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<p>For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where&nbsp;\(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.<span class="inline-marks">[7]</span></p>

For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where \(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.

General Binomial Theorem

Recovering a rational exponent

Binomial Theorem

<p>Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.<span class="inline-marks">[7]</span></p>

Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.

Combining two validity conditions

<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.

A negative integer power

<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

<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 \[\frac{(x+1)(x-\sqrt3)}{x+2}&lt;0,\] giving your answer in exact form.<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, \[\frac{(\tan\theta+1)(\tan\theta-\sqrt3)}{\tan\theta+2}&lt;0,\qquad -\frac{\pi}{2}&lt;\theta&lt;\frac{\pi}{2}.\]<span class="inline-marks">[2]</span></div></li></ol>

Solve the inequality \[\frac{(x+1)(x-\sqrt3)}{x+2}<0,\] giving your answer in exact form.

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

Tangent 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>Solve the inequality \(\displaystyle \frac4{x-\lambda}&lt;x-\lambda\), where&nbsp;\(\lambda\) is a real constant. Give your answer in terms of \(\lambda\).<span class="inline-marks">[4]</span></p>

Solve the inequality \(\displaystyle \frac4{x-\lambda}<x-\lambda\), where \(\lambda\) is a real constant. Give your answer in terms of \(\lambda\).

2026 TJC P1 Q3

<p>The sum, \(S_n\), of the first \(n\) terms of a sequence \(u_1,u_2,u_3,\ldots\) is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q05-item01" data-list-item-path="nyjc-q05-item01"><div class="list-item-paragraph">Give a reason why the series \(\displaystyle \sum_{r=1}^{\infty}u_r\) converges and write down the value of the sum to infinity.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item02" data-list-item-path="nyjc-q05-item02"><div class="list-item-paragraph">Show that&nbsp;\(\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item03" data-list-item-path="nyjc-q05-item03"><div class="list-item-paragraph">It is given that&nbsp;\(\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r\). Find the value of \(k\).<span class="inline-marks">[3]</span></div></li></ol>

The sum, \(S_n\), of the first \(n\) terms of a sequence \(u_1,u_2,u_3,\ldots\) is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]

Give a reason why the series \(\displaystyle \sum_{r=1}^{\infty}u_r\) converges and write down the value of the sum to infinity.

Show that \(\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}\).

It is given that \(\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r\). Find the value of \(k\).

Change of Base

Sigma Notation

\(u_n = S_n - S_{n-1} \)

2026 NYJC P1 Q5

APGP

Recurrence Relations and Summation

<ol class="abc-list"><li data-list-item-id="nyjc-q02-item01" data-list-item-path="nyjc-q02-item01"><div class="list-item-paragraph">Without using a calculator, solve the inequality \(\displaystyle \frac5{x^2-9}\leq\frac{x+1}{x+3}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="nyjc-q02-item02" data-list-item-path="nyjc-q02-item02"><div class="list-item-paragraph">Hence, solve exactly the inequality \(\displaystyle \frac5{e^{2x}-9}\leq\frac{e^x+1}{e^x+3}\).<span class="inline-marks">[2]</span></div></li></ol>

Without using a calculator, solve the inequality \(\displaystyle \frac5{x^2-9}\leq\frac{x+1}{x+3}\).

Hence, solve exactly the inequality \(\displaystyle \frac5{e^{2x}-9}\leq\frac{e^x+1}{e^x+3}\).

2026 NYJC P1 Q2

<p>A curve \(C\) has parametric equations \[x=4t^2+1,\qquad y=2t.\]</p>
<p>The points \(P\) and \(Q\) on \(C\) have parameters \(p\) and \(3p\) respectively. The point \(M\) is the midpoint of \(PQ\).</p>
<ol class="abc-list"><li data-list-item-id="q2-a" data-list-item-path="q2-a"><div class="list-item-paragraph">Show that the coordinates of \(M\) are \((20p^2+1,4p)\), and hence find the Cartesian equation of the locus of \(M\) as&nbsp;\(p\) varies.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q2-b" data-list-item-path="q2-b"><div class="list-item-paragraph">The line \(L\) has equation \[x=15y-39.\] It intersects the locus of \(M\) at two points.</div><div class="list-item-paragraph">Find the corresponding values of \(p\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q2-c" data-list-item-path="q2-c"><div class="list-item-paragraph">For the smaller value of \(p\), denote the corresponding points on \(C\) by \(P_1\) and \(Q_1\). For the larger value, denote them by \(P_2\) and \(Q_2\).</div><div class="list-item-paragraph"><p>The four distinct points \(P_1,Q_1,P_2,Q_2\) form a quadrilateral. Find the exact area of the quadrilateral.</p><span class="inline-marks">[5]</span></div></li></ol>

A curve \(C\) has parametric equations \[x=4t^2+1,\qquad y=2t.\]

The points \(P\) and \(Q\) on \(C\) have parameters \(p\) and \(3p\) respectively. The point \(M\) is the midpoint of \(PQ\).

