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<p>Expand $(1-2x)^{-3}$ up to and including the term in $x^3$, and state the open range of validity.<span class="inline-marks">[5]</span></p>

Expand $(1-2x)^{-3}$ up to and including the term in $x^3$, and state the open range of validity.

General Binomial Theorem

A negative integer power

Binomial Theorem

<p>The function $\mathrm{f}$ is defined by</p>
<p>\[\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,\]</p>
<p>where&nbsp;$a,b,c$ are constants.</p>
<p>The curve $y=\mathrm{f}(x)$ passes through $(0,3)$ and has a stationary point at $\left(\frac\pi6,4\right)$.</p>
<ol class="i-list"><li data-list-item-id="5051fe58-1e08-48fd-acb0-ee1b28acfb85" data-list-item-path="5051fe58-1e08-48fd-acb0-ee1b28acfb85"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their exact values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="f3299ee3-8d23-4137-be13-da91485900f1" data-list-item-path="f3299ee3-8d23-4137-be13-da91485900f1"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)+c$, where&nbsp;$R&gt;0$ and $0&lt;\alpha&lt;\frac\pi2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="13c4684e-6601-4b6f-bd4a-f416e8b586bc" data-list-item-path="13c4684e-6601-4b6f-bd4a-f416e8b586bc"><div class="list-item-paragraph">Hence solve $\mathrm{f}(x)=3$, giving all answers in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,\]

The curve $y=\mathrm{f}(x)$ passes through $(0,3)$ and has a stationary point at $\left(\frac\pi6,4\right)$.

Form three linear equations in $a,b,c$, and hence find their exact values.

Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)+c$, where $R>0$ and $0<\alpha<\frac\pi2$.

Hence solve $\mathrm{f}(x)=3$, giving all answers in exact form.

A trigonometric curve determined by a stationary point

System of Linear Equations

<p>The curve $C$ has equation</p>
<p>\[y=ax^2+bx+c,\]</p>
<p>where&nbsp;$a,b,c$ are constants. It passes through $P\left(1,\frac92\right)$.</p>
<p>The tangents to $C$ at the points where&nbsp;$x=1$ and $x=3$ have angles of inclination $\frac{\pi}{4}$ and $\frac{3\pi}{4}$, respectively. Angles of inclination are measured anticlockwise from the positive $x$-axis.</p>
<ol class="i-list"><li data-list-item-id="ac8a0ec9-c518-4df5-ae16-bc947f2a20bd" data-list-item-path="ac8a0ec9-c518-4df5-ae16-bc947f2a20bd"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="fdcca7ff-8d9a-4bce-afd6-954da5555f97" data-list-item-path="fdcca7ff-8d9a-4bce-afd6-954da5555f97"><div class="list-item-paragraph">The two tangents intersect at $T$. Find the exact area of the triangle bounded by the two tangents and the line $y=\frac92$.<span class="inline-marks">[3]</span></div></li></ol>

where $a,b,c$ are constants. It passes through $P\left(1,\frac92\right)$.

The tangents to $C$ at the points where $x=1$ and $x=3$ have angles of inclination $\frac{\pi}{4}$ and $\frac{3\pi}{4}$, respectively. Angles of inclination are measured anticlockwise from the positive $x$-axis.

Form three linear equations in $a,b,c$, and hence find their values.

The two tangents intersect at $T$. Find the exact area of the triangle bounded by the two tangents and the line $y=\frac92$.

