A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The complex numbers $z$ and $w$ satisfy the simultaneous equations $\vert{}z\vert{} - w^* = -3 - \sqrt{2}\mathrm{i}$ and $w^* + w + 5z = 1 + 20\mathrm{i}$, where&nbsp;$w^*$ is the complex conjugate of $w$. Find the value of $z$ and the corresponding value of $w$.<span class="inline-marks">[4]</span></p>

The complex numbers $z$ and $w$ satisfy the simultaneous equations $\vert{}z\vert{} - w^* = -3 - \sqrt{2}\mathrm{i}$ and $w^* + w + 5z = 1 + 20\mathrm{i}$, where $w^*$ is the complex conjugate of $w$. Find the value of $z$ and the corresponding value of $w$.

Complex Numbers

The total delivery time $T$, in seconds, follows the distribution $T\sim N(1800,225+\sigma^2)$, where $\sigma>0$. (a) Given that $\sigma=40$, find $P(1680<T<1920)$. (b) The delivery company requires the delivery time to differ from its mean by less than $120$ seconds with probability at least $0.90$. Find the greatest possible value of $\sigma$. (c) The company introduces the additional requirement that $P(T>1900)\leq0.05$. Find the range of values of $\sigma$ for which both requirements are satisfied.

The total delivery time $T$, in seconds, follows the distribution $T\sim N(1800,225+\sigma^2)$, where $\sigma>0$.

Normal Probabilities and Quantiles

Solving Normal Probability Inequalities

Delivery time under two probability requirements

Continuous Random Variables

A curve is defined parametrically by $x=3\tan t$ and $y=2\sec t$, where $-\frac{\pi}{2}<t<\frac{\pi}{2}$.

Tangents and Normals to Parametric Curves

Converting Parametric Equations to Cartesian Form

Normal and branch restriction of a trigonometric curve

Applications of Differentiation

Graphing Techniques

A curve is defined parametrically by $x=\ln(t+1)$ and $y=\ln(5-t)$, where $t$ takes all real values for which both expressions are defined.

Normal and coordinate ranges of a logarithmic curve

<p>A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-26" data-list-item-path="p1421-item-26"><div class="list-item-paragraph">Find $z$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-28" data-list-item-path="p1421-item-28"><div class="list-item-paragraph">Find $z-z^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-30" data-list-item-path="p1421-item-30"><div class="list-item-paragraph">Use the given sum and product to find $\dfrac1z+\dfrac1{z^*}$, without evaluating the two reciprocals separately.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z+z^*=6$, $zz^*=25$ and $\operatorname{Im}(z)>0$.

Applying Complex Conjugate Identities

Solving Equations Involving Complex Conjugates

Recovering a number from its sum and product

<p>The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-42" data-list-item-path="p1421-item-42"><div class="list-item-paragraph">Find $z^*w^*$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-44" data-list-item-path="p1421-item-44"><div class="list-item-paragraph">Find $(zz^*)(ww^*)$, explaining how you regroup the factors.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-46" data-list-item-path="p1421-item-46"><div class="list-item-paragraph"><p>Find $\operatorname{Re}(zw)$ and $\operatorname{Im}(zw)$.</p><span class="inline-marks">[2]</span></div></li></ol>

The complex numbers $z$ and $w$ satisfy $zw=5-12i$. Do not find $z$ and $w$ separately.

Using a product and its conjugate

<p>A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-71" data-list-item-path="p1421-item-71"><div class="list-item-paragraph">Find $(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-73" data-list-item-path="p1421-item-73"><div class="list-item-paragraph">Find $z^3+(z^*)^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p1421-item-75" data-list-item-path="p1421-item-75"><div class="list-item-paragraph">Find $(z^*)^6$ in Cartesian form.<span class="inline-marks">[2]</span></div></li></ol>

A complex number $z$ satisfies $z^3=-2+11i$. Use conjugate properties without finding $z$.

Using conjugates of powers

<p>Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-59" data-list-item-path="p1421-item-59"><div class="list-item-paragraph">$\dfrac{w^*}{z^*}$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-61" data-list-item-path="p1421-item-61"><div class="list-item-paragraph">$\left(\dfrac{z+w}{z-w}\right)^*$. Explain why the denominator is nonzero.<span class="inline-marks">[4]</span></div></li></ol>

Nonzero complex numbers $z$ and $w$ satisfy $\dfrac zw=2-3i$. Use conjugate properties to find the following in Cartesian form without finding $z$ or $w$.

Conjugating ratios without finding the numbers

<p>Let&nbsp;$z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-91" data-list-item-path="p1421-item-91"><div class="list-item-paragraph">Show that&nbsp;$A^*=z^*w$, and hence that&nbsp;$zw^*+z^*w$ is real.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-93" data-list-item-path="p1421-item-93"><div class="list-item-paragraph">Show that&nbsp;$zw^*-z^*w$ has zero real part. Explain why it need not be a nonzero purely imaginary number.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1421-item-95" data-list-item-path="p1421-item-95"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{zw^*-z^*w}{i}$ is real.<span class="inline-marks">[2]</span></div></li></ol>

Let $z$ and $w$ be arbitrary complex numbers. Put $A=zw^*$. Use conjugate laws; do not expand $z$ and $w$ into Cartesian form.

Properties of Complex Conjugates

Recognising real and imaginary expressions

<p>The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.</p>
<ol class="abc-list"><li data-list-item-id="p1421-item-7" data-list-item-path="p1421-item-7"><div class="list-item-paragraph">$(3z-2w^*)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-9" data-list-item-path="p1421-item-9"><div class="list-item-paragraph">$(z^*+w)^*$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p1421-item-11" data-list-item-path="p1421-item-11"><div class="list-item-paragraph">$(iz)^*$ and $iz^*$. Explain why these values differ.<span class="inline-marks">[3]</span></div></li></ol>

The complex numbers $z=2-3i$ and $w=-1+4i$ are given. Use conjugate properties to find each expression in the form $a+bi$, showing the property used before substituting.

Conjugating a linear combination

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="ri-p2-q4-2-content-items-1" data-list-item-path="ri-p2-q4-2-content-items-1"><div class="list-item-paragraph">The curve $y=\mathrm{f}(x)$ passes through the origin and has gradient given by</div><div class="list-item-paragraph">\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x+y}{1+xy}.\]</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\displaystyle(1+xy)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2+(y-1)\dfrac{\mathrm{d}y}{\mathrm{d}x}=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-2"><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)(i)</strong>, show that</div><div class="list-item-paragraph">\[(1+xy)\frac{\mathrm{d}^3y}{\mathrm{d}x^3}+\mathrm{p}(x)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)+\mathrm{q}(y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+r\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=0,\]</div><div class="list-item-paragraph">where the functions $\mathrm{p}(x)$, $\mathrm{q}(y)$ and the constant $r$ are to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-3" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-3"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="ri-p2-q4-2-content-items-2" data-list-item-path="ri-p2-q4-2-content-items-2"><div class="list-item-paragraph">The equation</div><div class="list-item-paragraph">\[2\sqrt{2}\cos\left(x-\frac\pi4\right)+3\sin2x-3=0\]</div><div class="list-item-paragraph">has a root $\theta$, which is close to zero.</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\theta^2+a\theta+b\approx0$, where&nbsp;$a$ and $b$ are integers to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-2"><div class="list-item-paragraph">Hence find an approximation for&nbsp;$\theta$ in surd form.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

**Do not use a calculator in answering this question.**

Small Angle Approximations

Approximations Using Power Series

Maclaurin Series from Differential Equations

Higher Derivatives by Implicit Differentiation

2026 RI P2 Q4

Techniques of Differentiation

Maclaurin Series and Power Series

<p>Given&nbsp;$h(x)=1-|x-2|$, where&nbsp;$0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set&nbsp;$R$ such that&nbsp;$R=\{y:y\ge a\text{ or}\hspace{0.5em}y\le b\}$ where&nbsp;$a,b\in\mathbb Z$.</p>
<p>Determine the values of $a$ and $b$.</p>

Given $h(x)=1-|x-2|$, where $0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set $R$ such that $R=\{y:y\ge a\text{ or }y\le b\}$ where $a,b\in\mathbb Z$.

Modulus Functions

Range of a Function

Reciprocal Graph, $y=\frac{1}{f(x)}$

Graphs of Modulus Function

Two branches in a reciprocal modulus range

Functions

Modulus Function

Transformations of Graphs

<p>For&nbsp;$-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or}\hspace{0.5em}y\ge b\}$. Find the exact values of $a$ and $b$.<span class="inline-marks">[4]</span></p>

For $-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or }y\ge b\}$. Find the exact values of $a$ and $b$.

Reciprocal range of a stretched modulus trough

<p>A curve $C$ has equation $y=x^{-3}\ln x$.</p>
<ol class="abc-list"><li data-list-item-id="p1q7-q-b2-i1" data-list-item-path="p1q7-q-b2-i1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q7-q-b2-i2" data-list-item-path="p1q7-q-b2-i2"><div class="list-item-paragraph">Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has equation $y=x^{-3}\ln x$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.

Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.

