IB SL Cumulative Exam – Part 1



IB SL Cumulative Exam Question Pool – Part 1

1. Find the coefficient of x3 in the expansion of (2 – x)5.

(Total 6 marks)

2. Given that log5 x = y, express each of the following in terms of y.

(a) log5 x2

(b) log5[pic]

(c) log25 x

(Total 6 marks)

3. 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)

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

(a) Express f (x) in the form a (x – p)2 + q, where a, p, q ∈ [pic].

(b) Find the minimum value of f (x).

(Total 6 marks)

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

(Total 6 marks)

6. In triangle ABC, AC = 5, BC = 7, [pic] = 48°, as shown in the diagram.

[pic]

Find [pic] giving your answer correct to the nearest degree.

(Total 6 marks)

7. Given that sin x = [pic], where x is an acute angle, find the exact value of

(a) cos x;

(b) cos 2x.

(Total 6 marks)

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

[pic]

Find the value of

(a) p;

(b) q.

(Total 6 marks)

9. Calculate the acute angle between the lines with equations

r = [pic] + s[pic] and r = [pic] + t[pic]

(Total 6 marks)

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

(a) Show that [pic]

(b) Show that, to three significant figures, cos[pic]

(Total 6 marks)

11. 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) Write down the value of

(i) r;

(ii) s.

(b) The mean of the population is 10. Find the value of t.

(Total 6 marks)

12. 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.

(Total 6 marks)

13. 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?

(Total 6 marks)

14. The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and the lines y = 0 and x = [pic]

[pic]

Given that sin [pic] = [pic] and cos [pic] = – [pic], calculate the exact area of the shaded region.

(Total 6 marks)

15. 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).

(Total 6 marks)

16. It is claimed that the masses of a population of lions are normally distributed with a mean mass of 310 kg and a standard deviation of 30 kg.

(a) Calculate the probability that a lion selected at random will have a mass of 350 kg or more.

(2)

(b) The probability that the mass of a lion lies between a and b is 0.95, where a and b are symmetric about the mean. Find the value of a and of b.

(3)

(Total 5 marks)

17. 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.

(Total 6 marks)

IB SL Cumulative Exam Question Pool – Part 2

1. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by

h = 2 + 20t – 5t2, t ≥ 0

(a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released).

(2)

(b) Show that the height of the ball after one second is 17 metres.

(2)

(c) At a later time the ball is again at a height of 17 metres.

(i) Write down an equation that t must satisfy when the ball is at a height of 17 metres.

(ii) Solve the equation algebraically.

(4)

(d) (i) Find [pic].

(ii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released).

(iii) Find when the ball reaches its maximum height.

(iv) Find the maximum height of the ball.

(7)

(Total 15 marks)

2. The function f (x) is defined as f (x) = –(x – h)2 + k. The diagram below shows part of the graph of f (x). The maximum point on the curve is P (3, 2).

[pic]

(a) Write down the value of

(i) h;

(ii) k.

(2)

(b) Show that f (x) can be written as f (x) = –x2 + 6x – 7.

(1)

(c) Find f ′ (x).

(2)

The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q.

(d) (i) Calculate the gradient of L.

(ii) Find the equation of L.

(iii) The line L intersects the curve again at R. Find the x-coordinate of R.

(8)

(Total 13 marks)

3. The diagram shows a triangular region formed by a hedge [AB], a part of a river bank [AC] and a fence [BC]. The hedge is 17 m long and [pic] is 29°. The end of the fence, point C, can be positioned anywhere along the river bank.

(a) Given that point C is 15 m from A, find the length of the fence [BC].

[pic]

(3)

(b) The farmer has another, longer fence. It is possible for him to enclose two different triangular regions with this fence. He places the fence so that [pic] is 85°.

(i) Find the distance from A to C.

(ii) Find the area of the region ABC with the fence in this position.

(5)

(c) To form the second region, he moves the fencing so that point C is closer to point A.

Find the new distance from A to C.

(4)

(d) Find the minimum length of fence [BC] needed to enclose a triangular region ABC.

(2)

(Total 14 marks)

4. Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB).

(a) Find

(i) [pic]

(ii) [pic], giving your answer in terms of k.

(3)

(b) Show that k = 7.

(3)

The point C is such that [pic] = [pic]

(c) Find the position vector of C.

(4)

(d) Find cos [pic].

(3)

(Total 13 marks)

5. Points A, B, and C have position vectors 4i + 2j, i – 3j and – 5i – 5j. Let D be a point on the x-axis such that ABCD forms a parallelogram.

(a) (i) Find [pic].

(ii) Find the position vector of D.

(4)

(b) Find the angle between [pic] and [pic].

(6)

The line L1 passes through A and is parallel to i + 4j. The line L2 passes through B and is parallel to 2i + 7j. A vector equation of L1 is r = (4i + 2j) + s(i + 4j).

(c) Write down a vector equation of L2 in the form r = b + tq.

(1)

(d) The lines L1 and L2 intersect at the point P. Find the position vector of P.

(4)

(Total 15 marks)

6. A supermarket records the amount of money d spent by customers in their store during a busy period. The results are as follows:

Money in $ (d) |0–20 |20–40 |40–60 |60–80 |80–100 |100–120 |120–140 | |Number of customers (n) |24 |16 |22 |40 |18 |10 |4 | | (a) Find an estimate for the mean amount of money spent by the customers, giving your answer to the nearest dollar ($).

(2)

(b) Copy and complete the following cumulative frequency table and use it to draw a cumulative frequency graph. Use a scale of 2 cm to represent $20 on the horizontal axis, and 2 cm to represent 20 customers on the vertical axis.

(5)

Money in $ (d) | ................
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