<p>I've been doing some past SAT math papers that my school teacher gave the class, and (as expected) I got a few questions wrong. For most of them I've managed to figure out why I was wrong (through research etc) and thus what the correct answer to the questions was, but with these 4 questions below, I still have no idea what the correct answer is or why. </p>
<p>Can someone please help (as I've been desperately clawing my hair out but to no avail) with these questions and importantly, explain why any particular answer is correct?</p>
<p>Question 1:</p>
<p>H(x)=(x^2)+1 and K(x)=-(x^2)+4. If K(H)=0, what are the roots/solutions?
a. ±i√3
b. ±i
c. ±2
d. ±1
e. ±1, ±i√3</p>
<p>Question 2:</p>
<p>3 points lie on an xy plane, with each point representing where a particular person lives. Amy lives at point (2,3); Brian lives at point (0,7); and Claire lives at point (-2,3). They all wish to meet up with each other for lunch, but at a location that is equidistant from each of their respective houses. At what point do they meet?
a. (0,4.5)
b. (0,13/3)
c. (0,3)
d. (1,5)
e. (-1,5)</p>
<p>Question 3: </p>
<p>(X/100) + (Y/66) = (3X+5Y)/Z. What is Z?
a. 8
b. 176
c. 330
d. 3630
e. 7260</p>
<p>I believe Question 1 is from a 2006 SAT and Question 2 and 3 from 2010 and 2011 respectively; in other words, they’re from the CollegeBoard. Also with regards to imaginary numbers, they appear on the SAT (though very rarely obviously) and don’t require you to have anything more than a very basic understanding of them. </p>
<p>OP, I’ll try and find the answer schemes to these papers and thus help out where I can.</p>
<h1>1. The K function has its roots at 2 and -2. So first set H(x) = 2 and you will get one pair of solutions, then set H(x) = -2 to get the other pair.</h1>
<h1>2. Well, you could do trial and error using the distance formula to check whether the three distances are equal. But try drawing the diagram neatly. You will see that only a or b are even plausible…and as is often the case on the SAT, there are hidden 3-4-5 triangles that will lead you to the answer.</h1>
<p>Can you double check that you typed #3 correctly?<br>
Was it “100y” or should it have been “110y”?</p>
<p>110y is a more reasonable question and it leads to answer C.</p>
<p>No idea about the 1st. What are those ±i√3, ±i? </p>
<p>2nd requires distance between 2 points formula: √[(x₂- x₁)^2 + (y₂- y₁)^2)] First draw an xy plane and mark all three points. You get an isosceles triangle. You will see that only A and B makes sense according to the drawing. Only A and B makes sense. I guess A will be correct. Let’s see… From your drawing you can see that the distance between point 0,4.5 and B(0,7) is 2.5. Now let’s see if the distance is the same to the point C.
√(0-(-2))^2+(4.5-3)^2=√2^2+1.5^2= √6.25=2.5 The distance from point A will be the same as from point C because the triangle is isosceles.
So yes, A is correct (0,4.5). The distance is equally 2.5 from all the triangle points.
Maybe there is some shortcut rule to get this but I don’t know it.</p>
<p>3rd answer is most likely C. I had the same problem like pckeller. Something is written incorrectly.
First you should add up the first part of the equation. 66x+100y/6600= 3x+5y/Z
Now multiply in X manner (I don’t know actual math language).
6600(3x+5y) = (66x+100y)Z
19800x + 33000y = 66xz + 100yz
19800x/66x = 33000y/100y = Z
number beside x must be wrong…because you get 300, not 330.</p>
<h1>1 is definitely E. What I did was pretty unnecessary but there you go:</h1>
<p>K(x)=-(x^2)+4 K(h)=0 so i substituted H for "x"in the K equation as in<br>
H(x)=(x^2)+1 So, K(H)= -[(x^2)+1]^2+4
Simplify:
-[x^4+2(x^2)+1]+4=0
-(x^4)-(2(x^2))+3=0
(x^4)+2(x^2)-3=0 (easier to deal with :D)
Then substitute the answer choices and see… You’ll find it’s E that matches.</p>
<h1>2 is A</h1>
<p>I tried roughly graphing it… It’s an isosceles triangle with 2√5 between Amy and Brian, and Claire and Brian. and of coarse 4 between Claire and Amy. I got stuck there, however.-_-" So I decided to just substitute the answer choices in the distance formula of each person and the point luckily it was A! :)</p>
<h1>3 I agree with the others… It’s prob a typo… Ask your teacher… If not… plz let us know!</h1>
<p>Btw, where can someone get those “past papers”??!! I thought they were nonexistent! I need those!
