The differential equation problem is on page 30 of the file below. Solve the equation step-by- step using an appropriate method such as substitution or integrating factors. Link: [https://drive.google.com/file/d/1RQ20 ZK-LSxp RyejKEMg 1t2q15dbpVLCS/view? usp=sharing]

Refer to page 22 of the following file for the trigonometric proof question. Prove the identity step-by-step using appropriate trigonometric properties. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing]

The differential equation question is on page 25 of the file linked below. Solve it using a suitable method, such as separation of variables or integrating factors. Link: [https://drive.google.com/file/d/1RQ 20 ZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing]

2. Triple Integral Applications 2a. Find the volume of the solids in the first octant which bounded by xy-plane, yz-plane, plane x+y=4 and z = x²+6. 2b. Given the region bounded in between z = r² and z =1, side by the cylinder r² ≤ 4, and in the first and second octant. Determine its volume by using cylindrical coordinate system. 2c. Solving Using Spherical Coordinates 2c. Calculate the volume of region which is bounded above by sphere of 2 x² + y²+z² = 81 and below by cone z = √x² + y² in the first octant.

(b) An otherwise fair six-sided die has been tampered with in an attempt to cheat at a dice game. The effect is that the 1 and 6 faces have a different probability of occurring than the 2, 3, 4 and 5 faces. Let θ be the probability of obtaining a 1 on this biased die. Then the outcomes of rolling the biased die have the following probability mass function. Table 1 The p.m.f. of outcomes of rolls of a biased die Outcome 1 2 3 4 5 6 Probability θ 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) θ (i) By consideration of the p.m.f. in Table 1, explain why it is necessary for θ to be such that 0 < θ < 1/2. [2] (ii) The value of θ is unknown. Data from which to estimate the value of θ were obtained by rolling the biased die 1000 times. The result of this experiment is shown in Table 2. Table 2 Outcomes of 1000 independent rolls of a biased die Outcome 1 2 3 4 5 6 Frequency 205 154 141 165 145 190 Show that the likelihood of θ based on these data is L(θ) = C θ395 (1 − 2θ) 605 ,…

Not use ai please

O State Jordan's lemma. Calculate the following integrals using complex variable techniques. 2π do (i) 1+8 cos²0' I COS x dx x²-2x+5° Leave your answers in terms of real functions of real variables.

Refer to page 138 of the shared file for the numerical methods problem. Use Newton's method to approximate the solution of the given nonlinear system of equations. Start with an initial guess provided in the problem and iterate for at least two steps. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Show the Jacobian matrix, iteration formulas, and each step of the computation.

Refer to page 128 for the heat equation problem. Solve the one-dimensional heat equation with the given initial and boundary conditions. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Use Fourier series or other appropriate methods.