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<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

<p>A curve has equation $y=x^n\sqrt{a^2-x^2}$, where&nbsp;$a$ and $n$ are integer constants such that&nbsp;$a \ge 2$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$, $a$ and $n$, expressing your answer as a single fraction in a fully factorised form.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the curve has the same gradient value at both&nbsp;$x=1$ and $x=-1$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find a condition that&nbsp;$n$ must satisfy.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the curve has a gradient of $0$ at $x=\sqrt{2}$, find the values of $a$ and $n$. Justify your answers clearly.</span><span class="inline-marks">[4]</span></div></li></ol>

A curve has equation $y=x^n\sqrt{a^2-x^2}$, where $a$ and $n$ are integer constants such that $a \ge 2$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$, $a$ and $n$, expressing your answer as a single fraction in a fully factorised form.

It is given that the curve has the same gradient value at both $x=1$ and $x=-1$.

Find a condition that $n$ must satisfy.

Given that the curve has a gradient of $0$ at $x=\sqrt{2}$, find the values of $a$ and $n$. Justify your answers clearly.

NJC IP4 Techniques of Differentiation Tutorial Q3

Techniques of Differentiation

<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></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Prove the identity $\frac{\sin x}{1-\cos x}+\frac{1-\cos x}{\sin x}=2 \operatorname{cosec} x$<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence solve the equation $\frac{\sin x}{1-\cos x}+\frac{1-\cos x}{\sin x}=5 $ for&nbsp;$0\leq x\leq 2\pi$<span class="inline-marks">[3]</span></div></li></ol>

2025 MSHS PRELIMS P1 Q8

Trigonometric Equations and Identities

<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

<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

<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

<div class="graph-block" data-graph-config="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" 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

<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

<p>Find the values of the constant $p$ such that&nbsp;$y=px^2-4x+p-3$ is always positive for all real values of $x$.<span class="inline-marks">[5]</span></p>

Find the values of the constant $p$ such that $y=px^2-4x+p-3$ is always positive for all real values of $x$.

2025 MSHS PRELIMS P1 Q3

Quadratic Expressions, Equations, and Functions

<p>A curve has equation $y = \frac{x^{n+1}}{n+1} + \frac{a}{x}$ for some non-zero constants $a$ and $n$. The curve has a common gradient value of $\frac{5}{2}$ at both&nbsp;$x = \sqrt{2}$ and $x = \frac{1}{\sqrt{2}}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">By using the substitution $b = 2^{\frac{n}{2}}$, find the equation of the curve.</span><span class="inline-marks">[7]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By considering the expansion of $\left( x - \frac{1}{x} \right)^2$, show that the gradient of the curve at all points on the curve is at least $2$.</span><span class="inline-marks">[3]</span></div></li></ol>

A curve has equation $y = \frac{x^{n+1}}{n+1} + \frac{a}{x}$ for some non-zero constants $a$ and $n$. The curve has a common gradient value of $\frac{5}{2}$ at both $x = \sqrt{2}$ and $x = \frac{1}{\sqrt{2}}$.

By using the substitution $b = 2^{\frac{n}{2}}$, find the equation of the curve.

By considering the expansion of $\left( x - \frac{1}{x} \right)^2$, show that the gradient of the curve at all points on the curve is at least $2$.

NJC IP4 Techniques of Differentiation Tutorial Q1

<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>The equation of a curve is $y=x^3$. The distinct points $P$ and $Q$ lying on the curve have&nbsp;$x$-coordinates $p$ and $q$ respectively, where&nbsp;$p$ and $q$ are real constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the tangent to the curve at $Q$ passes through point $P$, show that\[p^2 + pq + cq^2 = 0\] for some constant $c$ to be determined. Hence, find $p$ in terms of $q$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.</span><span class="inline-marks">[4]</span></div></li></ol>

The equation of a curve is $y=x^3$. The distinct points $P$ and $Q$ lying on the curve have $x$-coordinates $p$ and $q$ respectively, where $p$ and $q$ are real constants.

Given that the tangent to the curve at $Q$ passes through point $P$, show that\n\[p^2 + pq + cq^2 = 0\]\nfor some constant $c$ to be determined. Hence, find $p$ in terms of $q$.

Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.

NJC IP4 Tangents and Normals Tutorial Q2

<p></p>
<p></p>
<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

<p>The equation of a curve is $y=\frac{ \ln x}{x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773466434665-oon3h3sb1"><div class="list-item-paragraph">Find the coordinates of the stationary point of this curve. <br>Leave your answer in terms of e.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773466436825-f0ta69f8n"><div class="list-item-paragraph">Determine the nature of the stationary point.<span class="inline-marks">[3]</span></div></li></ol>

The equation of a curve is $y=\frac{ \ln x}{x}$.

2025 MSHS PRELIMS P2 Q2

Coordinate Geometry

<p>The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a factor of $\mathrm{f}(x)$.</span></div></li></ol>
<p>It is further given that&nbsp;$\mathrm{f}'(a)=0$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">What can be deduced about the factorised form of $\mathrm{f}(x)$?</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.

Write down a factor of $\mathrm{f}(x)$.

It is further given that $\mathrm{f}'(a)=0$.

What can be deduced about the factorised form of $\mathrm{f}(x)$?

NJC IP4 Techniques of Differentiation Tutorial Q4

<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.

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