a) Solve: dydx + 1 + 2y1 2x = 0 . [2] b) Calculate the mean deviation from mean of the data: 3, 5, 9, 11, 7, 6. [2] c) The chance that A can solve the problem is 13 and the chance that B can solve…

a) Solution of the Differential Equation

Given: $$\frac{dy}{dx}+\frac{1+\cos 2y}{1-\cos 2x}=0$$

Step 1: Rewrite the equation $$\frac{dy}{dx}=-\frac{1+\cos 2y}{1-\cos 2x}$$

Step 2: Use trigonometric identities Recall: $$1+\cos 2y=2{\cos }^{2}y,1-\cos 2x=2{\sin }^{2}x$$ Substitute these into the equation: $$\frac{dy}{dx}=-\frac{2{\cos }^{2}y}{2{\sin }^{2}x}=-\frac{{\cos }^{2}y}{{\sin }^{2}x}$$

Step 3: Separate variables $$\frac{dy}{{\cos }^{2}y}=-\frac{dx}{{\sin }^{2}x}$$

Step 4: Integrate both sides $$\int {\sec }^{2}ydy=-\int {\csc }^{2}xdx$$ $$\tan y=\cot x+C$$ where $C$ is the constant of integration.

Final Answer: $$\boxed{\tan y=\cot x+C}$$

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b) Mean Deviation from Mean

Given data: 3, 5, 9, 11, 7, 6

Step 1: Compute the mean ($\bar{x}$) $$\bar{x}=\frac{3+5+9+11+7+6}{6}=\frac{41}{6}\approx 6.833$$

Step 2: Compute absolute deviations from the mean $$|3-6.833|=3.833,|5-6.833|=1.833,|9-6.833|=2.167,$$ $$|11-6.833|=4.167,|7-6.833|=0.167,|6-6.833|=0.833$$

Step 3: Compute the mean deviation $$\text{Mean Deviation}=\frac{3.833+1.833+2.167+4.167+0.167+0.833}{6}$$ $$=\frac{12.8}{6}\approx 2.133$$

Final Answer: $$\boxed{2.133}$$

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c) Probability that the Problem is Solved by A and B

Given:

• Probability that A can solve the problem, $P(A)=\frac{1}{3}$
• Probability that B can solve the problem, $P(B)=\frac{2}{3}$

Assumption: A and B are independent events (their abilities are independent).

Step 1: Find the probability that both solve the problem Since A and B are independent: $$P(A\text{ and }B)=P(A)\times P(B)=\frac{1}{3}\times \frac{2}{3}=\frac{2}{9}$$

Step 2: Find the probability that the problem is solved by at least one of them $$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)=\frac{1}{3}+\frac{2}{3}-\frac{2}{9}=\frac{9}{9}-\frac{2}{9}=\frac{7}{9}$$

Step 3: Find the probability that the problem is solved by both The question asks for the probability that the problem is solved by A and B, which is already computed in Step 1.

Final Answer: $$\boxed{\frac{2}{9}}$$