Question

Consider the following statements:
$S_1$ : If the roots of $x^2-bx+c=0$ are two consecutive integers, then value of $b^2-4c$ is equal to 1.
$S_2$ : If $\alpha ,\beta$ are roots of $x^2$ - x + 3 = 0 then value of ${\alpha }^4+{\beta }^4$ is equal to 7.
$S_3$ : If $\alpha$ , $\beta$, $\gamma$ are the roots of $x^3-7x^2+16x-12=0$ then value of ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to 17.

• A. TTT
• B. FTF
• C. TFT
• D. FTT

Answer 1

Answer: (A)

TTT

$S_1=x^2-bx+c=0$

Let roots be $n$ and $n+1$

$2n+1=b$

$n(n+1)=c$

$b^2-4c={(2n+1)}^2-4n(n+1)={4n}^2+4n+1-{4n}^2-4n$

$=1$ $\text{t}rue$

$S_2=x^2-x+3=0\alpha +\beta =1\alpha \beta =3{\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta =1-6=-5{\alpha }^4+{\beta }^4={({\alpha }^2+{\beta }^2)}^2-2{\alpha }^2{\beta }^2=25-18$

$=7$ $\text{t}rue$

$S_3=x^3-7x^2+16x-12=0\alpha +\beta +\gamma =7\alpha \beta +\beta \gamma +\gamma \alpha =16{\alpha }^2+{\beta }^2+{\gamma }^2={(\alpha +\beta +\gamma )}^2-2(\alpha \beta +\beta \gamma +\gamma \alpha )=7^2-2.16=49-32=17$