{"id":180,"date":"2013-06-05T17:57:56","date_gmt":"2013-06-05T17:57:56","guid":{"rendered":"http:\/\/studentsuvidha.in\/?p=180"},"modified":"2013-06-05T17:57:56","modified_gmt":"2013-06-05T17:57:56","slug":"maths-3-solved-papers-mdu-btech-sample-papers","status":"publish","type":"post","link":"https:\/\/studentsuvidha.com\/site\/maths-3-solved-papers-mdu-btech-sample-papers\/","title":{"rendered":"MATHS 3 SOLVED PAPERS MDU BTECH SAMPLE PAPERS"},"content":{"rendered":"

CLICK HERE TO DOWNLOAD MATHS 3 SOLVED PAPERS
\n<\/a><\/strong><\/p>\n

 <\/p>\n

QUESTION PAPER<\/strong>
\nSubject with Code: Mathematics-IV(MATH-201-F)
\nTime:- 3 hours\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 Max Marks:- 100
\nNote: Q No-1 is compulsory. Attempt one question from each Section. All questions of Sections A, B,C and D carry equal marks.<\/strong>
\nQ1 . Objective type\/ short-answer type questions.(2.5 x 8 = 20 marks)<\/strong>
\nFind the Fourier Series to represent f(x)=x^2-2,when-2\u2264x\u22642.
\nFind the Fourier sine transform of e^(-ax)
\nDetermine the analytic function whose real part is x3-3xy2+3×2-3y2+1
\nExpand f(x) = |x|\u00a0\u00a0 as a fourier series-\u03c0<x<\u03c0.
\nSuppose that X has a Poisson Distribution. If P(X=2)=2P(X=1) Find P(X=3)
\nTaylor\u2019s series expansion of 1\/(z-2)\u00a0 in |z|<1 is\u2026\u2026\u2026\u2026\u2026\u2026
\nShow that log(6+8i)= log 10 + i\u3016\u00a0 tan\u3017^(-1)\u00a0 4\/3<\/p>\n

\nA 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.\u00a0 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.\n<\/p>\n

\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 SECTION-A<\/strong><\/p>\n

\nQ2\u00a0 (i) Show that for -\u03c0<x<\u03c0<\/p>\n

cos c x=sin\u2061(c\u03c0)\/\u03c0 [1\/c-(2c cos\u2061x)\/(c^2-1^2 )+(2c cos\u2061x)\/(c^2-2^2 )-\u2026\u2026\u2026]
\nWhere c is non\u00a0 integral. Hence deduce that
\n\u03c0 cosec (c\u03c0)=\u2211_(n=0)^\u221e\u2592\u3016\u3016(-1)\u3017^n [1\/(n+c)+1\/(n+1-c)] \u3017<\/p>\n

\n(ii)Expand f(x) = |cosx|\u00a0\u00a0 as a fourier series-\u03c0<x<\u03c0.<\/p>\n

\nQ3 (i ) Find the Fourier Transform of the Function f(x)= e^(-x^2\/2)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ,-\u221e<x<\u221e
\n(ii) Solve the following integral equation \u222b_0^\u221e\u2592\u3016f(x)\u00a0 cos\u2061\u3016\u03bbx dx =e^(-x)\u00a0 ,\u3017 \u3017 \u03bb>0<\/p>\n

SECTION-B<\/strong><\/p>\n

\nQ4 (i) If u=log\u2061tan\u2061\u3016(\u03c0\/4\u3017\u00a0 +\u03b8\/2) , then prove that
\ntanh\u2061\u3016u\/2\u3017=tan\u2061\u3016\u03b8\/2\u3017
\ncosh\u2061\u3016u=sec\u2061\u03b8 \u3017<\/p>\n

