Find three numbers a, b,c between 2 & 18 such that; O their sum is 25 @ the numbers 2, a, b are consecutive terms of an AP & Q.3 the numbers b?c?18 are consecutive terms ofa GP

Options

A. 500
B. 450
C. 720
D. 480

Answer

D

Explanation

We have,
a+b+c=25
2a=b+2
`c^(2)=18b`
Elminating a from (1) and (2), we have
`b=16-(2c)/3`
Then from (3),
`c^(2)=18(16-(2c)/3)`
or `c^(2)+12c-18xx16=0`
or `(c-12)(c+24)=0`
Now, c=-24 is not possible since it does not lie between 2 and 18. Hence, c=12. Then from (3), b=8 and finally from (2),a = 5.
Thus, a=5,b=8 and c=12. Hence, abc`=5xx8xx12=480`.
Also,equation `ax^(2)+bx+c=0` is `5x^(2)+8x+12=0`, which has imaginary roots.
If a,b,c are roots of the equation `x^(3)+qx^(2)+rx+s=0`, then the sum of product of roots taken two at a time is `r=5xx8+5xx12+8xx12=196`.