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Counting principle and binomial theorem

1.       Mr Blue, Mr Black, Mr Green, Mrs White, Mrs Yellow and Mrs Red sit around a circular table for a meeting. Mr Black and Mrs White must not sit together.

          Calculate the number of different ways these six people can sit at the table without Mr Black and Mrs White sitting together.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

2.       In how many ways can six different coins be divided between two students so that each student receives at least one coin?

Working:

 

 

Answer:

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(Total 3 marks)

 


 

3.       How many four-digit numbers are there which contain at least one digit 3?

Working:

 

 

Answer:

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(Total 3 marks)

 

 

4.       (a)     At a building site the probability, P(A), that all materials arrive on time is 0.85. The probability, P(B), that the building will be completed on time is 0.60. The probability that the materials arrive on time and that the building is completed on time is 0.55.

(i)      Show that events A and B are not independent.

(ii)     All the materials arrive on time. Find the probability that the building will not be completed on time.

(5)

(b)     There was a team of ten people working on the building, including three electricians and two plumbers. The architect called a meeting with five of the team, and randomly selected people to attend. Calculate the probability that exactly two electricians and one plumber were called to the meeting.

(2)

(c)     The number of hours a week the people in the team work is normally distributed with a mean of 42 hours. 10% of the team work 48 hours or more a week. Find the probability that both plumbers work more than 40 hours in a given week.

(8)

(Total 15 marks)

 


 

5.       A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the number of ways the committee may be chosen.

Working:

 

 

Answer:

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(Total 6 marks)

 


 

6.       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

(a)     two girls and two boys;

(b)     students all of the same gender.

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(Total 6 marks)

 


 

7.       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

(a)     only medical students are chosen;

(b)     all three law students are chosen.

(Total 6 marks)

 

 

8.       There are 25 disks in a bag. Some of them are black and the rest are white. Two are simultaneously selected at random. Given that the probability of selecting two disks of the same colour is equal to the probability of selecting two disks of different colour, how many black disks are there in the bag?

(Total 6 marks)

 


 

9.       There are 10 seats in a row in a waiting room. There are six people in the room.

(a)     In how many different ways can they be seated?

(b)     In the group of six people, there are three sisters who must sit next to each other.

In how many different ways can the group be seated?

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(Total 6 marks)

 


 

10.     Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number of ways in which the remaining nine people may be allocated to the cars. (The arrangement of people within a particular car is not relevant).

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(Total 6 marks)

 


 

11.     Given that

(1 + x)5 (1 + ax)6 º 1 + bx + 10x2 + ............... + a6 x11,

          find the values of a, b Î *.

Working:

 

 

Answer:

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(Total 4 marks)

 

 

12.     Find the coefficient of x7 in the expansion of (2 + 3x)10, giving your answer as a whole number.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

13.     The coefficient of x in the expansion of  is . Find the possible values of a.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

14.     Find the coefficient of x3 in the binomial expansion of .

Working:

 

 

Answer:

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(Total 6 marks)

 


 

15.     (a)     Prove, using mathematical induction, that for a positive integer n,

(cosq + i sinq)n = cos nq + i sin nq  where i2 = –1.

(5)

(b)     The complex number z is defined by z = cosq + i sinq.

(i)      Show that  = cos (–q) + i sin (–q).

(ii)     Deduce that zn + z–n = 2 cos .

(5)

(c)     (i)      Find the binomial expansion of (z + z–l)5.

(ii)     Hence show that cos5q =  (a cos 5q + b cos 3q + c cos q),
where a, b, c are positive integers to be found.

(5)

(Total 15 marks)

 

 

16.     (a)     Find the expansion of (2 + x)5, giving your answer in ascending powers of x.

(b)     By letting x = 0.01 or otherwise, find the exact value of 2.015.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 


 

17.     Consider the complex number z = cosq + i sinq.

(a)     Using De Moivre’s theorem show that

zn +  = 2 cos nq.

(2)

(b)     By expanding  show that

cos4q = (cos 4q + 4 cos 2q + 3).

(4)

(c)     Let g (a) = .

(i)      Find g (a).

(ii)     Solve g (a) = 1

(5)

(Total 11 marks)

 

 

18.     (a)     Write down the term in xr in the expansion of (x + h)n, where 0 £ r £ n, nÎ +.

(1)

(b)     Hence differentiate xn, nÎ +, from first principles.

(5)

(c)     Starting from the result xn ´ x–n = 1, deduce the derivative of x–n, nÎ +.

(4)

(Total 10 marks)

 


 

19.     Express  in the form  where a, bÎ .

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(Total 6 marks)

 


 

20.     Let y = cosq + i sinq.

(a)     Show that  = iy.

          [You may assume that for the purposes of differentiation and integration, i may be treated in the same way as a real constant.]

(3)

(b)     Hence show, using integration, that y = eiq.

(5)

(c)     Use this result to deduce de Moivre’s theorem.

(2)

(d)     (i)      Given that  = a cos5q + b cos3q + c cosq, where sinq  0, use de Moivre’s theorem with n = 6 to find the values of the constants a, b and c.

(ii)     Hence deduce the value of .

(10)

(Total 20 marks)

 

 

21.     Prove by induction that 12n + 2(5n−1) is a multiple of 7 for n Î +.

(Total 10 marks)

 


 

22.     Find the coefficient of the x3 term in the expansion of

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(Total 6 marks)

 


 

23.     Express  in the form  where a, bÎ.

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(Total 5 marks)

 


 

24.     Determine the first three terms in the expansion of (1− 2x)5 (1+ x)7 in ascending powers of x.

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(Total 5 marks)

 

 

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