\n(ii) If f(z) is an analytic function of z , prove that
\n( \u2202^(2 )\/\u3016\u2202x\u3017^2\u00a0 +\u2202^(2 )\/\u3016\u2202y\u3017^2 )| f(z)|^2=4|f^’ (z)|^2
\nQ5 (i) Show that\u00a0\u00a0 Evaluate the integral\u00a0\u00a0 \u222e_c\u2592(\u3016(cos\u3017\u2061\u3016\u03c0z^2 \u3017+sin\u2061\u3016\u03c0z^2 \u3017)dz)\/((z-2)\u3016(z-1)\u3017^2 )\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 c:\u00a0 |z|=3 by Cauchy\u2019s integral formula.
\n(ii) If tan\u2061\u3016(\u03b8+i\u03c6)= tan\u2061\u3016\u03b1+i sec\u03b1\u3017 \u3017\u00a0 show
\ne^(2\u03c6 )=\u2213cot\u2061\u3016\u03b1\/2\u3017\u00a0 and\u00a0 2\u03b8=(n+1\/2)\u03c0+\u03b1<\/p>\n

SECTION-C<\/strong><\/p>\n

\nQ6 (i) Evaluate the given integral \u222b_0^2\u03c0\u2592\u3016cos\u20613\u03b8\/(5-4cos\u03b8) d\u03b8\u3017 by Contour Integration.
\n(ii)Evaluate the integral by Cauchy integral formula \u222b_C\u2592(4-3z)\/(z(z-1)(z-2)) dz
\nwhere C is the Circle |z|=3\/2.<\/p>\n

\nQ7 (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,
\nare defective bolts. A bolt is chosen at random from the factory\u2019s production and is found defective. What is the probability that the bolt was manufactured by machine A or machine D?
\n(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.<\/p>\n

SECTION-D<\/strong><\/p>\n

\nQ8(i) Using graphical method, solving the following LPP
\nMaximize\u00a0 Z=2x_1+3x_2
\nx_1-x_2\u00a0 \u22642 ,\u00a0\u00a0 x_1+x_2\u22644,
\nx_1,x_2\u22650<\/p>\n

\n(ii) Solve the LPP by simplex method:
\nMaximize z = 10x_1+x_2+2x_3
\nSubject to the constraints
\nx_1+x_2-2x_(3 )\u00a0 \u226410 ,\u00a0\u00a0 4x_1+x_2+x_3\u00a0 \u226420
\nx_1,x_2,x_3\u22650<\/p>\n

\nQ9 (i)Solve the LPP by dual simplex method:
\nMinimize z=\u00a0\u00a0 2x_(1 )+2x_2+ 4x_3
\nSubject to the constraints:
\n\u30162x\u3017_1+3x_(2\u00a0 )+5x_3\u00a0 \u22652 ,
\n3x_1+ x_2+7x_3\u22643,\u3016\u00a0\u00a0\u00a0\u00a0 x\u3017_1+4x_2+6x_3\u00a0 \u22645\u00a0\u00a0\u00a0\u00a0 \u3016\u00a0\u00a0\u00a0\u00a0\u00a0 x\u3017_1,x_(2 ,\u00a0\u00a0 ) x_3\u22650<\/p>\n

\n(ii) Obtain the Dual of
\nMaximize z = 20x_1+30x_2
\nSubject to the constraints
\n3x_1+3x_2\u00a0 \u226436 ,\u00a0\u00a0 5x_1+2x_2\u226450,\u00a0 \u30162x\u3017_1+\u30166x\u3017_2\u226460
\nx_1,x_2\u22650<\/p>\n