Derivatives of Natural Logarithmic Functions

Product Rule

Area Bounded by Curves

Stationary Points and Their Nature

N2022 P1 Q7

Applications of Integration

Techniques of Integration

A small company makes 50 glass ornaments each working day. Some of the ornaments turn out to be faulty. State, in the context of the question, two assumptions needed for the number of faulty ornaments made in a day to be well modelled by a binomial distribution. Assume now that the number of faulty ornaments produced each day has the distribution $\mathrm{B}(50,0.04)$. Show that the numerical values of the mean and variance of this distribution differ by $0.08$. Find the probability that no more than 2 faulty ornaments are produced on a randomly chosen working day. Find the probability that no more than 2 faulty ornaments are produced on at least 3 days in a randomly chosen 5-day working week. State the distribution you use. Find the probability that no more than 10 faulty items are produced in a randomly chosen 5-day working week. State the distribution you use. The company also makes pens which are sold in randomly packed boxes of one hundred pens. The probability of a pen being not faulty is $p$, where $0<p<1$. For quality control purposes, a random sample of pens from each box is tested. Mr Lu and Mrs Ming carry out the tests but they use different methods. Mr Lu tests a random sample of 6 pens from a box. If there are no faulty pens or only 1 faulty pen the box is accepted. Mrs Ming tests a random sample of 3 pens from a box. If there are no faulty pens in her sample the box is accepted. If there are 2 or 3 faulty pens in her sample the box is rejected. If there is 1 faulty pen in her sample she takes a second random sample of 3 pens. She accepts the box if there are no faulty pens in this second sample. Show algebraically that Mrs Ming accepts a greater proportion of boxes than Mr Lu does.

A small company makes 50 glass ornaments each working day. Some of the ornaments turn out to be faulty.

Law of Total Probability

N2023 P2 Q10

Probability

<p class="paragraph-header">Do not use a calculator in answering this question.</p><p>The complex numbers $z$ and $w$ satisfy the following equations.</p><p style="text-align: center">\[\mathrm{i}z+2w=-1\]</p><p style="text-align: center">\[(2-\mathrm{i})z+\mathrm{i}w=6\]</p><p>Find $z$ and $w$, giving your answers in the form $a+\mathrm{i}b$ where $a$ and $b$ are real numbers.<span class="inline-marks">[4]</span></p>

Do not use a calculator in answering this question.

The complex numbers $z$ and $w$ satisfy the following equations.

Find $z$ and $w$, giving your answers in the form $a+\mathrm{i}b$ where $a$ and $b$ are real numbers.

Solving Simultaneous Complex Equations

N2022 P1 Q1

Find the complex numbers $z$ and $w$ which satisfy the simultaneous equations

\[(3+2\mathrm i)z-3w=3-11\mathrm i\quad\text{and}\quad(1-\mathrm i)z-6\mathrm iw=10+8\mathrm i.\]

2026 EJC P2 Q1

The masses of mangoes in kg have the distribution $\mathrm N(0.3,0.1^2)$.

Find the probability that two randomly selected mangoes weigh at least 0.5 kg in total.

Mark sells fruit boxes, each of which contain 10 randomly selected blueberries and 2 randomly selected mangoes. Let the number of unblemished blueberries in a randomly selected fruit box be $U$. On average, the probability of a blueberry being unblemished is $p$.

State a reason, in the context of the question, why $U$ might not be well-modelled by a binomial distribution.

Assume now that $U$ has the distribution $\mathrm B(10,p)$. The probability that a randomly selected fruit box contains at least 8 unblemished blueberries is 0.9.

Each fruit box is classified as grade A, B or C according to the following criteria:

Show that the probability that a randomly chosen fruit box is classified as grade C is 0.316, correct to 3 significant figures.

Hence find the probability that a randomly selected grade C fruit box contains at least 8 unblemished blueberries.

The selling prices of a grade A, grade B and grade C fruit box are $\$12$, $\$10$ and $\$6$ respectively. Find the probability that the total selling price of 5 randomly selected fruit boxes is less than $\$37$.

Mark decides to amend the criteria for grade A boxes as follows:

**Each** mango in the box weighs at least 0.3 kg **and** the box contains at least 9 unblemished blueberries.

State how the percentage of fruit boxes classified as grade A will change, and explain your answer.

Binomial Models and Assumptions

Symmetry and Properties of the Normal Distribution

Applying Probability Laws

Linear Combinations of Independent Normal Variables

Determining an Unknown Binomial Success Probability

2026 EJC P2 Q11

Discrete Random Variables (DRV)

