Question 1:

The diagram shows the cross-section of a 12 metre wide pond. The depths are taken every 3 metres. Use Simpson’s rule with five function values to find an approximate value for the area of the cross-section.

Question 2:

Consider the function y=1+3x+x3

(i) Find the coordinates of the stationary points and determine their nature. 

(ii) Find the coordinates of any points of inflexion. 

(iii) Draw a sketch of the curve y=1+3x+x3 clearly showing all its essential features.

Question 3:

An ambulance is delivering a patient to the hospital who is unconscious from a drug overdose. The doctor on duty does not know how much of the drug the unconscious patient has taken. The rate of change of the concentration of the drug in the blood is proportional to the concentration, i.e.dCdt=kC where C mg/L is the concentration of the drug in the blood, t hours after the drug was initially taken. 

(i) Show that C=C0ekt is a solution to dCdt=kC

(ii) Three hours after the patient took the overdose, the blood concentration of the drug was 2.45 mg/L. Half an hour later the concentration was 1.84 mg/L. Show that the initial concentration of the drug in the patient’s blood is 13.65 mg/L, correct to two decimal places. 

(iii) If the doctor on duty does not give the patient any further medication when will the drug concentration fall below the critical value of 0.5 mg/L? Answer correct to one decimal place.

Question 4:

From a circular disc of metal whose area is 100π m2 a sector is cut and used to make a right cone. The radius of the disc is R metres.

(i) If the right cone has base radius r metres and height h metres, show that the volume of the cone is given by V=πr2100-r23

(ii) Show that the maximum volume of the cone occurs when r=2003

Question 5:

Differentiate w.r.t x

(i) 2x3+4x-1 (ii) cosxx

 

 

Given the equation 3x2+4x-3=0 has the roots αand β, evaluate the following without finding α or β

(i) α+β (ii) αβ (iii) 2α2+2β2

Question 6:

The table below gives the of f(t) for 0t2

t

0

0.5l1.52
f(t)

0

0-300 370 330 27

Use the Trapezoidal Rule with 5 function values to evaluate

02f(t)dt  correct to 1 decimal place

 

 

The diagram represents the span of a bridge, 10 metres high and 6 metres wide. The curved part of the span is a parabola With vertex 9 metres above the ground. Using the axes shown in the diagram, find:

(i)the equation of the arc ABC;

(ii)the shaded area ABCDE.

Question 7:

for y=2x2ex

(i) Find the x and intercepts, if any.        

(ii) What happens to the function as x    

( iii) Show that stationary points exist at 0,0 and -2,8e2

(iv) Determine the nature of the stationary points.  

(v) It is known that 2 points of inflexion exist on this curve at x=-2±2 . Sketch the curve.  

Question 8:

The point Pt,1t lies on the hyperbola y =1x  (i) Given dyx=-1x2write down the gradient at x=t ii Hence show the equation of the tangent at the point P is x2+t2y=2t (iii) Find the coordinates of the points A and crosses the x and y axes respectively.   (iv) Show the area of the triangle AOB is a constant and so independent of the position P.

Question 9:

A right circular cone is inscribed in a sphere of radius R.                                                    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(i) Show that the volume of the cone can be found by   

V=13π2Rh-h2h

 (ii) Calculate the volume of the largest right circular cone      

inscribed in a sphere of radius R. Write your answer in terms of R

Question 10:

How much will $6000 accumulate to at the end of five years if it is invested in a fund which pays an interest rate of 4% p.a. compounded quarterly?

 

 

Beginning with a circular piece of fabric of radius 5 cm, Le sewed together circular strips of different coloured fabrics which increased in width to make a circular tablecloth. The finished width of the first strip was 10 cm, the second was 15 cm, the third was 20 cm and so on.

Show that the width of the tenth strip was 55 cm.                                                             

The radius of the table cloth was 455 cm. How many strips were sewn to the edge of the first circular piece?

 

When Jack left school, he borrowed $15 000 to buy his first car. The interest rate on the loan was 18% p.a. and Jack planned to pay back the loan in 60 equal monthly installments of $M

(i)Show that immediately after making his first monthly installment, Jack owed

$(15 000 > l .015 — M].

(ii)Show that immediately after making his third monthly installment, Jack owed                      2

$[15 000 x 1.0153}  M(I + 1 015 + 1-0152)].

(iii)Calculate the value of M.