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 3 a =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 $11,8,5$.