1. Let f(x) = 7 – 2x and g(x) = x + 3. (a) Find (g ° f)(x). (2) (b) Write down g–1(x). (1) (c) Find (f ° g–1)(5). (2) (Total 5 marks)
2. A line L passes through A(1, –1, 2) and is parallel to the line r =
2 3 1 5 1 2 s .
(a) Write down a vector equation for L in the form r = a + tb.
(2)
The line L passes through point P when t = 2. (b) Find (i) OP ; (ii) OP . (4) (Total 6 marks)
3. The probability distribution of a discrete random variable X is given by P(X = x) =
14 2
(b) Show that k = 3. (4) (c) Find E(X). (2) (Total 7 marks) 4. Let g(x) = ln2 x x , for x > 0.
(a) Use the quotient rule to show that
3 ln 2 1 ) ( x x x g . (4)
(b) The graph of g has a maximum point at A. Find the x-coordinate of A.
(3) (Total 7 marks)
5. Solve the equation 2cos x = sin 2x, for 0 ≤ x ≤ 3π.
(Total 7 marks)
6. Consider f(x) = 2kx2 – 4kx + 1, for k ≠ 0. The equation f(x) = 0 has two equal roots. (a) Find the value of k.
(5)
(b) The line y = p intersects the graph of f. Find all possible values of p.
(2) (Total 7 marks)
7. In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values p, q, r and s represent numbers of students.
(a) (i) Write down the value of s. (ii) Find the value of q.
(iii) Write down the value of p and of r.
(5)
(b) (i) A student is selected at random. Given that the student takes music, write down the probability the student takes art.
(ii) Hence, show that taking music and taking art are not independent events.
(4)
(c) Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.
(4) (Total 13 marks)
8. The following diagram shows the obtuse-angled triangle ABC such that 6 2 2 AC and 4 0 3 AB .
diagram not to scale
(a) (i) Write down BA . (ii) Find BC .
(3)
(b) (i) Find cosABˆC. (ii) Hence, find sinABC.
(7)
The point D is such that
p 5 4 CD , where p > 0.
(c) (i) Given that CD 50, show that p = 3. (ii) Hence, show that CD is perpendicular to BC .
(6) (Total 16 marks)
9. The velocity v m s–1 of a particle at time t seconds, is given by v = 2t + cos2t, for 0 ≤ t ≤ 2. (a) Write down the velocity of the particle when t = 0.
(1)
When t = k, the acceleration is zero. (b) (i) Show that k =
4 π
.
(ii) Find the exact velocity when t = 4 π . (8) (c) When t < 4 π , t v d d > 0 and when t > 4 π , t v d d > 0. Sketch a graph of v against t.
(4)
(d) Let d be the distance travelled by the particle for 0 ≤ t ≤ 1. (i) Write down an expression for d.
(ii) Represent d on your sketch.
(3) (Total 16 marks)
10. In an arithmetic sequence, u1 = 2 and u3 = 8. (a) Find d. (2) (b) Find u20. (2) (c) Find S20. (2)
11. The Venn diagram below shows events A and B where P(A) = 0.3, P(A B) = 0.6 and P(A ∩ B) = 0.1. The values m, n, p and q are probabilities.
(a) (i) Write down the value of n.
(ii) Find the value of m, of p, and of q.
(4) (b) Find P(B′). (2) (Total 6 marks)
12. The following diagram shows quadrilateral ABCD, with
4 4 AC and 1 3 AB , BC AD .
diagram not to scale
(c) Show that vectors BDandCA are perpendicular.
(3) (Total 7 marks) 13. Let h(x) = x x cos 6 . Find h′(0). (Total 6 marks) 14. Let f(x) = 3 ln x and g(x) = ln 5x3.
(a) Express g(x) in the form f(x) + ln a, where a +.
(4)
(b) The graph of g is a transformation of the graph of f. Give a full geometric description of this transformation.
(3) (Total 7 marks)
15. A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These are shown in the following box and whisker diagrams.
(a) Find the range of the lengths of all 200 fish.
(3)
(b) Four cumulative frequency graphs are shown below.
Which graph is the best representation of the lengths of the female fish?
(2) (Total 5 marks)
16. The following diagram shows part of the graph of the function f(x) = 2x2.
diagram not to scale The line T is the tangent to the graph of f at x = 1.
(a) Show that the equation of T is y = 4x – 2.
(5)
(b) Find the x-intercept of T.
(2)
(c) The shaded region R is enclosed by the graph of f, the line T, and the x-axis. (i) Write down an expression for the area of R.
(ii) Find the area of R.
(9) (Total 16 marks)
17. The following diagram shows part of the graph of a quadratic function f.
The x-intercepts are at (–4, 0) and (6, 0) and the y-intercept is at (0, 240).
(a) Write down f(x) in the form f(x) = –10(x – p)(x – q).
(2)
(b) Find another expression for f(x) in the form f(x) = –10(x – h)2 + k.
(4)
(c) Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x2.
(2)
A particle moves along a straight line so that its velocity, v m s–1, at time t seconds is given by v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
(d) (i) Find the value of t when the speed of the particle is greatest. (ii) Find the acceleration of the particle when its speed is zero.
(7) (Total 15 marks)
18. The following diagram represents a large Ferris wheel, with a diameter of 100 metres.
Let P be a point on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant rate, in an anticlockwise (counterclockwise) direction. One
revolution takes 20 minutes.
(a) Write down the height of P above ground level after (i) 10 minutes;
(ii) 15 minutes.
(2)
Let h(t) metres be the height of P above ground level after t minutes. Some values of h(t) are given in the table below.
t h(t) 0 0.0 1 2.4 2 9.5 3 20.6 4 34.5 5 50.0
(c) Sketch the graph of h, for 0 ≤ t ≤ 40.
(3)
(d) Given that h can be expressed in the form h(t) = a cos bt + c, find a, b and c.
(5) (Total 14 marks)
19. Let f(x) = 8x – 2x2. Part of the graph of f is shown below.
(a) Find the x-intercepts of the graph.
(4)
(b) (i) Write down the equation of the axis of symmetry. (ii) Find the y-coordinate of the vertex.
