Revision 12 IB Paper 1

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.

  2  1      A line L passes through A(1, –1, 2) and is parallel to the line r =  1   s 3  .  5    2    

(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) =

(a)

x2 , x  {1, 2, k}, where k > 0. 14

Write down P(X = 2). (1)

IB Questionbank Maths SL

1

(b)

Show that k = 3. (4)

(c)

Find E(X). (2) (Total 7 marks)

4.

Let g(x) =

(a)

ln x x2

, for x > 0.

Use the quotient rule to show that g ( x) 

1  2 ln x x3

. (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)


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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)


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8.

The following diagram shows the obtuse-angled triangle ABC such that   3   2     AB   0  and AC   2  .   4   6    

diagram not to scale

(a)

(i)

Write down BA .

(ii)

Find BC . (3)

(b)

(i)

ˆC. Find cos AB

(ii)

 Hence, find sin ABC . (7)

  4   The point D is such that CD   5  , where p > 0.  p  

(c)

(i)

Given that CD  50 , show that p = 3.

(ii)

Hence, show that CD is perpendicular to BC . (6) (Total 16 marks)


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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)

π . 4

(i)

Show that k =

(ii)

Find the exact velocity when t =

π . 4 (8)

(c)

When t <

π dv π dv , > 0 and when t > , > 0. 4 dt 4 dt

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) (Total 6 marks)


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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.

 3  4 The following diagram shows quadrilateral ABCD, with AD  BC, AB    and AC    . 1  4


(a)

Find BC . (2)

(b)

Show that BD   22  .   (2)


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(c)

Show that vectors BD and CA are perpendicular. (3) (Total 7 marks)

13.

Let h(x) =

6x . Find h′(0). cos x (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.


7

(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)


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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)


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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)


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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.

(b)

(i)

Show that h(8) = 90.5.

(ii)

Find h(21).

t

h(t)

0

0.0

1

2.4

2

9.5

3

20.6

4

34.5

5

50.0

(4)


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(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)

20.

(a)

Expand (2 + x)4 and simplify your result. (3)


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(b)

1   Hence, find the term in x2 in (2 + x)4 1  2  .  x  (3) (Total 6 marks)

21.

The straight line with equation y = (a)

3 x makes an acute angle θ with the x-axis. 4

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.


13

(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)


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23.

The graph of f(x) =

16  4 x 2 , 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. (1)


15

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) =

1 3 x  x 2  3x . Part of the graph of f is shown below. 3

There is a maximum point at A and a minimum point at B(3, –9).

(a)

Find the coordinates of A. (8)


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(b)

Write down the coordinates of (i)

the image of B after reflection in the y-axis;

(ii)

  2 the image of B after translation by the vector   ;  5 

(iii)

the image of B after reflection in the x-axis followed by a horizontal stretch with 1 scale factor . 2 (6) (Total 14 marks)

26.

Let f(x) =

(a)

cos x , for sin x ≠ 0. sin x

Use the quotient rule to show that f′(x) =

1 sin 2 x

. (5)

(b)

Find f′′(x). (3)

π π In the following table, f′   = p and f′′   = q. The table also gives approximate values of 2 2 π f′(x) and f′′(x) near x = . 2

(c)

x

π  0.1 2

π 2

π  0.1 2

f′(x)

–1.01

p

–1.01

f″(x)

0.203

q

–0.203

Find the value of p and of q. (3)


17

(d)

Use information from the table to explain why there is a point of inflexion on the graph of π f where x = . 2 (2) (Total 13 marks)

27.

 3    The line L1 is represented by the vector equation r =   1     25  

 2    p 1  .   8  

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)

 5   7     A third line L3 is perpendicular to L1 and is represented by r =  0   q  2  .  3  k     

(b)

Show that k = –2. (5)

The lines L1 and L3 intersect at the point A. (c)

Find the coordinates of A. (6)


18

 6    The lines L2 and L3 intersect at point C where BC   3  .   24  

(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)

(c)

Find the value of p. (3) (Total 6 marks)


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29.

 6    2 Let AB    2  and AC    3  .  2   3     

(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)

π Find f   . 2 (2)

(b)

π Find (g ° f)   . 2 (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 1 y =  x . Find the value of k. 8 (Total 6 marks)

32.

Solve log2x + log2(x – 2) = 3, for x > 2. (Total 7 marks)


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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  4 358  point at A(2, 4) and a maximum point at B   , .  3 27  (a)

Use the second derivative to justify that B is a maximum. (3)

(b)

Given that f′ =

3 2 x – x + p, show that p = –4. 2 (4)


21

(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 If he misses his bus, the probability that he will be late for school is

1 . 3

7 . 8

3 . 8 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. If he does not miss his bus, the probability that he will be late is

(a)

Find (i)

P(E ∩ F);

(ii)

P(F). (4)


22

(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

P (X)

1 9

3

6

(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.

The shaded region is enclosed by the curve of f, the x-axis, and the y-axis.


23

(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)

π  Let g(x) = 6 + 6sin  x   . The graph of f is transformed to the graph of g. 2  (d)

Give a full geometric description of this transformation. (2)

(e)

Given that



p

p

3π 2

g ( x)dx = 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)

(c)

Find the sum to infinity of this sequence. (2) (Total 5 marks)


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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. ˆ B is Points A, B and C are on the circumference of the larger circle such that AO

(a)

π radians. 3

Find the length of the arc ACB. (2)

(b)

Find the area of the shaded region. (4) (Total 6 marks)


25

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)


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42.

3  The graph of the function y = f(x) passes through the point  , 4  . The gradient function of f is 2  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).


(a)

(i)

 . Show that AC   4  2

(ii)

Find BD .

(iii)

Show that AC is perpendicular to BD . (5)

The line (AC) has equation r = u + sv. (b)

(i)

Write down vector u and vector v.

(ii)

Find a vector equation for the line (BD). (4)

The lines (AC) and (BD) intersect at the point P(3, k). IB Questionbank Maths SL

27

(c)


(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)

 3 The vector   translates the graph of (f ° g) to the graph of h.  1

(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)


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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 2  the point Q  , 0  . This tangent intersects the graph of f at the point R(–2, –8). 3 

(a)

(i)

Show that the gradient of [PQ] is

(ii)

Find f′(a).

(iii)

Hence show that a = 1.

a3 2 a 3

.

(7)


29

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.


(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. Let B be the event the second card drawn is the letter B.

(a)

Find P(A). (1)


30

(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

displacement acceleration (4)

(b)

Write down the value of t when the velocity is greatest. (2)


31

(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)

 3  The graph of g is translated by the vector   to give the graph of h.   2 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)

1 Solve the equation f–1(x) = ln  .  x (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. (Total 7 marks)


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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) π and the x-axis is θ radians, where 0 ≤ θ ≤ . 2

(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. (3)


33

(c)

dA . d

(i)

Find

(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  4  3 6       OP    3 , OQ    1 and OR    1 .  1  2 5      

(a)

Find (i)

PQ ;

(ii)

PR . (3)

(b)

1 Show that cos RPˆ Q  . 2 (7)

(c)

(i)

Find sin RPˆ Q .

(ii)

Hence, find the area of triangle PQR, giving your answer in the form a 3 . (6) (Total 16 marks)


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54.

Let f(x) =

ax 2

x 1

, –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) =

2ax( x 2  3) ( x 2  1) 3

, find the coordinates of all points of inflexion. (7)

(c)

It is given that

a

 f ( x)dx  2 ln(x

2

 1)  C .

(i)

Find the area of the shaded region, giving your answer in the form p ln q.

(ii)

Find the value of

8

 2 f ( x  1)dx . 4

(7) (Total 16 marks)


35

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.

10 The fifth term in the expansion of the binomial (a + b)n is given by   p 6 (2q) 4 . 4

(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. (1)


36

(b)

 32 x Given that log2  y  8

  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) =

3e 2 x sin x + e2x cos x, for 0 ≤ x ≤ π. Given that tan

π 1  , solve the equation 6 3

f(x) = 0. (Total 6 marks)

61.

π  Let f(x) = e–3x and g(x) = sin  x   . 3  (a)

Write down (i)

f′(x);

(ii)

g′(x). (2)


37

(b)

π  π Let h(x) = e–3x sin  x   . Find the exact value of h′   . 3  3 (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. (3)


38

(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.

8   The line L1 is parallel to the z-axis. The point P has position vector  1  and lies on L1. 0  

(a)

Write down the equation of L1 in the form r = a + tb. (2)

2 2     The line L2 has equation r =  4   s  1 . The point A has position vector   1  5     

(b)

6    2 . 9  

Show that A lies on L2. (4)

Let B be the point of intersection of lines L1 and L2.

(c)

(i)

9   Show that OB   1  . 14  

(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)


39

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 =

8 . 5 (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)


40

66.

(a)

 2   Let u =  3  and w =  1  

 3     1 . Given that u is perpendicular to w, find the value of p.  p   (3)

(b)

1   Let v =  q  . Given that v  42 , find the possible values of q. 5   (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)


41

68.

The diagram below shows the graph of a function f(x), for –2 ≤ x ≤ 4.


42

(a)

Let h(x) = f(–x). Sketch the graph of h on the grid below.

(2)

(b)

1 f(x – 1). The point A(3, 2) on the graph of f is transformed to the point P on 2 the graph of g. Find the coordinates of P. Let g(x) =

(3) (Total 5 marks)

69.

Consider f(x) = x2 + (a)

p , x ≠ 0, where p is a constant. x

Find f′(x). (2)


43

(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)

2 Find f–1   . 3 (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. (4)


44

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 +

20 2

x 4

, 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. (7)


45

(b)

The second derivative f′′(x) =

40(3x 2  4) ( x 2  4) 3

. 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) = (a)

x . Line L is the normal to the graph of f at the point (4, 2).

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. (2)


46

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. (3)

(c)

Find an expression for tan140. (1) (Total 6 marks)


47

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. y 4 3 2 1 0 –1 –2 –3 –4

π 2

π

3π 2

2π

x

(3)

(c)

Write down the number of solutions to the equation g (x) = 2, for 0  x  2. (2) (Total 6 marks)


48

78.

(a)

Find

1

 2 x  3 dx. (2)

(b)

Given that



3

0

1 dx = ln 2x  3

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,



49

81.

