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<p>Points $A$, $B$, $C$ and $D$ lie on a circle. The tangent to the circle at $A$ meets $BD$ produced at $E$. $AE$ is parallel to $BC$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1781502875383-n3numwirfas.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODE1MDI4NzUzODMtbjNudW13aXJmYXMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwODIxOSwiZXhwIjoxNzg1NzA4NTE5fQ.NNqhRTuhvq_e37v-lWnfl56dWZljOWioY8NTkNZAm24" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Prove that</p>
<ol class="i-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">$AB = AC$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">$\angle CDE = 2\angle ABC$.<span class="inline-marks">[3]</span></div></li></ol>

Points $A$, $B$, $C$ and $D$ lie on a circle. The tangent to the circle at $A$ meets $BD$ produced at $E$. $AE$ is parallel to $BC$.

2024 FMSS PRELIMS P1 Q3

Proofs in Plane Geometry

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Prove the identity $\ \operatorname{cosec} 4x + \cot 4x = \cot 2x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Hence, find, for&nbsp;$-180^\circ \le \theta \le 180^\circ$, the values of $\theta$ for which $\ \operatorname{cosec} 2\theta + \cot 2\theta = -\sqrt{3}$.<span class="inline-marks">[4]</span></div></li></ol>

Trigonometric Equations and Identities - General

2024 GESS PRELIMS P2 Q7

Trigonometry

<p>The polynomial $\mathrm{f}(x)$ is such that&nbsp;$\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where&nbsp;$a$ and $b$ are integers. It is given that&nbsp;$\mathrm{f}(x)$ is divisible by $2x - 3$ and that&nbsp;$\mathrm{f}'(1) = 6$.</p>
<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Find the values of $a$ and $b$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Using your values of $a$ and $b$, solve the equation $\mathrm{f}(x) = 0$.<span class="inline-marks">[3]</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ is such that $\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where $a$ and $b$ are integers. It is given that $\mathrm{f}(x)$ is divisible by $2x - 3$ and that $\mathrm{f}'(1) = 6$.

Applications of Differentiation — General

Remainder and Factor Theorems

2024 FMSS PRELIMS P1 Q7

Differentiation

Polynomials, Cubic Equations and Partial Fractions

<p>Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 &lt; 5x + 3$</p>

Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 < 5x + 3$

Quadratic Functions - General

Sketching Quadratic Graphs

Solving Quadratic Inequalities

Quadratic Functions, Equations and Inequalities

<p>Express $\frac{x + 7}{(x^2 - 9)(x - 3)}$ as the sum of $3$ partial fractions.<span class="inline-marks">[6]</span></p>

Express $\frac{x + 7}{(x^2 - 9)(x - 3)}$ as the sum of $3$ partial fractions.

Partial Fractions

2024 GESS PRELIMS P1 Q3

<p>Show that the equation $2(\mathrm{e}^x - 3) = \mathrm{e}^{\frac{1}{2}x}$ has only one solution and find this value correct to $3$ significant figures.<span class="inline-marks">[4]</span></p>

Show that the equation $2(\mathrm{e}^x - 3) = \mathrm{e}^{\frac{1}{2}x}$ has only one solution and find this value correct to $3$ significant figures.

Solving Exponential Equations Using Substitution

Quadratic Equations and Inequalities — General

Conversion between Exponential and Logarithm Form

2024 FMSS PRELIMS P2 Q1

Exponential and Logarithmic Functions

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that the solution of the equation $2^{3x+4} \times 5^{2x-1} = 16^x \times 5^{3x}$ is $\lg \frac{16}{5}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Express $2\log_2 x - \log_2(x - 4) = 3$ as a quadratic equation in $x$ and explain why there are no real solutions.<span class="inline-marks">[5]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q8

<p>A cubic polynomial satisfies $P(1) = P(2) = P(3) = 7$. Also $ P(0) = 1$<br>Find $ P(4)$</p>
<p></p>

A cubic polynomial satisfies $P(1) = P(2) = P(3) = 7$. Also $ P(0) = 1$
Find $ P(4)$

<p>Find the derivative of each of the following w.r.t $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>$y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} $</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>$y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) $</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>$ y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }$</p></div></li></ol>

Find the derivative of each of the following w.r.t $x$:

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

Techniques of Differentiation

<p>The diagram shows a prism in which its cross-section is a regular hexagon of side $ \left(5 + \sqrt{3} \right)$ cm.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782642566475-hoebx3c7nq.jpg?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI2NDI1NjY0NzUtaG9lYngzYzducS5qcGciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzA4MjE5LCJleHAiOjE3ODU3MDg1MTl9.a88nBpmiX0n8DnwbfYRkq8MnZ8n9EvDa_zfj02zNtHc" style="max-width: min(225px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find an expression for the cross-sectional area of the prism in the form $q\sqrt{3} + p$ cm$^2$, where&nbsp;$p$ and $q$ are integers.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Given that the volume of the prism is $3 \left(32\sqrt{3} + 138 \right)$ cm$^3$, find the height of the prism in the form $ \left(a\sqrt{3} + b \right)$ cm, where&nbsp;$a$ and $b$ are integers.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a prism in which its cross-section is a regular hexagon of side $ \left(5 + \sqrt{3} \right)$ cm.

