Home
Class 12
MATHS
If the roots of the equation x^2-2a x+a^...

If the roots of the equation `x^2-2a x+a^2-a-3=0` are ra and less than 3, then

A

`a lt 2`

B

`2 le a le 3`

C

`3 lt a le 4`

D

`a gt4`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `f(x)=x^(2)-2ax+a^(2)+a-3`
Since, both root are less than 3.
`implies a lt 3, betalt3`
`implies Sum, S=alpha+beta lt6`
`implies(alpha+beta)/(2)lt3`
`implies(2a)/(2)lt3`
`impliesa lt3" "...(i)`
Again, product, `P= alpha beta`
`impliesP lt9 impliesalpha beta lt9`
`impliesa^(2)+a-3lt9`
`impliesa^(2)+a-12lt0`
`implies(a-3)(a+4)lt0`
`implies-4 lta lt2" "..(ii)`
Again, `D=B^(2)-4ACge0`

`implies(-2a)^(2)-4.1(a^(2)+a-3)ge0`
`implies4a^(2)-4a^(2)-4a+12 ge0`
`implies-4a+12ge0impliesa le3" "...(iii)`
Again, `=" "a f (3)gt0`
`implies1.[(3)^(2)-2a(3)+a^(2)+a-3]gt0`
`implies9-6a+a^(2)+a-3gt0`
`impliesa^(2)-5a+6gt0`
`implies(a-2)(a-3)gt0`
`thereforea in (-oo,2)uu?(2,oo)" "...(iv)`
From Eqs. (i),(ii),(iii), and (iv), we get `a in (-4,2).`
Note There is correction in answer `lt 2 "should be" -4ltalt2.`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the roots of the equation x^(2)-2ax+a^(2)-a-3=0 are real and less than 3, then (a)a<2 b.2<-a<=3 c.34^(@)

If the roots of the equation x^(2)-2ax+a^(2)+a -3=0 are real and less than 3, then which one of the following is correct ?

If the roots of the equation x^(2)-4ax+4a^(2)+2a-3=0 are real and less than 3, then

If roots of the equation x^(2)-2ax+a^(2)+a-3=0 are real and less than 3 then a)a 4

Match the following {:("Column-I" , " Column -II"),("(A) if the roots of the equation" x^(2) -2ax +a^(2)+a-3=0 " are real and less than 3, then a belongs to " , "(p) a does not exist" ),("(B) If the roots of the equation " x^(2) -2ax +a^(2)+a-3=0 " are real and greater than , 3 then a belongs to " , "(q) " (3/2","3)),("(C) If exactly one root of the above equation lies in the interval (1,3) then a belongs to " , "(r) " (-1","2) cup (2","3)),("(D) IF the roots of the above equation are such that one root is greater than 3 and other smaller than 1, then a belongs to " , "(s) " (-oo","2)):}

Find the set of all values of a for which the roots of the equation x^(2)-2ax+a^(2)+a-3=0 are less than 3