CLICK HERE TO DOWNLOAD MATHS 3 SOLVED PAPERS <\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

CLICK HERE TO DOWNLOAD MATHS 3 SOLVED PAPERS   QUESTION PAPER Subject with Code: Mathematics-IV(MATH-201-F) Time:- 3 hours\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 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\u2264x\u22642. 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|\u00a0\u00a0 as a fourier series-\u03c0<x<\u03c0. Suppose that X has a Poisson Distribution. If P(X=2)=2P(X=1) Find P(X=3) Taylor\u2019s series expansion of 1\/(z-2)\u00a0 in |z|<1 is\u2026\u2026\u2026\u2026\u2026\u2026 Show that log(6+8i)= log 10 + i\u3016\u00a0 tan\u3017^(-1)\u00a0 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.\u00a0 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. \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 SECTION-A Q2\u00a0 (i) Show that for -\u03c0<x<\u03c0 cos c x=sin\u2061(c\u03c0)\/\u03c0 [1\/c-(2c cos\u2061x)\/(c^2-1^2 )+(2c cos\u2061x)\/(c^2-2^2 )-\u2026\u2026\u2026] Where c is non\u00a0 integral. Hence deduce that \u03c0 cosec (c\u03c0)=\u2211_(n=0)^\u221e\u2592\u3016\u3016(-1)\u3017^n [1\/(n+c)+1\/(n+1-c)] \u3017 (ii)Expand f(x) = |cosx|\u00a0\u00a0 as a fourier series-\u03c0<x<\u03c0. Q3 (i ) Find the Fourier Transform of the Function f(x)= e^(-x^2\/2)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ,-\u221e<x<\u221e (ii) Solve the following integral equation \u222b_0^\u221e\u2592\u3016f(x)\u00a0 cos\u2061\u3016\u03bbx dx =e^(-x)\u00a0 ,\u3017 \u3017 \u03bb>0 SECTION-B Q4 (i) If u=log\u2061tan\u2061\u3016(\u03c0\/4\u3017\u00a0 +\u03b8\/2) , then prove that tanh\u2061\u3016u\/2\u3017=tan\u2061\u3016\u03b8\/2\u3017 cosh\u2061\u3016u=sec\u2061\u03b8 \u3017 (ii) If f(z) is an analytic function of z , prove that ( \u2202^(2 )\/\u3016\u2202x\u3017^2\u00a0 +\u2202^(2 )\/\u3016\u2202y\u3017^2 )| f(z)|^2=4|f^’ (z)|^2 Q5 (i) Show that\u00a0\u00a0 Evaluate the integral\u00a0\u00a0 \u222e_c\u2592(\u3016(cos\u3017\u2061\u3016\u03c0z^2 \u3017+sin\u2061\u3016\u03c0z^2 \u3017)dz)\/((z-2)\u3016(z-1)\u3017^2 )\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 c:\u00a0 |z|=3 by Cauchy\u2019s integral formula. (ii) If tan\u2061\u3016(\u03b8+i\u03c6)= tan\u2061\u3016\u03b1+i sec\u03b1\u3017 \u3017\u00a0 show e^(2\u03c6 )=\u2213cot\u2061\u3016\u03b1\/2\u3017\u00a0 and\u00a0 2\u03b8=(n+1\/2)\u03c0+\u03b1 SECTION-C Q6 (i) Evaluate the given integral \u222b_0^2\u03c0\u2592\u3016cos\u20613\u03b8\/(5-4cos\u03b8) d\u03b8\u3017 by Contour Integration. (ii)Evaluate the integral by Cauchy integral formula \u222b_C\u2592(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\u2019s 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\u00a0 Z=2x_1+3x_2 x_1-x_2\u00a0 \u22642 ,\u00a0\u00a0 x_1+x_2\u22644, x_1,x_2\u22650 (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 )\u00a0 \u226410 ,\u00a0\u00a0 4x_1+x_2+x_3\u00a0 \u226420 x_1,x_2,x_3\u22650 Q9 (i)Solve the LPP by dual simplex method: Minimize z=\u00a0\u00a0 2x_(1 )+2x_2+ 4x_3 Subject to the constraints: \u30162x\u3017_1+3x_(2\u00a0 )+5x_3\u00a0 \u22652 , 3x_1+ x_2+7x_3\u22643,\u3016\u00a0\u00a0\u00a0\u00a0 x\u3017_1+4x_2+6x_3\u00a0 \u22645\u00a0\u00a0\u00a0\u00a0 \u3016\u00a0\u00a0\u00a0\u00a0\u00a0 x\u3017_1,x_(2 ,\u00a0\u00a0 ) x_3\u22650 (ii) Obtain the Dual of Maximize z = 20x_1+30x_2 Subject to the constraints 3x_1+3x_2\u00a0 \u226436 ,\u00a0\u00a0 5x_1+2x_2\u226450,\u00a0 \u30162x\u3017_1+\u30166x\u3017_2\u226460 x_1,x_2\u22650 CLICK HERE TO DOWNLOAD MATHS 3 SOLVED PAPERS<\/p>\n","protected":false},"author":1,"featured_media":597,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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