River Valley High School H2 Math papers, with video solutions

The diagram below shows a pin-board with $90$ pins on a wooden board. The horizontal and vertical distances between the pins are equal.

A rubber band goes around the pins to form a geometrical shape. Congruent shapes formed with different pins are considered different.

1. Rectangles are formed such that their sides are parallel to that of the board. How many rectangles can be formed?

   [1]

2. Triangles are formed such that at least one of the sides are parallel to that of the board. How many triangles of area $16$ square units can be formed?

   [4]

The diagram shows the sketch of the curve $C$, ${{\left( y-1 \right)}^{2}}=x\sqrt{{{x}^{2}}-1}$, with the vertex at $\left( 1,1 \right)$.

1. Write down the equation of the graph when $C$ is translated $1$ unit in the negative $y$-direction.

   [1]

2. The shaded region $R$, bounded by $C$ and the vertical line, $x=a$, is rotated through $\pi $ radians about the line $y=1$. By using the substitution $u=\sqrt{{{x}^{2}}-1}$, or otherwise, find the exact volume obtained in terms of $a$.

   [5]

1. For events \(A\) and \(B\), it is given that \(\mathrm{P}(B)=0.4\) and \(\mathrm{P}(\,\,A|B'\,\,)=0.15\). Find \(\mathrm{P}(\,\,A\cup B\,\,)\).[3]
2. In the last stage of a variety game show, the contestant aims to open a treasure chest containing the grand prize. The chest can be opened only when its two locks are unlocked. Each lock requires a unique key to be unlocked.
   The contestant is given up to three rounds to select the two correct keys among six different keys on a tray. For each round, he can only select one key which will be tried on both locks. The selected key of a particular round will be removed from the tray after that round.
   Show that the probability that a contestant wins the grand prize in the second round is \(\frac{\mathrm{1}}{\mathrm{15}}\).[1]
   Find the probability that
   1. the grand prize is won and the contestant selects the correct key in the first round,[2]
   2. the contestant selects the correct key in the first round given that he wins the grand prize.[2]

The variety game show runs for five days a week, over a period of \(60\) weeks. Find the probability that the mean number of shows per week for which the grand prize is won in the second round is more than \(0.4\).[3]

1. Given that the first, third and fourth terms of an arithmetic progression are three consecutive terms of a geometric progression, and that the sum of the first twenty even-numbered terms of the arithmetic progression is $960$, find the common difference of the arithmetic progression.

   [4]

2. Annie puts $\$\,x$ on 1 January 2010 into a bank account which pays compound interest at a rate of $3%$ per month on the last day of each month. On the first day of each subsequent month, she puts in an amount which is $\$\,x$ more than the amount she puts in the previous month. For example, she puts in $\$\,x$ on 1 January, $\$\,2x$ on 1 February and so on.
   1. Show that the amount of money in the bank account on the last day of March is $\$\frac{103x}{9}\left[103{{\left(1.03\right)}^{3}}-112\right]$.

      [4]

   2. Find the least integer value of $x$ so that the amount of interest earned for the first three months of the year 2010 exceeds $\$100$.

      [2]