1 3 x + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum 3 point at M, and a point of inflexion at N. Consider f (x) =

(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)


50

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  p a translation of  . q  p Find   and the value of t. q (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). (2)


51

(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) =

(ii)

Find P (X = 2).

3 . 16 (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)


52

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.

(a)

Number

1

2

3

4

5

6

Frequency

26

10

20

k

29

11

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)


53

86.

(a)

1 π and 0  A  , find cos 2A. 2 3

Given that cos A =

(3)

(b)

Given that sin B =

2 π and  B  , find cos B. 3 2 (3) (Total 6 marks)

87.

Part of the graph of a function f is shown in the diagram below. y 4 3 2 1 –2

–1

0

1

2

3

4

x

–1 –2 –3 –4

(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. (4) (Total 6 marks)


54

88.

There are 20 students in a classroom. Each student plays only one sport. The table below gives their sport and gender.

(a)

Football

Tennis

Hockey

Female

5

3

3

Male

4

2

3

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 (a)



5

1

3 f ( x) dx  12.

Show that



1 5

f ( x) dx   4. (2)

(b)

Find the value of

2

5

 x  f ( x)  dx   x  f ( x)  dx. 1

2

(5) (Total 7 marks)


55

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)


The point C is such that BC = (c)

1 AD. 2

Find the position vector of C. (4)

(d)

ˆC. Find cosAB (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)

1

(c)

π . Find the volume generated when the curve 2 of g is revolved through 2 about the x-axis. Let g (x) =

3 sin x (cos x) 2 for 0  x 

(7) (Total 14 marks)


56

92.

The following diagram shows a semicircle centre O, diameter [AB], with radius 2. ˆ B =  radians. Let P be a point on the circumference, with PO

(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. (1)


57

(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)

95.

Let f be the function given by f(x) = e0.5x, 0 ≤ x ≤ 3.5. The diagram shows the graph of f.

(a)

On the same diagram, sketch the graph of f–1. (3)


58

(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)


59

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)

98.

Let f(x) = sin3 x + cos3 x tan x, (a)

π < x < π. 2

Show that f(x) = sin x. (2)


60

(b)

Let sin x =

2 4 5 . Show that f(2x) =  . 3 9 (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

Score on first die

1

2

3

4

5

6

1

•

•

•

•

•

•

2

•

•

•

•

•

•

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)


61

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

3

 vdt , giving your answer in the form p – q cos 3. 0

(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.

When t = 3, there is a maximum value of 29, at M. When t = 9 , there is a minimum value of 15.


62

(a)

(i)

Find the value of a.

(ii)

Show that b =

(iii)

Find the value of d.

(iv)

Write down a value for c.

π . 6

(7)

The transformation P is given by a horizontal stretch of a scale factor of

1 , followed by a 2

 3   . translation of   10 

(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. (4)


63

(b)

Find S10. (3) (Total 7 marks)

103. Let un = 3 – 2n. (a)

Write down the value of u1, u2, and u3. (3)

20

(b)

Find

 (3  2n) . n 1

(3) (Total 6 marks)

104. Consider f(x) = (a)

x 5 .

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)


64

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)


65

107. Given that (a)

π 12    π and that cosθ =  , find 2 13

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 AB AC .

(ii)

Find the sine of the angle between AB and AC . (3) (Total 6 marks)


66

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. (4)


67

(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.

(a)

Score

1

2

3

4

Probability

1 5

2 5

1 10

x

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

(3x 2  2) dx. (4)


68

(b)

Find

1

 2e

2x

0

dx . (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 = 

1 2 (3)

(d)

log2 x + log2(x – 7) = 3 (5) (Total 13 marks)


69

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 Find the value of p, of q, and of r.

p q r

, where p, q, r 


118. Let f(x) = (a)

1 , x ≠ 0. x

Sketch the graph of f. (2)

 2 The graph of f is transformed to the graph of g by a translation of   .  3

(b)

Find an expression for g(x). (2)


70

(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)

 9    2     120. Two lines L1 and L2 are given by r1 =  4   s 6  and r2 =   6   10     

(a)

 1    6      20  t  10  .  2    2    

Let θ be the acute angle between L1 and L2. Show that cosθ =

52 . 140 (5)


71

(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)

 6    A third line L3 has direction vector  x  . If L1 and L3 are parallel, find the value of x.   30    (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 (a)

1 . 3

Sammy tosses the coin three times. Find the probability of getting (i)

three heads;

(ii)

two heads and one tail. (5)


72

(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)

(b)

Hence or otherwise write down the value of m. (2)


73

(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

(c)

(i)

the 10th term;

(ii)

an expression for the nth term.

Find the sum of the infinite sequence. (Total 6 marks)

126. Consider the events A and B, where P(A) = (a)

Write down P(B).

(b)

Find P(A  B).

(c)

Find P(A  B).

2 1 7 , P(B′) = and P(A  B) = . 5 4 8

(Total 6 marks)

127. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate (a)

ˆR; the size of PQ

(b)

the area of triangle PQR. (Total 6 marks)


74

128. The cumulative frequency graph below shows the heights of 120 girls in a school. 130 120 110

Cumulative frequency

100 90 80 70 60 50 40 30 20 10 0

150 155 160 165 170 175 180 185 Height in centimetres

(a)

(b)

Using the graph (i)

write down the median;

(ii)

find the interquartile range.

Given that 60 of the girls are taller than a cm, find the value of a. (Total 6 marks)

129. Given that p = loga 5, q = loga 2, express the following in terms of p and/or q. (a)

loga 10


75

(b)

loga 8

(c)

loga 2.5 (Total 6 marks)

ˆ B is 90, AB = 3, AC = 2 and BA ˆC 130. The following diagram shows a triangle ABC, where AC is .

(a)

Show that sin  =

5 . 3

(b)

Show that sin 2 =

4 5 . 9

(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). (Total 6 marks)


76

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)


f ′ (1);

(ii)

f ′ (3).


77

(b)

On the diagram below, draw the graph of y = 3 f (−x).

(c)

On the diagram below, draw the graph of y = f (2x).

(Total 6 marks)


78

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



3

f (x)dx = 5.

1



3

(a)

Write down

(b)

Find the value of

1

2 f (x)dx.



3

1

(3x2 + f (x))dx. (Total 6 marks)


79

138. (a) (b)

Given that (2x)2 + (2x) −12 can be written as (2x + a)(2x + b), where a, b  value of a and of b.

, find the

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) =

x  4 , 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.

.

(Total 6 marks)


80

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 q u =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.

(b)

Of the group 95 weigh less than k kilograms. Find the value of k.


81

(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. A = ......

(b)

B = ......

C= ......

D = ......

E = ......

Find the interquartile range. (Total 6 marks)


82

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.

20 2r . r

(a)

Show that  =

(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. (Total 6 marks)


83

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

π 3π x , one normal to the graph of f has equation x = k. 4 4

Find the value of k. (Total 6 marks)


84

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)


85

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)


86

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)

(b)

Calculate the area of A. (1)


87

(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 20

(i)

 3n ; n 1

100

(ii)

 3n .

n  21

(5) (Total 6 marks)


88

159. Let f (x) = loga x, x  0. (a)


f (a);

(ii)

f (1);

(iii)

f (a4 ). (3)

(b)

The diagram below shows part of the graph of f. y 2

1

–2

–1

0

f

1

2

x

–1

–2

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. (Total 6 marks)


89

161. Differentiate each of the following with respect to x. (a)

y = sin 3x (1)

(b)

y = x tan x (2)

(c)

y=

ln x x (3) (Total 6 marks)

 x 162. The diagram below shows the graph of f (x) = 1 + tan   for −360  x  360. 2

(a)

On the same diagram, draw the asymptotes. (2)

(b)

Write down (i)

the period of the function;

(ii)

the value of f (90). (2)


90

(c)

Solve f (x) = 0 for −360  x  360. (2) (Total 6 marks)

163. The following diagram shows part of the graph of f (x) = 5 − x2 with vertex V (0, 5). h Its image y = g (x) after a translation with vector   has vertex T (3, 6). k 

(a)


h;

(ii)

k. (2)

(b)

Write down an expression for g (x). (2)

(c)

On the same diagram, sketch the graph of y = g (−x). (2) (Total 6 marks)


91

164. A discrete random variable X has a probability distribution as shown in the table below.

(a)

x

0

1

2

3

P(X = x)

0.1

a

0.3

b

Find the value of a + b. (2)

(b)

Given that E(X) =1.5, find the value of a and of b. (4) (Total 6 marks)

4

165. (a)

 1 Expand  e   in terms of e.  e (4) 4

(b)

4

 1  1 Express  e   +  e   as the sum of three terms.  e  e (2) (Total 6 marks)

166. The area A km2 affected by a forest fire at time t hours is given by A = A0 ekt. When t = 5, the area affected is 1 km2 and the rate of change of the area is 0.2 km2 h−1. (a)

Show that k = 0.2. (4)

(b)

Given that A0 =

1 , find the value of t when 100 km2 are affected. e (2) (Total 6 marks)


92

167. On the axes below, sketch a curve y = f (x) which satisfies the following conditions. x

f (x)

−2  x  0 0

–1

0  x 1 1 1x2

2

f ′ (x)

f ′′ (x)

negative

positive

0

positive

positive

positive

positive

0

positive

negative

(Total 6 marks)


93

168. Let A and B be independent events such that P(A) = 0.3 and P(B) = 0.8. (a)

Find P(A  B).

(b)

Find P(A  B).

(c)

Are A and B mutually exclusive? Justify your answer. (Total 6 marks)


94

169. The following diagram shows part of the graph of f (x).

Consider the five graphs in the diagrams labelled A, B, C, D, E below.

(a)

Which diagram is the graph of f (x + 2) ?

(b)

Which diagram is the graph of – f (x) ?


95

(c)

Which diagram is the graph of f (–x) (Total 6 marks)

170. The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm. (a)

A student is chosen at random. Find the probability that the student’s height is greater than 180 cm.

(b)

In this group of students, 11.9 have heights less than d cm. Find the value of d. (Total 6 marks)

171. (a)

Let f (x) = e5x. Write down f ′ (x).

(b)

Let g (x) = sin 2x. Write down g′ (x).

