Quadratic Equations
Newton sums and reciprocal roots
nta_pyq_2025_apr
Grade 12
Question:
Let P n =$\alpha n +$$\beta n,n$$\i_n N. If P$10 = 123$,$$P_{9}$= 76$,$P = 47$and$P = 1$, then the quadratic equation having roots 8 1 1$$\alpha and 1$$\beta is$:
$x - x + 1 = 0 2$
$x + x - 1 = 0 2$
$x - x - 1 = 0 2$
$x + x + 1 = 0 2$
Step-by-Step Solution
Key Concept: Use$Newton-sum$recurrence to find$\alpha+\beta$and$\alpha\beta$, then form the quadratic for$1/\alpha$and$1/\beta$.
$\alpha 10 10 +$$\$beta = 123$(2)$$\alpha +$$\$beta = 1$9 9$$\alpha +$$\$beta = 76$8 8$$\alpha +$$\$beta = 47$$P_{10} =$P_{9}$+ P_{8}$2 2$x = x$+ 1$$\Rightarrow$$x - x - 1 = 0$$$\alpha +$$$\$beta = 1$,$$$\alpha$$$\beta = -1 1 1$$$\alpha +$$$\beta 1 1 + = = = -1, = -1$$$\alpha$$$\beta$$$\alpha$$$\beta -1$$$\alpha$$$$\beta$$
Correct Answer: 2