DieselShip's Online Content Library

Q1. (a) By applying Kirchoff’s Laws in a circuit the following equations were obtained:

24(I₁ - I₂) + 48 I1 = 4.2

16 I₂ - 4(I₁ - I₂) = 0.7

Calculate the values of the currents I₁ and I₂. (8)

(b) Pump A can fill an empty tank in 1 hour 40 minutes. A second more powerful pump, B, can fill the same tank in 40 minutes.

Calculate the overall time to fill the empty tank if pump A runs alone for 30 minutes and then pump B is used to assist pump A. (8)

Q2. (a) Solve for x in the following equation: (6)

(2x+3)/4 = (x-3)/5+2

(b) Make D the subject of the following formula: (6)

T = 12.5D/(D + 4d)

(c) The volumes of two solid spheres are in the ratio 2197 : 512.

Determine the ratio of their surface areas. (4)

Q3. (a) The sag, s metres, in a wire of length L metres stretched between two supports x metres apart, as illustrated in Fig Q3(a), is given by the formula:

L = x + 8 s²/3x

Calculate the distance x when L is 200 m and s is 8 m. (8)

(b) Given: R = ((27y - 18x) (4x²+ 12xy + 9y² ))/(4x²- 9y²)(10x + 15y)

Express R as a fraction in its simplest form. (8)

Q4. (a) Given: n = 10 log₁₀ (P₂/P₁ )

Calculate the value of P₁ when n = 2.5 and P₂ = 2.8 (4)

(b) Calculate the value of t such that ln(3-2/t) = -0.2 (6)

(c) Use laws of indices to fully simplify: (6)

∛((125h^5/2))/((27n^7/4) )×n^13/5/h

Q5. Table Q5 indicates the deflection, d mm, of a beam under loads, L Newtons.

The deflection is related to the load by the formula: L = kdⁿ where k and n are constants.

(a) Draw a graph to verify this relationship. (10)

(b) Determine approximate values of k and n. (6)

Suggested scales:      horizontal axis 2 cm = 0.1

vertical axis 2 cm = 0.04

Q6. (a) A roller of diameter 25 mm is placed in a V block as shown in Fig Q6(a).

The distance from the top of the roller to the top of the V block is 4.64 mm.

Calculate the width W of the block. (10)

(b) Given: H(t) = 6 5sin[(π/6)t + π/2]

(i) State the maximum value of H(t); (1)

(ii) Calculate the first positive value of t when this occurs. (5)

Q7. (a) The temperature T°C at a certain location t hours after 9 a.m. is given by the function:

T = t³/3 - 3t² + 8t + 10

Calculate the time when the temperature starts to fall. (8)

(b) Given: S = 5 + 2sinθ + 3cosθ

(i) Determine the value of dS/dθ when θ = 2π/3 radians (4)

(ii) Solve dS/dθ=0 for θ in the range 0 ≤ θ ≤π/2 (4)

Q8. (a) The average value, y̅, of a function y = f (x) in the range x = a to x = b is given by:

y̅ = 1/(b-a) ∫^b_af (x)dx

Determine the average value of the function y = 5x⁴ -4x in the range x = 0 to x = 2. (6)

(b) Fig Q8(b) shows a sketch of the function y = 3x² -x³

Calculate the volume of the solid of revolution obtained when the shaded area is rotated once about the x axis. (10)

Q9. Fig Q9 shows three heavy spheres lying inside a hollow cylinder. The diameter of the

cylinder is 250 mm. The diameters of EACH of the three spheres is 150 mm.

Calculate the volume of water, in cm³, to just cover the top sphere. (16)