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Question 1

Topic-3.1 – Algebra

Expand $(9-3x)^{-\frac{1}{2}}$ in ascending powers of x, up to and including the term in $x^{2}$, simplifying the
coefficients.

▶️Answer/Explanation

Solution :-

Obtain a correct unsimplified version of the x or x² term of the expansion of

$(9-3x)^{-\frac{1}{2}} or \left(1-\frac{x}{3}\right)^{-\frac{1}{2}}$

State correct first term 3

Obtain the next two terms $\frac{1}{2}x-\frac{1}{24}x^{2}$

Question 2

(a) Topic-3.2 – Trigonometric Functions

(b) Topic-3.6 – Numerical Methods

(a) By sketching a suitable pair of graphs, show that the equation $cot~2x=sec~x$ has exactly one root
in the interval $0<x<\frac{1}{2}\pi$.

(b) Show that if a sequence of real values given by the iterative formula

$x_{n+1}=\frac{1}{2}tan^{-1}(cos~x_{n})$

converges, then it converges to the root in part (a).

▶️Answer/Explanation

Solution :-

(a) Sketch a relevant graph, e.g., $y=cot~2x$

Sketch a second relevant graph on the same axes, e.g. $y=sec~x$ and justify the given
statement

(b) State $x=\frac{1}{2}tan^{-1}(cos~x)$

and rearrange to the given equation $cot~2x=sec~x$

Note: If using the alternative approach in (a), can stop at $tan~2x=cos~x$

Question 3

Topic-3.7 – Complex Numbers

The square roots of 6 – 8i can be expressed in the Cartesian form x + iy, where x and y are real and exact.

By first forming a quartic equation in x or y, find the square roots of 6 – 8i in exact Cartesian form.

▶️Answer/Explanation

Solution :-

Square x + iy and equate real and imaginary parts to 6 and -8 respectively

Obtain equations $x^{2}-y^{2}=6$ and $2xy=-8$

Eliminate one variable and find an equation in the other (from 2 equations each in 2
unknowns)

Obtain $x^{4}-6x^{2}-16=0$ or $y^{4}+6y^{2}-16=0$

Obtain answers $\pm(2\sqrt{2}-\sqrt{2}i)$ or exact equivalents

Question 4

Topic-3.2 – Logarithmic and exponential functions

Solve the equation $5^{x}=5^{x+2}-10$. Give your answer correct to 3 decimal places.

▶️Answer/Explanation

Solution :-

Use laws of indices correctly and solve for $5^{x}$

Use a correct method for solving an equation of the form $5^{x}=a$, where a > 0

Obtain answer -0.544

Question 5

(a) Topic-3.7 – Complex Numbers

(b) Topic-3.7 – Complex Numbers

(a) The complex number u is given by

$u=\frac{(cos\frac{1}{7}\pi+i~sin\frac{1}{7}\pi)^{4}}{cos\frac{1}{7}\pi-i~sin\frac{1}{7}\pi}.$

Find the exact value of arg u.

(b) The complex numbers u and $u^{*}$ are plotted on an Argand diagram.

Describe the single geometrical transformation that maps u onto $u^{*}$ and state the exact value of
arg $u^{*}$.

▶️Answer/Explanation

Solution :-

(a)$ \frac{4\pi}{7} and/ or -\frac{\pi}{7}$

Obtain arg u = $\frac{5}{7}\pi$

(b) Reflection (in the) real axis

arg u= $-\frac{5}{7}\pi$

Question 6

Topic-3.2 – Logarithmic and exponential functions

The variables x and y satisfy the equation $ay = bx$, where a and b are constants. The graph of $ln~y$ against x is a straight line passing through the points (0.50, 2.24) and (3.40, 8.27), as shown in the diagram.

Find the values of a and b. Give each value correct to 1 significant figure.

▶️Answer/Explanation

Solution :-

Form a pair of equations in a and b

Carry out a correct method for finding ln a or ln b or a or b

Obtain value $a=0.3$

Obtain value $b=8$

Alternative Method for Question 6:

Carry out a correct method for finding ln b or b
(Need to link the gradient to ln b at some point)

Obtain value $b=8$

Correct method to find ±ln a or a

Obtain value $a=0.3$

Question 7

a) Topic-3.2 – Trigonometric Functions

(b) Topic-3.2 – Trigonometric Functions

(a) Show that the equation $tan^{3}x+2~tan~2x-tan~x=0$ may be expressed as

$tan^{4}x-2~tan^{2}x-3=0$

for $tan~x\ne0$.

