Quadratic Equations JEE 2024: Checking a Flawed Solution

Q: Is 30 the correct answer to the supplied quadratic equations JEE 2024 problem?

The supplied solution does not establish 30 as the answer. Its proposed roots, 35.5 and 34.5, have product 1224.75, not 1222.75, and substitution gives a value greater than 12597.5. The original paper, complete options and a corrected solution are needed before presenting an official answer.

Q: How do you minimise the product of two roots with a fixed sum?

Use αβ = ((α + β)² − (α − β)²)/4: for a fixed sum, a larger root separation gives a smaller product. Determine the permitted domain and check whether the largest separation is actually allowed. A fixed real sum alone, without lower bounds on the roots, does not give a minimum product.

Q: Do α/2 and β/3 being non-integers mean the roots must be non-integers?

No. The restrictions exclude particular multiples, not all integer roots. The labelled pair α = 69, β = 1 satisfies both restrictions, but reversing the labels fails because 69/3 is an integer.

Q: What domain restrictions apply to the expression in this problem?

The radicals √(α − 1) and √(β − 1) require α ≥ 1 and β ≥ 1 for real values. The denominator |α − β| also requires distinct roots. These conditions must be checked alongside the root sum and quotient restrictions.

Can the supplied quadratic equations JEE 2024 answer be trusted?

The supplied quadratic equations JEE 2024 solution does not establish its asserted answer, C) 30: its chosen roots do not minimise the product, and its multiplication is wrong. The record says Year 2024 but gives 2023-01-24, Morning, so the paper attribution is unverified.

The task is to minimise the product of two labelled real roots satisfying:

x^{2} - 70 x + λ = 0, \frac{α}{2} \notin Z, \frac{β}{3} \notin Z .

At that minimum, evaluate:

\frac{(\sqrt{α - 1} + \sqrt{β - 1}) (λ + 35)}{| α - β |} .

The asserted answer is not mathematically verified. The complete option list is missing, so no other option can be quoted or assigned a letter.

What do Vieta’s formulas establish here?

For a fixed root sum, minimising the product means maximising the permitted root separation. This is the valid opening of the supplied solution, but the largest permitted separation still needs proof. Both the quotient restrictions and the expression’s domain matter.

Write the monic quadratic through its roots and compare coefficients:

(x - α) (x - β) & = x^{2} - (α + β) x + α β & = x^{2} - 70 x + λ .

Therefore:

α + β = 70, α β = λ .

Expand the separation identity:

(α - β)^{2} & = α^{2} - 2 α β + β^{2} & = (α + β)^{2} - 4 α β & = 4900 - 4 λ .

Rearranging gives:

λ = 1225 - \frac{(α - β)^{2}}{4} .

Equivalently, write the unordered root pair as:

{α, β} = {35 - t, 35 + t}, t \geq 0 .

Then:

λ = (35 - t) (35 + t) = 1225 - t^{2} .

Every pair with sum 70 is symmetric about 35. Symmetry alone does not identify a minimum.

For the target expression to be real and defined, we also need:

α \geq 1, β \geq 1, α \neq β .

These are expression-domain requirements, not consequences of merely having real roots. Without lower bounds on the roots, a fixed real sum alone does not give a minimum product.

Why does the supplied continuation fail?

The proposed roots are admissible, but they do not minimise the product. The continuation fails three checks:

Its multiplication is incorrect.
An admissible pair gives a smaller product.
Substituting the proposed pair cannot produce 30.

The supplied choice is:

α = 35.5, β = 34.5 .

Its sum and restrictions check out:

35.5 + 34.5 = 70, \frac{35.5}{2} = 17.75 \notin Z, \frac{34.5}{3} = 11.5 \notin Z .

Both radicals are defined and the denominator is nonzero. However, the exact product is:

35.5 \times 34.5 & = (35 + 0.5) (35 - 0.5) & = 1225 - 0.25 & = 1224.75,

not the supplied 1222.75.

Now test a different labelled pair:

α = 69, β = 1 .

It satisfies the sum and quotient restrictions:

α + β = 70, \frac{α}{2} = 34.5 \notin Z, \frac{β}{3} = \frac{1}{3} \notin Z .

Its radicals are defined, its roots are distinct, and its product is smaller: $λ = 69 \times 1 = 69 < 1224.75 .$

This counterexample disproves the claimed minimisation. No replacement solution method is needed to reject that claim.

Finally, substitute the proposed half-integer roots using their correct product:

| α - β | = 1, λ + 35 = 1259.75 .

The expression becomes:

(\sqrt{34.5} + \sqrt{33.5}) \times 1259.75 .

Each square root exceeds 5, so:

(\sqrt{34.5} + \sqrt{33.5}) \times 1259.75 > 10 \times 1259.75 = 12597.5 > 30 .

Publication gate: the requested full official-method solution cannot be completed honestly from this record. Obtain the original paper, all options and a corrected worked solution before publishing this as a solved PYQ.

Why is choosing roots close to 35 the wrong minimisation argument?

Close roots give a large fixed-sum product, not a small one. Choosing 34.5 and 35.5 places the roots only one unit apart, so their product is close to its maximum, not its minimum.

At zero separation:

t = 0 ⟹ λ = 1225 .

This is the largest fixed-sum product, although equal roots make the target denominator zero. Increasing the magnitude of the displacement decreases the product: $λ = 1225 - t^{2} .$

The non-integer restrictions exclude particular multiples. They neither require both roots to be non-integers nor force half-integer roots.

Root labels also matter. The counterexample’s ordering is allowed, but reversing it fails:

α = 1, β = 69 ⟹ \frac{β}{3} = 23 \in Z .

Linking this method error to an actual wrong option remains blocked until the missing options are supplied. No distractor should be invented.

Which two practice questions reinforce the correct method?

Use a specified-separation problem to check the identity, then a bounded-root problem to practise finding permitted endpoints. Both questions below are original practice, not historical PYQs.

How do you find the product when the roots differ by 6?

The product is 91, and the roots are 13 and 7 for Practice 1 below. The fixed-sum identity gives the product before you solve for either root.

Practice 1: Find the coefficient and both real roots when the roots differ by 6: $x^{2} - 20 x + k = 0 .$

Apply the separation identity: $36 = 20^{2} - 4 k .$

Therefore:

4 k = 400 - 36 = 364, k = 91 .

Label the larger root first:

a + b = 20, a - b = 6 .

Adding the equations and then using the sum gives:

2 a = 26, a = 13, b = 7 .

Check the product: $13 \times 7 = 91 .$

What are the product limits when both roots are at least 2?

The minimum is 16 and the maximum is 25 for Practice 2 below. The lower bound on the roots limits their separation; repeated roots remain allowed.

Practice 2: Determine the minimum and maximum coefficient when both roots are at least 2: $x^{2} - 10 x + k = 0 .$

Write the roots symmetrically about half their sum:

5 - t, 5 + t, t \geq 0 .

The lower-root condition fixes the permitted interval:

5 - t \geq 2 ⟹ 0 \leq t \leq 3 .

Hence:

k = (5 - t) (5 + t) = 25 - t^{2}, k \in [16, 25] .

Check both endpoints:

Minimum: roots 2 and 8 give product 16.
Maximum: repeated roots 5 and 5 give product 25.

For your next quadratic, use this checking order: fixed sum, product–separation identity, admissible domain, endpoint checks, substitution.

Next step: the past-paper archive on JEEnius and search every JEE Main paper from 2002 and every Advanced paper from 2007, by year, subject or chapter, each with a worked solution (free).

Frequently asked questions

Is 30 the correct answer to the supplied quadratic equations JEE 2024 problem?