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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>If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by </p>
<p style="text-align: center">$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$</p>
<p>This formula for solving quadratic equations is usually used when the <strong>quadratic expression cannot be factorised easily. </strong></p>
<p style="margin-top: 1.5rem"><strong>Tip: </strong></p>
<p>Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.</p>

If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by 

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

This formula for solving quadratic equations is usually used when the **quadratic expression cannot be factorised easily. **

Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.

Solving Quadratic Equations Using the Quadratic Formula

Solving Quadratic Equations by using Formula

Quadratic Functions, Equations and Inequalities

<p>The probability distribution of a discrete random variable $X$ is given by</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 390px;"><colgroup><col style="width: 38.4615%;"><col style="width: 17.9487%;"><col style="width: 17.9487%;"><col style="width: 25.6410%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$0$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$4$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$P(X=x)$</td><td style="text-align: center; --cell-justify: center">$p$</td><td style="text-align: center; --cell-justify: center">$p$</td><td style="text-align: center; --cell-justify: center">$1-2p$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="vm-q2-i" data-list-item-path="vm-q2-i"><div class="list-item-paragraph">State the range of possible values of $p$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="vm-q2-ii" data-list-item-path="vm-q2-ii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q2-iii" data-list-item-path="vm-q2-iii"><div class="list-item-paragraph">Show that&nbsp;$\operatorname{Var}(X)=4p(5-9p)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q2-iv" data-list-item-path="vm-q2-iv"><div class="list-item-paragraph">Hence find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.<span class="inline-marks">[3]</span></div></li></ol>

The probability distribution of a discrete random variable $X$ is given by

Constructing and Using Discrete Probability Distributions

Expectation and Variance of Discrete Random Variables

Determining Unknowns in Discrete Probability Distributions

Discrete Random Variables (DRV)

<p>A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that&nbsp;$P(X=5)=p$ and $P(X=-3)=1-p$, where&nbsp;$0\le p\le 1$.</p>
<ol class="i-list"><li data-list-item-id="vm-q1-i" data-list-item-path="vm-q1-i"><div class="list-item-paragraph">Show that&nbsp;$E(X)=-3+8p$ and $\operatorname{Var}(X)=64p(1-p)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="vm-q1-ii" data-list-item-path="vm-q1-ii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $E(X)$, stating the corresponding value of $p$ in each case.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="vm-q1-iii" data-list-item-path="vm-q1-iii"><div class="list-item-paragraph">Find the maximum and minimum possible values of $\operatorname{Var}(X)$, stating all corresponding values of $p$.<span class="inline-marks">[3]</span></div></li></ol>

A discrete random variable $X$ takes only the values $-3$ and $5$. It is given that $P(X=5)=p$ and $P(X=-3)=1-p$, where $0\le p\le 1$.

<p>Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.</p>
<ol class="i-list"><li data-list-item-id="gp-q2-i" data-list-item-path="gp-q2-i"><div class="list-item-paragraph">The first four possible winning sequences for Kai are $KK$, $LKK$, $KLKK$ and $LKLKK$, where&nbsp;$K$ denotes a round won by Kai and $L$ denotes a round won by Lina. Find the probability of each sequence.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="gp-q2-ii" data-list-item-path="gp-q2-ii"><div class="list-item-paragraph">Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="gp-q2-iii" data-list-item-path="gp-q2-iii"><div class="list-item-paragraph">Given that Lina wins the first round, find the probability that Kai wins the match.<span class="inline-marks">[2]</span></div></li></ol>

Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.

Conditional Probability

Repeated Trials with Infinitely Many Possible Outcomes

Probabilities of Successive Events

Application of Sum to Infinity

Waiting-Time and Stopping-Rule Probabilities

Probability

Sequence and Series

<p>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is $\dfrac{2}{5}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

Law of Total Probability

<p>The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.</p>
<ol class="i-list"><li data-list-item-id="prob-b-item-i" data-list-item-path="prob-b-item-i"><div class="list-item-paragraph">Find $P(A'\cap B')$ and hence prove that the events $A'$ and $B'$ are independent.<span class="inline-marks">[4]</span></div></li></ol>
<p>A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that&nbsp;$B$ and $C$ are independent, while $A$ and $C$ are not independent.</p>
<ol class="i-list" start="2"><li data-list-item-id="prob-b-item-ii" data-list-item-path="prob-b-item-ii"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[5]</span></div></li></ol>

The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.

A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that $B$ and $C$ are independent, while $A$ and $C$ are not independent.

