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Q1. (a) Express y as a single fraction in its simplest form:

y = (2x² + 8x)/(2x²+ 13x + 20)-(2x²- 3x)/(2x²+ 3x - 9)

(b) Solve for Z in the following equation:

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

Q2. (a) The minimum diameter d of a shaft with rotational frequency f and subjected to a bending moment M and torque T can be derived from the formula:

D² = 16/πf (M+√(M²+ T² ))

Transport this formula to make M the subject.

(b) Factories EACH of the following as fully as possible:

(i) x² -y²-6x+9

(ii) 20a³b³ - 36a²b² - 8ab

Q3. U235 is a radioactive isotope used in a nuclear propulsion system.

It decays into lead according to the law:

m = m₀ e^kt Where m_0 the original mass of U235 present and m is the mass present after t years.

The time taken for half of the isotope to decay is 7.04 × 10⁸ years.

(a) Determine EACH of the following:

(i) The value of K;

(ii) The time taken for a 100 mg sample of U235 to reduce to 99 mg.

(b) Solve the following equation for x;

4x × 10^2x+1 = 3^x+1

Q4. (a)The cost of pumping crude oil to a refinery is related to the radius of the transport pipeline.

For a pipeline of radius r centimetres, the cost per day, C in £thousands, is given by

C = r + 2025/r + 6

Calculate EACH of the following

(i) The radius which minimizes the cost;

(ii) The minimum cost per day.

(b) Determine the first and second derivatives of the function:

y = x³+ √x + 2e^x