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## Mathematics

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- Question 1 of 80
##### 1. Question

**Without using a calculator or a pencil and paper, which of these is a perfect square?**Correct**This may seem impossible to do, but three of these choices can be eliminated based on the digits. 2893467827688677 and 9134751098236810 can both be eliminated because a square number must end in a 1, 4, 5, 6, 9, or 00. Finally, 5916378240699596 is eliminated because a square number’s digital root must be 1, 4, 7, or 9. A digital root is calculated by adding the digits of the number and then adding the digits of the resulting number and repeating the process until we are left with a 1-digit number (e.g. to find the digital root of 89…8+9=17, 1+7=8). This leaves us only one choice. Hehehe – Mark Llego**Incorrect**This may seem impossible to do, but three of these choices can be eliminated based on the digits. 2893467827688677 and 9134751098236810 can both be eliminated because a square number must end in a 1, 4, 5, 6, 9, or 00. Finally, 5916378240699596 is eliminated because a square number’s digital root must be 1, 4, 7, or 9. A digital root is calculated by adding the digits of the number and then adding the digits of the resulting number and repeating the process until we are left with a 1-digit number (e.g. to find the digital root of 89…8+9=17, 1+7=8). This leaves us only one choice. Hehehe – Mark Llego** - Question 2 of 80
##### 2. Question

**What is the sum of the infinite series 1 + 0.5 + 0.25 + 0.125 + 0.0625 + … ?**Correct**The formula for finding an infinite sum is A/(1-R), where A is the first term and R is the common ratio, or the number by which you have to multiply the first term to get the second term. In this case, A = 1 and R = 0.5. A/(1-R) = 1/(1-0.5) = 1/0.5 = 2.**Incorrect**The formula for finding an infinite sum is A/(1-R), where A is the first term and R is the common ratio, or the number by which you have to multiply the first term to get the second term. In this case, A = 1 and R = 0.5. A/(1-R) = 1/(1-0.5) = 1/0.5 = 2.**##### Hint

(A number, not a word)

- Question 3 of 80
##### 3. Question

**Let f(x) = 3x/2. Let g(x) = 2x + 3. What is f(g(f(g(f(g(f(2)))))))? Write your answer as a fraction, not a decimal.**Correct**It just takes some work to do this one. Start like this: f(2) = 3(2)/2 = 3. g(3) = 2(3) + 3 = 9, and so on.**Incorrect**It just takes some work to do this one. Start like this: f(2) = 3(2)/2 = 3. g(3) = 2(3) + 3 = 9, and so on.**##### Hint

Type in a common fraction in the form x/y.

- Question 4 of 80
##### 4. Question

**An ant starts at the origin and moves one unit to the right, 1/2 unit up, 1/4 unit left, 1/8 unit down, 1/16 unit right, and so on. If it continues infinitely, on what point is it converging?**Correct**Take the coordinates separately. First, the x-coordinate. The x-coordinate of where the ant converges can be expressed as an infinite sum 1 – 1/4 + 1/16 – 1/64 + …. Using the formula given earlier, the sum is 1/(1-(-1/4)) = 4/5, so the x-coordinate is 4/5. Similarly solving for the y-coorindate of the point, we get 2/5.**Incorrect**Take the coordinates separately. First, the x-coordinate. The x-coordinate of where the ant converges can be expressed as an infinite sum 1 – 1/4 + 1/16 – 1/64 + …. Using the formula given earlier, the sum is 1/(1-(-1/4)) = 4/5, so the x-coordinate is 4/5. Similarly solving for the y-coorindate of the point, we get 2/5.** - Question 5 of 80
##### 5. Question

**I am somewhere on earth. I go one mile south, one mile east, and one mile north and I am back where I started. Where must I be?**CorrectYes, the North Pole is one possible place, but there are others. The “widest” part of the Earth is at the equator. However, if you go far enough north or south, the “circumference” of the earth will get smaller, eventually, near the poles, the circumference is only a mile. If you are one mile above such a place, then you can go one mile south and one mile east, circling the earth to end up right where you were, and then one mile north to end up back where you started. This is sort of hard to explain without a diagram.

IncorrectYes, the North Pole is one possible place, but there are others. The “widest” part of the Earth is at the equator. However, if you go far enough north or south, the “circumference” of the earth will get smaller, eventually, near the poles, the circumference is only a mile. If you are one mile above such a place, then you can go one mile south and one mile east, circling the earth to end up right where you were, and then one mile north to end up back where you started. This is sort of hard to explain without a diagram.

- Question 6 of 80
##### 6. Question

**An easy one. Solve for x:****7x – 2 = 4x + 1****Write your answer as a number.**Correct**Subtract 4x from both sides and you get 3x – 2 = 1. Then, add 2 to both sides to get 3x = 3. Finally, divide both sides by 3 and you get x = 1.**Incorrect**Subtract 4x from both sides and you get 3x – 2 = 1. Then, add 2 to both sides to get 3x = 3. Finally, divide both sides by 3 and you get x = 1.** - Question 7 of 80
##### 7. Question

**Another easy one:**

Factor:

x^3 – y^3Correct**If you do not already know how to factor the difference of two cubes, then just multiply all of the choices and you will get the right answer.**Incorrect**If you do not already know how to factor the difference of two cubes, then just multiply all of the choices and you will get the right answer.** - Question 8 of 80
##### 8. Question

**What is the slope of the line tangent to the parabola f(x) = x^2 + 1 at x = 2?**Correct**To find the slope of the line tangent to a curve at x = a, you use limits. The expression is the limit as h approaches 0 of the expression (f(a + h) – f(h))/h. Evaluating the limit, we get 4.**Incorrect**To find the slope of the line tangent to a curve at x = a, you use limits. The expression is the limit as h approaches 0 of the expression (f(a + h) – f(h))/h. Evaluating the limit, we get 4.** - Question 9 of 80
##### 9. Question

**1 + 2 + 3 + 4 + 5 + … + 100**CorrectThere is a formula for the sum of 1 + 2 + 3 + 4 + 5 + … + n. It is n(n+1)/2. Substituting 100 for n, we get 5050. If you don’t know this formula, you can find the sum quickly by grouping the numbers like this:

1 + 2 + 3 + 4 + … + 100 = 1 + 100 + 2 + 99 + … + 50 + 51. This is equal to 101 + 101 + 101 + 101 + … 50 times. Multiplying, 50 x 101 = 5050.IncorrectThere is a formula for the sum of 1 + 2 + 3 + 4 + 5 + … + n. It is n(n+1)/2. Substituting 100 for n, we get 5050. If you don’t know this formula, you can find the sum quickly by grouping the numbers like this:

1 + 2 + 3 + 4 + … + 100 = 1 + 100 + 2 + 99 + … + 50 + 51. This is equal to 101 + 101 + 101 + 101 + … 50 times. Multiplying, 50 x 101 = 5050. - Question 10 of 80
##### 10. Question

**What is the maximum number of regions into which 100 straight lines can divide a plane?**Correct**The formula for finding the number of regions that n lines can divide a plane into is (n(n+1)/2) + 1. Substituting 100 for n, we get 5051.**Incorrect**The formula for finding the number of regions that n lines can divide a plane into is (n(n+1)/2) + 1. Substituting 100 for n, we get 5051.** - Question 11 of 80
##### 11. Question

**According to the Pythagorean Theorem, the square of the hypotenuse of a right triangle is equal to what?**Correct**The Pythagoreans were a strict secret society in Ancient Greece. It is not known whether Pythagoras himself discovered this famous theorem, since the Pythagoreans (even after his death) attributed all results to him.**Incorrect**The Pythagoreans were a strict secret society in Ancient Greece. It is not known whether Pythagoras himself discovered this famous theorem, since the Pythagoreans (even after his death) attributed all results to him.** - Question 12 of 80
##### 12. Question

**The Pythagorean Theorem only applies to right-angled triangles. However, there is a more general “law” that governs all triangles in a relationship similar to that of the Pythagorean Theorem. What is the name of this law?**Correct**The cosine law states that c^2=a^2+b^2-2ab*cos C (where capital C is the angle). Note that this reduces to the Pythagorean Theorem when C=90 degrees, because cos 90 = 0. There are sine and tangent laws, they describe ratios in the triangle. There is no “triangle law” that I know of.**Incorrect**The cosine law states that c^2=a^2+b^2-2ab*cos C (where capital C is the angle). Note that this reduces to the Pythagorean Theorem when C=90 degrees, because cos 90 = 0. There are sine and tangent laws, they describe ratios in the triangle. There is no “triangle law” that I know of.** - Question 13 of 80
##### 13. Question

**Moving on several hundred years, in Italy in the 1500s there arose a great dispute between two leading mathematicians named Cardano and Tartaglia over a new method. What was this method?**Correct**The method was for the general solution of polynomial equations of degree 3. The general solution for degree 4 curves was found not long after, and in the 1800s Abel proved that equations of degree 5 and higher cannot be solved in general. The “exhaustion” method was used by Archimedes (a primitive version of the integral calculus), “fluxions” was Sir Isaac Newton’s invention, and the “Erlangen Programme” was stressed by the German Felix Klein in the late 1800s to unify group theory and geometry.**Incorrect**The method was for the general solution of polynomial equations of degree 3. The general solution for degree 4 curves was found not long after, and in the 1800s Abel proved that equations of degree 5 and higher cannot be solved in general. The “exhaustion” method was used by Archimedes (a primitive version of the integral calculus), “fluxions” was Sir Isaac Newton’s invention, and the “Erlangen Programme” was stressed by the German Felix Klein in the late 1800s to unify group theory and geometry.** - Question 14 of 80
##### 14. Question

**Moving on to the great Swiss mathematician Euler, he was one of the first to really explore infinite series. It has been known for a long time that the harmonic series (1+1/2+1/3+1/4+…) diverges, but that the infinite sum of the reciprocals of the squares (1+1/4+1/9+1/16+1/25…) converges. Euler was the first to determine the exact value of convergence, what is it?**CorrectThis is one of the most beautiful and interesting results in infinite series: that the sum of the reciprocals of the squares “closes in” on the irrational number (pi squared)/6. This result can also be used to provide a proof that there are an infinite number of prime numbers.

IncorrectThis is one of the most beautiful and interesting results in infinite series: that the sum of the reciprocals of the squares “closes in” on the irrational number (pi squared)/6. This result can also be used to provide a proof that there are an infinite number of prime numbers.

- Question 15 of 80
##### 15. Question

**Fermat’s Last Theorem is undoubtedly the most famous theorem in all of mathematics, first being proposed in the 1600s but not fully solved until the mid 1990s. What method was used to finally solve this centuries-old problem?**Correct**The method of infinite descent was used by Fermat in order to solve lower degree specific cases of the theorem, and is essentially a proof by contradiction. This method was proven not to work in general. Direct proofs also exist for specific cases (I have seen n=3 and n=4) but no general direct proof is known. Wiles finally cracked Fermat’s enigma using modular forms of elliptic curves.**Incorrect**The method of infinite descent was used by Fermat in order to solve lower degree specific cases of the theorem, and is essentially a proof by contradiction. This method was proven not to work in general. Direct proofs also exist for specific cases (I have seen n=3 and n=4) but no general direct proof is known. Wiles finally cracked Fermat’s enigma using modular forms of elliptic curves.** - Question 16 of 80
##### 16. Question

**Another famous theorem was first conjectured in the 1800s, but was not solved until 1976 in a highly controversial way: the proof depends on the use of a computer. To which theorem am I referring?**Correct**To this day there are mathematicians that debate the validity of the proof, as it cannot be manually checked by man. The computer program used, however, can be checked and reproduced. Rolle’s theorem is a simple result from the calculus, the marriage theorem is a combinatorial result that describes matchings, and Kuratowski’s theorem is a test for planarity of graphs.**Incorrect**To this day there are mathematicians that debate the validity of the proof, as it cannot be manually checked by man. The computer program used, however, can be checked and reproduced. Rolle’s theorem is a simple result from the calculus, the marriage theorem is a combinatorial result that describes matchings, and Kuratowski’s theorem is a test for planarity of graphs.** - Question 17 of 80
##### 17. Question

An example of a deceptively simple-stated theorem with an extraordinarily difficult proof is the Jordan curve theorem. What is the fundamental result of this theorem?