(c)

Let h (x) = e5x sin 2x. Find h′ (x). (Total 6 marks)

172. Let f (x) = a (x − 4)2 + 8. (a)

Write down the coordinates of the vertex of the curve of f.

(b)

Given that f (7) = −10, find the value of a.

(c)

Hence find the y-intercept of the curve of f. (Total 6 marks)

173. Let f (x) = x3 − 4 and g (x) = 2x. (a)

Find (g ◦ f ) (−2).

(b)

Find f −1 (x). (Total 6 marks)


96

174. Consider the four numbers a, b, c, d with a  b  c ≤ d, where a, b, c, d  The mean of the four numbers is 4. The mode is 3. The median is 3. The range is 6.

.

Find the value of a, of b, of c and of d. (Total 6 marks)

175. (a)

(b)

Let logc 3 = p and logc 5 = q. Find an expression in terms of p and q for (i)

log c 15;

(ii)

log c 25.

Find the value of d if log d 6 =

1 . 2 (Total 6 marks)


97

176. The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lie on the curve, where B is a minimum point and D is a maximum point.

(a)

Complete the following table, noting whether ƒ′(x) is positive, negative or zero at the given points. A

B

E

f ′ (x)

(b)

Complete the following table, noting whether ƒ′′(x) is positive, negative or zero at the given points. A

C

E

ƒ′′ (x) (Total 6 marks)

177. The velocity, v m s−1, of a moving object at time t seconds is given by v = 4t3 − 2t. When t = 2, the displacement, s, of the object is 8 metres. Find an expression for s in terms of t. (Total 6 marks)


98

178. The following diagram shows a circle with radius r and centre O. The points A, B and C are on ˆ C = . the circle and AO

The area of sector OABC is

4 2  and the length of arc ABC is . 3 3

Find the value of r and of . (Total 6 marks)

179. Let ƒ (x) = a sin b (x − c). Part of the graph of ƒ is given below.

Given that a, b and c are positive, find the value of a, of b and of c. (Total 6 marks)


99

180. Let ƒ (x) = 3 sin 2x, for 1  x  4 and g (x) = −5x2 + 27x − 35 for 1  x  4. The graph of ƒ is shown below.

(a)

On the same diagram, sketch the graph of g.

(b)

One solution of ƒ (x) = g (x) is 1.89. Write down the other solution.

(c)

Let h (x) = g (x) − ƒ (x). Given that h (x)  0 for p  x  q, write down the value of p and of q. (Total 6 marks)

181. Consider the infinite geometric series 405 + 270 + 180 +.... (a)

For this series, find the common ratio, giving your answer as a fraction in its simplest form.

(b)

Find the fifteenth term of this series.

(c)

Find the exact value of the sum of the infinite series. (Total 6 marks)


100

182. The points P(−2, 4), Q (3, 1) and R (1, 6) are shown in the diagram below.

(a)

Find the vector PQ .

(b)

Find a vector equation for the line through R parallel to the line (PQ). (Total 6 marks)

183. The population below is listed in ascending order. 5, 6, 7, 7, 9, 9, r, 10, s, 13, 13, t The median of the population is 9.5. The upper quartile Q3 is 13. (a)

(b)


r;

(ii)

s.

The mean of the population is 10. Find the value of t. (Total 6 marks)

184. Solve the following equations. (a)

ln (x + 2) = 3.

(b)

102x = 500. (Total 6 marks)


101

185. The probability distribution of the discrete random variable X is given by the following table. x

1

2

3

4

5

P(X = x)

0.4

p

0.2

0.07

0.02

(a)

Find the value of p.

(b)

Calculate the expected value of X. (Total 6 marks)

186. The graph of a function g is given in the diagram below.

The gradient of the curve has its maximum value at point B and its minimum value at point D. The tangent is horizontal at points C and E. (a)

Complete the table below, by stating whether the first derivative g′ is positive or negative, and whether the second derivative g′′ is positive or negative. Interval

g′

g′′

axb exƒ


102

(b)

Complete the table below by noting the points on the graph described by the following conditions. Conditions

Point

g′ (x) = 0, g′′ (x)  0 g′ (x)  0, g′′ (x) = 0 (Total 6 marks)

187. (a)

Express y = 2x2 – 12x + 23 in the form y = 2(x – c)2 + d.

The graph of y = x2 is transformed into the graph of y = 2x2 – 12x + 23 by the transformations a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units. (b)


k;

(ii)

p;

(iii)

q. (Total 6 marks)


103

ˆ B = . 188. The diagram below shows a circle of radius r and centre O. The angle AO

The length of the arc AB is 24 cm. The area of the sector OAB is 180 cm2. Find the value of r and of . (Total 6 marks)

189. A part of the graph of y = 2x – x2 is given in the diagram below.

The shaded region is revolved through 360 about the x-axis. (a)

Write down an expression for this volume of revolution.

(b)

Calculate this volume. (Total 6 marks)


104

190. Consider the function ƒ : x  3x2 – 5x + k. (a)

Write down ƒ′ (x).

The equation of the tangent to the graph of ƒ at x = p is y = 7x – 9. Find the value of (b)

p;

(c)

k. (Total 6 marks)

191. In a class, 40 students take chemistry only, 30 take physics only, 20 take both chemistry and physics, and 60 take neither. (a)

Find the probability that a student takes physics given that the student takes chemistry.

(b)

Find the probability that a student takes physics given that the student does not take chemistry.

(c)

State whether the events “taking chemistry” and “taking physics” are mutually exclusive, independent, or neither. Justify your answer. (Total 6 marks)


105

192. The diagram below shows the graph of ƒ (x) = x2 e–x for 0  x  6. There are points of inflexion at A and C and there is a maximum at B.

(a)

Using the product rule for differentiation, find ƒ′ (x).

(b)

Find the exact value of the y-coordinate of B.

(c)

The second derivative of ƒ is ƒ′′ (x) = (x2 – 4x + 2) e–x. Use this result to find the exact value of the x-coordinate of C. (Total 6 marks)

193. The displacement s metres at time t seconds is given by s = 5 cos 3t + t2 + 10, for t  0. (a)

Write down the minimum value of s.

(b)

Find the acceleration, a, at time t.

(c)

Find the value of t when the maximum value of a first occurs. (Total 6 marks)


106

194. The four populations A, B, C and D are the same size and have the same range. Frequency histograms for the four populations are given below.

(a)

Each of the three box and whisker plots below corresponds to one of the four populations. Write the letter of the correct population under each plot.

......

(b)

......

......

Each of the three cumulative frequency diagrams below corresponds to one of the four populations. Write the letter of the correct population under each diagram.

(Total 6 marks)


107

195. Let ln a = p, ln b = q. Write the following expressions in terms of p and q. (a)

ln a3b

(b)

 a  ln   b    (Total 6 marks)

196. The box and whisker diagram shown below represents the marks received by 32 students.

(a)

Write down the value of the median mark.

(b)

Write down the value of the upper quartile.

(c)

Estimate the number of students who received a mark greater than 6. (Total 6 marks)


108

197. The following diagram shows the graph of a function f.

Consider the following diagrams.

Complete the table below, noting which one of the diagrams above represents the graph of (a)

f ′(x);

(b)

f ′′(x). Graph (a)

f ′ (x)

(b)

f " (x)

Diagram

(Total 6 marks)


109

198. Events E and F are independent, with P(E) = (a)

P(F);

(b)

P(E  F).

2 1 and P(E  F) = . Calculate 3 3

(Total 6 marks)

199. Part of the graph of the function y = d (x −m)2 + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, 2).

(a)

(b)


m;

(ii)

p.

Find d. (Total 6 marks)

200. The line L passes through the points A (3, 2, 1) and B (1, 5, 3). (a)

Find the vector AB .

(b)

Write down a vector equation of the line L in the form r = a + tb. (Total 6 marks)


110

201. Find the exact value of x in each of the following equations. (a)

5x+1 = 625

(b)

loga (3x + 5) = 2 (Total 6 marks)

202. Let g (x) = 3x – 2, h (x) =

5x , x  4. x4

(a)

Find an expression for (h ◦ g) (x). Simplify your answer.

(b)

Solve the equation (h ◦ g) (x) = 0. (Total 6 marks)

203. The velocity v in m s−1 of a moving body at time t seconds is given by v = e2t−1. When t = 0 5. the displacement of the body is 10 m. Find the displacement when t =1. (Total 6 marks)

204. The line L passes through A (0, 3) and B (1, 0). The origin is at O. The point R (x, 3 − 3x) is on L, and (OR) is perpendicular to L. (a)

Write down the vectors AB and OR .

(b)

Use the scalar product to find the coordinates of R. (Total 6 marks)


111

205. A fair coin is tossed five times. Calculate the probability of obtaining (a)

exactly three heads;

(b)

at least one head. (Total 6 marks)

206. The function f is defined by f : x  30 sin 3x cos 3x, 0  x 

π . 3

(a)

Write down an expression for f (x) in the form a sin 6x, where a is an integer.

(b)

Solve f (x) = 0, giving your answers in terms of . (Total 6 marks)

207. The heights of certain flowers follow a normal distribution. It is known that 20 of these flowers have a height less than 3 cm and 10 have a height greater than 8 cm. Find the value of the mean  and the standard deviation . (Total 6 marks)

208. The shaded region in the diagram below is bounded by f (x) = x , x = a, and the x-axis. The shaded region is revolved around the x-axis through 360. The volume of the solid formed is 0.845.

Find the value of a. (Total 6 marks)


112

209. A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.

800 700 600 Number of candidates

500 400 300 200 100 10

20

30

40

50

60

70

80

90

100

Mark

(a)

Write down the number of students who scored 40 marks or less on the test.

(b)

The middle 50% of test results lie between marks a and b, where a < b. Find a and b. (Total 6 marks)

210. A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row. (a)

Calculate the number of seats in the 20th row.

(b)

Calculate the total number of seats. (Total 6 marks)


113

211. The function f is given by f (x) = 2sin (5x – 3). (a)

Find f " (x).

