Raffles Institution H2 Math papers, with video solutions

An arithmetic series has first term \(\frac{3}{2}\) and fourth term \({{u}_{4}}\).

A geometric series also has first term \(\frac{3}{2}\) and fourth term \({{u}_{4}}\).

Given that the common ratio of the geometric series is non-negative and the sum of its first \(3\) terms is \(\frac{21}{2}\), find the sum of the first \(10\) odd-numbered terms of the arithmetic series.[5]

The diagram shows the graph of Folium of Descartes with cartesian equation

${{x}^{3}}+{{y}^{3}}=3axy$,

where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation $y=-x-a$.

1. Given that $\left( 0,0 \right)$ is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.

   [5]

2. Sketch the graph of
   ${{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y$,
   including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the $x$-axis.

   [3]

A bag contains \(20\) blue balls and \(15\) red balls. In a game, Joe draws \(12\) balls one by one, at random and without replacement. The number of blue balls Joe draws from the bag is denoted by \(X\).

1. Find \(\mathrm{E}\left( X \right)\) and \(\mathrm{Var}\left( X \right)\).[4]
2. If \(\mathrm{P}\left( X=x \right)<\mathrm{P}\left( X=x+1 \right)\), show that \(x\) satisfies the inequality
   \(\left( x+a \right)\left( x+b \right)<\left( c-x \right)\left( d-x \right)\)
   where \(a\), \(b\), \(c\) and \(d\) are constants to be determined. Use this inequality to find the number of blue balls that Joe is most likely to draw.[5]

Joe plays this game \(30\) times.

3. Find the probability that Joe draws at least \(10\) blue balls in more than \(4\) of his \(30\) games.[3]

Prove that $\frac{1+\sin \frac{3\pi }{8}+\mathrm{i}\cos \frac{3\pi }{8}}{1+\sin \frac{3\pi }{8}-\mathrm{i}\cos \frac{3\pi }{8}}=\cos \frac{\pi }{8}+\mathrm{i}\sin \frac{\pi }{8}$.

1. 1. Expand $\mathrm{f}\left( x \right)=\frac{2}{2-x}-\frac{1}{{{\left( 1+x \right)}^{2}}}$ as a series in ascending powers of $x$ up to and including the term in ${{x}^{2}}$.

      [3]

   2. State the equation of the tangent to the curve $y=\mathrm{f}\left( x \right)$ at the origin.

      [1]

2. Using the standard series given in the List of Formulae (MF15) or otherwise, show that the first three non- zero terms in the Maclaurin series for ${{\mathrm{e}}^{\sqrt{1+x}}}$ can be expressed as $\mathrm{e}\left( 1+px+q{{x}^{3}}+... \right)$ where $p$ and $q$ are constants to be determined.

   [6]