Topic 3 Part [449 marks] a. Find all values of x for 0. x such that sin( x ) = 0. b. Find n n+ x sin( x )dx, showing that it takes different integer values when n is even and when n is odd. c. Evaluate sin( ) dx. 0. x x a. Express 4x 4x + 5 a(x h k Q. ) in the form + k where a, h, b. The graph of y = x is transformed onto the graph of y = 4x 4x + 5. Describe a sequence of transformations that does this, making the order of transformations clear. The function f is defined by f(x) = 4x 4x+5. c. Sketch the graph of y = f(x). d. Find the range of f. e. By using a suitable substitution show that 4 u + f(x)dx = du. f. Prove that 3.5 4x 4x+5 dx =. 6 [7 marks]
3a. Given that arctan( ) + arctan( ) = arctan( ), where 5 p Z +, find p. 8 p 3b. Hence find the value of arctan( ) + arctan( ) + arctan( ). 5 8 4a. Use the identity cosθ = cos θ to prove that cos x =, 0 x. +cos x 4b. Find a similar expression for sin x, 0 x. 4c. Hence find the value of ( 0 + cosx + cosx )dx. [4 marks] 5. The triangle ABC is equilateral of side 3 cm. The point D lies on [BC] such that BD = cm. Find cosdac. [5 marks] 6. Given that sin x + cosx =, find 3 cos4x.
The shaded region S is enclosed between the curve y = x + cosx, for 0 x, and the line y = x, as shown in the diagram below. 7a. Find the coordinates of the points where the line meets the curve. 7b. The region S is rotated by about the x-axis to generate a solid. (i) Write down an integral that represents the volume V of the solid. (ii) Find the volume V. [5 marks] 8a. Sketch the graph of x y = cos( ) for 4 0 x 8. 8b. Solve x cos( ) = 4 0 x 8. for 9. The first three terms of a geometric sequence are sin x, sin x and 4 sin xcos x, < x < (a) Find the common ratio r. (b). Find the set of values of x for which the geometric series sin x + sin x + 4 sin xcos x + converges. Consider x = arccos( ), x > 0. (c) 5. 4 Show that the sum to infinity of this series is [7 marks]
Consider the following functions: h(x) = arctan(x), x R g(x) =, x x R, x 0 0a. Sketch the graph of y = h(x). 0b. Find an expression for the composite function h g(x) and state its domain. 0c. Given that f(x) = h(x) + h g(x), (i) f (x) (ii) find in simplified form; show that f(x) = for x > 0. [7 marks] 0d. Nigel states that f is an odd function and Tom argues that f is an even function. (i) State who is correct and justify your answer. (ii) Hence find the value of f(x) for x < 0.. The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B. Find: (a) BOC; (b) the area of the shaded region.
. (a) Prove the trigonometric identity sin(x + y)sin(x y) = sin x sin y. (b) Given f(x) = sin(x + )sin(x ), x [0, ], find the range of f. (c) g. Given 6 6 6 6 g(x) = csc(x + )csc(x ), x [0, ], x, x 6 5 6, find the range of [8 marks] The diagram below shows a semi-circle of diameter 0 cm, centre O and two points A and B such that AOB = θ, where θ is in radians. 3a. Show that the shaded area can be expressed as 50θ 50 sin θ. 3b. Find the value of θ for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures. 4a. (i) Show that ( + i tan θ ) n + ( i tan θ ) n cos nθ =, cos θ 0. cos n θ i tan 3 8 ( + z ) 4 + ( z ) 4 = 0, z C (ii) Hence verify that is a root of the equation. [0 marks] ( + z ) 4 + ( z ) 4 = 0, z C (iii) State another root of the equation.
4b. (i) Use the double angle identity tan θ = tan θ to show that tan =. tan θ 8 cos 4x = 8cos 4 x 8cos x + (ii) Show that. 8 cos 4x (iii) Hence find the value of 0 dx. cos x Compactness is a measure of how compact an enclosed region is. The compactness, C A d, of an enclosed region can be defined by, where d is the area of the region and is the maximum distance between any two points in the region. C = For a circular region,. C = 4A Consider a regular polygon of n sides constructed such that its vertices lie on the circumference of a circle of diameter units. x n 5a. If n > and even, show that C = sin. n
n sin 5b. n If n > and odd, it can be shown that C =. (+cos ) Find the regular polygon with the least number of sides for which the compactness is more than. n 0.99 [4 marks] PQR QPR = 30 PQ = (x + ) cm PR = (5 x ) cm < x < 5 Consider the triangle where, and, where. 6a. Show that the area, A cm, of the triangle is given by A = ( 8 + 5x + 50). 4 x3 x 6b. (i) da State. dx (ii) da Verify that dx = 0 when x =. 3
d 6c. (i) Find A and hence justify that gives the maximum area of triangle P QR. dx x = 3 P QR (ii) State the maximum area of triangle. QR P QR (iii) Find when the area of triangle is a maximum. [7 marks] ABC In triangle, 3 sin B + 4 cos C = 6 4 sin C + 3 cos B =. and sin(b + C) = 7a. Show that. 7b. Robert conjectures that CAB can have two possible values. [5 marks] Show that Robert s conjecture is incorrect by proving that CAB has only one possible value.
