NDA 1 2024 Maths, question 8

Topic: Algebra

Q8

If $-\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+x^{4}=0$ where $a_{0},a_{1},a_{2},a_{3}$ are integers, then which one of the following is correct ?

(a) $a_{2}=a_{3}=0$
(b) $a_{2}=0\text{ and }a_{3}=-5$
(c) $a_{0}=6,a_{3}=0$ Correct answer
(d) $a_{1}=0\text{ and }a_{2}=5$

Solution.

Answer: (c) $a_{0}=6,a_{3}=0$

Integer coefficients require the conjugate irrational roots √2 and −√3 as well.
The monic quartic is therefore (x²−2)(x²−3)=x⁴−5x²+6.
Thus a₀=6, a₁=a₃=0 and a₂=−5, matching (c).