Question 8
We have
$x^4 + x^3 + 2x^2 + x + 1 = 0$
$\Leftrightarrow (x^4 + x^3 + x^2) + (x^2 + x + 1) = 0$
$\Leftrightarrow x^2(x^2 + x + 1) + (x^2 + x + 1) = 0$
$\Leftrightarrow (x^2+x+1)(x^2+1) = 0$
We have $x^2 + 1 \geq 1 > 0$ for all $x$ and
$x^2 + x + 1 = \left( x + \dfrac{1}{2} \right)^2 + \dfrac{3}{4} > 0$ for all $x$
Hence the given equation does not have any solution. So the answer is D.
Question 7
The number of students who have a dog or a cat is
$52 + 31 - 19 = 64$
So the number of students who have neither a dog or a cat is
$120 - 64 = 56$
The answer is B.
