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<p>$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$</p>

$ \displaystyle\int{x\sqrt{2x-1}\mathrm{d}x}$

Integration by Parts

Techniques of  Integration

<p>The roots of the quadratic equation $x^2+4x-1=0$ are $\alpha$ and $\beta$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the values of $\alpha+\beta$ and of $\alpha\beta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Without solving the equation, find the value of each of the following.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">$\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}$</div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">$\left(\alpha-\dfrac{2}{\beta^2}\right)\left(\beta-\dfrac{2}{\alpha^2}\right)$</div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Determine the quadratic equation with integer coefficients which has roots $\alpha-\dfrac{2}{\beta^2}$ and $\beta-\dfrac{2}{\alpha^2}$.</div></li></ol>

The roots of the quadratic equation $x^2+4x-1=0$ are $\alpha$ and $\beta$.

Write down the values of $\alpha+\beta$ and of $\alpha\beta$.

Without solving the equation, find the value of each of the following.

Determine the quadratic equation with integer coefficients which has roots $\alpha-\dfrac{2}{\beta^2}$ and $\beta-\dfrac{2}{\alpha^2}$.

Sum and Product of Roots

Structured chain to an equation with adjusted roots

Quadratic Functions, Equations and Inequalities

<p>The roots of the equation $2x^4-4x^3-5x^2+6x-10=0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the values of $\sum\alpha$, $\sum\alpha\beta$, $\sum\alpha\beta\gamma$ and $\alpha\beta\gamma\delta$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence find the value of $\alpha^2+\beta^2+\gamma^2+\delta^2$.</div></li></ol>

The roots of the equation $2x^4-4x^3-5x^2+6x-10=0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$.

Root sums and products of a quartic

Polynomials, Cubic Equations and Partial Fractions

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<p>The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.

Find the equation of the perpendicular bisector of $AB$.

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Find $\int x\left[ (1-3x^2)^5 + \mathrm{e}^{x^2+1} \right] \mathrm{d}x$.

Find $\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}(x^2)$. Hence, or otherwise, find the exact value of $\int_{0}^{\frac{1}{\sqrt{2}}} \frac{x}{\sqrt{(1-x^4)}} \mathrm{d}x$.

2016 DHS C1BT Q2

<p>Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.</p>

Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.

Partial fractions with a repeated linear factor

<p>Given that&nbsp;$\mathrm{f}(x) = 3^x$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(-|x|)$ on separate diagrams. Clearly indicate the intercept(s) on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = 3^x$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(-|x|)$ on separate diagrams. Clearly indicate the intercept(s) on the graph.

Transformations of Graphs - General

Transformations of Graphs

<p>Given that&nbsp;$\mathrm{f}(x) = x(x + 3)(x - 2)$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = x(x + 3)(x - 2)$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.

<p>Given that&nbsp;$\mathrm{f}(x) = \dfrac{x + 3}{x - 2}$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.<span class="inline-marks">[4]</span></p>

Given that $\mathrm{f}(x) = \dfrac{x + 3}{x - 2}$, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}(|x|)$ on separate diagrams. Clearly indicate the intercepts and asymptotes on the graph.

<p>The diagram below shows the graph of $y = \mathrm{f}(x)$.</p>
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<p>Sketch each of the following on a separate diagram, clearly indicating the $y$-intercept, turning point(s) and asymptote(s).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$y = 2\mathrm{f}(x - 2) + 1$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$y = -\mathrm{f}(x - 1) + 2$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$y = \mathrm{f}(-x + 1) - 2$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">$y = \mathrm{f}(2x - 2) + 1$<span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows the graph of $y = \mathrm{f}(x)$.

Sketch each of the following on a separate diagram, clearly indicating the $y$-intercept, turning point(s) and asymptote(s).

Reciprocal Graph, y=1⁄f(x)

<ol class="i-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where&nbsp;$r &gt; 0$ and $y \le r$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A hemispherical bowl of fixed radius $r\text{ cm}$ is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate.</p>
<p>Use your graph in part <strong>(i)</strong> to show that when the depth of water is $h\text{ cm}$ (where&nbsp;$h \le r$), the volume of water in the bowl is given by</p>
<p style="text-align: center">\[V = \frac{\pi h^2}{3}(3r - h),\]</p>
<p style="text-align: right">[4]</p>
<ol class="i-list" start="2"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Given that a full bowl of water would become empty in $24\text{ s}$, find the rate of decrease, in terms of $r$ and $h$, of the depth of water in the bowl at the instant when the depth of water is $h\text{ cm}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Without any differentiation, determine, in terms of $r$, the slowest rate at which the depth of water is decreasing.<span class="inline-marks">[2]</span></div></li></ol>

Sketch the graph with equation $x^2 + (y - r)^2 = r^2$, where $r > 0$ and $y \le r$.

A hemispherical bowl of fixed radius $r\text{ cm}$ is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate.

Use your graph in part **(i)** to show that when the depth of water is $h\text{ cm}$ (where $h \le r$), the volume of water in the bowl is given by

Given that a full bowl of water would become empty in $24\text{ s}$, find the rate of decrease, in terms of $r$ and $h$, of the depth of water in the bowl at the instant when the depth of water is $h\text{ cm}$.

Without any differentiation, determine, in terms of $r$, the slowest rate at which the depth of water is decreasing.

2016 VJC P1 Q9

Applications of Differentiation

The polynomial P(x)=x3+ax2-5x+6 is divisible by x-2. Find a and solve P(x)=0.

The polynomial $P(x)=x^3+ax^2-5x+6$ is divisible by $x-2$. Find $a$ and solve $P(x)=0$.

Unknown coefficient and remaining roots

A drink cools according to T=22+58e-0.18t, where T is in degrees Celsius and t is in minutes. Find when T=35.

A drink cools according to $T=22+58e^{-0.18t}$, where $T$ is in degrees Celsius and $t$ is in minutes. Find when $T=35$.

Math Question Bank