpha$ and $\beta$ , where $\alpha$ is an integer and $\beta$ is not an integer.

[ 2 ]

(b)

State the value of $\alpha$ and show that $\beta$ satisfies the equation $x=\sqrt[3]{-2 x-4.5}$ .

[ 4 ]

(c)

Show by calculation that $-1.4<\beta<-1.0$ .

[ 2 ]

(d)

Use an iterative formula, based on the equation in part (b), to find the value of $\beta$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

[ 3 ]

[Maximum number: 7]

The polynomials f(x) and g(x) are defined by

\mathrm{f}(x)=4 x^{3}+a x^{2}+8 x+15 \quad \text { and } \quad \mathrm{g}(x)=x^{2}+b x+18

where a and b are constants.

(a)

Given that (x+3) is a factor of f(x), find the value of a.

[ 2 ]

(b)

Given that the remainder is 40 when g(x) is divided by (x-2), find the value of b.

[ 2 ]

(c)

Hence solve the equation $\mathrm{f}(\operatorname{cosec} \theta)-\mathrm{g}(\operatorname{cosec} \theta)=0$ for $0<\theta<2 \pi$ .

[ 3 ]

[Maximum number: 9]

The diagram shows a tank for holding water. The tank is in the shape of a cube of side 50 cm. At time t seconds, the depth of water in the tank is $h \mathrm{~cm}$ . Water is poured into the tank at a rate of $5000 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ . Water pours out of the tank through a hole in the bottom at a rate proportional to $h^{2}$ .

When h=20, the depth of the water is increasing at a rate of $0.4 \mathrm{cms}^{-1}$ .

(a)

Show that $\frac{\mathrm{d} h}{\mathrm{~d} t}=\frac{500-h^{2}}{250}$ .

[ 4 ]

(b)

Given that h=0 when t=0, find the time taken for the depth of the water in the tank to reach 20 cm.

[ 5 ]

Covered syllabus points

Question 3(a)

Question 3

Question 3(a)

Question 3(b)

Question 4

Question 4(a)

Question 4(b)

Question 4(a)

Question 5

Question 5(a)

Question 5(b)

Question 4

Question 7

Question 7(a)

Question 7(b)

Question 7

Question 7(a)

Question 7(b)

Question 7(c)

Question 7(d)

Question 6

Question 6(a)

Question 6(b)

Question 6(d)

Question 10

Question 10(a)

Question 10(b)