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PSLE

PSLE Foundation Mathematics

<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

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="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

<p>A metal cube is heated to a temperature of $205$ ${}^\circ\mathrm{C}$ before being dropped into a liquid. As the cube cools, its temperature, $T$ ${}^\circ\mathrm{C}$, $t$ minutes after it enters the liquid is given by $T = K + 175\mathrm{e}^{-mt}$, where&nbsp;$K$ and $m$ are constants.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Show that the value of $K$ is $30$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>When&nbsp;$t = 3$, the temperature of the cube reaches $128$ ${}^\circ\mathrm{C}$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Find the value of $m$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1772934012813-bhcrcd5uv" data-list-item-path="item-1772934012813-bhcrcd5uv"><div class="list-item-paragraph">Find the rate at which the temperature of the cube is decreasing at the instant when&nbsp;$t=8$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1772934227228-uph8ba7oq" data-list-item-path="item-1772934227228-uph8ba7oq"><div class="list-item-paragraph">Explain why the temperature of the cube can never fall below $30^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = (4 + p)x^2 - 2x + p$, where $p$ is a constant.

The curve has a maximum point and intersects the line $y = 2x - 1$.

Interpreting Quadratic Graphs

2025 SST PRELIMS P1 Q3

<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>The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.<br>Find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">the equation of the straight line in the form of $y = mx + c$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">the coordinates of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">the exact area of triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770806346981-qa5ypeh48l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDYzNDY5ODEtcWE1eXBlaDQ4bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODY3NzE2LCJleHAiOjE3OTI4NjgwMTZ9.DpivXzs-ijOap__XdrJsQ5WrGpERkmuzu9sNWtTafCg" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.
Find

Area of Rectilinear Figures

Equation of a Straight Line

IP3 Class Test (May 2025) Bonus Question

Coordinate Geometry

<p>Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.<span class="inline-marks">[4]</span></p>

Find the values of the constant $c$ for which the line $y=4x+c$ is a tangent to the curve $y={{x}^{2}}+cx+\frac{21}{4}$.

IP3 Class Test (May 2025) Q1

<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>Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.<span class="inline-marks">[5]</span></p>

Find the range of values of $m$ for which the line $y = mx - 5$ meets the curve $2y = x^2 + 10x - 9$ at two distinct points.

2023 CSS AM S3 EOY Q3

<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

Exponential and Logarithmic Functions

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}\left( x \right)=27{{x}^{3}}-18{{x}^{2}}+6nx+2n+3$, show that&nbsp;$x=-1$ is a solution to $\mathrm{f}\left( \frac{x}{3} \right)=0$ for all values of $n$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Show that for any real number $m$ the curve $y={{x}^{3}}+5{{x}^{2}}-m$ and the line $y=6-mx$ meet at the point where&nbsp;$x=1$.</div></li><li><div class="list-item-paragraph">Find the range of values of $m$ such that the curve $y={{x}^{3}}+5{{x}^{2}}-m$ and the line $y=6-mx$ meet only at the point $x=1$.</div></li></ol></li></ol>

Cubic Expressions and Equations

Remainder and Factor Theorems

Polynomials, Cubic Equations and Partial Fractions

<ol class="abc-list"><li data-list-item-id="item-1765999146208-p7fb3c4lh" data-list-item-path="item-1765999146208-p7fb3c4lh"><div class="list-item-paragraph">Calculate the values of $c$ for which $2{{x}^{2}}-3x+c$ is always greater than $-2$.</div></li><li data-list-item-id="item-1765999146792-6sq9w1io2" data-list-item-path="item-1765999146792-6sq9w1io2"><div class="list-item-paragraph">Find the range of values of $m$ for which ${{x}^{2}}+mx+3m$ is always greater than $8$ for all real values of $x$.</div></li></ol>

<ol class="abc-list"><li><div class="list-item-paragraph">Find the range of values of $k$ for which $5{{x}^{2}}-3x+k$ is always positive.</div></li><li><div class="list-item-paragraph">Find the range of values of $k$ for which $2{{x}^{2}}+{{k}^{2}}+2-2(k+2)x$ is always positive.</div></li></ol>

Quadratic Expression that is always positive

<ol class="abc-list"><li><div class="list-item-paragraph">Find the range of values of $p$ such that the equation ${{x}^{2}}-2x-px-\frac{1}{4}p=0$ has real roots.</div></li><li><div class="list-item-paragraph">Find the range of values of $k$ for which $(1-k){{x}^{2}}+9=4kx$ has real and distinct roots</div></li></ol>

Quadratic Inequality involving Roots of an Equation

<p>Show that the equation ${{x}^{2}}+\mathrm{(2}a-\mathrm{2)}x\ +\mathrm{(2}a-\mathrm{3)}\ =0$ has real roots for all real values of $a$.</p>

Show that the equation ${{x}^{2}}+\mathrm{(2}a-\mathrm{2)}x\ +\mathrm{(2}a-\mathrm{3)}\ =0$ has real roots for all real values of $a$.

Quadratic Equation with Real Roots

<p>The equation $(k+2)\,{{x}^{2}}+4k=(4k+2)\,x$ has no real roots.  Find the range of values of $k$.</p>

The equation $(k+2)\,{{x}^{2}}+4k=(4k+2)\,x$ has no real roots.  Find the range of values of $k$.

Quadratic Equation with No Real Roots

<p>Find the set of values of $k$ such that the equation ${{x}^{2}}+(k-2)x+(2k-4)=0$ has $2$ real and distinct roots.</p>

Find the set of values of $k$ such that the equation ${{x}^{2}}+(k-2)x+(2k-4)=0$ has $2$ real and distinct roots.

Quadratic Equation with Real and Distinct Roots

<p>Find the values of $m$ for which $6({{x}^{2}}+1)+m(2m-7x)=0$ has equal real roots.</p>

Find the values of $m$ for which $6({{x}^{2}}+1)+m(2m-7x)=0$ has equal real roots.

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