A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IP Math

<p>A sequence \(u_1, u_2, u_3, \dots\) is given by</p>
<p style="text-align: center">\[u_1 = 2 \text{ and } u_n = \frac{3u_{n-1} - 1}{6} \text{ for } n \ge 2.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact values of \(u_2\) and \(u_3\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;\(u_n \to l\) as&nbsp;\(n \to \infty\). Showing your working, find the exact value of \(l\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">For this value of \(l\), verify that&nbsp;\(u_n = \frac{14}{3}\left(\frac{1}{2}\right)^n + l\) satisfies both the initial condition and the recurrence relation given in the question.</span></div></li></ol>

A sequence \(u_1, u_2, u_3, \dots\) is given by

\[u_1 = 2 \text{ and } u_n = \frac{3u_{n-1} - 1}{6} \text{ for } n \ge 2.\]

Find the exact values of \(u_2\) and \(u_3\).

It is given that \(u_n \to l\) as \(n \to \infty\). Showing your working, find the exact value of \(l\).

For this value of \(l\), verify that \(u_n = \frac{14}{3}\left(\frac{1}{2}\right)^n + l\) satisfies both the initial condition and the recurrence relation given in the question.

Limit of an expression

N2012 P1 Q3 [Modified]

Recurrence Relations and Summation

Video 1

<p>The sequence \(\{u_n\}\) is a geometric sequence with first term \(a\) and common ratio \(\frac{3}{2}\), where&nbsp;\(a &gt; 0\).</p>
<p>Another sequence \(\{v_n\}\) is defined by \(v_n = \frac{9}{u_n}\) for all positive integers \(n\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(\{v_n\}\) is also a geometric sequence.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(v_2 = u_2\), find the value of \(a\).</span><span class="inline-marks">[2]</span></div></li></ol>

The sequence \(\{u_n\}\) is a geometric sequence with first term \(a\) and common ratio \(\frac{3}{2}\), where \(a > 0\).

Another sequence \(\{v_n\}\) is defined by \(v_n = \frac{9}{u_n}\) for all positive integers \(n\).

Show that \(\{v_n\}\) is also a geometric sequence.

Given that \(v_2 = u_2\), find the value of \(a\).

2025 TJC P1 Q1

APGP

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express \(\frac{1}{Q(500 - Q)}\) in partial fractions.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A group of scientists is monitoring the population of a particular species of birds on an island. At time \(t\) years after the start of the monitoring, the number of birds on the island is \(Q\). The scientists observe that the birth rate is proportional to the bird's population and the birds are dying at a rate proportional to the square of the bird's population. There were \(1000\) birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are \(500\) birds.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that the differential equation relating \(Q\) and \(t\) is given by\[\frac{\mathrm{d}Q}{\mathrm{d}t} = kQ(500 - Q)\]where \(k\) is a positive constant.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the differential equation, expressing \(Q\) in terms of \(k\) and \(t\).</span><span class="inline-marks">[5]</span></div></li></ol></li></ol>

Express \(\frac{1}{Q(500 - Q)}\) in partial fractions.

A group of scientists is monitoring the population of a particular species of birds on an island. At time \(t\) years after the start of the monitoring, the number of birds on the island is \(Q\). The scientists observe that the birth rate is proportional to the bird's population and the birds are dying at a rate proportional to the square of the bird's population. There were \(1000\) birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are \(500\) birds.

2025 HCI P1 Q10

Differential Equations

<p>A sequence \(\{u_n\}\) of numbers are defined by \(u_1 = 1\) and</p>
<p style="text-align: center">\[u_{n+1} = 2u_n + 4, \quad n \in \mathbb{N}.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of \(u_2, u_3\) and \(u_4\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using GDC, find the least integral value of \(n\) for which \(u_n &gt; 256\).</span></div></li></ol>

A sequence \(\{u_n\}\) of numbers are defined by \(u_1 = 1\) and

\[u_{n+1} = 2u_n + 4, \quad n \in \mathbb{N}.\]

Find the values of \(u_2, u_3\) and \(u_4\).

Using GDC, find the least integral value of \(n\) for which \(u_n > 256\).

