Location of Roots

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JEE Main 2006

All the values of $m$ for which both roots of the equation $x^{2} - 2 m x + m^{2} - 1 = 0$ are greater than - 2 but less than 4, lie in the interval

- 2 < m < 0

(B)

m > 3

(C)

- 1 < m < 3

(D)

1 < m < 4

JEE Main 2005

LEVELJEE Main

If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval

(A)

(5, 6]

(B)

(6, \infty)

(C)

(- \infty, 4)

(D)

[4, 5]

JEE Advanced 2000

LEVELJEE Main

If $b > a$ , then the equation $(x - a) (x - b) - 1 = 0$ has

(A)

both roots in

(a, b)

(B)

both roots in

(- \infty, a)

(C)

both roots in

(b, \infty)

(D)

one root in

(- \infty, a)

and the other in

(b, \infty)

JEE Advanced 1999

LEVELJEE Main

If the roots of the equation $x^{2} - 2 a x + a^{2} + a - 3 = 0$ are real and less than 3, then

(A)

a < 2

(B)

2 \leq a \leq 3

(C)

3 < a \leq 4

(D)

a > 4

JEE Advanced 1989

LEVELJEE Main

Let $a, b, c$ be real numbers, $a \neq = 0$ . If $α$ is a root of $a^{2} x^{2} + b x + c = 0$ . $β$ is the root of $a^{2} x^{2} - b x - c = 0$ and $0 < α < β$ , then the equation $a^{2} x^{2} + 2 b x + 2 c = 0$ has a root $γ$ that always satisfies

(A)

γ = \frac{α + β}{2}

(B)

γ = α + \frac{β}{2}

(C)

γ = α

(D)

α < γ < β

JEE Advanced 2009

LEVELJEE Main

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is .........

JEE Advanced 1995

LEVELJEE Main

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

Consider the quadratic equation $(c - 5) x^{2} - 2 c x + (c - 4) = 0, c \neq = 5$ . Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is :

JEE Main 2019 (9 January)

LEVELJEE Main

If both the roots of the quadratic equation $x^{2} - m x + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ , then $m$ lies in the interval:

JEE Main 2020 - 3 Sep (Evening)

LEVELJEE Main

The set of all real values of $λ$ for which the quadratic equations, $(λ^{2} + 1) x^{2} - 4 λ x + 2 = 0$ always have exactly one root in the interval (0,1) is :

Let the set of all values of $p \in R$ , for which both the roots of the equation $x^{2} - (p + 2) x + (2 p + 9) = 0$ are negative real numbers, be the interval $(α, β]$ . Then $β - 2 α$ is equal to

JEE Main 2026 (24 January Shift 2)

LEVELJEE Main

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to

Let $∣ M ∣$ denote the determinant of a square matrix $M$ . Let $g : [0, \frac{π}{2}] \to R$ be the function defined by $g (θ) = f (θ) - 1 + f (\frac{π}{2} - θ) - 1$ where $f (θ) = \frac{1}{2} 1 - sin θ - 1 sin θ 1 - sin θ 1 sin θ 1 + sin π sin (θ - \frac{π}{4}) cot (θ + \frac{π}{4}) cos (θ + \frac{π}{4}) - cos \frac{π}{2} lo g_{e} (\frac{π}{4}) tan (θ - \frac{π}{4}) lo g_{e} (\frac{4}{π}) tan π$ . Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g (θ)$ , and $p (2) = 2 - 2$ . Then, which of the following is/are TRUE ?

* Multiple Correct Options

(A)

p (\frac{3 + 2}{4}) < 0

(B)

p (\frac{1 + 3 2}{4}) > 0

(C)

p (\frac{5 2 - 1}{4}) > 0

(D)

p (\frac{5 - 2}{4}) < 0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } All the values of m for which both roots of the equation x2−2mx+m2−1=0 are greater than - 2 but less than 4, lie in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If both the roots of the quadratic equation x2−2kx+k2+k−5=0 are less than 5, then k lies in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If b>a, then the equation (x−a)(x−b)−1=0 has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the equation x2−2ax+a2+a−3=0 are real and less than 3, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest value of k, for which both the roots of the equation x2−8kx+16(k2−k+1)=0 are real, distinct and have values at least 4, is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If both the roots of the quadratic equation x2−mx+4=0 are real and distinct and they lie in the interval [1,5], then m lies in the interval:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all real values of λ for which the quadratic equations, (λ2+1)x2−4λx+2=0 always have exactly one root in the interval (0,1) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest positive integral value of a, for which all the roots of x4−ax2+9=0 are real and distinct, is equal to

All the values of $m$ for which both roots of the equation $x^{2} - 2 m x + m^{2} - 1 = 0$ are greater than - 2 but less than 4, lie in the interval

If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval

If $b > a$ , then the equation $(x - a) (x - b) - 1 = 0$ has

If the roots of the equation $x^{2} - 2 a x + a^{2} + a - 3 = 0$ are real and less than 3, then

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is .........

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

If both the roots of the quadratic equation $x^{2} - m x + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ , then $m$ lies in the interval:

The set of all real values of $λ$ for which the quadratic equations, $(λ^{2} + 1) x^{2} - 4 λ x + 2 = 0$ always have exactly one root in the interval (0,1) is :

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to