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

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Without using a calculator, solve the inequality \(\frac{4}{x + 2} \geqslant \frac{x - 3}{x}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">Hence, solve the inequality \(\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}\).<span class="inline-marks">[2]</span></div></li></ol>

Without using a calculator, solve the inequality \(\frac{4}{x + 2} \geqslant \frac{x - 3}{x}\).

Hence, solve the inequality \(\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}\).

Inequalities involving Substitution

Inequalities involving Rational Functions

Inequalities involving Modulus Functions

N2024 P1 Q4

Inequalities

<p>The letters of the word \(\mathrm{PROBABILITY}\) are written on \(11\) cards: \(\mathrm{P, R, O, B, A, B, I, L, I, T, Y}\). Two cards are drawn at random without replacement. Let&nbsp;\(A\) be the event that the first card drawn is \(A\), and \(B\) the event that the second card drawn is \(B\).</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q016-part-1-003" data-list-item-path="ibsl-q016-part-1-003"><div class="list-item-paragraph">Find \(P(A)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q016-part-2-004" data-list-item-path="ibsl-q016-part-2-004"><div class="list-item-paragraph">Find \(P(B|A)\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q016-part-3-005" data-list-item-path="ibsl-q016-part-3-005"><div class="list-item-paragraph">Find \(P(A ∩ B)\).<span class="inline-marks">[3]</span></div></li></ol>

The letters of the word \(\mathrm{PROBABILITY}\) are written on \(11\) cards: \(\mathrm{P, R, O, B, A, B, I, L, I, T, Y}\). Two cards are drawn at random without replacement. Let \(A\) be the event that the first card drawn is \(A\), and \(B\) the event that the second card drawn is \(B\).

Conditional Probability

Dependent Events

Calculate ordered draws without replacement and show dependence.

Probability

<p>The diagram shows a circle divided into three sectors A, B and C with central angles \(90^\circ\), \(120^\circ\) and \(150^\circ\), respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let&nbsp;\(A\), \(B\), \(C\) and \(S\) denote landing in sector A, B, C and a shaded region, respectively.</p>
<div class="graph-block" data-graph-config="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" data-width="330" data-height="300" style="width:330px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q077-part-1-013" data-list-item-path="ibsl-q077-part-1-013"><div class="list-item-paragraph">\(P(B)\);</div></li><li data-list-item-id="ibsl-q077-part-2-014" data-list-item-path="ibsl-q077-part-2-014"><div class="list-item-paragraph">\(P(S)\);</div></li><li data-list-item-id="ibsl-q077-part-3-015" data-list-item-path="ibsl-q077-part-3-015"><div class="list-item-paragraph">\(P(A|S)\).</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

The diagram shows a circle divided into three sectors A, B and C with central angles \(90^\circ\), \(120^\circ\) and \(150^\circ\), respectively. Sectors A and B are shaded. The arrow is spun and cannot land on a boundary. Let \(A\), \(B\), \(C\) and \(S\) denote landing in sector A, B, C and a shaded region, respectively.

Set Theory and Probability — Combined Problems

Probability of a Single Event

Calculate spinner-sector, shaded-region and conditional probabilities.

<p>Let&nbsp;\(A\) and \(B\) be independent events, where&nbsp;\(P(A) = 0.6\) and \(P(B) = x\).</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q033-part-1-003" data-list-item-path="ibsl-q033-part-1-003"><div class="list-item-paragraph">Write down an expression for&nbsp;\(P(A ∩ B)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q033-part-2-004" data-list-item-path="ibsl-q033-part-2-004"><div class="list-item-paragraph"><p>Given that&nbsp;\(P(A ∪ B) = 0.8\),</p><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q033-part-2-1-005" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-1-005"><div class="list-item-paragraph"><p>find \(x\);</p></div></li><li data-list-item-id="ibsl-q033-part-2-2-006" data-list-item-path="ibsl-q033-part-2-004/ibsl-q033-part-2-2-006"><div class="list-item-paragraph">find \(P(A ∩ B)\).</div></li></ol></li><li data-list-item-id="ibsl-q033-part-3-007" data-list-item-path="ibsl-q033-part-3-007"><div class="list-item-paragraph"><p>Hence, explain why \(A\) and \(B\) are not mutually exclusive.</p><span class="inline-marks">[1]</span></div></li></ol>

Let \(A\) and \(B\) be independent events, where \(P(A) = 0.6\) and \(P(B) = x\).

