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<p>Prove, by induction, $(1+a)^n \ge 1+na$, where&nbsp;$a \in \mathbb{R}^+$, for all positive integers of $n$.<span class="inline-marks">[5]</span></p>

Prove, by induction, $(1+a)^n \ge 1+na$, where $a \in \mathbb{R}^+$, for all positive integers of $n$.

Mathematical Induction

<p>Prove by mathematical induction that&nbsp;$\displaystyle\sum_{r=1}^{n}(2r-1)=n^2$ for every $n\in\mathbb Z^+$.</p>

Prove by mathematical induction that $\displaystyle\sum_{r=1}^{n}(2r-1)=n^2$ for every $n\in\mathbb Z^+$.

Sum of the First Odd Integers

Recurrence Relations and Summation

<p>Disprove the statement “the quotient of any two irrational numbers is irrational”.</p>

Disprove the statement “the quotient of any two irrational numbers is irrational”.

A Quotient of Irrational Numbers

Inequalities

<p>Prove that the square of any odd integer leaves a remainder of $1$ when divided by $8$.</p>

Prove that the square of any odd integer leaves a remainder of $1$ when divided by $8$.

Odd Squares Modulo Eight

<p>Let&nbsp;$a$, $b$ and $d$ be integers. Prove that if&nbsp;$d\mid a$ and $d\mid b$, then&nbsp;$d\mid(5a-3b)$.</p>

Let $a$, $b$ and $d$ be integers. Prove that if $d\mid a$ and $d\mid b$, then $d\mid(5a-3b)$.

Divisibility of an Integer Linear Combination

<p>Prove that the product of any two consecutive integers is even.</p>

Prove that the product of any two consecutive integers is even.

Product of Consecutive Integers

<p>The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto (x-4)^2 + 1$ for&nbsp;$x \in \mathbb{R}, x &gt; 4$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph"><p>Sketch the graph of $y = \mathrm{f}(x)$. Your sketch should indicate the position of the graph in relation to the origin.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$, stating the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph">On the same diagram as in part <strong>(i)</strong>, sketch the graph of $y = \mathrm{f}^{-1}(x)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-1-item-3" data-list-item-path="imported-item-block-1-item-3"><div class="list-item-paragraph">Write down the equation of the line in which the graph of $y = \mathrm{f}(x)$ must be reflected in order to obtain the graph of $y = \mathrm{f}^{-1}(x)$, and hence find the exact solution of the equation $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto 3 + \frac{2}{2x-1}$ for&nbsp;$x \in \mathbb{R}, x \neq \frac{1}{2}$.</p>
<ol class="i-list" start="5"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find the range of $\mathrm{gf}$.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto (x-4)^2 + 1$ for $x \in \mathbb{R}, x > 4$.

Find $\mathrm{f}^{-1}(x)$, stating the domain of $\mathrm{f}^{-1}$.

On the same diagram as in part **(i)**, sketch the graph of $y = \mathrm{f}^{-1}(x)$.

Write down the equation of the line in which the graph of $y = \mathrm{f}(x)$ must be reflected in order to obtain the graph of $y = \mathrm{f}^{-1}(x)$, and hence find the exact solution of the equation $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.

The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto 3 + \frac{2}{2x-1}$ for $x \in \mathbb{R}, x \neq \frac{1}{2}$.

