A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<ol class="abc-list"><li data-list-item-id="item-a"><ol class="i-list"><li data-list-item-id="item-a-i"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{p} \cdot \mathbf{q} = 0$, what can be deduced about the vectors $\mathbf{p}$ and $\mathbf{q}$?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii"><div class="list-item-paragraph">With respect to the origin on a $xy$-plane, a fixed point $G$ has position vector $\mathbf{g}$ and a variable point $R$ has position vector $\mathbf{r}$. Describe the locus of $R$ if&nbsp;$\mathbf{r} \cdot (\mathbf{r} - \mathbf{g}) = 0$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With respect to the origin, plane $p$ has an equation given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})$, $\lambda \in \mathbb{R}$, $\mu \in \mathbb{R}$. Points $Q$ and $R$ have position vectors $3\mathbf{i} + 9\mathbf{j} + 5\mathbf{k}$ and $4\mathbf{i} + 11\mathbf{j} + 7\mathbf{k}$ respectively. Find the length of the projection of $\overrightarrow{QR}$ onto $p$.<span class="inline-marks">[4]</span></div></li></ol>

2024 NJC Promo Q7

Vectors

<p>A committee of $10$ people is chosen at random from a group consisting of $18$ women and $12$ men. The number of women on the committee is denoted by $R$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that&nbsp;$R = 4$.</span></div></li><li><div class="list-item-paragraph">The most probable number of women on the committee is denoted by $r$. By using the fact that&nbsp;$ \mathrm{P}(R = r) &gt; \mathrm{P}(R = r + 1)$, show that&nbsp;$r$ satisfies the inequality \[(r + 1)! (17 - r)! (9 - r)! (r + 3)! &gt; r! (18 - r)! (10 - r)! (r + 2)!\] and use this inequality to find the value of $r$.</div></li><li><div class="list-item-paragraph">Use the calculator to find $\mathrm{E}(R)$ and $\mathrm{Var}(R)$.</div></li></ol>

A committee of $10$ people is chosen at random from a group consisting of $18$ women and $12$ men. The number of women on the committee is denoted by $R$.

The most probable number of women on the committee is denoted by $r$. By using the fact that $P(R = r) > P(R = r + 1)$, show that $r$ satisfies the inequality\n\[(r + 1)! (17 - r)! (9 - r)! (r + 3)! > r! (18 - r)! (10 - r)! (r + 2)!\]\nAnd use this inequality to find the value of $r$.

Use the calculator to find $E(R)$ and $Var(R)$.

Discrete Random Variables (DRV)

<p>Relative to the origin $O$, the position vectors of $A$, $B$ and $C$  are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where&nbsp;$\mathbf{c}=5\mathbf{a}-\mathbf{b}$. Given that&nbsp;$\mathbf{a}$ is a unit vector, $\left| \mathbf{b} \right|=\sqrt{3}$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{1}{3}\pi $, find the exact area of triangle $ABC$.</p>

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$  are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where $\mathbf{c}=5\mathbf{a}-\mathbf{b}$. Given that $\mathbf{a}$ is a unit vector, $\left| \mathbf{b} \right|=\sqrt{3}$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{1}{3}\pi $, find the exact area of triangle $ABC$.

<p>Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered $0$, $1$, $2$, and $3$ respectively. The dice are rolled, and the numbers faced down on the two dice are recorded. The random variable $T$ is defined as the score on the red die multiplied by the score on the blue die.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability distribution of $T$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{E}(T)$ and show that&nbsp;$\mathrm{Var}(T) = \frac{115}{16}$. Show your workings clearly.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $\mathrm{P}(|T - 2\mu| &gt; \sigma)$, where&nbsp;$\mu = \mathrm{E}(T)$ and $\sigma^2 = \mathrm{Var}(T)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered $0$, $1$, $2$, and $3$ respectively. The dice are rolled, and the numbers faced down on the two dice are recorded. The random variable $T$ is defined as the score on the red die multiplied by the score on the blue die.

Find the probability distribution of $T$.

Find $\mathrm{E}(T)$ and show that $\mathrm{Var}(T) = \frac{115}{16}$. Show your workings clearly.

Evaluate $\mathrm{P}(|T - 2\mu| > \sigma)$, where $\mu = \mathrm{E}(T)$ and $\sigma^2 = \mathrm{Var}(T)$.

