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<p>Let&nbsp;\(f(x) = p(x - q)(x - r)\). Part of the graph of \(f\) is shown below.</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>The graph passes through the points \( (-2,0), (0, -4) \) and \( (4, 0) \).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the value of \(q\) and \(r\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Write down the <strong>equation</strong> of the axis of symmetry.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the value of \(p\).<span class="inline-marks">[3]</span></div></li></ol>

Let \(f(x) = p(x - q)(x - r)\). Part of the graph of \(f\) is shown below.

The graph passes through the points \( (-2,0), (0, -4) \) and \( (4, 0) \).

Quadratic Equations and Inequalities — General

Quadratic Functions, Equations and Inequalities

Functions

<p>Solve the equation \(\log_{9}{81} + \log_{9}{\frac{1}{9}} + \log_{9}{3} = \log_{9}{x}\).<span class="inline-marks">[4]</span></p>

Solve the equation \(\log_{9}{81} + \log_{9}{\frac{1}{9}} + \log_{9}{3} = \log_{9}{x}\).

Exponential and Logarithmic Functions

<p>Let&nbsp;\(p = \log_{10}{x}\), \(q = \log_{10}{y}\)and \(r = \log_{10}{z}\)<span class="inline-marks">[6]</span></p>
<p>Write the expression \(\log_{10}{ \left( \frac{x}{y^2\sqrt{z}} \right) }\) in terms of \(p, q\) and \(r\).</p>

Let \(p = \log_{10}{x}\), \(q = \log_{10}{y}\)and \(r = \log_{10}{z}\)

Write the expression \(\log_{10}{ \left( \frac{x}{y^2\sqrt{z}} \right) }\) in terms of \(p, q\) and \(r\).

<p>Let&nbsp;\(\log_{10}{P} = x\), \(\log_{10}{Q} = y\) and \(\log_{10}{R} = z\). Express \(\log_{10} \left( { \frac{P}{QR^3}} \right)^2\) in terms of \(x, y\) and \(z\).<span class="inline-marks">[4]</span></p>

Let \(\log_{10}{P} = x\), \(\log_{10}{Q} = y\) and \(\log_{10}{R} = z\). Express \(\log_{10} \left( { \frac{P}{QR^3}} \right)^2\) in terms of \(x, y\) and \(z\).

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;\(\log_3{x} - \log_3{(x - 5)} = \log_3{A}\), express \(A\) in terms of \(x\).<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Hence or otherwise, solve the equation \(\log_3{x} - \log_3{(x - 5)} = 1\).</div></li></ol>

<p>Let&nbsp;\(a = \log{x}\), \(b = \log{y}\), and \(c = \log{z}\).<span class="inline-marks">[6]</span></p>
<p>Write \(\log { \left( \frac{x^2 \sqrt{y}}{z^3} \right) }\) in terms of \(a, b\) and \(c\).</p>

Let \(a = \log{x}\), \(b = \log{y}\), and \(c = \log{z}\).

Write \(\log { \left( \frac{x^2 \sqrt{y}}{z^3} \right) }\) in terms of \(a, b\) and \(c\).

<p>Let&nbsp;\(f(x) = 8x - 2x^2\). Part of the graph of \(f\) is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="200" data-height="250" style="width:200px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the \(x\)-intercepts of the graph.<span class="inline-marks">[4]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Write down the equation of the axis of symmetry.</div></li><li><div class="list-item-paragraph">Find the \(y\)-coordinate of the vertex.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Let \(f(x) = 8x - 2x^2\). Part of the graph of \(f\) is shown below.

<p>A cubic polynomial satisfies \(P(1) = P(2) = P(3) = 7\). Also \( P(0) = 1\)<br>Find \( P(4)\)</p>
<p></p>

A cubic polynomial satisfies \(P(1) = P(2) = P(3) = 7\). Also \( P(0) = 1\)
Find \( P(4)\)

Polynomials, Cubic Equations and Partial Fractions

<p>Let&nbsp;\(\ln{a} = p\), \(\ln{b} = q\). Write the following expressions in terms of \(p\) and \(q\).<span class="inline-marks">[6]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\ln{a^3b}\)</div></li><li><div class="list-item-paragraph">\(\ln {\left( \frac{\sqrt{a}}{b} \right)}\)</div></li></ol>

Let \(\ln{a} = p\), \(\ln{b} = q\). Write the following expressions in terms of \(p\) and \(q\).

