This paper concentrates on the primary theme of X, Y and Z are binary integers as follows: X = 101101 Y = 101 Z = 10101 Showing all stages of working, perform the following calculations IN BINARY. You are advised to space out numbers so that they are in strict columns. a) Find X + Y + Z. in which you have to explain and evaluate its intricate aspects in detail. In addition to this, this paper has been reviewed and purchased by most of the students hence; it has been rated 4.8 points on the scale of 5 points. Besides, the price of this paper starts from £ 40. For more details and full access to the paper, please refer to the site.

NUMBER
& LOGIC

** **

**Instructions to candidates:**

a) Time
allowed: Three hours (plus an extra ten minutes’ reading time at the start – do
not write anything during this time)

b)
Answer any FIVE questions

c) All
questions carry equal marks. Marks for parts of questions are shown in [ ]

d) In
numerical questions, candidates must show and explain the method of working to
obtain full marks

e)
Non-programmable calculators are permitted in this examination. Calculators
should be used in all the non-binary questions. However, ensure that you write
down all intermediate values obtained from a calculator. Always explain in
words what you are calculating

f)
Ensure that you leave numeric answers in the format required by the question

g)
Ensure that you pay particular attention to words underlined, in CAPITALS or in
**bold**. FEW OR NO

MARKS
will be awarded to any question where these are ignored

h) No computer
equipment, books or notes may be used in this examination

1. X, Y
and Z are binary integers as follows:

X =
101101 Y = 101 Z = 10101

Showing
all stages of working, perform the following calculations IN BINARY. You are
advised to space out numbers so that they are in strict columns.

a) Find
X + Y + Z. [3]

b) Find
X – Z. [2]

c) Find
Y × Z. [4]

d) Find
X / Z. Leave your answer accurate to 3 binary places. [4]

e)
Convert X and Z to decimal and show that your answers to b) and d) are correct.
[4]

f) IF the
decimal number 24.68 is to be converted to binary as accurately as possible,
how many

BINARY
places would be needed? Explain your answer. [3]

2. a)
Consider the decimal values of the contents of the three memory locations P, Q
and R.

P =
0110 0000 0011 1000

Q =
1011 0000 1100 1000

R =
0110 1010 0000 1010

i.
Calculate the decimal value of the contents of P if it holds a midpoint
fractional value with the binary point assumed to be in the middle. Give the
fractional part as a fraction in its lowest cancelled form. [4]

ii.
Calculate the decimal value of the contents of Q if the value is held in two`s
complement binary form. [4]

iii.
Calculate the decimal value of the contents of R if the value is held in
floating point form with the left 10 bits used for the mantissa and the right 6
bits for the exponent. [4]

i.
Calculate the start and end addresses for program B if the three programs are
held one after another without any gaps. [2]

ii.
Calculate the size of program B in both hexadecimal AND decimal. [4]

iii. A
second copy of program B is loaded immediately after program C. Determine the hexadecimal
address of the end of that program. [2]

*
Continued overleaf*

3. a)
Write down an algorithm (for example pseudocode or flowchart) for converting **any **decimal number to binary. [8]

b)
Showing relevant working, convert the following:

i. Hex
value 17FB into decimal [4]

ii.
Decimal 4321 into hex [4]

iii.
Binary string 00110010 into decimal [2]

iv. Hex
value E9 into binary [2]

4. a)
The equation x4 + x2 – 80 = 0 has three solutions.

i. Using
a calculator and the iterative formula given below, determine the solution near
to x = 3 to five decimal places. You should show all working and all intermediate
values towards the final answer. x1 = x0 – (x04 + x02 – 80) / (4x03 + 2x0)

ii.
Explain how you know that this answer is correct to five decimal places. [15]

b)
Explain why iterative methods to solve equations are beneficial to the
mathematician/scientist. [5]

8. a) 200 holidaymakers were asked whether they had visited any of these
three countries – Austria,

Belgium and Croatia.

• 30 people have not visited any of these countries

• 10 people have been to all three countries

• 25 people have been to Belgium and Austria

• 20 people have been to Croatia and Austria but not to Belgium

• 65 people have been to exactly two of the countries

• 165 people have been to Croatia or to Belgium

• 120 people have been to Belgium or to Austria

i Draw a Venn diagram to represent this data. [4]

ii Use your diagram to calculate: [6]

• the number of people who had only visited Austria

• the number of people who had only visited Belgium

• the number of people who had only visited Croatia

b) If a coin is flipped 3 times, what is the probability of getting two
tails and one head? Show your

working. [6]

c) Two dice are rolled and the results are added together.

i Find the probability that their sum is an odd number. [2]

ii Find the probability that their sum is greater than 6.

6. a)
An item incurs a **loss **of 5% by
selling for £114. At what price should the item be sold to earn 5% **profit**? Show your working. [6]

b) A
company uses large quantities of “widgets” and buys a consignment several times
each year.

Every
year 30,000 widgets are used. One widget costs £20 and, in addition, there is a
handling charge of £40 for each order. A handling/carrying cost of 12% per item
is also applied. The company wishes to order as economically as possible.

i.
Explain what is meant by EOQ (Economic Order Quantity), stating what factors
are taken into account. [2]

ii.
Write down the formula for calculating EOQ and calculate the EOQ for this
product. [6]

c)
£1,000 is invested for 3 years at an investment rate of 5%.

i.
Calculate the simple interest on this amount. [2]

ii.
Calculate the compound interest on the same investment. [4]

*
Continued
overleaf*

8. a) 200 holidaymakers were asked whether they had visited any of these
three countries – Austria,

Belgium and Croatia.

• 30 people have not visited any of these countries

• 10 people have been to all three countries

• 25 people have been to Belgium and Austria

• 20 people have been to Croatia and Austria but not to Belgium

• 65 people have been to exactly two of the countries

• 165 people have been to Croatia or to Belgium

• 120 people have been to Belgium or to Austria

i Draw a Venn diagram to represent this data. [4]

ii Use your diagram to calculate: [6]

• the number of people who had only visited Austria

• the number of people who had only visited Belgium

• the number of people who had only visited Croatia

b) If a coin is flipped 3 times, what is the probability of getting two
tails and one head? Show your

working. [6]

c) Two dice are rolled and the results are added together.

i Find the probability that their sum is an odd number. [2]

ii Find the probability that their sum is greater than 6.

8. a)
200 holidaymakers were asked whether they had visited any of these three
countries – Austria, Belgium and Croatia.

• 30
people have not visited any of these countries

• 10
people have been to all three countries

• 25
people have been to Belgium and Austria

• 20
people have been to Croatia and Austria but not to Belgium

• 65
people have been to exactly two of the countries

• 165
people have been to Croatia or to Belgium

• 120
people have been to Belgium or to Austria

i. Draw
a Venn diagram to represent this data. [4]

ii. Use
your diagram to calculate: [6]

• The
number of people who had only visited Austria

• The
number of people who had only visited Belgium

• The
number of people who had only visited Croatia

b) If a
coin is flipped 3 times, what is the probability of getting two tails and one
head? Show your working. [6]

c) Two
dice are rolled and the results are added together.

i. Find
the probability that their sum is an odd number. [2]

ii.
Find the probability that their sum is greater than 6.