IB Math 11 Assignment: Chapters 1 & 2 (A) NAME: (Functions, Sequences and Series)

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1 IB Math 11 Assignment: Chapters 1 & 2 (A) NAME: (Functions, Sequences and Series) 1. Let f(x) = 7 2x and g(x) = x + 3. Find (g f)(x). Write down g 1 (x). (c) Find (f g 1 )(5). (Total 5 marks) 2. Consider f(x) = x 5. Find (i) f(11); (ii) f(5). Find the values of x for which f is undefined. (c) Let g(x) = x 2. Find (g f)(x).

2 3. The function f is given by f (x) = x 2 6x + 13, for x 3. Write f (x) in the form (x a) 2 + b. Find the inverse function f 1. (3) (c) State the domain of f In an arithmetic sequence, u 1 = 2 and u 3 = 8. Find d. Find u 20. (c) Find S Consider the arithmetic sequence 3, 9, 15,..., Write down the common difference. Find the number of terms in the sequence. (3) (c) Find the sum of the sequence.

3 6. Write down the first three terms of the sequence u n = 3n, for n 1. Find 20 (i) 3n ; n= (ii) 3n. n= 21 (5) 7. 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. Calculate the number of seats in the 20th row. Calculate the total number of seats. 8. The first three terms of an infinite geometric sequence are 32, 16 and 8. Write down the value of r. Find u 6. (c) Find the sum to infinity of this sequence. (Total 5 marks)

4 In a geometric series, u 1 = and u4 = Find the value of r. (3) Find the smallest value of n for which S n > 40. (Total 7 marks) 10. The first four terms of a sequence are 18, 54, 162, 486. Use all four terms to show that this is a geometric sequence. (i) Find an expression for the n th term of this geometric sequence. (ii) If the n th term of the sequence is , find the value of n. 11. A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum. Write down an expression for the value of the investment after n full years. What will be the value of the investment at the end of five years? (c) The value of the investment will exceed $ after n full years. (i) Write down an inequality to represent this information. (ii) Calculate the minimum value of n.

5 IB Math 11 Assignment: Chapters 1 & 2 (B) NAME: (Functions, Sequences and Series) 1. Let f(x) = 8 3x and g(x) = x + 2. Find (g f)(x). Write down g 1 (x). (c) Find (f g 1 )(5). (Total 5 marks) 2. Consider f(x) = x 5. Find (i) f(12); (ii) f(5). Find the values of x for which f is undefined. (c) Let g(x) = x 2. Find (g f)(x).

6 3. The function f is given by f (x) = x 2 8x + 19, for x 4. Write f (x) in the form (x a) 2 + b. Find the inverse function f 1. (3) (c) State the domain of f In an arithmetic sequence, u 1 = 3 and u 3 = 11. Find d. Find u 20. (c) Find S Consider the arithmetic sequence 4, 9, 14,..., Write down the common difference. Find the number of terms in the sequence. (3) (c) Find the sum of the sequence.

7 6. Write down the first three terms of the sequence u n = 3n, for n 1. Find 20 (i) 3n ; n= (ii) 3n. n= 21 (5) 7. A theatre has 30 rows of seats. There are 10 seats in the first row, 12 seats in the second row, and each successive row of seats has two more seats in it than the previous row. Calculate the number of seats in the 30th row. Calculate the total number of seats. 8. The first three terms of an infinite geometric sequence are 81, 27 and 9. Write down the value of r. Find u 6. (c) Find the sum to infinity of this sequence. (Total 5 marks)

8 In a geometric series, u 1 = and u4 = Find the value of r. (3) Find the smallest value of n for which S n > 40. (Total 7 marks) 10. The first four terms of a sequence are 18, 36, 72, 144. Use all four terms to show that this is a geometric sequence. (i) Find an expression for the n th term of this geometric sequence. (ii) If the n th term of the sequence is 36864, find the value of n. 11. A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum. Write down an expression for the value of the investment after n full years. What will be the value of the investment at the end of five years? (c) The value of the investment will exceed $ after n full years. (i) Write down an inequality to represent this information. (ii) Calculate the minimum value of n.

