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Q1. (a) A 2400 litre water tank can be filled by two supply pipes A and B working together in 30 minutes.

On its own, pipe A can fill the tank in 32 minutes less than pipe B.

Calculate the rate of flow of water from each pipe. (10)

(b) Make R the subject of the formula: (6)

T = 2π√(L/g (L + R²/r² ) )

Q2. (a) Determine the value of z, (z > 0), which satisfies the equation: (8)

3z/(z + 1) + 2/(z + 2) = 1

(b) Solve the following systems of equations for x and y: (8)

x² + y² + 6x – 6y – 7 = 0

y + 1 = 2x

Q3. (a) The modulus of rigidity, G, is given by:

G = (R⁴ θ)/L, where R is the radius, θ the angle of twist and L the length.

Calculate the percentage error in G when R is measured 1.5% too small, θ is measured 1% too small, and L is measured 1.6% too large. (8)

(b) Given y = 32, solve the following equation for x, correct to 3 decimal places: (8)

y = 25/3x+x

Q4. (a) Solve the following equation for t, 0 < t < 2 : (6)

ln (12 – 3t² ) = - 0.78

(b) Express the following in its simplest form: (6)

5a√(9b⁴ ) + 5a∛(8a³ b³ )-7∜(a⁴ b⁸ )

(c) Evaluate the following, without the using mathematical tables or calculator: (4)

( Log⁡27 - log⁡9)/(2 log⁡3 )

Q7. (a) Use differential calculus to determine EACH of the following for the function

y = x³ - 3x² - 9x + 10

(i) the coordinates of the turning points; (7)

(ii) the nature of the turning points. (3)

(b) The area, A cm², of a pool of oil under a leaking sump is given by

A = t + t^2/16 where t is the time in minutes.

Calculate EACH of the following for the pool of oil after 20 minutes:

(i) the area; (2)

(ii) the rate the area is growing. (4)

Q8. (a) The velocity v, in ms-1, of a particle at time t, in seconds, is given by

v = ds/dt = 30 – 8t

Given s = 0 when t = 0, determine EACH of the following:

(i) s in terms of t; (5)

(ii) the distance travelled in 4 seconds from t = 0. (2)

(b) Integrate EACH of the following functions, with respect to the given variable:

(i) 6x² + 2/√x-3 (2)

(ii) 2 θ + 3cos θ - 4sin θ. (3)

(c) Evaluate ∫₁²4/x² dx (4)