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Mathematics TMA Questions

## Question 1 – Solving Inequalities

Your solutions should not involve the use of Maxima, except in those parts of questions where this is explicitly required or suggested. Your solutions should not involve the use of any other mathematical software. Your work should be written in a good mathematical style, as described in Section 6 of Unit 1, and as demonstrated by the example and activity solutions in the study units. Five marks (referred to as good mathematical communication, or GMC, marks) on this TMA are allocated for how well you do this.

Your score out of 5 for GMC will be recorded against Question 10. You do not have to submit any work for Question

Question 1 – 5 marks

You should be able to answer this question after studying Unit 3. Use a table of signs to solve the inequality

y − 3

2y + 7 ≥ 0.

Question 2 – 5 marks

You should be able to answer this question after studying Unit 3. The population growth of bacteria can in some circumstances be modelled
by an exponential growth function. This growth can be measured by checking the bacterial density (the number of bacteria in a given volume).
This is done by finding how much light the bacteria in a test tube will absorb, measured in absorbance units (AU). Suppose that f (t) is the density of bacteria in a particular sample at time t (in hours). Assume that the density of bacteria, in AU, is modelled by the exponential growth function

f (t) = Bekt (t ≥ 0),

Where B and k are constants. After 2 hours the density was 0.0436 AU, and after 6 hours the density was 0.0978

(a) Show that the value of the constant k is 0.202 to three significant figures and find the value of the constant B, correct to three significant figures. [4]

(b) What is the bacterial density, in AU, predicted by the model after 9 hours? Give your answer to three significant figures. [1]

Hint: make sure you use accurate values of B and k, if you found these in part (a). You may use rounded values if you do not have accurate values but you will lose marks for this.

## Question 2 – Exponential Growth Functions

Question 3 – 16 marks

You should be able to answer this question after studying Unit 3.

(a) This part of the question concerns the graph of the function f (x) = (x − 3)2 − 2.

(i) Explain how the graph of f can be obtained from the graph of y = x2 by using appropriate translations. (You are not asked to sketch any graphs in this part, but you may find it helpful to do so.) [2]

(ii) Write down the image set of the function f , in interval notation. [2]

(b) This part of the question concerns the function g(x) = (x − 3)2 − 2 (3 ≤ x ≤ 6). The function g has the same rule as the function f in part

(a), but a smaller domain.

(i) Sketch the graph of g, using equal scales on the axes. (You should draw this by hand, rather than using any software.) Mark the coordinates of the endpoints of the graph. [3]

(ii) Give the image set of g, in interval notation. [1]

(iii) Show that the inverse function g−1 has the rule g−1(x) = 3 + √x + 2, justifying each step clearly, and also give its domain and image set. [5]

(iv) Add a sketch of y = g−1(x) to the graph that you produced in part (b)(i). Mark the coordinates of the endpoints of the graph of g−1. [3]

Question 4 – 10 marks

You should be able to answer this question after studying Unit 4. In triangle ABC (with the usual notation, as in Figure 44 of Unit 4), side a = 9 cm, side b = 11 cm and angle C = 32?.

(a) Use this information to calculate the area of triangle ABC, giving your answer to two decimal places. [2]

(b) (i) Use the cosine rule to show that the length of side c is 5.84 cm, rounded to two decimal places. [2]

(ii) Using only the lengths of the sides of the triangle, find the area of triangle ABC, giving your answer to two decimal places, and check that it is the same as the area you found in part (a). [2]

Hint: find Heron’s formula in Unit 4 (use the index for Book B).

(c) Without using the cosine rule again, find the remaining angles A and B, giving your answers to the nearest degree. [4]

## Question 3 – Function Translations and Inverses

Question 5 – 10 marks

You should be able to answer this question after studying Unit 4.

(a) Using the exact values for the sine and cosine of both 3π/4 and π/3, and the angle difference identity for sine, show that an exact value of sin(5π/12) is given by

√2 + √6

4 . [4]

(b) Use the exact value of cos(5π/6) and the half-angle identity for cosine to show that an exact value of cos(5π/12) is given by √2 − √3
2 . [4]

(c) Show that the exact values from parts (a) and (b) satisfy the standard trigonometric identity sin2 θ + cos2 θ = 1. [2]

Question 6 – 10 marks

You should be able to answer this question after studying Unit 5 and also Section 7 of the Computer Algebra Guide. Use Maxima to plot the parabola y = 3x2 − 6x − 1 and the ellipse

4x2 + 25y2 − 32x − 100y + 64 = 0 on the same graph. Plotting an ellipse in Maxima is done in the same way as plotting a circle. Use Maxima to find the coordinates of the points of intersection between the parabola and the ellipse. State the values of the coordinates rounded to two decimal places, but do not attempt to use Maxima for rounding. Include a printout or screenshot of your Maxima worksheet with your solutions. You are not expected to annotate your Maxima worksheet with explanations. [10]

Question 7 – 15 marks

You should be able to answer this question after studying Unit 5. A new high-speed helicopter is travelling at an air speed (that is, a speed
relative to the surrounding air) of 400 km h−1 with a heading of 70?. There is a wind blowing at a speed of 70 km h−1 from the south-east.
Take unit vectors i to point east and j to point north.

(a) Express the velocity h of the helicopter relative to the air, and the velocity w of the wind, in component form, giving the numerical values in km h−1 to one decimal place. [7]

(b) Express the resultant velocity v of the helicopter, relative to the ground, in component form, giving numerical values in km h−1 to one decimal place. [3]

(c) Hence find the magnitude and direction of the resultant velocity v of the helicopter, giving the magnitude in km h−1 to one decimal place and the direction as a bearing to the nearest degree. [5]

Question 8 – 18 marks

You should be able to answer this question after studying Unit 6. This question concerns the function

f (x) = x3 + 9

4 x2 − 3x − 2.

(a) Find the exact x- and y-coordinates of the stationary points of f . [5]

(b) Use the first derivative test to classify the stationary points that you found in part (a). [5]

(c) Sketch the graph of f , indicating the y-intercept and the points that you found in part (a). (You should draw this by hand, rather than using any software, and you can use different scales on the two axes if appropriate.) [5]

(d) Find the greatest and least values taken by f on the interval [−3, 2]. [3]