Functions

<p>A function $\mathrm{f}$ is said to be self-inverse if&nbsp;$\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.</p>
<p>It is given that&nbsp;$\mathrm{g}$ is a self-inverse function and is defined by $\mathrm{g} : x \mapsto \frac{x+a}{3x+b}$, for&nbsp;$x \in \mathbb{R},\, x &gt; -\frac{b}{3}$,</p>
<p>where&nbsp;$a$ and $b$ are constants and $\mathrm{g}(1) = 5$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Show that&nbsp;$a = 9$ and $b = -1$.</div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">Find $\mathrm{g}^{2021}(1)$.</div></li></ol>
<p>The function $\mathrm{h}$ is defined by</p>
<p style="text-align: center">\[\mathrm{h} : x \mapsto \left\vert{} 1-x \right\vert{}(x+5),\text{ for}\hspace{0.5em} x \in \mathbb{R},\, x \ge 2.\]</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathrm{h}^{-1}$ exists, find $\mathrm{h}^{-1}$ in similar form.</div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{gh}$ exists and find the exact range of $\mathrm{gh}$.</div></li></ol>

A function $\mathrm{f}$ is said to be self-inverse if $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.

It is given that $\mathrm{g}$ is a self-inverse function and is defined by $\mathrm{g} : x \mapsto \frac{x+a}{3x+b}$, for $x \in \mathbb{R},\, x > -\frac{b}{3}$,

where $a$ and $b$ are constants and $\mathrm{g}(1) = 5$.

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

\[\mathrm{h} : x \mapsto \left\vert{} 1-x \right\vert{}(x+5),\text{ for } x \in \mathbb{R},\, x \ge 2.\]

Given that $\mathrm{h}^{-1}$ exists, find $\mathrm{h}^{-1}$ in similar form.

Show that $\mathrm{gh}$ exists and find the exact range of $\mathrm{gh}$.

Modulus Functions

<p>The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]</p>
<p>where&nbsp;$k$ is a constant.</p>
<p>Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Determine the possible value(s) of $k$ such that&nbsp;$l_1$ and $l_2$ are skew.<span class="inline-marks">[3]</span></div></li></ol>
<p>Assume that&nbsp;$k = 4$ for the rest of this question.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that&nbsp;$\overrightarrow{AB}$ is perpendicular to both lines. Show that&nbsp;$\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where&nbsp;$p$ is an integer to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given further that&nbsp;$l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.</p>
<ol class="i-list" start="5"><li data-list-item-id="imported-item-block-8-item-0" data-list-item-path="imported-item-block-8-item-0"><div class="list-item-paragraph">Deduce the shortest distance from $P$ to $l_1$.<span class="inline-marks">[2]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]

Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.

Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.

Determine the possible value(s) of $k$ such that $l_1$ and $l_2$ are skew.

Assume that $k = 4$ for the rest of this question.

Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that $\overrightarrow{AB}$ is perpendicular to both lines. Show that $\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where $p$ is an integer to be determined.

Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.

It is given further that $l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.

Deduce the shortest distance from $P$ to $l_1$.

2024 NJC P1 Q12

Vectors

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x &gt; 2.\]</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Find $\mathrm{f}^{-1}(x)$ and write down the domain of $\mathrm{f}^{-1}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}$ and $\mathrm{f}^{-1}$ on the same diagram.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-2" data-list-item-path="imported-item-block-2-item-2"><div class="list-item-paragraph">Explain why the $x$-coordinate of the point of intersection of the curves in part <strong>(b)</strong> satisfies the equation</div><div class="list-item-paragraph" style="text-align: center">\[x^2 - 5x + 5 = 0\]</div><div class="list-item-paragraph">and find the exact value of this $x$-coordinate.<span class="inline-marks">[3]</span></div></li></ol>

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

\[\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x > 2.\]

Find $\mathrm{f}^{-1}(x)$ and write down the domain of $\mathrm{f}^{-1}$.

Sketch the graph of $\mathrm{f}$ and $\mathrm{f}^{-1}$ on the same diagram.

<p>The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Find the position vector of the foot of perpendicular from $A$ to the line $l$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Show that the equation of $p_1$ is $2x + y = 1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Show that&nbsp;$L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Show that&nbsp;$a = 4$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-4-item-2" data-list-item-path="imported-item-block-4-item-2"><div class="list-item-paragraph">Find the angle between $l$ and $p_2$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another point $C$ lies in the plane $p_1$ such that&nbsp;$BC$ is perpendicular to the line $L$.</p>
<ol class="i-list" start="6"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Show that&nbsp;$\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.<span class="inline-marks">[4]</span></div></li></ol>

The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.