Correct**Seems obvious, doesn’t it? The proof, however, is not. The reason is due to topological generalizations of the notions of “closed” and “open”.**Incorrect**Seems obvious, doesn’t it? The proof, however, is not. The reason is due to topological generalizations of the notions of “closed” and “open”.** - Question 18 of 80
##### 18. Question

**Ernst Kummer tried in vain to prove Fermat’s Last Theorem, and his method (similar to Fermat’s own) broke down because not all rings have the “nice” property of unique factorization. Studying these unique factorization domains did not lead to a proof of Fermat’s Last Theorem, but it did lead to the study of these:**Correct**Kummer called a number “ideal” if it behaved like a prime number in any arbitrary ring. The other three answers are algebraic terms, but were around long before Kummer.**Incorrect**Kummer called a number “ideal” if it behaved like a prime number in any arbitrary ring. The other three answers are algebraic terms, but were around long before Kummer.** - Question 19 of 80
##### 19. Question

**Cantor made waves in the traditional mathematical community by proving what?**CorrectThe other three answers are variations of the same result. Cantor’s most violent opponent was Kronecker, who strongly disagreed with the notion of different “sizes” of infinity. Cantor had several nervous breakdowns and eventually died in an asylum.

IncorrectThe other three answers are variations of the same result. Cantor’s most violent opponent was Kronecker, who strongly disagreed with the notion of different “sizes” of infinity. Cantor had several nervous breakdowns and eventually died in an asylum.

- Question 20 of 80
##### 20. Question

**Finally, a practical application. There is a result that says the following: if you climb up a mountain in a certain amount of time, and then climb down the mountain the next day in the exact same amount of time, there is at least one location on the mountain that you were at both days at exactly the same time (from when you started). Note that you can change your speed, stop to rest, take different paths, etc; you will always have one fixed point in your journey. What result guarantees this?**Correct**If we were to paramatize the curves, the graphs of ascent and descent would be automorphic (since the time for each of the ascent and descent is the same). Brouwer’s theorem says that any plane automorphism has a fixed point. Chebyshev’s theorem deals with number theory, the Minkowski theorem deals with optimization, and the Lebesgue theorem refines the notion of an integral.**Incorrect**If we were to paramatize the curves, the graphs of ascent and descent would be automorphic (since the time for each of the ascent and descent is the same). Brouwer’s theorem says that any plane automorphism has a fixed point. Chebyshev’s theorem deals with number theory, the Minkowski theorem deals with optimization, and the Lebesgue theorem refines the notion of an integral.** - Question 21 of 80
##### 21. Question

**The young mathematician is presented with this problem:****(13 + 7) x (5 – 3) / 4 x (1 + 1) =****Which operation would the young mathematician do first?**Correct**Parentheses take precedence, always – they are only necessary in an equation for that purpose. Ignore them to your peril.****So the answer would be:****(13 + 7) x (5 – 3) / 4 x (1 + 1) =**

20 x 2 / 4 X 2 =

40/8 = 5Incorrect**Parentheses take precedence, always – they are only necessary in an equation for that purpose. Ignore them to your peril.****So the answer would be:****(13 + 7) x (5 – 3) / 4 x (1 + 1) =**

20 x 2 / 4 X 2 =

40/8 = 5 - Question 22 of 80
##### 22. Question

**The young math student wants to learn more. He proceeds with his lessons. One of the most basic properties of math is demonstrated here. What is it called?****2 + 3 = 3 + 2**Correct**This is the commutative property. Something that commutes (like a commutator or commuter) goes back and forth, changing direction returning to its original state/position. The math example demonstrates that same property, since writing the equation in either direction (as “3 + 2” and “2 + 3”) results in the same answer, 5.****Note that addition and multiplication are both commutative, while subtraction and division are not. Try a few examples and show it to yourself.**Incorrect**This is the commutative property. Something that commutes (like a commutator or commuter) goes back and forth, changing direction returning to its original state/position. The math example demonstrates that same property, since writing the equation in either direction (as “3 + 2” and “2 + 3”) results in the same answer, 5.****Note that addition and multiplication are both commutative, while subtraction and division are not. Try a few examples and show it to yourself.** - Question 23 of 80
##### 23. Question

**The young student is taught another basic, yet important, property of mathematic operations, and it’s demonstrated below. Which property is it?****(5 + 2) + 7 = 5 + (2 + 7)**Correct**Note the order of the operands does not change as it would in a demonstration of commutativity, just the arrangement of the parentheses.****Think of the three operands, 5, 2 and 7 as classmates. In the example, the left side shows classmates 5 and 2 being friends (in other words, “associating”) leaving ‘7’ by itself. The right side of the equation shows the same three classmates, but this time 2 and 7 are associating, and 5 is alone.**Incorrect**Note the order of the operands does not change as it would in a demonstration of commutativity, just the arrangement of the parentheses.****Think of the three operands, 5, 2 and 7 as classmates. In the example, the left side shows classmates 5 and 2 being friends (in other words, “associating”) leaving ‘7’ by itself. The right side of the equation shows the same three classmates, but this time 2 and 7 are associating, and 5 is alone.** - Question 24 of 80
##### 24. Question

**Which mathematical property is demonstrated below?****3 + 0 = 3**

CorrectAdding zero to any number does not change the number’s value. So, by definition, zero is the identity element for addition (and subtraction). Likewise, the number one is the identity element for multiplication and division, as in 7 x 1 = 7 .

While this property may seem obvious, it is good to understand what it means and how one can use it to simplify otherwise (possibly complex) equations.

IncorrectAdding zero to any number does not change the number’s value. So, by definition, zero is the identity element for addition (and subtraction). Likewise, the number one is the identity element for multiplication and division, as in 7 x 1 = 7 .

While this property may seem obvious, it is good to understand what it means and how one can use it to simplify otherwise (possibly complex) equations.

