If α be a repeated root of a quadratic...

MathematicsProperties of DeterminantsJEE Advanced 1984Moderate

If $α$ be a repeated root of a quadratic equation $f (x) = 0$ and $A (x), B (x)$ and $C (x)$ be polynomials of degree 3, 4 and 5 respectively, then show that $A (x) A (α) A^{'} (α) B (x) B (α) B^{'} (α) C (x) C (α) C^{'} (α)$ is divisible by $f (x)$ , where prime denotes the derivatives.

Visualized Solution (English)

Defining the Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } F (x)

Let the given determinant be defined as a function $F (x)$ :
$F (x) = A (x) A (α) A^{'} (α) B (x) B (α) B^{'} (α) C (x) C (α) C^{'} (α)$
We are given that $α$ is a repeated root of the quadratic equation $f (x) = 0$ .

The Nature of a Repeated Root

Since $α$ is a repeated root of the quadratic $f (x) = 0$ , we can express $f (x)$ as:
$f (x) = k (x - α)^{2}$ , where $k$ is a constant.
To show $F (x)$ is divisible by $f (x)$ , we must show it is divisible by $(x - α)^{2}$ .

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } F (x) at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = α

Substitute $x = α$ into $F (x)$ :
$F (α) = A (α) A (α) A^{'} (α) B (α) B (α) B^{'} (α) C (α) C (α) C^{'} (α)$
Since Row 1 ( $R_{1}$ ) and Row 2 ( $R_{2}$ ) are identical, the determinant vanishes.
$∴ F (α) = 0$

Differentiating the Determinant

To find $F^{'} (x)$ , we differentiate row by row. Since $R_{2}$ and $R_{3}$ are constant with respect to $x$ :
$F^{'} (x) = A^{'} (x) A (α) A^{'} (α) B^{'} (x) B (α) B^{'} (α) C^{'} (x) C (α) C^{'} (α) + 0 + 0$
Only the derivative of the first row remains.

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } F^{'} (x) at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x = α

Substitute $x = α$ into $F^{'} (x)$ :
$F^{'} (α) = A^{'} (α) A (α) A^{'} (α) B^{'} (α) B (α) B^{'} (α) C^{'} (α) C (α) C^{'} (α)$
Since Row 1 ( $R_{1}$ ) and Row 3 ( $R_{3}$ ) are identical, the determinant vanishes.
$∴ F^{'} (α) = 0$

Applying the Factor Theorem

According to the factor theorem for repeated roots:
If $F (α) = 0$ and $F^{'} (α) = 0$ , then $(x - α)^{2}$ is a factor of $F (x)$ .
This means $F (x) = (x - α)^{2} \cdot Q (x)$ for some polynomial $Q (x)$ .

Final Conclusion

We know $f (x) = k (x - α)^{2}$ .
Since $F (x)$ is divisible by $(x - α)^{2}$ , it must also be divisible by $f (x)$ .
Key Takeaway: For a polynomial $P (x)$ to be divisible by $(x - α)^{n}$ , we must have $P (α) = P^{'} (α) = ... = P^{(n - 1)} (α) = 0$ .

00:00 / 00:00

Conceptually Similar Problems

MathematicsLocation of RootsJEE Advanced 1989Moderate

View in:English Hindi

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

Answer:D

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1983Easy

View in:English Hindi

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 2001Easy

View in:English Hindi

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

Answer:

α^{2} β, α β^{2}

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2000Easy

View in:English Hindi

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 $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

MathematicsProperties of DeterminantsJEE Advanced 1985Easy

View in:English Hindi

If $f_{r} (x), g_{r} (x), h_{r} (x), r = 1, 2, 3$ are polynomials in $x$ such that $f_{r} (a) = g_{r} (a) = h_{r} (a), r = 1, 2, 3$ and $F (x) = f_{1} (x) g_{1} (x) h_{1} (x) f_{2} (x) g_{2} (x) h_{2} (x) f_{3} (x) g_{3} (x) h_{3} (x)$ then $F^{'} (x)$ at $x = a$ is $\dots\dots\dots$

Answer:0

MathematicsProperties of DeterminantsJEE Advanced 1986Easy

View in:English Hindi

The determinant $a b a α + b b c b α + c a α + b b α + c 0$ is equal to zero, if

* Multiple Correct Options

(A)

a, b, c

are in A. P.

(B)

a, b, c

are in G. P.

(C)

a, b, c

are in H. P.

(D)

α

is a root of the equation

a x^{2} + b x + c = 0

(E)

(x - α)

is a factor of

a x^{2} + 2 b x + c

Answer:B, E

MathematicsLocation of RootsJEE Advanced 1995Easy

View in:English Hindi

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 $1 + c / a + ∣ b / a ∣ < 0$ .

MathematicsMean Value TheoremsJEE Main 2005Moderate

View in:English Hindi

If the equation $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x = 0$ , $a_{1} \neq = 0, n \geq 2$ , has a positive root $x = α$ , then the equation $n a_{n} x^{n - 1} + (n - 1) a_{n - 1} x^{n - 2} + \dots + a_{1} = 0$ has a positive root, which is

(A)

greater than

α

(B)

smaller than

α

(C)

greater than or equal to

α

(D)

equal to

α

Answer:B

MathematicsCommon RootsJEE Advanced 1979Moderate

View in:English Hindi

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 2010Easy

View in:English Hindi

Let $p$ and $q$ be real numbers such that $p \neq = 0, p^{3} \neq = - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $α / β$ and $β / α$ as its roots is

(A)

(p^{3} + q) x^{2} - (p^{3} + 2 q) x + (p^{3} + q) = 0

(B)

(p^{3} + q) x^{2} - (p^{3} - 2 q) x + (p^{3} + q) = 0

(C)

(p^{3} - q) x^{2} - (5 p^{3} - 2 q) x + (p^{3} - q) = 0

(D)

(p^{3} - q) x^{2} - (5 p^{3} + 2 q) x + (p^{3} - q) = 0

Answer:B