A Math

A Math (IGCSE)

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H1 Math

H2 Math

IB AA HL Math

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IB AI SL Math

IP Math

<p>Brand \(X\) and Brand \(Y\) batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is \(20\) metres.</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">When the toy car used Brand \(X\) battery, it completed the first lap in \(15\) seconds and each subsequent lap took \(3\) seconds longer than the previous lap.</div><ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0-sub-0" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-0"><div class="list-item-paragraph">In which lap did the toy car first have a lap timing of more than \(312.3\) seconds.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-0-sub-1" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-1"><div class="list-item-paragraph">Given that the toy car ran for&nbsp;\(23\) hours before stopping, find the average speed of the car during the last incomplete lap.<span class="inline-marks">[4]</span></div></li></ol></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">When the toy car used Brand \(Y\) battery, it completed the first lap in \(7.68\) seconds and the time taken for each subsequent lap was \(2.5\%\) more than the time for the previous lap. Given that the toy car stopped upon completing the \(230\text{th}\) lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.<span class="inline-marks">[2]</span></div></li></ol>
<p>In another test using Brand \(Y\) battery, the toy car was made to stop for&nbsp;\(5\) seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still \(7.68\) seconds. However, the time taken for the toy car to complete each subsequent lap was \(2.5\%\) more than the total time taken for the previous lap and the \(5\) seconds stop time. For example the time taken for the toy car to complete the second lap was \(1.025 \times (7.68 + 5)\) seconds.</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 the time taken for the toy car to complete the \(n\text{th}\) lap is \((212.68)(1.025)^{n-1} - 205\) seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.<span class="inline-marks">[4]</span></div></li></ol>

Brand \(X\) and Brand \(Y\) batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is \(20\) metres.

When the toy car used Brand \(X\) battery, it completed the first lap in \(15\) seconds and each subsequent lap took \(3\) seconds longer than the previous lap.

When the toy car used Brand \(Y\) battery, it completed the first lap in \(7.68\) seconds and the time taken for each subsequent lap was \(2.5\%\) more than the time for the previous lap. Given that the toy car stopped upon completing the \(230\text{th}\) lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.

In another test using Brand \(Y\) battery, the toy car was made to stop for \(5\) seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still \(7.68\) seconds. However, the time taken for the toy car to complete each subsequent lap was \(2.5\%\) more than the total time taken for the previous lap and the \(5\) seconds stop time. For example the time taken for the toy car to complete the second lap was \(1.025 \times (7.68 + 5)\) seconds.

Show that the time taken for the toy car to complete the \(n\text{th}\) lap is \((212.68)(1.025)^{n-1} - 205\) seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.

Problems involving inequalities

Geometric Progression

Financial Arithmetic or Iterative Pattern Recognition

Arithmetic Progression

2026 ACJC17 C1BT1 Q8

APGP

<p>With reference to the origin \(O\), the coordinates of \(A\), \(B\) and \(C\) are \((2,8,-a)\), \((1,-10,3)\) and \((5,6,a)\) respectively. The plane \(\pi\) contains points \(O\), \(A\), \(B\) and \(C\).</p>
<ol class="abc-list"><li data-list-item-id="ri-q11-l2-item01" data-list-item-path="ri-q11-l2-item01"><div class="list-item-paragraph">Show that&nbsp;\(a=1\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ri-q11-l2-item02" data-list-item-path="ri-q11-l2-item02"><div class="list-item-paragraph">Find the acute angle between \(\pi\) and the \(yz\)-plane.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item03" data-list-item-path="ri-q11-l2-item03"><div class="list-item-paragraph">\(\pi\) and the \(yz\)-plane meet in the line \(l\). By evaluating an appropriate cross product, find a vector equation for&nbsp;\(l\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item04" data-list-item-path="ri-q11-l2-item04"><div class="list-item-paragraph">Write down the distance from \(A\) to the \(yz\)-plane. Hence or otherwise, find the distance from \(A\) to \(l\).<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin \(O\), the coordinates of \(A\), \(B\) and \(C\) are \((2,8,-a)\), \((1,-10,3)\) and \((5,6,a)\) respectively. The plane \(\pi\) contains points \(O\), \(A\), \(B\) and \(C\).

Find the acute angle between \(\pi\) and the \(yz\)-plane.

