CAIE A-Level Mathematics A2 3 Pure Mathematics 3 Questions

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 3.9 Complex numbers

[Maximum number: 4]

Solve the inequality |2 x+3|>3|x+2|.

Square both sides:
$(2x+3)^2>9(x+2)^2.$
Then
$4x^2+12x+9>9x^2+36x+36,$
so
$5x^2+24x+27<0.$
Factorising,
(5x+9)(x+3)<0.
The critical values are x=-3 and $x=-\frac95$ . The product is negative between them, hence
$-3<x<-\frac95.$

B1 for the non-modular squared inequality or equivalent linear equations. M1 for solving the quadratic/equations. A1 for critical values -3 and $-\frac95$ . A1 for $-3<x<-\frac95$ , with strict inequalities.

Figure for Answer to Question 1 — CAIE A-Level Mathematics A2

[Maximum number: 4]

Solve the equation $\ln (2 x-1)=2 \ln (x+1)-\ln x$ . Give your answer correct to 3 decimal places.

Use law for the logarithm of a product, quotient or power
Remove logarithms and state a correct equation, e.g. $x(2 x-1)=(x+1)^{2}$
Solve a 3-term quadratic obtaining at least one root
Obtain answer 3.303 only

[Maximum number: 5]

(a)

By sketching a suitable pair of graphs, show that the equation $\operatorname{cosec} x=1+\mathrm{e}^{-\frac{1}{2} x}$ has exactly two roots in the interval $0<x<\pi$ .

[ 2 ]

Sketch a relevant graph, e.g. $y=\operatorname{cosec} x$ $\operatorname{cosec} x$ , U shaped, roughly symmetrical about $x=\frac{\pi}{2}, y\left(\frac{\pi}{2}\right)=1$
and domain at least $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$ .
Sketch a second relevant graph, e.g. $y=1+\mathrm{e}^{-\frac{1}{2} x}$ , and justify the
given statement
Exponential graph needs y(0)=2, negative gradient, always increasing, and $y(\pi)>1$
Needs to mark intersections with dots, crosses, or say roots at points of intersection, or equivalent

(b)

The sequence of values given by the iterative formula

x_{n+1}=\pi-\sin ^{-1}\left(\frac{1}{\mathrm{e}^{-\frac{1}{2} x_{n}}+1}\right),

with initial value $x_{1}=2$ , converges to one of these roots.
Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

Use the iterative formula correctly at least twice
2, 2.3217, 2.2760, 2.2824...
Need to see 2 iterations and following value inserted correctly
Obtain final answer 2.28
Must be supported by iterations
Show sufficient iterations to at least 4 d.p. to justify 2.28 to
$2 \mathrm{~d} . \mathrm{p}$ ., or show there is a sign change in the interval (2.275,
2.285)

[Maximum number: 8]

The variables x and $\theta$ satisfy the differential equation

x \sin ^{2} \theta \frac{\mathrm{~d} x}{\mathrm{~d} \theta}=\tan ^{2} \theta-2 \cot \theta

for $0<\theta<\frac{1}{2} \pi$ and x>0. It is given that x=2 when $\theta=\frac{1}{4} \pi$ .

(a)

Show that $\frac{\mathrm{d}}{\mathrm{d} \theta}\left(\cot ^{2} \theta\right)=-\frac{2 \cot \theta}{\sin ^{2} \theta}$ .
(You may assume without proof that the derivative of $\cot \theta$ with respect to $\theta$ is $-\operatorname{cosec}^{2} \theta$ .)

[ 1 ]

Show sufficient working to justify the given statement
e.g. see $2 \cot \theta \times-\operatorname{cosec}^{2} \theta$ in the working
or express in terms of $\sin \theta$ and $\cos \theta$ and use quotient
rule to obtain the given result. Solution must have $\theta$
present throughout and must reach the given answer.

(b)

Solve the differential equation and find the value of x when $\theta=\frac{1}{6} \pi$ .

[ 7 ]

Separate variables correctly

Check for relevant working in (a)
$\int x \mathrm{~d} x=\int \frac{\tan ^{2} \theta}{\sin ^{2} \theta}-\frac{2 \cot \theta}{\sin ^{2} \theta} \mathrm{~d} \theta$

Condone incorrect notation e.g. missing d x.

Need either the integral sign or the $\mathrm{d} x, \mathrm{~d} \theta$ .

Obtain term $\frac{1}{2} x^{2}$
Obtain terms $\tan \theta+\cot ^{2} \theta$

B1 + B1

Alternative: $\int \frac{2 \cot \theta}{\sin ^{2} \theta} \mathrm{~d} \theta=\int \frac{2 \cos \theta}{\sin ^{3} \theta} \mathrm{~d} \theta=-\frac{1}{\sin ^{2} \theta}(+C)$

Form an equation for the constant of integration, or use limits $x=2, \theta=\frac{1}{4} \pi$ , in
a solution with at least two correctly obtained terms of the form $a x^{2}, b \tan \theta$
and $\cot ^{2} \theta$ , where $a b c \neq 0$
Need to have 3 terms. Constant of correct form.

State correct solution in any form, e.g. $\frac{1}{2} x^{2}=\tan \theta+\cot ^{2} \theta$
or $\frac{1}{2} x^{2}=\tan \theta+\operatorname{cosec}^{2} \theta-1$

If everything else is correct, allow a correct final answer to imply this A1.

Substitute $\theta=\frac{1}{6} \pi$ and obtain answer x=2.67
$2.6748 \ldots \sqrt{\frac{18+2 \sqrt{3}}{3}}$ If see a correctly rounded value ISW.

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Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question 4

Question (a)

Question (b)