- Question 25 of 80
##### 25. Question

**Which property of math is the young student learning with the following equation?****3 x (4 + 7) = (3 x 4) + (3 x 7)**CorrectNote how the three is “distributed” to each of the elements in the parentheses. Knowing this characteristic is important for the proper solving of certain algebraic equations.

IncorrectNote how the three is “distributed” to each of the elements in the parentheses. Knowing this characteristic is important for the proper solving of certain algebraic equations.

- Question 26 of 80
##### 26. Question

**By convention, what is the correct order of operation for the following equation, written as such?****5 + 2 x 8 / 4 – 3 =**Correct**We scan the equation and perform operations in the following order:**

Parentheses, Exponents and Roots, Multiplication and Division, Addition and Subtraction (sometimes abbreviated as PEMDAS or PEDMAS). If there are competing operators, work from left to right. So we would do the following:**2 x 8 = 16, then 16 / 4 = 4, then add 5, then subtract 3 and we get 6.****Imagine if we tossed aside convention and simply performed the math in pure left-to-right order?****5 + 2 = 7, then 7 x 8 = 56, then 56 / 4 = 14, then 14 – 3 = 10!**Incorrect**We scan the equation and perform operations in the following order:**

Parentheses, Exponents and Roots, Multiplication and Division, Addition and Subtraction (sometimes abbreviated as PEMDAS or PEDMAS). If there are competing operators, work from left to right. So we would do the following:**2 x 8 = 16, then 16 / 4 = 4, then add 5, then subtract 3 and we get 6.****Imagine if we tossed aside convention and simply performed the math in pure left-to-right order?****5 + 2 = 7, then 7 x 8 = 56, then 56 / 4 = 14, then 14 – 3 = 10!** - Question 27 of 80
##### 27. Question

**Which of the following DOES NOT indicate multiplication of the number 4 and the variable, ‘Y’?**Correct**Another is to place a dot (a little higher than a period) between the ‘4’ and the ‘Y’. Note how confusing it would get to use the 4 x Y approach we all learned as youngsters in Algebra where we often use the letter ‘x’ to represent “the unknown quantity”: 4 x X is a bit confusing.****Most mathematicians use the 4Y syntax – it’s unambiguous, succinct and easy to write.****Programmers use the asterisk: 4 * Y**Incorrect**Another is to place a dot (a little higher than a period) between the ‘4’ and the ‘Y’. Note how confusing it would get to use the 4 x Y approach we all learned as youngsters in Algebra where we often use the letter ‘x’ to represent “the unknown quantity”: 4 x X is a bit confusing.****Most mathematicians use the 4Y syntax – it’s unambiguous, succinct and easy to write.****Programmers use the asterisk: 4 * Y** - Question 28 of 80
##### 28. Question

**In mathematics, what is a binary operation?**Correct**Addition and multiplication are two examples of binary operations.****The equations 2 + 3 and 2 x 3 each have two operands (the ‘2’ and the ‘3’) and one binary operator (the ‘+’ or the ‘x’)****There are “unary” operations such as placing a minus sign in front of a number to make it negative: -41****One is more likely to run into unary operators in computer programming, however.**Incorrect**Addition and multiplication are two examples of binary operations.****The equations 2 + 3 and 2 x 3 each have two operands (the ‘2’ and the ‘3’) and one binary operator (the ‘+’ or the ‘x’)****There are “unary” operations such as placing a minus sign in front of a number to make it negative: -41****One is more likely to run into unary operators in computer programming, however.** - Question 29 of 80
##### 29. Question

**When multiplying two negative numbers, what kind of number does one always get for an answer?**Correct**Two negative numbers, when multiplied, always yield a positive number.**Incorrect**Two negative numbers, when multiplied, always yield a positive number.** - Question 30 of 80
##### 30. Question

**Generally speaking, dividing some non-zero number by zero yields what result?**Correct**Depending on the context, dividing by zero can be quite problematic. In the simplest case, the answer is undefined – after all how does one physically divide something zero times? However, in some more advanced cases it is useful to consider the answer to be infinity.**Incorrect**Depending on the context, dividing by zero can be quite problematic. In the simplest case, the answer is undefined – after all how does one physically divide something zero times? However, in some more advanced cases it is useful to consider the answer to be infinity.** - Question 31 of 80
##### 31. Question

**Arithmetic:****What is the value of 3 + 2*7 ? Here * denotes multiplication.**Correct**By order of operations, you need to do the multiplication first, then the addition:****3 + 2*7 = 3 + 14 = 17**Incorrect**By order of operations, you need to do the multiplication first, then the addition:****3 + 2*7 = 3 + 14 = 17** - Question 32 of 80
##### 32. Question

**Basic Algebra:****Which of the following is equal to (2x – 7)^2 ?**Correct**Recall that (A – B)^2 = A^2 – 2AB + B^2. Put A = 2x, B = 7:****(2x – 7)^2 = (2x)^2 – 2*(2x)*7 + 7^2 = 4x^2 – 28x + 49.**Incorrect**Recall that (A – B)^2 = A^2 – 2AB + B^2. Put A = 2x, B = 7:****(2x – 7)^2 = (2x)^2 – 2*(2x)*7 + 7^2 = 4x^2 – 28x + 49.** - Question 33 of 80
##### 33. Question

**Analytic Geometry:****What can you say about the eccentricity e of a hyperbola?**Correct**All conic sections can be described in terms of a fixed point, called a focus, and a fixed line, called a directrix. Let P be a point on the conic, let F be the focus, and let l denote the directrix. Then the eccentricity e can be defined as the following ratio:****e = (the distance from P to F)/(the distance from P to l)****This ratio in fact is always a constant for a conic section. We can use the concept of eccentricity to distinguish all the conic sections. If e is greater than 1, the conic is a hyperbola. If e = 1, the conic is a parabola. If e is between 0 and 1, the conic is an ellipse. Finally, we will define the eccentricity of a circle to be 0. The eccentricity can also be computed from the standard form of the equations of an ellipse or hyperbola.**Incorrect**All conic sections can be described in terms of a fixed point, called a focus, and a fixed line, called a directrix. Let P be a point on the conic, let F be the focus, and let l denote the directrix. Then the eccentricity e can be defined as the following ratio:****e = (the distance from P to F)/(the distance from P to l)****This ratio in fact is always a constant for a conic section. We can use the concept of eccentricity to distinguish all the conic sections. If e is greater than 1, the conic is a hyperbola. If e = 1, the conic is a parabola. If e is between 0 and 1, the conic is an ellipse. Finally, we will define the eccentricity of a circle to be 0. The eccentricity can also be computed from the standard form of the equations of an ellipse or hyperbola.** - Question 34 of 80
##### 34. Question