(b)

Write down

 f ( x)dx . (Total 6 marks)

212. The diagram shows a cube, OABCDEFG where the length of each edge is 5cm. Express the following vectors in terms of i, j and k. z D

E

G y

F C k

j

O

B

i A x

(a)

OG ;

(b)

BD ;

(c)

EB . (Total 6 marks)


114

213. The graph of a function of the form y = p cos qx is given in the diagram below. y 40 30 20 10 /2



x

–10 –20 –30 –40

(a)

Write down the value of p.

(b)

Calculate the value of q. (Total 6 marks)

214. 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.

(b)

The initial displacement s is 40 metres. Find an expression for s in terms of t. (Total 6 marks)

215. The functions f and g are defined by f :  3x, g : x  x  2 . (a)

Find an expression for (f  g) (x).

(b)

Show that f –l (18) + g–l (18) = 22. (Total 6 marks)


115

216. A sum of $5000 is invested at a compound interest rate of 6.3% per annum. (a)

Write down an expression for the value of the investment after n full years.

(b)

What will be the value of the investment at the end of five years?

(c)

The value of the investment will exceed $10000 after n full years, (i)

Write down an inequality to represent this information.

(ii)

Calculate the minimum value of n. (Total 6 marks)

217. The function f is defined by f ( x) 

3 9 – x2

, for –3 < x < 3.

(a)

On the grid below, sketch the graph of f.

(b)

Write down the equation of each vertical asymptote.

(c)

Write down the range of the function f. (Total 6 marks)


116

218. A triangle has its vertices at A(–1, 3), B(3, 6) and C(–4, 4). (a)

Show that AB  AC  – 9

(b)

ˆ C  – 0.569. Show that, to three significant figures, cos BA (Total 6 marks)

219. A factory makes calculators. Over a long period, 2% of them are found to be faulty. A random sample of 100 calculators is tested. (a)

Write down the expected number of faulty calculators in the sample.

(b)

Find the probability that three calculators are faulty.

(c)

Find the probability that more than one calculator is faulty. (Total 6 marks)

220. 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.

(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. (Total 6 marks)

221. The speeds of cars at a certain point on a straight road are normally distributed with mean  and standard deviation . 15% of the cars travelled at speeds greater than 90 km h–1 and 12% of them at speeds less than 40 km h–1. Find  and . (Total 6 marks)


117

222. Let Sn be the sum of the first n terms of an arithmetic sequence, whose first three terms are u1, u2 and u3. It is known that S1 = 7, and S2 = 18. (a)

Write down u1.

(b)

Calculate the common difference of the sequence.

(c)

Calculate u4.

Working:

Answers: (a) .................................................................. (b) .................................................................. (c) .................................................................. (Total 6 marks)


118

223. Consider the line L with equation y + 2x = 3. The line L1 is parallel to L and passes through the point (6, –4). (a)

Find the gradient of L1.

(b)

Find the equation of L1 in the form y = mx + b.

(c)

Find the x-coordinate of the point where line L1 crosses the x-axis.

Working:



119

224. Consider the expansion of (x2 – 2)5. (a)

Write down the number of terms in this expansion.

(b)

The first four terms of the expansion in descending powers of x are x10 – 10x8 + 40x6 + Ax4 + ... Find the value of A.

Working:

Answers: (a) .................................................................. (b) .................................................................. (Total 6 marks)

225. The following diagram shows a circle of centre O, and radius r. The shaded sector OACB has ˆ B = θ = 1.5 radians. an area of 27 cm2. Angle AO A C B r O


120

(a)

Find the radius.

(b)

Calculate the length of the minor arc ACB.

Working:



121

226. Two unbiased 6-sided dice are rolled, a red one and a black one. Let E and F be the events E : the same number appears on both dice; F : the sum of the numbers is 10. Find (a)

P(E);

(b)

P(F);

(c)

P(E  F).

Working:



122

227. Let f (x) = (3x + 4)5. Find (a)

f (x);

(b)

f (x)dx.

Working:


228. Find the exact solution of the equation 92x = 27(1–x). Working:

Answer: ....…………………………………….......... (Total 6 marks)


123

229. The curve y = f (x) passes through the point (2, 6). Given that

dy = 3x2 – 5, find y in terms of x. dx

Working:


230. (a)

Given that log3 x – log3 (x – 5) = log3 A, express A in terms of x.

(b)

Hence or otherwise, solve the equation log3 x – log3 (x – 5) = 1.

Working:



124

 3   2 231. Find the cosine of the angle between the two vectors   and   .  4  1

Working:


232. The 45 students in a class each recorded the number of whole minutes, x, spent doing experiments on Monday. The results are x = 2230. (a)

Find the mean number of minutes the students spent doing experiments on Monday.

Two new students joined the class and reported that they spent 37 minutes and 30 minutes respectively. (b)

Calculate the new mean including these two students.

Working:


233. The function f is given by f (x) = e(x–11) –8.


125

(a)

Find f –1(x).

(b)

Write down the domain of f –l(x).

Working:


234. The graph of y = f (x) is shown in the diagram. y 2 1 –2

–1

0

1

2

3

4

5

6

7

8

x

–1 –2


126

(a)

On each of the following diagrams draw the required graph, (i)

y = 2 f (x); y 2 1 –2

0

–1

1

2

3

4

5

6

7

8

x

1

2

3

4

5

6

7

8

x

–1 –2

(ii)

y = f (x – 3). y 2 1 –2

–1

0 –1 –2


127

(b)

The point A (3, –1) is on the graph of f. The point A is the corresponding point on the graph of y = –f (x) + 1. Find the coordinates of A.

Working:

Answer: (b) .................................................................. (Total 6 marks)

235. Consider y = sin  x    . 9  (a)

(b)

The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where 0  x  π. Solve the equation sin  x    = – 1 , for 0  x  2. 2 9 

Working:



128

236. The table below shows some values of two functions, f and g, and of their derivatives f  and g . x

1

2

3

4

f (x)

5

4

–1

3

g (x)

1

–2

2

–5

f (x)

5

6

0

7

g (x)

–6

–4

–3

4

Calculate the following. (a)

d (f (x) + g (x)), when x = 4; dx

(b)

 g'( x)  6dx .

3

1

Working:



129

237. In triangle PQR, PQ is 10 cm, QR is 8 cm and angle PQR is acute. The area of the triangle is 20 ˆ R. cm2. Find the size of angle PQ Working:

Answers: ........................................................ (Total 6 marks)

238. A class contains 13 girls and 11 boys. The teacher randomly selects four students. Determine the probability that all four students selected are girls. Working:



130



239. Given that 3  (a)

p;

(b)

q.

7



3

= p + q 7 where p and q are integers, find

Working:

Answers: (a) ................................................. (b) ................................................. (Total 6 marks)


131

240. The following table shows the mathematics marks scored by students. Mark

1

2

3

4

5

6

7

Frequency

0

4

6

k

8

6

6

The mean mark is 4.6. (a)

Find the value of k.

(b)

Write down the mode.

Working:



132

241. Part of the graph of f (x) = (x – p) (x – q) is shown below.

The vertex is at C. The graph crosses the y-axis at B. (a)

Write down the value of p and of q.

(b)

Find the coordinates of C.

(c)

Write down the y-coordinate of B.

Working:

Answers: (a) ..................................................... (b) ..................................................... (c) ..................................................... (Total 6 marks)


133

242. Consider the functions f (x) = 2x and g (x) = (a)

Calculate (f ◦ g) (4).

(b)

Find g−1(x).

(c)

Write down the domain of g−1.

1 , x  3. x 3

Working:



134

243. The first term of an infinite geometric sequence is 18, while the third term is 8. There are two possible sequences. Find the sum of each sequence. Working:

Answers: ......................................................... ......................................................... (Total 6 marks)

244. Given



k

3

1 dx = ln 7, find the value of k. x2

Working:



135

245. A machine was purchased for $10000. Its value V after t years is given by V =100000e−0.3t. The machine must be replaced at the end of the year in which its value drops below $1500. Determine in how many years the machine will need to be replaced. Working:



136

246. Let f (x) = 6 sin x , and g (x) = 6e–x – 3 , for 0  x  2. The graph of f is shown on the diagram below. There is a maximum value at B (0.5, b). y B

0

1

(a)

Write down the value of b.

(b)

On the same diagram, sketch the graph of g.

(c)

Solve f (x) = g (x) , 0.5  x  1.5.

2

x

Working:



137

247. Let f (x) = (2x + 7)3 and g (x) cos2 (4x). Find (a)

f ′ (x);

(b)

g′ (x).

Working:


248. The events A and B are independent such that P(B) = 3P(A) and P(AB) = 0.68. Find P(B) Working:



138

249. A boat B moves with constant velocity along a straight line. Its velocity vector is given by  4 v =   .  3 At time t = 0 it is at the point (−2, 1). (a)

Find the magnitude of v.

(b)

Find the coordinates of B when t = 2.

(c)

Write down a vector equation representing the position of B, giving your answer in the form r = a + tb.

Working:



139

250. Consider the equation 3 cos 2x + sin x = 1 (a)

Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r , and p , q , r  .

(b)

Factorize f (x).

(c)

Write down the number of solutions of f (x) = 0, for 0  x  2.

Working:



140

251. The following diagram shows a rectangular area ABCD enclosed on three sides by 60 m of fencing, and on the fourth by a wall AB.

Find the width of the rectangle that gives its maximum area. Working:



141

252. Let f (x) = x3 – 2x2 – 1. (a)

Find f (x).

(b)

Find the gradient of the curve of f (x) at the point (2, –1).

Working:

Answers: (a) ………………………………………… (b) ………………………………………… (Total 6 marks)

253. The diagram below shows two circles which have the same centre O and radii 16 cm and 10 cm respectively. The two arcs AB and CD have the same sector angle  = 1.5 radians. A

B D

C

O


142

Find the area of the shaded region. Working:

Answer: ………………………………………….. (Total 6 marks)


143

254. The cumulative frequency curve below shows the marks obtained in an examination by a group of 200 students. 200 190 180 170 160 150 140 Number of 130 students 120 110 100 90 80 70 60 50 40 30 20 10 0


10

20

30

40 50 60 Mark obtained

70

80

90

100

144

(a)

Use the cumulative frequency curve to complete the frequency table below. Mark (x) Number of students

(b)

0  x < 20

20  x < 40

40  x < 60

22

60  x < 80

80  x < 100 20

Forty percent of the students fail. Find the pass mark.

