Quadratic Equations Questions

MathematicsQuadratic Equations90 Questions Found

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1983Easy

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If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1992Moderate

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Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

MathematicsLocation of RootsJEE Advanced 1995Easy

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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 $1 + c / a + ∣ b / a ∣ < 0$ .

MathematicsMaximum and Minimum Values of Quadratic ExpressionsJEE Advanced 1998Moderate

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Let $f (x) = A x^{2} + B x + C$ where $A, B, C$ are real numbers. Prove that if $f (x)$ is an integer whenever $x$ is an integer, then the numbers $2 A, A + B$ and $C$ are all integers. Conversely, prove that if the numbers $2 A, A + B$ and $C$ are all integers then $f (x)$ is an integer whenever $x$ is an integer.

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2000Easy

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If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

MathematicsCommon RootsJEE Advanced 1979Moderate

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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.

Answer:

(q - s)^{2} - (p - r) (p s - q r); Condition: (q - s)^{2} - (p - r) (p s - q r) = 0

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1982Easy

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If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Answer:(-4, 7)

MathematicsSolution of Quadratic EquationsJEE Advanced 1983Easy

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Find all real values of $x$ which satisfy $x^{2} - 3 x + 2 > 0$ and $x^{2} - 2 x - 4 \leq 0$

Answer:

x \in [1 - 5, 1) \cup (2, 1 + 5]

MathematicsTransformation of EquationsJEE Advanced 1985Easy

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Solve for $x : (5 + 26)^{x^{2} - 3} + (5 - 26)^{x^{2} - 3} = 10$

Answer:

\pm 2, \pm 2

MathematicsSolution of Quadratic EquationsJEE Advanced 1986Moderate

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For $a \leq 0$ , determine all real roots of the equation $x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0$

Answer:

{a \pm a 2, - a \pm a 6}

...9

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the quadratic equation ax2+bx+c=0 is equal to the n-th power of the other, then show that (acn)1/(n+1)+(anc)1/(n+1)+b=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p≥3 be an integer and α,β be the roots of x2−(p+1)x+1=0 using mathematical induction show that αn+βn. (i) is an integer and (ii) is not divisible by p

.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 α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 2+i3​ is a root of the equation x2+px+q=0, where p and q are real, then (p,q)=(…,…).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all real values of x which satisfy x2−3x+2>0 and x2−2x−4≤0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for x:(5+26​)x2−3+(5−26​)x2−3=10

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a≤0, determine all real roots of the equation x2−2a∣x−a∣−3a2=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the quadratic equation ax2+bx+c=0 is equal to the n-th power of the other, then show that (acn)1/(n+1)+(anc)1/(n+1)+b=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p≥3 be an integer and α,β be the roots of x2−(p+1)x+1=0 using mathematical induction show that αn+βn. (i) is an integer and (ii) is not divisible by p

.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 α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 2+i3​ is a root of the equation x2+px+q=0, where p and q are real, then (p,q)=(…,…).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all real values of x which satisfy x2−3x+2>0 and x2−2x−4≤0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for x:(5+26​)x2−3+(5−26​)x2−3=10

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a≤0, determine all real roots of the equation x2−2a∣x−a∣−3a2=0

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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 $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Find all real values of $x$ which satisfy $x^{2} - 3 x + 2 > 0$ and $x^{2} - 2 x - 4 \leq 0$

Solve for $x : (5 + 26)^{x^{2} - 3} + (5 - 26)^{x^{2} - 3} = 10$

For $a \leq 0$ , determine all real roots of the equation $x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0$

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $p \geq 3$ be an integer and $α, β$ be the roots of $x^{2} - (p + 1) x + 1 = 0$ using mathematical induction show that $α^{n} + β^{n}$ . (i) is an integer and (ii) is not divisible by $p$

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 $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If $2 + i 3$ is a root of the equation $x^{2} + p x + q = 0$ , where $p$ and $q$ are real, then $(p, q) = (\dots, \dots)$ .

Find all real values of $x$ which satisfy $x^{2} - 3 x + 2 > 0$ and $x^{2} - 2 x - 4 \leq 0$

Solve for $x : (5 + 26)^{x^{2} - 3} + (5 - 26)^{x^{2} - 3} = 10$

For $a \leq 0$ , determine all real roots of the equation $x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0$