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A Math (IGCSE)

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H1 Math

H2 Math

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IP Math

<p>It is given that&nbsp;$3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve $3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $3p^3-7p^2-22p+8=0$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.

Remainder and Factor Theorems

Polynomials, Cubic Equations and Partial Fractions

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the equations of the tangent to the curve $y=x^2$ at the point $(t, t^2)$, expressing the coefficients in terms of $t$ where necessary.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equations of all lines that are tangents to both the curves $y=x^2$ and $y=-x^2+8x-16$.</span><span class="inline-marks">[4]</span></div></li></ol>

Find the equations of the tangent to the curve $y=x^2$ at the point $(t, t^2)$, expressing the coefficients in terms of $t$ where necessary.

Find the equations of all lines that are tangents to both the curves $y=x^2$ and $y=-x^2+8x-16$.

NJC IP4 Tangents and Normals Tutorial Q3

Applications of Differentiation

<p>It is given that the solution of the equation $x\sqrt{24} = x\sqrt{3} + \sqrt{6}$ is $\frac{a + \sqrt{b}}{7}$. <br>Find the values of the integers $a$ and $b$.</p>

It is given that the solution of the equation $x\sqrt{24} = x\sqrt{3} + \sqrt{6}$ is $\frac{a + \sqrt{b}}{7}$. 
Find the values of the integers $a$ and $b$.

Applications of Surds

Equation Involving Surds

Surds

<ol class="num-list" start="2"><li><div class="list-item-paragraph">The polynomial $P(x) = ax^3 + bx^2 - 19x + 4$, where&nbsp;$a$ and $b$ are constants, has a factor $x + 4$ and is such that&nbsp;$2P(1) = 5P(0)$.</div><ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $a$ and of $b$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence, factorise $P(x)$ completely.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

CGSS 4AM 2021 Prelims P1 Q2

<p>In the diagram, $\triangle ABC$ and $\triangle ACE$ are right-angled triangles. $AC = 12$ cm, $BC = 13$ cm, $EC = 2AB$ and $\angle BCE = x$ in degrees.</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="200" style="width:400px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact value of $\cos x$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, show that the length of $BD$ is $\frac{13\sqrt{11}}{12 + \sqrt{11}}$ cm.</span><span class="inline-marks">[2]</span></div></li></ol>

In the diagram, $\triangle ABC$ and $\triangle ACE$ are right-angled triangles. $AC = 12$ cm, $BC = 13$ cm, $EC = 2AB$ and $\angle BCE = x$ in degrees.

Hence, show that the length of $BD$ is $\frac{13\sqrt{11}}{12 + \sqrt{11}}$ cm.

Trigonometric Equations and Identities - General

NJC IP4 Further Trigonometric Identities Tutorial Q2

Trigonometric Equations and Identities

<p>The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Find the perimeter of the square.</div></li></ol>

The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.

Properties of Circle

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Prove that&nbsp;$\frac{\cos^3x + \sin x \cos^2x + \cos x \sin^2x + \sin^3x}{1 + \cos 2x + \sin 2x} = \frac{1}{2 \cos x}$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence or otherwise, find all the angles between $0{}^\circ$ and $360{}^\circ$ inclusive which satisfy the equation $\frac{\cos^3x + \sin x \cos^2x + \cos x \sin^2x + \sin^3x}{1 + \cos 2x + \sin 2x} = 5 \cos x$.</span><span class="inline-marks">[5]</span></div></li></ol>

Prove that $\frac{\cos^3x + \sin x \cos^2x + \cos x \sin^2x + \sin^3x}{1 + \cos 2x + \sin 2x} = \frac{1}{2 \cos x}$.

Hence or otherwise, find all the angles between $0{}^\circ$ and $360{}^\circ$ inclusive which satisfy the equation $\frac{\cos^3x + \sin x \cos^2x + \cos x \sin^2x + \sin^3x}{1 + \cos 2x + \sin 2x} = 5 \cos x$.