Write down an expression for \(P(A ∩ B)\).

Independent Events

Mutually Exclusive Events

Use independence to solve an unknown event probability and distinguish mutual exclusivity.

<p>In any given season, a soccer team plays \(65 \%\) of their games at home. When the team plays at home, they win \(83 \%\) of their games. When they play away from home, they win \(26 \%\) of their games. The team plays one game.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q019-part-1-003" data-list-item-path="ibsl-q019-part-1-003"><div class="list-item-paragraph">Find the probability that the team wins the game.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ibsl-q019-part-2-004" data-list-item-path="ibsl-q019-part-2-004"><div class="list-item-paragraph">If the team does not win the game, find the probability that the game was played at home.<span class="inline-marks">[4]</span></div></li></ol>

In any given season, a soccer team plays \(65 \%\) of their games at home. When the team plays at home, they win \(83 \%\) of their games. When they play away from home, they win \(26 \%\) of their games. The team plays one game.

Find the probability that the team wins the game.

If the team does not win the game, find the probability that the game was played at home.

Use total and conditional probability for home/away match outcomes.

<p>In a group of \(16\) students, \(12\) take art and \(8\) take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values \(p, q, r\) and \(s\) represent numbers of students.</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q002-part-1-014" data-list-item-path="ibsl-q002-part-1-014"><div class="list-item-paragraph"><span class="inline-marks">[5]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-1-1-015" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of \(s\).</p></div></li><li data-list-item-id="ibsl-q002-part-1-2-016" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of \(q\).</p></div></li><li data-list-item-id="ibsl-q002-part-1-3-017" data-list-item-path="ibsl-q002-part-1-014/ibsl-q002-part-1-3-017"><div class="list-item-paragraph"><p>Write down the value of \(p\) and of \(r\).</p></div></li></ol></li><li data-list-item-id="ibsl-q002-part-2-018" data-list-item-path="ibsl-q002-part-2-018"><div class="list-item-paragraph"><span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q002-part-2-1-019" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-1-019"><div class="list-item-paragraph">A student is selected at random. Given that the student takes music, write down the probability the student takes art.</div></li><li data-list-item-id="ibsl-q002-part-2-2-020" data-list-item-path="ibsl-q002-part-2-018/ibsl-q002-part-2-2-020"><div class="list-item-paragraph">Hence, show that taking music and taking art are not independent events.</div></li></ol></li><li data-list-item-id="ibsl-q002-part-3-021" data-list-item-path="ibsl-q002-part-3-021"><div class="list-item-paragraph">Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.<span class="inline-marks">[4]</span></div></li></ol>

In a group of \(16\) students, \(12\) take art and \(8\) take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values \(p, q, r\) and \(s\) represent numbers of students.

Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.

Complete a Venn diagram, calculate conditional probability, test independence and sample without replacement.

<p>A company uses two machines, \(A\) and \(B\), to make boxes. Machine \(A\) makes \(60 \%\) of the boxes. \(80 \%\) of the boxes made by machine \(A\) pass inspection. \(90 \%\) of the boxes made by machine \(B\) pass inspection. \(A\) box is selected at random.</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q006-part-1-003" data-list-item-path="ibsl-q006-part-1-003"><div class="list-item-paragraph">Find the probability that it passes inspection.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ibsl-q006-part-2-004" data-list-item-path="ibsl-q006-part-2-004"><div class="list-item-paragraph"><p>The company would like the probability that a box passes inspection to be \(0.87\). Find the percentage of boxes that should be made by machine \(B\) to achieve this.</p><span class="inline-marks">[4]</span></div></li></ol>

A company uses two machines, \(A\) and \(B\), to make boxes. Machine \(A\) makes \(60 \%\) of the boxes. \(80 \%\) of the boxes made by machine \(A\) pass inspection. \(90 \%\) of the boxes made by machine \(B\) pass inspection. \(A\) box is selected at random.

Find the probability that it passes inspection.

Use total probability for two manufacturing machines and solve an inverse mixture proportion.