Working:

Answer: (b) ………………………………………….. (Total 6 marks)


145

255. Let f (x) = 2x + 1. (a)

On the grid below draw the graph of f (x) for 0  x  2.

(b)

Let g (x) = f (x +3) –2. On the grid below draw the graph of g (x) for –3  x  –1. y 6 5 4 3 2 1

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6 x

–1 –2 –3 –4 –5 –6


146

Working:

(Total 6 marks)

256. Let A and B be events such that P(A) =

1 3 7 , P(B) = and P(A  B) = . 2 4 8

(a)

Calculate P(A  B).

(b)

Calculate P(AB).

(c)

Are the events A and B independent? Give a reason for your answer.

Working:

Answers: (a) ………………………………………….. (b) ………………………………………….. (c) …………………………………….......... (Total 6 marks)


147

257. Let f (x) = sin (2x + 1), 0  x  π. (a)

Sketch the curve of y = f (x) on the grid below. y 2 1.5 1 0.5 0

0.5

1

1.5

2

2.5

3

3.5 x

–0.5 –1 –1.5 –2

(b)

Find the x-coordinates of the maximum and minimum points of f (x), giving your answers correct to one decimal place.

Working:



148

258. Let p = log10 x, q = log10 y and r = log10 z.  x   in terms of p, q and r. Write the expression log10  2 y z  

Working:


259. In a triangle ABC, AB = 4 cm, AC = 3 cm and the area of the triangle is 4.5 cm2. ˆC. Find the two possible values of the angle BA

Working:



149

260. Solve the equation 2 cos2 x = sin 2x for 0  x  π, giving your answers in terms of π. Working:


261. A car starts by moving from a fixed point A. Its velocity, v m s–1 after t seconds is given by v = 4t + 5 – 5e–t. Let d be the displacement from A when t = 4. (a)

Write down an integral which represents d.

(b)

Calculate the value of d.

Working:

Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks)


150

262. The following table shows four series of numbers. One of these series is geometric, one of the series is arithmetic and the other two are neither geometric nor arithmetic. (a)

Complete the table by stating the type of series that is shown. Series (i)

1  11  111  1111  11111  …

(ii)

1

(iii) (iv) (b)

Type of series

3 27 9   … 4 16 64 0.9  0.875  0.85  0.825  0.8  … 1 2 3 4 5      2 3 4 5 6

The geometric series can be summed to infinity. Find this sum.

Working:



151

263. The table below shows the marks gained in a test by a group of students. Mark

1

2

3

4

5

Number of students

5

10

p

6

2

The median is 3 and the mode is 2. Find the two possible values of p. Working:


264. Two lines L1 and L2 have these vector equations. L1 : r = 2i + 3j + t(i– 3j) L2 : r = i + 2j + s(i – j) The angle between L1 and L2 is . Find the cosine of the angle . Working:



152

265. The equation x2 – 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of k. Working:


266. When the expression (2 + ax)10 is expanded, the coefficient of the term in x3 is 414 720. Find the value of a. Working:



153

ˆ = 60°, Cˆ = 40°. 267. The following diagram shows a triangle ABC, where BC = 5 cm, B A

B

(a)

Calculate AB.

(b)

Find the area of the triangle.

40°

60° 5 cm

C

Working:



154

268. Let f (x) = 6 3 x 2 . Find f (x). Working:

Answer: …………………………………………........ (Total 6 marks)


155

269. The cumulative frequency curve below shows the heights of 120 basketball players in centimetres. 120 110 100 90 80 70 60 Number of players 50 40 30 20 10 0 160

165

170

175

180

185

190

195

200

Height in centimetres


156

Use the curve to estimate (a)

the median height;

(b)

the interquartile range.

Working:


270. Find the term containing x3 in the expansion of (2 – 3x)8. Working:



157

271. The following diagram shows a circle divided into three sectors A, B and C. The angles at the centre of the circle are 90°, 120° and 150°. Sectors A and B are shaded as shown.

A C

150°

90° 120° B

The arrow is spun. It cannot land on the lines between the sectors. Let A, B, C and S be the events defined by A: B: C: S:


Arrow lands in sector A Arrow lands in sector B Arrow lands in sector C Arrow lands in a shaded region.

158

Find (a)

P(B);

(b)

P(S);

(c)

P(AS).

Working:

Answers: (a) ………………………………………….. (b) ………………………………………….. (c) ………………………………………….. (Total 6 marks)

272. Let a = log x, b = log y, and c = log z.  x2 y  Write log  3  in terms of a, b and c.  z   

Working:



159

273. Let a, b, c and d be integers such that a < b, b < c and c = d. The mode of these four numbers is 11. The range of these four numbers is 8. The mean of these four numbers is 8. Calculate the value of each of the integers a, b, c, d. Working:

Answers: a = ............................., b = ............................. c = ............................., d = ............................. (Total 6 marks)


160

274. Let f (x) = 2x + 1 and g (x) = 3x2 – 4. Find (a)

f –1(x);

(b)

(g  f ) (–2);

(c)

( f  g) (x).

Working:



161

275. The displacement s metres of a car, t seconds after leaving a fixed point A, is given by s = 10t – 0.5t2. (a)

Calculate the velocity when t = 0.

(b)

Calculate the value of t when the velocity is zero.

(c)

Calculate the displacement of the car from A when the velocity is zero.

Working:


276. Let f (x) = 2 + cos (2x) – 2 sin (0.5x) for 0  x  3, where x is in radians. (a)

On the grid below, sketch the curve of y = f (x), indicating clearly the point P on the curve where the derivative is zero. y 4 3 2 1 0

0.5

1

1.5

2

2.5

3 x

–1 –2 –3 –4


162

(b)

Write down the solutions of f (x) = 0.

Working:


277. The population p of bacteria at time t is given by p = 100e0.05t. Calculate (a)

the value of p when t = 0;

(b)

the rate of increase of the population when t = 10.

Working:



163

278. The derivative of the function f is given by f (x) = e–2x +

1 , x < 1. 1 x

The graph of y = f (x) passes through the point (0, 4). Find an expression for f (x). Working:



164

279. Let f be a function such that (a)

(b)



3

0

f ( x ) dx  8 .

Deduce the value of 3

(i)



(ii)

  f ( x)  2dx.

If

0

2 f ( x) dx;

3

0



d

c

f ( x  2)dx  8 , write down the value of c and of d.

Working:

Answers: (a) (i) ........................................................ (ii) ....................................................... (b) c = ......................., d = ....................... (Total 6 marks)

280. The diagram below shows a circle of radius 5 cm with centre O. Points A and B are on the ˆ B is 0.8 radians. The point N is on [OB] such that [AN] is perpendicular to circle, and AO [OB]. A 5 cm O


0.8

N

B

165

Find the area of the shaded region. Working:



166

281. Part of the graph of the periodic function f is shown below. The domain of f is 0  x  15 and the period is 3. f(x) 4 3 2 1 0

(a)

(b)

0

1

2

3

4

5

6

7

8

9

10

x

Find (i)

f (2);

(ii)

f (6.5);

(iii)

f (14).

How many solutions are there to the equation f (x) = 1 over the given domain?

Working:

Answers: (a) (i) ……………………………………… (ii) ……………………………………… (iii) ……………………………………… (b) …………………………………………… (Total 6 marks)


167

282. Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s. Calculate s. Working:

Answer: .................................................................. (Total 6 marks)

283. Find the term containing x10 in the expansion of (5 + 2x2)7. Working:



168

284. The number of hours of sleep of 21 students are shown in the frequency table below. Hours of sleep

Number of students

4

2

5

5

6

4

7

3

8

4

10

2

12

1

Find (a)

the median;

(b)

the lower quartile;

(c)

the interquartile range.

Working:



169

285. Part of the graph of y = p + q cos x is shown below. The graph passes through the points (0, 3) and (, –1). y

3

2

1

0



2

x

–1

Find the value of (a)

p;

(b)

q.

Working:



170

x

286. Let f (x) = e 3 + 5 cos 2x. Find f (x). Working:


287. Find all solutions of the equation cos 3x = cos (0.5x), for 0  x  . Working:



171

288. The vector equations of two lines are given below.  5  3  r1 =   +    ,  – 2  1

 – 2  4 r2 =   + t    2  1

The lines intersect at the point P. Find the position vector of P. Working:



172

289. Consider events A, B such that P (A)  0, P (A)  1, P (B)  0, and P (B)  1. In each of the situations (a), (b), (c) below state whether A and B are mutually exclusive (M); independent (I); neither (N). (a)

P(A|B) = P(A)

(b)

P(A  B) = 0

(c)

P(A  B) = P(A)

Working:



173

290. Given that

(a)



3

(b)



3

1

1



3

1

g ( x)dx = 10, deduce the value of

1 g ( x)dx; 2 ( g ( x)  4)dx.

Working:



174

291. Given that log5 x = y, express each of the following in terms of y. (a)

log5 x2

(b)

log5  1   x

(c)

log25 x

Working:



175

292. Let f (x) = e–x, and g (x) = (a)

f –1 (x);

(b)

(g ° f ) (x).

x , x  –1. Find 1 x

Working:


293. Consider the vectors c = 3i + 4j and d = 5i – 12j. Calculate the scalar product cd. Working:



176

294. A family of functions is given by f (x) = x2 + 3x + k, where k  {1, 2, 3, 4, 5, 6, 7}. One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis. Working:



177

295. The diagram below shows a triangle and two arcs of circles. The triangle ABC is a right-angled isosceles triangle, with AB = AC = 2. The point P is the midpoint of [BC]. The arc BDC is part of a circle with centre A. The arc BEC is part of a circle with centre P. E B D 2

A

P

C

2

(a)

Calculate the area of the segment BDCP.

(b)

Calculate the area of the shaded region BECD.

Working:



178

296. Consider the following relations between two variables x and y. A.

y = sin x

B.

y is directly proportional to x

C.

y = 1 + tan x

D.

speed y as a function of time x, constant acceleration

E.

y = 2x

F.

distance y as a function of time x, velocity decreasing

Each sketch below could represent exactly two of the above relations on a certain interval. (i) y

(ii)

(iii)

y

x

y

x

x

Complete the table below, by writing the letter for the two relations that each sketch could represent. sketch

relation letters

(i) (ii) (iii)


179

297. A student measured the diameters of 80 snail shells. His results are shown in the following cumulative frequency graph. The lower quartile (LQ) is 14 mm and is marked clearly on the graph. 90

Cumulative frequency

80 70 60 50 40 30 20 10 0

(a)

(b)

5

0

10

15 LQ = 14

20

25

30

35

40

45

Diameter (mm)

On the graph, mark clearly in the same way and write down the value of (i)

the median;

(ii)

the upper quartile.

