Location of Roots

IMPORTANT

Location of Roots: Overview

This topic covers concepts, such as Quadratic Equation with Both the Roots Less than 'k', Quadratic Equation with Both the Roots Greater than 'k', Quadratic Equation with 'k' Lying between the Roots, etc.

Important Questions on Location of Roots

MEDIUM
IMPORTANT

For how many integral values of k, the equation x2-4x+k=0, where k is an integer has real roots and both of them lie in the interval 0, 5?

MEDIUM
IMPORTANT

If α+1α, β+1β are zeros of fx=x2-5x-a, where α, β0, for all xR, then the complete set of values of a is

MEDIUM
IMPORTANT

Let a,b,cR and a0 be such that a+c2<b2, then the quadratic equation ax2+bx+c=0 has 

HARD
IMPORTANT

The value of a for which the quadratic expression ax2+2a-3x-6 is positive for exactly three integral values of x is

MEDIUM
IMPORTANT

If the quadratic equation f(x)=px2-qx+r=0 has two distinct roots in (0, 2) where p, q, rN and f(1)=-1then the minimum value of p is

MEDIUM
IMPORTANT

The values of k for which each root of the equation x2-6kx+2-2k+9k2=0 is greater than 3 , always satisfy the inequality

HARD
IMPORTANT

Given fx=x2+a+2x+a2-a+2 and f-2<0.  Then a lies in the interval:

EASY
IMPORTANT

The values of 'a' for which the roots of the equation x2+x+a=0 are real and exceed 'a' are -

MEDIUM
IMPORTANT

For the given equation 2x3+3x+k=0, what are the values of k so that it contains two distinct real roots in the interval 0,1:

HARD
IMPORTANT

The range of a for which the equation x2+ax-4=0 has its smaller root in the interval -1, 2 is

HARD
IMPORTANT

Let the values of a for which one root of the equation a-5x2-2ax+a-4=0 is smaller than 1 and the other greater than 2 be (m, n). Find n-m.

MEDIUM
IMPORTANT

If p<q<r<s then the equation x-px-r+5x-qx-s=0 has

MEDIUM
IMPORTANT

If 2 lies between both the roots of x2-λ+1x+λ2+λ-8=0, then 

MEDIUM
IMPORTANT

Consider the function f(x)=x34-sinπx+3

HARD
IMPORTANT

If the roots of the quadratic equation 4p-p2-5x2-(2p-1)x+3p=0 lie on either side of unity, then the number of integral values of p is

HARD
IMPORTANT

The values of a for which 2x2-22a+1x+aa+1=0 may have one root less than a and other root greater than a is

MEDIUM
IMPORTANT

If x2-2x+2sinα=0 has unique root in (-1,1), then length of largest continuous interval of α in [0,2π] is

HARD
IMPORTANT

If e1 and e2 are the roots of the equation x2-ax+2=0 where e1,e2 are the eccentricities of an ellipse and hyperbola respectively then the value of a belongs to

HARD
IMPORTANT

If roots of equation k2-4k+5x2+2k-1x-3k=0 lie on either sides of unity, then number of integral values of k is

EASY
IMPORTANT

Let fx=x2+ax+a. If the equation fx=0 has exactly one root in [1,2], then number of possible integral values of a is