(3) (Total 7 marks)
(b) Hence, find the term in x2 in (2 + x)4 12 1 x . (3) (Total 6 marks)
21. The straight line with equation y = x 4 3
makes an acute angle θ with the x-axis. (a) Write down the value of tan θ.
(1)
(b) Find the value of (i) sin 2θ; (ii) cos 2θ.
(6) (Total 7 marks)
22. Consider the events A and B, where P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3. The Venn diagram below shows the events A and B, and the probabilities p, q and r.
(a) Write down the value of (i) p; (ii) q; (iii) r. (3)
(b) Find the value of P(A | B′).
(2)
(c) Hence, or otherwise, show that the events A and B are not independent.
(1) (Total 6 marks)
23. The graph of f(x) = 164x2 , for –2 ≤ x ≤ 2, is shown below.
The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis. Find the volume of the solid formed.
(Total 6 marks)
24. Let f(x) = log3 x , for x > 0. (a) Show that f–1(x) = 32x.
(2)
(b) Write down the range of f–1.
Let g(x) = log3 x, for x > 0.
(c) Find the value of (f –1 ° g)(2), giving your answer as an integer.
(4) (Total 7 marks) 25. Let f(x) = x x 3x 3 1 3 2
. Part of the graph of f is shown below.
There is a maximum point at A and a minimum point at B(3, –9).
(a) Find the coordinates of A.
(b) Write down the coordinates of
(i) the image of B after reflection in the y-axis; (ii) the image of B after translation by the vector
5 2 ;
(iii) the image of B after reflection in the x-axis followed by a horizontal stretch with scale factor 2 1 . (6) (Total 14 marks) 26. Let f(x) = x x sin cos , for sin x ≠ 0.
(a) Use the quotient rule to show that f′(x) = x 2 sin 1 . (5) (b) Find f′′(x). (3)
In the following table, f′ 2 π = p and f′′ 2 π
= q. The table also gives approximate values of f′(x) and f′′(x) near x = 2 π . x 0.1 2 π 2 π 1 . 0 2 π f′(x) –1.01 p –1.01 f″(x) 0.203 q –0.203
(d) Use information from the table to explain why there is a point of inflexion on the graph of f where x = 2 π . (2) (Total 13 marks)
27. The line L1 is represented by the vector equation r =
8 1 2 25 1 3 p .
A second line L2 is parallel to L1 and passes through the point B(–8, –5, 25).
(a) Write down a vector equation for L2 in the form r = a + tb.
(2)
A third line L3 is perpendicular to L1 and is represented by r =
k q 2 7 3 0 5 . (b) Show that k = –2. (5)
The lines L1 and L3 intersect at the point A. (c) Find the coordinates of A.
The lines L2 and L3 intersect at point C where 24 3 6 BC . (d) (i) Find AB .
(ii) Hence, find |AC |.
(5) (Total 18 marks)
28. Let f(x) = p(x – q)(x – r). Part of the graph of f is shown below.
The graph passes through the points (–2, 0), (0, –4) and (4, 0).
(a) Write down the value of q and of r.
(2)
(b) Write down the equation of the axis of symmetry.
(1)
29. Let AB 32 6 and AC 23 2 . (a) Find BC. (2)
(b) Find a unit vector in the direction of AB .
(3)
(c) Show that AB is perpendicular to AB .
(3) (Total 8 marks)
30. Let f(x) = cos 2x and g(x) = 2x2 – 1.
(a) Find 2 π f . (2) (b) Find (g ° f) 2 π . (2)
(c) Given that (g ° f)(x) can be written as cos (kx), find the value of k, k .
(3) (Total 7 marks)
31. Let f(x) = kx4. The point P(1, k) lies on the curve of f. At P, the normal to the curve is parallel to y = x
8 1
. Find the value of k.
33. A function f is defined for –4 ≤ x ≤ 3. The graph of f is given below.
The graph has a local maximum when x = 0, and local minima when x = –3, x = 2.
(a) Write down the x-intercepts of the graph of the derivative function, f′.
(2)
(b) Write down all values of x for which f′(x) is positive.
(2)
(c) At point D on the graph of f, the x-coordinate is –0.5. Explain why f′′(x) < 0 at D.
(2) (Total 6 marks)
34. Consider the function f with second derivative f′′(x) = 3x – 1. The graph of f has a minimum point at A(2, 4) and a maximum point at B
27 358 , 3 4 .
(a) Use the second derivative to justify that B is a maximum.
(3)
(c) Find f(x).
(7) (Total 14 marks)
35. José travels to school on a bus. On any day, the probability that José will miss the bus is 3 1
. If he misses his bus, the probability that he will be late for school is
8 7
. If he does not miss his bus, the probability that he will be late is
8 3
.
Let E be the event “he misses his bus” and F the event “he is late for school”. The information above is shown on the following tree diagram.
(a) Find
(i) P(E ∩ F); (ii) P(F).
(b) Find the probability that
(i) José misses his bus and is not late for school;; (ii) José missed his bus, given that he is late for school.
(5)
The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and Tuesday.
(c) Copy and complete the probability distribution table.
X (cost in euros) 0 3 6 P (X) 9 1 (3)
(d) Find the expected cost for José for both days.
(2) (Total 14 marks)
36. Let f(x) = 6 + 6sinx. Part of the graph of f is shown below.
(a) Solve for 0 ≤ x < 2π. (i) 6 + 6sin x = 6; (ii) 6 + 6 sin x = 0.
(5)
(b) Write down the exact value of the x-intercept of f, for 0 ≤ x < 2.
(1)
(c) The area of the shaded region is k. Find the value of k, giving your answer in terms of π.
(6) Let g(x) = 6 + 6sin 2 π
x . The graph of f is transformed to the graph of g. (d) Give a full geometric description of this transformation.
(2)
(e) Given that
2 π 3 d ) ( pp g x x = k and 0 ≤ p < 2π, write down the two values of p.
(3) (Total 17 marks)
37. The first three terms of an infinite geometric sequence are 32, 16 and 8. (a) Write down the value of r.
(1)
(b) Find u6.
(2)
38. Let g(x) = 2x sin x. (a) Find g′(x).
(4)
(b) Find the gradient of the graph of g at x = π.