2024 ASRJC P2 Q6

<p>Things to note,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">${{\mathrm{D}}_{\mathrm{f}{{\mathrm{f}}^{-1}}}}={{\mathrm{D}}_{{{\mathrm{f}}^{-1}}}}$</div></li><li><div class="list-item-paragraph">${{\mathrm{D}}_{{{\mathrm{f}}^{-1}}\mathrm{f}}}={\mathrm{D}_{\mathrm{f}}}$</div></li></ol>
<p>$\mathrm{f}{{\mathrm{f}}^{-1}}$ and ${{\mathrm{f}}^{-1}}\mathrm{f}$ have the same rule, i.e. $y=x$ but may not have the same <strong>domain</strong>. </p>

$\mathrm{f}{{\mathrm{f}}^{-1}}$ and ${{\mathrm{f}}^{-1}}\mathrm{f}$ have the same rule, i.e. $y=x$ but may not have the same **domain**. 

Functions

<p>In a particular city, the traffic flow is as shown in the diagram below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1774549314613-yf5e31efz09.png" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>$A$, $B$, $C$ and $D$ are the four traffic junctions. The average hourly volume of vehicles entering and leaving $A$, $B$, $C$ and $D$ during the peak hour in the morning is given by the number next to the respective arrow. At each of the traffic junction, the number of vehicles entering <strong>must be the same</strong> as the number of vehicles leaving.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Formulate a system of $4$ linear equations in terms of $x_1$, $x_2$, $x_3$ and $x_4$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By solving the system of linear equations in (i), find the solution(s) for&nbsp;$x_1$, $x_2$, $x_3$ and $x_4$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It was observed that the average hourly volume of vehicles between junctions $C$ and $D$ is $150$. Using this information and the solution(s) you have obtained in (ii), write down the average hourly volume of vehicles between the junctions $A$ and $D$, $A$ and $B$, and $B$ and $C$.</span><span class="inline-marks">[1]</span></div></li></ol>

In a particular city, the traffic flow is as shown in the diagram below.

$A$, $B$, $C$ and $D$ are the four traffic junctions. The average hourly volume of vehicles entering and leaving $A$, $B$, $C$ and $D$ during the peak hour in the morning is given by the number next to the respective arrow. At each of the traffic junction, the number of vehicles entering **must be the same** as the number of vehicles leaving.

Formulate a system of $4$ linear equations in terms of $x_1$, $x_2$, $x_3$ and $x_4$.

By solving the system of linear equations in (i), find the solution(s) for $x_1$, $x_2$, $x_3$ and $x_4$.

It was observed that the average hourly volume of vehicles between junctions $C$ and $D$ is $150$. Using this information and the solution(s) you have obtained in (ii), write down the average hourly volume of vehicles between the junctions $A$ and $D$, $A$ and $B$, and $B$ and $C$.

2010 RI Promo Q2

System of Linear Equations

<p>Without using a calculator, solve the inequality</p>
<p style="text-align: center">\[\frac{x+3}{x^2+x-2} \ge -1.\]<span class="inline-marks">[4]</span></p>

Without using a calculator, solve the inequality

Inequalities - General

2024 NJC Promo Q1

Inequalities

<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>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779426366202-yu4qd1rzjpn.png" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<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.

Vector Algebra

2024 ASRJC Promo Q5

<p>A curve $C$ has equation $x^2 y^2 + 3xy + 2y^2 - x - 2 = 0$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the gradients of the tangents at $x = 0$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Using an algebraic method, show that there is no part of the curve for which $x &lt; -4$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $x^2 y^2 + 3xy + 2y^2 - x - 2 = 0$.

