MAT Exam 2021 Paper

Concept:Consecutive integers must have a common difference of exactly $1$ between successive terms.Explanation:From Statement I, the average of $x$, $y$ and $z$ is $y$.Therefore, $\frac{x+y+z}{3} = y$, which gives $x + y + z = 3y$.Simplifying, $x + z = 2y$.This only shows that $y$ is the arithmetic mean of $x$ and $z$, so $x, y, z$ form an arithmetic progression.But an arithmetic progression need not be consecutive integers.For example, $1, 3, 5$ satisfies $x+z=2y$ but is not consecutive.Similarly, Statement II gives $y - x = z - y$, which also simplifies to $2y = x + z$.Thus, Statement II provides exactly the same information as Statement I.Even when both statements are taken together, no information about the actual values or common difference is available.Therefore, the data are insufficient to conclude whether $x, y$ and $z$ are consecutive.Answer:Option (d): Statements I and II together are not sufficient, and additional data is needed to answer the question.