(3) (Total 7 marks)
39. The diagram shows two concentric circles with centre O.
diagram not to scale The radius of the smaller circle is 8 cm and the radius of the larger circle is 10 cm.
Points A, B and C are on the circumference of the larger circle such that AOˆB is 3 π
radians.
(a) Find the length of the arc ACB.
(2)
(b) Find the area of the shaded region.
(4) (Total 6 marks)
40. The diagram below shows the probabilities for events A and B, with P(A′) = p.
(a) Write down the value of p.
(1) (b) Find P(B). (3) (c) Find P(A′ | B). (3) (Total 7 marks)
41. (a) Show that 4 – cos 2θ + 5 sin θ = 2 sin2 θ + 5 sin θ + 3.
(2) (b) Hence, solve the equation 4 – cos 2θ + 5 sin θ = 0 for 0 ≤ θ ≤ 2π.
(5) (Total 7 marks)
42. The graph of the function y = f(x) passes through the point ,4 2 3
. The gradient function of f is given as f′(x) = sin (2x – 3). Find f(x).
(Total 6 marks)
43. The diagram shows quadrilateral ABCD with vertices A(1, 0), B(1, 5), C(5, 2) and D(4, –1).
diagram not to scale
(a) (i) Show that
24 AC . (ii) Find BD .
(iii) Show that AC is perpendicular to BD .
(5)
The line (AC) has equation r = u + sv.
(c) Show that k = 1.
(3)
(d) Hence find the area of triangle ACD.
(5) (Total 17 marks) 44. Let f(x) = x2 + 4 and g(x) = x – 1. (a) Find (f ° g)(x). (2) The vector 1 3
translates the graph of (f ° g) to the graph of h. (b) Find the coordinates of the vertex of the graph of h.
(3)
(c) Show that h(x) = x2 – 8x + 19.
(2)
(d) The line y = 2x – 6 is a tangent to the graph of h at the point P. Find the x-coordinate of P. (5) (Total 12 marks)
45. Let f(x) = x3. The following diagram shows part of the graph of f.
diagram not to scale The point P (a, f(a)), where a > 0, lies on the graph of f. The tangent at P crosses the x-axis at the point Q ,0 3 2
. This tangent intersects the graph of f at the point R(–2, –8).
(a) (i) Show that the gradient of [PQ] is 3 2 3 a a .
(ii) Find f′(a).
(iii) Hence show that a = 1.
The equation of the tangent at P is y = 3x – 2. Let T be the region enclosed by the graph of f, the tangent [PR] and the line x = k, between x = –2 and x = k where –2 < k < 1. This is shown in the diagram below.
diagram not to scale
(b) Given that the area of T is 2k + 4, show that k satisfies the equation k4 – 6k2 + 8 = 0.
(9) (Total 16 marks)
46. The letters of the word PROBABILITY are written on 11 cards as shown below.
Two cards are drawn at random without replacement. Let A be the event the first card drawn is the letter A.
(b) Find P(B│A). (2) (c) Find P(A ∩ B). (3) (Total 6 marks)
47. Let f(x) = ex cos x. Find the gradient of the normal to the curve of f at x = π.
(Total 6 marks)
48. The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as functions of time, t.
(a) Complete the following table by noting which graph A, B or C corresponds to each function.
Function Graph
(Total 6 marks)
49. Let f(x) = x2 and g(x) = 2(x – 1)2.
(a) The graph of g can be obtained from the graph of f using two transformations. Give a full geometric description of each of the two transformations.
(2)
(b) The graph of g is translated by the vector 2 3
to give the graph of h. The point (–1, 1) on the graph of f is translated to the point P on the graph of h. Find the coordinates of P.
(4) (Total 6 marks)
50. Let f(x) = ex+3.
(a) (i) Show that f–1(x) = ln x – 3. (ii) Write down the domain of f–1.
(3)
(b) Solve the equation f–1(x) = x 1 ln . (4) (Total 7 marks)
51. The graph of y = x between x = 0 and x = a is rotated 360° about the x-axis. The volume of the solid formed is 32π. Find the value of a.
52. A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.
The point P(x, y) is a vertex of the rectangle and also lies on the circle. The angle between (OP) and the x-axis is θ radians, where 0 ≤ θ ≤
2 π
.
(a) Write down an expression in terms of θ for (i) x;
(ii) y.
(2)
Let the area of the rectangle be A. (b) Show that A = 18 sin 2θ.
(c) (i) Find d dA.
(ii) Hence, find the exact value of θ which maximizes the area of the rectangle. (iii) Use the second derivative to justify that this value of θ does give a maximum.
(8) (Total 13 marks)
53. The vertices of the triangle PQR are defined by the position vectors
5 1 6 OR and 2 1 3 OQ , 1 3 4 OP . (a) Find (i) PQ ; (ii) PR . (3) (b) Show that 2 1 Q Pˆ R cos . (7)
(c) (i) Find sinRPˆQ.
(ii) Hence, find the area of triangle PQR, giving your answer in the form a 3.
(6) (Total 16 marks)
54. Let f(x) = 1 2
xax , –8 ≤ x ≤ 8, a . The graph of f is shown below.
The region between x = 3 and x = 7 is shaded.
(a) Show that f(–x) = –f(x).
(2) (b) Given that f′′(x) = 2 3 2 ) 1 ( ) 3 ( 2 x x ax
, find the coordinates of all points of inflexion.
(7) (c) It is given that
f x x aln(x 1)C 2 d ) ( 2 .(i) Find the area of the shaded region, giving your answer in the form p ln q. (ii) Find the value of
8 42f(x 1)dx.
(7) (Total 16 marks)
55. Let f(x) = x2 and g(x) = 2x – 3. (a) Find g–1(x). (2) (b) Find (f ° g)(4). (3) (Total 5 marks)
56. Find the cosine of the angle between the two vectors 3i + 4j + 5k and 4i – 5j – 3k.
(Total 6 marks)
57. The fifth term in the expansion of the binomial (a + b)n is given by 6(2 )4 4 10 q p . (a) Write down the value of n.
(1)
(b) Write down a and b, in terms of p and/or q.
