Correct option is (B) 3, 5, 7
Let a - d, a, a+d are required three numbers in A.P.
Given that their sum is 15.
\(\therefore\) (a - d) + a + (a+d) = 15
\(\Rightarrow3a=15\)
\(\Rightarrow a=\frac{15}3=5\)
Also given that the sum of the squares of the extremes is 58.
\(\therefore(a-d)^2+(a+d)^2=58\)
\(\Rightarrow(5-d)^2+(5+d)^2=58\) \((\because a=5)\)
\(\Rightarrow 2(5^2+d^2)=58\) \((\because(a-b)^2+(a+b)^2=2(a^2+b^2))\)
\(\Rightarrow5^2+d^2=\frac{58}2=29\)
\(\Rightarrow d^2=29-5^2=29-25\)
\(=4=2^2\)
\(\Rightarrow d=2\)
\(\therefore a-d=5-2=3\)
& \(a+d=5+2=7\)
Hence, the required numbers are 3, 5 and 7.