Limit of a Sequence and Behaviour of a Sequence

Recurrence relation: linear

<p><strong>A graphing calculator is not allowed for this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the square roots of \(2\mathrm{i}\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation \(w^2 = 3 - 4\mathrm{i}\). Hence find the roots of the equation \(z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0\), giving your answers in Cartesian form.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\(w = \frac{1}{2 + \mathrm{i}\alpha}\) where&nbsp;\(\alpha \in \mathbb{R}\), and \(w^*\) denotes the complex conjugate of \(w\), find \(ww^*\) and hence prove that&nbsp;\(w + w^* = 4ww^*\) ---(1).</div><div class="list-item-paragraph">The complex number \(w\) is represented in an Argand diagram by the point \(A\). By using (1) and considering \(w = x + y\mathrm{i}\), where&nbsp;\(x, y \in \mathbb{R}\), show that, \(A\) lies on a circle as&nbsp;\(\alpha\) varies. Find the Cartesian equation of this circle, stating clearly its centre and radius.<span class="inline-marks">[6]</span></div></li></ol>

**A graphing calculator is not allowed for this question.**

Find the square roots of \(2\mathrm{i}\).

Solve the equation \(w^2 = 3 - 4\mathrm{i}\). Hence find the roots of the equation \(z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0\), giving your answers in Cartesian form.

Given that \(w = \frac{1}{2 + \mathrm{i}\alpha}\) where \(\alpha \in \mathbb{R}\), and \(w^*\) denotes the complex conjugate of \(w\), find \(ww^*\) and hence prove that \(w + w^* = 4ww^*\) ---(1).

2020 ASRJC P1 Q5(a) | 2024 SAJC P1 Q8b

Complex Numbers

<p>A sequence is defined by \(T_{n+2} = T_{n+1} + T_n\) with \(T_1 = 3\), \(T_2 = 1\) for&nbsp;\(n \ge 1\). Another sequence is defined by \(r_n = \frac{T_{n+1}}{T_n}\) for&nbsp;\(n \ge 1\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find \(T_4\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(r_{n+1} = 1 + \frac{1}{r_n}\) for all for&nbsp;\(n \ge 1\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;\(r_n\) converges to a limit \(L\). Find the exact value of \(L\).</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A third sequence \(u_n\), for&nbsp;\(n \ge 1\), is an arithmetic sequence with first term \(5\) and common difference \(2\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the least value of \(n\) such that&nbsp;\(\sum_{k=1}^n u_k &gt; 930\).</span><span class="inline-marks">[2]</span></div></li></ol>

A sequence is defined by \(T_{n+2} = T_{n+1} + T_n\) with \(T_1 = 3\), \(T_2 = 1\) for \(n \ge 1\). Another sequence is defined by \(r_n = \frac{T_{n+1}}{T_n}\) for \(n \ge 1\).

Show that \(r_{n+1} = 1 + \frac{1}{r_n}\) for all for \(n \ge 1\).

It is given that \(r_n\) converges to a limit \(L\). Find the exact value of \(L\).

A third sequence \(u_n\), for \(n \ge 1\), is an arithmetic sequence with first term \(5\) and common difference \(2\).

Find the least value of \(n\) such that \(\sum_{k=1}^n u_k > 930\).

2025 MI P2 Q4

<p>The position vectors of the points \(A\) and \(B\) are \(\mathbf{i}+2\mathbf{j}+3\mathbf{k}\) and \(2\mathbf{i}+3\mathbf{j}+p\mathbf{k}\) respectively, where&nbsp;\(p\) is a constant. The point \(C\) is such that&nbsp;\(OABC\) is a rectangle, where&nbsp;\(O\) is the origin.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(p=2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Write down the position vector of \(C\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of the line \(BC\).</span></div></li></ol>
<p>The equation of line \(l\) is given by \(\frac{x-1}{3}=\frac{y-1}{2}\), \(z=1\).</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Show that the lines \(BC\) and \(l\) are skew.</span></div></li></ol>

The position vectors of the points \(A\) and \(B\) are \(\mathbf{i}+2\mathbf{j}+3\mathbf{k}\) and \(2\mathbf{i}+3\mathbf{j}+p\mathbf{k}\) respectively, where \(p\) is a constant. The point \(C\) is such that \(OABC\) is a rectangle, where \(O\) is the origin.

