Category: QotD

AA HL · Integration Find the indefinite integral:     Hint First complete the square:     Then use the substitution     Solution First complete the square:     Therefore     Let     Then     Also,     Hence     Using the identity     we obtain     Since…

AA HL · Calculus · Related Rates Water is evaporating from a cup in the shape of an inverted cone. The rate of evaporation is proportional to the area of the surface of the water. Show that the depth of the water decreases at a constant rate that does not depend on the dimensions of…

AA SL · Sequences · Logarithms An arithmetic sequence has first term and common difference . The 13th term of the sequence is . Find the value of . Hint Use the formula for the th term of an arithmetic sequence:     You will also need to simplify . Solution Using     we…

AI HL · Probability · Poisson Distribution The number of marking errors made by a teacher on a piece of homework follows a Poisson distribution with mean 1.6. The teacher marks one piece of homework from each of 13 pupils. Find the probability that fewer than half of the pieces of homework contain at least…

AA HL · Complex Numbers · Trigonometry (a) Use de Moivre’s theorem to prove that \[ \cos 5\theta = \cos\theta \left( 16\cos^4\theta-20\cos^2\theta+5 \right). \] (b) By considering the equation \(\cos 5\theta=0\), show that \[ \cos^2\left(\frac{\pi}{10}\right) = \frac{5+\sqrt5}{8}. \]