Instructions: Do not use a calculator. You have 25 minutes. If you get stuck on the algebra, step back and ask yourself: "What is the graphical meaning of this question?"

Q1:

<aside> ❓

Find the range of values of $k$ for which the line $y = kx - 5$ does not intersect the curve $y = x^2 - 3x + 4$.

</aside>

• Answer:

Q2:

<aside> ❓

Solve the following equation for $x$:

$$ 4^x - 5(2^x) + 4 = 0 $$

</aside>

• Answer:

Q3:

<aside> ❓

Find the range of values for $p$ such that the quadratic expression $px^2 - 4x + p$ is strictly positive for all real values of $x$.

</aside>

• Answer:

Q4:

<aside> ❓

The roots of the equation $2x^2 - 5x + 1 = 0$ are $\alpha$ and $beta$. Find a new quadratic equation with integer coefficients whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$

</aside>

• Answer:

Q5: The Double Inequality

<aside> ❓

Solve the inequality:

$$ x^4 - 13x^2 + 36 \le 0 $$

</aside>

• Answer::