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In order for a proof by mathematical induction to be valid, the basis statement must be true for
46. “Theorem:” For every integer
“Proof (by mathematical induction): Suppose the theorem is true for an integer k, where
Now
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Discrete Mathematics With Applications
- Please help ASAP on all asked questions. Please show all work and steps.arrow_forwardPlease help ASAP on all asked questions. Please show all work and steps. Please circle the final answer.arrow_forward8d6 عدد انباء Q/ Design a rectangular foo A ing of B-2.75m to support a column of dimensions (0.46 x 0.46) m, dead load =1300kN, live load = 1300kN, qa-210kPa, fc' 21 MPa, fy- 400 MPa. =arrow_forward
- 5 of 5 (i) Let a discrete sample space be given by Ω = {ω1, 2, 3, 4}, Total marks 12 and let a probability measure P on be given by P(w1) 0.2, P(w2) = 0.2, P(w3) = 0.5, P(w4) = 0.1. = Consider the random variables X1, X2 → R defined by X₁(w3) = 1, X₁(4) = 1, X₁(w₁) = 1, X₁(w2) = 2, X2(w1) = 2, X2(w2) = 2, X2(W3) = 1, X2(w4) = 2. Find the joint distribution of X1, X2. (ii) [4 Marks] Let Y, Z be random variables on a probability space (N, F, P). Let the random vector (Y, Z) take on values in the set [0,1] × [0,2] and let the joint distribution of Y, Z on [0,1] × [0,2] be given by 1 dPy,z(y, z) (y²z + y²²) dy dz. Find the distribution Py of the random variable Y. [8 Marks]arrow_forwardRefer to page 40 for solving a time-optimal control problem. Instructions: • Formulate the problem by minimizing the time to reach a target state. • Apply Pontryagin's Maximum Principle to derive the optimal control and switching conditions. • Solve explicitly for the control and state trajectories. Include clear diagrams to visualize the solution. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardQ1/ Two plate load tests were conducted in a C-0 soil as given belo Determine the required size of a footing to carry a load of 1250 kN for the same settlement of 30 mm. Size of plates (m) Load (KN) Settlement (mm) 0.3 x 0.3 40 30 0.6 x 0.6 100 30 Qx 0.6zarrow_forward
- Total marks 16 5. Let (,,P) be a probability space and let X : → R be a random variable whose probability density function is given by f(x) = }}|x|e¯|×| for x Є R. (i) (ii) Find the characteristic function of the random variable X. [8 Marks] Using the result of (i), calculate the first two moments of the random variable X, i.e., E(X") for n = 1, 2. (iii) What is the variance of X? [6 Marks] [2 Marks]arrow_forwardRefer to page 12 for a problem on solving a homogeneous differential equation. Instructions: • Simplify the equation into a homogeneous form. Use appropriate substitutions to reduce complexity. Solve systematically and verify the final result with clear back-substitutions. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 36 for solving a bang-bang control problem. Instructions: • Formulate the problem, identifying the control constraints. • • Apply Pontryagin's Maximum Principle to derive the switching conditions. Clearly illustrate the switching points in the control trajectory. Verify the solution satisfies the optimality criteria. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
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