<p>Yes, of course, for you guys it is never difficult enough! Unless the question is out of scope, which is kind of unfair.</p>
<p>Suppose a and b are even numbers. Let c = 2a^2 + 4b^2. The greatest even factor of c is:</p>
<p>A) 2
B) 4
C) 6
D) 8
E) 16</p>
<p>c = 2a^2 + 4b^2 = 2 (a^2 + 2.b^2)
Since a and b are even numbers,
Let a = 2k
and b = 2m
c = 2 (a^2 + 2.b^2) = 2 [(2k)^2 + 2.(2m)^2]
c = 2[ 4k^2 + 8m^2]
c = 8 [ k^2 + 2m^2]
So, in general, the greatest even factor is 8.</p>
<p>However, what if k is even?
Then 2 can be taken out common from inside the brackets, and c can be expressed as a multiple of 16, in which case 16 will become the greatest even factor.</p>
<p>So, a bit vague.</p>
<p>Learn matrices! I had to plug them into my calculator to them and wasted a lot of time.</p>
<p>Matrices?</p>
<p>Suppose x > 0.</p>
<p>For what value of x does the inverse of the 3x3 matrix below NOT exist?</p>
<p>1 0 1
x 1 5
-1 x -7</p>
<p>A) 0 B) 1 C) 2 D) 4 E) 6</p>
<p>Inverse of a matrix A = Adjoint A / determinant of A
If determinant is 0, then matrix won't be invertible.</p>
<p>Determinant A = 1<em>(-7-5x) + 0 + 1</em>(x^2 +1) = -7-5x + x^2 + 1
= x^2 - 5x - 6 = (x-6)(x+1)
So x cannot be 6 or -1.</p>
<p>And out of the options we have 6 as the answer.</p>
<p>I concur with spidey. We don't know what what specific even number K is. Thus we don't know many factors of 2^x it has out of the given answer the largest <i>could</i> be 16.</p>
<p>^You guys are correct. What I meant to say was: "what is the greatest even number that must be a factor of c?"</p>
<p>Ya, wording is important lol</p>
<p>OK, a hard percentages problem. Soon to be attacked by many spiders, I'm sure.</p>
<p>An item in a store is priced in dollars and cents. With 4% sales tax, the final price is exactly p dollars where p is an integer (no rounding necessary). The smallest possible final price p is:</p>
<p>A) 1
B) 13
C) 25
D) 26
E) 50</p>
<p>not if mike gets there first lol</p>
<p>I'm not sure if I understand this problem. You ask for the smallest p is the difference between the 4% increase and the original 1.04x - 1x = .04x. So if my understanding if correct, then the answer is the smallest value that divides eventally .04 which is 1.</p>
<p>Let x be the initial price be D dollars + c cents
1.04 (D dollars + c cents) = p
p / 1.04 = D dollars + C cents.</p>
<p>Now, 1/1.04 is a rational number but non terminating, so the number of cents wont be a definite number (1/1.04 = 0.96153846153846153846153846153846 and counting)</p>
<p>13/1.04 however, is 12.5 which is 12 dollars and 5 cents .
So 13 is the smallest final possible price.</p>
<p>What is the length of the transverse axis of xy=-24?</p>
<p>A) sqrt(48)
B) sqrt(112)
C) sqrt(172)
D) sqrt(192)
E)sqrt(212)</p>
<p>xy = -24 is a rectangular hyperbola lying in 2nd and 4th quadrants.</p>
<p>We need to find the points on the hyperbola that are closest to the origin (0,0).
That can be found by the intersection of the hyperbola with the line y = -x
so
-x^2 = -24
x = +-sqrt(24)
Thus y also = +-sqrt(24)
So the points are (sqrt(24), - sqrt(24) ) & (-sqrt(24), sqrt(24) )</p>
<p>The length of the transverse axis is the distance between these two points:
= sqrt[ (rt(24) + rt(24))^2 + (rt(24) + rt(24))^2]
= sqrt[ 4<em>24 + 4</em>24]
= sqrt(192)</p>
<p>f(x) = [-(x-a)^2 * (x-b)(x-d)^10 * x^2]/[(x-c)^3 * (x-e)]
a<b<0<c<d<e
Which of the follow represent the intervals at which f(x) is negative?</p>
<p>A) (b,0) U (0,c) U (e, infinity)
B) (-infinity, a) U (a,b) U (c,d) U (d,e)
C) (a,b) U (0,c) U (d,e)
D) (-infinity,a) U (b,0) U (e,infinity)
E) None of the above</p>
<p>(I think spidey will still get this lol)</p>
<p>The even powered terms don't matter, since they will always be non - negative.
So what matters is:
-(x-b) / (x-c)^3 . (x-e)</p>
<p>So for f(x) to be negative:
-(x-b) / (x-c)^3 . (x-e) < 0
OR
(x-b) / (x-c)^3 . (x-e) > 0
where x not equal to b, c, or e.
Now plot b, c and e on the number line.
b will be the leftmost, then c on the right, and e will be further right.
So for (x-b) / (x-c)^3 . (x-e) to be greater than 0, use the wavy curve method...
thus x E (b,c) U (e, infinity)
But in the original expression, we also require x not equal to a, b, d, c, e or 0.
So x E (b,0) U (0,c) U (e, infinity)
option A.</p>
<p>Nice one :)</p>
<p>I'm out of questions for now, but I'll make some up later if you're interested lol.</p>
<p>Got some more!
This one's not multiple choice :P</p>
<p>State the specific and complete requirements for a,b,c,d,e so the following figures are formed:</p>
<p>ax^2 + by^2 + cx + dy + e = 0
for a circle and an ellipse</p>
<p>Actually u missed an 'xy' term which is required for ellipse.
Btw, the actual equation for conic section is :
ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0</p>
<p>Delta = abc + 2fgh - af^2 - bg^2 - ch^2</p>
<p>For circle: Delta not equal to 0 & h=0 & a=b
For ellipse : Delta not equal to 0 & h^2 < ab</p>
<p>And now you're just having fun lmao, I can't waste time answering such questions dude ;) Ur just messing.</p>
<p>I did something different <_<
If you have ax^2 + cx + by^2 + dy = -e
you need to put it in the form of an ellipse, so you complete the square on the LHS and get</p>
<p>a(x^2 + cx/a + (c/2a)^2) + b(y^2 + dy/b + (d/2b)^2) = -e + c^2/4a + d^2/4b (need to add extra terms on this side)</p>
<p>then factor the LHS...</p>
<p>a(x+(c/2a)^2) + b(y+(d/2b)^2 = -e + c^2/4a + d^2 / 4b</p>
<p>Then manipulate the above to create a circle or ellipse:
Circle: a=b, a and b don't equal 0, and -e + c^2/4a + d^2/4b > 0</p>
<p>Ellipse: a does not equal b, a is the same sign as b, -e + c^2/4a + d^2/4b > 0, and a and b don't equal 0</p>
<p>If you're wondering where I got these questions, I had a precalc teacher with an interest in obscure questions and these came from my old tests.</p>
<p>Have fun trying to follow that <_<</p>
<p>What's the equation of the circle that has its center at the origin and is tangent to the line with equation 3x-4y = 10?</p>