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PSLE

PSLE Foundation Mathematics

<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 $x+10y=4$.

Form three linear equations in $a,b,c$, and hence find their values.

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.

Tangents and Normals at a Point

Increasing and Decreasing Functions

Quadratic Expressions Positive or Negative for All Real Values

A cubic determined by a tangent and a normal

System of Linear Equations

Inequalities

Applications of Differentiation

<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-0" data-list-item-path="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;$x^2 - x + 1 &gt; 0$ for all real values of $x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-1" data-list-item-path="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-1"><div class="list-item-paragraph"><p>Using the result in part <strong>(i)</strong>, solve the inequality $\frac{1}{x-1} + \frac{3}{x+1} \leq 2 - 2x$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-2" data-list-item-path="h2-2025-sajc-promo-q2-sajc-q2-item-block-0-item-2"><div class="list-item-paragraph">Hence, solve the inequality $\frac{1}{\ln x - 1} + \frac{3}{\ln x + 1} \leq 2 - \ln x^2$, giving your answer in exact form.<span class="inline-marks">[2]</span></div></li></ol>

Show that $x^2 - x + 1 > 0$ for all real values of $x$.

Hence, solve the inequality $\frac{1}{\ln x - 1} + \frac{3}{\ln x + 1} \leq 2 - \ln x^2$, giving your answer in exact form.

Inequalities involving Substitution

Inequalities involving Rational Functions

2025 SAJC Promo Q2

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-0" data-list-item-path="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$x^2-x+3$ is always positive for all real values of $x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-1" data-list-item-path="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-1"><div class="list-item-paragraph"><p>Solve the inequality,</p></div><div class="list-item-paragraph"><p style="text-align: center;">$\frac{x\left(x^2-x+3\right)}{10-x-3x^2}\leq 0.$</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-2" data-list-item-path="h2-2025-rvhs-promo-q5-rvhs-q5-item-block-1-item-2"><div class="list-item-paragraph"><p>Hence solve the inequality,</p></div><div class="list-item-paragraph"><p style="text-align: center;">$\frac{|x|\left(x^2-|x|+3\right)}{10-|x|-3x^2}\leq 0.$</p><span class="inline-marks">[3]</span></div></li></ol>

**Do not use a calculator in answering this question.**

Show that $x^2-x+3$ is always positive for all real values of $x$.

Inequalities involving Modulus Functions

2025 RVHS Promo Q5

<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

2025 JVSS PRELIMS P1 Q2

Quadratic Functions, Equations and Inequalities

<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,

Intersection of a Line and a Curve

2024 FMSS PRELIMS P2 Q4

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">The curve $y = mx^2 - 12x + 2(x^2 + m - 1)$ lies entirely below the $x$-axis for all real values of $x$. Find the largest integer value of $m$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Show that the roots of the equation $2x^2 - 3(1 - x) = -p$ are real if&nbsp;$p \le 4\frac{1}{8}$.<span class="inline-marks">[4]</span></div></li></ol>

2024 CGSS P2 Q1

<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>

Equality of Indices

2024 CCHY P1 Q3

Exponential and Logarithmic Functions

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;$x^2 + 4x + 9$ is always positive for all real values of $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, without using a calculator, solve $\frac{(x^2 + 4x + 9)(x + 2)^2}{x^2 - 6x + 7} \le 0$.<span class="inline-marks">[3]</span></div></li></ol>

2024 NYJC Promo Q2

<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 range of 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

<p>Given the curve $y = kx^2 + 2kx - 3$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">By expressing $y$ in the form $k(x+b)^2 - 1$, determine the value of $k$ and of $b$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-content"><span class="list-item-text">Hence, show that the curve is always below the $x$-axis.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1772957050027-bldwukdfi"><div class="list-item-paragraph">State the turning point of the curve and determine the nature of this point.<span class="inline-marks">[2]</span></div></li></ol>

Given the curve $y=kx^2+2kx-3$, where $k$ is a constant.

By expressing $y$ in the form $k(x+b)^2-1$, determine the value of $k$ and of $b$.

Hence, show that the curve is always below the $x$-axis.

State the turning point of the curve and determine the nature of this point.

Completing the Square and Vertex Form

Maximum and Minimum Values of General Quadratic Functions

2025 AHS PRELIMS P1 Q5

<ol class="abc-list"><li data-list-item-id="item-a-7"><div class="list-item-content"><span class="list-item-text">Find the range of values of $p$ for which $y = x^2 - x - 2$ lies entirely above the line $y = p(x + 2)$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1772956600479-dlgb1j5s6"><div class="list-item-paragraph">Hence, deduce, without finding the discriminant, the number of intersection points when&nbsp;$p = 3$.</div></li></ol>

Find the range of values of $p$ for which $y=x^2-x-2$ lies entirely above the line $y=p(x+2)$.

Hence, deduce, without finding the discriminant, the number of intersection points when $p=3$.

2025 AHS PRELIMS P1 Q1

<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph">A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h = 7 + 20t - 5t^2$.<br>Find the maximum height attained by the ball and the time at which this occurs.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-content"><span class="list-item-text">Find the set of values of the constant $k$ for which the curve $y = x^2 + 6x + k$ lies entirely above the line $y = k(x - 1)$.</span><span class="inline-marks">[4]</span></div></li></ol>

A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h=7+20t-5t^2$. Find the maximum height attained by the ball and the time at which this occurs.

Find the set of values of the constant $k$ for which the curve $y=x^2+6x+k$ lies entirely above the line $y=k(x-1)$.