A parabola determined by tangent inclinations

<p>The curve $C$ has equation</p>
<p>\[y=x^3+ax^2+bx+c,\]</p>
<p>where&nbsp;$a,b,c$ are constants.</p>
<p>The tangent to $C$ at the point where&nbsp;$x=1$ has equation $y=2x+1$. The normal to $C$ at the point where&nbsp;$x=-1$ is parallel to the line $x+10y=4$.</p>
<ol class="i-list"><li data-list-item-id="cb773b6e-3834-477c-be26-b4675e5a49e1" data-list-item-path="cb773b6e-3834-477c-be26-b4675e5a49e1"><div class="list-item-paragraph">Form three linear equations in $a,b,c$, and hence find their values.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="6331ed95-e7fb-4e5e-a814-3ab1a8c4631b" data-list-item-path="6331ed95-e7fb-4e5e-a814-3ab1a8c4631b"><div class="list-item-paragraph">Show that the gradient of $C$ is always positive. Hence explain why $C$ intersects the $x$-axis exactly once, and find the $x$-coordinate of this intersection correct to 3 significant figures.<span class="inline-marks">[3]</span></div></li></ol>

The tangent to $C$ at the point where $x=1$ has equation $y=2x+1$. The normal to $C$ at the point where $x=-1$ is parallel to the line $x+10y=4$.

Show that the gradient of $C$ is always positive. Hence explain why $C$ intersects the $x$-axis exactly once, and find the $x$-coordinate of this intersection correct to 3 significant figures.

A cubic determined by a tangent and a normal

<ol class="i-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">Solve the inequality \[\frac{(x-2)(x+1)}{x-1}\geq0.\]<span class="inline-marks">[3]</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, in exact form, \[\frac{(\sec\theta-2)(\sec\theta+1)}{\sec\theta-1}\geq0,\qquad 0\leq\theta\leq2\pi,\] where&nbsp;$\sec\theta$ is defined.<span class="inline-marks">[3]</span></div></li></ol>

Solve the inequality \[\frac{(x-2)(x+1)}{x-1}\geq0.\]

Hence solve, in exact form, \[\frac{(\sec\theta-2)(\sec\theta+1)}{\sec\theta-1}\geq0,\qquad 0\leq\theta\leq2\pi,\] where $\sec\theta$ is defined.

Inequalities involving Substitution

Inequalities involving Rational Functions

Secant substitution in a rational inequality

Inequalities

<ol class="i-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">Solve the inequality \[\frac{(2x-1)(x+2)}{x-\frac{\sqrt3}{2}}\leq0.\]<span class="inline-marks">[3]</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, in exact form, \[\frac{(2\sin\theta-1)(\sin\theta+2)}{\sin\theta-\frac{\sqrt3}{2}}\leq0,\qquad 0\leq\theta\leq2\pi.\]<span class="inline-marks">[3]</span></div></li></ol>

Solve the inequality \[\frac{(2x-1)(x+2)}{x-\frac{\sqrt3}{2}}\leq0.\]

Hence solve, in exact form, \[\frac{(2\sin\theta-1)(\sin\theta+2)}{\sin\theta-\frac{\sqrt3}{2}}\leq0,\qquad 0\leq\theta\leq2\pi.\]

Sine substitution in a rational inequality

<ol class="i-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">Solve the inequality \[\frac{(2x+1)(x-2)}{x-1}\leq0.\]<span class="inline-marks">[3]</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, in exact form, \[\frac{(2\cos2\theta+1)(\cos2\theta-2)}{\cos2\theta-1}\leq0,\qquad 0&lt;\theta&lt;\pi.\]<span class="inline-marks">[2]</span></div></li></ol>

Solve the inequality \[\frac{(2x+1)(x-2)}{x-1}\leq0.\]

Hence solve, in exact form, \[\frac{(2\cos2\theta+1)(\cos2\theta-2)}{\cos2\theta-1}\leq0,\qquad 0<\theta<\pi.\]

Double-angle cosine substitution in a rational inequality

<p>Prove by induction that&nbsp;$n! \le n^n$ for&nbsp;$n \in \mathbb{Z}^+$.<span class="inline-marks">[7]</span></p>

Prove by induction that $n! \le n^n$ for $n \in \mathbb{Z}^+$.

