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Tim Gan Math

1984 A-Level H2 Mathematics

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Question 1Techniques of Differentiation10 marks
  1. The sum, ${{S}_{n}}$, of the first $n$ terms of an arithmetic progression is given by ${{S}_{n}}=pn+q{{n}^{2}}$.
    Given also that ${{S}_{3}}=6$ and ${{S}_{5}}=11$,
    1. Find the values of $p$ and $q$.

      [3]

    2. Deduce, or find otherwise, an expression for the $n$-th term and the value of the common difference.

      [3]

  2. Find the set of values of $\theta $ lying in the interval $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi $ such that the sum to infinity of the geometric series $1+\sin \theta +{{\sin }^{2}}\theta +\ldots $ is greater than $2$.

    [4]

Solution substep 1, image 1
Solution substep 2, image 1
Solution substep 2, image 2
Solution step 2, image 1
Solution step 2, image 2

Final Answer:

(a)(i) \(p = \frac{17}{10}\), \(q = \frac{1}{10}\) (a)(ii) \(u_n = \frac{1}{5}n + \frac{8}{5}\), \(d = \frac{1}{5}\) (b) \(\left\{\theta \in \mathbb{R}: 0.524 < \theta < \frac{\pi}{2}\right\}\)