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’s 3 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.
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Official Gujarat Technological University (GTU) examination paper and step-by-step solutions 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.
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