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PSLE

PSLE Foundation Mathematics

<p>The tangent to the curve \(y=2\ln(x+1)\) at \(A\), where&nbsp;\(x=3\), meets the \(y\)-axis at \(B\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q17-parts-item-1" data-list-item-path="amarea-q17-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-q17-parts-item-2" data-list-item-path="amarea-q17-parts-item-2"><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 \(y=2\ln(x+1)\) at \(A\), where \(x=3\), 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

Area under Logarithmic Curve

Logarithmic curve and its tangent beside the y-axis

Integration

Differentiation

<p>The tangent to the curve \(y=3\ln(x-1)\) at \(A\), where&nbsp;\(x=4\), meets the \(x\)-axis at \(B\). 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-q18-parts-item-1" data-list-item-path="amarea-q18-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-q18-parts-item-2" data-list-item-path="amarea-q18-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-q18-parts-item-3" data-list-item-path="amarea-q18-parts-item-3"><div class="list-item-paragraph">Hence find the exact area enclosed by the curve, the tangent and the \(x\)-axis.<span class="inline-marks">[5]</span></div></li></ol>

The tangent to the curve \(y=3\ln(x-1)\) at \(A\), where \(x=4\), meets the \(x\)-axis at \(B\). The curve meets the \(x\)-axis at \(C\).

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

Logarithmic curve with a tangent and the x-axis

<p>In the diagram, the curve \(y=3\ln(x-2)\) cuts the \(x\)-axis at \(Q\). A line which meets the curve at \(P(5, p)\) cuts the \(x\)-axis at \((8, 0)\).</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/1767949777613-dr8vc92ptia.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDk3Nzc2MTMtZHI4dmM5MnB0aWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ1MywiZXhwIjoxNzkyNzMxNzUzfQ.wFnG0mIInSlxYqBGZwCTRp_ERoyofdndjIS0iYFkXw0" 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">State the exact coordinates of \(Q\) and \(P\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the exact area bounded by the curve, the line and the \(x\)-axis.</span></div></li></ol>

In the diagram, the curve \(y=3\ln(x-2)\) cuts the \(x\)-axis at \(Q\). A line which meets the curve at \(P(5, p)\) cuts the \(x\)-axis at \((8, 0)\).

State the exact coordinates of \(Q\) and \(P\).

Calculate the exact area bounded by the curve, the line and the \(x\)-axis.

<p>In the diagram, the curve \(y=2\ln(x+3)\) cuts the \(y\)-axis at \((0, q)\). A line, which meets the curve at \((-1, p)\) cuts the \(y\)-axis at \((0, 0.5)\).</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/1767949498766-kszumh0ngxc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDk0OTg3NjYta3N6dW1oMG5neGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ1MywiZXhwIjoxNzkyNzMxNzUzfQ.3AviJ_2XTq6_qWbmbhuLDDAdFXILMYZKX3cR3VwaD-4" style="max-width: min(250px, 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">State the exact value of \(p\) and of \(q\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate the exact area bounded by the curve, the line and the \(y\)-axis.</div></li></ol>

In the diagram, the curve \(y=2\ln(x+3)\) cuts the \(y\)-axis at \((0, q)\). A line, which meets the curve at \((-1, p)\) cuts the \(y\)-axis at \((0, 0.5)\).

State the exact value of \(p\) and of \(q\).

Calculate the exact area bounded by the curve, the line and the \(y\)-axis.

<p>In the diagram, the curve \(y = \ln\left(\frac{x}{2}\right)\) cuts the \(x\)-axis at \(Q(q, 0)\) and passes through the point \(P(5, p)\).</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/1767949218084-enbwz6g5yvp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDkyMTgwODQtZW5id3o2ZzV5dnAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQ1MywiZXhwIjoxNzkyNzMxNzUzfQ.VsH7HlwRZlZwNXvwE46iR4VvX7nj35HxQRV4fRWugKY" 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">State the exact value of \(p\) and of \(q\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the exact area of the shaded region.</span></div></li></ol>

In the diagram, the curve \(y = \ln\left(\frac{x}{2}\right)\) cuts the \(x\)-axis at \(Q(q, 0)\) and passes through the point \(P(5, p)\).

Calculate the exact area of the shaded region.

<p>In the diagram, the curve \(y = \ln(x - 1)\) cuts the \(x\)-axis at \(Q(q, 0)\) and passes through the point \(P(3, p)\).</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/1767948920557-k59avhz9u4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDg5MjA1NTctazU5YXZoejl1NC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzMxNDUzLCJleHAiOjE3OTI3MzE3NTN9.CZuUv7zjD_u-fs8CtUpwKrnDRdCy0Z_zj_pgycAz4p0" 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">State the exact value of \(p\) and of \(q\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the exact area of the shaded region.</span></div></li></ol>

In the diagram, the curve \(y = \ln(x - 1)\) cuts the \(x\)-axis at \(Q(q, 0)\) and passes through the point \(P(3, p)\).

Math Question Bank