A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<p>It is given that&nbsp;$y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ and label clearly the equations of the asymptotes and intercepts if any.</div></li><li><div class="list-item-paragraph">The graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ cannot lie between values $p$ and $m$, where&nbsp;$p&gt;m$ . State the value of $p$ and $m$ .</div></li><li><div class="list-item-paragraph">By means of a graphical argument, state the maximum number of real roots the equation $kx\left( x-2 \right)-2{{x}^{2}}=3$ have.</div></li></ol>

It is given that $y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.

Parametric Equations

ACJC Tutorial 2 Q8

Graphing Techniques

<p>Given that&nbsp;$b$ is a real constant such that&nbsp;$0 &lt; b &lt; 4$, describe fully a sequence of transformations that transforms the curve $y=x^2$ to the curve $y=4x^2+bx+1$.<span class="inline-marks">[4]</span></p>
<p>Sketch the curve $y=\frac{1}{4x^2+bx+1}$. Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of $b$ where appropriate.<span class="inline-marks">[3]</span></p>

Given that $b$ is a real constant such that $0 < b < 4$, describe fully a sequence of transformations that transforms the curve $y=x^2$ to the curve $y=4x^2+bx+1$.

Sketch the curve $y=\frac{1}{4x^2+bx+1}$. Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of $b$ where appropriate.

2025 HCI P1 Q6

Transformations of Graphs

<p>The curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{t}^{2}}+4t$, $y={{t}^{3}}+{{t}^{2}}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the curve for $-2\le t\le 1$.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>The tangent to the curve at point $P$ where $t=2$ is denoted by $l$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of $l$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.</span></div><p style="text-align: right;">[4]</p></li></ol>

$x={{t}^{2}}+4t$, $y={{t}^{3}}+{{t}^{2}}$.

The tangent to the curve at point $P$ where $t=2$ is denoted by $l$.

The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.

N2009 P2 Q1

<p>A curve $C$ has parametric equations $x=\tan \theta +1$, $y=\sec \theta $.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find a cartesian equation of $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, giving the coordinates of any points where $C$ crosses the $x$- and $y$-axes and the equations of any asymptotes.</span></div></li></ol>

A curve $C$ has parametric equations $x=\tan \theta +1$, $y=\sec \theta $.

Sketch $C$, giving the coordinates of any points where $C$ crosses the $x$- and $y$-axes and the equations of any asymptotes.

NYJC Advanced Graphing Techniques Self-Review 7

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{2-{{x}^{2}}}{2x+1}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{2-{{x}^{2}}}{2x+1}$ and sketch the curve.

<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$a$ is a positive constant, sketch the curve with equation $y=\frac{x+a}{x-a}$. State the equations of any asymptotes and the coordinates of the points where the curve crosses the axes.<span class="inline-marks">[3]</span></div></li><li><ol class="abc-list"><li><div class="list-item-paragraph">Solve the inequality $\left| \frac{x+a}{x-a} \right|&gt;1$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Solve the inequality $\frac{\left| x \right|+a}{\left| x \right|-a}&gt;1$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

2021 VJC P1 Q2

<p>A curve has equation $y = x + 1 + \frac{3}{x-1}$. Using differentiation, find the set of values of $x$ where the curve is strictly decreasing. Give your answers in exact form.<span class="inline-marks">[4]</span></p>

A curve has equation $y = x + 1 + \frac{3}{x-1}$. Using differentiation, find the set of values of $x$ where the curve is strictly decreasing. Give your answers in exact form.

2025 HCI P1 Q1

Techniques of Differentiation

<p>The curve $C$ has equation $\frac{9{{\left( y+3 \right)}^{2}}}{4}-\frac{{{\left( x-2 \right)}^{2}}}{4}=1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$, stating the equations of any asymptotes and the coordinates of any turning points.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence find the set of values of $m$ such that the straight line with gradient $m$ that passes through the point $\left( 2,-3 \right)$ intersects $C$ at least once.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $\frac{9{{\left( y+3 \right)}^{2}}}{4}-\frac{{{\left( x-2 \right)}^{2}}}{4}=1$.

Sketch $C$, stating the equations of any asymptotes and the coordinates of any turning points.

Hence find the set of values of $m$ such that the straight line with gradient $m$ that passes through the point $\left( 2,-3 \right)$ intersects $C$ at least once.

2022 VJC Promo Q3

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{5}{{{x}^{2}}+2x+5}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{5}{{{x}^{2}}+2x+5}$ and sketch the curve.

<p>The parametric equations of a curve are $x=\frac{1}{2}{{\mathrm{e}}^{t}}-2{{\mathrm{e}}^{-t}}$ and $y={{\mathrm{e}}^{t}}+2{{\mathrm{e}}^{-t}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Using calculus, find the exact gradient of the normal to the curve at the point where&nbsp;$t=\ln 4$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that the equation of the normal is $5x+7y=39$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $t$ where the normal meets the curve again.<span class="inline-marks">[4]</span></div></li></ol>

The parametric equations of a curve are $x=\frac{1}{2}{{\mathrm{e}}^{t}}-2{{\mathrm{e}}^{-t}}$ and $y={{\mathrm{e}}^{t}}+2{{\mathrm{e}}^{-t}}$.

2023 NYJC Promo Q7

<p>The curve $C$ has equation ${\frac{{{\left( x-2 \right)}^{2}}}{2}+{{\left( y-3 \right)}^{2}}=1}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760868495354-qu7i0wz74" data-list-item-path="item-1760868495354-qu7i0wz74"><div class="list-item-paragraph">Sketch $C$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760868495916-nthzlmc19" data-list-item-path="item-1760868495916-nthzlmc19"><div class="list-item-paragraph">Hence state the range of positive values of $k$ such that&nbsp;${{\left( x-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}={{k}^{2}}$ intersects $C$ at twice.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation ${\frac{{{\left( x-2 \right)}^{2}}}{2}+{{\left( y-3 \right)}^{2}}=1}$.

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}-3x+3}{1-x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760426225255-w8z8w80qh" data-list-item-path="item-1760426225255-w8z8w80qh"><div class="list-item-paragraph">Sketch $C$, stating the coordinates of any turning points and points of intersection with the axes, and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760426225968-2ct1ijudn" data-list-item-path="item-1760426225968-2ct1ijudn"><div class="list-item-paragraph">Determine the set of exact values of $k$ such that&nbsp;$C$ and the line $y=kx$ has two points of intersection for all real values of $x$.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}-3x+3}{1-x}$.

2023 DHS P1 Q2

<p>A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the curve.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\displaystyle\int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\mathrm{t}\,\,\mathrm{d}t}$, and use the substituition $\sin t=u$ to find this area.</div><p style="text-align: right;">[5]</p></li></ol>

A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.

N2007 P1 Q11

<p>A curve is given by the parametric equations $x={{t}^{2}}-3,\,\,y=t({{t}^{2}}-3)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the Cartesian equation.</div></li><li><div class="list-item-paragraph">Prove that the curve is symmetrical about the $x$-axis.</div></li><li><div class="list-item-paragraph">Show that there are no points on the curve for which $x < -3$.</div></li></ol>

A curve is given by the parametric equations $x={{t}^{2}}-3,\,\,y=t({{t}^{2}}-3)$.

ACJC Tutorial Q2

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{2{{x}^{2}}+1}{{{x}^{2}}+3}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{2{{x}^{2}}+1}{{{x}^{2}}+3}$ and sketch the curve.

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=t-\frac{10}{t}$, $\qquad$ $y=6t-{{t}^{2}}$, $\qquad$ for&nbsp;$\sqrt{10}\le t\le 6$.</p>
<ol class="i-list"><li data-list-item-id="item-1760262090022-a6ijrsd1t" data-list-item-path="item-1760262090022-a6ijrsd1t"><div class="list-item-paragraph">Sketch the curve $C$, labelling the coordinates of any points of intersection with the axes.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that a plot of flat land is represented by the region bounded by $C$ and both axes, where units are measured in kilometres.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760262103663-jn03lcj6x" data-list-item-path="item-1760262103663-jn03lcj6x"><div class="list-item-paragraph">Find the area of the plot of land.<span class="inline-marks">[3]</span></div></li></ol>
<p>The landowner considers using a portion of this land to rear cattle. This portion is a rectangular piece of land, $OPRQ$, where&nbsp;$O$ is the origin, $R$ is a point on the curve, $P$ and $Q$ are the points on the $x$-axis and $y$-axis respectively.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760262117075-h1fxfk9pr" data-list-item-path="item-1760262117075-h1fxfk9pr"><div class="list-item-paragraph">By using differentiation, find the value of $t$ that maximises the area of rectangle $OPRQ$. (You need not show that your answer gives a maximum.)<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given instead that the landowner decides to divide the plot of flat land equally into two by building a straight fence through the land, parallel to the $y$-axis. It is assumed that the thickness of the fence is negligible.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760262139188-doonvxy5a" data-list-item-path="item-1760262139188-doonvxy5a"><div class="list-item-paragraph">Using the result from part <strong>(ii)</strong>, find the equation of the fence that divides the plot of land equally into two parts.<span class="inline-marks">[4]</span></div></li></ol>

$x=t-\frac{10}{t}$, $\qquad$ $y=6t-{{t}^{2}}$, $\qquad$ for $\sqrt{10}\le t\le 6$.

It is given that a plot of flat land is represented by the region bounded by $C$ and both axes, where units are measured in kilometres.

The landowner considers using a portion of this land to rear cattle. This portion is a rectangular piece of land, $OPRQ$, where $O$ is the origin, $R$ is a point on the curve, $P$ and $Q$ are the points on the $x$-axis and $y$-axis respectively.

It is given instead that the landowner decides to divide the plot of flat land equally into two by building a straight fence through the land, parallel to the $y$-axis. It is assumed that the thickness of the fence is negligible.

2023 MI P1 Q12

<p>The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-paragraph">It is given that&nbsp;$C$ has a tangent which is parallel to the $y$-axis. Show that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that&nbsp;$k \leq \frac{1}{54}$.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-c-9"><div class="list-item-content"><span class="list-item-text">Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.</span><span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where $k$ is a constant.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

It is given that $C$ has a tangent which is parallel to the $y$-axis.\n\nShow that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that $k \leq \frac{1}{54}$.

Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.

Math Question Bank