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1. Paper: QUANTITATIVE TECHNIQUE (1C)
Specific Instructions:

Answer all the four questions.

Marks allotted 100.
General Instructions:

The Student should submit this assignment in the handwritten form (not in the typed format)

The Student should submit this assignment within the time specified by the exam dept

Each Question mentioned in this assignment should be answered within the word limit specified

The student should only use the Rule sheet papers for answering the questions.

The student should attach this assignment paper with the answered papers.
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Failure to comply with the above Five instructions would lead to rejection of assignment.

2. Question No - 1
a.
Lim
X →0
Prove that
eax - ebx
x
b.
Lim
x→a
Evaluate the following limit:
lx – al
x-a
c.
If MR = AR (1 – 1/ nd ) , where nd is price elasticity of demand, find the limit of MR when the
elasticity tends to infinity and interpret phenomenon.
1–x
1+x
prove that (1 - x2)dy/dx + y = 0
d.
If y =
e.
If xy = ex - y , prove that dy/dx = log x / (1 + log x)2
f.
Differentiate xx with respect to x.

3. g.
For the equation a x2 + 2hxy + b y2 = 1 , verify that (dy/dx) x (dx/dy) = 1
Question No – 2
a.
Examine for maximum or minimum values of the function (1 - x) 2 ex
b.
The difference of two numbers is 100. The square of the larger one exceeds five times the
square of the smaller one by an amount which is maximum. Find the numbers.
c.
A monopolist’s demand curve is: p = 200 – 5q. Find marginal revenue function. What is the
relationship between the slopes of the average and marginal revenue curves? At what price is marginal
revenue is zero?
d.
Find the elasticity of demand and supply at equilibrium price for demand function p =
and supply function x = 2p – 10, where p is price and x is quantity.
e.
100 – x 2
If n1 and n2 be the price elasticities of supply laws p = e x and p = (ex /x) respectively. Show that
n1 n2 = n2 – n1
Question No – 3
a.
A steel plant is capable of producing x tones per day of a low-grade steel and y tones per day of
a high-grade steel, where y = (40 – 5x)/(10 – x). If the fixed market price of low-grade steel is half of that
of high-grade, show that about 5.5 tones of low-grade steel are produced per day for maximum total
revenue.
b.
If f(x, y) = (x3 + y3)/(x – y), (x, y) ≠ (0, 0) and f(0, 0) = 0, then show that f xy ≠ fyx at origin.
c.
If xy + yx = ab , find dy/dx
d.
Verify Euler’s theorem for the function z= x3 log(y/x)
Question No – 4
a.
≠0.
Using calculus, show that: MR = AR (1 – 1/ nd ) and verify for the demand function x = (a-p)/b , b

4. b.
The production function of a firm is given as: Q = 8 L K – L 2 –K2, L >0, K >0. Find the marginal
productivities of labour (L) and Capital (C). and also show that
L δQ/δL + K δQ/δK = 2Q.
c.
Evaluate the following integrals:
∫dx/(x2 – 16) and ∫(x+1)dx/(3+2x-x2)