NJC IP4 Further Trigonometric Identities Tutorial Q1

<p>The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where&nbsp;$x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.</p>
<p>The graph of the function models the curved surface of the half-pipe.</p>
<ol class="i-list"><li data-list-item-id="item-1779863970301-8i7w7kccu"><div class="list-item-paragraph">Show that the equation of the curve can be written in the form $y =   \frac{1}{2}(x - 4)^2 + 1$.<br>Hence, state the coordinates of the turning point.</div></li><li data-list-item-id="item-1779864013798-i6efiat6a"><div class="list-item-paragraph">Explain what the turning point represents in the context of the half-pipe.</div></li><li data-list-item-id="item-1779864053174-je0mqtrit"><div class="list-item-paragraph">Sketch the graph of the quadratic function clearly showing:</div><ul class="bullet-list"><li data-list-item-id="subitem-1779864068279-6voek6d7l"><div class="list-item-paragraph">the turning point,</div></li><li data-list-item-id="subitem-1779864074574-hmuf5w09s"><div class="list-item-paragraph">the intercept with the y-axis,</div></li><li data-list-item-id="subitem-1779864074994-t5ucj8kvb"><div class="list-item-paragraph">the axis of symmetry.</div></li></ul></li><li data-list-item-id="item-1779864093967-t3qlykxyu"><div class="list-item-paragraph">If the right end of the half pipe has a height of $13.5$ m from the ground, find the width of the half pipe from one end to the other.</div></li></ol>

The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where $x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

Quadratic Expressions, Equations, and Functions

<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjQwMCwiaGVpZ2h0IjozNTAsImVsZW1lbnRzIjpbeyJpZCI6IjJjOWIxYzRhLWZlYTktNDA1Ny05MjlkLTQxZDg1NmJjMDYwZiIsInR5cGUiOiJmdW5jdGlvbmdyYXBoIiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJ5PWYoeCkiLCJleHByZXNzaW9uIjoiLTQqKHgrMileMiooeC0zKSIsImxhYmVsT2Zmc2V0IjpbMCwwXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjotMiwieSI6MCwiaWQiOiIyMThhOGY2My05ODczLTQ3MWItOWU5My0wZTM0MTM4M2FlN2UiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IigtMiwwKSIsInBvaW50U2l6ZSI6MSwibGFiZWxPZmZzZXQiOlswLC0xMF19LHsieCI6LTMsInkiOjI0LCJpZCI6IjUwYTNiNjY4LTE1YzYtNDI4OS1iZmU2LTgzYWM2ZjVjYzBkMiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiKC0zLDI0KSIsInBvaW50U2l6ZSI6MSwibGFiZWxPZmZzZXQiOls1LC0xMF19LHsieCI6MywieSI6MCwiaWQiOiJiYmI0ZTNkZi1jNDU3LTQyNzctOThkMS1kOGI5NmJiMjI0ZjMiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IigzLDApIiwicG9pbnRTaXplIjoxLCJsYWJlbE9mZnNldCI6WzUsLTEwXX0seyJ4IjoyLjUsInkiOjUwLCJpZCI6IjM1MDA5NDg3LThiYjMtNGQ1Yi04YTkxLWU5MWE0ODdhNjU4OSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJ5PWYoeCkifV0sInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6Wy01LDEwMCw1LC01MF0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsInNob3dBeGlzTGFiZWxzIjpmYWxzZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOmZhbHNlLCJ4QXhpc0xhYmVsT2Zmc2V0IjpbMiwtMTFdLCJ5QXhpc0xhYmVsT2Zmc2V0IjpbLTE0LDVdLCJvcmlnaW5MYWJlbE9mZnNldCI6Wy01LC0yXX0=" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<p>It is given that&nbsp;$\mathrm{f}(x)$ is a cubic function. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the factorised form.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find the value of $\mathrm{f}(1)$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}(x)$ is a cubic function. 

Graphs of Power Functions and Exponential Functions

Cubic Expressions and Equations

Functions and Graphs

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Given that&nbsp;$ \mathrm{f}(x) = \frac{2x + 3}{\sqrt{4x - 3}}$, find $ \mathrm{f}'(x)$.</div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the set of values of $x$ for&nbsp;$y = \ln\left(\frac{3 + 8x}{8x - 5}\right)$ to be a decreasing function.</div></li></ol>

2024 CCHY P1 Q4

Techniques of Differentiation

<p>If&nbsp;$\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.</p>

If $\frac{m^x}{n^y} \times \frac{n^{4-3x}}{(m^{y+1})^2} = \frac{n^7}{m^{10}}$, find the value(s) of $x$ and $y$.

