# Research Papers On Mathematics Extension

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NEWCASTLE GRAMMAR SCHOOL

Student Number: ______________________________

[pic]
2011
TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics Extension 2

Examination Date: Wednesday 17th August

Examiner: Mr. M. Brain

Question 1 (Start a new booklet) Marks

a) Find 3

(i) [pic]

(ii) [pic]

b) Evaluate 8

(i) [pic]

(ii) [pic]

c) Use the substitution [pic] to find [pic] 4

Question 2 (Start a new booklet) Marks

a) (i) Express [pic] in mod-arg form 3

(ii) Hence express [pic] in surd form

b) Evaluate [pic] given that [pic] 1

c) (i) On an Argand diagram shade the region where both 4 [pic] and [pic]

(ii) Find the sets of values of [pic] and [pic] for the points in the shaded region

d) [pic] and [pic] are two complex numbers such that [pic] 7

(i) On an Argand diagram show vectors representing [pic], [pic], [pic] and [pic]

(ii) Show that [pic]

(iii) If [pic] is the angle between the vectors representing [pic] and [pic] show that [pic]

(iv) Show that [pic]

Question 3 (Start a new booklet) Marks

a) A is a point outside a circle with centre O. P is a second point 6 outside the circle such that PT=PA where PT is a tangent to the circle at T. AO cuts the circle at D and C. PC cuts the circle at B. AB cuts the circle at E.

[pic]

(i) Show that [pic] is similar to [pic]

(ii) Show that [pic] is similar to [pic]

(iii) Hence show that DE is parallel to AP

b) (i) On the same number plane sketch the graphs of [pic] 3 and [pic]

(ii) Hence, or otherwise, solve [pic]

Question 3 continued on next page

Question 3 continued Marks

c) The area between [pic] and [pic], from the y-axis 4 to the point of intersection, A, is rotated about the line [pic]

[pic]

(i) Find the co-ordinates of point A

(ii) Calculate the generated volume of revolution

d) Solve the equation [pic] given that 2 it has a triple root

Question 4 (Start a new booklet) Marks

a) Find all the roots of [pic] 4 given that [pic]is one of the roots

b) The line through the origin which is perpendicular to the tangent 7

at [pic] to the rectangular hyperbola [pic] meets the

tangent at N.

[pic]

Show that the locus of N has the equation [pic]

c) Give a possible equation for the graph below: 4

[pic]

Question 5 ( Start a new booklet) Marks

a) The equation of a curve is [pic] 7

(i) Show that the numerical value of y satisfies [pic]

(ii) Find the equations of the asymptotes

(iii) Show that [pic]

(iv) Sketch the curve

b) Sketch the graph of each equation on a separate number plane: 8

(i) (1) [pic] (2) [pic]

(ii) (1) [pic] (2) [pic]

(iii) (1) [pic] (2) [pic]

Question 6 (Start a new booklet) Marks

a) The inequality [pic] is true for any real positive 5 numbers a,b and c. Given that [pic] show:

(i) [pic]

(ii) [pic]

(iii) [pic]

b) (i) Show that the area, A, of a regular pentagon of side 4 length P is given by

[pic]

(ii) The area enclosed by [pic] and [pic] is the base of a solid. Cross-sections