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Question 1

Topic-3.1 – Algebra

The polynomial $4x^{3} + ax^{2} + 5x + b$, where a and b are constants, is denoted by p(x). It is given that $(2x+1)$ is a factor of p(x). When p(x) is divided by (x-4) the remainder is equal to 3 times the remainder when p(x) is divided by (x-2).

Find the values of a and b.

▶️Answer/Explanation

Solution :-

Substitute $x=-\frac{1}{2}$ and equate the result to zero

Obtain a correct equation, e.g. $-\frac{4}{8}+\frac{a}{4}-\frac{5}{2}+b=0$

Substitute $x=2$ and $x=4$ and use $p(4)=3p(2)$

Obtain a correct equation, e.g. $3(32+4a+10+b)=256+16a+20+b$

Obtain $a=-32$ and $b=11$

Question 2

Topic-3.5 – Integration

Find the exact value of $\int_{1}^{e^{2}}x^{2}ln~3x~dx$. Give your answer in the form $a~ln~b+c$, where a and c are rational and b is an integer.

▶️Answer/Explanation

Solution :-

Integrate to obtain $px^{3}ln~3x+q\int{x^{2}}dx$

Obtain $\frac{1}{3}x^{3}ln~3x-\frac{1}{3}\int{x^{2}}dx$

Complete integration to obtain $\frac{1}{3}x^{3}ln~3x-\frac{1}{9}x^{3}$

Use correct limits correctly in an expression of the form $rx^{3}ln~3x+sx^{3}$

Obtain $\frac{53}{3}ln~3-\frac{26}{9}$

Question 3

Topic-3.4 – Differentiation

The equation of a curve is $ln(x+y)=3x^{2}y$.

Find the gradient of the curve at the point (1, 0).

▶️Answer/Explanation

Solution :-

Integrate to obtain $px^{3}ln~3x+q\int{x^{2}}dx$

Obtain $\frac{1}{3}x^{3}ln~3x-\frac{1}{3}\int{x^{2}}dx$

Complete integration to obtain $\frac{1}{3}x^{3}ln~3x-\frac{1}{9}x^{3}$

Use correct limits correctly in an expression of the form $rx^{3}ln~3x+sx^{3}$

Obtain $\frac{53}{3}ln~3-\frac{26}{9}$

Question 4

(a) Topic-3.2 – Trigonometric Functions

(b) Topic-3.2 – Trigonometric Functions

(a) Show that $sec^{4}\theta-tan^{4}\theta\equiv1+2~tan^{2}\theta$.

(b) Hence or otherwise solve the equation $sec^{4}2\alpha-tan^{4}2\alpha=2~tan^{2}2\alpha~sec^{2}2\alpha$ for $0^{\circ}<\alpha<180^{\circ}$.

▶️Answer/Explanation

Solution :-

(a) Factorise an expression in $cos^{2}\theta$ or $sec^{2}\theta$

Or obtain an expression in $cos^{2}\theta$ or $sec^{2}\theta$

Use of $1+tan^{2}\theta=sec^{2}\theta$ (anywhere)

Obtain $1 \times (1+tan^{2}\theta+tan^{2}\theta)=1+2~tan^{2}\theta$

(b) Form an equation in $tan~2\alpha.$

Or multiply through by $cos^{4}2\alpha$ to form an equation in $sin~2\alpha$ or $cos~2\alpha$

Solve for $tan~2\alpha$ or equivalent

Obtain one correct solution for a, e.g. $20.0(30..)^{\circ}$

Obtain a second correct value for a, e.g. $70.0^{\circ}(69.9698…^{\circ})$

Obtain solutions $110^{\circ}$ (110.0) and $160^{\circ}$ (160.0) for a, and no others in range

Question 5

(a) Topic-3.2 – Trigonometric Functions

(b) Topic-3.2 – Trigonometric Functions

(a) By sketching a suitable pair of graphs, show that the equation $2+e^{-0.2x}=ln(1+x)$ has only one root.

(b) Show by calculation that this root lies between 7 and 9.

