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 real 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^2-8x+a^2-6a=0 are real and distinct, then find all possible value of adot

If the both the roots of the quadratic equation x^(2)-2kx+k^(2)+k-5=0 are less than 5, then k

If a, b are real then show that the roots of the equation (a-b)x^(2)-6(a+b)x-9(a-b)=0 are real and unequal.

How many roots of the equation 3x^4+6x^3+x^2+6x+3=0 are real ?

Find the real roots of the equation . x^2+5|x|+6=0

If the roots of the equation (c^(2)-ab)x^(3)-2(a^(2)-bc)x+b^(2)-ac=0 are real and equal prove that either a=0 (or) a^(3)+b^(3)+c^(3)=3"abc" .

For what real values of a do the roots of the equation x^2-2x-(a^2-1)=0 lie between the roots of the equation x^2-2(a+1)x+a(a-1)=0.

If roots of equation x^2+2c x+a b=0 are real and unequal, then prove that the roots of x^2-2(a+b)x+a^2+b^2+2c^2=0 will be imaginary.

Find a so that the sum and product of the roots of the equation 2x^(2)+(a-3)x+3a-5=0 are equal is ……