The logo, for a company that makes chocolate, is a sector of a circle of radius 3 cm The area of the logo is. cm, shown as shaded in the diagram. Find, in radians, the value of the angle, as indicated on the diagram. 8a. θ Find the total length of the perimeter of the logo. 8b. 9. Find all solutions to the equation tan x + tan x = 0 where 0 x < 360.
In triangle ABC, BC = 3 cm, ABC = θ and BCA =. 3 AB = 3 0a. Show that length. 3 cos θ+sin θ [4 marks] 0b. Given that AB has a minimum value, determine the value of θ for which this occurs. [4 marks] Consider the functions f(x) = tan x, 0 x < x+ and g(x) =, x R, x. x Find an expression for a. g f(x), stating its domain.
sin x+cos x b. Hence show that g f(x) =. sin x cos x y = g f(x) c. Let, find an exact value for dy at the point on the graph of y = g f(x) where x =, expressing your answer in the form a + b 3, a, b Z. dx 6 y = g f(x) x x = 0 x = 6 Show that the area bounded by the graph of, the -axis and the lines and is d.. ln( + 3)
A triangle. ABC has A = 50, and AB = 7 cm BC = 6 cm 0 cm. Find the area of the triangle given that it is smaller than. 3a. (i) Use the binomial theorem to expand. (ii) Hence use De Moivre s theorem to prove (cos θ + i sin θ) 5 sin 5θ = 5cos 4 θ sin θ 0cos θsin 3 θ + sin 5 θ. cos 5θ cos θ sin θ (iii) State a similar expression for in terms of and. Let z = r(cos α + i sin α), where α is measured in degrees, be the solution of z 5 = 0 which has the smallest positive argument. Find the value of and the value of. 3b. r α [4 marks]
3c. Using (a) (ii) and your answer from (b) show that 6sin 4 α 0sin α + 5 = 0. [4 marks] Hence express 3d. sin 7 a+b c in the form where a, b, c, d Z. d [5 marks] ABC AB = 5 cm BC = cm ABC = 00 In triangle,, and. Find the area of the triangle. 4a.
4b. Find AC. Farmer Bill owns a rectangular field, 0 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and attaches the other end to his goat Gruff. 5a. Given that the rope is 5 m long, calculate the percentage of Bill s field that Gruff is able to graze. Give your answer correct to the nearest integer. [4 marks] Bill replaces Gruff s rope with another, this time of length 5b. one half of Bill s field. Show that a satisfies the equation a, 4 < a < 0 a 4 arcsin( ) + 4 a 6 = 40. a, so that Gruff can now graze exactly [4 marks]
5c. Find the value of a. The diagram shows a tangent, (TP), to the circle with centre O and radius r. The size of POA is θ radians. 6a. Find the area of triangle AOP in terms of r and θ. [ mark] 6b. Find the area of triangle POT in terms of r and θ. 6c. Using your results from part (a) and part (b), show that sin θ < θ < tanθ. 7a. (i) Sketch the graphs of y = sin x and y = sin x, on the same set of axes, for 0 x. (ii) Find the x-coordinates of the points of intersection of the graphs in the domain 0 x. (iii) Find the area enclosed by the graphs. [9 marks]
7b. Find the value of x dx using the substitution 0 4 x x = 4sin θ. [8 marks] 7c. The increasing function f satisfies f(0) = 0 and f(a) = b, where a > 0 and b > 0. (i) By reference to a sketch, show that a f(x)dx = ab b (x)dx. 0 0 f (ii) Hence find the value of 0 x 4 arcsin( )dx. [8 marks] 8. The points P and Q lie on a circle, with centre O and radius 8 cm, such that POQ = 59. [5 marks] Find the area of the shaded segment of the circle contained between the arc PQ and the chord [PQ]. 9. The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r. Find an expression for P A k r in the form, where k Z +.
30. The diagram below shows a curve with equation [5 marks] y = + ksin x, defined for 0 x 3. The point A (, ) 6 lies on the curve and B(a, b) is the maximum point. (a) Show that k = 6. (b) Hence, find the values of a and b. 3. (a) (b) Show that arctan( ) + arctan( ) = Hence, or otherwise, find the value of arctan() + arctan(3). 3 4. [5 marks] 3. [5 marks] The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius R and the inner circle has radius r. Consider the shaded regions with areas A and B. Given that A : B = :, find the exact value of the ratio R : r.
