Question 9

xy equals II and III

Given the equations 1/m + 1/r = x and m + r = y, we can express xy in terms of m and r. By manipulating the first equation, we find that x can be rewritten as (m + r) / (mr), which leads to yx = (m + r)² / (mr). This shows that the product xy leads us to two significant results, corresponding to options II and III.

A) None

This option is incorrect as there are valid relationships derived from the given equations. The product xy can indeed be expressed in terms of m and r, leading to meaningful results.

B) I only

Option I alone does not capture the complete relationships produced by the equations. While it may seem plausible, it fails to account for the additional results obtained when considering both II and III.

C) II only

While II is one of the correct outcomes derived from the equations, stating only II does not encompass the full scope of the results. The relationship defined in III is also valid and must be included in the answer.

D) III only

Similar to option C, stating only III does not provide a complete picture. Although III is a correct result, it ignores the fact that II is also a consequence of the given equations.

E) II and III

This option is correct as both II and III are valid relationships derived from the expressions for x and y. The manipulation of the given equations confirms that xy indeed leads to both outcomes.

Conclusion

The correct answer is E, as both II and III accurately reflect the relationships established by the equations involving x and y. All other options fail to recognize the full extent of the results that can be derived from the original equations, thereby making E the only comprehensive choice.