(2)
(c) Write down an expression for the sixth term in the expansion.
(3) (Total 6 marks)
58. (a) Find log2 32.
(b) Given that log2 y x 8 32
can be written as px + qy, find the value of p and of q.
(4) (Total 5 marks)
59. A function f has its first derivative given by f′(x) = (x – 3)3. (a) Find the second derivative.
(2)
(b) Find f′(3) and f′′(3).
(1)
(c) The point P on the graph of f has x-coordinate 3. Explain why P is not a point of inflexion.
(2) (Total 5 marks)
60. Let f(x) = 3e2x sin x + e2x cos x, for 0 ≤ x ≤ π. Given that
3 1 6 π
tan , solve the equation f(x) = 0. (Total 6 marks) 61. Let f(x) = e–3x and g(x) = 3 π sin x .
(a) Write down (i) f′(x); (ii) g′(x).
(b) Let h(x) = e–3x sin 3 π
x . Find the exact value of h′ 3 π . (4) (Total 6 marks)
62. Two boxes contain numbered cards as shown below.
Two cards are drawn at random, one from each box.
(a) Copy and complete the table below to show all nine equally likely outcomes. 3, 9
3, 10 3, 10
(2)
Let S be the sum of the numbers on the two cards. (b) Write down all the possible values of S.
(2)
(c) Find the probability of each value of S.
(2)
(d) Find the expected value of S.
(e) Anna plays a game where she wins $50 if S is even and loses $30 if S is odd.
Anna plays the game 36 times. Find the amount she expects to have at the end of the 36 games.
(3) (Total 12 marks)
63. The line L1 is parallel to the z-axis. The point P has position vector 0 1 8 and lies on L1.
(a) Write down the equation of L1 in the form r = a + tb.
(2)
The line L2 has equation r =
5 1 2 1 4 2
s . The point A has position vector 9 2 6 .
(b) Show that A lies on L2.
(4)
Let B be the point of intersection of lines L1 and L2.
(c) (i) Show that
14 1 9 OB . (ii) Find AB . (7)
(d) The point C is at (2, 1, –4). Let D be the point such that ABCD is a parallelogram. Find OD .
(3) (Total 16 marks)
64. In this question s represents displacement in metres and t represents time in seconds.
The velocity v m s–1 of a moving body is given by v = 40 – at where a is a non-zero constant. (a) (i) If s = 100 when t = 0, find an expression for s in terms of a and t.
(ii) If s = 0 when t = 0, write down an expression for s in terms of a and t.
(6)
Trains approaching a station start to slow down when they pass a point P. As a train slows down, its velocity is given by v = 40 – at, where t = 0 at P. The station is 500 m from P. (b) A train M slows down so that it comes to a stop at the station.
(i) Find the time it takes train M to come to a stop, giving your answer in terms of a. (ii) Hence show that a =
5 8
.
(6)
(c) For a different train N, the value of a is 4.
Show that this train will stop before it reaches the station.
(5) (Total 17 marks)
65. Let f(x) = 2x3 + 3 and g(x) = e3x – 2. (a) (i) Find g(0).
(ii) Find (f ° g)(0). (5) (b) Find f–1(x). (3) (Total 8 marks)
66. (a) Let u = 1 3 2 and w = p 1 3
. Given that u is perpendicular to w, find the value of p.
(3) (b) Let v = 5 1
q . Given that v 42, find the possible values of q.
(3) (Total 6 marks)
67. Let X be normally distributed with mean 100 cm and standard deviation 5 cm. (a) On the diagram below, shade the region representing P(X > 105).
(2)
(b) Given that P(X < d) = P(X > 105), find the value of d.
(2)
(c) Given that P(X > 105) = 0.16 (correct to two significant figures), find P(d < X < 105).
(2) (Total 6 marks)
(a) Let h(x) = f(–x). Sketch the graph of h on the grid below. (2) (b) Let g(x) = 2 1
f(x – 1). The point A(3, 2) on the graph of f is transformed to the point P on the graph of g. Find the coordinates of P.
(3) (Total 5 marks) 69. Consider f(x) = x2 + x p , x ≠ 0, where p is a constant. (a) Find f′(x). (2)
(b) There is a minimum value of f(x) when x = –2. Find the value of p.
(4) (Total 6 marks)
70. Solve cos 2x – 3 cos x – 3 – cos2 x = sin2 x, for 0 ≤ x ≤ 2π.
(Total 7 marks)
71. Let f(x) = k log2 x.
(a) Given that f–1(1) = 8, find the value of k.
(3) (b) Find f–1 3 2 . (4) (Total 7 marks)
72. In a class of 100 boys, 55 boys play football and 75 boys play rugby. Each boy must play at least one sport from football and rugby.
(a) (i) Find the number of boys who play both sports. (ii) Write down the number of boys who play only rugby.
(3)
(b) One boy is selected at random.
(i) Find the probability that he plays only one sport.
(ii) Given that the boy selected plays only one sport, find the probability that he plays rugby.
Let A be the event that a boy plays football and B be the event that a boy plays rugby. (c) Explain why A and B are not mutually exclusive.
(2) (d) Show that A and B are not independent.
(3) (Total 12 marks) 73. Let f(x) = 3 + 4 20 2
x , for x ≠ ±2. The graph of f is given below.
diagram not to scale The y-intercept is at the point A.
(a) (i) Find the coordinates of A. (ii) Show that f′(x) = 0 at A.
(b) The second derivative f′′(x) = 2 3 2 ) 4 ( ) 4 3 ( 40 x x . Use this to (i) justify that the graph of f has a local maximum at A;
(ii) explain why the graph of f does not have a point of inflexion.
(6)
(c) Describe the behaviour of the graph of f for large │x│.
(1)
(d) Write down the range of f.
(2) (Total 16 marks)
74. Let f(x) = x . Line L is the normal to the graph of f at the point (4, 2). (a) Show that the equation of L is y = –4x + 18.
(4)
(b) Point A is the x-intercept of L. Find the x-coordinate of A.
In the diagram below, the shaded region R is bounded by the x-axis, the graph of f and the line L.
(c) Find an expression for the area of R.
