Nature of Roots

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JEE Main 2004

LEVELBoard

If one root of the equation $x^{2} + p x + 12 = 0$ is 4, while the equation $x^{2} + p x + q = 0$ has equal roots, then the value of 'q' is

Product of real roots of the equation $x^{2} + ∣ x ∣ + 9 = 0$

Let $S$ be the set of all non-zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $∣ x_{1} - x_{2} ∣ < 1$ . Which of the following intervals is(are) a subset(s) of $S$ ?

* Multiple Correct Options

(A)

(- 1/2, - 1/ 5)

(B)

(- 1/ 5, 0)

(C)

(0, 1/ 5)

(D)

(1/ 5, 1/2)

JEE Advanced 2014

LEVELJEE Main

The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has

(A)

one purely imaginary root

(B)

all real roots

(C)

two real and two purely imaginary roots

(D)

neither real nor purely imaginary roots

JEE Advanced 2006

LEVELJEE Main

Let $a, b, c$ be the sides of a triangle where $a \neq = b \neq = c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (ab + b c + c a) = 0$ are real, then

(A)

λ < 4/3

(B)

λ > 5/3

(C)

λ \in (1/3, 5/3)

(D)

λ \in (4/3, 5/3)

JEE Advanced 1994

LEVELJEE Main

If $p, q, r$ are +ve and are in A.P., the roots of quadratic equation $p x^{2} + q x + r = 0$ are all real for

(A)

∣ r / p - 7∣ \geq 43

(B)

∣ p / r - 7∣ \geq 43

(C)

all

p

and

r

(D)

p

and

r

JEE Advanced 1994

LEVELJEE Main

Let $p, q \in {1, 2, 3, 4}$ . The number of equations of the form $p x^{2} + q x + 1 = 0$ having real roots is

The equation $(cos p - 1) x^{2} + (cos p) x + sin p = 0$ in the variable $x$ , has real roots. Then $p$ can take any value in the interval

(A)

(0, 2 π)

(B)

(- π, 0)

(C)

(- π /2, π /2)

(D)

(0, π)

JEE Advanced 1989

LEVELJEE Main

If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $α^{4}, β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the equation $x^{2} - 4 q x + 2 q^{2} - r = 0$ has always

one positive and one negative root

JEE Advanced 1980

LEVELJEE Main

Both the roots of the equation $(x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0$ are always

If $l, m, n$ are real, $l \neq = m$ , then the roots by the equation: $(l - m) x^{2} - 5 (l + m) x - 2 (l - m) = 0$ are

Let $a > 0, b > 0$ and $c > 0$ . Then the roots of the equation $a x^{2} + b x + c = 0$

(A)

are real and negative

(B)

have negative real parts

If $p$ and $q$ are chosen randomly from the set ${1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ , with replacement, determine the probability that the roots of the equation $x^{2} + p x + q = 0$ are real.

JEE Advanced 2003

LEVELJEE Main

If $x^{2} + (a - b) x + (1 - a - b) = 0$ where $a, b \in R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$ .

JEE Main 2019 (9 January)

LEVELJEE Main

The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is :

JEE Main 2019 (08 April Shift 2)

LEVELBoard

The number of integral values of m for which the equation $(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0$ has no real root is :

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

Let $p, q \in R$ . If $2 - 3$ is a root of the quadratic equation, $x^{2} + p x + q = 0$ , then :

(A)

q^{2} + 4 p + 14 = 0

(B)

p^{2} - 4 q - 12 = 0

(C)

q^{2} - 4 p - 16 = 0

(D)

p^{2} - 4 q + 12 = 0

JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

The least positive value of 'a' for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is ____________.

JEE Main 2020 - 8 Jan (Morning)

LEVELJEE Main

The least positive value of $a$ for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is

JEE Main 2020 - 3 Sep (Morning)

LEVELJEE Main

Consider the two sets: $A = {m \in R : both the roots of x^{2} - (m + 1) x + m + 4 = 0 are real}$ and $B = [- 3, 5)$ . Which of the following is not true?

(A)

A \cap B = {- 3}

(B)

B - A = (- 3, 5)

(C)

A \cup B = R

(D)

A - B = (- \infty, - 3) \cup (5, \infty)

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Main

The set of all values of $k > - 1$ , for which the equation $(3 x^{2} + 4 x + 3)^{2} - (k + 1) (3 x^{2} + 4 x + 3) (3 x^{2} + 4 x + 2) + k (3 x^{2} + 4 x + 2)^{2} = 0$ has real roots is :

(A)

(1, \frac{5}{2}]

(B)

[2, 3)

(C)

[\frac{- 1}{2}, 1)

(D)

(\frac{1}{2}, \frac{3}{2}] - {1}

JEE Main 2021 (26 Aug Shift 1)

LEVELJEE Main

The sum of all integral values of $k (k \neq = 0)$ for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is .

JEE Main 2022 (26 July Shift 1)

LEVELJEE Main

If for some $p, q, r \in R$ , not all have same sign, one of the roots of the equation $(p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0$ is also a root of the equation $x^{2} + 2 x - 8 = 0$ , then $\frac{q ^{2} + r ^{2}}{p ^{2}}$ is equal to ______.

