Level 2 Algebra

Merit

(a)

Simplify the following expressions fully, leaving your answer with positive indices.

(i) $\displaystyle\left(\frac{\displaystyle x^6}{\displaystyle x^2 y^{-3}}\right)^2 $

(ii)$ \sqrt[3]{\frac{\displaystyle 8k^5 \times (k^{-2})^4}{\displaystyle k}} $

Merit

(b)

Prove that the roots of the equation $(x-n)^2 - k^2 = 0$ are always $x = n + k$ and $x = n - k$.

Excellence

(c)

The circumference of a tree trunk grows by $15\%$ each year. The initial circumference is $2\pi \text{ cm}$.

(i) Show that an expression for the cross-sectional area of the tree trunk, $A \text{ cm}^2$, after $t$ years is given by $A = \pi(1.3225)^t$.

(ii) Find how long it will take for the cross-sectional area of the trunk to first exceed $15\pi \text{ cm}^2$.

Excellence

(d)

A tangent to the curve $y = x^2 + x + 1$ passes through the origin $(0, 0)$. Use algebra to find the equations of the two possible tangent lines.

Merit

(a)

(i) Solve the following exponential equation for $x$.

$$ e^{2x+1} = 3 $$

(ii) Simplify the following expression fully, writing your answer as a single logarithm.

$$ \log(4a) + 2 \log\left(\frac{\displaystyle a}{\displaystyle 8}\right) $$

Merit

(b)

Express the function $g(x) = 4x^2 + 8x - 1$ in the form $g(x) = a(x + h)^2 + k$, and hence state the coordinates of the vertex of the parabola.

Excellence

(c)

Show that the roots of the equation $-2x^2 + 3k(x+1) + 2 = 0$ are $x = -1$ and $x = \frac{3}{2}k + 1$.

Excellence

(d)

Two children are standing underneath a rope bridge in a playground. The first child is 1.1m tall and the second is 1.3m tall. They are each standing symmetrically 1m either side of the centre of the bridge. The first child is standing 1m from the left tower (the y-axis). The bridge is attached to the left tower 2.2m above the ground. The bridge can be modelled by the equation $f(x) = 0.03x^2 + bx + c$ where $x$ is the horizontal distance from the left tower and $y$ is the vertical distance from the ground. Find the value of $b$ and hence find the height of the bridge above each child's head.

Achieved

(a)

Use the quadratic formula to find the solutions to the equation $3x^2 + 5x = 3$.

Merit

(b)

(i) The value, $V$, of a piece of industrial equipment in dollars is modelled by the function $V = 80000 \times (0.92)^t$, where $t$ is the number of years since the beginning of 2010. Explain what the $0.92$ represents in the context of this function.

(ii) Find the year in which the value of the equipment will drop to less than half of its initial value.

Merit

(c)

Simplify the following algebraic expression into a single fraction:

$$\frac{\displaystyle x - 1}{\displaystyle x} + \frac{\displaystyle x + 5}{\displaystyle x^2} - \frac{\displaystyle 2x - 1}{\displaystyle 3x}$$

Excellence

(d)

(i) Solve the equation$$x^4 - 20x^2 + 84 = 0,$$ and state how many roots the equation has. Give your answers in the form $\pm \sqrt{n}$.

(ii) Consider the equation $$x^4 - 20x^2 + (k^2 - 1) = 0$$ where $k$ is a constant. Find the exact value(s) of $k$ that ensure the equation has exactly three distinct real roots.