33. A triangle has sides of length ( n + n + ), (n + ) and ( n ) n >. (a) (b) where Explain why the side ( n + n + ) must be the longest side of the triangle. Show that the largest angle, θ, of the triangle is 0. [8 marks] 34. (a) Show that [0 marks] sin nx = sin((n + )x)cosx cos((n + )x)sin x. (b) Hence prove, by induction, that sin nx cosx + cos3x + cos5x + + cos((n )x) =, sin x for all n, sin x 0. Z + (c) Solve the equation cosx + cos3x =, 0 < x <. 35. The graph below shows [4 marks] y = acos(bx) + c. Find the value of a, the value of b and the value of c.
36. In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO of the cone is 8 cm and the angle BOC is 60. Calculate the size of the angle BAC.
37. Points A, B and C are on the circumference of a circle, centre O and radius [ marks] r. A trapezium OABC is formed such that AB is parallel to OC, and the angle AOC is θ, θ. (a) Show that angle BOC θ. is (b) Show that the area, T, of the trapezium can be expressed as T = sin θ sin θ. r r (c) (i) Show that when the area is maximum, the value of θ satisfies cosθ = cosθ. (ii) Hence determine the maximum area of the trapezium when r =. (Note: It is not required to prove that it is a maximum.) 38. If x satisfies the equation 3 3 sin(x + ) = sin xsin( ), show that tanx = a + b 3, where a, b Z +.
39. Consider the triangle ABC where BAC = BD DC =. 70, AB = 8 cm and AC = 7 cm. The point D on the side BC is such that Determine the length of AD. 40. [ marks] The interior of a circle of radius cm is divided into an infinite number of sectors. The areas of these sectors form a geometric sequence with common ratio k. The angle of the first sector is θ radians. (a) Show that θ = ( k). (b) The perimeter of the third sector is half the perimeter of the first sector. Find the value of k and of θ. 4. Triangle ABC has AB = 5 cm, BC = 6 cm and area 0 cm. (a) Find sin B. (b) Hence, find the two possible values of AC, giving your answers correct to two decimal places. Given that f(x) = + sin x, 0 x 3, 4a. sketch the graph of [ mark] f; 4b. show that [ mark] (f(x)) = 3 + sin x cosx;
4c. find the volume of the solid formed when the graph of f is rotated through [4 marks] radians about the x-axis. The diagram below shows a circular lake with centre O, diameter AB and radius km. Jorg needs to get from A to B as quickly as possible. He considers rowing to point P and then walking to point B. He can row at 3 km h and walk at 6 km h. Let PAB = θ radians, and t be the time in hours taken by Jorg to travel from A to B. 43a. Show that 3 t = ( cosθ + θ). 43b. Find the value of θ for which dt dθ = 0. 43c. What route should Jorg take to travel from A to B in the least amount of time? Give reasons for your answer. 44a. Given that 3 arctan arctan = arctana, a Q +, find the value of a. 44b. Hence, or otherwise, solve the equation arcsin x = arctana. Show that 45a. sin θ = tanθ. +cos θ
Hence find the value of 45b. cot in the form 8 a + b, where a,b Z. Write down the expansion of 46a. (cosθ + isin θ) 3 in the form a + ib, where a and b are in terms of sin θ and cosθ. Hence show that 46b. cos3θ = 4cos 3 θ 3 cosθ. Similarly show that 46c. cos5θ = 6cos 5 θ 0cos 3 θ + 5 cosθ. 46d. Hence solve the equation cos5θ + cos3θ + cosθ = 0, where θ [, ]. By considering the solutions of the equation 46e. cos5θ = 0, show that 5+ 5 cos = and state the value of 0 8 cos 7. 0 [8 marks] 47. Given ΔABC, with lengths shown in the diagram below, find the length of the line segment [CD]. [5 marks]
48. The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB. [7 marks] The diagram below shows two concentric circles with centre O and radii cm and 4 cm. The points P and Q lie on the larger circle and POQ = x, where 0 < x <. (a) Show that the area of the shaded region is 49. 8 sin x x. (b) Find the maximum area of the shaded region. [7 marks]
Two non-intersecting circles C, containing points M and S, and C, containing points N and R, have centres P and Q where PQ = 50. The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is α, the size of the obtuse angle NQR is β, and the size of the angle MPQ is θ. The arc length MS is l and the arc length NR is l. This information is represented in the diagram below. The radius of C is x, where x 0 and the radius of C is 0. 50. (a) Explain why x < 40. (b) Show that cosθ = x 0 50. (c) (i) Find an expression for MN in terms of x. (ii) Find the value of x that maximises MN. (d) Find an expression in terms of x for (i) α ; (ii) β. (e) The length of the perimeter is given by l + l + MN + SR. (i) Find an expression, b(x), for the length of the perimeter in terms of x. (ii) Find the maximum value of the length of the perimeter. (iii) Find the value of x that gives a perimeter of length 00. [8 marks] International Baccalaureate Organization 07 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for Sierra High School