<p>Consider the events A and B, where&nbsp;\(P(A) = 0.5\), \(P(B) = 0.7\) and \(P(A ∩ B) = 0.3\). The Venn diagram below shows the events \(A\) and \(B\), and the probabilities \(p, q\) and \(r\).</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q009-part-1-014" data-list-item-path="ibsl-q009-part-1-014"><div class="list-item-paragraph">Write down the value of<span class="inline-marks">[3]</span></div><ol class="i-list"><li data-list-item-id="ibsl-q009-part-1-1-015" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-1-015"><div class="list-item-paragraph"><p>\(p\);</p></div></li><li data-list-item-id="ibsl-q009-part-1-2-016" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-2-016"><div class="list-item-paragraph"><p>\(q\);</p></div></li><li data-list-item-id="ibsl-q009-part-1-3-017" data-list-item-path="ibsl-q009-part-1-014/ibsl-q009-part-1-3-017"><div class="list-item-paragraph"><p>\(r\).</p></div></li></ol></li><li data-list-item-id="ibsl-q009-part-2-018" data-list-item-path="ibsl-q009-part-2-018"><div class="list-item-paragraph"><p>Find the value of \(P(A | B')\).</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q009-part-3-019" data-list-item-path="ibsl-q009-part-3-019"><div class="list-item-paragraph"><p>Hence, or otherwise, show that the events \(A\) and \(B\) are not independent.</p><span class="inline-marks">[1]</span></div></li></ol>

Consider the events A and B, where \(P(A) = 0.5\), \(P(B) = 0.7\) and \(P(A ∩ B) = 0.3\). The Venn diagram below shows the events \(A\) and \(B\), and the probabilities \(p, q\) and \(r\).

Complementary Events

Complete Venn-region probabilities, calculate a conditional complement and test independence.

<p>The Venn diagram below shows events A and B where&nbsp;\(P(A) = 0.3\), \(P(A ∪ B) = 0.6\) and \(P(A ∩ B) = 0.1\). The values \(m, n, p\) and \(q\) are probabilities.</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q003-part-1-014" data-list-item-path="ibsl-q003-part-1-014"><ol class="i-list"><li data-list-item-id="ibsl-q003-part-1-1-015" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-1-015"><div class="list-item-paragraph"><p>Write down the value of \(n\).</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="ibsl-q003-part-1-2-016" data-list-item-path="ibsl-q003-part-1-014/ibsl-q003-part-1-2-016"><div class="list-item-paragraph"><p>Find the value of \(m\), of \(p\), and of \(q\).</p></div></li></ol></li><li data-list-item-id="ibsl-q003-part-2-017" data-list-item-path="ibsl-q003-part-2-017"><div class="list-item-paragraph"><p>Find \(P(B')\).</p><span class="inline-marks">[2]</span></div></li></ol>

The Venn diagram below shows events A and B where \(P(A) = 0.3\), \(P(A ∪ B) = 0.6\) and \(P(A ∩ B) = 0.1\). The values \(m, n, p\) and \(q\) are probabilities.

Complete Venn-region probabilities and calculate a complement.

<p>The Venn diagram below shows information about \(120\) students in a school. Of these, \(40\) study Chinese (C), \(35\) study Japanese (J), and \(30\) study Spanish (S). A student is chosen at random from the group. Find the probability that the student</p>
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<ol class="abc-list"><li data-list-item-id="ibsl-q050-part-1-016" data-list-item-path="ibsl-q050-part-1-016"><div class="list-item-paragraph">studies exactly two of these languages;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q050-part-2-017" data-list-item-path="ibsl-q050-part-2-017"><div class="list-item-paragraph">studies only Japanese;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ibsl-q050-part-3-018" data-list-item-path="ibsl-q050-part-3-018"><div class="list-item-paragraph">does not study any of these languages.<span class="inline-marks">[3]</span></div></li></ol>

The Venn diagram below shows information about \(120\) students in a school. Of these, \(40\) study Chinese (C), \(35\) study Japanese (J), and \(30\) study Spanish (S). A student is chosen at random from the group. Find the probability that the student

Read a three-set Venn diagram and calculate exact-two, only-one and neither probabilities.

<p>Consider the events \(A\) and \(B\), where&nbsp;\(P(A)=\frac25\), \(P(B')=\frac14\) and \(P(A\cup B)=\frac78\).</p>
<ol class="abc-list"><li data-list-item-id="ibsl-q044-part-1-003" data-list-item-path="ibsl-q044-part-1-003"><div class="list-item-paragraph">Write down \(P(B)\).</div></li><li data-list-item-id="ibsl-q044-part-2-004" data-list-item-path="ibsl-q044-part-2-004"><div class="list-item-paragraph">Find \(P(A ∩ B)\).</div></li><li data-list-item-id="ibsl-q044-part-3-005" data-list-item-path="ibsl-q044-part-3-005"><div class="list-item-paragraph">Find \(P(A | B)\).</div></li></ol>
<p><span class="inline-marks">[6]</span></p>

Consider the events \(A\) and \(B\), where \(P(A)=\frac25\), \(P(B')=\frac14\) and \(P(A\cup B)=\frac78\).