Show that the equation of $p_1$ is $2x + y = 1$.

The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.

Show that $L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.

Find the angle between $l$ and $p_2$.

Another point $C$ lies in the plane $p_1$ such that $BC$ is perpendicular to the line $L$.

Show that $\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.

<p>An isosceles triangle $ABC$, with $AB = AC$, is inscribed in a fixed circle of radius 1 unit and centre $O$. It is given that angle $BOC = 2\theta$, where&nbsp;$\theta$ is acute. Using calculus, show that the area of the triangle $ABC$ is a maximum when it is equilateral. State the maximum value of this area exactly.<span class="inline-marks">[8]</span></p>

An isosceles triangle $ABC$, with $AB = AC$, is inscribed in a fixed circle of radius 1 unit and centre $O$. It is given that angle $BOC = 2\theta$, where $\theta$ is acute. Using calculus, show that the area of the triangle $ABC$ is a maximum when it is equilateral. State the maximum value of this area exactly.

2024 NYJC P1 Q7

Applications of Differentiation

<p>The boundary of a wildlife reserve is quadrilateral $KLMN$.</p>
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<p>$KL=6.1\text{ km}$, $LM=8.4\text{ km}$, $MN=7.5\text{ km}$, $\angle LMN=68^\circ$ and $\angle KNL=36^\circ$.</p>
<p>The angle $\angle LKN$ is acute. Give non-exact numerical answers correct to $3$ significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q06-q-a" data-list-item-path="emath-aot-q06-q-a"><div class="list-item-paragraph">Find diagonal $LN$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q06-q-b" data-list-item-path="emath-aot-q06-q-b"><div class="list-item-paragraph">Find $\angle LKN$.<span class="inline-marks">[2]</span></div></li></ol>

The boundary of a wildlife reserve is quadrilateral $KLMN$.

$KL=6.1\text{ km}$, $LM=8.4\text{ km}$, $MN=7.5\text{ km}$, $\angle LMN=68^\circ$ and $\angle KNL=36^\circ$.

The angle $\angle LKN$ is acute. Give non-exact numerical answers correct to $3$ significant figures.

Applications of Trigonometry — General

Boundary of a wildlife reserve

Trigonometry

<p>A vertical lookout tower has base $B$ and top $T$. Observation posts $P$ and $Q$ lie on the same straight, level path from $B$.</p>
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<p>$P$ is between $B$ and $Q$, $PQ=35\text{ m}$, and the angles of elevation of $T$ from $P$ and $Q$ are $31^\circ$ and $18^\circ$, respectively.</p>
<p>Give non-exact numerical answers correct to $3$ significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q08-q-a" data-list-item-path="emath-aot-q08-q-a"><div class="list-item-paragraph">Find $BP$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q08-q-b" data-list-item-path="emath-aot-q08-q-b"><div class="list-item-paragraph">Find the height $BT$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q08-q-c" data-list-item-path="emath-aot-q08-q-c"><div class="list-item-paragraph">Find $QT$.<span class="inline-marks">[2]</span></div></li></ol>

A vertical lookout tower has base $B$ and top $T$. Observation posts $P$ and $Q$ lie on the same straight, level path from $B$.

$P$ is between $B$ and $Q$, $PQ=35\text{ m}$, and the angles of elevation of $T$ from $P$ and $Q$ are $31^\circ$ and $18^\circ$, respectively.

Give non-exact numerical answers correct to $3$ significant figures.