2024 NJC Promo Q6

Applications of Differentiation

<p>An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.</p>
<p>In the diagram below, the labelled points are defined relative to an origin $O$ with coordinates $(0,0,0)$ on the airfield. The $x$-, $y$- and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ correspond to the east, north and vertically upwards directions respectively. The point $A$ has coordinates $(2000,500,0)$ and the point $B$ has coordinates $(1400,2900,0)$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1779351220037-qkw6j2hys4.png" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Flight paths relative to the airfield origin" class="aligncenter custom-width" /></figure></div>
<p>Aircraft Alpha takes off from Point $A$ in the direction $4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$ as shown.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find the initial take-off angle $\theta$, the angle that Aircraft Alpha's flight path makes with the horizontal ground.<span class="inline-marks">[2]</span></div></li></ol>
<p>The peak of a majestic mountain has coordinates $(18000,4500,2000)$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the coordinates of the closest point on the flight path of Aircraft Alpha to the peak. Hence find the closest distance between the flight path and the peak, giving your answer to the nearest metre.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Aircraft Bravo takes off from Point $B$. For aviation safety reasons, the flight paths of Aircraft Alpha and Aircraft Bravo should never cross each other. Given that Aircraft Bravo will take off in the direction $4\mathbf{i} + 2\mathbf{j} + h\mathbf{k}$, find the value of $h$ which will result in the flight paths crossing each other, giving your answer to $3$ decimal places.<span class="inline-marks">[2]</span></div></li></ol>
<p>The value of $h$ is now given to be $1$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-iv"><div class="list-item-paragraph">Find the cartesian equation of the plane containing point $B$ and the flight path of Aircraft Alpha. Using this answer, provide another mathematical explanation as to why the flight path of Aircraft Bravo will not cross the flight path of Aircraft Alpha.<span class="inline-marks">[4]</span></div></li></ol>

An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.

In the diagram below, the labelled points are defined relative to an origin $O$ with coordinates $(0,0,0)$ on the airfield. The $x$-, $y$- and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ correspond to the east, north and vertically upwards directions respectively. The point $A$ has coordinates $(2000,500,0)$ and the point $B$ has coordinates $(1400,2900,0)$.

Aircraft Alpha takes off from Point $A$ in the direction $4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$ as shown.

The peak of a majestic mountain has coordinates $(18000,4500,2000)$.

The value of $h$ is now given to be $1$.

Vectors - General

2023 ACJC P1 Q12

<p>Relative to the origin $O$, the position vectors of $A$, $B$ and $C$  are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where&nbsp;$\mathbf{c}=3\mathbf{a}-5\mathbf{b}$. Given that&nbsp;$\left| \mathbf{a} \right|=\sqrt{6}$, $\mathbf{b}$ is a unit vector and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{2}{3}\pi $, find the exact area of triangle $ABC$.</p>

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$  are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where $\mathbf{c}=3\mathbf{a}-5\mathbf{b}$. Given that  $\left| \mathbf{a} \right|=\sqrt{6}$, $\mathbf{b}$ is a unit vector and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{2}{3}\pi $, find the exact area of triangle $ABC$.

<p>Physicists are investigating the reflective property of a particular reflective surface. The diagram below shows the set-up of a particular experiment, where a laser emitting device was placed at the point with coordinates $(1, 2, 3)$. A laser beam was emitted in the direction parallel to $\mathbf{i} + \mathbf{k}$. The path of the emitted laser beam and its reflected path make the same angle $\theta$ with the reflective surface. The plane containing these two paths is perpendicular to the reflective surface.</p>
<p>Write down the vector equation of the path of the emitted laser beam.<span class="inline-marks">[1]</span></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1774500717235-6nupuw71o97.png" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Diagram of laser reflection" class="aligncenter custom-width" /></figure></div>
<p>It is known that the reflective surface has equation $x + y + z = 4$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\theta$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the laser beam meets the reflective surface at the point $(0, 2, 2)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the vector equation of the path of the reflected laser beam.</span><span class="inline-marks">[5]</span></div></li></ol>

Physicists are investigating the reflective property of a particular reflective surface. The diagram below shows the set-up of a particular experiment, where a laser emitting device was placed at the point with coordinates $(1, 2, 3)$. A laser beam was emitted in the direction parallel to $\mathbf{i} + \mathbf{k}$. The path of the emitted laser beam and its reflected path make the same angle $\theta$ with the reflective surface. The plane containing these two paths is perpendicular to the reflective surface.

Write down the vector equation of the path of the emitted laser beam.

It is known that the reflective surface has equation $x + y + z = 4$.

Show that the laser beam meets the reflective surface at the point $(0, 2, 2)$.

Find the vector equation of the path of the reflected laser beam.

2017 VJC P1 Q11

<p>A bag containing $(n+5)$ numbered balls. Two of the balls are numbered $3$, three of the balls are numbered $4$ and $n$ of the balls are numbered $5$. Two balls are taken, at random and without replacement, from the bag. The random variable $S$ is the sum of the numbers on the two balls taken.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Determine the probability distribution of $S$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">For the case where&nbsp;$n=1$, find $P(S=10)$ and explain this result.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$E(S)=\frac{10n+36}{n+5}$ and $\text{Var}(S)=\frac{g(n)}{(n+5)^{2}(n+4)}$ where&nbsp;$g(n)$ is a quadratic polynomial to be determined.</span><span class="inline-marks">[6]</span></div></li></ol>

A bag containing $(n+5)$ numbered balls. Two of the balls are numbered $3$, three of the balls are numbered $4$ and $n$ of the balls are numbered $5$. Two balls are taken, at random and without replacement, from the bag. The random variable $S$ is the sum of the numbers on the two balls taken.

