Topics Included to Keep in Shape :

Transcription

1 Name: Keeping In Shape Packet #1 (Year 1 Material) WE WILL USE THIS PACKET TO KEEP IN SHAPE AND PRACTICE SOME OF THE MATERIAL LEARNED IN IB MATH SL YEAR 1. YOU ARE EXPECTED TO WORK ON THIS IN CLASS (OR AT HOME, WHEN ASSIGNED). YOU WILL HAVE AN OPPORTUNITY TO CHECK YOUR WORK WITH AN ANSWER KEY AND ASK ANY QUESTIONS AT EXTRA HELP! Topics Included to Keep in Shape : Arithmetic and Geometric Sequences and Series Solving Logarithmic Equations Substituting for Logarithmic Expressions Coefficients of a Binomial Expansion Sum to Infinity of a Geometric Series Solving Exponential Equations using Change of Base Solving for Common Ratio Solving For parts of a parabola Composition of functions (with inverse functions) Discriminant Application of an Exponential Model Complete the square for vertex form Tips to be successful when working to keep in shape : 1. After this packet is completed, check the key. If there are questions you do not understand come to extra help! 2. Practice WITH YOUR FORMULA BOOKLET OUT 4. Use your calculator! 5. Do your best and believe in yourself!

2 1. Portable telephones are first sold in the country Cellmania in During 1990, the number of units sold is 160. In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360. In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360 respectively. What is the common ratio of this sequence? Assume that this trend in sales continues. How many units will be sold during 2002? (c) In what year does the number of units sold first exceed 5000? Between 1990 and 1992, the total number of units sold is 760. (d) What is the total number of units sold between 1990 and 2002? During this period, the total population of Cellmania remains approximately (e) Use this information to suggest a reason why the geometric growth in sales would not continue. 2. In an arithmetic sequence, the first term is 2, the fourth term is 16, and the n th term is Find the common difference d. Find the value of n, AND find the sum of the first 20 terms

3 3. Solve the equation log 27 x = 1 log 27 (x 0.4). 4. Find the coefficient of x 3 in the expansion of (2 x) Gwendolyn added the multiples of 3, from 3 to 3750 and found that Calculate s = s. 6. Given that log 5 x = y, express each of the following in terms of y. log 5 x 2 log 1 5 x (c) log 25 x

4 7. 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. Complete the table by stating the type of series that is shown. Series (i) Type of series (ii) (iii) (iv) The geometric series can be summed to infinity. Find this sum. 8. Find the exact solution of the equation = 27 (1 x). 9. Factorize x 2 3x 10. Solve the equation x 2 3x 10 = 0.

5 10. Consider the sequence x 3, x +1, 2x + 8,. When x = 5, the sequence is geometric. (i) (ii) Write down the first three terms. Find the common ratio. (c) Consider the general sequence again. Write two ratios of expressions that can represent the common difference. (d) Using a proportion from the ratios in (c) solve for the other value of x for which the sequence is geometric. (e) For this value of x, find (i) (ii) the common ratio; the sum of the infinite sequence.

6 11. 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. 12. Let f (x) = x, and g (x) = 2 x. Solve the equation (f 1 g)(x) = The quadratic equation 4x 2 + 4kx + 9 = 0, k > 0 has exactly one solution for x. Find the value of k.

7 14. Initially a tank contains litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given by V = (0.933 t ). (c) Find the value of V after 5 minutes. Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out of the tank. The tank is regarded as effectively empty when 95% of the liquid has flowed out. Show that it takes almost three-quarters of an hour for this to happen. 15. Express f (x) = x 2 6x + 14 in the form f (x) = (x h) 2 + k, where h and k are to be determined. Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y x 2 6x + 14.