A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<p>The function \(\mathrm{f}\) is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \neq \frac{a}{b},\]</p>
<p>where&nbsp;\(a\) and \(b\) are non-zero constants.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find \(\mathrm{f}^{-1}(x)\). Hence or otherwise find \(\mathrm{f}^2(x)\) and state the range of \(\mathrm{f}^2\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">The function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \frac{1}{x}\) for all real non-zero \(x\). State whether the composite function \(\mathrm{fg}\) exists, justifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Solve the equation \(\mathrm{f}^{-1}(x) = x\).<span class="inline-marks">[3]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by

\[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for } x \in \mathbb{R}, \ x \neq \frac{a}{b},\]

where \(a\) and \(b\) are non-zero constants.

N2009 P2 Q3

Functions

<ol class="abc-list" start="2"><li data-list-item-id="item-1780997990538-ccqpro9r7"><div class="list-item-paragraph">The position vectors of the variable point \(R\) and the fixed points \(A\) and \(B\) are given by</div><div class="list-item-paragraph" style="text-align: center">\(\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{b} = \begin{pmatrix} -1 \\ 5 \\ 3 \end{pmatrix} \text{ respectively.}\)</div><div class="list-item-paragraph">It is given that&nbsp;\(\mathbf{r} \cdot (\mathbf{a} \times \mathbf{b}) = 0\).</div><ol class="i-list"><li data-list-item-id="subitem-1780997993251-00a0o1uz1"><div class="list-item-paragraph">Describe geometrically the set of points, denoted by \(S\), to which \(R\) belongs.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1780997993490-ia1hb1u89"><div class="list-item-paragraph">Find the exact shortest distance between the point \((-8, -2, 9)\) and \(S\).<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2021 DHS P2 Q5 Modified

Vectors

<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto 1 - (x + 3)^2\), \(x \in \mathbb{R}\), \(x \ge -2\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that the inverse function of \(\mathrm{f}\) exists. Find \(\mathrm{f}^{-1}(x)\) and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">On the same diagram, sketch the graphs of \(y = \mathrm{f}(x)\), \(y = \mathrm{f}^{-1}(x)\) and \(y = \mathrm{f}^{-1}\mathrm{f}(x)\), showing clearly the relationship between the graphs.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Hence, find the exact solution of \(\mathrm{f}^{-1}\mathrm{f}(x) \le \mathrm{f}(x)\).<span class="inline-marks">[3]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto 1 - (x + 3)^2\), \(x \in \mathbb{R}\), \(x \ge -2\).

2011 NJC P1 Q7

<p>Given that&nbsp;\(\mathrm{g}(x) = 2 + \mathrm{e}^x\), \(x \in \mathbb{R}\) and \(\mathrm{f}(x) = -\ln x\), \(x \in \mathbb{R},x&gt;0\).</p>
<p>Find the exact range of</p>
<ol class="abc-list"><li><div class="list-item-paragraph"> \(\mathrm{fg}\),</div></li><li><div class="list-item-paragraph"> \(\mathrm{gf}\).</div></li></ol>

Given that \(\mathrm{g}(x) = 2 + \mathrm{e}^x\), \(x \in \mathbb{R}\) and \(\mathrm{f}(x) = -\ln x\), \(x \in \mathbb{R},x>0\).

Composite Functions

<p>Evaluate the following: </p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(4^{2x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\ln(2x)}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)\)</div></li></ol>

Techniques of Differentiation

<p>Given that</p>
<p style="text-align: center">\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]</p>
<p>where&nbsp;\(A\) and \(B\) are constants with \(x &gt; -1\).</p>
<p>Show that&nbsp;\((x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\).<span class="inline-marks">[5]</span></p>

\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]

where \(A\) and \(B\) are constants with \(x > -1\).

Show that \((x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\).

2018 AJC C1BT1 Q3

<p>The cross-section of a skateboard half-pipe is modelled by the quadratic function \(y =  \frac{1}{2}x^2 - 4x + 9\), where&nbsp;\(x\) metres represents the horizontal distance from left end of the half-pipe, and \(y\) metres represents the height above the ground.</p>
<p>The graph of the function models the curved surface of the half-pipe.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the equation of the curve can be written in the form \(y =   \frac{1}{2}(x - 4)^2 + 1\).<br>Hence, state the coordinates of the turning point.</div></li><li><div class="list-item-paragraph">Explain what the turning point represents in the context of the half-pipe.</div></li><li><div class="list-item-paragraph">Sketch the graph of the quadratic function clearly showing:</div><ul class="bullet-list"><li><div class="list-item-paragraph">the turning point,</div></li><li><div class="list-item-paragraph">the intercept with the y-axis,</div></li><li><div class="list-item-paragraph">the axis of symmetry.</div></li></ul></li><li><div class="list-item-paragraph">If the right end of the half pipe has a height of \(13.5\) m from the ground, find the width of the half pipe from one end to the other.</div></li></ol>