(b) Hence solve the equation $tan^{3}2\theta+2~tan~4\theta-tan~2\theta=0$ for $0<\theta<\pi$. Give your answers in exact form.

▶️Answer/Explanation

Solution :-

(a) Use correct double angle formula to obtain an equation in $tan~x$

Obtain a correct equation in $tan~x$ in any form without fractions

Reduce to the given answer of $tan^{4}x-2~tan^{2}x-3=0$ correctly

(b) A complete correct method to solve the equation to obtain a value for $θ$

Obtain two of $(θ=)\frac{1}{6}\pi,\frac{1}{3}\pi,\frac{2}{3}\pi$ and
$\frac{5}{6}\pi$

Obtain the other two of $(θ=)\frac{1}{6}\pi,\frac{1}{3}\pi,\frac{2}{3}\pi$ and
$\frac{5}{6}\pi$ and no others in the interval

Question 8

(a) Topic-3.4 – Differentiation

(b) Topic-3.4 – Differentiation

The parametric equations of a curve are

$x=tan^{2}2t, \quad y=cos~2t,$

for $0<t<\frac{1}{4}\pi.$

(a) Show that
$\frac{dy}{dx}=-\frac{1}{2}cos^{3}2t.$

(b) Hence find the equation of the normal to the curve at the point where $t=\frac{1}{8}\pi$. Give your answer in the form $y=mx+c$.

▶️Answer/Explanation

Solution :-

(a) Use correct product rule or chain rule to find derivative of x with respect to t

Obtain $\frac{dx}{dt}=4~tan~2t~sec^{2}2t$ oe

$\frac{dy}{dt}=-2~sin~2t$

Use $\frac{dy}{dx}=\frac{dy}{dt}\times\frac{dt}{dx}$
to obtain given answer
$\frac{dy}{dx}=-\frac{1}{2}cos^{3}2t$

(b) Obtain $x=1$ and
$y=\frac{\sqrt{2}}{2}$

State or imply gradient of tangent is
$\frac{-\sqrt{2}}{8}$
or gradient of normal is $4\sqrt{2}$

Use correct method to find equation of normal using their values

Obtain equation of normal is
$y=4\sqrt{2}x-\frac{7\sqrt{2}}{2}$
or equivalent 3 term equation

Question 9

(a) Topic-3.8 – Vectors

(b) Topic-3.8 – Vectors

With respect to the origin O, the points A, B and C have position vectors given by

$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix}, \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}$

(a) The point D is such that ABCD is a trapezium with $\overrightarrow{DC} = 3\overrightarrow{AB}.$

Find the position vector of D.

(b) The diagonals of the trapezium intersect at the point P.

Find the position vector of P.

▶️Answer/Explanation

Solution :-

(a) Use a correct method to find $\overrightarrow{OD}$

Obtain position vector of D is 3i – 11j – 10k

(b) Carry out correct method for finding a vector equation for $\overrightarrow{AC}$ or $\overrightarrow{BD}$

Both diagonal equations correct.

Equate at least two pairs of corresponding components and solve for λ or for μ

Obtain $λ = \frac{1}{4} or μ = \frac{1}{4}$

Obtain position vector of P is $\frac{3}{4}i+\frac{1}{4}j-\frac{7}{4}k$

(b) Alternative Method for Question 9(b):

State or imply $\overrightarrow{AC}=5i-3j+5k$

Identify similar triangles with ratio 1 : 3

Use similar triangles to obtain $\overrightarrow{OP}$, e.g. $\overrightarrow{OP}=\overrightarrow{OA}+\frac{1}{4}\overrightarrow{AC}$

Obtain position vector of P is $\frac{3}{4}i+\frac{1}{4}j-\frac{7}{4}k$

Carry out correct process for evaluating the scalar product of two relevant vectors

Using the correct process for the moduli, divide their scalar product by the product
of their moduli and evaluate the inverse cosine of the result to obtain an angle

Obtain answer 77.3° (or 1.35 radians)

Question 10

(a) Topic-3.7 – Differential Equations

(b) Topic-3.1 – Algebra

(d) Topic-3.7 – Differential Equations

A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a
constant rate of 40π starting when time t = 0 and r = 0. At the same time, air begins to flow out of the
balloon at a rate of 0.8πr. The balloon remains a sphere at all times.

