If the equation has four positive roots, then is:

Explanation: Let the four positive real roots of the given biquadratic equation be and . Step 1: Using Relations Between Roots and Coefficients (Vieta's Formulas) For the polynomial equation , Vieta's formulas give: Sum of the roots: Product of the roots: Step 2: Applying the AM-GM Inequality For any positive real numbers, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM) : Let us calculate the AM and GM for our four positive roots : Arithmetic Mean (AM): Geometric Mean (GM): Step 3: Finding the Roots Since we found that and , we have: The equality condition for the AM-GM inequality holds if and only if all the terms are equal to each other. Therefore: So, the equation has four identical roots, all equal to . Step 4: Finding the Values of and Since all roots are equal

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MAH-CET 2026 — Mathematics PYQ

MAH-CET | Mathematics | 2026

If the equation $x^{4} - 4 x^{3} + a x^{2} + b x + 1 = 0$ has four positive roots, then $(a, b)$ is:

Choose the correct answer:

Explanation

Let the four positive real roots of the given biquadratic equation be $α, β, γ,$ and $δ$ .

Step 1: Using Relations Between Roots and Coefficients (Vieta's Formulas)

For the polynomial equation $x^{4} - 4 x^{3} + a x^{2} + b x + 1 = 0$ , Vieta's formulas give:

Sum of the roots:
$α + β + γ + δ = - \frac{Coefficient of x ^{3}}{Coefficient of x ^{4}} = - \frac{- 4}{1} = 4$
Product of the roots:
$α \cdot β \cdot γ \cdot δ = \frac{Constant term}{Coefficient of x ^{4}} = \frac{1}{1} = 1$

Step 2: Applying the AM-GM Inequality

For any positive real numbers, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM):

$AM \geq GM$

Let us calculate the AM and GM for our four positive roots $α, β, γ, δ$ :

Arithmetic Mean (AM):
$AM = \frac{α + β + γ + δ}{4} = \frac{4}{4} = 1$
Geometric Mean (GM):
$GM = (α \cdot β \cdot γ \cdot δ)^{\frac{1}{4}} = (1)^{\frac{1}{4}} = 1$

Step 3: Finding the Roots

Since we found that $AM = 1$ and $GM = 1$ , we have:

$AM = GM$

The equality condition for the AM-GM inequality holds if and only if all the terms are equal to each other. Therefore:

$α = β = γ = δ = 1$

So, the equation has four identical roots, all equal to $1$ .

Step 4: Finding the Values of $a$ and $b$

Since all roots are equal to $1$ , the polynomial can be written in its factored form as:

$(x - 1)^{4} = 0$

Expanding $(x - 1)^{4}$ using the Binomial Theorem gives:

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 = 0$

Now, compare this expanded equation with the original given equation:

$x^{4} - 4 x^{3} + a x^{2} + b x + 1 = 0$

By comparing the corresponding coefficients, we find:

$a = 6$
$b = - 4$

Thus, the ordered pair $(a, b) = (6, - 4)$ .