Independent Events

Complementary Events

Applying Probability Laws

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

<p>For events $A$ and $B$, it is given that&nbsp;$P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.</p>
<ol class="i-list"><li data-list-item-id="prob-a-item-i" data-list-item-path="prob-a-item-i"><div class="list-item-paragraph">Find $P(B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-ii" data-list-item-path="prob-a-item-ii"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="prob-a-item-iii" data-list-item-path="prob-a-item-iii"><div class="list-item-paragraph">Find $P(A'\cap C)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-iv" data-list-item-path="prob-a-item-iv"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.

For a third event $C$, it is given that $P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.

<p>A wildlife researcher is planning five observation visits in a reserve with eight distinct observation stations.</p>
<ol class="i-list"><li data-list-item-id="observation-item-i" data-list-item-path="observation-item-i"><div class="list-item-paragraph">In how many ways can five different stations be selected if the order of the visits is irrelevant?</div></li><li data-list-item-id="observation-item-ii" data-list-item-path="observation-item-ii"><div class="list-item-paragraph">In how many ways can a route through five different stations be arranged if the order of the visits matters?</div></li><li data-list-item-id="observation-item-iii" data-list-item-path="observation-item-iii"><div class="list-item-paragraph">How many ordered five-visit schedules can be formed if stations may be revisited?</div></li><li data-list-item-id="observation-item-iv" data-list-item-path="observation-item-iv"><div class="list-item-paragraph">In how many ways can five station visits be selected if stations may be revisited and the order of the visits is irrelevant?</div></li></ol>

A wildlife researcher is planning five observation visits in a reserve with eight distinct observation stations.

Arrangements with Repetition Allowed

Permutations of Distinct Objects

Selections with Identical Objects

Combinations and Selections

Permutation and Combination

<ol class="i-list"><li data-list-item-id="item-similar-i" data-list-item-path="item-similar-i"><div class="list-item-paragraph">In how many ways can six copies of a book be distributed among nine people if no one gets more than one copy?</div></li><li data-list-item-id="item-similar-ii" data-list-item-path="item-similar-ii"><div class="list-item-paragraph">In how many ways can six different books be distributed among nine people if no one gets more than one copy?</div></li><li data-list-item-id="item-similar-iii" data-list-item-path="item-similar-iii"><div class="list-item-paragraph">In how many ways can six different books be distributed among nine people if each person can get any number of books?</div></li><li data-list-item-id="item-similar-iv" data-list-item-path="item-similar-iv"><div class="list-item-paragraph"><p>In how many ways can four copies of a book be distributed among nine people if each person can get any number of books?</p></div></li></ol>

Distributing Identical Objects

Distributing Distinct Objects

Distribution of Objects

<p>Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 &lt; 5x + 3$</p>

Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 < 5x + 3$

Quadratic Inequalities

Solving Quadratic Inequalities

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1783577066686-nwkpkpnl7cj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzcwNjY2ODYtbndrcGtwbmw3Y2oucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MDk0MywiZXhwIjoxNzkzNTQxMjQzfQ.jvzmuWzjKhYsfkJEav1SYwRTZm9OU_c1Sy2qN8eZh5o" style="max-width: min(180px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The origin $O$ and the points $A$, $B$ and $C$ lie in the same plane, where&nbsp;$\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$. It is given that&nbsp;$\angle AOC = \angle BOC = \theta$, where&nbsp;$\theta$ is an acute angle.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}$ where&nbsp;$\mathbf{\hat{a}}$ and $\mathbf{\hat{b}}$ are unit vectors in the directions of vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">If&nbsp;$\mathbf{c}$ can be written as&nbsp;$m\mathbf{\hat{a}} + n\mathbf{\hat{b}}$, where&nbsp;$m$ and $n$ are constants, use the result from <strong>(a)</strong> to show that&nbsp;$m = n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Write down the equation of the line passing through the points $A$ and $B$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;$|\mathbf{a}| = 3$, $|\mathbf{b}| = 2$ and $m = n$, show that the position vector of the point of intersection of the line passing through $A$ and $B$ and the line passing through $O$ and $C$ is $t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)$, where&nbsp;$t$ is a constant to be determined.<span class="inline-marks">[3]</span></div></li></ol>

The origin $O$ and the points $A$, $B$ and $C$ lie in the same plane, where $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$. It is given that $\angle AOC = \angle BOC = \theta$, where $\theta$ is an acute angle.

