Quadratic and Integral mixed BITSAT 2021 Ques

Question

f(x)= x²+bx+c,  \(\int_ α^ β f (x) d x\) =0 and \(\alpha+\beta=0\), then which of the following must be true?

(a) f(x) has a positive and a negative root

(b) b+c = -1

(c) b² = 4c

(d) \(\int_ α^ β f (x) d x\) > 0

Answer 1

\(\int\limits^\beta_\alpha f(x) dx = 0\)

⇒ \(\int\limits^\beta_\alpha (x^2 + bx + c) dx = 0\)

⇒ \(\left[\frac{x^3}3 + \frac{bx^2}2 + cx\right]^\beta_\alpha = 0\)

⇒ \(\frac13(\beta^3- \alpha^3) + \frac b2(\beta^2- \alpha^2) + c(\beta - \alpha) = 0\)

⇒ \((\beta - \alpha) \left(\frac{\beta^2 + \alpha^2 + \alpha\beta}{3} + \frac b2 (\beta + \alpha) + c\right)=0\)

⇒ \((\beta - \alpha) \left(\frac{(\beta+ \alpha)^2 - \alpha\beta}{3} + \frac b2 \times 0+ c\right)=0\)   \((\because \beta + \alpha = 0)\)

⇒ \((\beta- \alpha )\left(\frac{-\alpha\beta}{3}+c\right)=0\)

⇒ \(\frac{(\beta - \alpha)(3c- \alpha\beta)}{3}=0\)

⇒ \(\beta - \alpha = 0\;or\;3c- \alpha\beta =0\)

⇒ \(\beta = \alpha \;or\; \alpha\beta = 3c.\)