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<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>Sketch $ (3y-1)^2+(x+3)^2=18 $ indicating any turning points and axial intercept(s).</p>

Sketch $ (3y-1)^2+(x+3)^2=18 $ indicating any turning points and axial intercept(s).

Conics Graphs

<p>Sketch $x^2+3x-y^2+12y=5$ indicating any turning points and axial intercept(s).</p>

Sketch $x^2+3x-y^2+12y=5$ indicating any turning points and axial intercept(s).

<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

Timothy Gan/Question/H2 Math/Permutation and Combination/Q017112 [Q017112]

<p>Sketch $ (y+5)^2-(x+2)^2=9  $ indicating any turning points and axial intercept(s).</p>

Sketch $ (y+5)^2-(x+2)^2=9  $ indicating any turning points and axial intercept(s).

<p style="text-align: center">$ \frac{ \left(x-a \right)^2}{p^2}+\frac{ \left(y-b \right)^2}{q^2} = 1  $</p>
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<p>Centre: </p>

$ \frac{ \left(x-a \right)^2}{p^2}+\frac{ \left(y-b \right)^2}{q^2} = 1  $

Ellipse

<p>Sketch ${{(2x-3)}^{2}}+{{(y+3)}^{2}}=15$ indicating any turning points and axial intercept(s).</p>

Sketch ${{(2x-3)}^{2}}+{{(y+3)}^{2}}=15$ indicating any turning points and axial intercept(s).

<p>Sketch $x^2+6x-y^2+3y=25$ indicating any turning points and axial intercept(s).</p>

Sketch $x^2+6x-y^2+3y=25$ indicating any turning points and axial intercept(s).

<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>Sketch the following curve, labelling the coordinates of any axial intercepts and turning points:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$6y-y^2=2x $</div></li><li><div class="list-item-paragraph">$ (y-3)^2=4x+6 $</div></li></ol>

Sketch the following curve, labelling the coordinates of any axial intercepts and turning points:

<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>$12$ students consisting of $4$ from Class $A$, $5$ from Class $B$ and $3$ from Class $C$ participate in a team bonding activity where they are divided into $3$ groups of $4$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of ways such that each group has at most two students from Class $A$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>During lunchtime, all the students from Class $A$ and Class $B$ are seated at a round table with $10$ chairs. Find the number of different seating arrangements if no two students from Class $A$ are seated next to each other if the chairs are</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">all identical,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">numbered.</span><span class="inline-marks">[2]</span></div></li></ol>

$12$ students consisting of $4$ from Class $A$, $5$ from Class $B$ and $3$ from Class $C$ participate in a team bonding activity where they are divided into $3$ groups of $4$.

Find the number of ways such that each group has at most two students from Class $A$.

During lunchtime, all the students from Class $A$ and Class $B$ are seated at a round table with $10$ chairs. Find the number of different seating arrangements if no two students from Class $A$ are seated next to each other if the chairs are

<p>A student wishes to walk from the corner $X$ to the corner $Y$ through streets as given in the street map as shown below. He does not move to the left or down.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of different routes from $X$ to $Y$ the student can take.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The student wishes to visit $P$ on his way from $X$ and $Y$ as indicated in the diagram below. Find the number of different routes he can take.</span></div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1771922792396-8ka47n1mhlo.png" style="max-width: min(320px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

A student wishes to walk from the corner $X$ to the corner $Y$ through streets as given in the street map as shown below. He does not move to the left or down.

Find the number of different routes from $X$ to $Y$ the student can take.

The student wishes to visit $P$ on his way from $X$ and $Y$ as indicated in the diagram below. Find the number of different routes he can take.

<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