QUESTIONS AND SOLUTIONS

 

I.   PART 1

 

QUESTION 01

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

 

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QUESTION 02

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

 

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QUESTION 03

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

 

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QUESTION 04

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

 

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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?

 

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QUESTION 06

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

 

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QUESTION 07

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

 

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QUESTION 08

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

 

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QUESTION 09

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

 

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QUESTION 10

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

 

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QUESTION 11

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

 

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QUESTION 12

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

 

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II.   PART 2

For \(A = \begin{pmatrix} \dfrac 12 & 0 \\ 0 & \dfrac 12 \end{pmatrix}\) answer the following questions.

 

QUESTION 01

Calculate \(A^n\).

 

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QUESTION 02

Calculate \(\displaystyle S = \sum_{k = 1}^{n} A^k\).

 

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QUESTION 03

Calculate the inverse \(S^{-1}\) of the matrix \(\displaystyle S = \sum_{k = 1}^{n} A^k\).

 

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III.   PART 3

For any natural number \(k > 0\), let \(I_{2k + 1} = \dfrac {2k}{2k + 1} \:.\: \dfrac {2k - 2}{2k - 1} \dotso \dfrac 45 \:.\: \dfrac 23\) and \(I_{2k} = \dfrac {2k - 1}{2k} \:.\: \dfrac {2k - 3}{2k - 2} \dotso \dfrac 34 \:.\: \dfrac 12 \:.\: \dfrac {\pi}{2}\). Answer the following questions.

 

QUESTION 01

Caluculate \(\displaystyle \int_0^{\dfrac {\pi}{2}} \sin^3 x \: dx\)

 

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QUESTION 02

Find \(a_k\) which satisfies \(I_{2k + 1} \:.\: I_{2k} = \dfrac {\pi}{2} \:.\: a_k\).

 

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QUESTION 03

Find \(b_k\) which satisfies \(I_{2k - 1} \:.\: I_{2k} = \dfrac {\pi}{2} \:.\: b_k\)

 

QUESTION 04

Calculate \(\displaystyle \lim_{k \rightarrow \sim} \dfrac 1k \left \{ \dfrac {(2k)(2k - 2) \dotso 4 \:.\: 2}{(2k - 1)(2k - 3) \dotso 3 \:.\: 1}  \right \}^2 \) by assuming \(I_{2k + 1} < I_{2k} < I_{2k - 1}\).

 

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