Question 1: If y is inversely proportional to x and y = 6 when x = 4, find the constant of proportionality k and write the general formula for y in terms of x.

Answer: k=24; y=24/x

Question 2: Calculate y when x=8, given that y=12 when x=4, assuming y ∝ 1/x.

Answer: 6

Question 3: Plot the graph of y = 36/x on the provided grid. Describe the key features of the graph.

Answer: The graph is a rectangular hyperbola with asymptotes at x=0 and y=0, decreasing in both quadrants, passing through points like (1,36), (2,18), (3,12), etc.

Question 4: A machine's power consumption y is inversely proportional to its speed x. If at x=3, y=20, what is y when x=6? Show your working.

Answer: y=10

Question 5: Explain why the statement 'y is directly proportional to x' is incorrect for inverse proportion problems.

Answer: Because in inverse proportion, y varies as 1/x, not directly with x; when x increases, y decreases.

Question 6: A student claims that if y ∝ 1/x and y=8 when x=2, then y must be 4 when x=4. Is this correct? Explain and verify.

Answer: Yes, because y=k/x. Find k: 8= k/2 → k=16. When x=4, y=16/4=4, so the claim is correct.

Question 7: Identify the common mistake made when students assume y ∝ x instead of y ∝ 1/x in inverse proportion problems. How can this error be avoided?

Answer: A common mistake is assuming y increases as x increases, which contradicts inverse variation. To avoid this, remember y decreases as x increases in inverse proportion.

Question 8: Given the inverse proportionality y= k/x, determine the value of k if y=5 when x=2, then find y when x=10.

Answer: k=10; y=1

Question 9: A real-world example involves a car traveling at a certain speed where the travel time is inversely proportional to the speed. If the time is 4 hours at 60 km/h, what is the time at 120 km/h? Explain your reasoning.

Answer: Since time ∝ 1/speed, t1 * s1 = t2 * s2. 4*60= t2*120 → t2= (4*60)/120=2 hours.

Question 10: Challenge: Derive the formula for y in terms of x if y ∝ 1/x and y=10 when x=5, then verify with a different x value.

Answer: k=50; y=50/x. For x=10, y=50/10=5.

Question 11: Identify and correct the misconception: 'y = 1/x is always positive.' Is this true? Why or why not?

Answer: No, y=1/x is positive when x>0, but negative when x<0. The statement is only true for x>0.