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PSLE Foundation Mathematics

<p>The curve \(y=3\cos\left(x-\dfrac{\pi}{3}\right)\) meets the \(y\)-axis at \(P\) and first meets the positive \(x\)-axis at \(Q(b,\ 0)\). The chord \(PQ\) is drawn.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q19-parts-item-1" data-list-item-path="amarea-q19-parts-item-1"><div class="list-item-paragraph">Find the exact value of \(b\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q19-parts-item-2" data-list-item-path="amarea-q19-parts-item-2"><div class="list-item-paragraph">Find the equation of the chord \(PQ\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q19-parts-item-3" data-list-item-path="amarea-q19-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve and the chord.<span class="inline-marks">[5]</span></div></li></ol>

The curve \(y=3\cos\left(x-\dfrac{\pi}{3}\right)\) meets the \(y\)-axis at \(P\) and first meets the positive \(x\)-axis at \(Q(b,\ 0)\). The chord \(PQ\) is drawn.

Hence find the exact area enclosed by the curve and the chord.

Area Bounded by a Curve and Straight Lines

Equation of a Straight Line

Shifted cosine curve and its chord

Integration

Coordinate Geometry

<p>The tangent to the curve \(y=2\sin x\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{3}\), meets the \(y\)-axis at \(B\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q22-parts-item-1" data-list-item-path="amarea-q22-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at \(A\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q22-parts-item-2" data-list-item-path="amarea-q22-parts-item-2"><div class="list-item-paragraph">Find the coordinates of \(B\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amarea-q22-parts-item-3" data-list-item-path="amarea-q22-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve, the tangent and the \(y\)-axis.<span class="inline-marks">[5]</span></div></li></ol>

The tangent to the curve \(y=2\sin x\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{3}\), meets the \(y\)-axis at \(B\).

Find the equation of the tangent at \(A\).

Hence find the exact area enclosed by the curve, the tangent and the \(y\)-axis.

Tangents and Normals at a Point

Sine curve and its tangent beside the y-axis

Differentiation

<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 diagram shows the curve \(y=x^2\) and the line \(y=4\), which meet at \(A\) in the first quadrant.</p>
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<p>Find the area of the shaded region bounded by the curve, the line and the \(y\)-axis.<span class="inline-marks">[4]</span></p>

The diagram shows the curve \(y=x^2\) and the line \(y=4\), which meet at \(A\) in the first quadrant.

Find the area of the shaded region bounded by the curve, the line and the \(y\)-axis.

Region between a parabola and a horizontal line

<p>The curve \(y=\dfrac{12}{x^2}\) meets the line \(y=3\) at \(A\), where the \(x\)-coordinate of \(A\) is \(p&gt;0\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q29-parts-item-1" data-list-item-path="amarea-q29-parts-item-1"><div class="list-item-paragraph">Find the value of \(p\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q29-parts-item-2" data-list-item-path="amarea-q29-parts-item-2"><div class="list-item-paragraph">Hence find the area bounded by the curve, the line and the line \(x=1\).<span class="inline-marks">[5]</span></div></li></ol>

The curve \(y=\dfrac{12}{x^2}\) meets the line \(y=3\) at \(A\), where the \(x\)-coordinate of \(A\) is \(p>0\).

Hence find the area bounded by the curve, the line and the line \(x=1\).

Reciprocal-square curve above a horizontal line

<p>The curve \(y=5-\mathrm{e}^{x}\) has a tangent at \(A\), where the \(x\)-coordinate of \(A\) is \(\ln2\). The tangent meets the \(y\)-axis at \(B\), and the curve meets the \(x\)-axis at \(C\).</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-q16-parts-item-1" data-list-item-path="amarea-q16-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at \(A\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q16-parts-item-2" data-list-item-path="amarea-q16-parts-item-2"><div class="list-item-paragraph">Find the coordinates of \(B\) and \(C\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q16-parts-item-3" data-list-item-path="amarea-q16-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 curve \(y=5-\mathrm{e}^{x}\) has a tangent at \(A\), where the \(x\)-coordinate of \(A\) is \(\ln2\). The tangent meets the \(y\)-axis at \(B\), and the curve meets the \(x\)-axis at \(C\).

