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<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

Vectors

<p>Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}+2\mathbf{j}+5\mathbf{k}\), \(\mathbf{i}-\mathbf{k}\) and \(2\mathbf{i}+\mathbf{j}\) respectively.</p>
<p>The line \(l_1\) passes through \(A\) and is parallel to the vector \(3\mathbf{i}+2\mathbf{j}\). The line \(l_2\) passes through the points \(B\) and \(C\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find a vector equation of the line \(l_1\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Show that the lines \(l_1\) and \(l_2\) intersect and that the position vector \(\mathbf{OX}\) of the point of intersection is \(7\mathbf{i}+6\mathbf{j}+5\mathbf{k}\).<span class="inline-marks">[3]</span></div></li></ol>
<p>The vector \(a\mathbf{i}+b\mathbf{j}+\mathbf{k}\) is perpendicular to both&nbsp;\(l_1\) and \(l_2\). Find the values of \(a\) and \(b\).</p>
<p>Hence give an equation of the line \(l_3\) that is concurrent with \(l_1\) and \(l_2\) and is perpendicular to both&nbsp;\(l_1\) and \(l_2\).</p>
<p>If&nbsp;\(V\) is a point on \(l_3\) such that&nbsp;\(VABX\) is a tetrahedron with base \(ABX\) and height \(5\sqrt{14}\) units, find the coordinates of \(V\).<span class="inline-marks">[7]</span></p>

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}+2\mathbf{j}+5\mathbf{k}\), \(\mathbf{i}-\mathbf{k}\) and \(2\mathbf{i}+\mathbf{j}\) respectively.

The line \(l_1\) passes through \(A\) and is parallel to the vector \(3\mathbf{i}+2\mathbf{j}\). The line \(l_2\) passes through the points \(B\) and \(C\).

The vector \(a\mathbf{i}+b\mathbf{j}+\mathbf{k}\) is perpendicular to both \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\).

Hence give an equation of the line \(l_3\) that is concurrent with \(l_1\) and \(l_2\) and is perpendicular to both \(l_1\) and \(l_2\).

If \(V\) is a point on \(l_3\) such that \(VABX\) is a tetrahedron with base \(ABX\) and height \(5\sqrt{14}\) units, find the coordinates of \(V\).

<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>

Techniques of Differentiation

<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>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>

<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 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

<p>The line \(l\) has vector equation</p>
<p style="text-align: center">\[l : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} , \quad \lambda \in \mathbb{R}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find the position vector of the point \(N\), the foot of perpendicular from the point \(A(6, 4, 4)\) to the line \(l\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the position vector of the point \(A'\), the reflection of \(A\) in the line \(l\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the coordinates of the points on the line \(l\) that are \(3\sqrt{3}\) units away from point \(A\).<span class="inline-marks">[2]</span></div></li></ol>

\[l : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} , \quad \lambda \in \mathbb{R}.\]

<p>The points \(A\) and \(B\) have coordinates \((2, 1, -1)\) and \((5, -1, 3)\) respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Find a vector equation of the line \(AB\).<span class="inline-marks">[1]</span></div></li></ol>
<p>The point \(C\) has coordinates \((10, -2, 6)\).</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-2"><div class="list-item-paragraph">Find the position vector of the foot of perpendicular from \(C\) to the line \(AB\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3"><div class="list-item-paragraph">Find a vector equation of the line which is a reflection of the line \(AC\) in the line \(AB\).<span class="inline-marks">[3]</span></div></li></ol>

The points \(A\) and \(B\) have coordinates \((2, 1, -1)\) and \((5, -1, 3)\) respectively.

The point \(C\) has coordinates \((10, -2, 6)\).

<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 line \(l\) passes through \(A(3, 1, 4)\) and is parallel to \(2\mathbf{i} + \mathbf{j} + \mathbf{k}\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">State the vector equation of \(l\).<span class="inline-marks">[1]</span></div></li></ol>
<p>The plane \(p\) has equation</p>
<p style="text-align: center">\[\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} = 7.\]</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the acute angle between \(l\) and \(p\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the point of intersection of \(l\) and \(p\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">Find the vector equation of the reflection of \(l\) in \(p\).<span class="inline-marks">[6]</span></div></li></ol>

The line \(l\) passes through \(A(3, 1, 4)\) and is parallel to \(2\mathbf{i} + \mathbf{j} + \mathbf{k}\).

\[\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} = 7.\]

<p>The curve \(C\) has equation \(y = \frac{3x^2 + ax + 3}{x + 1}\), where&nbsp;\(a\) is a constant, \(a \neq 6\).</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find the set of values of \(a\) such that&nbsp;\(C\) does not cut the \(x\)-axis.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">It is given that the equation of \(C\) is \(y = \frac{3x^2 - 6x + 3}{x + 1}\). Sketch \(C\) and give the equations of any asymptotes. Also, state the coordinates of any points where&nbsp;\(C\) crosses the axes and of any turning points.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Let&nbsp;\(k\) be a positive constant. By sketching a suitable graph in the same diagram in part <strong>(b)</strong>, find an inequality satisfied by \(k\) such that the equation</div><div class="list-item-paragraph" style="text-align: center">\[k^2(x - 1)^2 + \left(\frac{3x^2 - 6x + 3}{x + 1} + 12\right)^2 = k^2\]</div><div class="list-item-paragraph">has 2 real and distinct roots.<span class="inline-marks">[2]</span></div></li></ol>

The curve \(C\) has equation \(y = \frac{3x^2 + ax + 3}{x + 1}\), where \(a\) is a constant, \(a \neq 6\).

Finding Unknowns in Equations

2023 VJC Promo Q10

<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>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

<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>The position vectors of points \(A\), \(B\) and \(C\), relative to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, where</p>
<p style="text-align: center">\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find \(\mathbf{a} \times \mathbf{b}\) and deduce the exact area of triangle \(OAB\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, find the volume of the tetrahedron \(OABC\).</div><div class="list-item-paragraph">[The volume of a tetrahedron is \(\frac{1}{3} \times \text{base area} \times \text{perpendicular height}\).]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the perpendicular distance from \(B\) to line \(OA\).<span class="inline-marks">[2]</span></div></li></ol>

The position vectors of points \(A\), \(B\) and \(C\), relative to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, where

\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]

2024 NYJC Promo Q5

<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>The line \(l\) passes through \(A(1, 2, -1)\) and is parallel to \(3\mathbf{i} + 3\mathbf{k}\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">State the vector equation of \(l\).<span class="inline-marks">[1]</span></div></li></ol>
<p>The plane \(p\) has equation</p>
<p style="text-align: center">\[\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} = 13.\]</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the acute angle between \(l\) and \(p\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the point of intersection of \(l\) and \(p\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">Find the vector equation of the reflection of \(l\) in \(p\).<span class="inline-marks">[6]</span></div></li></ol>

The line \(l\) passes through \(A(1, 2, -1)\) and is parallel to \(3\mathbf{i} + 3\mathbf{k}\).

\[\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} = 13.\]

<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>

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