STEP Easter School Examination
STEVE -- SEE ALSO STEP Easter school fluency test
Numbers
Write down or compute the first 20 square numbers
Write down or compute the the 20 prime numbers
Write down or compute the first 10 cube numbers
(2000, I0
Show that the coefficient of $x^{-12}$ is $-15$ in the expansion of
$$\left(x^4-\frac{1}{x^2}\right)^5\left(x-\frac{1}{x}\right)^6$$
Express a simple fraction in lowest form
$$\frac{1}{3}\cdot\left(1-\frac{1}{9}\right)^2+2\cdot\left(1-\frac{1}{3}\right)\cdot\frac{1}{9}\cdot \left(1-\frac{1}{9}\right)$$
Algebraic fluency
Expand and simplify $$\left(\sqrt{2}+\sqrt{3} + \sqrt{5}\right)^4$$
(02III2)
If $a=\frac{n}{n+2}$ and $b=\frac{n+1}{n+3}$ compute and simplify
$$\frac{a+b}{1-ab}$$
If $x^2+y^2+2axy=0$, find $y'$.
2006 I14
Find the maximum value of
$$P(n)=\left(\frac{n}{n+1}\right)^{r-1}\frac{1}{n+1}$$ where $r$ is a constant greater than 1.
07 I 4
Write $x^3-3xbc+b^3+c^3$ as a product of $(x+b+c)$ with another factor.
07 III 1
Expand $(1-4x)^{-1}{2}$, finding the coeffients of the the first 5 terms as fractions in their lowest terms
07
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)
00 I 4
Find the maximum value of $f(x) = \frac{x^6}{(x^2+1)^4}$
04 I III
i Express $\left(3+2\sqrt{5}\right)^3$ in the form $a+b\sqrt{5}$, where $a$ and $b$ are integers.
04 I 3
Let
$$P(x) = x^3-5x^2+2x^2y+xy^2-8xy-3y^2+6x+6y$$
Given that $P(3)=0$, write $P(x)$ as a product of 3 linear factors.