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<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>Find \(\int \frac{1}{\mathrm{e}^x + 1} \mathrm{d}x\).</p>

Find \(\int \frac{1}{\mathrm{e}^x + 1} \mathrm{d}x\).

Integration

<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto \sqrt{5 + 2x}\), for&nbsp;\(x \in \mathbb{R}\), \(x &gt; -\frac{5}{2}\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the value of \(a\) such that&nbsp;\(\mathrm{f}^2(a) = 5\).<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|\), for&nbsp;\(x \in \mathbb{R}\), \(x \le k\).</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the largest value of \(k\) for&nbsp;\(\mathrm{g}^{-1}\) to exist.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that&nbsp;\(k = -1\).</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find \(\mathrm{g}^{-1}(x)\) and state its domain.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Explain why the composite function \(\mathrm{fg}\) exists. State the exact range of \(\mathrm{fg}\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto \sqrt{5 + 2x}\), for \(x \in \mathbb{R}\), \(x > -\frac{5}{2}\).

Another function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|\), for \(x \in \mathbb{R}\), \(x \le k\).

2024 ASRJC Promo Q7

Functions

<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">Express \(\frac{1}{x^3 + x}\) as the sum of its partial fractions.</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Hence find \(\int \frac{1}{x^3 + x} \mathrm{d}x\)</span></div></li></ol>

Express \(\frac{1}{x^3 + x}\) as the sum of its partial fractions.

Hence find \(\int \frac{1}{x^3 + x} \mathrm{d}x\)

<p>Find \(\int e^x \sin(e^x) \mathrm{d}x\).</p>

<p>The diagram shows part of the graph \(y = x^{\cos 2x}\), for&nbsp;\(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.</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/1783573222728-q8fnjduy6xq.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzMyMjI3MjgtcThmbmpkdXk2eHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMjg5NywiZXhwIjoxNzg1NzEzMTk3fQ.HIZtb40_KoeF9Hy99xsxdasVSCHLRvQy1lyLJEhHim8" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">At the point on the graph where&nbsp;\(x = \pi\), find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of \(8\) units per hour.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle that the tangent to the graph where&nbsp;\(x = \pi\) makes with the horizontal.<span class="inline-marks">[1]</span></div></li></ol>

The diagram shows part of the graph \(y = x^{\cos 2x}\), for \(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.

2024 ACJC P2 Q1

Techniques of Differentiation

Applications of Differentiation

<p>Find \(\int \mathrm{e}^{\cos(t)} \sin(t) \mathrm{d}t\).</p>

Find \(\int \mathrm{e}^{\cos(t)} \sin(t) \mathrm{d}t\).

<p>The curve \(C\) has equation \(\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1\).</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch \(C\), stating the equations of the asymptotes and their point of intersection.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence state the exact range of values of \(m\) such that the straight line \(y=m(x+2)+3\) intersect \(C\).<span class="inline-marks">[2]</span></div></li></ol>

The curve \(C\) has equation \(\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1\).

Graphing Techniques

<p>In triangle \(PQR\), the vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) represent the vectors \(\overrightarrow{PQ}\), \(\overrightarrow{QR}\) and \(\overrightarrow{PR}\) respectively.</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/1779426366202-yu4qd1rzjpn.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0MjYzNjYyMDIteXU0cWQxcnpqcG4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMjg5NywiZXhwIjoxNzg1NzEzMTk3fQ.l73CZSXY4jiTm20OwUGPGWgt1tRRm8aAxx-h_9y-SsQ" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">State a geometrical interpretation of \(|\mathbf{c} \times \mathbf{\hat{a}}|\) and hence show that the area of the triangle \(PQR\) can be written as&nbsp;\(\frac{1}{2}|\mathbf{c} \times \mathbf{{a}}|\) units\(^2\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With clear explanations, show that&nbsp;\(|\mathbf{c} \times \mathbf{a}| = |\mathbf{b} \times \mathbf{a}|\) and deduce that&nbsp;\(\frac{|\mathbf{c}|}{\sin \angle PQR} = \frac{|\mathbf{b}|}{\sin \angle QPR}\) (Sine Rule).<span class="inline-marks">[2]</span></div></li></ol>

In triangle \(PQR\), the vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) represent the vectors \(\overrightarrow{PQ}\), \(\overrightarrow{QR}\) and \(\overrightarrow{PR}\) respectively.

Vector Algebra

2024 ASRJC Promo Q5

Vectors

<p>Find \(\int \frac{\sin(\ln(x))}{x} \mathrm{d}x\).</p>

Find \(\int \frac{\sin(\ln(x))}{x} \mathrm{d}x\).

Scatter Diagrams and Correlation

Correlation & Regressions

<p>Find \(\int \sec(2\theta) \tan(2\theta) \, \mathrm{d}\theta\).</p>

Find \(\int \sec(2\theta) \tan(2\theta) \, \mathrm{d}\theta\).