Write down the interquartile range.

Working:



180

298. The graph of the function f (x) = 3x – 4 intersects the x-axis at A and the y-axis at B. (a)

(b)

Find the coordinates of (i)

A;

(ii)

B.

Let O denote the origin. Find the area of triangle OAB.

Working:

Answers: (a) (i) ........................................................... (ii) ........................................................... (b) .................................................................. (Total 6 marks)

299. The equation kx2 + 3x + 1 = 0 has exactly one solution. Find the value of k. Working:



181

300. A painter has 12 tins of paint. Seven tins are red and five tins are yellow. Two tins are chosen at random. Calculate the probability that both tins are the same colour. Working:


301. Complete the following expansion. (2 + ax)4 = 16 + 32ax + … Working:



182

302. Arturo goes swimming every week. He swims 200 metres in the first week. Each week he swims 30 metres more than the previous week. He continues for one year (52 weeks). (a)

How far does Arturo swim in the final week?

(b)

How far does he swim altogether?

Working:



183

 4 303. A vector equation for the line L is r =   + t  4

 3   .  1

Which of the following are also vector equations for the same line L? A.

 4 r =   + t  4

 2   . 1

B.

 4 r =   + t  4

 6   .  2

C.

 0 r =   + t 1

 1   .  3

7 r =   + t  5 Working:

 3   .  1

D.

Answer: …….................................................................. …….................................................................. (Total 6 marks)


184

304. (a)

The diagram shows part of the graph of the function f (x) =

q . The curve passes x– p

through the point A (3, 10). The line (CD) is an asymptote. y 15

C

10

A

5

–15

–10

–5

0

5

10

15 x

-5

-10

-15

D

Find the value of (i)

p;

(ii)

q.


185

(b)

The graph of f (x) is transformed as shown in the following diagram. The point A is transformed to A (3, –10). y 15

C

10

5

–15

–10

–5

5

0

10

15 x

–5

–10

–15


A

D

186

Give a full geometric description of the transformation. Working:

Answers: (a) (i) ........................................................... (ii) ........................................................... (b) .................................................................. .................................................................. (Total 6 marks)

305. (a) (b)

 – 30  60  . Find the scalar product of the vectors   and   40   25

Two markers are at the points P (60, 25) and Q (–30, 40). A surveyor stands at O (0, 0) and looks at marker P. Find the angle she turns through to look at marker Q.

Working:



187

306. The mass m kg of a radio-active substance at time t hours is given by m = 4e–0.2t. (a)

Write down the initial mass.

(b)

The mass is reduced to 1.5 kg. How long does this take?

Working:


307. It is given that

dy = x3+2x – 1 and that y = 13 when x = 2. dx

Find y in terms of x. Working:



188

308. The function f is given by f (x) = x2 – 6x + 13, for x  3. (a)

Write f (x) in the form (x – a)2 + b.

(b)

Find the inverse function f –1.

(c)

State the domain of f –1.

Working:



189

309. (a) (b)

Factorize the expression 3 sin2 x – 11 sin x + 6. Consider the equation 3 sin2 x – 11 sin x + 6 = 0. (i)

Find the two values of sin x which satisfy this equation,

(ii)

Solve the equation, for 0°  x  180°.

Working:

Answers: (a) .................................................................. (b) (i) ........................................................... (ii) ...........................................................

(Total 6 marks)


190

310. (a) (b)

Find (1 + 3 sin (x + 2))dx. The diagram shows part of the graph of the function f (x) = 1 + 3 sin (x + 2).



The area of the shaded region is given by

a 0

f ( x)dx .

y

4

2

–4

–2

0

2

4

x

–2

Find the value of a. Working:



191

311. The diagram shows the graph of y = f (x). y

0


x

192

On the grid below sketch the graph of y = f (x). y

0

x

(Total 6 marks)

312. From January to September, the mean number of car accidents per month was 630. From October to December, the mean was 810 accidents per month. What was the mean number of car accidents per month for the whole year? Working:

Answer: ...................................................................... (Total 6 marks)


193

313. In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998. (a)

Find the common difference d.

(b)

Find the value of n.

Working:



194

314. Let f (x) = 2x, and g (x) =

x , (x  2). x–2

Find (a)

(g  f ) (3);

(b)

g–1 (5).

Working:



195

315. The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends an angle of 2 radians at the centre O. C A

B 15 cm

Diagram not to scale 2 rad O

AÔB = 2 radians OA = 15 cm

Find (a)

the length of the arc ACB;

(b)

the area of the shaded region.

Working:



196

 x   1  – 2 316. A vector equation of a line is       t   , t   y   2  3 

.

Find the equation of this line in the form ax + by = c, where a, b, and c 

.

Working:


317. Two boats A and B start moving from the same point P. Boat A moves in a straight line at 20 km h–1 and boat B moves in a straight line at 32 km h–1. The angle between their paths is 70°. Find the distance between the boats after 2.5 hours. Working:



197

9

1  318. Consider the expansion of  3x 2 –  . x 

(a)

How many terms are there in this expansion?

(b)

Find the constant term in this expansion.

Working:



198

319. Let f (x) = sin 2x and g (x) = sin (0.5x). (a)

(b)

Write down (i)

the minimum value of the function f ;

(ii)

the period of the function g.

Consider the equation f (x) = g (x). Find the number of solutions to this equation, for 0  x 

3π . 2

Working:

Answers: (a) (i) .......................................................... (ii) .......................................................... (b) ................................................................. (Total 6 marks)


199

320. Solve the equation log27 x = 1 – log27 (x – 0.4). Working:


321. The derivative of the function f is given by f (x) =

1 – 0.5 sin x, for x  –1. 1 x

The graph of f passes through the point (0, 2). Find an expression for f (x). Working:



200

322. A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. (a)

The first two apples are green. What is the probability that the third apple is red?

(b)

What is the probability that exactly two of the three apples are red?

Working:



201

323. The diagram shows part of the graph of y = a (x – h)2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0). y P

2

1

A –1

(a)

(b)

0

1

x


h;

(ii)

k.

Calculate the value of a.

Working:

Answers: (a) (i) .......................................................... (ii) .......................................................... (b) ................................................................. (Total 6 marks)


202

324. Figure 1 shows the graphs of the functions f1, f2, f3, f4. Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg the derivative of f1 is shown in diagram (d). Figure 1

Figure 2

y

y (a)

f1

O O

x

x

y

y (b)

f2 O

O

x

y

x

y (c)

f3 O

x

O

y

x

y (d)

f4 O

O

x

x

y (e) O


x

203

Complete the table below by matching each function with its derivative. Function

Derivative diagram

f1

(d)

f2 f3 f4

Working:

(Total 6 marks)

325. Consider the following statements A:

log10 (10x) > 0.

B:

–0.5  cos (0.5x)  0.5.

C:

–

(a)

Determine which statements are true for all real numbers x. Write your answers (yes or no) in the table below.

π π  arctan x  . 2 2

Statement

(a) Is the statement true for all real numbers x? (Yes/No)

(b)

If not true, example

A B C


204

(b)

If a statement is not true for all x, complete the last column by giving an example of one value of x for which the statement is false.

Working:

(Total 6 marks)

326. Let f (x) =

x 3 . Find

(a)

f (x);

(b)

 f ( x)dx.

Working:



205

ˆ = 48°, as shown in the diagram. 327. In triangle ABC, AC = 5, BC = 7, A C 5 A

48°

7


B

ˆ , giving your answer correct to the nearest degree. Find B Working:



206

328. Consider the function f (x) = 2x2 – 8x + 5. (a)

Express f (x) in the form a (x – p)2 + q, where a, p, q 

(b)

Find the minimum value of f (x).

.

Working:


329. Find the coefficient of x3 in the expansion of (2 – x)5. Working:



207

330. Solve the equation ex = 5 – 2x, giving your answer correct to four significant figures. Working:


331. Given that sin x = (a)

cos x;

(b)

cos 2x.

1 , where x is an acute angle, find the exact value of 3

Working:



208

332. For events A and B, the probabilities are P (A) =

3 4 . , P (B) = 11 11

Calculate the value of P (A  B) if

6 ; 11

(a)

P (A  B) =

(b)

events A and B are independent.

Working:


333. The graph of y = x3 – 10x2 +12x + 23 has a maximum point between x = –1 and x = 3. Find the coordinates of this maximum point. Working:



209

334. Three positive integers a, b, and c, where a < b < c, are such that their median is 11, their mean is 9 and their range is 10. Find the value of a. Working:


335. Consider the functions f : x  4(x – 1) and g : x  (a)

Find g–1.

(b)

Solve the equation ( f ° g–1) (x) = 4.

6–x . 2

Working:



210

336. Calculate the acute angle between the lines with equations  4  4 r =   + s   and  – 1  3

 2 1 r =   + t    4  – 1

Working:



211

337. The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and 3π the lines y = 0 and x = . 4 y

3 4

Given that sin

3π = 4

3π 2 and cos =– 2 4



x

2 , calculate the exact area of the shaded 2

region. Working:



212

338. $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000. Working:


339. Consider the trigonometric equation 2 sin2 x = 1 + cos x. (a)

Write this equation in the form f (x) = 0, where f (x) = a cos2 x + b cos x + c, and a, b, c  .

(b)

Factorize f (x).

(c)

Solve f (x) = 0 for 0°  x  360°.

Working:



213

340. The sketch shows part of the graph of y = f (x) which passes through the points A(–1, 3), B(0, 2), C(l, 0), D(2, 1) and E(3, 5).

8 7 6 E

5 4 A 3 B 2 D 1 C –4

–3

–2

–1

0

1

2

3

4

5

–1 –2

A second function is defined by g (x) = 2f (x – 1). (a)

Calculate g (0), g (1), g (2) and g (3).

(b)

On the same axes, sketch the graph of the function g (x).

