students all of the same gender. (Total 6 marks)

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Ethel Harrison
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1 January Exam Review: Math 11 IB HL 1. A team of five students is to be chosen at random to take part in a debate. The team is to be chosen from a group of eight medical students and three law students. Find the probability that only medical students are chosen; all three law students are chosen. 2. There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a committee. Calculate the probability that the committee contains two girls and two boys; students all of the same gender. 3. Given that (a + i)(2 bi) = 7 i, find the value of a and of b, where a, b.. Let P(z) = z 3 + az 2 + bz + c, where a, b, and c. Two of the roots of P(z) = 0 are 2 and ( 3 + 2i). Find the value of a, of b and of c. 5. The sum of the first n terms of a series is given by S n = 2n 2 n, where n +. Find the first three terms of the series. Find an expression for the n th term of the series, giving your answer in terms of n. 6. The sum of the first n terms of an arithmetic sequence {u n } is given by the formula S n = n 2 2n. Three terms of this sequence, u 2, u m and u 32, are consecutive terms in a geometric sequence. Find m. 7. Consider f (x) = x 3 2x 2 5x + k. Find the value of k if (x + 2) is a factor of f (x). 8. The polynomial f (x) = x 3 + 3x 2 +ax + b leaves the same remainder when divided by (x 2) as when divided by (x +1). Find the value of a. 1

2 9. Find the range of values of m such that for all x. m(x + 1) x Find the total area of the two regions enclosed by the curve y = x 3 3x 2 9x +27 and the line y = x Box A contains 6 red balls and 2 green balls. Box B contains red balls and 3 green balls. A fair cubical die with faces numbered 1, 2, 3,, 5, 6 is thrown. If an even number is obtained, a ball is selected from box A; if an odd number is obtained, a ball is selected from box B. Calculate the probability that the ball selected was red. Given that the ball selected was red, calculate the probability that it came from box B. 12. In a game a player pays an entrance fee of $n. He then selects one number from 1, 2, 3,, 5, 6 and rolls three standard dice. If his chosen number appears on all three dice he wins four times his entrance fee. If his number appears on exactly two of the dice he wins three times the entrance fee. If his number appears on exactly one die he wins twice the entrance fee. If his number does not appear on any of the dice he wins nothing. Copy and complete the probability table below. Profit ($) n n 2n 3n Probability Show that the player s expected profit is $ 17 n. 216 What should the entrance fee be so that the player s expected loss per game is 3 cents? (Total 8 marks) 13. Bag A contains 2 red and 3 green balls. Two balls are chosen at random from the bag without replacement. Find the probability that 2 red balls are chosen. Bag B contains red and n green balls. Two balls are chosen without replacement from this bag. If the probability that two red balls are chosen is 15 2, show that n = 6. 2

3 A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B. Calculate the probability that two red balls are chosen. (d) Given that two red balls are chosen, find the probability that a 1 or a 6 was obtained on the die. (3) (Total 13 marks) 1. Robert travels to work by train every weekday from Monday to Friday. The probability that he catches the train on Monday is The probability that he catches the train on any other weekday is A weekday is chosen at random. Find the probability that he catches the train on that day. Given that he catches the train on that day, find the probability that the chosen day is Monday. 15. Jack and Jill play a game, by throwing a die in turn. If the die shows a 1, 2, 3 or, the player who threw the die wins the game. If the die shows a 5 or 6, the other player has the next throw. Jack plays first and the game continues until there is a winner. Write down the probability that Jack wins on his first throw. Calculate the probability that Jill wins on her first throw. Calculate the probability that Jack wins the game. (1) (3) 16. The figure below shows part of the curve y = x 3 7x 2 + 1x 7. The curve crosses the x-axis at the points A, B and C. 3

4 y 0 A B C x Find the x-coordinate of A. Find the x-coordinate of B. Find the area of the shaded region. 17. Consider the tangent to the curve y = x 3 + x 2 + x 6. Find the equation of this tangent at the point where x = 1. Find the coordinates of the point where this tangent meets the curve again. 18. The line y = 16x 9 is a tangent to the curve y = 2x 3 + ax 2 + bx 9 at the point (1,7). Find the values of a and b. 19. When the polynomial P (x) = x 2 + px 2 + qx + 1 is divided by (x 1) the remainder is 2. When 13 P (x) is divided by (2x 1) the remainder is. Find the value of p and of q. 20. Given functions f : x x + 1 and g : x x 3, find the function (f g) l. 21. The functions f and g are defined by f (x) = 2x 1,

5 x g (x) =,x 1. x 1 Find the values of x for which ( f g) (x) (g f ) (x). 3x 22. The function f is defined as f (x) =, x 2. x 2 Find an expression for f 1 (x). Write down the domain of f Evaluate (1 + i) 2, where i = 1. Prove, by mathematical induction, that (1 + i) n = ( ) n, where n *. Hence or otherwise, find (1 + i) 32. (Total 10 marks) 5

Find the value of n in order for the player to get an expected return of 9 counters per roll.

Find the value of n in order for the player to get an expected return of 9 counters per roll. . A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters