CAIE A-Level Mathematics 2 Pure Mathematics 2 Question Bank

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: 7]

(a)

Sketch, on the same diagram, the graphs of y=|3 x-5| and y=2 x+7.

[ 2 ]

(b)

Solve the equation |3 x-5|=2 x+7.

[ 3 ]

Solve 3 x-5=2 x+7 to obtain x=12

B1

Attempt solution of linear equation where signs of 3 x and 2 x are
different

M1

3 x-5=-2 x-7 OE.

Obtain $x=-\frac{2}{5}$

A1

Alternative solution for question 4(b)

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

B1

Must be working with $(3 x-5)^{2}=(2 x+7)^{2}$ .

Attempt solution of 3-term quadratic equation

M1

Obtain $-\frac{2}{5}$ and 12

A1

(c)

Hence solve the equation $\left|3^{y+1}-5\right|=2 \times 3^{y}+7$ , giving your answer correct to 3 significant figures.

[ 2 ]

Apply logarithms and use power law for $3^{y}=k$ where k>0 or
correct equivalent

M1

Using their positive answer from part (b)
or greater accuracy; and no other values.

Obtain 2.26

A1

[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
DM1
Confirm $x=\sqrt{\frac{9}{x+2}}$
A1
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
M1
Obtain $-0.1 \ldots$ and 0.5 or equivalents and justify conclusion
A1
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 $\sin 2 \theta=2 \sin \theta \cos \theta$
B1
Obtain $\sin \theta=\frac{1}{6}$
B1

(b)

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

[ 2 ]

Use correct identity or identities to find value of $\sec \theta$
M1
Obtain $\frac{6}{\sqrt{35}}$ or exact equivalent
A1

(c)

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

[ 2 ]

Use correct identity or identities to find value of $\cos 2 \theta$
M1
Obtain $\frac{17}{18}$ or exact equivalent
A1

[Maximum number: 8]

Figure for Question 2. Pure Mathematics 2 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 $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$

M1

Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$

A1

Substitute x=0 to obtain -10

A1

(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

M1

$8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}=0$ .

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$

A1

AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$

B1

Use limits 0 and $\ln 2$ correctly to find area

M1

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 $\frac{5}{2}$

A1

OE

All question bank results loaded

Covered syllabus points

2. Pure Mathematics 2 question 1

Question (a)

Question (b)

Question (c)

2. Pure Mathematics 2 question 2

Question (a)

Question (b)

Question (c)

2. Pure Mathematics 2 question 3

Question (a)

Question (b)

Question (c)

2. Pure Mathematics 2 question 4

Question (a)

Question (b)