\(\pi\) and the \(yz\)-plane meet in the line \(l\). By evaluating an appropriate cross product, find a vector equation for \(l\).

Write down the distance from \(A\) to the \(yz\)-plane. Hence or otherwise, find the distance from \(A\) to \(l\).

Length of Projection

Equation of Planes

Relationships Between Two Planes

2026 RI P1 Q11

Vectors

<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

<p>The curve \(C\) has equation</p>
<p>\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]</p>
<p>The line \(x=1\) intersects \(C\) at \(A(1,0)\) and at a second point \(B\).</p>
<ol class="i-list"><li data-list-item-id="b818afb7-41c3-4440-814f-cebde95ce4c2" data-list-item-path="b818afb7-41c3-4440-814f-cebde95ce4c2"><div class="list-item-paragraph">Using differentiation, find the exact coordinates of the point on \(C\) where&nbsp;\(x\) has its greatest value.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="35fabeb6-60ee-401c-9944-333d45b502e4" data-list-item-path="35fabeb6-60ee-401c-9944-333d45b502e4"><div class="list-item-paragraph">Sketch \(C\) and the line \(x=1\) on the same axes. State the equation of the asymptote of \(C\) and find the coordinates of \(B\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92" data-list-item-path="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92"><div class="list-item-paragraph">Find the area of the finite region enclosed by \(C\) and the line \(x=1\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="9a5b99e2-d202-43e6-9ba3-93603dc40d0d" data-list-item-path="9a5b99e2-d202-43e6-9ba3-93603dc40d0d"><div class="list-item-paragraph">Find the volume generated when this region is rotated through \(2\pi\) radians about the \(y\)-axis.<span class="inline-marks">[2]</span></div></li></ol>

\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]

The line \(x=1\) intersects \(C\) at \(A(1,0)\) and at a second point \(B\).

Using differentiation, find the exact coordinates of the point on \(C\) where \(x\) has its greatest value.

Sketch \(C\) and the line \(x=1\) on the same axes. State the equation of the asymptote of \(C\) and find the coordinates of \(B\).

Volume of Revolution about the y-axis

Area bounded by 2 curves

An enclosed region beside a sideways exponential curve

Applications of Integration

<p>The function \(\mathrm{f}\) is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]</p>
<ol class="abc-list"><li data-list-item-id="rf-33" data-list-item-path="rf-33"><div class="list-item-paragraph">Express \(\mathrm{f}(x)\) in the form \(R\cos(x+\alpha)\), where&nbsp;\(R&gt;0\) and \(0&lt;\alpha&lt;\frac{\pi}{2}\), giving the exact values of \(R\) and \(\alpha\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-34" data-list-item-path="rf-34"><div class="list-item-paragraph">Hence state the range of \(\mathrm{f}\) and sketch the graph of \(y=\mathrm{f}(x)\), indicating the coordinates of the turning points and the intersections with the coordinate axes.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function \(\mathrm{g}\) is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where&nbsp;\(b&gt;-\alpha\).</p>
<ol class="abc-list" start="3"><li data-list-item-id="rf-37" data-list-item-path="rf-37"><div class="list-item-paragraph">Find the largest value of \(b\) for which \(\mathrm{g}^{-1}\) exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-38" data-list-item-path="rf-38"><div class="list-item-paragraph">Using this value of \(b\), find \(\mathrm{g}^{-1}(x)\), stating its domain and range.<span class="inline-marks">[3]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]

Express \(\mathrm{f}(x)\) in the form \(R\cos(x+\alpha)\), where \(R>0\) and \(0<\alpha<\frac{\pi}{2}\), giving the exact values of \(R\) and \(\alpha\).

Hence state the range of \(\mathrm{f}\) and sketch the graph of \(y=\mathrm{f}(x)\), indicating the coordinates of the turning points and the intersections with the coordinate axes.

The function \(\mathrm{g}\) is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where \(b>-\alpha\).

Find the largest value of \(b\) for which \(\mathrm{g}^{-1}\) exists.

Using this value of \(b\), find \(\mathrm{g}^{-1}(x)\), stating its domain and range.

Inverse Functions

Range of a Function

Restricting a shifted cosine for an inverse

Functions

<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

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

<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 \(p\).

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