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QUESTION PAPER
Subject with Code: Mathematics-IV(MATH-201-F)
Time:- 3 hours Max Marks:- 100
Note: Q No-1 is compulsory. Attempt one question from each Section. All questions of Sections A, B,C and D carry equal marks.
Q1 . Objective type/ short-answer type questions.(2.5 x 8 = 20 marks)
Find the Fourier Series to represent f(x)=x^2-2,when-2≤x≤2.
Find the Fourier sine transform of e^(-ax)
Determine the analytic function whose real part is x3-3xy2+3×2-3y2+1
Expand f(x) = |x| as a fourier series-π<x<π.
Suppose that X has a Poisson Distribution. If P(X=2)=2P(X=1) Find P(X=3)
Taylor’s series expansion of 1/(z-2) in |z|<1 is………………
Show that log(6+8i)= log 10 + i〖 tan〗^(-1) 4/3
A Company Produce two type of model M1 and M2. Each M1 model require 4 hours of grinding and 2 hours of polishing where as each M2 model requires 2 hours of grinding and 5 hours of polishing. The company has 2 grinders and 3 polishers. Each grinder works for 40 hours a week and each polisher works for 60 hours a week. Profit on an M1 model is Rs. 3 and on an M2 model is Rs. 4. Whatever is produce in a week is sold in the market. How should the company allocate its production capacity to the two types of models so that it may make the maximum profit in a week? Formulate the problem as an LPP.
SECTION-A
Q2 (i) Show that for -π<x<π
cos c x=sin(cπ)/π [1/c-(2c cosx)/(c^2-1^2 )+(2c cosx)/(c^2-2^2 )-………]
Where c is non integral. Hence deduce that
π cosec (cπ)=∑_(n=0)^∞▒〖〖(-1)〗^n [1/(n+c)+1/(n+1-c)] 〗
(ii)Expand f(x) = |cosx| as a fourier series-π<x<π.
Q3 (i ) Find the Fourier Transform of the Function f(x)= e^(-x^2/2) ,-∞<x<∞
(ii) Solve the following integral equation ∫_0^∞▒〖f(x) cos〖λx dx =e^(-x) ,〗 〗 λ>0
SECTION-B
Q4 (i) If u=logtan〖(π/4〗 +θ/2) , then prove that
tanh〖u/2〗=tan〖θ/2〗
cosh〖u=secθ 〗
(ii) If f(z) is an analytic function of z , prove that
( ∂^(2 )/〖∂x〗^2 +∂^(2 )/〖∂y〗^2 )| f(z)|^2=4|f^’ (z)|^2
Q5 (i) Show that Evaluate the integral ∮_c▒(〖(cos〗〖πz^2 〗+sin〖πz^2 〗)dz)/((z-2)〖(z-1)〗^2 ) c: |z|=3 by Cauchy’s integral formula.
(ii) If tan〖(θ+iφ)= tan〖α+i secα〗 〗 show
e^(2φ )=∓cot〖α/2〗 and 2θ=(n+1/2)π+α
SECTION-C
Q6 (i) Evaluate the given integral ∫_0^2π▒〖cos3θ/(5-4cosθ) dθ〗 by Contour Integration.
(ii)Evaluate the integral by Cauchy integral formula ∫_C▒(4-3z)/(z(z-1)(z-2)) dz
where C is the Circle |z|=3/2.
Q7 (i) In a bolt factory, there are four machines A, B, C and D manufacturing 20%, 15%, 25% and 40% of the total output respectively. Of their outputs 5%,4%,3% and 2% in the same order,
are defective bolts. A bolt is chosen at random from the factory’s production and is found defective. What is the probability that the bolt was manufactured by machine A or machine D?
(ii) If the variance of the Poisson distribution is 2, find the probabilities for r=1, 2, 3, 4 from the recurrence relation of the Poisson distribution.
SECTION-D
Q8(i) Using graphical method, solving the following LPP
Maximize Z=2x_1+3x_2
x_1-x_2 ≤2 , x_1+x_2≤4,
x_1,x_2≥0
(ii) Solve the LPP by simplex method:
Maximize z = 10x_1+x_2+2x_3
Subject to the constraints
x_1+x_2-2x_(3 ) ≤10 , 4x_1+x_2+x_3 ≤20
x_1,x_2,x_3≥0
Q9 (i)Solve the LPP by dual simplex method:
Minimize z= 2x_(1 )+2x_2+ 4x_3
Subject to the constraints:
〖2x〗_1+3x_(2 )+5x_3 ≥2 ,
3x_1+ x_2+7x_3≤3,〖 x〗_1+4x_2+6x_3 ≤5 〖 x〗_1,x_(2 , ) x_3≥0
(ii) Obtain the Dual of
Maximize z = 20x_1+30x_2
Subject to the constraints
3x_1+3x_2 ≤36 , 5x_1+2x_2≤50, 〖2x〗_1+〖6x〗_2≤60
x_1,x_2≥0