Mathematical Induction

<p>At a game show, Ming was given&nbsp;$20$ multiple-choice questions. He gets $5$ points if he answers a question correctly, gets a point if he does not answer a question and gets $2$ points deducted if he answers a question wrongly. If Ming answered at least a question wrongly and scored $52$ points at the game show, find the number of questions he answered wrongly.<span class="inline-marks">[4]</span></p>

At a game show, Ming was given $20$ multiple-choice questions. He gets $5$ points if he answers a question correctly, gets a point if he does not answer a question and gets $2$ points deducted if he answers a question wrongly. If Ming answered at least a question wrongly and scored $52$ points at the game show, find the number of questions he answered wrongly.

Inequalities - General

2026 HCI P1 Q1

<p>Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.</p>

Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.

Permutation and Combination

<ol class="abc-list"><li data-list-item-id="h2-2025-yijc-promo-q4-yijc-q4-item-block-0-item-0" data-list-item-path="h2-2025-yijc-promo-q4-yijc-q4-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$\sum_{r=1}^{n} r^2=\frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=1}^{n}(4r+1)(6r+1)$ in terms of $n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-yijc-promo-q4-yijc-q4-item-block-0-item-1" data-list-item-path="h2-2025-yijc-promo-q4-yijc-q4-item-block-0-item-1"><div class="list-item-paragraph">Hence find $\sum_{r=8}^{2n}(4r+1)(6r+1)$ in terms of $n$.<span class="inline-marks">[3]</span></div></li></ol>

Given that $\sum_{r=1}^{n} r^2=\frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=1}^{n}(4r+1)(6r+1)$ in terms of $n$.

Hence find $\sum_{r=8}^{2n}(4r+1)(6r+1)$ in terms of $n$.

Sigma Notation

2025 YIJC Promo Q4

Recurrence Relations and Summation

<ol class="abc-list"><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-0" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-0"><div class="list-item-paragraph">With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $P$ lies on $AB$, such that&nbsp;$AP:PB=2:3$. Given that&nbsp;$OP$ and $AB$ are perpendicular, show that&nbsp;$\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1"><div class="list-item-paragraph">The plane $\pi$ has equation $\mathbf{r}=\begin{pmatrix}0\\2\\1\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\3\end{pmatrix}+\mu\begin{pmatrix}3\\0\\-1\end{pmatrix},\ \lambda,\mu\in\mathbb{R}.$</div><ol class="i-list"><li data-list-item-id="item-1789031430179-n1av0zoyd" data-list-item-path="item-1789031430179-n1av0zoyd"><div class="list-item-paragraph"><p>Show that a cartesian equation of $\pi$ is $x-4y+3z=-5$.</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-paragraph"><p>The line $l_1$ has equation $\mathbf{r}=\begin{pmatrix}6\\-1\\4\end{pmatrix}+t\begin{pmatrix}1\\1\\1\end{pmatrix},\ t\in\mathbb{R}.$</p></div><ol class="i-list" start="2"><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1-sub-4" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1/h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1-sub-4"><div class="list-item-paragraph"><p>Find the acute angle between $l_1$ and $\pi$.</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content"><p>The line $l_2$ passes through the point $Q(-2,-1,2)$ and is parallel to $2\mathbf{i}+3\mathbf{j}+\mathbf{k}$.</p></div><div class="list-item-after-content"><ol class="i-list" start="3"><li data-list-item-id="item-1789031789556-jh2eio6wg" data-list-item-path="item-1789031789556-jh2eio6wg"><div class="list-item-paragraph"><p>Find the position vector of the foot of the perpendicular from $Q$ to $\pi$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1789031809165-lneq3t01d" data-list-item-path="item-1789031809165-lneq3t01d"><div class="list-item-paragraph"><p>Find the distance between $l_2$ and $\pi$.</p><span class="inline-marks">[2]</span></div></li></ol></div></li></ol>

With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $P$ lies on $AB$, such that $AP:PB=2:3$. Given that $OP$ and $AB$ are perpendicular, show that $\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.

