Common Roots

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JEE Main 2013

LEVELJEE Main

If the equations $x^{2} + 2 x + 3 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R$ , have a common root, then $a : b : c$ is

The quadritic equations $x^{2} - 6 x + a = 0$ and $x^{2} - c x + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

A value of $b$ for which the equations $x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have one root in common is

(A)

- 2

(B)

- i 3

(C)

i 5

(D)

2

JEE Advanced 2008

LEVELJEE Main

STATEMENT-1: $(p^{2} - q) (b^{2} - a c) \geq 0$ and STATEMENT-2: $b \neq = p a$ or $c \neq = q a$

(A)

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

(B)

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

(C)

STATEMENT-1 is True, STATEMENT-2 is False

(D)

STATEMENT-1 is False, STATEMENT-2 is True

JEE Advanced 1985

LEVELJEE Main

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

If the quadratic equations $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ ( $a \neq = b$ ) have a common root, then the numerical value of $a + b$ is .........

JEE Advanced 1979

LEVELJEE Main

If $α, β$ are the roots of $x^{2} + p x + q = 0$ and $γ, δ$ are the roots of $x^{2} + r x + s = 0$ , evaluate $(α - γ) (α - δ) (β - γ) (β - δ)$ in terms of $p, q, r$ and $s$ . Deduce the condition that the equations have a common root.

JEE Main 2019 (12 April)

LEVELJEE Advanced

If $α, β$ and $γ$ are three consecutive terms of a non-constant G.P. such that the equations $x^{2} + 2 α x + β = 0$ and $x^{2} + β x - 1 = 0$ have a common root, then $α (β + γ)$ is equal to :

(A)

β γ

(B)

(C)

α γ

(D)

α β

JEE Main 2020 - 4 Sep (Evening)

LEVELJEE Main

Let $λ \neq = 0$ be in $R$ . If $α$ and $β$ are the roots of the equation, $x^{2} - x + 2 λ = 0$ and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to :

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Main

Let $λ \neq = 0$ be in $R$ . If $α$ and $β$ are the roots of the equation $x^{2} - x + 2 λ = 0$ , and $α$ and $γ$ are the roots of equation $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to .

JEE Main 2023 (30 January Shift 2)

LEVELJEE Main

If the value of real number $a > 0$ for which $x^{2} - 5 a x + 1 = 0$ and $x^{2} - a x - 5 = 0$ have a common real roots is $\frac{3}{2 β}$ , then $β$ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the equations x2+2x+3=0 and ax2+bx+c=0,a,b,c∈R, have a common root, then a:b:c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A value of b for which the equations x2+bx−1=0,x2+x+b=0 have one root in common is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } STATEMENT-1: (p2−q)(b2−ac)≥0 and STATEMENT-2: b=pa or c=qa

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the quadratic equations x2+ax+b=0 and x2+bx+a=0 (a=b) have a common root, then the numerical value of a+b is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β and γ are three consecutive terms of a non-constant G.P. such that the equations x2+2αx+β=0 and x2+βx−1=0 have a common root, then α(β+γ) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let λ=0 be in R. If α and β are the roots of the equation, x2−x+2λ=0 and α and γ are the roots of the equation, 3x2−10x+27λ=0, then λβγ​ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let λ=0 be in R. If α and β are the roots of the equation x2−x+2λ=0, and α and γ are the roots of equation 3x2−10x+27λ=0, then λβγ​ is equal to .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the value of real number a>0 for which x2−5ax+1=0 and x2−ax−5=0 have a common real roots is 2β​3​, then β is equal to

If the equations $x^{2} + 2 x + 3 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R$ , have a common root, then $a : b : c$ is

A value of $b$ for which the equations $x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have one root in common is

STATEMENT-1: $(p^{2} - q) (b^{2} - a c) \geq 0$ and STATEMENT-2: $b \neq = p a$ or $c \neq = q a$

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

If the quadratic equations $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ ( $a \neq = b$ ) have a common root, then the numerical value of $a + b$ is .........

If $α, β$ and $γ$ are three consecutive terms of a non-constant G.P. such that the equations $x^{2} + 2 α x + β = 0$ and $x^{2} + β x - 1 = 0$ have a common root, then $α (β + γ)$ is equal to :

Let $λ \neq = 0$ be in $R$ . If $α$ and $β$ are the roots of the equation, $x^{2} - x + 2 λ = 0$ and $α$ and $γ$ are the roots of the equation, $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to :

Let $λ \neq = 0$ be in $R$ . If $α$ and $β$ are the roots of the equation $x^{2} - x + 2 λ = 0$ , and $α$ and $γ$ are the roots of equation $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to .

If the value of real number $a > 0$ for which $x^{2} - 5 a x + 1 = 0$ and $x^{2} - a x - 5 = 0$ have a common real roots is $\frac{3}{2 β}$ , then $β$ is equal to