Equality of Indices

Index Notation

Indices

Laws of Indices

Exponential and Logarithmic Functions

<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that&nbsp;$3x^2 - 5x + 21$ is always positive for all real values of $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The curve $y = ax^n$, where&nbsp;$a$ and $n$ are constants, passes through $(2, 48)$, $(3, 108)$ and $(k, 192)$. Find the values of $a$, $n$ and $k$ where&nbsp;$k &gt; 0$.<span class="inline-marks">[3]</span></div></li></ol>

Nature of Roots of Quadratic Equations

2024 CCHY P1 Q3

Quadratic Equations and Inequalities

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Integrate $\frac{3(1 - x)^{3} - 2}{1 - x}$ with respect to $x$.</span></div></li></ol>

Integrate $\frac{3(1 - x)^{3} - 2}{1 - x}$ with respect to $x$.

SCGS 4AM Prelim 2022

Integration Techniques

<p>Find $\frac{\mathrm{d}}{\mathrm{d}x}\left[(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) \dots (x^{1024}+1)\right]$</p>

Find $\frac{\mathrm{d}}{\mathrm{d}x}\left[(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) \dots (x^{1024}+1)\right]$

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

<p>Find the total area enclosed by the curve $y = (3x - 2)(x + 2)$ and the $x$-axis.</p>

Find the total area enclosed by the curve $y = (3x - 2)(x + 2)$ and the $x$-axis.

Area Enclosed by the Curve and the X-Axis

Definite Integrals and Area under graph

<ol class="num-list" start="2"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Differentiate $\frac{5x^2 + 3}{10x^2 - 6}$ with respect to $x$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Differentiate $\left( \frac{5x^2 + 3}{10x^2 - 6} \right)^4$ with respect to $x$, leaving your answer in the form $\frac{kx(5x^2 + 3)^m}{(5x^2 - 3)^n}$ where&nbsp;$k$, $m$ and $n$ are constants.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

XHS 2022 4Express AM MYE 

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<ol class="i-list"><li data-list-item-id="item-1781083736022-kfskn68ky"><div class="list-item-paragraph">Given that the line $y = 2x - k$ meets the curve $y = \frac{k}{4}x^2 - 2kx + 1$, find the range of values of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1781083737079-h01r9msxr"><div class="list-item-paragraph">A student claimed that when&nbsp;$k = -\frac{1}{2}$, the curve does not intersect the line. By showing your workings clearly, explain whether the statement is valid.<span class="inline-marks">[2]</span></div></li></ol>

Intersection of a Line and a Curve

2024 CCHY P1 Q1

<ol class="num-list" start="8"><li><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^2+1\right)e^{2x-1}\right]=2e^{2x-1}\left(x^2+x+1\right)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, show that&nbsp;$\int -2e^{2x-1}\left(x^2+x\right) \mathrm{d}x = -x^2e^{2x-1}+c$.</span><span class="inline-marks">[4]</span></div></li></ol></li></ol>

TKSS 4AM 2022 Prelims P2 Q8

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">If&nbsp;$a \cos^2 \theta + b \sin^2 \theta = c$, show that&nbsp;$\tan^2 \theta = \frac{a - c}{c - b}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-bi"><ol class="i-list"><li data-list-item-id="subitem-1781085512961-qeke46jc7"><div class="list-item-paragraph">Given that&nbsp;$\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x$, show that&nbsp;$6 \sin^2 x + \sin x - 2 = 0$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1781085513516-spykf9uxq"><div class="list-item-paragraph">Hence, solve the equation $\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x$ for&nbsp;$0 \le \theta \le 2\pi$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2024 CCHY P1 Q11

<p>A triangle has a base of $\left(\frac{1}{\sqrt{2}} - \frac{\sqrt{32}}{2} + \frac{16}{\sqrt{48}}\right)$ m and a height of $h$ m. Given that the area of the triangle is $\left(2\sqrt{2} - \sqrt{3}\right)$ m$^2$, find, without using calculator, the value of $h$ in the form $\frac{a\sqrt{6} + b}{5}$ where&nbsp;$a$ and $b$ are integers.<span class="inline-marks">[5]</span></p>

A triangle has a base of $\left(\frac{1}{\sqrt{2}} - \frac{\sqrt{32}}{2} + \frac{16}{\sqrt{48}}\right)$ m and a height of $h$ m. Given that the area of the triangle is $\left(2\sqrt{2} - \sqrt{3}\right)$ m$^2$, find, without using calculator, the value of $h$ in the form $\frac{a\sqrt{6} + b}{5}$ where $a$ and $b$ are integers.

A Math Question Bank