The cross-section of a skateboard half-pipe is modelled by the quadratic function \(y =  \frac{1}{2}x^2 - 4x + 9\), where \(x\) metres represents the horizontal distance from left end of the half-pipe, and \(y\) metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

Quadratic Expressions, Equations, and Functions

<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\tan x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(3^{\ln x}\right)\)</div></li><li><div class="list-item-paragraph">\(\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)\)</div></li></ol>

<p>In hypothesis testing, the \(\mathbf{p}\)-value is generally defined as:</p>
<p>The probability, assuming the null hypothesis is true, of obtaining a result as extreme as or more extreme than the observed sample result.</p>
<p>In context, you should always mention:</p>
<ol class="num-list"><li><div class="list-item-paragraph">Assume \(H_0\) is true</div></li><li><div class="list-item-paragraph">Probability of getting the observed test statistic or something more extreme</div></li><li><div class="list-item-paragraph">Use the direction of the alternative hypothesis</div></li></ol>
<p>For example:</p>
<p class="paragraph-header">Right-tailed test</p>
<p>If</p>
<p style="text-align: center">\[H_0 : \mu = 50, \quad H_1 : \mu &gt; 50\]</p>
<p>and the sample mean is larger than 50, then the p-value is:</p>
<p>The probability, assuming the true mean is 50, of obtaining a sample mean at least as large as the one observed.</p>
<p>So "more extreme" means <strong>larger</strong>.</p>
<p class="paragraph-header">Left-tailed test</p>
<p>If</p>
<p style="text-align: center">\[H_0 : p = 0.3, \quad H_1 : p &lt; 0.3\]</p>
<p>then the p-value is:</p>
<p>The probability, assuming the true proportion is 0.3, of obtaining a sample proportion at most as small as the one observed.</p>
<p>So "more extreme" means <strong>smaller</strong>.</p>
<p class="paragraph-header">Two-tailed test</p>
<p>If</p>
<p style="text-align: center">\[H_0 : \mu = 20, \quad H_1 : \mu \neq 20\]</p>
<p>then the p-value is:</p>
<p>The probability, assuming the true mean is 20, of obtaining a sample mean as far from 20 as, or further than, the observed sample mean.</p>
<p>So "more extreme" means <strong>further away from the null value in either direction</strong>.</p>
<p>A good general template is:</p>
<p>The p-value is the probability, assuming \(H_0\) is true, of obtaining a test statistic at least as extreme as the observed value in the direction of \(H_1\).</p>
<p>For exam writing, avoid saying "the p-value is the probability that&nbsp;\(H_0\) is true." That is <strong>incorrect</strong>.</p>

In hypothesis testing, the \(\mathbf{p}\)-value is generally defined as:

The probability, assuming the null hypothesis is true, of obtaining a result as extreme as or more extreme than the observed sample result.

and the sample mean is larger than 50, then the p-value is:

The probability, assuming the true mean is 50, of obtaining a sample mean at least as large as the one observed.

The probability, assuming the true proportion is 0.3, of obtaining a sample proportion at most as small as the one observed.

\[H_0 : \mu = 20, \quad H_1 : \mu \neq 20\]

The probability, assuming the true mean is 20, of obtaining a sample mean as far from 20 as, or further than, the observed sample mean.

So "more extreme" means **further away from the null value in either direction**.

The p-value is the probability, assuming \(H_0\) is true, of obtaining a test statistic at least as extreme as the observed value in the direction of \(H_1\).

For exam writing, avoid saying "the p-value is the probability that \(H_0\) is true." That is **incorrect**.

Hypothesis Testing

<p>The planes \(p\) and \(q\) have equations</p>
<p style="text-align: center">\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]</p>
<p>where&nbsp;\(a\) and \(b\) are constants and \(\lambda\) and \(\mu\) are parameters.</p>
<p>The line \(l\) passes through the point \((5, 4, 0)\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). The planes \(p\) and \(q\) meet in the line \(l\).</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;\(a = 1\) and \(b = \frac{1}{2}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the exact acute angle between the planes \(p\) and \(q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the distance from the point \(A(2, 0, 3)\) to the plane \(q\). Hence deduce the shortest distance from \(A\) to \(l\).<span class="inline-marks">[4]</span></div></li></ol>
<p>The plane \(q\) is reflected about the plane \(p\) to obtain the plane \(q'\).</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d"><div class="list-item-paragraph">Find a cartesian equation of the plane \(q'\).<span class="inline-marks">[3]</span></div></li></ol>

The planes \(p\) and \(q\) have equations

\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]

where \(a\) and \(b\) are constants and \(\lambda\) and \(\mu\) are parameters.