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

Exponential curve with a tangent and the coordinate axes

<p>The curve \(y=\tan x\) meets the line \(y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{6}\). The line meets the positive \(x\)-axis at \(B\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q21-parts-item-1" data-list-item-path="amarea-q21-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q21-parts-item-2" data-list-item-path="amarea-q21-parts-item-2"><div class="list-item-paragraph">Find the exact area of the shaded region bounded by the curve, the line and the \(x\)-axis.<span class="inline-marks">[5]</span></div></li></ol>

The curve \(y=\tan x\) meets the line \(y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}\) at \(A\), where the \(x\)-coordinate of \(A\) is \(\dfrac{\pi}{6}\). The line meets the positive \(x\)-axis at \(B\).

Find the exact area of the shaded region bounded by the curve, the line and the \(x\)-axis.

Tangent curve and a descending line

<p>The points \(A\) and \(B\) lie on the curve \(y=8+2x-x^2\), where the \(x\)-coordinates of \(A\) and \(B\) are \(-1\) and \(3\), respectively.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q09-parts-item-1" data-list-item-path="amarea-q09-parts-item-1"><div class="list-item-paragraph">Find the equation of the chord \(AB\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q09-parts-item-2" data-list-item-path="amarea-q09-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the chord.<span class="inline-marks">[4]</span></div></li></ol>

The points \(A\) and \(B\) lie on the curve \(y=8+2x-x^2\), where the \(x\)-coordinates of \(A\) and \(B\) are \(-1\) and \(3\), respectively.

Hence find the area enclosed by the curve and the chord.

Area between a parabola and a chord

<p>The curve \(y=(x-1)^2\) intersects the line \(y=3-x\) at the points \(P\) and \(Q\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q07-parts-item-1" data-list-item-path="amarea-q07-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(P\) and \(Q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q07-parts-item-2" data-list-item-path="amarea-q07-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the line.<span class="inline-marks">[4]</span></div></li></ol>

The curve \(y=(x-1)^2\) intersects the line \(y=3-x\) at the points \(P\) and \(Q\).

Hence find the area enclosed by the curve and the line.

Intersection of a Line and a Curve

Shifted parabola below a descending line

Quadratic Functions, Equations and Inequalities

<p>The tangent to the curve \(y=x^3\) at \(P(1,\ 1)\) meets the curve again at \(Q\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q08-parts-item-1" data-list-item-path="amarea-q08-parts-item-1"><div class="list-item-paragraph">Find the equation of the tangent at \(P\) and the coordinates of \(Q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q08-parts-item-2" data-list-item-path="amarea-q08-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the tangent.<span class="inline-marks">[4]</span></div></li></ol>

The tangent to the curve \(y=x^3\) at \(P(1,\ 1)\) meets the curve again at \(Q\).

Find the equation of the tangent at \(P\) and the coordinates of \(Q\).

Hence find the area enclosed by the curve and the tangent.

Cubic curve and its tangent

<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>
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<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.

Region with a changing upper boundary

<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 \(x=(y-2)^2\) intersects the line \(x=4-y\) at the points \(P\) and \(Q\).</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-q25-parts-item-1" data-list-item-path="amarea-q25-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(P\) and \(Q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q25-parts-item-2" data-list-item-path="amarea-q25-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the line.<span class="inline-marks">[5]</span></div></li></ol>

The curve \(x=(y-2)^2\) intersects the line \(x=4-y\) at the points \(P\) and \(Q\).

Area Enclosed by the Curve and the Y-Axis

Sideways parabola and an oblique line

<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 \(PQ\).

Rational curve with a tangent and the coordinate axes

<p>The line \(x=1\) meets the curve \(x=\dfrac{(y-2)^2}{4}\) at \(P\) and \(Q\).</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>
<p>Find the area enclosed by the curve and the line.<span class="inline-marks">[5]</span></p>

The line \(x=1\) meets the curve \(x=\dfrac{(y-2)^2}{4}\) at \(P\) and \(Q\).

