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E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by </p>
<p style="text-align: center">$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$</p>
<p>This formula for solving quadratic equations is usually used when the <strong>quadratic expression cannot be factorised easily. </strong></p>
<p style="margin-top: 1.5rem"><strong>Tip: </strong></p>
<p>Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.</p>

If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by 

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

This formula for solving quadratic equations is usually used when the **quadratic expression cannot be factorised easily. **

Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.

Solving Quadratic Equations Using the Quadratic Formula

Solving Quadratic Equations by using Formula

Quadratic Functions, Equations and Inequalities

<p>Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.</p>

Yan is a bit behind and he has 18 tasks to complete on the weekend. Each task takes one hour to complete. Yan knows he has one task in each of Mathematics, Chemistry and Physics. He does not want to work on two of these tasks consecutively. There are no restrictions on the order in which Yan does the other tasks. Find the number of ways Yan can complete his 18 tasks.

Arrangements with Objects Separated

Permutation and Combination

<ol class="abc-list"><li data-list-item-id="tkss-p2-q11-q-0-content-items-0" data-list-item-path="tkss-p2-q11-q-0-content-items-0"><div class="list-item-paragraph">Prove $\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="tkss-p2-q11-q-0-content-items-1" data-list-item-path="tkss-p2-q11-q-0-content-items-1"><div class="list-item-paragraph">Solve $3\sin x\tan x-\sec x=1$ for&nbsp;$-180^\circ\leq x\leq 180^\circ$.<span class="inline-marks">[6]</span></div></li></ol>

Prove $\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta$.

Solve $3\sin x\tan x-\sec x=1$ for $-180^\circ\leq x\leq 180^\circ$.

Trigonometric Equations

Compound and Double-Angle Identities

Proving and Simplifying Trigonometric Identities

2025 TKSS PRELIMS P2 Q11

Trigonometry

Find $\dfrac{d}{dx}\left(\dfrac{1}{2e^x+1}\right)$.

Hence find $\displaystyle \int \dfrac{e^x}{(2e^x+1)^2}\, dx$

Integration as an Inverse Process of Differentiation

Derivatives of Exponential Functions

Chain Rule

2025 TKSS PRELIMS P1 Q1

Differentiation

Integration

<p>The first three terms in the expansion of $(1+ax)^n$ are $1-3x+\frac{33}{8}x^2$, where&nbsp;$a$ and $n$ are constants. Find the value of $a$ and of $n$.<span class="inline-marks">[5]</span></p>

The first three terms in the expansion of $(1+ax)^n$ are $1-3x+\frac{33}{8}x^2$, where $a$ and $n$ are constants. Find the value of $a$ and of $n$.

Finding Parameters in Binomial Expansions

2025 NASS PRELIMS P1 Q4

Binomial Theorem

The number of cells in an experiment can be modelled by a function $N(t)=50000\left(1.6^{\frac{t}{2}}\right)$,

where $t$ is the number of hours that have passed from the start of the experiment.

Find the number of cells present at the start of the experiment.

Find the time taken for the number of cells to be ten times the initial number of cells. Give your answer to the nearest hour.

It is given that the number of cells at time $t=t_2$ is double the number of cells at the time $t=t_1$. Show that the difference between the two timings $(t_2-t_1)$ is approximately 2.95 hours, corrected to 3 significant figures.

Applications of Exponential and Logarithmic Functions

2025 NASS PRELIMS P1 Q9

Exponential and Logarithmic Functions

It is given that $\sin A=-\frac{1}{\sqrt{5}}$ and $\tan B=-\frac{3}{4}$, where $A$ and $B$ are in the same quadrant. Without using a calculator, find the exact value of

Given that $\sin A=q$, where $A$ is an obtuse angle, express $\tan^2 A$ in terms of $q$.

Symmetry of Trigonometric Functions

Exact Trigonometric Values and the Unit Circle

2025 NASS PRELIMS P1 Q6

<ol class="abc-list"><li data-list-item-id="jvss-p1-q7-q-0-content-items-0" data-list-item-path="jvss-p1-q7-q-0-content-items-0"><div class="list-item-paragraph">Express $\frac{5x^2+3x+7}{(x+1)(x-2)^2}$ in partial fractions.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="jvss-p1-q7-q-0-content-items-1" data-list-item-path="jvss-p1-q7-q-0-content-items-1"><div class="list-item-paragraph">Hence find $\int \frac{5x^2+3x+7}{(x+1)(x-2)^2}\, dx$.<span class="inline-marks">[3]</span></div></li></ol>

Express $\frac{5x^2+3x+7}{(x+1)(x-2)^2}$ in partial fractions.

Hence find $\int \frac{5x^2+3x+7}{(x+1)(x-2)^2}\, dx$.