Determine the probability distribution of $S$.

For the case where $n=1$, find $P(S=10)$ and explain this result.

Show that $E(S)=\frac{10n+36}{n+5}$ and $\text{Var}(S)=\frac{g(n)}{(n+5)^{2}(n+4)}$ where $g(n)$ is a quadratic polynomial to be determined.

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">The points $P$, $Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. $P$ and $Q$ are fixed points and $R$ is a variable point. Describe geometrically the set of possible positions of $R$ such that</div><ol class="i-list"><li data-list-item-id="item-a-i"><div class="list-item-paragraph">$\mathbf{r} \cdot \mathbf{q} = \mathbf{p} \cdot \mathbf{q}$<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-ii"><div class="list-item-paragraph">$\mathbf{r} \times \mathbf{q} = \mathbf{p} \times \mathbf{q}$<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With respect to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. $C$ lies on $AB$ produced such that&nbsp;$AC : CB = 3 : 2$. Given that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are unit vectors, the length of $OC$ is $\sqrt{7}$ units and the angle $AOB$ is $60^\circ$, find $|2\mathbf{a} - \mathbf{c}|$ by using a suitable scalar product.<span class="inline-marks">[5]</span></div></li></ol>

2023 JPJC P2 Q5

<p>Tom has a bag of wooden rectangular blocks of identical size. The bag contains $1$ blue block, $m$ red blocks and $(m-1)$ yellow blocks, where&nbsp;$m &gt; 2$. Tom and Jerry play a game using Tom's bag of wooden blocks. Jerry draws $2$ blocks at random, one at a time, without replacement. $3$ points will be awarded if a yellow block is drawn, $2$ points will be awarded if a red block is drawn, but no points will be awarded if a blue block is drawn. Jerry's final score is the product of the points awarded for the $2$ blocks drawn.</p>
<p>Let the random variable $X$ denotes Jerry's final score.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$P(X=0) = \frac{1}{m}$ and hence, find the probability distribution of $X$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $m$ if Jerry's expected final score is $5$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Tom pays Jerry \$$5$ if Jerry's final score is at least $5$ and Jerry pays Tom \$$a$ if his final score is less than $5$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Using the value of $m$ found in <strong>(ii)</strong>, find the range of values of $a$ if Tom is expected to make a profit.<span class="inline-marks">[2]</span></div></li></ol>

Tom has a bag of wooden rectangular blocks of identical size. The bag contains $1$ blue block, $m$ red blocks and $(m-1)$ yellow blocks, where $m > 2$. Tom and Jerry play a game using Tom's bag of wooden blocks. Jerry draws $2$ blocks at random, one at a time, without replacement. $3$ points will be awarded if a yellow block is drawn, $2$ points will be awarded if a red block is drawn, but no points will be awarded if a blue block is drawn. Jerry's final score is the product of the points awarded for the $2$ blocks drawn.

Let the random variable $X$ denotes Jerry's final score.

Show that $P(X=0) = \frac{1}{m}$ and hence, find the probability distribution of $X$.

Find the value of $m$ if Jerry's expected final score is $5$.

Tom pays Jerry \$$5$ if Jerry's final score is at least $5$ and Jerry pays Tom \$$a$ if his final score is less than $5$.

Using the value of $m$ found in (ii), find the range of values of $a$ if Tom is expected to make a profit.

<p>HIMHEYS' Confectionery recently created three types of chocolate: Organic white, Organic milk and Organic dark, available in $250$ g bars. Cocoa butter, an essential ingredient in chocolate bars, makes up $25\%$, $20\%$ and $15\%$ of the mass of an Organic white, Organic milk and Organic dark chocolate bar respectively.</p>
<p>To prepare for the official launch of their chocolate bars at an upcoming Food Expo, HIMHEYS' decides to manufacture a total of $300$ bars for the event, with more than $70$ bars of each type.</p>
<p>The confectionary intends to use $14$ kg of cocoa butter in the production of the above batch of chocolate bars. If the number of milk chocolate bars is to be smaller than the number of white chocolate bars, determine how many organic chocolate bars of each type can be produced.<span class="inline-marks">[6]</span></p>

HIMHEYS' Confectionery recently created three types of chocolate: Organic white, Organic milk and Organic dark, available in $250$ g bars. Cocoa butter, an essential ingredient in chocolate bars, makes up $25\%$, $20\%$ and $15\%$ of the mass of an Organic white, Organic milk and Organic dark chocolate bar respectively.