(3)
(d) The region R is rotated 360° about the x-axis. Find the volume of the solid formed, giving your answer in terms of π.
(8) (Total 17 marks)
75. Let p = sin40 and q = cos110. Give your answers to the following in terms of p and/or q. (a) Write down an expression for
(i) sin140; (ii) cos70.
(2) (b) Find an expression for cos140.
76. Consider the arithmetic sequence 2, 5, 8, 11, .... (a) Find u101.
(3) (b) Find the value of n so that un = 152.
(3) (Total 6 marks)
77. Consider g (x) = 3 sin 2x.
(a) Write down the period of g.
(1)
(b) On the diagram below, sketch the curve of g, for 0 x 2. 4 3 2 1 0 –1 –2 –3 –4 π 2 32π 2π π y x (3)
(c) Write down the number of solutions to the equation g (x) = 2, for 0 x 2.
(2) (Total 6 marks)
78. (a) Find d . 3 2 1 x x
(2) (b) Given that x x 3d 2 1 3 0
= ln P , find the value of P.(4) (Total 6 marks)
79. A particle moves along a straight line so that its velocity, v m s−1 at time t seconds is given by
v = 6e3t + 4. When t = 0, the displacement, s, of the particle is 7 metres. Find an expression for s in terms of t. (Total 7 marks) 80. Let f (x) = ln (x + 5) + ln 2, for x –5. (a) Find f −1(x). (4) Let g (x) = ex.
(b) Find (g ◦ f) (x), giving your answer in the form ax + b, where a, b, .
(3) (Total 7 marks)
81. Consider f (x) = 3 1
x3 + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum point at M, and a point of inflexion at N.
(a) Find f′(x).
(3) (b) Find the x-coordinate of M.
(4) (c) Find the x-coordinate of N.
(3) (d) The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form
y = ax + b.
(4) (Total 14 marks)
82. Let f (x) = 3(x + 1)2 – 12.
(a) Show that f (x) = 3x2 + 6x – 9.
(2) (b) For the graph of f
(i) write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry; (iii) write down the y-intercept;
(iv) find both x-intercepts.
(8) (c) Hence sketch the graph of f.
(2) (d) Let g (x) = x2. The graph of f may be obtained from the graph of g by the two
transformations:
a stretch of scale factor t in the y-direction followed by a translation of . q p Find q p
and the value of t.
(3) (Total 15 marks)
83. A four-sided die has three blue faces and one red face. The die is rolled.
Let B be the event a blue face lands down, and R be the event a red face lands down. (a) Write down
(i) P (B); (ii) P (R).
(b) If the blue face lands down, the die is not rolled again. If the red face lands down, the die is rolled once again. This is represented by the following tree diagram, where p, s, t are probabilities.
Find the value of p, of s and of t.
(2)
Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is finished. If the red face lands down, he scores 1 and rolls one more time. Let X be the total score obtained.
(c) (i) Show that P (X = 3) = . 16
3
(ii) Find P (X = 2).
(3)
(d) (i) Construct a probability distribution table for X. (ii) Calculate the expected value of X.
(5)
(e) If the total score is 3, Guiseppi wins $10. If the total score is 2, Guiseppi gets nothing. Guiseppi plays the game twice. Find the probability that he wins exactly $10.
(4) (Total 16 marks)
84. A box contains 100 cards. Each card has a number between one and six written on it. The following table shows the frequencies for each number.
Number 1 2 3 4 5 6
Frequency 26 10 20 k 29 11
(a) Calculate the value of k.
(2) (b) Find
(i) the median;
(ii) the interquartile range.
(5) (Total 7 marks)
85. The following diagram shows part of the graph of f, where f (x) = x2 − x − 2.
(a) Find both x-intercepts.
(4) (b) Find the x-coordinate of the vertex.
(2) (Total 6 marks)
86. (a) Given that cos A = 3 1 and 0 A , 2 π
find cos 2A.
(3) (b) Given that sin B =
3 2 and 2 π B , find cos B. (3) (Total 6 marks)
87. Part of the graph of a function f is shown in the diagram below.
4 4 3 3 2 2 1 1 0 –1 –1 –2 –2 –3 –4 y x
(a) On the same diagram sketch the graph of y = − f (x).
(2) (b) Let g (x) = f (x + 3).
(i) Find g (−3).
(ii) Describe fully the transformation that maps the graph of f to the graph of g.
88. There are 20 students in a classroom. Each student plays only one sport. The table below gives their sport and gender.
Football Tennis Hockey
Female 5 3 3
Male 4 2 3
(a) One student is selected at random.
(i) Calculate the probability that the student is a male or is a tennis player.
(ii) Given that the student selected is female, calculate the probability that the student does not play football.
(4) (b) Two students are selected at random. Calculate the probability that neither student plays
football. (3) (Total 7 marks) 89. Let
5 1 3f (x)dx 12.(a) Show that
1 5 f (x)dx 4.(2) (b) Find the value of
2
1 5 2 ( ) d . d ) (x x x f x x f x (5) (Total 7 marks)
90. Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB). (a) Find
(i) AB ;
(ii) AD , giving your answer in terms of k.
(3) (b) Show that k = 7.
(3) The point C is such that BC = AD.
2 1
(c) Find the position vector of C.
(4) (d) Find cosABˆC. (3) (Total 13 marks) 91. Let f : x sin3 x.
(a) (i) Write down the range of the function f.
(ii) Consider f (x) =1, 0 x 2. Write down the number of solutions to this equation. Justify your answer.
(5) (b) Find f′ (x), giving your answer in the form a sinp x cosq x where a, p, q .
(2) (c) Let g (x) = 2 1 ) (cos sin 3 x x for 0 x 2 π
. Find the volume generated when the curve of g is revolved through 2 about the x-axis.
(7) (Total 14 marks)
92. The following diagram shows a semicircle centre O, diameter [AB], with radius 2. Let P be a point on the circumference, with POˆB = radians.
(a) Find the area of the triangle OPB, in terms of .
(2) (b) Explain why the area of triangle OPA is the same as the area triangle OPB.
(3) Let S be the total area of the two segments shaded in the diagram below.
(c) Show that S = 2( − 2 sin ).
(3) (d) Find the value of when S is a local minimum, justifying that it is a minimum.