**Calculus:****Suppose f(x) is a differentiable function for all real numbers x which also satisfies:****f ‘(4) = 0,**

f ‘(x) is negative for all x less than 4,

f ‘(x) is positive for all x greater than 4**What can be said about the critical value x = 4?**Correct**Since f ‘(x) is negative for all x less than 4, f(x) is decreasing for all x less than 4. Since f ‘(x) is positive for all x greater than 4, f(x) is increasing for all x greater than 4. Therefore, f(x) has a local minimum at x = 4 by the first derivative test. The first derivative test will determine the nature of a local extrema even when the second derivative test fails. So in that respect, it is a better test. However, the second derivative test is faster when the second derivative is easier to compute. Note that we couldn’t use the second derivative test here since I gave no information about it (in fact, it didn’t have to exist anywhere!)**Incorrect**Since f ‘(x) is negative for all x less than 4, f(x) is decreasing for all x less than 4. Since f ‘(x) is positive for all x greater than 4, f(x) is increasing for all x greater than 4. Therefore, f(x) has a local minimum at x = 4 by the first derivative test. The first derivative test will determine the nature of a local extrema even when the second derivative test fails. So in that respect, it is a better test. However, the second derivative test is faster when the second derivative is easier to compute. Note that we couldn’t use the second derivative test here since I gave no information about it (in fact, it didn’t have to exist anywhere!)** - Question 35 of 80
##### 35. Question

**Advanced Calculus:****Let f(x,y,z) = xyz. What is the gradient of f?**Correct**The components of the gradient are the partial of f with respect to x, the partial of f with respect to y, and the partial of f with respect to z. Partial derivatives are found by differentiating with respect to the variable, treating the other variables as constants.**Incorrect**The components of the gradient are the partial of f with respect to x, the partial of f with respect to y, and the partial of f with respect to z. Partial derivatives are found by differentiating with respect to the variable, treating the other variables as constants.** - Question 36 of 80
##### 36. Question

**Basic Set Theory:****How many 2 element subsets does a 4 element set have?**Correct**The number of 2 element subsets of a 4 element set is just C(4,2) = 6. You can also list them: Suppose S = {a, b, c, d}. These are the two element subsets of S:****{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}**Incorrect**The number of 2 element subsets of a 4 element set is just C(4,2) = 6. You can also list them: Suppose S = {a, b, c, d}. These are the two element subsets of S:****{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}** - Question 37 of 80
##### 37. Question

**Linear Algebra:****Let A be a square matrix with the property that there exists a positive integer k such that A^k is the zero matrix. A is said to be which of the following?**Correct**Nilpotent matrices are important in the study of canonical forms. An orthogonal matrix is one whose transpose is equal to its inverse, a symmetric matrix is one that equals its transpose, and an idempotent matrix is one that equals its square.**Incorrect**Nilpotent matrices are important in the study of canonical forms. An orthogonal matrix is one whose transpose is equal to its inverse, a symmetric matrix is one that equals its transpose, and an idempotent matrix is one that equals its square.** - Question 38 of 80
##### 38. Question

**Elementary Number Theory:****Which of the following numbers is congruent to 3 mod 7?**Correct**Let a and b be integers. We say that a is congruent to b mod n if n divides a – b, in other words, if there exists an integer k such that****a – b = kn.****17 is the answer because when we divide 17 by 7, the remainder is 3:****17 = 2*7 + 3 so 17 is congruent to 3 mod 7.****You can check that mod 7, 27 is congruent to 6, 37 is congruent to 2, and 47 is congruent to 5.**Incorrect**Let a and b be integers. We say that a is congruent to b mod n if n divides a – b, in other words, if there exists an integer k such that****a – b = kn.****17 is the answer because when we divide 17 by 7, the remainder is 3:****17 = 2*7 + 3 so 17 is congruent to 3 mod 7.****You can check that mod 7, 27 is congruent to 6, 37 is congruent to 2, and 47 is congruent to 5.** - Question 39 of 80
##### 39. Question

**Basic Probability:****A box contains 2 good and 2 bad light bulbs. 2 light bulbs are selected from the box without replacement. Find the probability that both light bulbs selected were good.**Correct**I’ll give two different solutions.**

Solution 1: The probability the first bulb selected is good is 2/4 = 1/2. Now out of the three bulbs remaining, only 1 is good. So the probability that the second is good given that the first is good is 1/3 (note this is a conditional probablility). Therefore, the probability that the first bulb is good and the second bulb is good is found by:**(1/2) * (1/3) = 1/6.****Solution 2: The sample space has C(4,2) elements (it’s the number of ways we can select 2 light bulbs from the 4 in the box). We need the number of ways we can select exactly 2 good bulbs: We can select 2 good bulbs from the 2 good bulbs C(2,2) ways, and we can select 0 bad bulbs from the 2 bad bulbs C(2,0) ways. Hence the probability is given by:****C(2,2) * C(2,0)/C(4,2) = 1*1/6 = 1/6.****Note that I include the extra term in the numerator to emphasize this is a hypergeometric probability (marbles in urns, committee problems, etc. have probabilities that take this form).**Incorrect**I’ll give two different solutions.**

