VCE Maths Methods External Assessment Practice Questions

Question 6 

Solve h(x) = 1 for x ∈ R. (2 marks)

Question 7

The following information applies to questions 7 and 8.

Consider the function: f:R\{-1} → R, f(x)= \frac{3}{1+x} -1.

Identify the asymptotes of f. (1 mark)

Question 8

For what values of x is f(x) < -2. (2 marks)

Question 9 

The following information applies to questions 9 and 10.

A web service company runs a number of servers. Each year, it is known that the probability a hard drive fails is \frac{9}{16}, the probability that the water cooling system fails is \frac{3}{16} and the probability that both fail is \frac{1}{16}.

What is the probability that a server suffers from only hard drive failure? (1 mark)

Question 10

If four servers are selected at random, what is the probability that exactly three servers suffer hard drive failure, given that at least one of these four servers have undergone hard drive failure? Leave all numbers in exponent form (i.e. do not calculate 9³). (2 marks)

Question 11 

Let f:[0,∞ ) → R, where f(x)=2 e^{-3x}+1. Determine the domain and rule of f^{-1}, the inverse function of f. (2 marks)

Question 12 

For the function g[0,∞ ) → R, g(x) = \frac{1}{2}x^{2}, find the area enclosed between g and the line y=x. (2 marks)

Question 13 

Now consider the functions g(x) = \frac{1}{2}x^{2} and f(x) = αx, where (0,1).

What value of ensures that the area enclosed between g and f is equal to \frac{1}{2}? (4 marks)

Question 14 

The following information concerns questions 14–18:

Define p(x) = ln(\frac{1}{x})+ln(1 - x)

State the maximal domain and range of p. (2 marks)

Question 15 

Find the gradient to p at the point x=a. (1 mark)

Question 16 

Find the equation of the tangent to p at x=\frac{1}{2}. (1 mark)

Question 17 

Find the equation of the line perpendicular to p at the point (1/2, p(1/2)). (1 mark)

Question 18 

Why is p a one-to-one function? (1 mark)

Question 19 

Consider a line y=b-x  which intersects the function f:[0,∞ ) → R, f(x)=x^{3}-2x. At which x values (correct to two decimal places) is the acute angle between this line and the tangent to f 50^o? (3 marks)

Question 20 

Consider the transformation defined by the following sequence of steps: a reflection about the y axis, a dilation by 2 in the y direction, a translation of +3 in the x direction, a translation by +1 in the y direction.

What is the image of the function y=x^{2}-x under this transformation? (3 marks)

Question 21 

Does the function above intersect the image you determined in question 20? If so, what are the intersections? (2 marks)

Question 22

The following information applies to questions 22–25.

A Zoologist is investigating the lifespan of Tigers in captivity. She has discovered that, on average, Tigers live for 15 years in captivity, and that only 10% of tigers live for more than 20 years.

Assuming that the lifespan of Tigers is normally distributed, what is the standard deviation of this distribution (to 2 decimal places)? (2 marks)

Question 23 

What is the probability, to two decimal places, that a tiger survives for at least 18 years, given that it has survived for 16 years? (2 marks)

Question 24 

Given a zoo population with 5 tigers that are older than 16 years, what is the probability that fewer than 3 of these tigers survive to reach their 18th birthday? (2 marks)

Question 25 

Assume there are n tigers (of any age) in a zoo.

Find the probability that at least one of these tigers survives to the age of 17, and hence determine the number of tigers the zoo must acquire to ensure that there is at least a 95% chance that at least one survives to the age of 21. (2 marks)

Answers for VCE Maths Methods Questions

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