JRAHS Yr11 Mathematics Half Yearly 2010 Re-Test Page 1 of 4 QUESTION 1 (15 Marks) Marks (a) Find the value of 56 . 2 2 . 3 56 . 2 2 . 3 2 2 , correct to the nearest tenth. 2 (b) Evaluate 1 – cosec 2 60 o . 2 (c) Differentiate with respect to x: (i) 3 2 ex + 1 1 (ii) 3 5 2 1 x x (Simplify your answer) 2 (d) Solve for x: 2(3x 2 – x) 2 + 3(x – 3x 2 ) +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 340 o T whilst Car B travels at 35 km/h on a bearing of 035 o T. 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 2 3 1 3 1 ) ( x x x f , then evaluate ) 1 ( ' f . 2 (ii) Find the solutions for x such that ) ( ) ( ' x f x f . 2 (d) (i) Draw a neat sketch of the function y = x x 6 2 . 2 (ii) Hence, find the solutions for x such that y ≤ 0. 1 (e) Solve for θ: cos 2 θ – 0.75 = 0 for 0 o ≤ θ ≤ 360 o . 2
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QUESTION 1 (15 Marks) Marks 2 - WordPress.com 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
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(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
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).
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