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<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

<p>Without using a calculator, solve the inequality $\frac{4x}{{{x}^{2}}-2x-4}&gt;-x$.	<span class="inline-marks">[5]</span></p>

Without using a calculator, solve the inequality $\frac{4x}{{{x}^{2}}-2x-4}>-x$.	

Inequalities - General

2015 NYJC P1 Q1

Inequalities

<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>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

<p>Without using a calculator, solve the inequality $\frac{5{{x}^{2}}-x-14}{2{{x}^{2}}+x-3}\le 3$.				<span class="inline-marks">[4]</span></p>

Without using a calculator, solve the inequality $\frac{5{{x}^{2}}-x-14}{2{{x}^{2}}+x-3}\le 3$.				

2015 SAJC P1 Q1

<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

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve the inequality $3|x^2-4|\le |x+2|$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence, solve $\frac{3}{x^2}\left|1-4x^2\right|\le \left|\frac{1+2x}{x}\right|$.<span class="inline-marks">[2]</span></div></li></ol>

Solve the inequality $3|x^2-4|\le |x+2|$.

Hence, solve $\frac{3}{x^2}|1-4x^2|\le |\frac{1+2x}{x}|$.

2025 SAJC P2 Q1

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, solve the inequality $\frac{2x - 25}{x^2 - 2x - 3} \leq 2$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve the inequality $\frac{2|x + 1| - 25}{(x + 1)^2 - 2|x + 1| - 3} \leq 2$.</span><span class="inline-marks">[3]</span></div></li></ol>

Without using a calculator, solve the inequality $\frac{2x - 25}{x^2 - 2x - 3} \leq 2$.

Hence solve the inequality $\frac{2|x + 1| - 25}{(x + 1)^2 - 2|x + 1| - 3} \leq 2$.

2025 TJC P1 Q5

<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>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}.

H2 Math Question Bank