The line \(l\) passes through the point \((5, 4, 0)\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). The planes \(p\) and \(q\) meet in the line \(l\).

The plane \(q\) is reflected about the plane \(p\) to obtain the plane \(q'\).

2024 RI Promo Q9

<p>Referring to the origin \(O\), two planes \(\pi_1\) and \(\pi_2\) are given by</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} = 18 \quad \text{and} \quad \pi_2 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} = 3.\]</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Given that point \(A(2, 5, 2)\) lies on \(\pi_2\), find \(\overrightarrow{OB}\), where&nbsp;\(B\) is the image of \(A\) when reflected in the plane \(\pi_1\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find a vector equation of the line of intersection of \(\pi_1\) and \(\pi_2\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1781324042384-buo6cn6yi"><div class="list-item-paragraph">Find the Cartesian equation of the plane which is the image of \(\pi_2\) when&nbsp;\(\pi_2\) is reflected in \(\pi_1\).<span class="inline-marks">[3]</span></div></li></ol>

Referring to the origin \(O\), two planes \(\pi_1\) and \(\pi_2\) are given by

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} = 18 \quad \text{and} \quad \pi_2 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} = 3.\]

<p>With respect to the origin \(O\), the fixed points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where&nbsp;\(\mathbf{a}\) and \(\mathbf{b}\) are non-zero and non-parallel.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">The variable point \(R\) has position vector \(\mathbf{r} = \lambda \mathbf{a} + (1 - \lambda)\mathbf{b}\), where&nbsp;\(\lambda\) is a real parameter. Describe geometrically the set of all possible positions of \(R\).<span class="inline-marks">[1]</span></div></li></ol>
<p>It is given that the angle \(AOB\) is \(90^\circ\).</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b"><div class="list-item-paragraph">Explain why \(\mathbf{a} \cdot \mathbf{b} = 0\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Among the set of all possible points \(R\), the point \(R^*\) is the closest to the origin \(O\). Find the position vector of \(R^*\). Hence state the ratio \(AR^* : BR^*\) in terms of magnitudes of \(\mathbf{a}\) and \(\mathbf{b}\).<span class="inline-marks">[4]</span></div></li></ol>

With respect to the origin \(O\), the fixed points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, where \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero and non-parallel.

It is given that the angle \(AOB\) is \(90^\circ\).

2024 RI Promo Q2

<p>The curve \(C_1\) has equation \(y = 2x + 3 + \frac{12}{x - 2}\), \(x \neq 2\).</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Sketch \(C_1\), showing clearly the equations of asymptotes, the coordinates of the axial intercepts and the turning points, if any.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">By completing the square for both&nbsp;\(x\) and \(y\), give a geometrical interpretation of the curve \(C_2\) with equation \(x^2 - 4x + y^2 + 6y - 3 = 0\).</div><div class="list-item-paragraph">Hence write down the number of real roots for the equation \[(x - 2)^2 + \left(2x + 6 + \frac{12}{x - 2}\right)^2 = 4^2.\]<span class="inline-marks">[3]</span></div></li></ol>

The curve \(C_1\) has equation \(y = 2x + 3 + \frac{12}{x - 2}\), \(x \neq 2\).

2024 TMJC Promo Q3

Graphing Techniques

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;\(x^2 + 4x + 9\) is always positive for all real values of \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, without using a calculator, solve \(\frac{(x^2 + 4x + 9)(x + 2)^2}{x^2 - 6x + 7} \le 0\).<span class="inline-marks">[3]</span></div></li></ol>

Inequalities - General

2024 NYJC Promo Q2

Inequalities

<ol class="abc-list"><li data-list-item-id="item-a"><ol class="i-list"><li data-list-item-id="item-a-i"><div class="list-item-paragraph">Given that&nbsp;\(\mathbf{p} \cdot \mathbf{q} = 0\), what can be deduced about the vectors \(\mathbf{p}\) and \(\mathbf{q}\)?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii"><div class="list-item-paragraph">With respect to the origin on a \(xy\)-plane, a fixed point \(G\) has position vector \(\mathbf{g}\) and a variable point \(R\) has position vector \(\mathbf{r}\). Describe the locus of \(R\) if&nbsp;\(\mathbf{r} \cdot (\mathbf{r} - \mathbf{g}) = 0\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With respect to the origin, plane \(p\) has an equation given by \(\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})\), \(\lambda \in \mathbb{R}\), \(\mu \in \mathbb{R}\). Points \(Q\) and \(R\) have position vectors \(3\mathbf{i} + 9\mathbf{j} + 5\mathbf{k}\) and \(4\mathbf{i} + 11\mathbf{j} + 7\mathbf{k}\) respectively. Find the length of the projection of \(\overrightarrow{QR}\) onto \(p\).<span class="inline-marks">[4]</span></div></li></ol>

