Question 1

Write the following in interval notation:

Question 2 (Technology free)

Consider the function

  1. Evaluate
  2. Find an expression for
  3. Find an expression for
  4. Find expressions for:
  1. Find:
  2. Find exact solutions for:
  3. State the domain and range of the function

Question 3 (Technology free)

Let

a. Find an expression in terms of c for: i. ii

b

c.

d.

e. Find an expression in terms of p for:

Question 4 (Technology free)

Consider the function:

  1. Sketch the graph of
  2. Write down the range of

Question 5 (CAS allowed)

The pollution level at x km from the CBD of the city Mathville is given by the function:

  1. Sketch a graph of the function
  2. State the range of the function . Give your answer where appropriate correct to two decimal places.
  3. Evaluate and interpret it within the context of this problem.
  4. What is the minimum level of pollution and where does it occur?
  5. What is the maximum level of pollution and where does it occur?
  6. At what distance from the CBD is the pollution level equal to 6 units? Give your answer correct to two decimal places.
  7. For what values of x is the pollution level above 6 units?

Question 6 (Technology free )

The graph of the function shows a straight line with a range of . The description of the function is where D Find the domain D

Question 7 (Technology free)

Find the implied domain of the functions:

Question 8 (Technology free)

  1. Solve the quadratic inequality:
  2. State the maximal domain of the function:
  3. State the maximal domain of the function :

Question 9 (Technology free)

  1. Express the quadratic: in the form: , where m and n are integers.
  2. Solve the quadratic inequality:
  3. State the implied domain of the function:

Question 10 (CAS allowed)

  1. Sketch the graph of the function defined by:
  1. State the range of this function.
  2. Evaluate: i.

ii.

iii.

Question 11(Technology free)

  1. Sketch the graph of the function:
  1. Evaluate:
  2. Evaluate:
  3. Evaluate:

*Question 12 (CAS allowed)

  1. Show that the function : is an odd function algebraically.
  2. Sketch the graph of
  3. Explain how you can recognize that it is odd geometrically.

*Question 13 (CAS allowed)

  1. Show that the function: is an even function algebraically.
  2. Sketch the graph of
  3. Explain how you can recognize geometrically that it is an even function.

*Question 14 (Technology free)

Consider the function: + 1

  1. Sketch the graph of this function.
  2. Even though this function displays symmetry, it is not an even function. Explain why not.

*Question 15 (Technology free)

Evaluate:

  1. |-7|

*Question 16 (Technology free)

Represent the following sets on a number line, and write each in interval notation:

*Question 17 (Technology free)

Sketch the graph of the functions:

*Question 18

Consider the function: Define as a hybrid function.

*Question 19 (Technology Free)

Consider the function:

  1. Sketch the graph of
  2. Sketch the graph of

*Question 20 (Technology Free)

Consider the function

  1. Sketch the graph of
  2. Sketch the graph of

*Question 21 (Technology free and CAS)

Let

  1. Sketch
  2. Sketch
  3. Confirm your answers to b and c by graphing on CAS.

*Question 22 (CAS allowed)

Let

  1. Define on CAS, and then graph the function . Sketch the graph of

y = .

  1. Graph the function Sketch the graph of y =

*Question 23 (Technology free)

Let and let

  1. Write down the function and state its domain.
  2. Evaluate:
  3. Write down the function and state its domain.
  4. Sketch the graph of the function .

*Question 24 (CAS allowed)

Consider the functions: and

  1. Write down the function and state its domain.
  2. Evaluate:
  3. Write down the function: and state its domain.
  4. Evaluate: (Give your answer as an exact value)
  5. Write down the function and state its domain.
  6. Evaluate: (Give your answer as an exact value)

Question 25 (Technology free)

Let and let

a. Find

b. Find

c. Find

d. Find

e. Find the composite function

f. Find the composite function

Question 26 (Technology free )

Let and let .

  1. Find
  2. Hence evaluate uoh(
  3. Find
  4. Hence evaluate hou(
  5. Find the composite function h ou(x)
  6. Find the composite function oh (x)

Question 27 (Technology free )

Let and let

  1. Evaluate
  2. Hence evaluate
  3. Evaluate
  4. Evaluate
  5. Find the composite function
  6. Find the composite function

*Question 28 (CAS allowed )

Let and let

  1. Evaluate
  2. Hence evaluate
  3. Evaluate
  4. Hence evaluate hog(1)
  5. Use CAS to find the rule of the function
  6. Use CAS to find the rule of the function

*Question 29 (CAS allowed )

Let and let +

  1. Use CAS to determine the function fog(x)
  2. Evaluate: fog(-2)
  3. Use CAS to determine the function g of(x)
  4. Evaluate gof (-2)

*Question 30 (Technology free )

*Question 31 (Technology free )

Let and let Write down the rule of:
a.

b.

*Question 32 (Technology free)

Let and let

  1. Evaluate
  2. Hence evaluate
  3. Evaluate
  4. Hence evaluate
  5. Evaluate
  6. Explain why cannot be evaluated.
  7. Explain why also cannot be evaluated.

*Question 33 (Technology free)

Let and let .

  1. Evaluate
  2. Hence evaluate fov(7).
  3. Evaluate and explain why ) is undefined.
  4. For what other value of x is ) undefined?

*Question 34 (Technology free)

Let

i.Evaluate:

ii. Evaluate:

iii. Evaluate

Question 35 (Technology free)

Let .

  1. Sketch the graph of and explain why an inverse function exists.
  2. Find the rule and domain of the inverse function
  3. Sketch the graph of on the same diagram.

Question 36(Technology free)

Let .

  1. Explain why the inverse function exists.
  2. Find the rule of the inverse function.

Question 37 (Technology free)

Let where

  1. Evaluate:
  2. Find the domain of .

Question 38 (CAS allowed)

Let . Find the smallest value of b for which an inverse function exists.

Question 39(Technology free)

Let .

  1. Sketch the graph of and explain why the inverse function exists.
  2. Find the domain and range of the inverse function, and sketch it on the same diagram as above.
  3. Find the rule of .
  4. Find: {

Question 40 (Technology free)

Let ,

  1. Find the range of
  2. State the domain and range of the inverse function
  3. Find the rule of
  4. Sketch the graphs of and on the same diagram.
  5. Find:

Question 41(Technology free)

Solve the following literal equations for x:

Question 42(Technology free)

Write the equation below with y as the subject:

Question 43 (Technology free)

If A is a matrix such that:

, find the matrix A.

Question 44 (Technology free )

If is a matrix such that:

= , find the matrix B.

Question 45 (Technology free)

  1. Express the simultaneous equations below in matrix form:
  1. Solve the simultaneous equations using matrix methods.

Question 46 (Technology free)

Let . Find the matrix , without using CAS.

Question 47(CAS allowed)

A cubic function passes through the three points:

(2, 8), (-1, -19) and (3, 21)

  1. Use this information to write down three simultaneous equations with the variables a, b and c.
  2. Write these simultaneous equations in the matrix form:

State the matrix A.

  1. Write down the inverse matrix .
  2. Determine the values of a, b and c.

Question 48 (CAS allowed)

Determine the values of m for which the simultaneous equations:

have no solution.

Question 49 (CAS allowed)

Determine the values of for which the simultaneous equations:

have: (i) no solution

(ii) infinitely many solutions.

Question 50 (CAS allowed)

Triangle ABC is isosceles with BC = AC. The co-ordinates of the vertices are A (6, 2) and B(2, 8).

Parade College Year 12 Maths Methods Headstart ProgramPage 1