Question : If $y$ is an integer, then $(y^{3}-y)$ is always multiple of _____.
Option 1: 5
Option 2: 7
Option 3: 9
Option 4: 6
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Correct Answer: 6
Solution : $y^{3}-y=y(y^{2}-1)=y(y-1)(y+1)$ This product includes three consecutive integers: $y-1$, $y$, and $y+1$ In any set of three consecutive integers, at least one number is always a multiple of 3 and one is always even. Hence, the correct answer is 6.
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Question : If $\left(3 y+\frac{3}{y}\right)=8$, then find the value of $\left(y^2+\frac{1}{y^2}\right)$.
Option 1: $5\frac{1}{9}$
Option 2: $4\frac{5}{6}$
Option 3: $7\frac{1}{9}$
Option 4: $9\frac{1}{9}$
Question : If $A:B=7:9$ and $B:C =3:5$, then $A:B:C$ is equal to:
Option 1: $7:9:5$
Option 2: $21:35:45$
Option 3: $7:9:15$
Option 4: $7:3:15$
Question : If $A:B=3:4$ and $B:C=6:5$, then $C:A$ is
Option 1: $10:9$
Option 2: $9:10$
Option 3: $8:9$
Option 4: $9:8$
Question : If $4x^2+y^2 = 40$ and $xy=6$, then the value of $2x-y$ is:
Option 1: 1
Option 2: 3
Option 3: 4
Option 4: 2
Question : If $X$ is 20% less than $Y$, then find the values of$\frac{Y–X}{Y}$ and $\frac{X}{X–Y}$.
Option 1: $\frac{1}{5}$ and $-4$
Option 2: $5$ and $-\frac{1}{4}$
Option 3: $\frac{2}{5}$ and $-\frac{5}{2}$
Option 4: $\frac{3}{5}$ and $-\frac{5}{3}$
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