Find a vector equation of the line \(BC\).

The equation of line \(l\) is given by \(\frac{x-1}{3}=\frac{y-1}{2}\), \(z=1\).

Show that the lines \(BC\) and \(l\) are skew.

2006 JJC P1 Q12

Vectors

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;\(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is \(\displaystyle \int\nolimits_1^3 \mathrm{f}(x) \, \mathrm{d}x\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence evaluate \(\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.<span class="inline-marks">[4]</span></div></li></ol>

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\n\n\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]\n\nis \(\displaystyle \int_1^3 \mathrm{f}(x) \, \mathrm{d}x\).

Hence evaluate \(\lim_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.

Riemann Sums

2025 NYJC P2 Q3

Functions

Applications of Intergration

<p>An arithmetic series has positive first term \(a\) and common difference \(d\). The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(d = -\frac{7}{16}a\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Give a reason why the geometric series converges.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that the sum to infinity of the geometric series is \(\frac{32}{7}\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the sum of the first twenty terms of the arithmetic series.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the sum of all the terms after the \(n\)th term of the geometric series is greater than \(2\)% of the sum of the first \(n\) terms of the geometric series, find the possible values of \(n\).</span><span class="inline-marks">[4]</span></div></li></ol>

An arithmetic series has positive first term \(a\) and common difference \(d\). The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.

Give a reason why the geometric series converges.

It is given that the sum to infinity of the geometric series is \(\frac{32}{7}\).

Find the sum of the first twenty terms of the arithmetic series.

Given that the sum of all the terms after the \(n\)th term of the geometric series is greater than \(2\)% of the sum of the first \(n\) terms of the geometric series, find the possible values of \(n\).

2025 RI Promo Q11

<p>Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively where&nbsp;\(\mathbf{a}\) is a non-zero vector and \(\mathbf{b} \neq \mathbf{c}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\), show that&nbsp;\(\mathbf{b} - \mathbf{c} = \lambda \mathbf{a}\), where&nbsp;\(\lambda\) is a non-zero constant.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that the area of the trapezium \(OABC\) is \(10\) units\({}^2\), \(|\mathbf{a}| = 3\), \(|\mathbf{c}| = 5\) and the angle between \(\mathbf{a}\) and \(\mathbf{c}\) is \(30{}^\circ \).</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(BC = 5\) units.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">State the value of \(k\) such that&nbsp;\(\overrightarrow{OA} = k\overrightarrow{CB}\), where&nbsp;\(k\) is a positive constant.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The line \(OC\) meets the line \(AB\) at point \(D\). Find \(\overrightarrow{OD}\) in terms of \(\mathbf{c}\).<span class="inline-marks">[5]</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 where \(\mathbf{a}\) is a non-zero vector and \(\mathbf{b} \neq \mathbf{c}\).

Given that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\), show that \(\mathbf{b} - \mathbf{c} = \lambda \mathbf{a}\), where \(\lambda\) is a non-zero constant.

It is given that the area of the trapezium \(OABC\) is \(10\) units\({}^2\), \(|\mathbf{a}| = 3\), \(|\mathbf{c}| = 5\) and the angle between \(\mathbf{a}\) and \(\mathbf{c}\) is \(30{}^\circ \).

State the value of \(k\) such that \(\vec{OA} = k\vec{CB}\), where \(k\) is a positive constant.

The line \(OC\) meets the line \(AB\) at point \(D\). Find \(\vec{OD}\) in terms of \(\mathbf{c}\).

2025 TMJC P2 Q5

<p>The lines \(l_1\) and \(l_2\) have equations</p>
<p style="text-align: center">\[l_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\0 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}, \\\quad l_2 : \mathbf{r} = \begin{pmatrix} 0 \\ 5 \\ -8 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\]</p>
<p>where&nbsp;\(\lambda\) and \(\mu\) are parameters.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(l_1\) and \(l_2\) are skew lines.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The point \(P\) has coordinates \((1, 5, 0)\) and the plane \(\pi_1\) has equation \(y - z - 2 = 0\).</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of \(P'\), the reflection of \(P\) in \(\pi_1\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane \(\pi_2\) containing \(P\) and \(l_1\).</span><span class="inline-marks">[3]</span></div></li></ol>

The lines \(l_1\) and \(l_2\) have equations

\[l_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\0 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}, \\\quad l_2 : \mathbf{r} = \begin{pmatrix} 0 \\ 5 \\ -8 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\]

where \(\lambda\) and \(\mu\) are parameters.

