Quadratic Equations
Limit of parameterized quadratic roots
nta_pyq_2025_apr
Grade 12
Question:
For t > -1, let$\alpha and$$\beta be the roots of the equation t t 1 1 1 2 (($t + 2)$$7 - 1)$$x + ((t + 2)$$6 - 1)$$x + ((t + 2)$$21 - 1) = 0$If lim t$$\$to-1$+$$\$alphat = a$and lim t$$\$to-1$+$$$$\$betat = b$, then 72($a + b)$is equal to ________. 2$
Step-by-Step Solution
Key Concept: Let$u=t+2$and use$u\to1^+$to approximate coefficients by$first-order$powers, then use Vieta on the limiting quadratic.
($t + 2)$$6 - 1$$a + b = lim$($\alpha +$$\beta) =$lim - (98)$t$$\$to-1$+ t$$\$to-1 + 1$($t + 2)$$7 - 1$let$t + 2 = y$$1/6$$y - 1$7$a + b = lim$= +$1/7$6 y$$\$to1$$y - 1$49 2 72($a + b) = 72 = 98$36$
Correct Answer: 98