Solution 1: The probability the first bulb selected is good is 2/4 = 1/2. Now out of the three bulbs remaining, only 1 is good. So the probability that the second is good given that the first is good is 1/3 (note this is a conditional probablility). Therefore, the probability that the first bulb is good and the second bulb is good is found by:**(1/2) * (1/3) = 1/6.****Solution 2: The sample space has C(4,2) elements (it’s the number of ways we can select 2 light bulbs from the 4 in the box). We need the number of ways we can select exactly 2 good bulbs: We can select 2 good bulbs from the 2 good bulbs C(2,2) ways, and we can select 0 bad bulbs from the 2 bad bulbs C(2,0) ways. Hence the probability is given by:****C(2,2) * C(2,0)/C(4,2) = 1*1/6 = 1/6.****Note that I include the extra term in the numerator to emphasize this is a hypergeometric probability (marbles in urns, committee problems, etc. have probabilities that take this form).** - Question 40 of 80
##### 40. Question

**Real Analysis:****Let f be a continuous function with domain the closed interval [0,1]. Which of the following is not necessary true?**Correct**Continuous functions do not have to be differentiable anywhere! For a proof of the existence of a nowhere differentiable function on the interval [0,1], see James R. Munkres, “Topology”, second ed., section 49. In fact, he proves something better: Given any continuous function f on [0,1] and any epsilon greater than 0, there exists a continuous function g on [0,1] with****|f(x) – g(x)| less than epsilon for all x****and g is nowhere differentiable.****Other answers: The fact that f attains its maximum value at some c in [0,1] is the extreme value theorem. Continuous functions are Riemann integrable. Finally, [0,1] is a closed and bounded set of real numbers, hence is compact by the Heine-Borel Theorem. Therefore, the image of f is compact (the continuous image of a compact set is compact).**Incorrect**Continuous functions do not have to be differentiable anywhere! For a proof of the existence of a nowhere differentiable function on the interval [0,1], see James R. Munkres, “Topology”, second ed., section 49. In fact, he proves something better: Given any continuous function f on [0,1] and any epsilon greater than 0, there exists a continuous function g on [0,1] with****|f(x) – g(x)| less than epsilon for all x****and g is nowhere differentiable.****Other answers: The fact that f attains its maximum value at some c in [0,1] is the extreme value theorem. Continuous functions are Riemann integrable. Finally, [0,1] is a closed and bounded set of real numbers, hence is compact by the Heine-Borel Theorem. Therefore, the image of f is compact (the continuous image of a compact set is compact).** - Question 41 of 80
##### 41. Question

**Group Theory:****Let G be a group. What is the name given to the subgroup that consists of all elements of G that commute with every element of G?**Correct**The center of G is in fact a subgroup: Let x, y be elements of the center, and let g be an element of G. Then****(xy)g = x(yg) by associativity****= x(gy) since y is an element of the center****= (xg)y by associativity****= (gx)y since x is an element of the center****= g(xy) by associativity. Thus xy is in the center of G.****Also, x^(-1)g = (g^(-1)x)^(-1) by the formula for the inverse of a product****= (xg^(-1))^(-1) since x is in the center of G and g^(-1) is an element of G****= gx^(-1) hence x^(-1) is in the center. Therefore, the center of G is a subgroup. Note that G doesn’t have to be finite.****Other answers: The commutator subgroup of G is the subgroup generated by elements of the form xyx^(-1)y^(-1) for x,y elements of G. The trivial subgroup is the subgroup consisting only of the identity element. Finally, if p is a prime such that p^k divides the order of G but p^(k+1) doesn’t divide the order of G, then a subgroup of order p^k is called a Sylow p-subgroup. Sylow p-subgroups exist by the famous “Sylow Theorems.” Moreover, any two Sylow p-subgroups are conjugate and the number of Sylow p-subgroups is congruent to 1 mod (order of G divided by p^k).**Incorrect**The center of G is in fact a subgroup: Let x, y be elements of the center, and let g be an element of G. Then****(xy)g = x(yg) by associativity****= x(gy) since y is an element of the center****= (xg)y by associativity****= (gx)y since x is an element of the center****= g(xy) by associativity. Thus xy is in the center of G.****Also, x^(-1)g = (g^(-1)x)^(-1) by the formula for the inverse of a product****= (xg^(-1))^(-1) since x is in the center of G and g^(-1) is an element of G****= gx^(-1) hence x^(-1) is in the center. Therefore, the center of G is a subgroup. Note that G doesn’t have to be finite.****Other answers: The commutator subgroup of G is the subgroup generated by elements of the form xyx^(-1)y^(-1) for x,y elements of G. The trivial subgroup is the subgroup consisting only of the identity element. Finally, if p is a prime such that p^k divides the order of G but p^(k+1) doesn’t divide the order of G, then a subgroup of order p^k is called a Sylow p-subgroup. Sylow p-subgroups exist by the famous “Sylow Theorems.” Moreover, any two Sylow p-subgroups are conjugate and the number of Sylow p-subgroups is congruent to 1 mod (order of G divided by p^k).** - Question 42 of 80
##### 42. Question

**Ring Theory:****Let R be a commutative ring with 1 and let m be a maximal ideal. Which of the following is FALSE?**Correct**Since m is maximal, there does not exist an ideal between it and R (with respect to inclusion). Note that m maximal implies that R/m is a field implies that R/m is an integral domain (fields are integral domains) which implies that m is a prime ideal (p is a prime ideal of R iff R/p is an integral domain).****I realize there’s a lot of terminology here, but I’ll give you a little idea of what this is about:****A ring R is a set with two operations: addition and multiplication. The sum of two elements of R is another element of R, and the product of two elements of R is another element of R. This ring R is commutative, which means the multiplication satisfies xy = yx for all x,y in R. It has 1 means there exists an element 1 in R such that x*1 = x for all x in R. The elements of a ring also satisfy a number of axioms, but they are all “logical” and what you would want to be true.****An integral domain is a commutative ring with 1 with the additional property that whenever a,b are elements of R such that ab = 0, then a = 0 or b = 0. This should look very familiar, since the set of real numbers is an integral domain.****A field has the additional property that every nonzero element has a multiplicative inverse.**Incorrect**Since m is maximal, there does not exist an ideal between it and R (with respect to inclusion). Note that m maximal implies that R/m is a field implies that R/m is an integral domain (fields are integral domains) which implies that m is a prime ideal (p is a prime ideal of R iff R/p is an integral domain).****I realize there’s a lot of terminology here, but I’ll give you a little idea of what this is about:****A ring R is a set with two operations: addition and multiplication. The sum of two elements of R is another element of R, and the product of two elements of R is another element of R. This ring R is commutative, which means the multiplication satisfies xy = yx for all x,y in R. It has 1 means there exists an element 1 in R such that x*1 = x for all x in R. The elements of a ring also satisfy a number of axioms, but they are all “logical” and what you would want to be true.****An integral domain is a commutative ring with 1 with the additional property that whenever a,b are elements of R such that ab = 0, then a = 0 or b = 0. This should look very familiar, since the set of real numbers is an integral domain.****A field has the additional property that every nonzero element has a multiplicative inverse.** - Question 43 of 80
##### 43. Question