Show that \(l_1\) and \(l_2\) are skew lines.

The point \(P\) has coordinates \((1, 5, 0)\) and the plane \(\pi_1\) has equation \(y - z - 2 = 0\).

Find the coordinates of \(P'\), the reflection of \(P\) in \(\pi_1\).

Find the cartesian equation of the plane \(\pi_2\) containing \(P\) and \(l_1\).

2025 SAJC P1 Q3

<p>State the derivative of \(\tan x\). Hence, find \(\displaystyle\int \sec^3 x \, \mathrm{d}x\).<span class="inline-marks">[5]</span></p>

State the derivative of \(\tan x\). Hence, find \(\displaystyle\int \sec^3 x \, \mathrm{d}x\).

2025 CJC CA1 Q3

Techniques of  Integration

<p>The acute angle between two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\) radians, and the vectors \(\mathbf{b}\) and \(2\mathbf{a}-\mathbf{b}\) are perpendicular. Find the value of \(\alpha\) if&nbsp;\(|\mathbf{b}| = \sqrt{2}|\mathbf{a}|\).<span class="inline-marks">[3]</span></p>

The acute angle between two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\) radians, and the vectors \(\mathbf{b}\) and \(2\mathbf{a}-\mathbf{b}\) are perpendicular. Find the value of \(\alpha\) if \(|\mathbf{b}| = \sqrt{2}|\mathbf{a}|\).

2025 YIJC P2 Q1

<p>The sum of the first \(n\) terms of a sequence, \(u_r\), is given by \[\sum_{r=1}^{n} u_r = 1 - \frac{n}{(n+1)!}.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find \(u_n\) in terms of \(n\), expressing your answer as a single algebraic fraction.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why \(\displaystyle \sum_{r=5}^{n} u_r &lt; \frac{1}{30}\), for all \(n \ge 5\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why \(\displaystyle \sum_{r=1}^{\infty} u_r\) is a convergent series.<span class="inline-marks">[1]</span></div></li></ol>

The sum of the first \(n\) terms of a sequence, \(u_r\), is given by \[\sum_{r=1}^{n} u_r = 1 - \frac{n}{(n+1)!}.\]

Find \(u_n\) in terms of \(n\), expressing your answer as a single algebraic fraction.

Explain why \(\sum_{r=5}^{n} u_r < \frac{1}{30}\), for all \(n \ge 5\).

Explain why \(\sum_{r=1}^{\infty} u_r\) is a convergent series.

Sigma Notation

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the expansion of \((8 - x)^{\frac{1}{3}}\) in ascending powers of \(x\) up to and including the term in \(x^2\). State the set of values of \(x\) for which this expansion is valid.</span></div></li><li><div class="list-item-content"><span class="list-item-text">By substituting \(x=-1\), find an approximation of \(\sqrt[3]{3}\), leaving your answer in the form \(\frac{m}{n}\), where \(m\) and \(n\) are integers to be determined.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The substitution \(x=5\) can also be used to find an approximation for \(\sqrt[3]{3}\). Explain whether this will give a better approximation than using \(x=-1\).</span></div></li></ol>

Find the expansion of \((8 - x)^{\frac{1}{3}}\) in ascending powers of \(x\) up to and including the term in \(x^2\). State the set of values of \(x\) for which this expansion is valid.

By substituting \(x=-1\), find an approximation of \(\sqrt[3]{3}\), leaving your answer in the form \(\frac{m}{n}\), where \(m\) and \(n\) are integers to be determined.

The substitution \(x=5\) can also be used to find an approximation for \(\sqrt[3]{3}\). Explain whether this will give a better approximation than using \(x=-1\).

