I. 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 \(\displaystyle \int_0^{\dfrac {\pi}{2}} x \cos x \: dx\)
II. PART 2
Let \(A = \begin{pmatrix} 3 & 2 \\ -1 & 0 \end{pmatrix}\), \(B = \begin{pmatrix} a & -2 \\ 1 & 2 \end{pmatrix}\) dan \(C = \begin{pmatrix} b & 2 \\ -1 & -1 \end{pmatrix}\). satisfying the following condition.
\(B^2 = B\)
\(BC = CB = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)
QUESTION 01
Find \(a\) and \(b\).
QUESTION 02
Suppose \(A = xB + yC\). Find \(x\) and \(y\).
QUESTION 03
Find \(A^5\).
III. PART 3
Let \(f(x) = \dfrac {\log_e x}{x} \quad (x > 0)\)
QUESTION 01
Find the maximum value \(M\) of \(f(x).
QUESTION 02
Find the tangent line \(l\) to the curve \(y = f(x)\) passing through the point \((0,0)\).
QUESTION 03
Calculate the area \(S\) among the curve \(y = f(x)\), the line \(l\), and the x-axis.