CAIE A-Level Mathematics AS 2 Pure Mathematics 2 Questions

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 Algebra 2.2 Logarithmic and exponential functions 2.3 Trigonometry 2.4 Differentiation 2.5 Integration 2.6 Numerical solution of equations

[Maximum number: 4]

Solve the inequality |5 x+7|>|2 x-3|.

Solve 5 x+7=2 x-3 to obtain $-\frac{10}{3}$
Or inequality.

Attempt solution of linear equation where 5 x and 2 x have
different signs
Or inequality.

Obtain $-\frac{4}{7}$
State $x<-\frac{10}{3}, x>-\frac{4}{7}$
A0 if ' ... and . . . ' used.

Alternative Method for Question 1

State or imply non-modulus equation $(5 x+7)^{2}=(2 x-3)^{2}$
Or inequality.

Attempt solution of three-term quadratic equation
Or inequality.

Obtain $-\frac{10}{3}$ and $-\frac{4}{7}$
State $x<-\frac{10}{3}, x>-\frac{4}{7}$
A0 if ' ... and . . .' used.

[Maximum number: 8]

(a)

$5 \quad$ (a) Given that $2 \ln (x+1)+\ln x=\ln (x+9)$ , show that $x=\sqrt{\frac{9}{x+2}}$ .

[ 3 ]

Use the power law correctly
*M1
Use correct process to obtain equation with no logarithms
Confirm $x=\sqrt{\frac{9}{x+2}}$
AG;
condone absence of justification for choice of positive root.

(b)

It is given that the equation $x=\sqrt{\frac{9}{x+2}}$ has a single root.

Show by calculation that this root lies between 1.5 and 2.0.

[ 2 ]

Consider sign of $x-\sqrt{\frac{9}{x+2}}$ or equivalent for 1.5 and 2
Obtain $-0.1 \ldots$ and 0.5 or equivalents and justify conclusion
AG

(c)

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

[ 3 ]

[Maximum number: 6]

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

(a)

Find the exact value of $\sin \theta$ .

[ 2 ]

Use $\sin 2 \theta=2 \sin \theta \cos \theta$
Obtain $\sin \theta=\frac{1}{6}$

(b)

Find the exact value of $\sec \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\sec \theta$
Obtain $\frac{6}{\sqrt{35}}$ or exact equivalent

(c)

Find the exact value of $\cos 2 \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\cos 2 \theta$
Obtain $\frac{17}{18}$ or exact equivalent

[Maximum number: 8]

Figure for Question 4 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

(a)

Find the gradient of the curve at A.

[ 3 ]

Differentiate to obtain form $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$
Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$
Substitute x=0 to obtain -10

(b)

Show that the x-coordinate of B is $\ln 2$ and hence find the area of the shaded region.

[ 5 ]

Attempt to find x-coordinate of B
$8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}=0$ .

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$
AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$
Use limits 0 and $\ln 2$ correctly to find area
For integral of form $k_{3} \mathrm{e}^{-x}+k_{4} \mathrm{e}^{2 x}$ where $k_{3} k_{4} \neq 0$ .
$k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $\frac{5}{2}$
OE

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)

Question (c)

Question 4

Question (a)

Question (b)