<p>An estimate is reliable when it is made under conditions that are well supported by the data. In regression, we should first check whether the value being used is within the range of the original data; this is called interpolation and is usually more reliable. </p>
<p>If the value is outside the data range, it is extrapolation, which is less reliable because the same pattern may not continue beyond the observed data. We should also check whether there is a strong linear correlation between the two variables, because regression estimates are only dependable when the data follows a clear linear trend. </p>
<p>So, a reliable estimate usually requires both conditions: the value is within the data range and the variables have a strong linear correlation.</p>
<p><strong>Interpolation</strong> - The process of carrying out estimation within the given data range.<br><strong>Extrapolation</strong> – The process of carrying out estimation beyond the given data range.</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/1775711500945-gfbv3ul3z96.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU3MTE1MDA5NDUtZ2ZidjN1bDN6OTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMjg5NywiZXhwIjoxNzg1NzEzMTk3fQ.996Hkl0f12enl7cnm4oxr3vk9k_Wliikfmjf9R-QVMI" style="max-width: min(600px, 100%); height: auto; display: block;" alt="Flowchart showing reliability of estimates based on interpolation and linear correlation" class="aligncenter custom-width" /></figure></div>

An estimate is reliable when it is made under conditions that are well supported by the data. In regression, we should first check whether the value being used is within the range of the original data; this is called interpolation and is usually more reliable. 

If the value is outside the data range, it is extrapolation, which is less reliable because the same pattern may not continue beyond the observed data. We should also check whether there is a strong linear correlation between the two variables, because regression estimates are only dependable when the data follows a clear linear trend. 

So, a reliable estimate usually requires both conditions: the value is within the data range and the variables have a strong linear correlation.

**Interpolation** - The process of carrying out estimation within the given data range.
**Extrapolation** – The process of carrying out estimation beyond the given data range.

Reliability of Estimates

<p>Differentiate each of the following expressions with respect to \(x\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(\cos^{-1}\left(3x^2\right)\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(\ln\left(\frac{\sqrt{2x+1}}{x^3}\right)\).<span class="inline-marks">[3]</span></div></li></ol>

Differentiate each of the following expressions with respect to \(x\).

2024 JPJC Promo Q1

<p>Find \(\int \frac{\mathrm{e}^x}{\mathrm{e}^x + 1} \mathrm{d}x\).</p>

Find \(\int \frac{\mathrm{e}^x}{\mathrm{e}^x + 1} \mathrm{d}x\).

<p>Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.</p>

Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.

Area Enclosed by the Curve and the X-Axis

<p>Find \(\int x^3 \sqrt{x^2 + 1} \mathrm{d}x\).</p>

Find \(\int x^3 \sqrt{x^2 + 1} \mathrm{d}x\).

<p>We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use \(x\) to estimate \(y\), the regression line of \(y\) on \(x\) is chosen because it makes the errors in the predicted \(y\)-values as small as possible. <br></p>
<p>On the other hand, when we use \(y\) to estimate \(x\), the regression line of \(x\) on \(y\) is used because it minimises the errors in the predicted \(x\)-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">Scenario</th><th style="text-align: center; --cell-justify: center">Given&nbsp;\(x\) to estimate \(y\)</th><th style="text-align: center; --cell-justify: center">Given&nbsp;\(y\) to estimate \(x\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y\) depends on \(x\)</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the \(y\) on \(x\) line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">\(x\) depends on \(y\)</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the \(x\) on \(y\) line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">No observable dependence</td><td style="text-align: center; --cell-justify: center">Use the \(y\) on \(x\) line equation.</td><td style="text-align: center; --cell-justify: center">Use the \(x\) on \(y\) line equation.</td></tr></tbody></table></div></div>

We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use \(x\) to estimate \(y\), the regression line of \(y\) on \(x\) is chosen because it makes the errors in the predicted \(y\)-values as small as possible. 


On the other hand, when we use \(y\) to estimate \(x\), the regression line of \(x\) on \(y\) is used because it minimises the errors in the predicted \(x\)-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.

Choice of Regression Lines

<p>In a school with \(1200\) students, an average of \(1\) out of \(20\) students is sick in a week.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the expected number of students who are sick in a week.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a class of \(25\) students, find the probability that at least \(2\) of them are sick in a week.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">In a sample of \(10\) classes with \(25\) students each, find the probability that less than five classes have at least \(2\) students who are sick in a week, given that there are at least two such classes.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the least value of \(n\) such that in a sample of \(n\) randomly selected students, the probability that at least \(6\) are sick in a week is more than \(0.95\).</span><span class="inline-marks">[3]</span></div></li></ol>

In a school with \(1200\) students, an average of \(1\) out of \(20\) students is sick in a week.

State, in this context, two assumptions needed for the number of students who are sick in a week to be well modelled by a binomial distribution. Explain why one of these assumptions may not be met.

For the rest of this question, assume that the number of students who are sick in a week has a binomial distribution.

Find the expected number of students who are sick in a week.

In a class of \(25\) students, find the probability that at least \(2\) of them are sick in a week.

In a sample of \(10\) classes with \(25\) students each, find the probability that less than five classes have at least \(2\) students who are sick in a week, given that there are at least two such classes.

Find the least value of \(n\) such that in a sample of \(n\) randomly selected students, the probability that at least \(6\) are sick in a week is more than \(0.95\).

H1 Math Question Bank