Quantitative Methods
Foundation Examination
6
March
201
2
Spring
201
2
100 marks
-
3 hours
Module A
Additional reading time
-
15 minutes
Q.1
(
a
)
Factorize f(x) by completing the square where
:
f(x)
=
x
4
–
2x
2
y
2
–
8y
4
Find the possible value of x, where f(x) = 0 and y = 5
(05 marks)
(
b
)
Find the sum of all integers between 170 and 1000 which are exactly divisible by 8.
(05 marks)
(
c
)
So
lve the following equation:
(2e
5x
+ 5e
x
)(2e
2x
–
11) =
–
55e
x
(06 marks)
Q.2
(a)
Ali
would require a sum of Rs. 300,000 after three years from now and a sum
of Rs. 500,000
after five years from now, for the purpose of education of his son. He is planning to deposit
quarterly amounts in an investment scheme to get the desired amounts at the required time. If
the rate of interest is 12% compounded quarterly, what amount should Ali deposit at the start
of each quarter? (05 marks)
(b)
Hayyan invested an amount of Rs. 400,000 in an investment scheme a
nd got Rs. 545,881 at
the end of three years. Find:
(i)
The effective annual rate o
f interest if interest was compounded
monthly.
(ii)
The nominal rate if interest was compounded quarterly.
(06 marks)
Q.3
(a)
Find dy/dx where
x
xx
y)2)(3( −+
= (05 marks)
(
b
)
The dem
and function for a firm’s product is q = 150,000
–
75p where q equals the number of
units demanded and p equals the price in rupees.
(
i
)
Determine the price which should be charged to maximize total revenue.
(
ii
)
What is the maximum revenue
that the fir
m can earn
?
(
iii
)
How many units are expected to be demanded when the firm is earning maximum
revenue? (07 marks)
Q.4
(
a
)
Sketch the feasibl
e region and identify the redundant constraints from the following set of
inequalities:
(
i
)
x + y ≤ 6
(
ii
)
5x + 3y ≤ 15
(
iii
)
x ≤ 2
(
iv
)
x , y ≥ 0
(07 marks)
(b)
If A=
=
32-
7-5
B and
52
73
Prove that B is an inverse of A.
(04 marks)
Q.5
Company A and B are cement manufactures. The quantity of cement in bags produced by company
A has a mean of 50 kg and standard deviation of 0.24 kg, while quantity in bags of company B has
a mean of 49.8 kg and standard deviation of 0.75 kg.
If random samples of 30 bags from each company are taken, find the probability that the difference
between means of the two samples is more than 0.25 kg. (07 marks)
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