(8) (e) Find a value of for which S has its greatest value.
(2) (Total 18 marks)
93. Consider the infinite geometric sequence 3, 3(0.9), 3(0.9)2, 3(0.9)3, … . (a) Write down the 10th term of the sequence. Do not simplify your answer.
(b) Find the sum of the infinite sequence.
(4) (Total 5 marks)
94. A particle is moving with a constant velocity along line L. Its initial position is A(6, –2, 10). After one second the particle has moved to B(9, –6, 15).
(a) (i) Find the velocity vector, AB . (ii) Find the speed of the particle.
(4)
(b) Write down an equation of the line L.
(2) (Total 6 marks)
(b) Write down the range of f–1. (1) (c) Find f–1(x). (3) (Total 7 marks)
96. Let A and B be independent events, where P(A) = 0.6 and P(B) = x. (a) Write down an expression for P(A ∩ B).
(1)
(b) Given that P(A B) = 0.8, (i) find x;
(ii) find P(A ∩ B).
(4)
(c) Hence, explain why A and B are not mutually exclusive.
(1) (Total 6 marks)
97. The diagram shows part of the graph of y = f′(x). The x-intercepts are at points A and C. There is a minimum at B, and a maximum at D.
(a) (i) Write down the value of f′(x) at C.
(ii) Hence, show that C corresponds to a minimum on the graph of f, i.e. it has the same x-coordinate.
(3)
(b) Which of the points A, B, D corresponds to a maximum on the graph of f?
(1)
(c) Show that B corresponds to a point of inflexion on the graph of f.
(3) (Total 7 marks)
(b) Let sin x = 3 2 . Show that f(2x) = 9 5 4 . (5) (Total 7 marks)
99. Two standard six-sided dice are tossed. A diagram representing the sample space is shown below.
Score on second die
1 2 3 4 5 6
1 • • • • • •
2 • • • • • •
Score on first die 3 • • • • • •
4 • • • • • •
5 • • • • • •
6 • • • • • •
Let X be the sum of the scores on the two dice. (a) Find (i) P(X = 6); (ii) P(X > 6); (iii) P(X = 7 | X > 5). (6)
(b) Elena plays a game where she tosses two dice. If the sum is 6, she wins 3 points.
If the sum is greater than 6, she wins 1 point. If the sum is less than 6, she loses k points.
Find the value of k for which Elena’s expected number of points is zero.
(7) (Total 13 marks)
100. The acceleration, a m s–2, of a particle at time t seconds is given by a = 2t + cost. (a) Find the acceleration of the particle at t = 0.
(2)
(b) Find the velocity, v, at time t, given that the initial velocity of the particle is 2 m s–1.
(5)
(c) Find
30v , giving your answer in the form p – q cos 3. dt
(7)
(d) What information does the answer to part (c) give about the motion of the particle?
(2) (Total 16 marks)
101. Let f(t) = a cos b (t – c) + d, t ≥ 0. Part of the graph of y = f(t) is given below.
(a) (i) Find the value of a. (ii) Show that b =
6 π
. (iii) Find the value of d. (iv) Write down a value for c.
(7)
The transformation P is given by a horizontal stretch of a scale factor of 2 1 , followed by a translation of 10 3 .
(b) Let M′ be the image of M under P. Find the coordinates of M′.
(2)
The graph of g is the image of the graph of f under P. (c) Find g(t) in the form g(t) = 7 cos B(t – C) + D.
(4)
(d) Give a full geometric description of the transformation that maps the graph of g to the graph of f.
(3) (Total 16 marks)
102. In an arithmetic sequence u21 = –37 and u4 = –3. (a) Find
(i) the common difference; (ii) the first term.
(b) Find S10.
(3) (Total 7 marks)
103. Let un = 3 – 2n.
(a) Write down the value of u1, u2, and u3.
(3) (b) Find
20 1 ) 2 3 ( n n . (3) (Total 6 marks) 104. Consider f(x) = x5. (a) Find (i) f(11); (ii) f(86); (iii) f(5). (3)(b) Find the values of x for which f is undefined.
(2)
(c) Let g(x) = x2. Find (g ° f)(x).
(2) (Total 7 marks)
105. The quadratic function f is defined by f(x) = 3x2 – 12x + 11. (a) Write f in the form f(x) = 3(x – h)2 – k.
(3)
(b) The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x) = 3(x – p)2 + q.
(3) (Total 6 marks)
106. The graph of a function of the form y = p cos qx is given in the diagram below.
(a) Write down the value of p.
(2)
(b) Calculate the value of q.
(4) (Total 6 marks)
107. Given that π 2
π
and that cosθ = 13 12 , find (a) sin θ; (3) (b) cos 2θ; (3) (c) sin (θ + π). (1) (Total 7 marks)
108. (a) Given that 2 sin2 θ + sinθ – 1 = 0, find the two values for sin θ.
(4)
(b) Given that 0° ≤ θ ≤ 360° and that one solution for θ is 30°, find the other two possible values for θ.
(2) (Total 6 marks)
109. Consider the points A(5, 8), B(3, 5) and C(8, 6). (a) Find
(i) AB ; (ii) AC .
(3)
(b) (i) Find ABAC.
(ii) Find the sine of the angle between AB and AC .
110. A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.
(a) Write down the number of students who scored 40 marks or less on the test.
(2)
(b) The middle 50 % of test results lie between marks a and b, where a < b. Find a and b.
(4) (Total 6 marks)
111. A random variable X is distributed normally with a mean of 100 and a variance of 100. (a) Find the value of X that is 1.12 standard deviations above the mean.
(b) Find the value of X that is 1.12 standard deviations below the mean.
(2) (Total 6 marks)
112. In a game a player rolls a biased four-faced die. The probability of each possible score is shown below. Score 1 2 3 4 Probability 5 1 5 2 10 1 x
(a) Find the value of x.
(2)
(b) Find E(X).
(3)
(c) The die is rolled twice. Find the probability of obtaining two scores of 3.