Length of Projection

Finding The Equations of Planes

Finding the Distance of a Point to the Plane

Relationship Between a Line and a Plane

Basic Results: Ratio Theorem

2025 YIJC Promo Q8

Vectors

<p>It is given that&nbsp;$ \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4} $.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-0" data-list-item-path="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-0"><div class="list-item-paragraph">Find $ \sum_{r=4}^{N+3} (r-2)^3 $ in terms of $N$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-1" data-list-item-path="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-1"><div class="list-item-paragraph">Hence, find the largest value of $N$ for which $ \sum_{r=4}^{N+3} \left[(2r-4)^3+2^r\right] &lt; 10^5 $.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $ \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4} $.

Find $ \sum_{r=4}^{N+3} (r-2)^3 $ in terms of $N$.

Hence, find the largest value of $N$ for which $ \sum_{r=4}^{N+3} \left[(2r-4)^3+2^r\right] < 10^5 $.

Sigma Notation By Substitution

Summations Involving APGP

2025 VJC Promo Q3

<ol class="abc-list"><li data-list-item-id="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-0" data-list-item-path="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-0"><div class="list-item-paragraph">The curve $C_1$ has equation $\frac{(x-1)^2}{16}-\frac{(y+3)^2}{4}=1$. Sketch $C_1$, giving the equations of the asymptotes and the coordinates of the centre and vertices.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-1" data-list-item-path="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-1"><div class="list-item-paragraph">The curve $C_2$ has equation $x^2-4x+y^2+6y+13=a^2$, where&nbsp;$a&gt;0$, intersects $C_1$ at exactly two points. Find the range of values of $a$.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C_1$ has equation $\frac{(x-1)^2}{16}-\frac{(y+3)^2}{4}=1$. Sketch $C_1$, giving the equations of the asymptotes and the coordinates of the centre and vertices.

The curve $C_2$ has equation $x^2-4x+y^2+6y+13=a^2$, where $a>0$, intersects $C_1$ at exactly two points. Find the range of values of $a$.

Range of unknowns in Equations

Conics Graphs

2025 VJC Promo Q2

Graphing Techniques

<p>With reference to the origin $O$, the points $A$, $B$ and $X$ have position vectors $3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $11\mathbf{i}+10\mathbf{j}+9\mathbf{k}$ respectively. The point $C$ lies on $OA$ produced such that&nbsp;$OA:AC=1:\alpha$, where&nbsp;$\alpha$ is a constant. The point $D$ lies on $OB$ produced such that&nbsp;$OB:BD=1:\beta$, where&nbsp;$\beta$ is a constant.</p>
<p>It is given that&nbsp;$OCXD$ is a parallelogram.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-0" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-0"><div class="list-item-paragraph">Write down the position vectors of $C$ and $D$ in terms of $\alpha$ and $\beta$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-1" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-1"><div class="list-item-paragraph">Show that&nbsp;$\beta=1$ and find the value of $\alpha$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-2" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-2"><div class="list-item-paragraph">Find the exact shortest distance from $X$ to line $OD$.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the points $A$, $B$ and $X$ have position vectors $3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $11\mathbf{i}+10\mathbf{j}+9\mathbf{k}$ respectively. The point $C$ lies on $OA$ produced such that $OA:AC=1:\alpha$, where $\alpha$ is a constant. The point $D$ lies on $OB$ produced such that $OB:BD=1:\beta$, where $\beta$ is a constant.

It is given that $OCXD$ is a parallelogram.

Write down the position vectors of $C$ and $D$ in terms of $\alpha$ and $\beta$.

Show that $\beta=1$ and find the value of $\alpha$.

Find the exact shortest distance from $X$ to line $OD$.

Foot of Perpendicular of a Point to a Line and Point of Reflection

2025 TJC Promo Q3

<p>Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that&nbsp;$|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0"><div class="list-item-paragraph">By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that&nbsp;$\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1"><div class="list-item-paragraph">By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2"><div class="list-item-paragraph">Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3"><div class="list-item-paragraph">Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that $|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.

By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that $\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.

By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.