Intersection of Two Lines

Angle Bisectors Using Vectors

Equations of Lines

Scalar (Dot) Product

2024 ACJC P2 Q4

Vectors

<p>The diagram shows part of the graph $y = x^{\cos 2x}$, for&nbsp;$0 \le x \le 5$, which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by $x$ units and its vertical distance travelled is denoted by $y$ units.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1783573222728-q8fnjduy6xq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzMyMjI3MjgtcThmbmpkdXk2eHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MDk0MywiZXhwIjoxNzkzNTQxMjQzfQ.BKxpSE0Qq_0HY8bjklug6zdsilNXTB1SC5UnZXBHgxQ" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">At the point on the graph where&nbsp;$x = \pi$, find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of $8$ units per hour.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle that the tangent to the graph where&nbsp;$x = \pi$ makes with the horizontal.<span class="inline-marks">[1]</span></div></li></ol>

The diagram shows part of the graph $y = x^{\cos 2x}$, for $0 \le x \le 5$, which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by $x$ units and its vertical distance travelled is denoted by $y$ units.

Derivatives of Trigonometric Functions

Logarithmic Differentiation

Derivative as the Gradient of a Curve

Connected Rates of Change

2024 ACJC P2 Q1

Applications of Differentiation

Techniques of Differentiation

<p>Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}$ is a vector perpendicular to the plane $\pi$ .<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Prove that the equation of plane $\pi$ can be written as</div><div class="list-item-paragraph" style="text-align: center">\[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]</div><div class="list-item-paragraph">explaining clearly the reason for any result that you use in your proof.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{a} = \mathbf{i}$, $\mathbf{b} = \mathbf{j}$ and $\mathbf{c} = \mathbf{k}$, show that the equation of the plane $\pi$ can be written as&nbsp;$\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1$. Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane $\pi$.<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .

Distance from a Point to a Plane

Vector (Cross) Product

Equations of Planes

2024 ACJC P2 Q2

<p>The curve $C$ has equation $\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch $C$, stating the equations of the asymptotes and their point of intersection.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence state the exact range of values of $m$ such that the straight line $y=m(x+2)+3$ intersect $C$.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1$.

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Graphs of Conic Sections

Graphing Techniques

<p>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\cos^{-1}\left(3x^2\right)$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\ln\left(\frac{\sqrt{2x+1}}{x^3}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

Differentiate each of the following expressions with respect to $x$.

Derivatives of Natural Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

Chain Rule

2024 JPJC Promo Q1

<p>The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for&nbsp;$x \in \mathbb{R}$, $x &gt; -\frac{5}{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the value of $a$ such that&nbsp;$\mathrm{f}^2(a) = 5$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|$, for&nbsp;$x \in \mathbb{R}$, $x \le k$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the largest value of $k$ for&nbsp;$\mathrm{g}^{-1}$ to exist.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that&nbsp;$k = -1$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find $\mathrm{g}^{-1}(x)$ and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Explain why the composite function $\mathrm{fg}$ exists. State the exact range of $\mathrm{fg}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for $x \in \mathbb{R}$, $x > -\frac{5}{2}$.

Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|$, for $x \in \mathbb{R}$, $x \le k$.

Composite Functions

Existence of Composite Functions

Modulus Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Inverse Functions

2024 ASRJC Promo Q7

Functions

<p>In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively.</p>
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<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">State a geometrical interpretation of $|\mathbf{c} \times \mathbf{\hat{a}}|$ and hence show that the area of the triangle $PQR$ can be written as&nbsp;$\frac{1}{2}|\mathbf{c} \times \mathbf{{a}}|$ units$^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With clear explanations, show that&nbsp;$|\mathbf{c} \times \mathbf{a}| = |\mathbf{b} \times \mathbf{a}|$ and deduce that&nbsp;$\frac{|\mathbf{c}|}{\sin \angle PQR} = \frac{|\mathbf{b}|}{\sin \angle QPR}$ (Sine Rule).<span class="inline-marks">[2]</span></div></li></ol>

In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively.

Areas of Triangles and Parallelograms Using Vectors

Distance from a Point to a Line

2024 ASRJC Promo Q5

<p>In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the expected number of students who are sick in a week.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.</span><span class="inline-marks">[3]</span></div></li></ol>

In a school with $1200$ students, an average of $1$ out of $20$ students is sick in a week.

State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.

For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.

Find the expected number of students who are sick in a week.

In a class of $25$ students, find the probability that at least $2$ of them are sick in a week.

In a sample of $10$ classes with $25$ students each, find the probability that less than five classes have at least $2$ students who are sick in a week, given that there are at least two such classes.

Find the least value of $n$ such that in a sample of $n$ randomly selected students, the probability that at least $6$ are sick in a week is more than $0.95$.

Minimum Number of Binomial Trials

Binomial Probabilities Using Technology

Binomial Modelling and Applications

<p>Find the total area enclosed by the curve $y = (3x - 2)(x + 2)$ and the $x$-axis.</p>

Find the total area enclosed by the curve $y = (3x - 2)(x + 2)$ and the $x$-axis.

H1 Math Question Bank