(2) (Total 7 marks)
113. Find the equation of the tangent to the curve y = e2x at the point where x = 1. Give your answer in terms of e2.
(Total 6 marks) 114. (a) Find
2 1 2 2) 3 ( x dx. (4)(b) Find
1 0 2 d e 2 x x. (3) (Total 7 marks)115. The velocity v m s–1 of a moving body at time t seconds is given by v = 50 – 10t. (a) Find its acceleration in m s–2.
(2)
(b) The initial displacement s is 40 metres. Find an expression for s in terms of t.
(4) (Total 6 marks)
116. Solve the following equations. (a) logx 49 = 2 (3) (b) log2 8 = x (2) (c) log25 x = 2 1 (3) (d) log2 x + log2(x – 7) = 3 (5) (Total 13 marks)
117. Let f (x) = 2x2 – 12x + 5.
(a) Express f(x) in the form f(x) = 2(x – h)2 – k.
(3)
(b) Write down the vertex of the graph of f.
(2)
(c) Write down the equation of the axis of symmetry of the graph of f.
(1)
(d) Find the y-intercept of the graph of f.
(2)
(e) The x-intercepts of f can be written as r
q p
, where p, q, r . Find the value of p, of q, and of r.
(7) (Total 15 marks) 118. Let f(x) = x 1 , x ≠ 0.
(a) Sketch the graph of f.
(2)
The graph of f is transformed to the graph of g by a translation of 3 2 . (b) Find an expression for g(x).
(c) (i) Find the intercepts of g.
(ii) Write down the equations of the asymptotes of g. (iii) Sketch the graph of g.
(10) (Total 14 marks)
119. A spring is suspended from the ceiling. It is pulled down and released, and then oscillates up and down. Its length, l centimetres, is modelled by the function l = 33 + 5cos((720t)°), where t is time in seconds after release.
(a) Find the length of the spring after 1 second.
(2)
(b) Find the minimum length of the spring.
(3)
(c) Find the first time at which the length is 33 cm.
(3)
(d) What is the period of the motion?
(2) (Total 10 marks)
120. Two lines L1 and L2 are given by r1 =
10 6 2 6 4 9 s and r2 = 2 10 6 2 20 1 t .
(a) Let θ be the acute angle between L1 and L2. Show that cosθ = 140
52 .
(b) (i) P is the point on L1 when s = 1. Find the position vector of P. (ii) Show that P is also on L2.
(8)
(c) A third line L3 has direction vector 30 6
x . If L1 and L3 are parallel, find the value of x.
(3) (Total 16 marks)
121. The heights of trees in a forest are normally distributed with mean height 17 metres. One tree is selected at random. The probability that a selected tree has a height greater than 24 metres is 0.06.
(a) Find the probability that the tree selected has a height less than 24 metres.
(2)
(b) The probability that the tree has a height less than D metres is 0.06. Find the value of D.
(3)
(c) A woodcutter randomly selects 200 trees. Find the expected number of trees whose height lies between 17 metres and 24 metres.
(4) (Total 9 marks)
122. The probability of obtaining heads on a biased coin is 3 1
.
(b) Amir plays a game in which he tosses the coin 12 times. (i) Find the expected number of heads.
(ii) Amir wins $ 10 for each head obtained, and loses $ 6 for each tail. Find his expected winnings.
(5) (Total 10 marks)
123. Let g(x) = x3 – 3x2 – 9x + 5.
(a) Find the two values of x at which the tangent to the graph of g is horizontal.
(8)
(b) For each of these values, determine whether it is a maximum or a minimum.
(6) (Total 14 marks)
124. The diagram below shows part of the graph of y = sin 2x. The shaded region is between x = 0 and x = m.
(a) Write down the period of this function.
(2)
(c) Find the area of the shaded region.
(6) (Total 10 marks)
125. Consider the infinite geometric sequence 25, 5, 1, 0.2, … . (a) Find the common ratio.
(b) Find
(i) the 10th term;
(ii) an expression for the nth term. (c) Find the sum of the infinite sequence.
(Total 6 marks)
126. Consider the events A and B, where P(A) = 5 2 , P(B′) = 4 1 and P(A B) = 8 7 . (a) Write down P(B).
(b) Find P(A B). (c) Find P(A B).
(Total 6 marks)
127. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate
(a) the size of PQˆR; (b) the area of triangle PQR.
128. The cumulative frequency graph below shows the heights of 120 girls in a school. 130 120 110 100 90 80 70 60 50 40 30 20 10 0 185 180 175 170 165 160 155 150 C um ul at iv e fr eq ue nc y Height in centimetres (a) Using the graph
(i) write down the median; (ii) find the interquartile range.
(b) Given that 60 of the girls are taller than a cm, find the value of a.
(Total 6 marks)
(b) loga 8
(c) loga 2.5
(Total 6 marks)
130. The following diagram shows a triangle ABC, where ACˆB is 90, AB = 3, AC = 2 and BAˆC is .
(a) Show that sin = 3
5 .
(b) Show that sin 2 = 9
5 4 .
(c) Find the exact value of cos 2.
(Total 6 marks)
131. The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x). (a) Write down f −1(x).
(b) (i) Find ( f ◦ g) (x). (ii) Find ( f ◦ g)−1 (x).
132. The velocity v of a particle at time t is given by v = e−2t + 12t. The displacement of the particle at time t is s. Given that s = 2 when t = 0, express s in terms of t.
(Total 6 marks)
133. The heights of boys at a particular school follow a normal distribution with a standard deviation of 5 cm. The probability of a boy being shorter than 153 cm is 0.705.
(a) Calculate the mean height of the boys.
(b) Find the probability of a boy being taller than 156 cm.
(Total 6 marks)
134. The graph of the function y = f (x), 0 x 4, is shown below.
(a) Write down the value of (i) f ′ (1);
(b) On the diagram below, draw the graph of y = 3 f (−x).
135. Consider the expansion of the expression (x3 − 3x)6. (a) Write down the number of terms in this expansion. (b) Find the term in x12.
(Total 6 marks)
136. Let f (x) = x3 − 3x2 − 24x +1.
The tangents to the curve of f at the points P and Q are parallel to the x-axis, where P is to the left of Q.
(a) Calculate the coordinates of P and of Q.
Let N1 and N2 be the normals to the curve at P and Q respectively.