JEE Main 2023 (25 January Shift 1)

LEVELJEE Main

Let $S = {α : lo g_{2} (9^{2 α - 4} + 13) - lo g_{2} (\frac{5}{2} \cdot 3^{2 α - 4} + 1) = 2}$ . Then the maximum value of $β$ for which the equation $x^{2} - 2 (\sum_{α \in S} α)^{2} x + \sum_{α \in S} (α + 1)^{2} β = 0$ has real roots, is

JEE Main 2024 (04 Apr Shift 2)

LEVELJEE Advanced

Let $S = {sin^{2} 2 θ : (sin^{4} θ + cos^{4} θ) x^{2} + (sin 2 θ) x + (sin^{6} θ + cos^{6} θ) = 0 has real roots}$ . If $α$ and $β$ be the smallest and largest elements of the set $S$ , respectively, then $3 ((α - 2)^{2} + (β - 1)^{2})$ equals _________

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the equation x2+px+12=0 is 4, while the equation x2+px+q=0 has equal roots, then the value of 'q' is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Product of real roots of the equation x2+∣x∣+9=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0 has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be the sides of a triangle where a=b=c and λ∈R. If the roots of the equation x2+2(a+b+c)x+3λ(ab+bc+ca)=0 are real, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p,q,r are +ve and are in A.P., the roots of quadratic equation px2+qx+r=0 are all real for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p,q∈{1,2,3,4}. The number of equations of the form px2+qx+1=0 having real roots is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation (cosp−1)x2+(cosp)x+sinp=0 in the variable x, has real roots. Then p can take any value in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β are the roots of x2+px+q=0 and α4,β4 are the roots of x2−rx+s=0, then the equation x2−4qx+2q2−r=0 has always

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Both the roots of the equation (x−b)(x−c)+(x−a)(x−c)+(x−a)(x−b)=0 are always

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If l,m,n are real, l=m, then the roots by the equation: (l−m)x2−5(l+m)x−2(l−m)=0 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a>0,b>0 and c>0. Then the roots of the equation ax2+bx+c=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p and q are chosen randomly from the set {1,2,3,4,5,6,7,8,9,10}, with replacement, determine the probability that the roots of the equation x2+px+q=0 are real.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x2+(a−b)x+(1−a−b)=0 where a,b∈R then find the values of a for which equation has unequal real roots for all values of b.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of all possible positive integral values of α for which the roots of the quadratic equation, 6x2−11x+α=0 are rational numbers is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral values of m for which the equation (1+m2)x2−2(1+3m)x+(1+8m)=0 has no real root is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p,q∈R. If 2−3​ is a root of the quadratic equation, x2+px+q=0, then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least positive value of 'a' for which the equation, 2x2+(a−10)x+233​=2a has real roots is ____________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least positive value of a for which the equation, 2x2+(a−10)x+233​=2a has real roots is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the two sets: A={m∈R:both the roots of x2−(m+1)x+m+4=0 are real} and B=[−3,5). Which of the following is not true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all values of k>−1, for which the equation (3x2+4x+3)2−(k+1)(3x2+4x+3)(3x2+4x+2)+k(3x2+4x+2)2=0 has real roots is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all integral values of k(k=0) for which the equation x−12​−x−21​=k2​ in x has no real roots, is .

If one root of the equation $x^{2} + p x + 12 = 0$ is 4, while the equation $x^{2} + p x + q = 0$ has equal roots, then the value of 'q' is

Product of real roots of the equation $x^{2} + ∣ x ∣ + 9 = 0$

The quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p (p (x)) = 0$ has

Let $a, b, c$ be the sides of a triangle where $a \neq = b \neq = c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (ab + b c + c a) = 0$ are real, then

If $p, q, r$ are +ve and are in A.P., the roots of quadratic equation $p x^{2} + q x + r = 0$ are all real for

Let $p, q \in {1, 2, 3, 4}$ . The number of equations of the form $p x^{2} + q x + 1 = 0$ having real roots is

The equation $(cos p - 1) x^{2} + (cos p) x + sin p = 0$ in the variable $x$ , has real roots. Then $p$ can take any value in the interval

If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $α^{4}, β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the equation $x^{2} - 4 q x + 2 q^{2} - r = 0$ has always

Both the roots of the equation $(x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0$ are always

If $l, m, n$ are real, $l \neq = m$ , then the roots by the equation: $(l - m) x^{2} - 5 (l + m) x - 2 (l - m) = 0$ are

Let $a > 0, b > 0$ and $c > 0$ . Then the roots of the equation $a x^{2} + b x + c = 0$

If $p$ and $q$ are chosen randomly from the set ${1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ , with replacement, determine the probability that the roots of the equation $x^{2} + p x + q = 0$ are real.

If $x^{2} + (a - b) x + (1 - a - b) = 0$ where $a, b \in R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$ .

The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is :

The number of integral values of m for which the equation $(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0$ has no real root is :

Let $p, q \in R$ . If $2 - 3$ is a root of the quadratic equation, $x^{2} + p x + q = 0$ , then :

The least positive value of 'a' for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is ____________.

The least positive value of $a$ for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is

Consider the two sets: $A = {m \in R : both the roots of x^{2} - (m + 1) x + m + 4 = 0 are real}$ and $B = [- 3, 5)$ . Which of the following is not true?

The set of all values of $k > - 1$ , for which the equation $(3 x^{2} + 4 x + 3)^{2} - (k + 1) (3 x^{2} + 4 x + 3) (3 x^{2} + 4 x + 2) + k (3 x^{2} + 4 x + 2)^{2} = 0$ has real roots is :

The sum of all integral values of $k (k \neq = 0)$ for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is .