Working:

Answers: (a) .................................................................. .................................................................. (Total 6 marks)

341. Given the following frequency distribution, find IB Questionbank Maths SL

214

(a)

the median;

(b)

the mean. Number (x)

1

2

3

4

5

6

Frequency (f )

5

9

16

18

20

7

Working:


342. The diagram shows part of the graph with equation y = x2 + px + q. The graph cuts the x-axis at –2 and 3. y 6 4 2 –3

–2

–1

0

1

2

3

4

x

–2 –4 –6


215


p;

(b)

q.

Working:


343. Each year for the past five years the population of a certain country has increased at a steady rate of 2.7% per annum. The present population is 15.2 million. (a)

What was the population one year ago?

(b)

What was the population five years ago?

Working:



216

344. The following diagram shows a triangle with sides 5 cm, 7 cm, 8 cm.

5

7

Diagram not to scale

8

Find (a)

the size of the smallest angle, in degrees;

(b)

the area of the triangle.

Working:



217

345. The point P ( 1 , 0 ) lies on the graph of the curve of y = sin (2x –1). 2 Find the gradient of the tangent to the curve at P. Working:

Answer: ....................................................................... (Total 4 marks)

346. Use the binomial theorem to complete this expansion. (3x + 2y)4 = 81x4 + 216x3 y +... Working:



218

347. A bag contains 10 red balls, 10 green balls and 6 white balls. Two balls are drawn at random from the bag without replacement. What is the probability that they are of different colours? Working:


348. Find (a)

 sin (3x  7)dx;

(b)

e

–4 x

dx .

Working:



219

349. Find the angle between the following vectors a and b, giving your answer to the nearest degree. a = –4i – 2j b = i – 7j Working:



220

350. (a)

On the following diagram, sketch the graphs of y = ex and y = cos x for –2  x  1. y 2 1 –2

–1

0 –1

1

x

–2

(b)

The equation ex = cos x has a solution between –2 and –1. Find this solution.

Working:



221

351. The function f is defined by

f :xa

3 x . 2

3 – 2x ,

Evaluate f –1(5). Working:


352. (a) (b)

Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. Hence or otherwise, solve the equation 3 sin2 x + 4 cos x – 4 = 0,

0  x  90.

Working:



222

353. The following diagram shows the graph of y = f (x). It has minimum and maximum points at 1

(0, 0) and ( 1, ). 2

y 3.5 3 2.5 2 1.5 1 0.5 –2

–1

0

1

2

3

x

–0.5 –1 –1.5 –2 –2.5

3 . 2

(a)

On the same diagram, draw the graph of y  f ( x – 1) 

(b)

What are the coordinates of the minimum and maximum points of 3 y  f ( x – 1)  ? 2

Working:

Answer: (b) ................................................................ (Total 4 marks)


223

354. In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T. T

O

A

Diagram not to scale If OA = 12 cm, and the circle has a radius of 6 cm, find the area of the shaded region. Working:



224

355. The first three terms of an arithmetic sequence are 7, 9.5, 12. (a)

What is the 41st term of the sequence?

(b)

What is the sum of the first 101 terms of the sequence?

Working:



225

356. The diagram below shows a line passing through the points (1, 3) and (6, 5). y (6,5)

(1,3)

x

0

Find a vector equation for the line, giving your answer in the form  x  a        y  b 

c  t   , where t is any real number. d 

Working:



226

357. The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2. y y = x2 8 6

y = 5 – 3(x–4)

2

4 2

–2


0

2

4

6

x

227

The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations. A reflection in the line y = 0 a vertical stretch with scale factor k a horizontal translation of p units a vertical translation of q units.

followed by followed by followed by

Write down the value of (a)

k;

(b)

p;

(c)

q.

Working:



228

358. The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle  at the centre of the circle is 2 radians. Diagram not to scale

B

A

O

(a)

Calculate the area of the sector AOB.

(b)

Calculate the area of the shaded region.

Working:



229

359. Solve the equation log9 81 + log9

1 + log9 3 = log9 x. 9

Working:



230

360. The following Venn diagram shows the universal set U and the sets A and B. U

B A

(a)

Shade the area in the diagram which represents the set B  A'.

n(U) = 100, n(A) = 30, n(B) = 50, n(A  B) = 65. (b)

Find n(B  A′).

(c)

An element is selected at random from U. What is the probability that this element is in B  A′ ?

Working:

Answers: (b) .................................................................. (c) .................................................................. (Total 4 marks)


231

361. Consider the function f (x) = k sin x + 3x, where k is a constant. (a)

Find f (x).

(b)

When x =

 , the gradient of the curve of f (x) is 8. Find the value of k. 3

Working:



232

 x 362. The diagram below shows the graph of y = x sin   , for 0  x < m, and 0  y < n, where x is in 3 radians and m and n are integers. y n n–1

0

m–1

m

x


m;

(b)

n.

Working:



233

363. Given that f (x) = 2e3x, find the inverse function f –1(x). Working:


 4  4  4 364. Consider the binomial expansion (1  x) 4 1    x    x 2    x 3  x 4 . 1  2  3

(a)

 4  4  4 By substituting x = 1 into both sides, or otherwise, evaluate         . 1   2   3 


234

(b)

 9  9   9  9   9  9  9   9 Evaluate                        . 1   2   3   4   5   6   7   8 

Working:


 2x   x  1  and   are perpendicular for two values of x. 365. The vectors   x – 3  5 

(a)

Write down the quadratic equation which the two values of x must satisfy.

(b)

Find the two values of x.

Working:



235

366. The diagrams below show two triangles both satisfying the conditions ˆ C = 50°. AB = 20 cm, AC = 17 cm, AB

Diagrams not to scale Triangle 1

Triangle 2

A

A

B

C

(a)

ˆ B in Triangle 2. Calculate the size of AC

(b)

Calculate the area of Triangle 1.

B

C

Working:



236

367. The events B and C are dependent, where C is the event “a student takes Chemistry”, and B is the event “a student takes Biology”. It is known that P(C) = 0.4, P(B | C) = 0.6, P(B | C) = 0.5. (a)

Complete the following tree diagram. Chemistry

Biology

B 0.4

C B B C B

(b)

Calculate the probability that a student takes Biology.

(c)

Given that a student takes Biology, what is the probability that the student takes Chemistry?

Working:

Answers: (b) .................................................................. (c) .................................................................. (Total 4 marks)


237

368. The depth, y metres, of sea water in a bay t hours after midnight may be represented by the function

 2  y  a  b cos  t  , where a, b and k are constants.  k  The water is at a maximum depth of 14.3 m at midnight and noon, and is at a minimum depth of 10.3 m at 06:00 and at 18:00. Write down the value of (a)

a;

(b)

b;

(c)

k.

Working:



238

369. (a)

Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)2 + k, where h and k are to be determined.

(b)

Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y – x2 – 6x + 14.

Working:


370. Let f (x) = 1 – x2. Given that f (3) = 0, find f (x). Working:

Answer: .................................................................... (Total 4 marks)


239

371. Town A is 48 km from town B and 32 km from town C as shown in the diagram. C 32km A

B

48km

ˆ B to the nearest degree. Given that town B is 56 km from town C, find the size of angle CA

Working:


372. In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20 males. (a)

Using this information, complete the table below. Males

Females

Totals

Unemployed Employed Totals


200

240

(b)

If a person is selected at random from this group of 200, find the probability that this person is (i)

an unemployed female;

(ii)

a male, given that the person is employed.

Working:

Answers: (b) (i) .......................................................... (ii) .......................................................... (Total 4 marks)

1   6 373. Find the size of the angle between the two vectors   and   . Give your answer to the – 8  2 nearest degree.

Working:



241

374. A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is modelled by N = 10 e0.4t. (a)

How many leopards are there after 2 years?

(b)

How long will it take for the number of leopards to reach 100? Give your answers to an appropriate degree of accuracy.

Give your answers to an appropriate degree of accuracy. Working:


375. Given the function f (x) = x2 – 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f (3) = 0. Working:



242

 2 376. A line passes through the point (4,–1) and its direction is perpendicular to the vector   . Find 3 the equation of the line in the form ax + by = p, where a, b and p are integers to be determined.

Working:


377. Consider the function f : x  x  1, x  – 1 (a)

Determine the inverse function f –1.

(b)

What is the domain of f –1?

Working:


 P   378. Let log10P = x , log10Q = y and log10R = z. Express log10  3   QR 


2

in terms of x , y and z.

243

Working:


379. The diagram shows the graph of y = f (x), with the x-axis as an asymptote. y

B(5, 4)

x

A(–5, –4)

(a)

On the same axes, draw the graph of y =f (x + 2) – 3, indicating the coordinates of the images of the points A and B.


244

(b)

Write down the equation of the asymptote to the graph of y = f (x + 2) – 3.

Working:

Answer: (b) ................................................................... . (Total 4 marks)

380. (a) (b)

Express 2 cos2 x + sin x in terms of sin x only. Solve the equation 2 cos2 x + sin x = 2 for x in the interval 0  x  , giving your answers exactly.

Working:



245

381. Each day a runner trains for a 10 km race. On the first day she runs 1000 m, and then increases the distance by 250 m on each subsequent day. (a)

On which day does she run a distance of 10 km in training?

(b)

What is the total distance she will have run in training by the end of that day? Give your answer exactly.

Working:


9

2   382. Determine the constant term in the expansion of  x – 2  . x  

Working:



246

383. (a)

Sketch, on the given axes, the graphs of y = x2 and y – sin x for –1  x  2. y

–1


–0.5

0

0.5

1

1.5

2

x

247

(b)

Find the positive solution of the equation x2 = sin x, giving your answer correct to 6 significant figures.

Working:

Answer: (b) ................................................................... . (Total 4 marks)

384. In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term. Working:



248

385. Two functions f, g are defined as follows: f : x  3x + 5 g : x  2(1 – x) Find (a)

f –1(2);

(b)

(g  f )(–4).

Working:



249

386. In a survey, 100 students were asked “do you prefer to watch television or play sport?” Of the 46 boys in the survey, 33 said they would choose sport, while 29 girls made this choice. Boys

Girls

Sport

33

29

Total

46

Total

Television

100

By completing this table or otherwise, find the probability that (a)

a student selected at random prefers to watch television;

(b)

a student prefers to watch television, given that the student is a boy.