Use complements, unions, intersections and conditional probability for two events.

<p>The diagram below shows the probabilities for events \(A\) and \(B\), with \(P(A') = p\).</p>
<div class="flowchart-block probability-tree-block" data-width="620" data-height="320" style="width:620px;max-width:100%;display:block;margin:0 auto;">
    <svg
      xmlns="http://www.w3.org/2000/svg"
      width="620"
      height="320"
      viewBox="0 0 620 320"
      role="img"
      aria-label="Probability tree"
    >
      <defs>
        
      </defs>
      <rect x="0" y="0" width="620" height="320" fill="#ffffff" />
      <g class="flowchart-edges">
        
      <path d="M 50,160L 255,82" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="87.02137433574791" y="92.91515049779898" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(\frac{1}{5}\)</div>
    </foreignObject>
  
      
    <foreignObject x="187.6213743357479" y="67.95515049779898" width="51" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(A\)</div>
    </foreignObject>
  
      
    
      <path d="M 50,160L 255,238" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="122.02137433574791" y="197.08484950220102" width="51" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(p\)</div>
    </foreignObject>
  
      
    <foreignObject x="184.1213743357479" y="222.044849502201" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(A&#39;\)</div>
    </foreignObject>
  
      
    
      <path d="M 255,82L 455,30" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="290.9771260752907" y="27.450484904964192" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(\frac{1}{4}\)</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="15" width="51" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(B\)</div>
    </foreignObject>
  
      
    
      <path d="M 255,82L 455,110" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="292.5589301765183" y="94.86478445344062" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(\frac{3}{4}\)</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="95" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(B&#39;\)</div>
    </foreignObject>
  
      
    
      <path d="M 255,238L 455,210" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="292.5589301765183" y="195.13521554655938" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(\frac{3}{8}\)</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="195" width="51" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(B\)</div>
    </foreignObject>
  
      
    
      <path d="M 255,238L 455,290" fill="none" stroke="#111827" stroke-width="1.6" stroke-linecap="round" stroke-linejoin="round" />
      
    <foreignObject x="290.9771260752907" y="262.5495150950358" width="121" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-probability-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:center;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(\frac{5}{8}\)</div>
    </foreignObject>
  
      
    <foreignObject x="467" y="275" width="58" height="30">
      <div xmlns="http://www.w3.org/1999/xhtml" class="flowchart-edge-label probability-tree-outcome-label" style="box-sizing:border-box;width:100%;height:100%;display:flex;align-items:center;justify-content:flex-start;overflow:visible;padding:2px 5px;border-radius:4px;background-color:transparent;color:#111827;font-family:Arial, sans-serif;font-size:12px;line-height:1.15;text-align:center;white-space:pre-wrap;overflow-wrap:anywhere">\(B&#39;\)</div>
    </foreignObject>
  
      
    
      </g>
      <g class="flowchart-nodes">
        
      </g>
    </svg>
  </div>
<ol class="abc-list"><li data-list-item-id="ibsl-q014-part-1-017" data-list-item-path="ibsl-q014-part-1-017"><div class="list-item-paragraph"><p>Write down the value of \(p\).</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="ibsl-q014-part-2-018" data-list-item-path="ibsl-q014-part-2-018"><div class="list-item-paragraph">Find \(P(B)\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ibsl-q014-part-3-019" data-list-item-path="ibsl-q014-part-3-019"><div class="list-item-paragraph"><p>Find \(P(A' | B)\).</p><span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows the probabilities for events \(A\) and \(B\), with \(P(A') = p\).

Constructing and Using Probability Trees

Read a probability tree and calculate a conditional probability.