2024 NJC Promo Q7

<p>Without using a calculator, solve the inequality</p>
<p style="text-align: center">\[\frac{x+3}{x^2+x-2} \ge -1.\]<span class="inline-marks">[4]</span></p>

Without using a calculator, solve the inequality

2024 NJC Promo Q1

<p>Relative to the origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The point \(P\) is on \(OA\) such that&nbsp;\(OP = 3PA\) and point \(Q\) is on \(AB\) such that&nbsp;\(AQ = kAB\), where&nbsp;\(k\) is a constant and \(0 &lt; k &lt; 1\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Express \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(k\), where applicable.<span class="inline-marks">[2]</span></div></li></ol>
<p>Point \(R\) is on \(PQ\) produced such that&nbsp;\(PQ : QR = 4 : 5\).</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Given that the points \(O\), \(B\) and \(R\) are collinear, find the value of \(k\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Show that the area of triangle \(OPR\) can be written as&nbsp;\(s\left| \mathbf{a}\times \mathbf{b} \right|\), where&nbsp;\(s\) is a constant to be found.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The point \(P\) is on \(OA\) such that \(OP = 3PA\) and point \(Q\) is on \(AB\) such that \(AQ = kAB\), where \(k\) is a constant and \(0 < k < 1\).

Point \(R\) is on \(PQ\) produced such that \(PQ : QR = 4 : 5\).

2024 MI PRU2 P2 Q3

<p>The function \(\mathrm{f}\) is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{1}{(x-2)^2+1},\quad x \in \mathbb{R}.\]</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(x)\), indicating the coordinates of the turning point and the equation of any asymptotes. Explain why \(\mathrm{f}^{-1}\) does not exist.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">If the domain of \(\mathrm{f}\) is restricted to \([k, \infty)\), state the smallest value of \(k\) for which the function \(\mathrm{f}^{-1}\) exists.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Using the value of \(k\) found in part <strong>(ii)</strong>, find \(\mathrm{f}^{-1}(x)\) and state its domain.<span class="inline-marks">[3]</span></div></li></ol>

\[\mathrm{f} : x \mapsto \frac{1}{(x-2)^2+1},\quad x \in \mathbb{R}.\]

Inverse Functions

2024 MI PRU2 P2 Q2

<p>A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is \(12\) years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than \(12\) years.</p>
<p>The durations \(x\), in years, of a random sample of \(45\) refrigerators are summarised as follows.</p>
<p style="text-align: center">\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate unbiased estimates of the population mean and variance of the lifespan of the refrigerators.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">State hypotheses that can be used to test if the mean lifespan of the refrigerators is less than \(12\) years, defining any parameters you use. Test, at the \(5\%\) level of significance, whether the mean lifespan of the refrigerators is less than \(12\) years.<span class="inline-marks">[4]</span></div></li></ol>
<p>The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from \(12\) years. He records the durations of a large random sample of \(n\) refrigerators and finds that their mean lifespan is \(12.4\) years and variance is \(4.1\) square years.</p>
<p>A test at \(5\%\) significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the range of possible values of \(n\).<span class="inline-marks">[4]</span></div></li></ol>

A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is \(12\) years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than \(12\) years.

The durations \(x\), in years, of a random sample of \(45\) refrigerators are summarised as follows.

\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]

The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from \(12\) years. He records the durations of a large random sample of \(n\) refrigerators and finds that their mean lifespan is \(12.4\) years and variance is \(4.1\) square years.

A test at \(5\%\) significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.

2024 ACJC P2 Q8

<p>An arithmetic series has first term \(a\) and common difference \(d\), where&nbsp;\(a\) is non-zero. The \(8\)th, \(4\)th and \(2\)nd term of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are increasing.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;\(a = d\).<span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the sum of the first \(20\) terms of the arithmetic series is \(630\).</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the value of \(a\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the exact sum of the first \(10\) terms of the geometric series.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Give a reason why the geometric series converges and find the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li></ol>

An arithmetic series has first term \(a\) and common difference \(d\), where \(a\) is non-zero. The \(8\)th, \(4\)th and \(2\)nd term of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are increasing.

It is given that the sum of the first \(20\) terms of the arithmetic series is \(630\).

Math Question Bank