2024 GESS PRELIMS P1 Q11

Surds

<p>Given that&nbsp;$\mathrm{f}(x) = x^2 - ax + 3$, where&nbsp;$a$ is a constant,</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">find the range of values of $a$ for which $f(x) &gt; x - 1$, for all real values of $x$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">find the value(s) of $a$ for which the line $y = a + 4$ is a tangent to the curve $y = f(x)$.<span class="inline-marks">[3]</span></div></li></ol>

Given that $\mathrm{f}(x) = x^2 - ax + 3$, where $a$ is a constant,

Quadratic Inequalities

Intersection of a Line and a Curve

Nature of Roots of Quadratic Equations

2024 FMSS PRELIMS P2 Q4

<p>Suppose $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4$, it is also given&nbsp;$P(5) = -19$.<br>Find $P(6)$.</p>

Suppose $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4$, it is also given $P(5) = -19$.
Find $P(6)$.

<p>The equation of a curve is $y = \frac{x}{e^{2x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Given that&nbsp;$x$ is changing at a constant rate of $0.18$ units per second, find the rate of change of $y$ when&nbsp;$x = 1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Explain if the curve has more than one stationary point.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = \frac{x}{e^{2x}}$.

2024 GESS PRELIMS P2 Q8

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Given that the expression $2x^3 + ax^2 + bx + c$ leaves the same remainder when divided by $(x - 5)$ and when divided by $(2x + 1)$, show that&nbsp;$9a + 2b = -91$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Given that the remainder is $-6$ when the expression $2x^3 + ax^2 + bx + c$ is divided by $(x - 1)$, and $(x + 1)$ is a factor, calculate the value of $c$.<span class="inline-marks">[4]</span></div></li></ol>

2024 GESS PRELIMS P2 Q1

<p>The equation of a curve is $y = ax^3 + b$, where&nbsp;$a$ and $b$ are constants. The equation of the normal to the curve at the point where&nbsp;$x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.<span class="inline-marks">[6]</span></p>

The equation of a curve is $y = ax^3 + b$, where $a$ and $b$ are constants. The equation of the normal to the curve at the point where $x = 1$ is $5y + 2x = 12$. Find the value of $a$ and of $b$.

Parallel and Perpendicular Lines

Equation of a Straight Line

2024 FMSS PRELIMS P2 Q2

Coordinate Geometry

<p>The equation of a curve is $y = -2x^2 + 3x + 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Write $y = -2x^2 + 3x + 5$ in the form $y = a(x - h)^2 + k$, where&nbsp;$a$, $h$ and $k$ are constants.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Using your results in part <strong>(a)</strong>, explain clearly why the maximum value of $y$ is $k$ when&nbsp;$x = h$.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = -2x^2 + 3x + 5$.

2024 GESS PRELIMS P1 Q1

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Write down the first three terms in the expansion, in descending powers of $x$, of $ \left(x^2 + \frac{m}{x} \right)^9$, where&nbsp;$m$ is an integer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Given that the coefficient of $x^3$ in the expansion of $ \left(x^2 + \frac{m}{x} \right)^9$ is $-126$,</div><ol class="i-list"><li data-list-item-id="subitem-1781507291750-bcufrqavo"><div class="list-item-paragraph">show that&nbsp;$m = -1$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1781507292113-kvd75ax8q"><div class="list-item-paragraph">hence, find the term independent of $x$ in the expression of $ \left(2 - \frac{1}{x^3} \right) \left(x^2 + \frac{m}{x} \right)^9$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2024 FMSS PRELIMS P2 Q7

Binomial Theorem

<p>Without expanding completely, prove that&nbsp;$ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.</p>

Without expanding completely, prove that $ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Solve the equation $15(16^x) - 4^x = 6$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Solve the equation $\lg(2y) + 2\ln(y) = 2$.<span class="inline-marks">[4]</span></div></li></ol>

2024 GESS PRELIMS P2 Q5

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find the term independent of $x$ in the expansion of $\left(2x^2 + \frac{3}{x}\right)^9$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><ol class="i-list"><li data-list-item-id="subitem-1781511615480-i4ty918y5" data-list-item-path="item-2/subitem-1781511615480-i4ty918y5"><div class="list-item-paragraph">Find the first three terms of the expansion of $(2 - 3x)^5$ in ascending powers of $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1781511646569-ijhb25enu" data-list-item-path="item-2/subitem-1781511646569-ijhb25enu"><div class="list-item-paragraph">Hence, find the value of $n$ such that the coefficient of $x$ in the expansion of $ (2 - 3x)^5(1 - 3x)^n$ is $-720$<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A Math (IGCSE) Question Bank