To prepare for the official launch of their chocolate bars at an upcoming Food Expo, HIMHEYS' decides to manufacture a total of $300$ bars for the event, with more than $70$ bars of each type.

The confectionary intends to use $14$ kg of cocoa butter in the production of the above batch of chocolate bars. If the number of milk chocolate bars is to be smaller than the number of white chocolate bars, determine how many organic chocolate bars of each type can be produced.

2012 TJC P1 Q3

<p>The plane $p$ passes through the points with coordinates $(4, -1, -3)$, $(-2, -5, 2)$ and $(4, -3, -2)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a Cartesian equation of $p$.</span></div></li></ol>
<p>The line $l_1$ has equation $\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}$ and the line $l_2$ has equation $\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}$, where&nbsp;$k$ is a constant. It is given that&nbsp;$l_1$ and $l_2$ intersect.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the value of $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$l_1$ lies in $p$ and find the coordinates of the point at which $l_2$ intersects $p$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $l_2$ and $p$.</span></div></li></ol>

The plane $p$ passes through the points with coordinates $(4, -1, -3)$, $(-2, -5, 2)$ and $(4, -3, -2)$.

The line $l_1$ has equation $\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}$ and the line $l_2$ has equation $\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}$, where $k$ is a constant. It is given that $l_1$ and $l_2$ intersect.

Show that $l_1$ lies in $p$ and find the coordinates of the point at which $l_2$ intersects $p$.

Find the acute angle between $l_2$ and $p$.

N2011 P1 Q11

<p>In a game of chance, a player has to spin a fair spinner. The spinner has&nbsp;$7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has&nbsp;$1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1774425191077-fmb79sg61yr.png" style="max-width: min(150px, 100%); height: auto; display: block;" alt="Spinner diagram" class="aligncenter custom-width" /></figure></div>
<p>The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it and the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a player wins a game.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that a player wins a game, find the probability that the spinner came to rest over $\mathbf{B}$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.</span><span class="inline-marks">[2]</span></div></li></ol>

In a game of chance, a player has to spin a fair spinner. The spinner has $7$ sections and an arrow which has an equal chance of coming to rest over any of the $7$ sections. The spinner has $1$ section labelled $\mathbf{R}$, $2$ sections labelled $\mathbf{B}$ and $4$ sections labelled $\mathbf{Y}$ (see diagram).

The player then has to throw one of three fair six-sided dice, coloured red, blue or yellow. If the spinner comes to rest over $\mathbf{R}$ the red die is thrown, if the spinner comes to rest over $\mathbf{B}$ the blue die is thrown and if the spinner comes to rest over $\mathbf{Y}$ the yellow die is thrown. The yellow die has one face with $*$ on it and the blue die has two faces with $*$ on it and the red die has three faces with $*$ on it. The player wins the game if the die thrown comes to rest with a face showing $*$ uppermost.

Find the probability that a player wins a game.

Given that a player wins a game, find the probability that the spinner came to rest over $\mathbf{B}$.

Find the probability that a player wins $3$ consecutive games, each time throwing a die of a different colour.

<p>A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$p = 25$, find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $\mathrm{f}(p)$. Show that&nbsp;$\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that&nbsp;$\mathrm{f}$ is an increasing function for&nbsp;$0 \leq p \leq 100$, and explain what this statement means in the context of the question.</div></li></ol>

A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.

Given that $p = 25$, find the probability that a randomly chosen component is faulty.

For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $f(p)$. Show that $f(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that $f$ is an increasing function for $0 \leq p \leq 100$, and explain what this statement means in the context of the question.

N2009 P2 Q7

Probability

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;$x^2 + 4x + 9$ is always positive for all real values of $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, without using a calculator, solve $\frac{(x^2 + 4x + 9)(x + 2)^2}{x^2 - 6x + 7} \le 0$.<span class="inline-marks">[3]</span></div></li></ol>

IB HL Math Question Bank