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Elgin Academy Mathematics Department Higher Unit 2 Revision Please do not write on these sheets. Return this booklet to your teacher when you are finished with it. Good Luck. Outcome 1 Use the Factor/Remainder Theorem and apply quadratic theory. PC(a): Apply the Factor/Remainder theorem to a polynomial function. Example Show that x 2 is a factor of f x x 3 4 x 2 x 6 and express f (x) in fully factorised form. x 2 2x 3 x 2 x 3 4x 2 x 6 (x 3 2x 2 ) 2x 2 x (2 x 2 4 x) 3x 6 2 1 OR 4 1 6 2 4 6 1 2 3 0 ( 3 x 6) 0 Hence ( x 2) is a factor of f(x) and f ( x) ( x 2)( x 2 2 x 3) ( x 2)( x 1)( x 3) Exercise 1 1. Show that x 1 is a factor of f x x 3 7 x 6 . Express f x in fully factorised form. 2. Show that x 2 is a factor of f x 3x 3 2 x 2 19 x 6 . Express f x in fully factorised form. 3. Show that x 3 is a factor of f x x 3 3x 2 16 x 48 . Express f x in fully factorised form. PC(b) Determine the nature of the roots of a quadratic equation using the discriminant. Example Use the discriminant to determine the nature of the roots of x 2 3x 5 0 . b 2 4ac (3) 2 4.1.5 11 0 so there are no real roots. Exercise 2 Use the discriminant to determine the nature of the roots of the following equations: 1. x 2 4 x 1 0 2. x 2 3x 4 0 3. 2 x 2 5 0 4. 3x 2 x 5 0 5. 2 x 2 2 x 3 0 6. 5 x 2 10 x 5 Outcome 2 Use basic integration. PC(a): Integrate functions reducible to sums of powers of x. Example Find dx 5x 2 dx 5x 2 1 dx 15 x 2 dx 15 x 1 C 2 5x Exercise 3 6 x 1. Find 2 2. Find dx 4. Find x 3x 4 dx 3 x 4 3. Find dx 5. Find 6 x dx 6. Find 2 x 3 dx x2 3 x dx PC(b): Find the area between a curve and the x-axis using integration. Example Calculate the area between the curve and the x-axis in the diagram: 4 3x x 2 0 y y 4 3x x 2 4 x 1 x 0 x 4 or 1 1 4 3x x 4 -4 1 x 2 4.1 4 x 32 x 2 13 x 3 20 56 3 2 1 4 .12 13 .13 4. 4 32 . 4 13 . 4 2 3 Exercise 4 Calculate the area between the following curves and the x-axis: 1. y 12 x 1 between x = 1 and x = 3. 2. y (3 x)(1 x) 3. y (2 x)( 2 x) 4. y (6 x)( 2 x) PC(c): Find the area between two curves using integration. Example Find the area enclosed by the graphs of the functions y = 2x and y = x2: First find the x coordinates of the points of contact: 2x x 2 x 2 2x 0 x( x 2) 0 x 0,2 y Use a sketch to decide which graph is uppermost: 2 0 4 y = 2x (2 x x 2 ) dx ( this is all that is needed for the unit test) 2 2 0 4 0 1 x 2 13 x 3 1 3 2 2 0 3 8 3 2 13 0 3 0 y = x2 2 1 3 Exercise 5 Find the area enclosed by the graphs of each pair of the following functions: 1. y = 2x2 and y = 4x 2. y = x2 and y = x + 2 3. y = 2 x – x2 and y = 5x – 4 x Outcome 3 Solve trigonometric equations and apply trigonometric formulae. PC(a): Solve a trigonometric equation in a given interval. Example Solve 2sin2x = 1, 0 x 2 sin 2 x 2x x 6 1 2 (30 O ), 5 (150 O ) 6 5 , 12 12 2 2 5 13 5 17 So x , , , 12 12 12 12 12 12 the period of sin2 x Exercise 6 Solve the following equations for 0 x 2: 1. 2cos2x = 1 2. tan2x = 3 3. 2sin2x = 3 4. 3tan2x = -1 PC(b): Apply trigonometric formulae in the solution of a geometric problem. D A Example The diagrams show two right angled triangles, ABC and DEF. E 12 C Write down the values of sin xo and cos yo. AC 12 2 5 2 13, sin x o ED 4 2 3 2 5, cos y o b) y x B a) 3 5 5 13 4 5 Show that the exact value of sin(x + y) o = sin( x y ) sin x cos y cos x sin y 12 3 cos x o and sin y o 13 5 5 4 12 3 56 sin( x y ) o 13 5 13 5 65 56 . 65 4 F Exercise 7 1. The diagrams show two right angled triangles, PQR and STV. S P 4 8 x Q 2. y T R 15 V 3 a) Write down the values of sin xo and cos yo. b) By expanding cos(x – y), show that cos(x – y) = 77 85 The diagrams show two right angled triangles, JKL and MNP. M J 4 7 x K y N 24 L P 3 a) Write down the values of sin xo and cos yo. b) By expanding cos(x + y), show that cos(x + y) = 44 125 PC(c): Solve a trigonometric equation involving an addition formula in a given interval. Example Express cos xocos 60o + sin xosin 60o in the form cos(A-B)o. Use this result to solve the equation cos xocos 60o + sin xosin 60o = 2 , 0 x 360. 