**Complex Analysis:****Which of the following is NOT a sixth root of 1?**Correct**Recall Euler’s formula: e^(i*theta) = cos(theta) + i*sin(theta).**

The sixth roots of 1 are:**e^(2*pi*i/6) = cos(2*pi/6) + i*sin(2*pi/6) = (1/2) + i*(sqrt(3)/2)****e^(4*pi*i/6) = cos(2*pi/3) + i*sin(2*pi/3) = -(1/2) + i*(sqrt(3)/2)****e^(6*pi*i/6) = cos(pi) + i*sin(pi) = -1****e^(8*pi*i/6) = cos(4*pi/3) + i*sin(4*pi/3) = -(1/2) – i*(sqrt(3)/2)****e^(10*pi*i/6) = cos(5*pi/3) + i*sin(5*pi/3) = (1/2) – i*(sqrt(3)/2)****e^(12*pi*i/6) = cos(2*pi) + i*sin(2*pi) = 1****The incorrect answer e^(i*pi/2) = cos(pi/2) + i*sin(pi/2) = i. Note that i^6 = i^2 = -1, so i is not a sixth root of 1.**Incorrect**Recall Euler’s formula: e^(i*theta) = cos(theta) + i*sin(theta).**

The sixth roots of 1 are:**e^(2*pi*i/6) = cos(2*pi/6) + i*sin(2*pi/6) = (1/2) + i*(sqrt(3)/2)****e^(4*pi*i/6) = cos(2*pi/3) + i*sin(2*pi/3) = -(1/2) + i*(sqrt(3)/2)****e^(6*pi*i/6) = cos(pi) + i*sin(pi) = -1****e^(8*pi*i/6) = cos(4*pi/3) + i*sin(4*pi/3) = -(1/2) – i*(sqrt(3)/2)****e^(10*pi*i/6) = cos(5*pi/3) + i*sin(5*pi/3) = (1/2) – i*(sqrt(3)/2)****e^(12*pi*i/6) = cos(2*pi) + i*sin(2*pi) = 1****The incorrect answer e^(i*pi/2) = cos(pi/2) + i*sin(pi/2) = i. Note that i^6 = i^2 = -1, so i is not a sixth root of 1.** - Question 44 of 80
##### 44. Question

**Basic Topology:****Let X be a Hausdorff topological space and C a compact subset of X. What must be true of the complement of C?**CorrectA compact subset of a Hausdorff space is closed, hence its complement is open. There is a lot of terminology here to digest:

First of all, a topology on a set X is a collection of sets, called open sets with the property that any union of open sets is open, any finite intersection of open sets is open, and X together with the empty set is open. The set X together with a topology on X is called a topological space.

A subset C of X is compact if whenever C is contained in a union of open sets, a union of a finite subcollection of those open sets contains C (we say that every open cover of C has a finite subcover).

A topological space is Hausdorff if whenever x and y are two points in X, there exists disjoint open sets U and V such that x is in U and y is in V.

Finally, closed sets are just complements of open sets.

For a proof of the fact that a compact subset of a Hausdorff space is closed, see Theorem 26.3 on p. 165 of “Topology”, second ed., by James R. Munkres.

IncorrectA compact subset of a Hausdorff space is closed, hence its complement is open. There is a lot of terminology here to digest:

First of all, a topology on a set X is a collection of sets, called open sets with the property that any union of open sets is open, any finite intersection of open sets is open, and X together with the empty set is open. The set X together with a topology on X is called a topological space.

A subset C of X is compact if whenever C is contained in a union of open sets, a union of a finite subcollection of those open sets contains C (we say that every open cover of C has a finite subcover).

A topological space is Hausdorff if whenever x and y are two points in X, there exists disjoint open sets U and V such that x is in U and y is in V.

Finally, closed sets are just complements of open sets.

For a proof of the fact that a compact subset of a Hausdorff space is closed, see Theorem 26.3 on p. 165 of “Topology”, second ed., by James R. Munkres.

- Question 45 of 80
##### 45. Question

**In trigonometry, angles are formed by the rotation of a ray about its endpoint from an initial position to a terminal position. The measure of an angle can be negative or positive, depending on the direction of its rotation. Which direction of rotation returns negative angles: counter-clockwise (ccw) or clockwise (cw)?**CorrectThis is often confusing to students, as angles in geometry are always positive.

IncorrectThis is often confusing to students, as angles in geometry are always positive.

##### Hint

Answer ccw or cw only

- Question 46 of 80
##### 46. Question

**The two main trigonometric functions, sine (sin) and cosine (cos) differ by the addition of the prefix “co” to “cosine.” From where does the “co” derive?**CorrectSine and cosine are known as cofunctions to each other; i.e., the sin of an angle is the same as the cosine of that angle’s complement [sin(A) = cos(90-A) and cos(A) = sin(90-A)].

Other cofunction pairs are tangent/cotangent and secant/cosecant.

IncorrectSine and cosine are known as cofunctions to each other; i.e., the sin of an angle is the same as the cosine of that angle’s complement [sin(A) = cos(90-A) and cos(A) = sin(90-A)].

Other cofunction pairs are tangent/cotangent and secant/cosecant.