Application of Dot Product: Angle Bisector

Application of Cross Product: Area of Parallelogram and Triangle

Vector Dot Product

2025 SAJC Promo Q8

<p>The planes $\pi_1$ and $\pi_2$ have equations</p>
<p style="text-align: center">${2x + 2y + z = 2,}$<br>${5x - 2y - z = 5,}$</p>
<p>respectively.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0"><div class="list-item-paragraph">Find the acute angle between $\pi_1$ and $\pi_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1"><div class="list-item-paragraph">Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2"><div class="list-item-paragraph">The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where&nbsp;$\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3"><div class="list-item-paragraph"><p>Hence or otherwise, find the cartesian equations of the two planes that are parallel to $\pi_1$ and at a distance of $6$ units from $\pi_1$.</p><span class="inline-marks">[4]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations

${2x + 2y + z = 2,}$
${5x - 2y - z = 5,}$

Find the acute angle between $\pi_1$ and $\pi_2$.

Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.

The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where $\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.

Finding Equations of Line

Relationships Between Two Planes

2025 SAJC Promo Q10

<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0"><div class="list-item-paragraph">The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that&nbsp;$3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where&nbsp;$3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1"><div class="list-item-paragraph"><p>The variable position vector $\mathbf{v}$ satisfies the equation</p></div><div class="list-item-paragraph"><p style="text-align: center;">$\mathbf{v}\times(-\mathbf{i}+2\mathbf{j}+\mathbf{k})=a\mathbf{i}+\mathbf{j}+\mathbf{k},$</p></div><div class="list-item-paragraph"><p>where $a$ is a non-zero constant.</p></div><ol class="i-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0"><div class="list-item-paragraph">Show that&nbsp;$a=3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Hence, by finding the set of position vectors $\mathbf{v}$, describe geometrically the set of points represented by the position vectors $\mathbf{v}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that $3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where $3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.

Cross Product

2025 RI Promo Q6

<p>Mr Tan decides to start a regular savings plan to save up for an overseas trip. At the start of the first month, he deposits $1000 and continues to deposit $800 at the start of every subsequent month. He withdraws $10\%$ of the amount in the account at the end of each month for investment starting from the first month. The account pays no interest.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-0" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-0"><div class="list-item-paragraph">Show that the amount of money in the savings account at the end of the $n$th month is $\$(7200 - 7000(0.9)^n)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-1" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-1"><div class="list-item-paragraph">Find the earliest possible month such that the amount of money in the savings account first reaches $5000. Explain whether this occurs at the start or end of the month.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-2" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-2"><div class="list-item-paragraph">State the amount of money that Mr Tan has in the savings account in the long run after withdrawing for his monthly investment.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-3" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-3"><div class="list-item-paragraph">Find the total amount of money Mr Tan withdraws for investment at the end of 1 year.<span class="inline-marks">[2]</span></div></li></ol>

Mr Tan decides to start a regular savings plan to save up for an overseas trip. At the start of the first month, he deposits $1000 and continues to deposit $800 at the start of every subsequent month. He withdraws $10\%$ of the amount in the account at the end of each month for investment starting from the first month. The account pays no interest.

Show that the amount of money in the savings account at the end of the $n$th month is $\$(7200 - 7000(0.9)^n)$.

Find the earliest possible month such that the amount of money in the savings account first reaches $5000. Explain whether this occurs at the start or end of the month.

State the amount of money that Mr Tan has in the savings account in the long run after withdrawing for his monthly investment.

Find the total amount of money Mr Tan withdraws for investment at the end of 1 year.