Maclaurin Series and Power Series

<p>A piece of vanguard sheet, \(ABCDEF\), is in the form of a regular hexagon of side \(a\) cm. A kite shape is cut out from each corner to form the shaded shape, as shown in Fig. \(1\). It is then folded to form the open hexagonal box of height \(h\) cm, as shown in Fig. \(2\).</p>
<p>Show that the volume \(V\) of the box in cm\(^3\) is given by \(V = 2\sqrt{3} h\left(\frac{\sqrt{3}}{2}a - h\right)^2.\)</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/1770518988667-j52aih1mvjo.png" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

A piece of vanguard sheet, \(ABCDEF\), is in the form of a regular hexagon of side \(a\) cm. A kite shape is cut out from each corner to form the shaded shape, as shown in Fig. \(1\). It is then folded to form the open hexagonal box of height \(h\) cm, as shown in Fig. \(2\).

Show that the volume \(V\) of the box in cm\(^3\) is given by \(V = 2\sqrt{3} h\left(\frac{\sqrt{3}}{2}a - h\right)^2.\)

2015 AJC P1 Q11 (Modified) IP2U11 HOT Q1

Applications of Differentiation

<p>The sum, \(S_n\), of the first \(n\) terms of the sequence \(\{u_n\}\) is given by</p>
<p style="text-align: center">\[S_n = 2n^2 - 3n, n = 1, 2, 3, \dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the sum of the terms from the \(10\)th term to the \(23\)rd term of \(\{u_n\}\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find \(u_n\) in terms of \(n\).</span></div></li></ol>

The sum, \(S_n\), of the first \(n\) terms of the sequence \(\{u_n\}\) is given by

Find the sum of the terms from the \(10\)th term to the \(23\)rd term of \(\{u_n\}\).

Method of Difference involving uₙ−uₙ₋₁.

Recovering term from sequence example

<p>Planes \(p\) and \(q\) are perpendicular. Plane \(p\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 6\). Plane \(q\) contains the line \(l\) with equation \(\mathbf{r} = \mathbf{j} - 12\mathbf{k} + \lambda(4\mathbf{i} + \mathbf{j} + 10\mathbf{k})\) where&nbsp;\(\lambda\) is a parameter. The point \(A\) has coordinates \((0, 1, -12)\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find a cartesian equation of \(q\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of the line \(m\), where&nbsp;\(p\) and \(q\) meet.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>\(l\) passes through \(p\) at point \(B\). Point \(C\) is the foot of the perpendicular of \(A\) to \(p\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the position vector of \(B\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the position vector of \(C\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact area of \(\Delta ABC\).</span><span class="inline-marks">[2]</span></div></li></ol>

Planes \(p\) and \(q\) are perpendicular. Plane \(p\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 6\). Plane \(q\) contains the line \(l\) with equation \(\mathbf{r} = \mathbf{j} - 12\mathbf{k} + \lambda(4\mathbf{i} + \mathbf{j} + 10\mathbf{k})\) where \(\lambda\) is a parameter. The point \(A\) has coordinates \((0, 1, -12)\).

Find a vector equation of the line \(m\), where \(p\) and \(q\) meet.

\(l\) passes through \(p\) at point \(B\). Point \(C\) is the foot of the perpendicular of \(A\) to \(p\).

2025 YIJC P1 Q10

<p>Myra washed three different pairs of gloves and wants to hang them up to dry on a clothes line. How many ways can she hang the gloves if all the right gloves are always on the right of their respective left gloves?</p>

Myra washed three different pairs of gloves and wants to hang them up to dry on a clothes line. How many ways can she hang the gloves if all the right gloves are always on the right of their respective left gloves?

Permutation and Combination

<p></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The non-zero vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are such that&nbsp;\(\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}\). Given that&nbsp;\(\mathbf{b} \neq -\mathbf{c}\), find a linear relationship between \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The variable position vector \(\mathbf{v}\) satisfies the equation \(\mathbf{v} \times (\mathbf{i} - 3\mathbf{k}) = 2\mathbf{j}\).</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the set of position vectors \(\mathbf{v}\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe the set of points represented by position vectors \(\mathbf{v}\) geometrically.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

The non-zero vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}\). Given that \(\mathbf{b} \neq -\mathbf{c}\), find a linear relationship between \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

The variable position vector \(\mathbf{v}\) satisfies the equation \(\mathbf{v} \times (\mathbf{i} - \(3\)\mathbf{k}) = \(2\)\mathbf{j}.

IB HL Math Question Bank