Hope this helped and I didn’t confuse you too much!</p>
<p>The thing about #2 is that know right away that the point you are looking for is on the y axis, somewhere above y = 3. While you could set up an algebraic equation, it really is quicker to check the answers… At y = 4.5, you are 2.5 units away from the upper point. And the distance to either side point: 2 horizontally and 1.5 vertically which just so conveniently happens to be a 3-4-5 triangle that’s been scaled down by a factor of 2.</p>
<p>I agree with pckeller that is there is typo in question#3. </p>
<p>It should instead be:</p>
<p>(X/110) + (Y/66) = (3X+5Y)/Z. What is Z?
a. 8
b. 176
c. 330
d. 3630
e. 7260</p>
<p>and the answer would then be C. </p>
<p>I have not seen this question in the 45 official SAT tests that I have gone through. My only guess is that these could be from SAT I or II Math Subject tests.</p>
<ol>
<li>It should read K(H(x)) = 0 or something like that. K(H) = 0 is unclear. Assuming K(H(x)) = 0, we set H(x) = 2, -2 since K(2) = K(-2) = 0. Then we just solve.</li>
</ol>
<p>H(x) = 2 → x = ±1
H(x) = -2 → x = ±i sqrt(3) → Ans is (E).</p>
<ol>
<li>They meet at the circumcenter of the circle determined by those three points. The triangle is isosceles, so the x-coordinate must be 0, by symmetry. If this were not a multiple choice question, we could find the circumradius by finding the area of the triangle ABC.</li>
</ol>
<p>[ABC] = 8 = (AB<em>BC</em>CA)/4R = 80/4R → R = 2.5 The y-coordinate of the circumcenter is 7 - 2.5 = 4.5, (A). Since this question is multiple choice, we could guess/check A,B,C.</p>
<ol>
<li>“100” should be “110” (since 1/110 and 1/66 are in the ratio 3:5). In this case the answer is 330, (C).</li>
</ol>
<p>I am not from USA and I have just answered all these questions. I am sharing with you guys !! They are not hard at all</p>
<p>1)
(x^2+1)^2+4 = x^4 + 2x^2 + 5 = 0
We can translate it to t
t = x^2 so it will be
t^2 + 2t + 5 = 0
( -b - √∆ ) / 2a and ( -b + √∆ ) / 2a are our roots then you can reach the answer</p>
<p>2)
The points are (2,3) , (0,7) , (-2,3) making a triangle and the question is asking us that triangle’s center of gravity which is G.</p>
<p>G=( (x1+x2+x3) / 3 , (y1+y2+y3) / 3 )</p>
<p>G=( (2+0-2) / 3 , (3+7+3) / 3 ) </p>
<p>G=(0 , 13/3)</p>
<p>3)
In this question there is an mistake. You wrote (x/100)+(y/66)=(3x+5y)/z but it must be
(x/110)+(y/66)=(3x+5y)/z I tried both questions so it must be the second</p>
<p>(66x+110y)/7260=(3x+5y)/z</p>
<p>(66x+110y) / (3x+5y) = 7260 / z</p>
<p>22 = 7260 / z
22z = 7260
z = 330
Which it is C</p>
<p>for 1, and 3, you can just plug the answers into the questions and see which ones work. For 1, just plug the roots into the two equations, and whichever answer makes both of the equations 0 is the correct answer. And for 3, you can just pick random values for x and y, then plug in the answers for z and see which one makes both sides equal. For 2 though, you have to find the center of the circle, which is a bit more complicated.</p>
<p>The first one is pretty easy. You just use what you know about composite functions to basically substitute H into K. Then you will get a fourth degree polynomial, and you can substitute U for x^2 and then solve with factoring. Then the roots take care of themselves nicely. You get x=±sqrt(3) = ±isqrt(3) and x=±1.</p>
<p>The second one is very easy if you have taken Physics and have ever done center or mass problems with discrete systems. Just like ugerduzel said you find the average of i the j vectors (or in the case x and y coordinates) and that is your center of mass (or in this case the mean distance between each point).
You get i component (or x in this case) = (-2+0+2)/3=0
You get j component (or y in this case) = (3+3+7)/3=13/3</p>
<p>The third one has to be a typo. There is one and only one number b that satisfies a/b=c and in this case there are two differing ones.</p>
luckily it was A! :)</p>