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Question 1 A given number in base x can be converted to any other base y

Math

Question 1

  1. A given number in base x can be converted to any other base y. According to the expansion method, if abc.de is any given number in base x, then

              write its value in base 10.                                                                   [3]

  1. Convert the following numbers using number system conversions, show

              your answer in details:                                                                         [6]

    1. (723)8 to hexa decimal system
    2. (0.ABDF)16to decimal system
    3. Convert 0.375 to binary system iv. Which digits from (0,1,2,3,4,5) are not allowed in Quinary system (base 5) representation.
    1. (11010.1011)2 to hexadecimal.
    2. (257)10 to the binary system.
  1. Consider the binary number 10.0011  [4]
    1. Convert the above number to the decimal system
    2. What are the place values of the digits 1 in the number 0.00112
    3. what is the sum of (1+1+1+1) in binary system iv. calculate 101 divided by 10 using long division.
  2. Which one is the correct representation of a binary number from the

following?           [2] i. 1101

    1. (214)2
    2. (0000)2 iv. (11)2

Question 2

 

 a general term of a sequence? Why?                         [2]

 

(b) Which term of the sequence with general term

 is

?            [2]

 

(c) An arithmetic sequence has its 4th term equal to 18 and its 12th term equal

 

to 50. Find its 99th term.

[4]

  1. State whether the following sequences are arithmetic, geometric or not any of them. Find the common ratio if it is a geometric sequence and find the common difference d if it is an arithmetic sequence. Then, find the

             next two terms.                                                                                       [6]

    1. −3,3,−3,3
    2. bn = n2 + 3 iii.
  1. Consider the geometric sequence (bn) with
     and q = 3. Is 243 a

              term of this sequence?                                                                        [3]

  1. The nineteenth term of a sequence is -52, and the fourth term is -7. The difference between consecutive terms in the sequence is constant. Find

             the 201st term.                                                                                         [3]

  1. Show whether the following sequence is convergent or divergent.

            

                                                                                           [2]

 

  1. Is the following numbers 1,-4,9,-16,... represent a sequence, if so, find a formula for the nth term of the sequence.

[2]

  1. Show by mathematical induction that for all positive integers n,

             

                                                                 [4]

 

  1. Find the remainder when 3123 is divided by 7. [2]

Question 3

  1. State whether the following statements are false or true, explain your

             answer:                                                                                                      [6]

    1. Given any integers a, b, c and any positive integer n

If a b(modn) and b c(modn), then a c(modn).

    1. Suppose a b(modn) and c d(modn),then a + c b + c(modn). iii. 7x ≡ 12(mod7).
  1. Find the least positive value of x such that:

             71 ≡ x(mod8)                                                                                         [3]

  1. Calculate the multiplicative inverse of 168 in modulo 83.        [3] (d) Calculate the inverse of 4 modulo 15. Show your steps.    [3]

Question 4

  1. A triangle has sides a = 2 and b = 3 and angle C = 60.

            Find                                                                                                              [4]

    1. the length of side c. ii. Find the sine of angle B using sine rules.
  1. If we have a triangle which has one of its side c = 2 and angles A = π/4 and B = π/3. Workout the length a of the side opposite A. [4]
  2. XY Z is a right angled triangle with Y = 90?. Given that
    ,

find z, Cos(Z) and the angle of Z.

[6]

  1. Let g be a function with its domain (0,∞), defined by
    .  [6]
    1. Sketch the graph of g. ii. Is g continuous at other points of its domain?

Question 5

  1. Consider the function f : Z Z given by

(

                                                                               n + 1,      if n is even

f(x) = n − 3, if n is odd

[6]

    1. Is f injective? Prove your answer
    2. Is f surjective? Prove your answer
  1. The function f : R R is defined as follows:

? −x, if x < 0

??

                                                              f(x) =        x2,         if 0 ≤ x ≤ 1.

                                                                           ??1,         if x > 1

Plot the function, and say whether it is Bijective function or not. Explain

             your answer.                                                                                            [5]

  1. The velocity of a particle moving along the x axis varies according to the expression vx = 40 − 5t2, where vx is in meters per second and t is in seconds.

      Find the average acceleration in the time interval t = 0 to t = 2 sec.                                                                                [4]

  1. An object moving with uniform acceleration has a velocity of 12cm/s in the positive x direction when its x coordinate is 3.00cm. If its x coordinate

                 2 seconds later is −5cm, what is its acceleration?                   [5]

END OF PAPER

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