7912
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2011-07-19T08:40:06
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<mdoxml version="1.0"><p>STEP Easter School Examination</p>
<p>STEVE -- SEE ALSO STEP Easter school fluency test</p>
<p> </p>
<p>Numbers</p>
<p>Write down or compute the first 20 square numbers</p>
<p>Write down or compute the the 20 prime numbers</p>
<p>Write down or compute the first 10 cube numbers</p>
<p> </p>
<p>(2000, I0</p>
<p>Show that the coefficient of $x^{-12}$ is $-15$ in the expansion of</p>
<p>$$\left(x^4-\frac{1}{x^2}\right)^5\left(x-\frac{1}{x}\right)^6$$</p>
<p> </p>
<p>Express a simple fraction in lowest form</p>
<p>$$\frac{1}{3}\cdot\left(1-\frac{1}{9}\right)^2+2\cdot\left(1-\frac{1}{3}\right)\cdot\frac{1}{9}\cdot \left(1-\frac{1}{9}\right)$$</p>
<p> </p>
<p>Algebraic fluency</p>
<p>Expand and simplify $$\left(\sqrt{2}+\sqrt{3} + \sqrt{5}\right)^4$$</p>
<p> </p>
<p> </p>
<p> </p>
<p>(02III2)</p>
<p>If $a=\frac{n}{n+2}$ and $b=\frac{n+1}{n+3}$ compute and simplify</p>
<p>$$\frac{a+b}{1-ab}$$</p>
<p> </p>
<p>If $x^2+y^2+2axy=0$, find $y'$.</p>
<p> </p>
<p> </p>
<p>2006 I14</p>
<p>Find the maximum value of</p>
<p>$$P(n)=\left(\frac{n}{n+1}\right)^{r-1}\frac{1}{n+1}$$ where $r$ is a constant greater than 1.</p>
<p> </p>
<p> </p>
<p>07 I 4</p>
<p>Write $x^3-3xbc+b^3+c^3$ as a product of $(x+b+c)$ with another factor.</p>
<p> </p>
<p>07 III 1</p>
<p>Expand $(1-4x)^{-1}{2}$, finding the coeffients of the the first 5 terms as fractions in their lowest terms</p>
<p> </p>
<p>07</p>
<p>IF $x=a(\cos t+\ln \tan\frac{1}{2}y)$ and $y=a\sin t$, find $\frac{dx}{dt}$, $\frac{dy}{dt}$ and thence $\frac{dy}{dx}$ (tan t)</p>
<p> </p>
<p>00 I 4</p>
<p>Find the maximum value of $f(x) = \frac{x^6}{(x^2+1)^4}$</p>
<p>04 I III</p>
<p> </p>
<p>i Express $\left(3+2\sqrt{5}\right)^3$ in the form $a+b\sqrt{5}$, where $a$ and $b$ are integers.</p>
<p> </p>
<p>04 I 3</p>
<p>Let</p>
<p>$$P(x) = x^3-5x^2+2x^2y+xy^2-8xy-3y^2+6x+6y$$</p>
<p>Given that $P(3)=0$, write $P(x)$ as a product of 3 linear factors.</p>
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<h3>Why do this problem?</h3>
<p>Proof is, of course, a central part of mathematics. However constructing proofs is often difficult for novices. This problem provides a bridge through the device of classic faulty proofs of 'obviously' wrong results. Analysing these faulty proofs will provide training in reading proofs, raise awareness of mathematical hazards, such as division by zero, and provide motivation for rigour.</p>
<p> </p>
<h3>Possible approach</h3>
<div>Make it a race to find errors in as many proofs as possible in the available time. Split the class into small groups and give them one of the first three proofs to work on. Let everyone know that the next proof will be given out when the group has convinced you of the location of the error.</div>
<div> </div>
<div>When a group wishes to try to convince you of an error choose one member of the group at random: only this member is permitted to speak or write. If you are convinced, give the group the next dodgy proof; otherwise give minimal feedback such as "Sorry, I'm not convinced" or "That explanation didn't seem clear to me" and leave them to attempt to tighten up their argument.</div>
<div> </div>
<div>Use common sense to judge the acceptable level of rigour.</div>
<div> </div>
<div>Be on the lookout for the tell-tale signs of 'um', 'er', 'the thing' and 'it' which so often indicate confused thinking.</div>
<div> </div>
<div> </div>
<div> </div>
<h3>Key questions</h3>
<div>Does that argument sound clear to you? (students will probably know when they are bluffing!)</div>
<div> </div>
<h3>Possible extension</h3>
<div>Students might be asked to invent their own dodgy proofs.</div>
<div> </div>
<h3>Possible support</h3>
<div>You could simply discuss the proofs as a group or you could draft in some helpers who assist you in deciding if explanations are acceptably clear.</div>
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