To solve this fourth cache, you will need to solve maths problems similar to those found on a GCE Advanced (A2) Level Maths paper. If you want, start at the first cache in the series (Geomaths #1) though there is no pre-requistic - you can do these in any order. The other caches in the series (so far) are Geomaths #2 and Geomaths #3.
The cache can be found at N50 ab.cde W 000 fg.hjk. The cache itself is a medium sized black taped clip-lock box.
Please do not include hints, spoilers etc in your logs about the questions or the cache location. This will be classed as cheating and may result in disqualification from the cache!
GEOCACHING EXAMINATIONS BOARD
General Certificate of Education
MATHEMATICS
Advanced Level Examination
Time allowed: who can be first to find?
You do not need to show clearly how you work out your answer.
Use of a calculator is permitted.
1.It is given that 
, where m is a positive constant.
i) Find an equation for m of the form 
ii) Use an iterative process, based on the equation in part (i), to find the value of m correct to 4 decimal places. Use a starting value of 0.4. The fourth decimal place of the answer gives clue c. i.e. #.###c
2.The equation of a curve is y = x2 ln(3x - 4). Find the exact value of 
at the point on the curve for which x = 3. Double the first digit of your answer to get e.
3.Lines L1, L2 and L3 have vector equations
L1: r = (5i - j - 2k) + s(-6i + 8j - 2k),
L2: r = (3i - 8j) + t(i + 3j + 2k),
L3: r = (2i + j + 3k) + u(3i + cj + k).
Calculate the acute angle between L1 and L2. The first digit of your answer gives g.
4.A water tank is designed with a horizontal base. At time t minutes, the depth of water in the tank is x cm. When t = 0, x = 72. In the side of the tank there is a tap which the water can flow through. When the tap is opened, the flow of water is modelled by the differential equation
.
The owner of the tank needs to drain it to a depth of 35cm for maintenance. How long (in minutes) does it take for the level of water to fall from a depth of 72cm to the desired depth?
The first digit of your answer (expressed as a fraction or decimal) gives b.
5.The equation
x3 - 6x + 2 = 0
may be solved by the Newton-Raphson method. Successive approximations to the root are given by x1, x2, ... , xn, ... .
Use the Newton-Raphson method to find the root of the equation which is close to 2. Complete sufficient approximations of the root to achieve a stable fifth decimal place. Let the fifth decimal place be h.
6.A car of mass 1500kg travels along a straight road inclined at 3˚ to the horizontal. The resistance to motion of the car from air and friction is pvN, where v ms-1 is the speed of the car and p is a constant. The car travels at a constant speed of 20 ms-1 up the slope and the engine of the car works at a constant rate of 21.4kW.
Calculate the value of p to the nearest whole number. The first digit of p gives clue k. i.e. k##
7.
A ball of mass 0.7kg is moving with speed 17 ms-1 in a straight line when it is struck by a bat. The impulse exerted by the bat has magnitude 20 Ns and the ball is deflected through an angle of 90˚ as shown on the diagram. Find the acute angle of the direction of the impulse with the x-axis to 1 decimal place. The difference between the first two digits of the angle gives j. i.e. if the answer is xy.# then x-y=j
8.Two continuous random variables S and T have probability density functions fS and fT given respectively by
where p and q are constants.
Find the value of p, and express it as a fraction.
Clue d is found by raising the denominator to the power of the numerator i.e. 2/5 would give 52 = 25.
9.Find the exact value of 
Your answer, in the form 
will give clue a and clue f.
You can check your answers for this puzzle on GeoChecker.com.
