Let λ ≠ 0 be a real number. Let α,β be the roots of the equation 14x² - 31x + 3λ = 0 and α,γ be the roots of the equation 35x² - 53x + 4λ = 0. Then 3α/β and 4α/γ are the roots of the equation

Solution

Maths sir · 5 steps

Kill λ first: scale the two equations so the λ terms match, subtract, and α falls out.

1. Remove lambda. Put the common root α into both equations. Multiply the first by 4 and the second by 3, so both have 12λ, then subtract.

   (105α² - 159α) - (56α² - 124α) = 0 ⇒ 49α² - 35α = 0

2. Find alpha and lambda. α = 0 would force λ = 0, which is not allowed, so α = 35/49 = 5/7. Putting this into the first equation gives 50/7 - 155/7 + 3λ = 0, so λ = 5.
3. Other roots. The first equation is 14x² - 31x + 15 = 0, with product of roots 15/14, so β = 15/14 ÷ 5/7 = 3/2. The second is 35x² - 53x + 20 = 0, with product 4/7, so γ = 4/7 ÷ 5/7 = 4/5.
4. New roots. 3α/β = (15/7)/(3/2) = 10/7 and 4α/γ = (20/7)/(4/5) = 25/7. Their sum is 5 and their product is 250/49.
5. So. The equation is x² - 5x + 250/49 = 0. Multiplying by 49 gives 49x² - 245x + 250 = 0, which is option (b).

The idea to keep. For a common root with an unknown parameter, scale the equations so the parameter terms match and subtract. Then use the product of roots to get each partner root quickly.

Answer from the NTA final answer key. Solution by PrepWiser.