Integration Using Partial Fractions

Partial Fractions

2025 JVSS PRELIMS P1 Q7

Polynomials, Cubic Equations and Partial Fractions

<p>Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.</p>
<p>A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.</p>
<ol class="abc-list"><li data-list-item-id="jvss-p1-q9-q-1-content-items-0" data-list-item-path="jvss-p1-q9-q-1-content-items-0"><div class="list-item-paragraph"><p>Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-1" data-list-item-path="jvss-p1-q9-q-1-content-items-1"><div class="list-item-paragraph"><p>Show that the $x$-coordinate of $P$ is $\mathrm{e}^{\frac{1}{6}}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-2" data-list-item-path="jvss-p1-q9-q-1-content-items-2"><div class="list-item-paragraph"><p>Find the equation of normal to the curve at the point $x=\mathrm{e}$. Leave your answer in terms of $\mathrm{e}$.</p><span class="inline-marks">[5]</span></div></li></ol>

Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.

A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.

Derivatives of Natural Logarithmic Functions

Tangents and Normals at a Point

Quotient Rule

Stationary Points and Their Nature

2025 JVSS PRELIMS P1 Q9

<p>Find the set of values of the constant $p$ for which the curve $y=px^2+(p-2)x+5$ lies entirely above the $x$-axis.<span class="inline-marks">[5]</span></p>

Find the set of values of the constant $p$ for which the curve $y=px^2+(p-2)x+5$ lies entirely above the $x$-axis.

Nature of Roots of Quadratic Equations

Quadratic Expressions Positive or Negative for All Real Values

2025 JVSS PRELIMS P1 Q2

Show that $x - 1$ is a factor of $3x^3 + 8x^2 - 5x - 6$.

Hence solve the equation $3x^3 + 8x^2 - 5x = 6$ completely.

Hence solve the equation $3(y - 2)^3 + 8(y - 2)^2 - 5y + 4 = 0$.

Cubic Expressions and Equations

Remainder and Factor Theorems

2025 CGS PRELIMS P2 Q2

<ol class="abc-list"><li data-list-item-id="cgs-p2-q1-q-0-content-items-0" data-list-item-path="cgs-p2-q1-q-0-content-items-0"><div class="list-item-paragraph">Show that the equation $3e^{2x} + 7e^x = 6$ has only one solution and find its value correct to 2 significant figures.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cgs-p2-q1-q-0-content-items-1" data-list-item-path="cgs-p2-q1-q-0-content-items-1"><div class="list-item-paragraph">Factorise $(3x + 1)^3 - 8$ completely.<span class="inline-marks">[2]</span></div></li></ol>

Show that the equation $3e^{2x} + 7e^x = 6$ has only one solution and find its value correct to 2 significant figures.

Solving Exponential Equations Using Substitution

Factorisation by Special Identities

2025 CGS PRELIMS P2 Q1

<p>Solve the equation $9 \sin x = -2 \sec x$ for&nbsp;$-\pi \le x \le 0$.<span class="inline-marks">[4]</span></p>

Solve the equation $9 \sin x = -2 \sec x$ for $-\pi \le x \le 0$.

2025 BVSS PRELIMS P1 Q2

<p>A particle starts from a fixed point $O$ with velocity $4\text{ m s}^{-1}$. For the first $3$ seconds its acceleration is $2t\text{ m s}^{-2}$. It then begins a second stage of motion; $T$ seconds after this stage begins, its acceleration is constant at $-5\text{ m s}^{-2}$.</p>
<ol class="i-list"><li data-list-item-id="amkin-q23-parts-item-1" data-list-item-path="amkin-q23-parts-item-1"><div class="list-item-paragraph">Find the velocity at the end of the first stage and the distance travelled during that stage.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amkin-q23-parts-item-2" data-list-item-path="amkin-q23-parts-item-2"><div class="list-item-paragraph">Find the time $T$ at which the particle first comes to rest.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amkin-q23-parts-item-3" data-list-item-path="amkin-q23-parts-item-3"><div class="list-item-paragraph">Find the total distance travelled from $O$ before it first comes to rest.<span class="inline-marks">[3]</span></div></li></ol>

A particle starts from a fixed point $O$ with velocity $4\text{ m s}^{-1}$. For the first $3$ seconds its acceleration is $2t\text{ m s}^{-2}$. It then begins a second stage of motion; $T$ seconds after this stage begins, its acceleration is constant at $-5\text{ m s}^{-2}$.

Find the velocity at the end of the first stage and the distance travelled during that stage.

Find the time $T$ at which the particle first comes to rest.

Find the total distance travelled from $O$ before it first comes to rest.

