4. (a) Divide 20 into 4 parts which are in A.P. and such that. the product of the first and fourth is to the product o the secondi.and third in the ratio 2:3.

Explanation

Let the four part be ` a - 3d , a - d , a + d , a + 3d `
Now , ` (a - 3d) + (a - d) + (a + d) + (a + 3d) = 20 `
` rArr 4a = 20 `
` rArr a = 5 `
Thus parts become ` 5 - 3d , 5 - d , 5 + d , 5 + 3d `
Also ` (( 5- 3d) (5 + 3d))/((5 - d)(5 + d)) = (2)/(3)`
` rArr (25 - 9d^(2))/(25 -d^(2)) = (2)/(3)`
` rArr 75 - 27d^(2) = 50 - 2d^(2)`
` rArr 25 = 25 d^(2)`
` rArr d = pm 1 `
Hence numbers are 2, 4, 6, 8.