JRAHS Yr11 Mathematics Half Yearly 2010 Re-Test Page 1 of 4

QUESTION 1 (15 Marks) Marks

(a) Find the value of56.22.3

56.22.3 22

, correct to the nearest tenth. 2

(b) Evaluate 1 – cosec260o. 2 (c) Differentiate with respect to x:

(i) 3

2ex + 1 1

(ii) 35

21

x

x (Simplify your answer) 2

(d) Solve for x: 2(3x2 – x)2 + 3(x – 3x2) +1 = 0 3 (e) Three sides of a right triangle are given by x, (2x + 2) and (3x – 4) units. 3 Given that (3x – 4) is the hypotenuse, find the value of x.

(f) (i) Find the gradient of the line that passes through the point (–1, 2) and (2

1, 3). 1

(ii) Hence find the equation of the line. 1 QUESTION 2 (15 Marks) Start a NEW page (a) Find the equation of the line passing through the point (1, –4), that is 1

perpendicular to a line with gradient 4

1.

(b) Two cars travel from O, at the same time. Car A travels at 90 km/h on a bearing of 340oT whilst Car B travels at 35 km/h on a bearing of 035oT. Draw a neat diagram including all of the given information and find : (i) the distance between the two cars after 2.5 hours. 3 (ii) the bearing from Car B to Car A. 2

(c) (i) If 23 131)( xxxf , then evaluate )1('f . 2 (ii) Find the solutions for x such that )()(' xfxf . 2

(d) (i) Draw a neat sketch of the function y = xx 62 . 2

(ii) Hence, find the solutions for x such that y ≤ 0. 1 (e) Solve for θ: cos2θ – 0.75 = 0 for 0o ≤ θ ≤ 360o. 2

JRAHS Yr11 Mathematics Half Yearly 2010 Re-Test Page 2 of 4

QUESTION 3 (15 Marks) Start a NEW page Marks (a) In the diagram below (not to scale), TVWTVU as shown. PT is tangent to 3 the circle. All measurements are in centimetres. Show that ΔTUV ≡ ΔPWT. (b) Differentiate with respect to x, giving your answer in simplest form.

(i) 62 33

xx 2

(ii) 11 2 xx 2 (c) Find the exact perimeter of the figure ABCDE below. The sectors are identical, 3 with AE=BE=BD=CD and BDCEBDAEB .

(d) Find the radius and centre of the circle x2 – 6x + y2 + 100y – 162 = 0 3 (e) Find the locus of the point P(x, y) such that it is always equidistant to the 2 points A(3, 1) and B(–4, 0).

A

B

C

D E

40 cm 40 cm

50o

50o

50o

Diagram not to scale

P

T

U

V

W

● ●

62

5

3

JRAHS Yr11 Mathematics Half Yearly 2010 Re-Test Page 3 of 4

QUESTION 4 (15 Marks) Start a NEW page Marks Use the answer sheet provided on the last page to answer parts (a) and (b). (a) Part of the function y = f(x) is drawn below. 1

If this is an ODD function, complete the rest of the graph on the answer sheet provided.

(b) (i) Draw a neat sketch of y = 3 x and y = 2–x, using the answer sheet provided 2 on the last page. (You only need to sketch within the first quadrant.) (ii) Hence, approximate the point of intersection of the two curves, correct 1 to 1 decimal place. (c) Solve 2tanA+ sinA – 2cosA – 4 = 0 over the domain 0o ≤ A ≤ 360o. 3

(d) Simplify

27

3333333

, giving your answer in EXACT form. 2

(e) Find the domain and range of the function: y =4

12 x

. 2

(f) Using First Principles of Differentiation, differentiate with respect to x: 4

f(x) = 1

32 x

x

y

O

JRAHS Yr11 Mathematics Half Yearly 2010 Re-Test Page 4 of 4

Answers to Question 4 Part (a)

Part (b) (i)

Part (b) (ii)

Point of intersection is at ( ____ , ____ )

         

         

         

         

         

         

         

         

         

         

         

         

         

x

y

O

x

y

1.00 0.5 • •

0.5 •

1.0 •