Theory of Equations 5 Question 12
12. The smallest value of $k$, for which both the roots of the equation $x^{2}-8 k x+16\left(k^{2}-k+1\right)=0$ are real, distinct and have values atleast 4 , is
(2009)
Then, the quadratic equation $a x^{2}+b x+c=0$ has
(a) no root in $(0,2)$
(b) atleast one root in $(1,2)$
(c) a double root in $(0,2)$
(d) two imaginary roots
Objective Questions II
(One or more than one correct option)
Show Answer
Solution:
- $\int _0^{1 / 2} f(x) d x<\int _0^{t} f(x) d x<\int _0^{3 / 4} f(x) d x$
Now, $\int f(x) d x=\int\left(1+2 x+3 x^{2}+4 x^{3}\right) d x$
$=x+x^{2}+x^{3}+x^{4}$
$\Rightarrow \quad \int _0^{1 / 2} f(x) d x=\frac{15}{16}>\frac{3}{4}, \quad \int _0^{3 / 4} f(x) d x=\frac{530}{256}<3$