Show that the coordinates of \(M\) are \((20p^2+1,4p)\), and hence find the Cartesian equation of the locus of \(M\) as \(p\) varies.

Area from a parametric midpoint locus

Functions

<p>A curve \(C\) has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]</p>
<p>The points \(P\) and \(Q\) on \(C\) have parameters \(p\) and \(2p\) respectively, where&nbsp;\(p&gt;0\). The tangents to \(C\) at \(P\) and \(Q\) meet at \(R\).</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph">Show that the tangent to \(C\) at the point with parameter \(t\) has equation \[3tx-2y=t^3+3t+4.\]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph">Find the coordinates of \(R\) in terms of \(p\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph"><p>Show that the area of triangle \(PQR\) is \(\dfrac{5}{3}p^5\).</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-d" data-list-item-path="q1-d"><div class="list-item-paragraph">Given that the area of triangle \(PQR\) is \(\dfrac{160}{3}\), find the value of \(p\).<span class="inline-marks">[2]</span></div></li></ol>

A curve \(C\) has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]

The points \(P\) and \(Q\) on \(C\) have parameters \(p\) and \(2p\) respectively, where \(p>0\). The tangents to \(C\) at \(P\) and \(Q\) meet at \(R\).

Show that the tangent to \(C\) at the point with parameter \(t\) has equation \[3tx-2y=t^3+3t+4.\]

Find the coordinates of \(R\) in terms of \(p\).

Given that the area of triangle \(PQR\) is \(\dfrac{160}{3}\), find the value of \(p\).

Differentiation of Parametric Equations

Area enclosed by two parametric tangents

Techniques of Differentiation

<p>The sum of the first \(n\) terms, where&nbsp;\(n\geq1\), of a geometric series is given by \[S_n=\frac54\left[1-\left(-\frac13\right)^n\right].\]</p>
<ol class="abc-list"><li data-list-item-id="hci-q04-item01" data-list-item-path="hci-q04-item01"><div class="list-item-paragraph">Write down the value of the sum to infinity of the geometric series and find the least value of \(n\) such that the sum of the first \(n\) terms and the sum to infinity differ by less than \(10^{-3}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q04-item02" data-list-item-path="hci-q04-item02"><div class="list-item-paragraph">Given that&nbsp;\(\displaystyle u_k=k\sum_{n=1}^{k}S_n\), find an expression for&nbsp;\(u_k\) in terms of \(k\) where&nbsp;\(k\in\mathbb{Z}^+\).<span class="inline-marks">[4]</span></div></li></ol>

The sum of the first \(n\) terms, where \(n\geq1\), of a geometric series is given by \[S_n=\frac54\left[1-\left(-\frac13\right)^n\right].\]

Write down the value of the sum to infinity of the geometric series and find the least value of \(n\) such that the sum of the first \(n\) terms and the sum to infinity differ by less than \(10^{-3}\).

Given that \(\displaystyle u_k=k\sum_{n=1}^{k}S_n\), find an expression for \(u_k\) in terms of \(k\) where \(k\in\mathbb{Z}^+\).

Sum to infinity

2026 HCI P1 Q4

<ol class="abc-list"><li data-list-item-id="item-1789207760802-5uijxw870" data-list-item-path="item-1789207760802-5uijxw870"><div class="list-item-paragraph"><p>Without the use of a calculator, solve the inequality \(\displaystyle \frac{2x^2-8x+3}{x^2-4x-1}\geq1\).</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1789207770914-dqpk5u17t" data-list-item-path="item-1789207770914-dqpk5u17t"><div class="list-item-paragraph"><p>Deduce the solution of the inequality \(\displaystyle \frac{2x^2+8x+3}{x^2+4x-1}\geq1\).</p><span class="inline-marks">[2]</span></div></li></ol>

2026 HCI P1 Q2

<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

<ol class="i-list"><li data-list-item-id="asrjc-q02-item01" data-list-item-path="asrjc-q02-item01"><div class="list-item-paragraph">Using an algebraic method, solve the inequality \(\displaystyle \frac{2x^2+17x+37}{x^2+3x-10}\leq1\) exactly.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="asrjc-q02-item02" data-list-item-path="asrjc-q02-item02"><div class="list-item-paragraph">Hence, solve \(\displaystyle \frac{2x^2+17|x|+37}{x^2+3|x|-10}\leq1\).<span class="inline-marks">[2]</span></div></li></ol>

Using an algebraic method, solve the inequality \(\displaystyle \frac{2x^2+17x+37}{x^2+3x-10}\leq1\) exactly.

Hence, solve \(\displaystyle \frac{2x^2+17|x|+37}{x^2+3|x|-10}\leq1\).

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