(b) Write down the coordinates of the points where (i) the tangent at P intersects N2;
(ii) the tangent at Q intersects N1.
(Total 6 marks)
137. It is given that
31 f (x)dx = 5.
(a) Write down
31 2 f (x)dx.
(b) Find the value of
3 1 (3x2 + f (x))dx.
138. (a) Given that (2x)2 + (2x) −12 can be written as (2x + a)(2x + b), where a, b , find the value of a and of b.
(b) Hence find the exact solution of the equation (2x)2 + (2x) −12 = 0, and explain why there is only one solution.
(Total 6 marks)
139. The population of a city at the end of 1972 was 250 000. The population increases by 1.3 per year.
(a) Write down the population at the end of 1973. (b) Find the population at the end of 2002.
(Total 6 marks)
140. One of the terms of the expansion of (x + 2y)10 is ax8 y2. Find the value of a.
(Total 6 marks)
141. Let f (x) = x4, x − 4 and g (x) = x2, x . (a) Find (g ◦ f ) (3).
(b) Find f −1(x).
(c) Write down the domain of f −1.
142. The eye colour of 97 students is recorded in the chart below.
Brown Blue Green
Male 21 16 9
Female 19 19 13
One student is selected at random.
(a) Write down the probability that the student is a male.
(b) Write down the probability that the student has green eyes, given that the student is a female.
(c) Find the probability that the student has green eyes or is male.
(Total 6 marks)
143. Let f ′ (x) = 12x2 − 2.
Given that f (−1) =1, find f (x).
(Total 6 marks)
144. Consider the vectors u = 2i + 3 j − k and v = 4i + j − pk. (a) Given that u is perpendicular to v find the value of p. (b) Given that qu=14, find the value of q.
(Total 6 marks)
145. The weights of a group of children are normally distributed with a mean of 22.5 kg and a standard deviation of 2.2 kg.
(a) Write down the probability that a child selected at random has a weight more than 25.8 kg.
(c) The diagram below shows a normal curve.
On the diagram, shade the region that represents the following information: 87 of the children weigh less than 25 kg
(Total 6 marks)
146. The velocity, v, in m s−1 of a particle moving in a straight line is given by v = e3t−2, where t is the time in seconds.
(a) Find the acceleration of the particle at t =1.
(b) At what value of t does the particle have a velocity of 22.3 m s−1? (c) Find the distance travelled in the first second.
(Total 6 marks)
147. A set of data is
18, 18, 19, 19, 20, 22, 22, 23, 27, 28, 28, 31, 34, 34, 36. The box and whisker plot for this data is shown below.
(a) Write down the values of A, B, C, D and E.
148. The following diagram shows a sector of a circle of radius r cm, and angle at the centre. The perimeter of the sector is 20 cm.
(a) Show that = r r 2 20 .
(b) The area of the sector is 25 cm2. Find the value of r.
(Total 6 marks)
149. Consider two different quadratic functions of the form f (x) = 4x2 − qx + 25. The graph of each function has its vertex on the x-axis.
(a) Find both values of q.
(b) For the greater value of q, solve f (x) = 0.
(c) Find the coordinates of the point of intersection of the two graphs.
(Total 6 marks)
150. Let f (x) = ln (x + 2), x −2 and g (x) = e(x−4), x 0. (a) Write down the x-intercept of the graph of f. (b) (i) Write down f (−1.999).
(ii) Find the range of f.
(c) Find the coordinates of the point of intersection of the graphs of f and g.
151. The graph of a function f is shown in the diagram below. The point A (–1, 1) is on the graph, and y = −1 is a horizontal asymptote.
(a) Let g (x) = f (x −1) + 2. On the diagram, sketch the graph of g. (b) Write down the equation of the horizontal asymptote of g.
(c) Let A′ be the point on the graph of g corresponding to point A. Write down the coordinates of A′.
(Total 6 marks)
152. Let f (x) = 3 cos 2x + sin2 x. (a) Show that f ′ (x) = −5 sin 2x. (b) In the interval
4 π x
4 3π
, one normal to the graph of f has equation x = k. Find the value of k.
153. The histogram below represents the ages of 270 people in a village.
(a) Use the histogram to complete the table below.
Age range Frequency Mid-interval value
0 age 20 40 10 20 ≤ age 40 40 ≤ age 60 60 ≤ age 80 80 ≤ age ≤100 (2)
(b) Hence, calculate an estimate of the mean age.
(4) (Total 6 marks)
154. The Venn diagram below shows information about 120 students in a school. Of these, 40 study Chinese (C), 35 study Japanese (J), and 30 study Spanish (S).
A student is chosen at random from the group. Find the probability that the student (a) studies exactly two of these languages;
(1) (b) studies only Japanese;
(2) (c) does not study any of these languages.
(3) (Total 6 marks)
155. A bag contains four apples (A) and six bananas (B). A fruit is taken from the bag and eaten. Then a second fruit is taken and eaten.
(a) Complete the tree diagram below by writing probabilities in the spaces provided.
(3) (b) Find the probability that one of each type of fruit was eaten.
(3) (Total 6 marks)
156. The following diagram shows part of the graph of y = cos x for 0 x 2. Regions A and B are shaded.
(a) Write down an expression for the area of A.
(1)
(c) Find the total area of the shaded regions.
(4) (Total 6 marks)
157. The first four terms of a sequence are 18, 54, 162, 486.
(a) Use all four terms to show that this is a geometric sequence.
(2)
(b) (i) Find an expression for the nth term of this geometric sequence. (ii) If the nth term of the sequence is 1062 882, find the value of n.
(4) (Total 6 marks)
158. (a) Write down the first three terms of the sequence un = 3n, for n 1.
(1) (b) Find (i)
20 1 ; 3 n n (ii)
100 21 3 n n . (5) (Total 6 marks)159. Let f (x) = loga x, x 0. (a) Write down the value of
(i) f (a); (ii) f (1); (iii) f (a4 ).
(3)
(b) The diagram below shows part of the graph of f.
2 0 –1 –2 2 1 –1 –2 x f y 1
On the same diagram, sketch the graph of f−1.
(3) (Total 6 marks)
160. Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the point where x =1.