<p>Find the derivative of each of the following w.r.t \(x\):</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>\(y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} \)</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>\(y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) \)</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>\( y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }\)</p></div></li></ol>

Find the derivative of each of the following w.r.t \(x\):

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

Techniques of Differentiation

Differentiation

<p>Suppose \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\) satisfies \(P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4\), it is also given&nbsp;\(P(5) = -19\).<br>Find \(P(6)\).</p>

Suppose \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\) satisfies \(P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4\), it is also given \(P(5) = -19\).
Find \(P(6)\).

<p>If&nbsp;\(\log_{a}{2} = x\) and \(\log_{a}{5} = y\), find in terms of \(x\) and \(y\), expressions for<span class="inline-marks">[4]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\log_{2}{5}\);</div></li><li><div class="list-item-paragraph">\(\log_{a}{20}\).</div></li></ol>

If \(\log_{a}{2} = x\) and \(\log_{a}{5} = y\), find in terms of \(x\) and \(y\), expressions for

<p>Given that&nbsp;\(\log_5{x} = y\), express each of the following in terms of \(y\).<span class="inline-marks">[6]</span></p>
<ol class="abc-list"><li><div class="list-item-paragraph">\( \log_5{x^2}\)</div></li><li><div class="list-item-paragraph">\( \log_5{ \left( \frac{1}{x} \right) }\)</div></li><li><div class="list-item-paragraph">\( \log_{25}{x}\)</div></li></ol>

Given that \(\log_5{x} = y\), express each of the following in terms of \(y\).

Let \(f(x) = 3x\), \(g(x) = 2x - 5\) and \(h(x) = (f \circ g)(x)\)

<p>Without expanding completely, prove that&nbsp;\( \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0\) for every positive integer \(n\).</p>

Without expanding completely, prove that \( \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0\) for every positive integer \(n\).

Binomial Theorem

<p>Solve the equation \(9^{x-1} = \left( \frac{1}{3} \right)^{2x}\).<span class="inline-marks">[4]</span></p>

Solve the equation \(9^{x-1} = \left( \frac{1}{3} \right)^{2x}\).

<p>Let&nbsp;\(f(x) = \cos{2x}\) and \(g(x) = 2x^2 - 1\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find \(f \left( \frac{\pi}{2} \right) \)<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find \((g \circ f) \left( \frac{\pi}{2} \right) \).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\( (g \circ f)(x)\) can be written as&nbsp;\(\cos{(kx)}\), find the value of \(k, k \in \mathbb{Z}\)<span class="inline-marks">[3]</span></div></li></ol>

Let \(f(x) = \cos{2x}\) and \(g(x) = 2x^2 - 1\).

Trigonometric Functions and Graphs - General

Trigonometric Equations and Identities - General

Trigonometry

<p>Find the <strong>exact</strong> solution of the equation \(9^{2x} = 27^{(1-x)}\).<span class="inline-marks">[6]</span></p>

Find the **exact** solution of the equation \(9^{2x} = 27^{(1-x)}\).

<p>Solve the equation \( \log_{27}{x} = 1 - \log_{27}{(x - 0.4)}\).<span class="inline-marks">[6]</span></p>

Solve the equation \( \log_{27}{x} = 1 - \log_{27}{(x - 0.4)}\).

<p>Let&nbsp;\(f(x) = A\mathrm{e}^{kx} + 3\). Part of the graph of \(f\) is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="262" style="width:300px;height:262px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The \(y\)-intercept is at \( (0, 13) \).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(A = 10\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\(f(15) = 3.49\) (correct to \(3\) significant figures), find the value of \(k\).<span class="inline-marks">[3]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Using your value of \(k\), find \(f'(x)\).<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence, explain why \(f\) is a decreasing function.</div></li><li><div class="list-item-paragraph">Write down the equation of the horizontal asymptote of the graph \(f\).</div></li></ol></li></ol>
<p>Let&nbsp;\(g(x) = -x^2 + 12x - 24\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the area enclosed by the graphs of \(f\) and \(g\).<span class="inline-marks">[6]</span></div></li></ol>

IB SL Math Question Bank