Question

If the sum of three numbers in A.P. is $24$ and their product is $440$ , then
I. The numbers are $5,8,11$.
II. The numbers are $11,8,5$.

• A. Both I and II are true
• B. I is true and II is false
• C. I is false and II is true
• D. Neither I nor II is true

Answer 1

Answer: (A)

Let the three numbers are $a-d,a,a+d$.
Given, sum of the numbers $=24$
$\therefore a-d+a+a+d=24$
$\Rightarrow 3a=24$
$\Rightarrow a=8$
and product of the numbers $=440$
$\therefore (a-d)a(a+d)=440$
$\Rightarrow a\left(a^2-d^2\right)=440$
$\Rightarrow 8\left(64-d^2\right)=440$
$\Rightarrow 64-d^2=55$
$\Rightarrow d^2=64-55$
$\Rightarrow d^2=9$
$\Rightarrow d=\pm 3$
When, $a=8$ and $d=3$, then numbers are
$a-d=8-3=5$
$a=8$ and $a+d=8+3=11$
$\Rightarrow 5,8,11$
When, $a=8$ and $d=-3$, then numbers are
$a-d=8+3=11$
$a=8$
and $a+d=8-3$
$=5$
$\Rightarrow 11,8,5$
Hence, the numbers are $5,8,11$ or $11,8,5$.