9 1. attempt to form composite (M1) e.g. g ( 7 2x), 7 2x + 3 (g f)(x) = 10 2x A1 N2 2 g 1 (x) = x 3 A1 N1 1 (c) METHOD 1 valid approach (M1) e.g. g 1 (5), 2, f (5) f = 3 A1 N2 2 METHOD 2 attempt to form composite of f and g 1 (M1) e.g. (f g 1 )(x) = 7 2(x 3), 13 2x (f g 1 )(5) = 3 A1 N2 2 [5] 2. (i) 6 A1 N1 (ii) 9 A1 N1 (iii) 0 A1 N1 x < 5 A2 N2 (c) (g f)(x) = ( x 5 ) 2 (M1) = x 5 A1 N2 [7] 3. a = 3, b = 4 (A1) f (x) = (x 3) A1 (C2) y = (x 3) METHOD 1 x = (y 3) x 4 = (y 3) 2 (M1) x 4 = y 3 (M1) y = x (A1) 3 METHOD 2 y 4 = (x 3) 2 (M1) y 4 = x 3 (M1) y = x y = x f 1 (x) = x (A1) 3 (c) x 4 (A1)(C1)

10 4. attempt to find d (M1) u3 u e.g. 1, 8 = 2 + 2d 2 d = 3 A1 N2 2 correct substitution (A1) e.g. u 20 = 2 + (20 1)3, u 20 = u 20 = 59 A1 N2 2 (c) correct substitution (A1) e.g. S 20 = 2 20 (2 + 59), S20 = 2 20 ( ) S 20 = 610 A1 N common difference is 6 A1 N1 evidence of appropriate approach (M1) e.g. u n = 1353 correct working e.g = 3 + (n 1)6, n = A1 A1 N2 (c) evidence of correct substitution A1 226 ( ) 226 e.g. S 226 =, ( ) 2 2 S 226 = (accept ) A1 N1 6. 3, 6, 9 A1 N1 (i) Evidence of using the sum of an AP M eg 2 3+ ( 20 1) 3 20 n = 1 3n = 630 A1 N1 (ii) METHOD Correct calculation for 3n n = 1 eg ( ), Evidence of subtraction eg (A1) (M1) 100 n = 21 3n = A1 N2 METHOD 2

11 Recognising that first term is 63, the number of terms is 80 (A1)(A1) eg ( ), ( ) 100 n = 21 3n = A1 N2 7. Recognizing an AP (M1) u 1 = 15 d = 2 n = 20 (A1) substituting into u 20 = 15 + (20 1) 2 M1 = 53 (that is, 53 seats in the 20th row) A1 N Substituting into S 20 = (2(15) + (20 1)2) (or into ( )) 2 2 M1 = 680 (that is, 680 seats in total) A1 N r = = 32 2 correct calculation or listing terms (A1) 1 e.g u 6 = , 8, 32,... 4, 2, 1 2 A1 N1 A1 N2 (c) evidence of correct substitution in S A e.g., S = 64 A1 N1 [5] 9. evidence of substituting into formula for nth term of GP (M1) 1 3 e.g. u 4 = r setting up correct equation r 3 = 81 3 A1 r = 3 A1 N2 METHOD 1 setting up an inequality (accept an equation) 1 n 1 n (3 1) (1 3 ) e.g ; 81 n > > 40; 3 > evidence of solving e.g. graph, taking logs n > n = 8 METHOD 2 if n = 7, sum = ; if n = 8, sum = n = 8 (is the smallest value) M1 M1 (A1) A1 N2 A2 A2 N2 [7]

12 10. For taking three ratios of consecutive terms (M1) = = ( = 3) A hence geometric AG N0 (i) r = 3 (A1) u n = 18 3 n 1 A1 N2 (ii) For a valid attempt to solve 18 3 n 1 = (M1) eg trial and error, logs n = 11 A1 N (1.063) n A1 N1 Value = $ 5000(1.063) 5 (= $ ) = $ 6790 to 3 s.f. (accept $ 6786, or $ ) A1 N1 (c) (i) 5000(1.063) n > or (1.063) n > 2 A1 N1 (ii) Attempting to solve the inequality nlog(1.063) > log2 (M1) n > (A1) 12 years A1 N3 Note: Candidates are likely to use TABLE or LIST on a GDC to find n. A good way of communicating this is suggested below. Let y = x When x = 11, y = , when x = 12, y = x = 12 i.e. 12 years (M1) (A1) A1 N3