Question 1:

Three towns, A, B and C form a triangle.

Town A is 80 km from Town B and Town C is 40 km from Town A as shown below: 

 

 

 

 

 

 

 

 

 

The bearing of Town B from Town A is 130° . The bearing of Town C from Town A is 240°

(a) Use this information to find the size of ∠CAB, and hence find the area of the triangle formed by the three towns to the nearest square kilometre.

(b) Using the cosine rule, find the distance between Town B and Town C, to the nearest kilometre.

 

 

 

Given f(x)4-x2complete this table of values, correct to 3 decimal places.

x00.511.52
f(x)     

 

Use the Trapezoidal rule, with sub intervals, to estimate the value of

024-x2dx

 

 

 

For the curve: y=x3-3x2-9x+4

(a) Find any stationary points and determine their nature.

(b) Find any points of inflexion.

 

 

 

 

Find the exact value of cotθ given that cosθ =0.6 and sinθ <0

 

 

 

 A geometric progression has 5th term 9 and 13th term 59049

(a) Find the first term and the common ratio.

(b) Find the 19th term.

 

 

 

The number of bacteria in a culture can be modelled by B=120000e0.4t

where t is the time in hours after the experiment started.

(a) How many bacteria are there after 6 hours have passed?

(b) How fast was the culture growing after 6 hours? 

(c) What was the average rate of increase over the first 6 hours?

(d) How long in hours and minutes, will it take until the number of bacteria doubles?

 

 

 

 

In an experiment, 2 balls are drawn at random and without replacement from an urn containing 4 red balls and 6 black balls. Let X be the number of red balls selected.

(a) 

OutcomeRRRBBRBB 
X2110 
p(X=x)215    
x.p(x)     
X2     

(b) What is the expected number of red balls drawn?

(c) What is the variance, V(x) of this distribution?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 2:

(b)The diagram shows the graph y=x3. A is the point (1, 1)  and the tangent at A cuts the x and Y axes at X and Y respectively.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(i) Show that the equation of the tangent at A is given by y=3x  2.  (ii) Write down the coordinates of X and Y (iii) Find the perpendicular distance from the point P(2, 1) to the tangent at A.  (iv) Find the area of PXY.

Question 3:

 

 

 

 

 

 

 

 

 

 

 

 

 

The graph of the function y=f(x) is shown in the diagram above. The area of shaded region B is twice the area of  he shaded region A. The areas of shaded regions C and D are equal. Write an alternative , equivalent expression for 012f(x)dx in terms of one  integral of f(x). 

Question 4:

The population P of ants in a colony is determined by P=500×e12 where t is the time in weeks since the colony was originally established. 

(i) Find the size of the colony after 10 weeks. 

(ii) How long would it take for the population to reach 10 000?

(iii) Find dPdt and explain what this means for the population trend? 

(iv) Sketch the graph of dPdt against t.

Question 5:

(b) In the diagram below, AE is parallel to BC. BDC = 40°, BCD = 25°, EAC = 65°. (i) Copy the diagram into your booklet.  (ii) Prove that ABC is isosceles.

Question 6:

Paul and Wendy borrow $20000 from the Miami Bank. This loan plus interest is to be repaid in equal monthly instalments of $399 over five years. Interest of 7.2% p.a is compounded monthly on the balance owing at the start of each month. Let $An be the amount owing after n months. 

(i) Over the five year repayment period, how much interest is charged? 

(ii) Show that  A1 =19721 

(iii) Clearly show that A2=20000×1.0062-3991+1.006

(iv) Deduce then that An=66500-46500×1.006n

(v) After two years of repayments Paul and Wendy decide on the very next day to repay the loan in one full payment. How much will this one payment be?

Question 7:

If the straight line y= mx is a tangent to the curve y=ex2 find the exact value of m. Clearly show your working. 

Question 8:

(a) A fishing boat located at P observes two islands B and C where B is in a westerly direction from C. The bearings of B and C from P are 295° and 058° respectively.  P is 7 km from B and 5 km from C.

 

 

 

 

 

 

 

 

 

 

 

 

Question 9:

Differentiate e3x2x

Question 10:

Find 1+sec23xdx