<p>The following Venn diagram shows a sample space \(U\) and events \(A\) and \(B\). It is known that&nbsp;\(n(U)=36\), \(n(A)=11\), \(n(B)=6\), and \(n((A\cup B)')=21\).</p>
<div class="graph-block" data-graph-config="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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="ibsl-q101-part-1-010" data-list-item-path="ibsl-q101-part-1-010"><div class="list-item-paragraph"><p>On the diagram, shade the region \((A ∪ B)′\).</p></div></li><li data-list-item-id="ibsl-q101-part-2-011" data-list-item-path="ibsl-q101-part-2-011"><div class="list-item-paragraph">Find</div><ol class="i-list"><li data-list-item-id="ibsl-q101-part-2-1-012" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-1-012"><div class="list-item-paragraph"><p>\(n(A ∩ B)\);</p></div></li><li data-list-item-id="ibsl-q101-part-2-2-013" data-list-item-path="ibsl-q101-part-2-011/ibsl-q101-part-2-2-013"><div class="list-item-paragraph">\(P(A ∩ B)\).</div></li></ol></li><li data-list-item-id="ibsl-q101-part-3-014" data-list-item-path="ibsl-q101-part-3-014"><div class="list-item-paragraph"><p>Explain why events \(A\) and \(B\) are not mutually exclusive.</p></div></li></ol>
<p><span class="inline-marks">[4]</span></p>

The following Venn diagram shows a sample space \(U\) and events \(A\) and \(B\). It is known that \(n(U)=36\), \(n(A)=11\), \(n(B)=6\), and \(n((A\cup B)')=21\).

Shade a Venn complement, calculate an intersection and explain non-mutual-exclusivity.

<p>Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that&nbsp;\(\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3\). It is given that&nbsp;\(\mathbf{a}\) is a unit vector and \(\mathbf{b} \cdot \mathbf{b} = 9\).</p>
<p>Show that&nbsp;\(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular.<span class="inline-marks">[3]</span></p>

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that \(\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3\). It is given that \(\mathbf{a}\) is a unit vector and \(\mathbf{b} \cdot \mathbf{b} = 9\).

Show that \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular.

Vector (Cross) Product

Perpendicular Vectors Using the Scalar Product

Scalar (Dot) Product

N2024 P1 Q2

Vectors

<p>The following is a cumulative frequency diagram for the time \(t\), in minutes, taken by \(80\) students to complete a task.</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/1786953828869-b82zdxfu1tw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODY5NTM4Mjg4NjktYjgyemR4ZnUxdHcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1OTEzMywiZXhwIjoxNzkzNTU5NDMzfQ.0VBWwNRPlCKoBILq3FYvYTYyGucjB2IEJGdgdiOxUYI" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1786953844612-exxddojo8" data-list-item-path="item-1786953844612-exxddojo8"><div class="list-item-paragraph"><p>Write down the median.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1786953845405-z953e55oq" data-list-item-path="item-1786953845405-z953e55oq"><div class="list-item-paragraph"><p>Find the interquartile range.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1786953869062-0evyr2i2x" data-list-item-path="item-1786953869062-0evyr2i2x"><div class="list-item-paragraph"><p>Complete the frequency table below.</p><span class="inline-marks">[2]</span></div><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Time (minutes)</strong></th><th style="text-align: center; --cell-justify: center"><strong>Number of students</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(0 \leq t \leq 10\)</td><td style="text-align: center; --cell-justify: center">\(5\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(10 \leq t \leq 20\)</td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(20 \leq t \leq 30\)</td><td style="text-align: center; --cell-justify: center">\(20\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(30 \leq t \leq 40\)</td><td style="text-align: center; --cell-justify: center">\(24\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(40 \leq t \leq 50\)</td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(50 \leq t \leq 60\)</td><td style="text-align: center; --cell-justify: center">\(6\)</td></tr></tbody></table></div></li></ol>

The following is a cumulative frequency diagram for the time \(t\), in minutes, taken by \(80\) students to complete a task.

Statistics

<p>In a school with \(125\) girl, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below.</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/1786953563513-vh1xvshocpb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODY5NTM1NjM1MTMtdmgxeHZzaG9jcGIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1OTEzMywiZXhwIjoxNzkzNTU5NDMzfQ.RQynrOupbcu-41pIAUKcg3kvySf-me2QmLD50jjUNyA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1786953569496-w2tis8wgc" data-list-item-path="item-1786953569496-w2tis8wgc"><ol class="i-list"><li data-list-item-id="subitem-1786953571712-7yawslq2c" data-list-item-path="item-1786953569496-w2tis8wgc/subitem-1786953571712-7yawslq2c"><div class="list-item-paragraph"><p>Write down the value of \(p\).</p></div></li><li data-list-item-id="subitem-1786953572112-hl8laf3jy" data-list-item-path="item-1786953569496-w2tis8wgc/subitem-1786953572112-hl8laf3jy"><div class="list-item-paragraph"><p>Find the value of \(q\)</p><span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1786953601329-1ejqzkozr" data-list-item-path="item-1786953601329-1ejqzkozr"><div class="list-item-paragraph"><p>Find the median number of sit-ups.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1786953602169-guainichd" data-list-item-path="item-1786953602169-guainichd"><div class="list-item-paragraph"><p>Find the mean number of sit-ups.</p><span class="inline-marks">[2]</span></div></li></ol>

In a school with \(125\) girl, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below.

