PART 1

QUESTION 01

Solve the equation \(x^3 + x^2 - 4x + 2 = 0\).

QUESTION 02

Solve the equation \(\cos 2x + 3 \cos x + 2 = 0 \quad (0 < x < 2 \pi)\).

QUESTION 03

Solve the equation \(3^{2x+1} + 5 \:.\: 3^x - 2 = 0\)

QUESTION 04

Solve the inequality \(4^{x+1} + 11 \:.\: 2^x - 3 > 0\).

QUESTION 05

Solve the equation \((\log_2 x )^2 = \log_4 x^4\).

QUESTION 06

Solve the inequality \(\log_3 (3 - x) + \log_3 (x + 1) < 1\).

QUESTION 07

Let \(\vec a\) and \(\vec b\) be two vectors such as \(\| a \| = 1, \| b \| = 3\) and \(\vec a \cdot \vec b = 2\). Calculate \(\| 2 \vec a - 3 \vec b \|\).

QUESTION 08

The line \(l\) passes through the intersection point of the line \(7x - y = 5\) with the line \(3x + 2y = 7\). The line \(l\) is perpendicular to the line \(x - 2y- 3 = 0\). Find the equation of the line \(l\).

QUESTION 09

The Nth partial sum \(S_N\) of the sequence \(\{ a_n \}\) satisfies the following condition:

\(S_N = 3^N + 2N -1\)

Find the nth term \(a_n\) of the sequence \(\{ a_n \}\).

QUESTION 10

Calculate \(\displaystyle \lim_{x \rightarrow \sim} \left(\sqrt{x^2 + 3x + 4} - x \right)\).

QUESTION 11

Let \(f(x) = \log_e \{x (x + e) \}\). Calculate \(f'(e)\).

QUESTION 12

Calculate \(\displaystyle \int_0^{\dfrac {\pi}{2}} x \cos x \: dx\)