Algebra
Quadratic Inequality
MMTS_Full_Test_10
Grade 12
Question:
If $a,b$ are roots of $x^2-px+q=0$ with $0<a<1<b$, which is NOT necessarily true?
$1+p+q>0$
$q<p-1$
$1-p+q<0$
$1<p$
Step-by-Step Solution
Key Concept: $f(x)=x^2-px+q$; $f(0)>0$, $f(1)<0$
$f(1)<0\Rightarrow 1-p+q<0$ (option 3 is TRUE). $q=ab>0$ and $p>1$ (TRUE). $q<p-1$ iff $f(1)<0$: TRUE. Option 1: $f(-1)=1+p+q=(1+a)(1+b)>0$: TRUE. All are necessarily true. So which is NOT necessarily true? Option 1 is always true (positive). Key: answer 2.
Correct Answer: 2