PART 1

QUESTION 01

Solve the equation \(x^3 - 2x^2 - x + 2 = 0\).

QUESTION 02

Solve the equation \(\cos x - 2 \cos 2x = 0 \quad (0 < x < \pi)\).

QUESTION 03

Express \(\left | \sqrt{8} - 3 \right | + \left | 2 - \sqrt{2} \right|\) without the absolute value symbols.

QUESTION 04

Solve the equation \(\log_2 (x - 1) = \log_4 (x - 1)\).

QUESTION 05

Find the maximum value \(m\) of the function \(f(x) = \cos x + \cos \left (x + \dfrac {\pi}{3} \right) \quad (0 \leq x < 2 \pi)\). Also, at what values of \(x\) does \(f(x)\) have the maximum?

QUESTION 06

By using \(\displaystyle \lim_{t \rightarrow 0} (1 + t)^{\dfrac 1t} = e\), calculate \(\displaystyle \lim_{h \rightarrow 0} (1 + 2h)^{\dfrac 1h} = e\).

QUESTION 07

Find the intersection point of the line \(\dfrac {x - 1}{6} = \dfrac {y - 1}{2} = \dfrac {z - 2}{3}\) and the plane \(x + 2y - 4z + 1 = 0\).

QUESTION 08

Find the tangent line to the curve \(y = \log_e x\) which goes through the point (0, 0).

QUESTION 09

Calculate \(\displaystyle \sum_{n = 1}^{\sim} \dfrac {1}{n(n + 2)}\)

QUESTION 10

Calculate \(\displaystyle \lim_{x \rightarrow - \sim} \dfrac {2x + 1}{\sqrt{x^2 + 1}}\)

QUESTION 11

Let \(f(x) = \log_e \dfrac{\sqrt{x - 1}}{x + 1}\). Calculate \(f'(x)\).

QUESTION 12

Calculate \(\displaystyle \int_{- \pi}^{\pi} \sin 3x \sin x \: dx\)