Applications of Quadratic Functions and Equations

2025 DSS PRELIMS P2 Q1

<p>Explain why $3x^2 + 15x + 20$ cannot be smaller than $1$.<span class="inline-marks">[3]</span></p>

Explain why $3x^2+15x+20$ cannot be smaller than 1.

2025 DSS PRELIMS P1 Q3

<p>Find the range of values of the constant $m$ for which the curve $y=(m-3)x^2 + mx + m$ lies completely above the $x$-axis.<span class="inline-marks">[5]</span></p>

Find the range of values of the constant $m$ for which the curve $y=(m-3)x^2+mx+m$ lies completely above the $x$-axis.

2025 CGSS PRELIMS P1 Q2

<p>The equation of a curve is $y = (m+1)x^2 - 8x + 3m$, where&nbsp;$m$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">In the case where&nbsp;$m = 1$, show that the line $y = 1 - 4x$ is a tangent to the curve. Find the coordinates of the point of contact.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Find the range of values of $m$ for which the curve is above the line $y = 5$.</span><span class="inline-marks">[5]</span></div></li></ol>

The equation of a curve is $y=(m+1)x^2-8x+3m$, where $m$ is a constant.

In the case where $m=1$, show that the line $y=1-4x$ is a tangent to the curve. Find the coordinates of the point of contact.

Find the range of values of $m$ for which the curve is above the line $y=5$.

2025 SCGS PRELIMS P1 Q5

<ol class="i-list"><li data-list-item-id="item-1772341626320-o5rhkfdq5" data-list-item-path="item-1772341626320-o5rhkfdq5"><div class="list-item-paragraph">Given that the curve $y = ax^2 + 2bx - 1$ lies entirely below the $x$-axis, determine the conditions that must be applied to the constants $a$ and $b$.</div></li><li data-list-item-id="item-1772341626891-9ovz55gkt" data-list-item-path="item-1772341626891-9ovz55gkt"><div class="list-item-paragraph">If&nbsp;$a$ and $b$ are both integers, state an example of the values of $a$ and $b$ which satisfy the conditions found in (i).</div></li></ol>

Given that the curve $y=ax^2+2bx-1$ lies entirely below the $x$-axis, determine the conditions that must be applied to the constants $a$ and $b$.

If $a$ and $b$ are both integers, state an example of the values of $a$ and $b$ which satisfy the conditions found in (i).

2025 CHS PRELIMS P2 Q2

<p>Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">It is given that the expression $b^2 - 4ac$ is called the <strong>discriminant</strong> of the equation $ax^2 + bx + c = 0$.</div><div class="list-item-paragraph">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the discriminant.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the relationship between the number of distinct <strong>real roots</strong> of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely above the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ is completely below the $x$-axis,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the graph of $y = ax^2 + bx + c$ and the $x$-axis have at least one common point.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>Note that&nbsp;$a \neq 0$ throughout this question.</p>

Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$.

Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.

State the relationship between the number of distinct **real roots** of the equation $ax^2 + bx + c = 0$ and the number of points of intersections between the graph of $y = ax^2 + bx + c$ and the $x$-axis.

Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that

Note that $a \neq 0$ throughout this question.

Solving Quadratic Equations Using the Quadratic Formula

IP2 Class Test (May 2025) Bonus Question

<p>Given that&nbsp;$x$ is real, prove that&nbsp;$2{{x}^{2}}-x+2$ is always positive.				<span class="inline-marks">[1]</span></p>
<p>Solve the inequality $\frac{3x-1}{2{{x}^{2}}-x+2}\ge \frac{1}{x}$, leaving your answers in exact form.					<span class="inline-marks">[3]</span></p>
<p>Hence, without the use of a calculator, find the set of values of $x$ for which $\frac{3\left| x \right|-1}{2{{x}^{2}}-\left| x \right|+2}\ge \frac{1}{\left| x \right|}$.	<span class="inline-marks">[2]</span></p>

Given that $x$ is real, prove that $2{{x}^{2}}-x+2$ is always positive.				

Solve the inequality $\frac{3x-1}{2{{x}^{2}}-x+2}\ge \frac{1}{x}$, leaving your answers in exact form.					

Hence, without the use of a calculator, find the set of values of $x$ for which $\frac{3\left| x \right|-1}{2{{x}^{2}}-\left| x \right|+2}\ge \frac{1}{\left| x \right|}$.	

2013 NJC P1 Q2

Transformations of Graphs

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The curve with equation $y = ax^2 + bx + a$, where $a$ and $b$ are constants, lies completely above the $x$-axis.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the conditions which must apply to $a$ and $b$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Give an example of possible values for $a$ and $b$ which satisfy the conditions in part <strong>(i)</strong>.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Using a suitable substitution, explain why the equation $3^{2x-1} = 3^{x+2} - k$ has no solution if $k &gt; 60.75$.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve with equation $y = ax^2 + bx + a$, where $a$ and $b$ are constants, lies completely above the $x$-axis.

Using a suitable substitution, explain why the equation $3^{2x-1} = 3^{x+2} - k$ has no solution if $k > 60.75$.

Solving Exponential Equations Using Substitution

2023 Ahmad Ibrahim Sec 3 AM EOY Q6

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the graph of $y = -4x^2 + 4kx - (k^2 + 1)$ lies entirely below the $x$-axis.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $k$ for which the line $y = kx + 2$ meets the curve $x^2 + y^2 = 2$.</span><span class="inline-marks">[4]</span></div></li></ol>

Explain why the graph of $y = -4x^2 + 4kx - (k^2 + 1)$ lies entirely below the $x$-axis.

Find the range of values of $k$ for which the line $y = kx + 2$ meets the curve $x^2 + y^2 = 2$.

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