Working:



250

387. The vectors u, v are given by u = 3i + 5j, v = i – 2j. Find scalars a, b such that a(u + v) = 8i + (b – 2)j. Working:


388. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for (a)

log2 5;

(b)

loga 20.

Working:



251

389. Solve the equation 3 cos x = 5 sin x, for x in the interval 0°  x  360°, giving your answers to the nearest degree. Working:


390. Find a vector equation of the line passing through (–1, 4) and (3, –1). Give your answer in the form r = p + td, where t  . Working:



252

391. Find the coordinates of the point on the graph of y = x2 – x at which the tangent is parallel to the line y = 5x. Working:


392. If f (x) = cos x, and f    = – 2, find f (x). 2 Working:



253

393. Find the sum of the infinite geometric series

2  4  8  16  ... 3 9 27 81 Working:


394. Find the coefficient of a5b7 in the expansion of (a + b)12. Working:



254

395. If A is an obtuse angle in a triangle and sin A = 5 , calculate the exact value of sin 2A. 13 Working:


396. The quadratic equation 4x2 + 4kx + 9 = 0, k > 0 has exactly one solution for x. Find the value of k. Working:



255

397. The diagram shows three graphs. y

B A C x

A is part of the graph of y = x. B is part of the graph of y = 2x. C is the reflection of graph B in line A.

Write down (a)

the equation of C in the form y =f (x);

(b)

the coordinates of the point where C cuts the x-axis.

Working:



256

398. Let f (x) = x3. f (5  h)  f (5) for h = 0.1. h

(a)

Evaluate

(b)

What number does

f (5  h)  f (5) approach as h approaches zero? h

Working:


399. Two ordinary, 6-sided dice are rolled and the total score is noted. (a)

Complete the tree diagram by entering probabilities and listing outcomes. 6

Outcomes ...............

....... 6 ....... .......

not 6 ............... 6

.......

...............

....... not 6 .......


not 6 ...............

257

(b)

Find the probability of getting one or more sixes.

Working:

Answer: (b) ............................................................... (Total 4 marks)

400. The table shows the scores of competitors in a competition. Score

10

20

30

40

50

Number of competitors with this score

1

2

5

k

3

The mean score is 34. Find the value of k. Working:



258

401. A curve with equation y =f (x) passes through the point (1, 1). Its gradient function is f (x) = –2x + 3. Find the equation of the curve. Working:


3 402. Given that sin θ = 1 , cos θ = – and 0° ≤ θ ≤ 360°, 2 2 (a)

find the value of θ;

(b)

write down the exact value of tan θ.

Working:



259

403. The line L passes through the origin and is parallel to the vector 2i + 3j. Write down a vector equation for L. Working:



260

404. The following Venn diagram shows a sample space U and events A and B.

U

A

B

n(U) = 36, n(A) = 11, n(B) = 6 and n(A  B)′ = 21. (a)

On the diagram, shade the region (A  B)′.

(b)

Find

(c)

(i)

n(A  B);

(ii)

P(A  B).

Explain why events A and B are not mutually exclusive.

Working:

Answers: (b) (i) ........................................................... (ii) ........................................................... (c) .................................................................. (Total 4 marks)


261

405. The diagram shows a vertical pole PQ, which is supported by two wires fixed to the horizontal ground at A and B. P

36 B 30 70

Q

A

BQ = 40 m ˆ Q = 36° PB ˆ Q = 70° BA ˆ Q = 30° AB

Find (a)

the height of the pole, PQ;

(b)

the distance between A and B.

Working:



262

406. Given that f (x) = (2x + 5)3 find (a)

f (x);

(b)

 f ( x)dx.

Working:



263

407. The diagrams show how the graph of y = x2 is transformed to the graph of y = f (x) in three steps. For each diagram give the equation of the curve. y

y

(a) 1 0 y=x2

0

x

x

1

y (b)

(c)

y

7

4 3

0

1

0

x

1

x

Working:



264

408. The diagram shows a circle of radius 5 cm. 1 radian

Find the perimeter of the shaded region. Working:



265

409. f (x) = 4 sin  3x    . 2  For what values of k will the equation f (x) = k have no solutions? Working:



266

410. $1000 is invested at the beginning of each year for 10 years. The rate of interest is fixed at 7.5% per annum. Interest is compounded annually. Calculate, giving your answers to the nearest dollar (a)

how much the first $1000 is worth at the end of the ten years;

(b)

the total value of the investments at the end of the ten years.

Working:



267

411. The triangle ABC is defined by the following information  2 OA =   ,   3

(a)

 3 AB =   , AB BC = 0,  4

0 AC is parallel to   . 1

On the grid below, draw an accurate diagram of triangle ABC. y 4 3 2 1

–2

–1

O

1

2

3

4

5

6

x

–1 –2 –3 –4


268

(b)

Write down the vector OC .

Working:


412. The diagram shows the graph of the function y = ax2 + bx + c. y x


269

Complete the table below to show whether each expression is positive, negative or zero. Expression

positive

negative

zero

a c b2 – 4ac b

Working:

(Total 4 marks)


270

413. The diagram shows the graph of the function y = 1 + 1 , 0 < x  4. Find the exact value of the x area of the shaded region.

4 3

1 y = 1+ – x

2 1 13 1

0

1

2

3

4

Working:



271

414. (a) (b)

Factorize x2 – 3x – 10. Solve the equation x2 – 3x – 10 = 0.

Working:


415. The diagram represents the graph of the function f : x  (x – p)(x – q). y

– 12

2

x

C

(a)

Write down the values of p and q.


272

(b)

The function has a minimum value at the point C. Find the x-coordinate of C.

Working:


416. Find the equation of the normal to the curve with equation y = x3 + 1 at the point (1, 2). Working:

Answer: ......................................................................... (Total 4 marks)


273

417. Find the sum of the arithmetic series 17 + 27 + 37 +...+ 417. Working:

Answer: ......................................................................... (Total 4 marks)

418. A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the largest angle. Working:



274

419. The graph represents the function f : x  p cos x, p 

.

y 3

 

x

–3

Find (a)

the value of p;

(b)

the area of the shaded region.

Working:



275

 2p 3  and det A = 14, find the possible values of p. 420. If A =   4 p p

Working:


421. O is the centre of the circle which has a radius of 5.4 cm.

O

A


B

276

The area of the shaded sector OAB is 21.6 cm2. Find the length of the minor arc AB. Working:


422. ABCD is a rectangle and O is the midpoint of [AB]. D

A

C

O

B

Express each of the following vectors in terms of OC and OD (a)

CD

(b)

OA


277

(c)

AD

Working:


2x

423. Solve the equation 9x–1 =  1  .  3 Working:



278

424. Find the coefficient of x5 in the expansion of (3x – 2)8. Working:


425. Differentiate with respect to x (a) (b)

3  4x esin x

Working:



279

  426. The vectors i , j are unit vectors along the x-axis and y-axis respectively.       The vectors u = – i + 2 j and v = 3 i + 5 j are given.

(a)

    Find u + 2 v in terms of i and j .

   A vector w has the same direction as u + 2 v , and has a magnitude of 26.

(b)

   Find w in terms of i and j .

Working:



280

427. Two functions f and g are defined as follows: f (x) = cos x,

0  x  2;

g (x) = 2x + 1,

x

.

Solve the equation (g  f)(x) = 0. Working:


428. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series. Working:



281

429. The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point. y

B

A

0

C

x

Find the coordinates of A, B and C. Working:



282

430. For the events A and B, p(A) = 0.6, p(B) = 0.8 and p(A  B) = 1. Find (a)

p(AB);

(b)

p( A  B).

Working:


431. At a conference of 100 mathematicians there are 72 men and 28 women. The men have a mean height of 1.79 m and the women have a mean height of 1.62 m. Find the mean height of the 100 mathematicians. Working:



283

432. The diagram shows part of the graph of y = 12x2(1 – x). y

0

x

(a)

Write down an integral which represents the area of the shaded region.

(b)

Find the area of the shaded region.

Working:



284

433. Differentiate with respect to x: (a)

(x2 + l)2.

(b)

1n(3x – 1).

Working:



285

25

434. The mean of the population x1, x2, ........ , x25 is m. Given that

x

i

= 300 and

i 1

25

 ( x – m) i

2

= 625, find

i 1

(a)

the value of m;

(b)

the standard deviation of the population.

Working:


435. Solve the equation 3 sin2 x = cos2 x, for 0°  x  180°. Working:



286

5 2 11 2 436. A and B are 2 × 2 matrices, where A =  and BA =    . Find B. 2 0 44 8 

Working:


437. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7). (a)

Find the vectors OB and AC .

(b)

Find the angle between the diagonals of the quadrilateral OABC.

Working:



287

438. Find the coefficient of a3b4 in the expansion of (5a + b)7. Working:


439. Three of the following diagrams I, II, III, IV represent the graphs of (a)

y = 3 + cos 2x

(b)

y = 3 cos (x + 2)

(c)

y = 2 cos x + 3.


288

Identify which diagram represents which graph. I

II

y

y

4

2 1

2  – – 

– x  – – 

–

 – 



 – 

4

1

3  – 

–3



5

2

–

x  – 

y

IV

3

 – – 



x  – 

–2

y

III

–1

 – 



x  – 

2 1

–

 – – 

 – 

Working:



289

440. Solve the equation 43x–1 = 1.5625 × 10–2. Working:


441. The diagram shows part of the graph of y = 1 . The area of the shaded region is 2 units. x y

0


1

a

x

290

Find the exact value of a. Working:


442. The function f is given by f (x) =

1n ( x  2) . Find the domain of the function.

Working:



291

443. A population of bacteria is growing at the rate of 2.3% per minute. How long will it take for the size of the population to double? Give your answer to the nearest minute. Working:


444. Let f (x) =

x , and g (x) = 2x. Solve the equation (f –1  g)(x) = 0.25.

Working:



292

445. The diagrams show a circular sector of radius 10 cm and angle θ radians which is formed into a cone of slant height 10 cm. The vertical height h of the cone is equal to the radius r of its base. Find the angle θ radians.

10cm

h

10cm r

Working:



293

Revision 12 IB Paper 1

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