The curve $$y = x^3 + px^2 + 2$$ has a stationary point at $x = 4$. Find $p$ and determine the nature of the stationary point.
$p = -6$, min
A line $l$ has gradient $-2$ and passes through the point $A(3,5)$. $B$ is a point on $l$ such that the distance $AB$ is $6\sqrt{5}$. Find the possible coordinates of $B$.
$(-3, 17)$ or $(9,-7)$
Prove by contradiction that $\sqrt[3]{2}$ is irrational.
$\sqrt[3]{2} = \dfrac{a}{b}$ and manipulate to find $a$ and $b$ are both even.
- Express $5x^2 + 20x - 8$ in the form $a(x+p)^2 + q$.
- State the equation of the line of symmetry of $y = 5x^2 + 20x - 8$.
- Calculate the discriminant of $5x^2 + 20x - 8$.
- State the number of real roots of the equation $5x^2 + 20x - 8 = 0$.
- $5(x+2)^2 - 28$
- $x = -2$
- $560$
- $2$
One of the terms in the binomial expansion of $(4 + ax)^6$ is $160x^3$. - Find $a$
- Hence find the first two terms in ascending powers of $x$ in the expansion.