$x_{n+1}=exp(2+e^{-0.2x_{n}})-1$

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[exp(x) is an alternative notation for $e^{x}$.]

▶️Answer/Explanation

Solution :-

(a) Sketch a relevant graph, e.g. $y=2+e^{-0.2x}$

For the sketches:

Correct curvature
Intersections with the y-axis approximately correct
Horizontal asymptote approximately correct – need not draw in
Allow scale not marked and implied by their sketch

Sketch a second relevant graph, e.g. $y=ln(1+x)$ and justify the given
statement

(b) Calculate the value of a relevant expression or values of a relevant pair of
expressions at $x=7$ and $x=9$

Complete the argument correctly with correct calculated values

Obtain final answer 8.03

Show sufficient iterations to at least 4 decimal places to justify 8.03 to 2
decimal places, or show that there is a sign change in the interval
(8.025, 8.035)

Question 6

(a) Topic-3.4 – Differentiation

(b) Topic-3.5 – Integration

The diagram shows the curve $y=sin~2x(1+sin~2x)$, for $0\le x\le\frac{3}{4}\pi$, and its minimum point M. The
shaded region bounded by the curve that lies above the x-axis and the x-axis itself is denoted by R.

(a) Given that the x-coordinate of M lies in the interval $\frac{1}{2}\pi<x<\frac{3}{4}\pi$, find the exact coordinates
of M.

(b) Find the exact area of the region R.

▶️Answer/Explanation

Solution :-

(a) Correct use of product rule to differentiate

Obtain $2~cos~2x+4~cos~2x~sin~2x$

Equate derivative to zero and solve for 2x or x

Obtain $x=\frac{7}{12}\pi, y=-\frac{1}{4}$

(b) Use correct double angle formula to integrate

Or use integration by parts and correct double angle formula

Obtain $-\frac{1}{2}cos~2x+\frac{x}{2}-\frac{1}{8}sin~4x(+C)$

Use limits 0 and $\frac{1}{2}\pi$ correctly in a solution containing $p~cos~2x$ and $q~sin~4x$

Obtain $\frac{1}{4}\pi-1$

Question 7

(a) Topic-3.1 – Algebra

(b) Topic-3.1 – Algebra

Let $f(x)=\frac{5x^{2}+8x+5}{(1+2x)(2+x^{2})}$.

(a) Express $f(x)$ in partial fractions.

(b) Hence find the coefficient of x³ in the expansion of f(x).

▶️Answer/Explanation

Solution :-

(a) State or imply the form
${A}{1+2x}+\frac{Bx+C}{2+x^{2}}$

Use a correct method to find a constant

Obtain one of $A=1 B=2$ and $C=3$

Obtain a second value

Obtain the third value

(b) State
${-1.-2.-3}{3!}(2x)^{3}$
or -8

Use a correct method to obtain the coefficient of $x^{2}$ in the expansion of
$(2+x^{2})^{-1}$
or the coefficient of $x^{2}$ in the expansion of
$(1+\frac{x^{2}}{2})^{-1}$ .

Obtain $(Bx+C)\times\frac{1}{2}x-\frac{1}{2}x^{2}$ or $-\frac{B}{4}x^{3}$ or
$-\frac{B}{4}$

Obtain final answer $-8\frac{1}{2}$ or $-8\frac{1}{2}x^{3}$

Question 8

(a) Topic-3.7 – Complex Numbers

(b) Topic-3.7 – Complex Numbers

(d) Topic-3.7 – Complex Numbers

(a) Given that $z=1+yi$ and that y is a real number, express $\frac{1}{z}$ in the form $a+bi$, where a and b are
functions of y.

(b) Show that $(\frac{a-1}{2})^{2}+b^{2}=\frac{1}{4}$, where a and b are the functions of y found in part (a).

(c) On a single Argand diagram, sketch the loci given by the equations $Re(z)=1$ and $\left|z-\frac{1}{2}\right|=\frac{1}{2}$, where z is a complex number.

