<p>For the math one, line AB has to be an integer. That means that the x value where the two parabolas intersect is an integer. The point of intersection of the parabolas is where their y values are equal. So set them equal.
x^2=k-x^2 → 2x^2=k<br>
Plug in values for x, say 1,2,3,4 and you get 2,8,18,32, respectively.
That means your answer is C, it cannot be 12.</p>
<p>PencilxBoxes’s solution is solid.
Here’s a variation:
x^2=k-x^2 → 2x^2=k
k/2= x^2
Since x is an integer, k/2 has to be a perfect square.
Divide the answers by 2:
A. 2/2 =1 <— a square
B. 8/2 = 4 <— a square
C. 12/2 = 6 <— not a square. That’s the answer.</p>
<p>I would have gone E for the second one? Just sounds better and I think its a case of ambiguous pronoun of “their”. Please correct me if I’m wrong because writing for me is just a case of hearing the mistakes</p>
<p>visualize the Y-axis, line AB and curve y = k-x^2 is part of new curve with integer increases in X. hence, the Y value of this imaginary curve (A to y) has to take on a perfect square value, and this value is half of K (0 to y). </p>
<p>hence,
1^2 * 2 = 2
2^2 *2 = 8
3^2 * 2 18 </p>
<p>12 doesnt fit.</p>
<p>for question 2, A B D has awkward construction and E is incorrect in meaning; people do not derive their preference/tastes from the unidentified products in question</p>