A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<p>The curve \(D\) has equation \(y = \frac{2x - 5}{x^2 + 2x - 3}.\)</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of \(D\), stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By considering a suitable curve, hence, find the exact range of values of \(k\), where&nbsp;\(k &gt; 0\), such that the equation \((x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2\) has at least one <strong>positive</strong> real root.<span class="inline-marks">[3]</span></div></li></ol>

The curve \(D\) has equation \(y = \frac{2x - 5}{x^2 + 2x - 3}.\)

Sketch the graph of \(D\), stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).

By considering a suitable curve, hence, find the exact range of values of \(k\), where \(k > 0\), such that the equation \[(x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2\] has at least one **positive** real root.

Inequality involving Circle

2024 NYJC Promo Q3

Graphing Techniques

<p>Find the number of arrangements of the \(6\) letters in the word ELEVEN in which</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">there are no restrictions,</span></div></li><li><div class="list-item-content"><span class="list-item-text">all the \(3\) letters E are consecutive,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the first letter is E and the last letter is N,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the first letter is E or the last letter is N or both.</span></div></li></ol>

Find the number of arrangements of the \(6\) letters in the word ELEVEN in which

the first letter is E and the last letter is N,

the first letter is E or the last letter is N or both.

N1986 P1 Q10

Permutation and Combination

Video 1

<p>Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">\(y = 2x^3 - 5x^2 + x - 2\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(y = 4x^2 e^x\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(y = \frac{\ln x}{2x}\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(y = |2x - x^2|\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(y = \frac{ax + 1}{x - b}\), where constants \(a\), \(b\) are given by</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(-1 &lt; a &lt; 0\) and \(0 &lt; b &lt; 1\);</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(a &lt; -1\) and \(b &gt; 1\).</span></div></li></ol></li></ol>
<p>What could be said about the graph if&nbsp;\(ab + 1 = 0\)?</p>

Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.

\(y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}\)

\(y = \frac{ax + 1}{x - b}\), where constants \(a\), \(b\) are given by

What could be said about the graph if \(ab + 1 = 0\)?

Sketching of Rational Functions

<p>Box \(A\) contains \(6\) balls numbered \(1, 2, 2, 2, 5, 7\).</p>
<p>Box \(B\) contains \(4\) balls numbered \(1, 4, 4, 7\).</p>
<p>Box \(C\) contains \(3\) balls numbered \(3, 4, 6\).</p>
<p>One ball is removed at random from each box.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that each of the three numbers obtained is greater than \(3\) is \(\frac{1}{6}\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that each of the three numbers obtained is greater than \(3\) or the sum of the three numbers obtained is \(13\) (or both).</span><span class="inline-marks">[4]</span></div></li></ol>

Box \(A\) contains \(6\) balls numbered \(1, 2, 2, 2, 5, 7\).

Box \(B\) contains \(4\) balls numbered \(1, 4, 4, 7\).

Box \(C\) contains \(3\) balls numbered \(3, 4, 6\).

One ball is removed at random from each box.

Show that the probability that each of the three numbers obtained is greater than \(3\) is \(\frac{1}{6}\).

Find the probability that each of the three numbers obtained is greater than \(3\) or the sum of the three numbers obtained is \(13\) (or both).

Probability

<p>A four-letter word can be formed from any of the \(26\) letters in the alphabet. Determine the number of such words with</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">repetitions of the letters allowed,</span></div></li><li><div class="list-item-content"><span class="list-item-text">different letters and containing exactly one vowel,</span></div></li><li><div class="list-item-content"><span class="list-item-text">repetitions allowed and containing at least one vowel.</span></div></li></ol>

A four-letter word can be formed from any of the \(26\) letters in the alphabet. Determine the number of such words with

different letters and containing exactly one vowel,

repetitions allowed and containing at least one vowel.

<p>Legox bricks come in different colours and sizes. Bricks of the same size and colour are identical. Jamie has&nbsp;\(9\) Legox bricks. The following table shows the various colours and sizes of Legox bricks that Jamie has.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 320px;"><colgroup><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"> </th><th style="text-align: center; --cell-justify: center">Small</th><th style="text-align: center; --cell-justify: center">Medium</th><th style="text-align: center; --cell-justify: center">Large</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Green</td><td style="text-align: center; --cell-justify: center">\(0\)</td><td style="text-align: center; --cell-justify: center">\(0\)</td><td style="text-align: center; --cell-justify: center">\(2\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Blue</td><td style="text-align: center; --cell-justify: center">\(1\)</td><td style="text-align: center; --cell-justify: center">\(1\)</td><td style="text-align: center; --cell-justify: center">\(1\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Red</td><td style="text-align: center; --cell-justify: center">\(2\)</td><td style="text-align: center; --cell-justify: center">\(2\)</td><td style="text-align: center; --cell-justify: center">\(0\)</td></tr></tbody></table></div></div>
<p>Jamie decides to use only \(4\) bricks to arrange in a row. Find the number of different possible arrangements if all \(3\) colours are used.<span class="inline-marks">[4]</span></p>