<p>Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface. Two underwater sonar buoys, buoy A and buoy B, are located at points $A$ and $B$ respectively. The underwater sonar buoys are in the water body.</p>
<p>Points are defined relative to an origin $(0,0,0)$ on the ocean bed, with units in metres.</p>
<p>The ocean surface is modelled as the plane with equation $3y+4z=240$. A submarine moves along a straight path with equation $\frac x2=y-20=\frac{-z+13}2$. $A$ and $B$ have coordinates $(-10,31,5)$ and $(4,17,19)$ respectively.</p>
<ol class="abc-list"><li data-list-item-id="rvhs-q12-l4-item01" data-list-item-path="rvhs-q12-l4-item01"><div class="list-item-paragraph">Find a cartesian equation of the plane containing the path of the submarine and buoy B.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="rvhs-q12-l4-item02" data-list-item-path="rvhs-q12-l4-item02"><div class="list-item-paragraph">Determine the acute angle between the path of the submarine and the ocean surface.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rvhs-q12-l4-item03" data-list-item-path="rvhs-q12-l4-item03"><div class="list-item-paragraph">An unidentified boat on the ocean surface is detected by buoys A and B. The boat is found to be equidistant from both buoys A and B. Describe geometrically the set of all possible positions of the location of the unidentified boat.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="rvhs-q12-l4-item04" data-list-item-path="rvhs-q12-l4-item04"><div class="list-item-paragraph">At low tide, the ocean surface falls below the current ocean surface at a perpendicular distance of $0.8$ metres. The ocean surface at low tide is also modelled as a plane. Find the cartesian equation of this plane.<span class="inline-marks">[2]</span></div></li></ol>

Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface. Two underwater sonar buoys, buoy A and buoy B, are located at points $A$ and $B$ respectively. The underwater sonar buoys are in the water body.

Points are defined relative to an origin $(0,0,0)$ on the ocean bed, with units in metres.

The ocean surface is modelled as the plane with equation $3y+4z=240$. A submarine moves along a straight path with equation $\frac x2=y-20=\frac{-z+13}2$. $A$ and $B$ have coordinates $(-10,31,5)$ and $(4,17,19)$ respectively.

Find a cartesian equation of the plane containing the path of the submarine and buoy B.

Determine the acute angle between the path of the submarine and the ocean surface.

An unidentified boat on the ocean surface is detected by buoys A and B. The boat is found to be equidistant from both buoys A and B. Describe geometrically the set of all possible positions of the location of the unidentified boat.

At low tide, the ocean surface falls below the current ocean surface at a perpendicular distance of $0.8$ metres. The ocean surface at low tide is also modelled as a plane. Find the cartesian equation of this plane.

Distance from a Point to a Plane

Angle Between a Line and a Plane

Equations of Planes

Geometrical Interpretation of Vector Equations

Line of Intersection of Two Planes

2026 RVHS P1 Q12

Vectors

<p>Brand $X$ and Brand $Y$ batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is $20$ metres.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">When the toy car used Brand $X$ battery, it completed the first lap in $15$ seconds and each subsequent lap took $3$ seconds longer than the previous lap.</div><ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0-sub-0" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-0"><div class="list-item-paragraph">In which lap did the toy car first have a lap timing of more than $312.3$ seconds.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-0-sub-1" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-1"><div class="list-item-paragraph">Given that the toy car ran for&nbsp;$23$ hours before stopping, find the average speed of the car during the last incomplete lap.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">When the toy car used Brand $Y$ battery, it completed the first lap in $7.68$ seconds and the time taken for each subsequent lap was $2.5\%$ more than the time for the previous lap. Given that the toy car stopped upon completing the $230\text{th}$ lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.<span class="inline-marks">[2]</span></div></li></ol>
<p>In another test using Brand $Y$ battery, the toy car was made to stop for&nbsp;$5$ seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still $7.68$ seconds. However, the time taken for the toy car to complete each subsequent lap was $2.5\%$ more than the total time taken for the previous lap and the $5$ seconds stop time. For example the time taken for the toy car to complete the second lap was $1.025 \times (7.68 + 5)$ seconds.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Show that the time taken for the toy car to complete the $n\text{th}$ lap is $(212.68)(1.025)^{n-1} - 205$ seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.<span class="inline-marks">[4]</span></div></li></ol>

Brand $X$ and Brand $Y$ batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is $20$ metres.

When the toy car used Brand $X$ battery, it completed the first lap in $15$ seconds and each subsequent lap took $3$ seconds longer than the previous lap.

When the toy car used Brand $Y$ battery, it completed the first lap in $7.68$ seconds and the time taken for each subsequent lap was $2.5\%$ more than the time for the previous lap. Given that the toy car stopped upon completing the $230\text{th}$ lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.

In another test using Brand $Y$ battery, the toy car was made to stop for $5$ seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still $7.68$ seconds. However, the time taken for the toy car to complete each subsequent lap was $2.5\%$ more than the total time taken for the previous lap and the $5$ seconds stop time. For example the time taken for the toy car to complete the second lap was $1.025 \times (7.68 + 5)$ seconds.

Show that the time taken for the toy car to complete the $n\text{th}$ lap is $(212.68)(1.025)^{n-1} - 205$ seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.

H2 Math Question Bank