<p>A box contains \(100\) cards. Each card has a number between one and six written on it. <br>The following table shows the frequencies for each number.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 375px;"><colgroup><col style="width: 20.0000%;"><col style="width: 13.3333%;"><col style="width: 13.3333%;"><col style="width: 13.3333%;"><col style="width: 13.3333%;"><col style="width: 13.3333%;"><col style="width: 13.3333%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Number</strong></th><th style="text-align: center; --cell-justify: center">\(1\)</th><th style="text-align: center; --cell-justify: center">\(2\)</th><th style="text-align: center; --cell-justify: center">\(3\)</th><th style="text-align: center; --cell-justify: center">\(4\)</th><th style="text-align: center; --cell-justify: center">\(5\)</th><th style="text-align: center; --cell-justify: center">\(6\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><strong>Frequency</strong></td><td style="text-align: center; --cell-justify: center">\(26\)</td><td style="text-align: center; --cell-justify: center">\(10\)</td><td style="text-align: center; --cell-justify: center">\(20\)</td><td style="text-align: center; --cell-justify: center">\(k\)</td><td style="text-align: center; --cell-justify: center">\(29\)</td><td style="text-align: center; --cell-justify: center">\(11\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1786953446103-s43zjnrfs" data-list-item-path="item-1786953446103-s43zjnrfs"><div class="list-item-paragraph"><p>Calculate the value of \(k\).</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1786953447431-qt07lqj2c" data-list-item-path="item-1786953447431-qt07lqj2c"><div class="list-item-paragraph"><p>Find</p><span class="inline-marks">[5]</span></div><ol class="i-list"><li data-list-item-id="subitem-1786953465832-zqt3vl5rn" data-list-item-path="item-1786953447431-qt07lqj2c/subitem-1786953465832-zqt3vl5rn"><div class="list-item-paragraph"><p>the median;</p></div></li><li data-list-item-id="subitem-1786953466249-2urvey9ug" data-list-item-path="item-1786953447431-qt07lqj2c/subitem-1786953466249-2urvey9ug"><div class="list-item-paragraph"><p>the interquartile range.</p></div></li></ol></li></ol>

A box contains \(100\) cards. Each card has a number between one and six written on it. 
The following table shows the frequencies for each number.

<p>The curve \(y=\tan x\) meets the line \(y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{6}\). The line meets the positive \(x\)-axis at \(B\).</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q21-parts-item-1" data-list-item-path="amarea-q21-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q21-parts-item-2" data-list-item-path="amarea-q21-parts-item-2"><div class="list-item-paragraph">Find the exact area of the shaded region bounded by the curve, the line and the \(x\)-axis.<span class="inline-marks">[5]</span></div></li></ol>

The curve \(y=\tan x\) meets the line \(y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{6}\). The line meets the positive \(x\)-axis at \(B\).

Find the exact area of the shaded region bounded by the curve, the line and the \(x\)-axis.

Area Bounded by a Curve and Straight Lines

Tangent curve and a descending line

Integration

<p>The diagram shows part of the curve \(y=\dfrac{4-x}{\sqrt{x+1}}\), which meets the positive \(x\)-axis at \(A\).</p>
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<p>Find the exact area of the shaded region bounded by the curve and the coordinate axes.<span class="inline-marks">[5]</span></p>

The diagram shows part of the curve \(y=\dfrac{4-x}{\sqrt{x+1}}\), which meets the positive \(x\)-axis at \(A\).

Find the exact area of the shaded region bounded by the curve and the coordinate axes.

Area Enclosed by the Curve and the X-Axis

Radical curve ending at the x-axis

<p>Find the derivative of each of the following w.r.t \(x\):</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>\(y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} \)</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>\(y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) \)</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>\( y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }\)</p></div></li></ol>

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