IB Mathematics AA Standard Level

1. [Maximum mark: 6] In a group of 20 girls, 13 take history and 8 take economics. Three girls take both history and economics, as shown in the following Venn diagram. The values p and q represent numbers of girls. [Diagram: A rectangle labelled U contains two overlapping circles labelled history and economics. The history-only region is labelled p, the intersection is labelled 3, and the region outside both circles is labelled q.] (a) Find the value of (i) p; (ii) q. [4] (b) A girl is selected at random. Find the probability that she takes economics but not history. [2]

1. [Maximum mark: 6] In an arithmetic sequence, the first term is 3 and the second term is 7. (a) Find the common difference. [2] (b) Find the tenth term. [2] (c) Find the sum of the first ten terms of the sequence. [2]

1. [Maximum mark: 6] A bag contains 5 green balls and 3 white balls. Two balls are selected at random without replacement. (a) Complete the following tree diagram. [3] [Tree diagram: First draw branches are green with probability 5/8 and white with probability 3/8. From green, second-draw branches are green and white; from white, second-draw branches are green and white. The missing conditional probabilities are to be completed.] (b) Find the probability that exactly one of the selected balls is green. [3]

2. [Maximum mark: 5] Let f(x) = 5x and g(x) = x² + 1, for x ∈ ℝ. (a) Find f⁻¹(x). [2] (b) Find (f ∘ g)(7). [3]

2. [Maximum mark: 7] The vectors a = (4, 2) and b = (k + 3, k) are perpendicular to each other. (a) Find the value of k. [4] (b) Given that c = a + 2b, find c. [3]

2. [Maximum mark: 6] In an arithmetic sequence, the first term is 8 and the second term is 5. (a) Find the common difference. [2] (b) Find the tenth term. [2] (c) Find the sum of the first ten terms. [2]

3. [Maximum mark: 6] The following diagram shows triangle PQR. [Diagram not to scale: Triangle PQR has ∠PQR = 30°, ∠QRP = 45°, and side PQ = 13 cm.] ∠PQR = 30°, ∠QRP = 45° and PQ = 13 cm. Find PR.

3. [Maximum mark: 6] The random variable X is normally distributed with a mean of 100. The following diagram shows the normal curve for X. [Diagram: A symmetric normal curve centred at x = 100. The region R under the curve to the right of x = 107 is shaded.] Let R be the shaded region under the curve, to the right of 107. The area of R is 0.24. (a) Write down P(X > 107). [1] (b) Find P(100 < X < 107). [3] (c) Find P(93 < X < 107). [2]

3. [Maximum mark: 6] The following diagram shows the graph of a function f, with domain −2 ≤ x ≤ 4. [Diagram: A smooth one-to-one increasing graph begins at (−2,0), passes through (0,2) and (2,3), and ends at (4,7). It is initially concave down and later concave up.] The points (−2, 0) and (4, 7) lie on the graph of f. (a) Write down the range of f. [1] (b) Write down (i) f(2); (ii) f⁻¹(2). [2] (c) On the grid opposite, sketch the graph of f⁻¹. [3]

4. [Maximum mark: 6] Jim heated a liquid until it boiled. He measured the temperature of the liquid as it cooled. The following table shows its temperature, d degrees Celsius, t minutes after it boiled. [Table: t (min): 0, 4, 8, 12, 16, 20; d (°C): 105, 98.4, 85.4, 74.8, 68.7, 62.1.] (a) (i) Write down the independent variable. (ii) Write down the boiling temperature of the liquid. [2] Jim believes that the relationship between d and t can be modelled by a linear regression equation. (b) Jim describes the correlation as very strong. Circle the value below which best represents the correlation coefficient. 0.992 0.251 0 −0.251 −0.992 [2] (c) Jim’s model is d = −2.24t + 105, for 0 ≤ t ≤ 20. Use his model to predict the decrease in temperature for any 2 minute interval. [2]

4. [Maximum mark: 6] The following figures consist of rows and columns of squares. The figures form a continuing pattern. Figure 1 has two rows and one column. Figure 2 has three rows and two columns. [Diagram: Figure n is a rectangular array with n columns and n + 1 rows; Figures 1 to 4 are shown.] Figure 5 has p rows and q columns. (a) Write down the value of (i) p; (ii) q. [2] Each small square has an area of 1 cm². Let Aₙ be the total area of Figure n. The following table gives the first five values of Aₙ. [Table: n: 1, 2, 3, 4, 5; Aₙ (cm²): 2, 6, 12, 20, k.] (b) Find the value of k. [2] (c) Find an expression for Aₙ in terms of n. [2]

4. [Maximum mark: 7] The following diagram shows triangle ABC, with AB = 3 cm, BC = 8 cm, and ∠ABC = π/3. [Diagram not to scale.] (a) Show that AC = 7 cm. [4] (b) The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle. [Diagram not to scale.] Find the exact perimeter of this shape. [3]

5. [Maximum mark: 7] (a) Find ∫ x e^(x²−1) dx. [4] (b) Find f(x), given that f′(x) = x e^(x²−1) and f(−1) = 3. [3]

5. [Maximum mark: 6] Let f′(x) = 3x²/(x³ + 1)⁵. Given that f(0) = 1, find f(x).

5. [Maximum mark: 6] Let f(x) = 1 + e^(−x) and g(x) = 2x + b, for x ∈ ℝ, where b is a constant. (a) Find (g ∘ f)(x). [2] (b) Given that lim_(x→+∞) (g ∘ f)(x) = −3, find the value of b. [4]

6. [Maximum mark: 6] The following diagram shows the graph of f′, the derivative of f. [Diagram: Coordinate axes with x shown from approximately −4 to 10 and y from approximately −12 to 8. A smooth curve y = f′ has a local minimum at A near (1.5, −5), then rises through the labelled point (4, −2), reaches a local maximum at B at approximately (6, 0), and then decreases.] The graph of f′ has a local minimum at A, a local maximum at B and passes through (4, −2). (a) The point P(4, 3) lies on the graph of the function f. (i) Write down the gradient of the curve of f at P. (ii) Find the equation of the normal to the curve of f at P. [4] (b) Determine the concavity of the graph of f when 4 < x < 5 and justify your answer. [2]

6. [Maximum mark: 5] The values of the functions f and g and their derivatives for x = 1 and x = 8 are shown in the following table. [Table with columns x, f(x), f′(x), g(x), g′(x). Row x=1: 2, 4, 9, −3. Row x=8: 4, −3, 2, 5.] Let h(x) = f(x)g(x). (a) Find h(1). [2] (b) Find h′(8). [3]

6. [Maximum mark: 7] Let f(x) = 15 − x², for x ∈ ℝ. The following diagram shows part of the graph of f and the rectangle OABC, where A is on the negative x-axis, B is on the graph of f, and C is on the y-axis. [Diagram not to scale: O is the origin; A=(x,0) with x<0; B=(x,f(x)); C=(0,f(x)); OABC is a rectangle under the parabola.] Find the x-coordinate of A that gives the maximum area of OABC.

7. [Maximum mark: 8] The first three terms of a geometric sequence are ln(x¹⁶), ln(x⁸), ln(x⁴), for x > 0. (a) Find the common ratio. [3] (b) Solve ∑_(k=1)^∞ 2^(5−k) ln x = 64. [5]

7. [Maximum mark: 7] Solve log₂(2 sin x) + log₂(cos x) = −1, for 2π < x < 5π/2.