Velocity from Acceleration

Piecewise Motion and Changing Time Origins

Distance Travelled and Average Speed

Motion Conditions, Rest and Direction

Two-stage acceleration and braking

Kinematics

<p>A particle $P$ moves in a straight line from a fixed point $O$. Its displacement, $s$ metres, from $O$ at time $t$ seconds is given by $s=2t^3-9t^2+12t$, where&nbsp;$t\geq0$.</p>
<ol class="i-list"><li data-list-item-id="amkin-q01-parts-item-1" data-list-item-path="amkin-q01-parts-item-1"><div class="list-item-paragraph">Find the displacement of $P$ when&nbsp;$t=3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amkin-q01-parts-item-2" data-list-item-path="amkin-q01-parts-item-2"><div class="list-item-paragraph">Find the positive values of $t$ for which $P$ is instantaneously at rest.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amkin-q01-parts-item-3" data-list-item-path="amkin-q01-parts-item-3"><div class="list-item-paragraph">Find the range of values of $t$ for which the acceleration of $P$ is positive.<span class="inline-marks">[2]</span></div></li></ol>

A particle $P$ moves in a straight line from a fixed point $O$. Its displacement, $s$ metres, from $O$ at time $t$ seconds is given by $s=2t^3-9t^2+12t$, where $t\geq0$.

Find the displacement of $P$ when $t=3$.

Find the positive values of $t$ for which $P$ is instantaneously at rest.

Find the range of values of $t$ for which the acceleration of $P$ is positive.

Differentiating Displacement and Velocity

Cubic displacement and acceleration sign

<p>The curve $y=kx-x^2$, where&nbsp;$k&gt;0$, meets the positive $x$-axis at $A(4,\ 0)$.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q04-parts-item-1" data-list-item-path="amarea-q04-parts-item-1"><div class="list-item-paragraph">Find the value of $k$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amarea-q04-parts-item-2" data-list-item-path="amarea-q04-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the $x$-axis.<span class="inline-marks">[4]</span></div></li></ol>

The curve $y=kx-x^2$, where $k>0$, meets the positive $x$-axis at $A(4,\ 0)$.

Hence find the area enclosed by the curve and the $x$-axis.

Area Enclosed by the Curve and the X-Axis

Parameter from an intercept and the enclosed area

<p>The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinate of $P$ is $3$. The tangent meets the $y$-axis at $Q$, and the curve meets the positive $x$-axis at $R$.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q30-parts-item-1" data-list-item-path="amarea-q30-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $P$ and the equation of the tangent $PQ$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q30-parts-item-2" data-list-item-path="amarea-q30-parts-item-2"><div class="list-item-paragraph">Find the coordinates of $Q$ and $R$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q30-parts-item-3" data-list-item-path="amarea-q30-parts-item-3"><div class="list-item-paragraph">Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.<span class="inline-marks">[5]</span></div></li></ol>

The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinate of $P$ is $3$. The tangent meets the $y$-axis at $Q$, and the curve meets the positive $x$-axis at $R$.

Find the coordinates of $P$ and the equation of the tangent $PQ$.

Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.

Area Bounded by a Curve and Straight Lines

Rational curve with a tangent and the coordinate axes

<p>The curve $y=\dfrac{4}{\sqrt{x+1}}$ meets the line $y=2$ at $A$.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q14-parts-item-1" data-list-item-path="amarea-q14-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q14-parts-item-2" data-list-item-path="amarea-q14-parts-item-2"><div class="list-item-paragraph">Hence find the area of the shaded region bounded by the curve, the line and the $y$-axis.<span class="inline-marks">[4]</span></div></li></ol>

The curve $y=\dfrac{4}{\sqrt{x+1}}$ meets the line $y=2$ at $A$.

Hence find the area of the shaded region bounded by the curve, the line and the $y$-axis.

Radical curve above a horizontal line

<p>The points $P(0,\ 2)$ and $Q(\pi,\ 2)$ lie on the curve $y=2+2\sin x$.</p>
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<p>Find the exact area enclosed by the curve and the chord $PQ$.<span class="inline-marks">[4]</span></p>

The points $P(0,\ 2)$ and $Q(\pi,\ 2)$ lie on the curve $y=2+2\sin x$.

Find the exact area enclosed by the curve and the chord $PQ$.

Sine arch above a horizontal chord

<p>The curve $y=8+2x-x^2$ meets the line $y=8$ at the positive $x$-coordinate point $A$, and meets the positive $x$-axis at $B$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q10-parts-item-1" data-list-item-path="amarea-q10-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $A$ and $B$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q10-parts-item-2" data-list-item-path="amarea-q10-parts-item-2"><div class="list-item-paragraph">Hence find the area of the shaded region bounded by the curve, the line and the coordinate axes.<span class="inline-marks">[5]</span></div></li></ol>

The curve $y=8+2x-x^2$ meets the line $y=8$ at the positive $x$-coordinate point $A$, and meets the positive $x$-axis at $B$.

Hence find the area of the shaded region bounded by the curve, the line and the coordinate axes.

A Math Question Bank