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Problems of the Day (29 May 2024)

  1. There are three dice A, B, and C, each with six faces. Die A has three faces marked with 2, two faces with 4, and one face marked with 6. Die B has one face marked with 1, three faces marked with 2, and two faces marked with 3. Die C has two faces marked with 1 and four faces marked with 2. When the three dice are thrown randomly once, let \( E \) be the event of getting a sum of the numbers appearing on top faces equal to 8. Find \(P(E)\).

    Solution:

    \( n(E) = \text{coefficient of } x^8 \text{ in } \)

    \( (3x^2 + 2x^4 + x^6)(x + 3x^2 + 2x^3)(2x + 4x^2) \)

    \( = \text{coefficient of } x^4 \text{ in } \)

    \( (3 + 2x^2 + x^4)(1 + 3x + 2x^2)(2 + 4x) \)

    \( = \text{coefficient of } x^4 \text{ in } \)

    \( (3 + 2x^2 + x^4)(2 + 10x + 16x^2 + 8x^3) \)

    \( = 2 \times 16 \)

    \( = 32 \)

    \( \therefore P(E) = \frac{32}{6^3} = \frac{4}{27} \)

  2. Let \( f(x) = \frac{\sin x + \sin 3x + \sin 5x + \sin 7x}{\cos x + \cos 3x + \cos 5x + \cos 7x} \)

    Find the fundamental period of f.

    Solution:

    \( f(1) = \frac{(\sin x + \sin 7x) + (\sin 3x + \sin 5x)}{(\cos x + \cos 7x) + (\cos 3x + \cos 5x)} \)

    \( = \frac{2 \sin 4x \cos 3x + 2 \sin 4x \cos x}{2 \cos 4x \cos 3x + 2 \cos 4x \cos x} \)

    \( = \frac{2 \sin 4x (\cos 3x + \cos x)}{2 \cos 4x (\cos 3x + \cos x)} \)

    \( = \tan 4x\frac{\cos 2x \cos x}{\cos 2x \cos x} \)

    So the given function is essentially equal to \( \tan 4x \) only. It looks like that the fundamental period is \( \frac{\pi}{4} \). When we look closely we find out that whenever \( \cos 2x \cos x = 0 \) then \( f \) is not defined. This results in a fundamental period \( \pi \) as can be seen on the graph.

  3. Let \( A \) be a square matrix of order 3 satisfies the matrix equation \( A^3 - 6A^2 + 7A - 81I = 0 \). The the value of \( \text{det}(\text{adj}(I - 2A^{-1})) \) is equal to ....

    Solution:

    Observe that \( \text{adj}(I - 2A^{-1}) \) = \( |I - 2A^{-1}|^2 \)

    \(= \frac{|A - 2I|^2}{|A|^2} \)

    The characteristic equation of \( A \) is given as \( \lambda^3 - 6\lambda^2 + 7\lambda - 8 = 0 \)

    \( \Rightarrow |A| = -(-8) = 8 \)

    \( A^3 - 6A^2 + 7A - 8I = 0 \)

    \( \Rightarrow (A - 2I)(A^2 - 4A - I) - 10I = 0 \)

    \( \Rightarrow (A - 2I) \left((A - 2I)^2 - 5I\right) - 10I = 0 \)

    \( \Rightarrow X(X^2 - 5I) - 10I = 0 \)

    Where \( X = A - 2I \)

    \( \Rightarrow X^3 - 5X - 10I = 0 \)

    Which is the characteristic eqn of \( X \)

    Thus \( |X| = |A - 2I| = -10 \)

    \( \therefore \text{det}(\text{adj}(I - 2A^{-1})) = \frac{|A - 2I|^2}{|A|^2} = \frac{100}{64} = \frac{25}{16} \)