Find the area enclosed by the curve and the line.

Sideways parabola cut by a vertical line

<p>The points \(A\) and \(B\) lie on the curve \(y=2\mathrm{e}^{-x}+1\), where their \(x\)-coordinates are \(0\) and \(2\), respectively. The chord \(AB\) is drawn.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q15-parts-item-1" data-list-item-path="amarea-q15-parts-item-1"><div class="list-item-paragraph">State the coordinates of \(A\) and \(B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q15-parts-item-2" data-list-item-path="amarea-q15-parts-item-2"><div class="list-item-paragraph">Find the exact area enclosed by the curve and the chord \(AB\).<span class="inline-marks">[5]</span></div></li></ol>

The points \(A\) and \(B\) lie on the curve \(y=2\mathrm{e}^{-x}+1\), where their \(x\)-coordinates are \(0\) and \(2\), respectively. The chord \(AB\) is drawn.

State the coordinates of \(A\) and \(B\).

Find the exact area enclosed by the curve and the chord \(AB\).

Exponential curve and a chord

<p>The diagram shows part of the curve \(x=y^2-3y\) and the line \(x+y=3\).</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/1768008012336-hpxi3pcmgut.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDgwMTIzMzYtaHB4aTNwY21ndXQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTYwNCwiZXhwIjoxNzkyNzMxOTA0fQ.f-gQyCj_7QOtvRvX-hBXnH-O-Dqzhljo-Xnv25EHAx4" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>If the line and the curve intersect at \(P\) and \(Q\), find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">coordinates of \(P\) and \(Q\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">area of the shaded region.</span></div></li></ol>

The diagram shows part of the curve \(x=y^2-3y\) and the line \(x+y=3\).

If the line and the curve intersect at \(P\) and \(Q\), find the

<p>The diagram shows part of the curve \(y=2x-x^2\) and the line \(PQ\) joining the point \(P(0,4)\) to \(Q\), the maximum point of the curve.</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/1768007757138-uphfiazexqb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDc3NTcxMzgtdXBoZmlhemV4cWIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTYwNCwiZXhwIjoxNzkyNzMxOTA0fQ.wBBmwGANkgMlWlL-UC-cUXCD43XBHwnPwoKZgw6QYaI" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of the point \(Q\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the shaded region.</span></div></li></ol>

The diagram shows part of the curve \(y=2x-x^2\) and the line \(PQ\) joining the point \(P(0,4)\) to \(Q\), the maximum point of the curve.

Stationary Points and Their Nature

<p>The diagram shows part of the curve \(y=4-\mathrm{e}^{\frac{1}{2}x}\) which cuts the axes at \(A\) and at \(B\).</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/1768007534603-21zzm2ivsu1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDc1MzQ2MDMtMjF6em0yaXZzdTEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTYwNCwiZXhwIjoxNzkyNzMxOTA0fQ.nb4YZiHHSBP8wbwvWKEZ1n-X6ymsWGUnOWSkU00iLO8" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of \(A\) and of \(B\).</span></div></li></ol>
<p>The tangent to the curve at \(A\) meets the \(x\)-axis at \(C\).</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of \(C\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the shaded region.</span></div></li></ol>

The diagram shows part of the curve \(y=4-\mathrm{e}^{\frac{1}{2}x}\) which cuts the axes at \(A\) and at \(B\).

Find the coordinates of \(A\) and of \(B\).

The tangent to the curve at \(A\) meets the \(x\)-axis at \(C\).

<p>The region \(R\) shown in the diagram below is bounded by the curve \(y=4-x^2\), the line \(y=3x\) and the \(x\)-axis. Find the area of region \(R\).</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/1768007342263-4oyo73179t2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDczNDIyNjMtNG95bzczMTc5dDIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTYwNCwiZXhwIjoxNzkyNzMxOTA0fQ.RlnywWQzcoGgubepb74zV9PkIpqMLKHqsvwAND1oqEo" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The region \(R\) shown in the diagram below is bounded by the curve \(y=4-x^2\), the line \(y=3x\) and the \(x\)-axis. Find the area of region \(R\).

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