Problems involving inequalities

Limit of a Sequence and Behaviour of a Sequence

Geometric Progression

Financial Arithmetic or Iterative Pattern Recognition

2025 NYJC Promo Q8

APGP

<div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjozNTIsImhlaWdodCI6MjU4LCJlbGVtZW50cyI6W3siaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTAiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAuMywtMl0sInBvaW50MiI6Wy00LjAzLC0wLjIxXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6ImltcG9ydGVkLXNlZ21lbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMSIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMC4zLC0yXSwicG9pbnQyIjpbMi42LDAuN10sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTIiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAuMywtMl0sInBvaW50MiI6WzEuMiwzLjRdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0zIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlstNC4wMywtMC4yMV0sInBvaW50MiI6WzEuMiwzLjRdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC00IiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlsyLjYsMC43XSwicG9pbnQyIjpbMS4yLDMuNF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTUiLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6Wy00LjAzLC0wLjIxXSwicG9pbnQyIjpbMi42LDAuN10sInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6LTIuMjYsInkiOjEuMDEsImlkIjoiaW1wb3J0ZWQtcG9pbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtNiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50U2l6ZSI6MS41LCJwb2ludFN0eWxlIjoiY2lyY2xlIiwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjowLjE1LCJ5IjotMi4yNywiaWQiOiJpbXBvcnRlZC10ZXh0LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTciLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoT1xcKSIsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6Mi42LCJ5IjowLjcsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC04IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKEFcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOi00LjM2LCJ5IjotMC4yMSwiaWQiOiJpbXBvcnRlZC10ZXh0LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTkiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoQlxcKSIsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6MS4wNCwieSI6My42MSwiaWQiOiJpbXBvcnRlZC10ZXh0LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTEwIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKENcXCkiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOi0yLjcsInkiOjEuMjQsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xMSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChEXFwpIiwic3Ryb2tlV2lkdGgiOjEuNX1dLCJzaG93WEF4aXMiOmZhbHNlLCJzaG93WUF4aXMiOmZhbHNlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstNC44NzIzLDQuMjA2NCw0LjY2MDIsLTIuNzgxNV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtYWpvckNvbG9yIjoiIzk0YTNiOCIsIm1ham9yU3RlcFgiOjEsIm1ham9yU3RlcFkiOjEsIm1pbm9yQ29sb3IiOiIjY2JkNWUxIiwibWFqb3JPcGFjaXR5IjowLjQ1LCJtaW5vck9wYWNpdHkiOjAuNDUsIm1pbm9yRWxlbWVudHNYIjo0LCJtaW5vckVsZW1lbnRzWSI6NCwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjUsInNob3dNYWpvckdyaWRMaW5lcyI6dHJ1ZSwic2hvd01pbm9yR3JpZExpbmVzIjp0cnVlfSwic2hvd0F4aXNMYWJlbHMiOmZhbHNlLCJzaG93TmF2aWdhdGlvbiI6ZmFsc2UsImtlZXBBc3BlY3RSYXRpbyI6dHJ1ZSwieFRpY2tMYWJlbEV2ZXJ5IjoxLCJ5VGlja0xhYmVsRXZlcnkiOjEsInhBeGlzTGFiZWxPZmZzZXQiOlsyLC0xMV0sInlBeGlzTGFiZWxPZmZzZXQiOlstMTQsNV0sIm9yaWdpbkxhYmVsT2Zmc2V0IjpbLTUsLTJdfQ==" data-width="352" data-height="258" style="width:352px;height:258px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and $C$ have position vectors with respect to the origin $O$ given by $\overrightarrow{OA}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, between $B$ and $C$, such that&nbsp;$BD:DC=1:2$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-0" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-0"><div class="list-item-paragraph">Find the position vector of $D$<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-1" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-1"><div class="list-item-paragraph">Find a cartesian equation of face $ABC$ and state its exact distance from $O$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-2" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-2"><div class="list-item-paragraph">Hence find the acute angle between $OD$ and face $ABC$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-3" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-3"><div class="list-item-paragraph">Find the coordinates of the point on $OD$ which is closest to $A$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and $C$ have position vectors with respect to the origin $O$ given by $\overrightarrow{OA}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, between $B$ and $C$, such that $BD:DC=1:2$.

Find a cartesian equation of face $ABC$ and state its exact distance from $O$.

Hence find the acute angle between $OD$ and face $ABC$.

Find the coordinates of the point on $OD$ which is closest to $A$.

Math Question Bank