Legox bricks come in different colours and sizes. Bricks of the same size and colour are identical. Jamie has \(9\) Legox bricks. The following table shows the various colours and sizes of Legox bricks that Jamie has.

Jamie decides to use only \(4\) bricks to arrange in a row. Find the number of different possible arrangements if all \(3\) colours are used.

Permutations of Identical Objects

2020 CJC P2 Q9 [Modified]

<p>Naval signals are made by arranging coloured flags in a vertical line and the flags are read from top to bottom. How many different signals can be made from \(1\) green, \(3\) red and \(2\) blue flags if</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">all \(6\) of the flags are used,</span></div></li><li><div class="list-item-content"><span class="list-item-text">at least \(5\) of them are used?</span></div></li></ol>

Naval signals are made by arranging coloured flags in a vertical line and the flags are read from top to bottom. How many different signals can be made from \(1\) green, \(3\) red and \(2\) blue flags if

<p>A curve \(C\) has parametric equations \(x = 3t^2\), \(y = 2t^3\), for&nbsp;\(t &lt; 0\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch \(C\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Points \(P\) and \(Q\) have parameters \(p\) and \(q\) respectively. Given that&nbsp;\(q = 2p\), find the cartesian equation of the locus of the mid-point of \(PQ\) as&nbsp;\(p\) varies.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A curve \(L\) has equation \(x = y^2 + 1\).</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">It is given that the curves \(C\) and \(L\) touch at the point \(M\). Show that&nbsp;\(4t^6 - 3t^2 + 1 = 0\) at \(M\).<br>Hence, or otherwise, find the exact coordinates of \(M\).<span class="inline-marks">[6]</span></div></li></ol>

A curve \(C\) has parametric equations \(x = 3t^2\), \(y = 2t^3\), for \(t < 0\).

Points \(P\) and \(Q\) have parameters \(p\) and \(q\) respectively. Given that \(q = 2p\), find the cartesian equation of the locus of the mid-point of \(PQ\) as \(p\) varies.

A curve \(L\) has equation \(x = y^2 + 1\).

It is given that the curves \(C\) and \(L\) touch at the point \(M\). Show that \(4t^6 - 3t^2 + 1 = 0\) at \(M\).\nHence, or otherwise, find the exact coordinates of \(M\).

Parametric Equations

N2013 P1 Q11

<p>The government of Country \(X\) uses a particular method to create a unique Identification Number (IN) for each of its citizens. The IN consists of \(4\) digits from \(0\) to \(9\) followed by one of the ten letters \(A - J\). The digits in the IN can be repeated.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the number of different Identification Numbers that can be created.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Suppose the letter at the end of the IN is determined by the following steps:</div><div class="list-item-paragraph">Step 1: Find the sum of all the digits in the IN.<br>Step 2: Divide the sum by \(10\) and note the remainder.<br>Step 3: Use the table below to determine the letter that corresponds to the remainder.</div><div class="table-container"><table style="table-layout: fixed; width: 480px;"><colgroup><col style="width: 80px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"></colgroup><thead><tr style="height: 40px;"><th style="height: 40px; min-height: 40px; text-align: left; --cell-justify: flex-start"><p>Remainder</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(0\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(1\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(2\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(3\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(4\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(5\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(6\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(7\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(8\)</p></th><th style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>\(9\)</p></th></tr></thead><tbody><tr style="height: 40px;"><td style="height: 40px; min-height: 40px; text-align: left; --cell-justify: flex-start"><p>Letter</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>A</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>B</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>C</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>D</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>E</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>F</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>G</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>H</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>I</p></td><td style="height: 40px; min-height: 40px; text-align: center; --cell-justify: center"><p>J</p></td></tr></tbody></table></div><div class="list-item-paragraph">It is known that all citizens who are born in the year \(1990\) must have the digit '\(9\)' appearing twice in their IN and the sum of all the digits in their IN is at least \(20\). Eric, who is born in the year \(1990\), has 'I' as the last letter of his IN. Find the number of INs that could possibly belong to Eric.<span class="inline-marks">[4]</span></div></li></ol>

The government of Country \(X\) uses a particular method to create a unique Identification Number (IN) for each of its citizens. The IN consists of \(4\) digits from \(0\) to \(9\) followed by one of the ten letters \(A - J\). The digits in the IN can be repeated.