Hill lookout and observation posts

<p>A warehouse is $18\text{ m}$ long and $12\text{ m}$ wide, with vertical walls $6\text{ m}$ high.</p>
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<p>Its symmetric pitched roof has sloping edges $7.5\text{ m}$ long in each end cross-section. The ridge is directly above the centre line of the warehouse.</p>
<p>Give non-exact numerical answers correct to $3$ significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q11-q-a" data-list-item-path="emath-aot-q11-q-a"><div class="list-item-paragraph">Find the vertical height of the roof ridge above the ground.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q11-q-b" data-list-item-path="emath-aot-q11-q-b"><div class="list-item-paragraph">Find the distance from a bottom corner to the opposite top corner of the vertical walls.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q11-q-c" data-list-item-path="emath-aot-q11-q-c"><div class="list-item-paragraph">From the same bottom corner, find the angle of elevation of the ridge point at the opposite end.<span class="inline-marks">[3]</span></div></li></ol>

A warehouse is $18\text{ m}$ long and $12\text{ m}$ wide, with vertical walls $6\text{ m}$ high.

Its symmetric pitched roof has sloping edges $7.5\text{ m}$ long in each end cross-section. The ridge is directly above the centre line of the warehouse.

Warehouse crane beneath a roof

<p>The diagram shows a coastal reserve bounded by four survey markers $P$, $Q$, $R$ and $S$ on level ground.</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>
<p>The bearing of $P$ from $S$ is $342^\circ$ and $SP=176\text{ m}$. Marker $Q$ is $80\text{ m}$ due west of $P$.</p>
<p>The angle $PQR$ is $117^\circ$ and the bearing of $P$ from $R$ is $052^\circ$.</p>
<p>Give non-exact numerical answers correct to $3$ significant figures unless a different degree of accuracy is stated.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q-a" data-list-item-path="emath-aot-q-a"><div class="list-item-paragraph">Find the bearing of $R$ from $P$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-b" data-list-item-path="emath-aot-q-b"><div class="list-item-paragraph">Find $\angle QPR$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-c" data-list-item-path="emath-aot-q-c"><div class="list-item-paragraph">Show that the distance $PR$ is $168.66\text{ m}$, correct to two decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-d" data-list-item-path="emath-aot-q-d"><div class="list-item-paragraph">Find the area of triangle $PQR$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q-e" data-list-item-path="emath-aot-q-e"><div class="list-item-paragraph">Find $\angle RPS$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-f" data-list-item-path="emath-aot-q-f"><div class="list-item-paragraph">Hence, calculate the distance $SR$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-g" data-list-item-path="emath-aot-q-g"><div class="list-item-paragraph">An observation tower, $24\text{ m}$ high, stands vertically at $P$.</div><div class="list-item-paragraph">Find the angle of elevation of the top of the tower from $S$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a coastal reserve bounded by four survey markers $P$, $Q$, $R$ and $S$ on level ground.

The bearing of $P$ from $S$ is $342^\circ$ and $SP=176\text{ m}$. Marker $Q$ is $80\text{ m}$ due west of $P$.

The angle $PQR$ is $117^\circ$ and the bearing of $P$ from $R$ is $052^\circ$.

Give non-exact numerical answers correct to $3$ significant figures unless a different degree of accuracy is stated.

Surveying a coastal reserve

<p>The first four terms in a sequence are $1, 4, 9$ and $16$.</p>
<ol class="abc-list"><li data-list-item-id="item-1785756533528-2co5fuelw" data-list-item-path="item-1785756533528-2co5fuelw"><div class="list-item-paragraph"><p>Write down the $6$th term of this sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756534704-9z7mggxiy" data-list-item-path="item-1785756534704-9z7mggxiy"><div class="list-item-paragraph"><p>Find an expression, in terms of $n$, for the $n$ th, $T_n$, of the sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756630861-b8kpl7gee" data-list-item-path="item-1785756630861-b8kpl7gee"><div class="list-item-paragraph"><p>Is the number $99$ part of the sequence above? Explain your answer.</p><span class="inline-marks">[1]</span></div></li></ol>

The first four terms in a sequence are $1, 4, 9$ and $16$.

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