No They didn't, not really. They did the observations. Now.
First let us set up the equation we are told—that the product of c and 3 is b.
3c=b
Now we must isolate c so that we can add its value to 3.
3c=b
c= b 3
Finally, let us add this value to 3.
b 3 +3
Our final answer is E, b 3 +3
[Note: because this problem uses variables in both the problem and in the answer choices—a key feature of a PIN question—you can always use the strategy of plugging in numbers to solve the question.]
Because this question uses variables in both the problem and in the answer choices, you can always use PIN to solve it. Simply assign a value for x and then find the corresponding answer in the answer choices. For this explanation, however, we’ll be using algebra.
First, distribute out one of your x’s in the denominator.
x+1 (x)(x2−1)
Now we can see that the (x2−1) can be further factored.
x+1 (x)(x−1)(x+1)
We now have two expressions of (x+1), one on the numerator and one on the denominator, which means we can cancel them out and simply put 1 in the numerator.
1 x(x−1)
And once we distribute the x back in the denominator, we will have:
1 x2−x
Our final answer is J, 1 x2−x .
Before doing anything else, make sure you convert all your measurements into the same scale. Because we are working mainly with inches, convert the table with a 3 foot diameter into a table with a (3)(12)=(36) inch diameter.
Now we know that the tablecloth must hang an additional 5+1 inches on EVERY side, so our full length of the tablecloth, in any straight line, will be:
1+5+36+5+1
48 inches.
Our final answer is K, 48.
The position of the a values (in front of the sine and cosine) means that they determine the amplitude (height) of the graphs. The larger the a value, the taller the amplitude.
Since each graph has a height larger than 0, we can eliminate answer choices C, D, and E.
Because y1 is taller than y2, it means that y1 will have the larger amplitude. The y1 graph has the amplitude of a1, which means that a1 will be larger thana2.
Our final answer is B, 0<a2<a1.
If you remember your trigonometry shortcuts, you know that 1−cos2x+cos2x=1. This means, then, that sin2x=1−cos2x (and that cos2x=1−sin2x).
So we can replace our 1−cos2x in our first numerator with sin2x. We can also replace our 1−sin2x in our second numerator with cos2x. Now our expression will look like this:
√sin2x sinx + √cos2x cosx
We also know that the square root of a value squared will cancel out to be the original value alone (for example,√22=2), so our expression will end up as:
sinx sinx + cosx cosx
Or, in other words:
=1+1
=2
Our final answer is H, 2.
7) We know from working with nested functions that we must work inside out. So we must use the equation for the function g(x) as our input value for function f(x).
f(g(x))=√7x+b
Now we know that this function passes through coordinates (4, 6), so let us replace our x and y values for these givens. (Remember: the name of the function—in this case f(g(x))—acts as our y value).
6=√7(4)+b
36=7(4)+b
36=28+b
8=b
Our final answer is A, b=8.
8) If you’ve brushed up on your log basics, you know that logb( m n )=logb(m)−logb(n). This means that we can work this backwards and convert our first expression into:
log2( 24 3 )
log2(8)
We also know that a log is essentially asking: “To what power does the base need to raised in order to achieve this certain value?” In this particular case, we are asking: “To which power must 2 be raised to equal 8?” To which the answer is 3. (23=8)
Now this expression is equal to log5(x), which means that we must ALSO raise our 5 to the power of 3 in order to achieve x. So:
53
125
Our final answer is J, 125.
9) Once we’ve slogged through the text of this question, we can see that we are essentially being asked to find the largest value of the square root of the sum of the squares of our coordinate points (√x2+y2). So let us estimate what the coordinate points are of our z’s.
Because we are working with squares, negatives are not a factor—we are looking for whichever point has the largest combination of coordinate point, since a negative square will be a positive. At a glance, the two points with the largest coordinates are z1 and z5.
Let us estimate and say that z1 looks to be close to coordinates (-4, 5), which would give us a modulus value of:
√−42+52
√16+25
6.4
Point z5 looks to be a similar distance along the x-axis in the opposite direction, but is considerably lower than point z1. This would probably put it around (4, 2), which would give us a modulus value of:
√42+22
√16+4
4.5
The larger (and indeed largest) modulus value is at point $z_1$
Our final answer is F, z1.
10) For a problem like this, you may not know what a rational number is, but you may still be able to solve it just by looking at whatever answer seems to fit with the others the least. Answer choices A, B, C, and D all produce non-integer values when we take their square root, but answer choice E is the exception.
√
Becomes:
√64 √49
8 7
A rational number is any number that can be expressed as the fraction of two integers, and this is the only option that fits the definition. Or, if you don’t know what a rational number is, you can simply see that this is the only answer that produces integer values once we have taken the root, which makes it stand out from the crowd.
Our final answer is E, √
11) Because we are working with numbers in the triple digits, our numbers with at least one 0 will have that 0 in either the units digit or the tens digit (or both, though they will only be counted once).
We know that our numbers are inclusive, so our first number will be 100, and will include every number from 100 though 109. That gives us 10 numbers so far.
From here, we can see that the first 10 numbers of 200, 300, 400, 500, 600, 700, 800, and 900 will be included as well, giving us a total of:
10*9
90 so far.
Now we also must include every number that ends in 0. For the first 100 (NOT including 100, which we have already counted!), we would have:
110, 120, 130, 140, 150, 160, 170, 180, 190
This gives us 9 more numbers, which we can also expand to include 9 more in the 200’s, 300’s, 400’s, 500’s, 600’s, 700’s, 800’s, and 900’s. This gives us a total of:
9*9
81
Now, let us add our totals (all the numbers with a units digit of 0 and all the numbers with a tens digit of 0) together:
90+81
171
There are a total of 900 numbers between 100 and 999, inclusive, so our final probability will be:
171 900
Our final answer is D, 171 900
12) First, turn our given equation for line q into proper slope-intercept form.
−2x+y=1
y=2x+1
Now, we are told that the angles the lines form are congruent. This means that the slopes of the lines will be opposites of one another [Note: perpendicular lines have opposite reciprocal slopes, so do NOT get these concepts confused!].
Since we have already established that the slope of line q is 2, line r must have a slope of -2.
Our final answer is F, -2
13) If you remember your trigonometry rules, you know that tan−1( a b ) is the same as saying tanΘ= a b . Knowing our mnemonic device SOH, CAH, TOA, we know that tan Θ = opposite/adjacent. If a is our opposite and b is our adjacent, this means that Θ will be our right-most angle.
Knowing that, we can find the cos of Θ as well. The cosine will be the adjacent over the hypotenuse. the adjacent still being b and the hypotenuse being √a2+b2. So cos[tan−1( a b )] will be:
b √a2+b2
Our final answer is D, b √a2+b2
14) By far the easiest way to solve this question is to use PIN and simply pick a number for our x and find its corresponding y value. After which, we can test out our answer choices to find the right one.
So if we said x was 24, (Why 24? Why not!), then our t value would be 2, our u value would be 4, and our y value would be 42. And x−y would be 24−42=−18
Now let us test out our answer choices.
At a glance, we can see that answer choices H and J would be positive and answer choice K is 0. We can therefore eliminate them all.
We can also see that (t−u) would be negative, but (u−t) would not be, so it is likely that F is our answer. Let us test it fully to be sure.
9(t−u)
9(2−4)
9(−2)
−18
Success!