Find the number of different Identification Numbers that can be created.

<p>Players \(A\) and \(B\) compete in a racquet match consisting of at most \(3\) sets. Each set is won by either Player \(A\) or \(B\), and the match is won by the first person to win two sets. Player \(A\) has a probability of \(\frac{2}{3}\) of winning each of the first two sets. If the match goes into the third set, Player \(A\) has a probability of \(p\) of winning this set.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">With the aid of a tree diagram, find in terms of \(p\), the probability that player \(A\) will win the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the range of values of the probability that player \(A\) wins the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the value of \(p\) in order for the match to be fair is \(\frac{1}{8}\).</span></div></li></ol>
<p>Given that Player \(A\) wins the match, find, in terms of \(p\), the probability that he won the \(2^{\text{nd}}\) set.</p>

Players \(A\) and \(B\) compete in a racquet match consisting of at most \(3\) sets. Each set is won by either Player \(A\) or \(B\), and the match is won by the first person to win two sets. Player \(A\) has a probability of \(\frac{2}{3}\) of winning each of the first two sets. If the match goes into the third set, Player \(A\) has a probability of \(p\) of winning this set.

With the aid of a tree diagram, find in terms of \(p\), the probability that player \(A\) will win the match.

Deduce the range of values of the probability that player \(A\) wins the match.

Show that the value of \(p\) in order for the match to be fair is \(\frac{1}{8}\).

Given that Player \(A\) wins the match, find, in terms of \(p\), the probability that he won the \(2^{\text{nd}}\) set.

2011 NYJC Prelim

<p>A bag contains \(10\) orange-flavoured, \(14\) strawberry-flavoured, and \(16\) cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.</p>
<p>Calculate the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,</span></div></li><li><div class="list-item-content"><span class="list-item-text">he selects an even number of sweets.</span></div></li></ol>

A bag contains \(10\) orange-flavoured, \(14\) strawberry-flavoured, and \(16\) cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.

the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,

<ol class="abc-list"><li><div class="list-item-paragraph">The sum and product of \(m\) and \(n\) are \(18\) and \(77\) respectively. Find the value(s) of \(m-n\).</div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;\(x^2-6x+1=0\). Without using a calculator, find the value of \(x^4+\frac{1}{x^4}\).</span></div></li></ol>

The sum and product of \(x\) and \(y\) are \(18\) and \(77\) respectively. Without finding the values of \(x\) and \(y\), determine the value(s) of \(x-y\).

It is given that \(x^2-6x+1=0\). Without using a calculator, find the value of \(x^4+\frac{1}{x^4}\).

Sum and Product of Roots

Quadratic Equations and Inequalities

Algebraic Identities

<p>A group of \(8\) people consists of \(4\) boys and \(4\) girls.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">They go to a theatre and sit in the last row of \(10\) adjacent seats at random.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of ways in which they can be seated if they are not all seated together.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that they are not all seated together, find the probability that the boys are seated together and the girls are also seated together.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">At a dinner, the group sits around a square table which has two seats on each side.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of ways in which they can be seated.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the number of ways in which they can be seated if there must be a boy and a girl on each side of the table.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">There are \(3\) different desserts on the menu and every one selects a dessert each. Find the number of different selections that can be made by the group.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

A group of \(8\) people consists of \(4\) boys and \(4\) girls.

They go to a theatre and sit in the last row of \(10\) adjacent seats at random.

At a dinner, the group sits around a square table which has two seats on each side.

2020 VJC P2 Q6

<p>Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.</p>

Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.

Area Enclosed by the Curve and the X-Axis

Definite Integrals and Area under graph

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express \(\frac{11x + 12}{x^2(x + 4)}\) in partial fractions.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find \(\int \frac{11x + 12}{x^2(x + 4)} \, \mathrm{d}x\).</span><span class="inline-marks">[2]</span></div></li></ol>

Express \(\frac{11x + 12}{x^2(x + 4)}\) in partial fractions.

Hence find \(\int \frac{11x + 12}{x^2(x + 4)} \, \mathrm{d}x\).