(d) The complex number z is such that $Re(z)=1$. Use your answer to part (b) to give a geometrical
description of the locus of $\frac{1}{z}$.

▶️Answer/Explanation

Solution :-

(a) Multiply numerator and denominator by $1-yi$

Obtain
\[\frac{1}{1+y^{2}}+\frac{-y}{1+y^{2}}i\]

(b) Express $\left(a-\frac{1}{2}\right)^{2}+b^{2}$ in terms of y and expand the bracket

Obtain
\[\left(\frac{1}{(1+y^{2})^{2}}-\frac{2}{(1+y^{2})}+\frac{1}{4}\right)+\frac{y^{2}}{(1+y^{2})^{2}}\]

Obtain $\frac{1}{4}$ from full and correct working

Show a circle centre $\frac{1}{2}+0i$

Show a circle with radius $\frac{1}{2}$ and centre not at the origin

(d) circle centre $\frac{1}{2}+0i$ with radius $\frac{1}{2}$

Question 9

(a) Topic-3.8 – Vectors

(b) Topic-3.8 – Vectors

9. The position vector of point A relative to the origin O is $\vec{OA}=8i-5j+6k.$
The line passes through A and is parallel to the vector $2i+j+4k.$

(a) State a vector equation for l.

(b) The position vector of point B relative to the origin O is $\vec{OB}=-ti+4tj+3tk$, where t is a constant.
The line also passes through B.

Find the value of t.

(c) The line m has vector equation $r=5i-j+2k+\mu(ai-j+3k)$. The acute angle between the directions of l and m is θ, where $cos~\theta=\frac{1}{\sqrt{6}}$.

Find the possible values of a.

▶️Answer/Explanation

Solution :-

(a) Use a correct method to form a vector equation

Obtain $r=8i-5j+6k+\lambda(2i+j+4k)$

(b) State the position vector of a point on in component form
Or at least 2 correct components seen

Equate to $-ti+4tj+3tk$ and solve for t

Obtain $t=-2$

Complete the process for finding the cosine of θ

Obtain
\[\frac{2a+11}{\sqrt{21}\sqrt{10+a^{2}}}=\pm\frac{1}{\sqrt{6}}\]

Form a 3-term quadratic equation in a and solve for a

Obtain $a=-2, a=-86$

Question 10

(a) Topic-3.7 – Differential Equations

(b) Topic-3.7 – Differential Equations

A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time
t minutes, the depth of the water in the tank is h metres. There is a tap at the bottom of the tank. When
the tap is open, water flows out of the tank at a rate proportional to the square root of the volume of
water in the tank.

(a) Show that $\frac{dh}{dt}=-\lambda\sqrt{h}$, where $\lambda$ is a positive constant.

(b) At time $t=0$ the tap is opened. It is given that $h=4$ when $t=0$ and that $h=2.25$ when $t=20$.

Solve the differential equation to obtain an expression for t in terms of h, and hence find the time
taken to empty the tank.

▶️Answer/Explanation

Solution :-

(a) $\frac{dV}{dt}=\pm k\sqrt{V}$ or
$\frac{dV}{dt}=16\pi\frac{dh}{dt}$

Correct use of chain rule and $V=16\pi h$

Obtain
$\frac{dh}{dt}=\frac{dV}{dt}\div\frac{dV}{dh}=\frac{-k\sqrt{16\pi h}}{16\pi}$

$=-\left(\frac{k}{4\sqrt{\pi}}\right)\sqrt{h}=-\lambda\sqrt{h}$
since
$\frac{k}{4\sqrt{\pi}}$
is constant

(b) Separate variables correctly and commence integration

Obtain $-\lambda t=2\sqrt{h}(+C)$

Use the boundary conditions in an equation containing pt and $q\sqrt{h}$ to form
an equation in $\lambda$ and/or C

Use the boundary conditions in an equation containing pt and $q\sqrt{h}$ to form a
second equation in $\lambda$ andor C and solve

Hence $t=80-40\sqrt{h}$

Time to empty the tank is 80 minutes

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