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Find the value of 𝑅𝑒(𝑓(𝑧)) and 𝐼𝑚(𝑓(𝑧)) for 𝑓(𝑧) = 1 1−𝑧 at 7 + 2𝑖.
Solve the following equation by Gauss Seidel method correct up to two decimal places. 20𝑥 + 2𝑦 + 𝑧 = 30 𝑥 − 40𝑦 − 3𝑧 = −75 2𝑥 − 𝑦 + 10𝑧 = 30
Find and plot all the roots of (1 + 𝑖)¹
If 𝑓(𝑧) = 𝑢 + 𝑖𝑣 is an analytic function of 𝑧 and 𝑢 + 𝑣 = 𝑥² − 𝑦² + 2𝑥𝑦, find 𝑓(𝑧).
Show that 𝑢(𝑥, 𝑦) = 2𝑥 − 𝑥³ + 3𝑥𝑦² is harmonic in some domain and find a harmonic conjugate 𝑣(𝑥, 𝑦).
Determine and sketch the image of |𝑧| = 1 under the transformation 𝑤 = 𝑧 + 𝑖.
Determine the Mobius transformation that maps 𝑧1 = 0, 𝑧2 = 1, 𝑧3 = ∞ onto 𝑤1 = −1, 𝑤2 = −𝑖, 𝑤3 = 1 respectively.
Find the radius of convergence of the series ∑ (1 + 1 𝑛2)𝑛3 𝑧𝑛 ∞ 𝑛=1 .
Expand 𝑓(𝑧) = 1 (𝑧+1)(𝑧+3) in Laurent’s series in the interval 1 < |𝑧| <
Determine the poles of 𝑓(𝑧) = 𝑧2 (𝑧−1)²(𝑧+2) and residue at each pole. Hence evaluate ∫ 𝑓(𝑧) 𝑑𝑧𝐶 , where 𝐶 is the circle |𝑧| = 3.
Using N-R method find an iterative formula to find 𝑞𝑡ℎ root of a positive number N.
Using Gauss Elimination method solve the following system of equations. 8𝑥2 + 2𝑥3 = −7 3𝑥1 + 5𝑥2 + 2𝑥3 = 8 6𝑥1 + 2𝑥2 + 8𝑥3 = 20
Prove that ∮ 𝑑𝑧 𝑧−𝑎 = 2𝜋𝑖𝐶 , where 𝐶 is the circle |𝑧 − 𝑎| = 𝑟.
𝑥: 20 23 26 29 𝑦: 0.3420 0.3907 0.4384 0.4848
Using Runge-Kutta method of fourth order to find 𝑦(0.1), 𝑦(0.2) given 𝑑𝑦 𝑑𝑥 = 2𝑥 + 𝑦, 𝑦(0) = 1.
Evaluate ∫ 1 1+𝑥2 1 0 𝑑𝑥 using trapezoidal rule with ℎ = 0.2.
Using Langrange’s interpolation formula fit a polynomial to the given data: 𝑥: 0 1 2 5 𝑓(𝑥): 2 3 12 147
Using Taylor’s series method, find correct to four decimal place the value at 𝑦(0.1) given 𝑑𝑦 𝑑𝑥 = 𝑥² + 𝑦² and 𝑦(0) = 1.
Evaluate ∫ 𝑑𝑥 1+𝑥 3 0 with 𝑛 = 6, by using Simpson’s³ 8 rule.
𝑥: 20 30 40 50 𝑦: 512 439 346 243
Using Secant method find the real root of the equation 𝑥𝑒𝑥 − 1 = 0 correct up to three decimal places between 0 and 1.
BE Common
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Official Gujarat Technological University (GTU) examination paper for Mathematics-4 (Summer 2024, B.E. · Common Engineering, Sem 4). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.
Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.
2140001 Mathematics-4
Official GTU PDF · Summer 2024
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