2025 CHHY PRELIMS P1 Q11

Polynomials, Cubic Equations and Partial Fractions

<p>Sketch the graph of the following pairs of parametric equations.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">\(x = e^t + 1, y = t + 1\), where&nbsp;\(t &gt; 1\)</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(x = \cos^3 t, y = \sin^3 t\), where&nbsp;\(0 &lt; t \leq \pi\)</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(x = 2\theta + \sin(2\theta), y = 3\sin\theta\), where&nbsp;\(-\frac{\pi}{2} \leq \theta &lt; \frac{\pi}{2}\)</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(x = 2 + 2\cos u, y = \tan u\), for&nbsp;\(0 \leq u &lt; \frac{\pi}{2}\)</span><span class="inline-marks">[2]</span></div></li></ol>

Sketch the graph of the following pairs of parametric equations.

\(x = e^t + 1, y = t + 1\), where \(t > 1\)

\(x = \cos^3 t, y = \sin^3 t\), where \(0 < t \leq \pi\)

\(x = 2\theta + \sin(2\theta), y = 3\sin\theta\), where \(-\frac{\pi}{2} \leq \theta < \frac{\pi}{2}\)

\(x = 2 + 2\cos u, y = \tan u\), for \(0 \leq u < \frac{\pi}{2}\)

<p>The curve \(C\) has parametric equations \[x = -\frac{2}{t - 1}, \quad y = \frac{4}{t + 1}, \quad t &lt; 1, \quad t \neq \pm 1.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch a clearly labelled diagram of \(C\), indicating any axial intercepts and asymptotes (if any) of this curve.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find also its cartesian equation, stating any restrictions where applicable.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve \(C\) has parametric equations \[x = -\frac{2}{t - 1}, \quad y = \frac{4}{t + 1}, \quad t < 1, \quad t \neq \pm 1.\]

Sketch a clearly labelled diagram of \(C\), indicating any axial intercepts and asymptotes (if any) of this curve.

Find also its cartesian equation, stating any restrictions where applicable.

2024 DHS Promo Q4

<p>\(4\) distinct green books, \(4\) distinct blue books and \(2\) distinct red books are arranged on a shelf. The green books are always placed together in the same order, but the red books are always separated. Calculate the number of ways in which this can be done.</p>

\(4\) distinct green books, \(4\) distinct blue books and \(2\) distinct red books are arranged on a shelf. The green books are always placed together in the same order, but the red books are always separated. Calculate the number of ways in which this can be done.

<p>A company buys \(p\%\) of its electronic components from supplier \(A\) and the remaining \((100 - p)\%\) from supplier \(B\). The probability that a randomly chosen component supplied by \(A\) is faulty is \(0.05\). The probability that a randomly chosen component supplied by \(B\) is faulty is \(0.03\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(p = 25\), find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of \(p\), the probability that a randomly chosen component that is faulty was supplied by \(A\) is denoted by \(\mathrm{f}(p)\). Show that&nbsp;\(\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}\). Prove by differentiation that&nbsp;\(\mathrm{f}\) is an increasing function for&nbsp;\(0 \leq p \leq 100\), and explain what this statement means in the context of the question.</div></li></ol>

A company buys \(p\%\) of its electronic components from supplier \(A\) and the remaining \((100 - p)\%\) from supplier \(B\). The probability that a randomly chosen component supplied by \(A\) is faulty is \(0.05\). The probability that a randomly chosen component supplied by \(B\) is faulty is \(0.03\).

Given that \(p = 25\), find the probability that a randomly chosen component is faulty.

For a general value of \(p\), the probability that a randomly chosen component that is faulty was supplied by \(A\) is denoted by \(f(p)\). Show that \(f(p) = \frac{0.05p}{0.02p + 3}\). Prove by differentiation that \(f\) is an increasing function for \(0 \leq p \leq 100\), and explain what this statement means in the context of the question.

N2009 P2 Q7

<p>The curve \(C_1\) is given by the equation</p>
<p style="text-align: center">\[y = \frac{ax^2 + bx}{x + c}\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The curve \(C_1\) has a vertical asymptote at \(x = 2\). The linear oblique asymptote of \(C_1\) passes through the points with coordinates \((0, 10)\) and \((-2, 0)\). Determine the values of the unknown constants \(a\), \(b\) and \(c\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence sketch \(C_1\), stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).</span></div></li></ol>

The curve \(C_1\) is given by the equation

The curve \(C_1\) has a vertical asymptote at \(x = 2\). The linear oblique asymptote of \(C_1\) passes through the points with coordinates \((0, 10)\) and \((-2, 0)\). Determine the values of the unknown constants \(a\), \(b\) and \(c\).

Hence sketch \(C_1\), stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).

H1 Math Question Bank