(a) Show that r and t satisfy the differential equation

$\frac{dr}{dt}=\frac{50-r}{5r^{2}}.$

(b) Find the quotient and remainder when $5r^{2}$ is divided by 50-r.

(d) Find the value of t when the radius of the balloon is 12.

▶️Answer/Explanation

Solution :-

(a) Obtain $\frac{dV}{dr}=40\pi-0.8\pi r$ or equivalent

Obtain $\frac{dV}{dr}=4\pi r^{2}$ or equivalent e.g. $\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}$

Use the chain rule to obtain given answer (including the derivative)

(b) Commence division and reach quotient of the form
$-5r\pm250$
or $5r^{2}=(50-r)(Ar+B)+C$ and reach $A=-5$ and $B=\pm250$

Obtain quotient $-5r-250$

Obtain remainder 12 500

Or express in the form $\frac{dt}{dr}=\frac{5r^{2}}{50-r}(=-(5r+250)+\frac{12500}{50-r})$

Obtain term t

Obtain terms $\frac{A}{2}r^{2}+Br-Cln(50-r)$

Obtain terms $-\frac{5}{2}r^{2}-250r-12500ln(50-r)$

Use t=0, r=0 to evaluate a constant or as limits in a solution containing terms of
the form $r^{2}$, r, $ln(50-r)$ and t

Obtain final answer $t=-\frac{5}{2}r^{2}-250r-12500ln(50-r)+12500ln50$

(d) Obtain r=70.5

Question 11

(a) Topic-3.4 – Differentiation

(b) Topic-3.5 – Integration

$Let f(x) = \frac{2e^{2x}}{e^{2x}-3e^{x}+2}.$

(a) Find f'(x) and hence find the exact coordinates of the stationary point of the curve with
equation y = f(x).

(b) Use the substitution $u = e^{x}$ and partial fractions to find the exact value of $\int_{ln~3}^{ln~5}f(x)dx$.

Give your answer in the form $ln~a$, where a is a rational number in its simplest form.

▶️Answer/Explanation

Solution :-

(a) Use correct quotient rule NB the question asks for f'(x) so need complete form

Obtain correct derivative in any form, e.g.,
$\frac{4e^{2x}(e^{2x}-3e^{x}+2)-2e^{2x}(2e^{2x}-3e^{x})}{(e^{2x}-3e^{x}+2)^{2}}$

Equate their derivative to zero

Solve for x to obtain x = ln a

Obtain $x=ln\frac{4}{3}$ and $y=-16$

(a) Alternative Method for Question 11(a)

Complete method to express f(x) in partial fractions

Differentiate to obtain $f'(x)=\frac{se^{x}}{(e^{x}-2)^{2}}+\frac{te^{x}}{(e^{x}-1)^{2}}$

Obtain $f'(x)=\frac{-8e^{x}}{(e^{x}-2)^{2}}+\frac{2e^{x}}{(e^{x}-1)^{2}}$

Equate derivative to zero and solve for x to obtain $x=lna$

Obtain $x=ln\frac{4}{3}$ and $y=-16$

(b) State or imply $\frac{du}{dx}=e^{x}$

Obtain $\int\frac{2u}{u^{2}-3u+2}du$ or equivalent

Or $\int(\frac{2}{u}+\frac{8}{u(u-2)}-\frac{2}{u(u-1)})du$

State or imply partial fractions of the form $\frac{A}{u-1}+\frac{B}{u-2}$

Or $_{1}\frac{C}{u}+\frac{D}{u-2}+\frac{E}{u-1}$
Or $_{2}\frac{2u-3}{u^{2}-3u+2}+\frac{F}{u-2}+\frac{G}{u-1}$

Use a correct method for finding a constant

Obtain correct $\frac{-2}{u-1}+\frac{4}{u-2}$

Or $_{2}\frac{2u-3}{u^{2}-3u+2}+\frac{3}{u-2}-\frac{3}{u-1}$

Integrate to obtain $a~ln(u-1)+b~ln(u-2)$ or equivalent

Obtain correct $-2~ln(u-1)+4~ln(u-2)$ or equivalent

Correctly use limits u=5 and 3 in an expression of the form $a~ln(u-1)+b~ln(u-2)$
or x=ln~5 and ln 3 in an expression of the form $a~ln(e^{x}-1)+b~ln(e^{x}-2)$

Obtain $ln\frac{81}{4}$

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