MATHEMATICS 3C/3D

TEST THREE – EQUATIONS, INEQUALITIES and PROOF

Friday 23rd May, 2014

Time Allowed: 55 minutes NAME:

TOTAL MARK: /20 (No calculator) + /35 (Calculator) =

TEACHER COMMENT:

Student self-assessment of outcome descriptions Not demonstrated

Marginal Demonstrated

Simplify algebraic fractions that involve sums and differences, products and quotients, with constants, linear expressions and quadratic terms in either the numerator or the denominator.Solve equations and inequations that involve algebraic fractions with constant, linear or quadratic terms in the numerator and/or denominator.Solve two variable linear programming problems, with sensitivity analysis.Solve systems of equations in three variables by a systematic elimination method.Make, test, prove algebraically and/or disprove by counter-example conjectures based on number patterns, divisibility and similar integer situations.Distinguish general geometric arguments from specific cases; follow and ascertain the validity of geometric arguments.Solve problems involving triangles (isosceles, right, similar and congruent), angles in circles and tangents to circles.Construct deductive proofs involving triangles (isosceles, right, similar and congruent), angles in circles and tangents to circles.

SECTION ONE – RESOURCE FREE (20 marks). Recommended time allocation is 20 minutes.

1. [10 marks]

(a) Convert to a form that has a common denominator [3]

(b) Hence, or otherwise, solve [3]

(c) Identify all the values of x for which [4]

2. [5 marks]

Determine the values of x, y and z which satisfy this set of simultaneous equations:

Spare working space:

3. [5 marks]

“If P is an odd prime, then at least one of or will also be prime.”

(a) Is this conjecture true for Explain how. [2]

(b) Is it true for ? Why? [1]

(c) Is it generally true? Provide either an explanation or a counter-example. [2]

Spare working space:

NAME:

SECTION TWO – CALCULATORS (CAS and scientific), 1 A4 page of NOTES and mathematical templates ALLOWED (35 marks). Suggested time 35 minutes.

4. [8 marks]

(a) Given that , as shown, prove that ΔABC ΔEDC. [2]

(b) Use a property of similar triangles to determine x and y [2]

(c) O is the centre of the circle, with P and R on the circumference. Show that ΔPOR is isosceles. [1]

(d) . Evaluate, with a brief reason or explanation:

(i) [1]

(ii) [1]

(iii) [1]

5. [15 marks]

This graph represents the initial feasible region for the numbers of small (x) and large (y) containers that are stored in the warehouse of Sentinel Storage Limited.Each small container requires 8 m2 of floorspace and each large container 12 m2.

(a) From the graph,

identify:

(i)

(i)

(i)

(i)

(i)

(i)

(i) the maximum number of containers that can be stored [2]

(ii) the equation of the other graphed inequality [2]

(b) Add these restrictions to the graph:

(i) the maximum available floorspace is 960 m2. [2]

(ii) the number of large containers cannot exceed the number of small containers by more than 30 [2]

(c) Clearly mark all the vertices of the revised feasible region. [2]

(d) Sentinel’s management charges $20 per week to store a small container. What range of charges for a large container will ensure that their preference to store 60 small and 40 large containers is maintained, whilst maximising revenue?What is this maximum revenue? [5]

6. [12 marks]

(a) For the circle, centre O and tangent drawn at X, as shown, evaluate θ, giving reasons for each step: [4]

Two circles, centres P and Q, intersect at points X and Y, as shown.

(b) Prove that [4]

(b) If XY = 8 cm, and the circles have radii 8.5 cm and 5 cm respectively, determine the distance PQ. [4]