5 cos x o cos 60 o sin x o sin 60 o cos( x 60) o 2 cos( x 60) o 5 x 60 66.4, 293.6 x 126.4, 353.6. Exercise 8 1. Express sin xocos 20o + cos xosin 20o in the form sin(A+B)o. Hence solve the equation sin xocos 20o + cos xosin 20o = 0.6, 0 x 360. 2. Express cos xocos 30o - sin xosin 30o in the form cos(A+B)o. Hence solve the equation cos xocos 30o + sin xosin 30o = 0.2, 0 x 360. Outcome 4 Use the equation of the circle. PC(a): Given the centre (a,b) and radius r, find the equation of the circle in the form (x - a)2 + (y - b)2 = r2. Example Write down the equation of the circle with centre (2, 7) and radius 4. x 22 y 72 16 Exercise 9 Write down the equations of the circles with these centres and radii: 1. radius 7 and centre (3,-1) 3. radius 1 and centre (-2, 5) 2. radius 2.5 and centre (-5,-7) PC(b): Find the radius and centre of a circle given the equation in the form x2 + y2 +2g x + 2f y + c = 0. Example Write down the centre and radius of the circle x 2 y 2 4 x 8 y 17 0 Centre g , f 2,4 Radius g 2 f 2 c 22 42 17 3 Exercise 10 Write down the centre and radius of each of the following circles: 1. x 2 y 2 6 x 10 y 9 0 3. x 2 y 2 2 x 2 y 5 0 2. x 2 y 2 8x 2 y 13 0 PC(c): Determine whether or not a given line is a tangent to a given circle. Show that the straight line y 2 x 3 is a tangent to the circle x 2 y 2 4x 6 y 7 0 . Example x 2 2 x 3 4 x 62 x 3 7 0 2 x 2 4 x 2 12 x 9 4 x 12 x 18 7 0 5 x 2 20 x 20 0 x 2 4x 4 0 x 2 2 0 OR b 2 4ac 4 4.1.4 0 2 Equal roots, so only one point of contact proving the line is a tangent to the circle. Exercise 11 1. Show that the straight line y 4 x 7 is a tangent to the circle x 2 y 2 6x 4 y 4 0 . 2. Show that the straight line y 2 x 4 is a tangent to the circle x 2 y 2 2x 6 y 5 0 . 3. Show that the straight line y 3x 20 is a tangent to the circle x 2 y 2 14 x 22 y 160 0 . PC(d): Determine the equation of the tangent to a given circle given the point of contact. Example The point A(13, 9) lies on the circle with equation x 2 y 2 2 x 4 y 152 0 . Find the equation of the tangent at A. Coordinates of centre = (1, 4) so m r 94 5 13 1 12 Since, at the point of contact, the tangent is perpendicular to the radius mT mr 1 so mT 12 2.4 . 5 Equation of tangent is therefore y 9 12 x 13 12 x 5 y 201 0 5 Exercise 12 1. Find the equation of the tangent to the circle x 2 y 2 8x 6 y 40 0 at the point B(3,-1). 2. Find the equation of the tangent to the circle x 2 y 2 2 x 4 y 1 0 at the point C(-1, 2). 3. Find the equation of the tangent to the circle x 2 y 2 4 x 2 y 3 0 at the point D(3,-2). Answers Exercise 1 1. (x - 1)(x - 2)(x + 3) 2. (x + 2)(x - 3)(3x + 1) 3. (x - 3)(x - 4)(x + 4) Exercise 2 b 2 4ac 0 , distinct real roots; Q1,4,5. b 2 4ac 0 , equal real roots; Q6. b 2 4ac 0 , no real roots; Q2,3. Exercise 3 1. 6 c x 1 1 3. 2 c c 3 x x 2 5 x 6 x c 5 2. 5. 4 x 3 c 6. Exercise 4 1. 4 2. Exercise 5 2 1. 0 4 x 2 x 2 dx 10 2/3 2. 1. 7 9 15 8 , 8 , 8 , 8 3. 10 2/3 x 2 x dx 2 2 3. 1 Exercise 6 2. , 3 , 6 , 1 3 1 2 x x 12 x c 3 2 4. 85 1/3 2x x 2 1 2 7 4 6 4. 3. 3 2 5x 4 dx x 2 3x 4 dx 2 1 5 11 17 23 , , , 12 12 12 12 Exercise 7 1. a) sin x = 8/17, cos y = 3/5 2. a) sin x = 7/25, cos y = 3/5 b) Proof. b) Proof. Exercise 8 1. sin(x + 20), x = 16.9, 123.1 2. cos(x + 30), x = 48.5, 251.5 Exercise 9 1. (x – 3)2 + (y +1)2 = 49 2. (x + 5)2 + (y +7)2 = 6.25 3. (x + 2)2 + (y +5)2 = 1 Exercise 10 1. C(-3, 5), R = 5. 2. C(-4, -1), R = 2. 3. C(1, -1), R = 7. Exercise 11 Proofs Exercise 12 1. y + 1 = 7/4(x – 3) or 4y = 7x – 25 3. y + 2 = 1(x – 3) or y = x – 5 2. y - 2 = 1/2(x + 1) or 2y = x + 5