- Question 47 of 80
##### 47. Question

**The functions cotangent, secant, and cosecant have what relationship to the functions tangent, cosine, and sine (respectively)?**Correct**Using reference triangle parameters, cot A = x/y, csc A = r/y, and sec A = r/x. These are reciprocals of the definitions for tangent (tan A = y/x), sine (sin A = y/r), and cosine (cos A = x/r).****This relationship can also be expressed as cot A = 1/tan A, csc A = 1/sin A, and sec A = 1/cos A.****Also, the terms inverse and reciprocal are sometimes used synonymously in other diciplines. Reciprocal is more correctly the multiplicitive inverse. A regular inverse is a relation that interchanges the domain and range of another relation. For example: y=2x+8 and x=2y+8 would be inverses of each other, as their domain and range variables are swapped.**Incorrect**Using reference triangle parameters, cot A = x/y, csc A = r/y, and sec A = r/x. These are reciprocals of the definitions for tangent (tan A = y/x), sine (sin A = y/r), and cosine (cos A = x/r).****This relationship can also be expressed as cot A = 1/tan A, csc A = 1/sin A, and sec A = 1/cos A.****Also, the terms inverse and reciprocal are sometimes used synonymously in other diciplines. Reciprocal is more correctly the multiplicitive inverse. A regular inverse is a relation that interchanges the domain and range of another relation. For example: y=2x+8 and x=2y+8 would be inverses of each other, as their domain and range variables are swapped.** - Question 48 of 80
##### 48. Question

**Angles are measured in degrees or in radians. The conversion factor with which to multiply to convert from degrees to radians is:**Correct**There are 360 degrees (2pi radians) in a full circular rotation. The ratio of radians to degrees is therefore 2pi/360, or pi/180.**Incorrect**There are 360 degrees (2pi radians) in a full circular rotation. The ratio of radians to degrees is therefore 2pi/360, or pi/180.** - Question 49 of 80
##### 49. Question

**The sine function is an odd function (as opposed to even). Which of these statements holds true for a general odd function, f?**Correct**Any function having the property that for each x in the domain, -x is also in the domain and f(-x) = -f(x), is called an odd function.**Incorrect**Any function having the property that for each x in the domain, -x is also in the domain and f(-x) = -f(x), is called an odd function.** - Question 50 of 80
##### 50. Question

**The cosine function is an even function (as opposed to odd). Which of these statements holds true for a general even function, f?**CorrectAny function having the property that for each x in the domain, -x is in the domain and f(-x) = f(x) is called an even function.

IncorrectAny function having the property that for each x in the domain, -x is in the domain and f(-x) = f(x) is called an even function.

- Question 51 of 80
##### 51. Question

**There are 8 trigonometric identities called Fundamental Identities. Three of these are called Pythagorean Identities, because they are based on the Pythagorean Theorem. Which of the following is NOT a Pythagorean Identity?**Correct**The Pythagorean Identities can be solved for any of their terms, then used in substitutions for trigonometric proofs.**Incorrect**The Pythagorean Identities can be solved for any of their terms, then used in substitutions for trigonometric proofs.** - Question 52 of 80
##### 52. Question

**Let’s start out with food! There is one chocolate mint bar with plenty of sprinkles on it. Obe Seguy and his 4 friends want to buy it. They buy it using their combined money that totaled to $2.50. If they decide to split it into equal parts for each one of them, what is the percentage of the whole chocolate bar each of the kids will receive?**Correct**There are five people (Obe with his 4 friends). Divide one (the whole chocolate bar) by five, and .2 is the answer. After converting this into a percent, the answer is 20%.**Incorrect**There are five people (Obe with his 4 friends). Divide one (the whole chocolate bar) by five, and .2 is the answer. After converting this into a percent, the answer is 20%.** - Question 53 of 80
##### 53. Question

**What is the y-intercept of a direct variation? A direct variation always passes through the origin.**CorrectA direct variation is a line that always passes through the origin (0,0)! The y-intercept is the point which the line crosses the y-axis. Other accepted answers: (0,0) 0,0 origin.

IncorrectA direct variation is a line that always passes through the origin (0,0)! The y-intercept is the point which the line crosses the y-axis. Other accepted answers: (0,0) 0,0 origin.

##### Hint

The hint is the second sentence in the question.

- Question 54 of 80
##### 54. Question

**The sum of two numbers is 74. The difference between the two numbers is 16. What are the two numbers?**CorrectSolve by using system of equations. The two equations are x+y=74 and x-y=16. Isolate either y or x in the second equation and substitute in the first. Other accepted answers: 45 and 29 29, 45 45, 29 29 45 45 29

IncorrectSolve by using system of equations. The two equations are x+y=74 and x-y=16. Isolate either y or x in the second equation and substitute in the first. Other accepted answers: 45 and 29 29, 45 45, 29 29 45 45 29

##### Hint

Two numbers with “and” between them.

- Question 55 of 80
##### 55. Question

**Mark Llego has a pattern of cans. The first can has 2 balls inside. The second has 5 markers. The third has 8 bananas, and the fourth has 11 children captured inside. When “n” represents the position of the can in his collection (the can with markers would be 2 since it is the second can), which of the following shows the amount of objects in each can?**Correct**The coefficient is 3, and the constant is 1. Setting up a chart is a time-saving way of solving this problem. If you multiply 3 by the position of the can and subtract 1 you will get the amount of objects inside. Let’s say it is the can with markers, so 3(2)-1=5. 5 markers are inside!**Incorrect**The coefficient is 3, and the constant is 1. Setting up a chart is a time-saving way of solving this problem. If you multiply 3 by the position of the can and subtract 1 you will get the amount of objects inside. Let’s say it is the can with markers, so 3(2)-1=5. 5 markers are inside!** - Question 56 of 80
##### 56. Question

**Diwata is drawing a graph representing the growth of his knees over the past year. He found that following equation, 4x+2y=6, could be used to represent the line in his graph. Using this information, which equation is perpendicular to his line?**CorrectFirst convert the equation in the problem, 4x+2y=6, into y-intercept form. y=-2x+3 is the result. Perpendicular equations will have the opposite reciprocal of the slope. 2’s opposite reciprocal is -1/2, so 3x-6y=5 is correct.

IncorrectFirst convert the equation in the problem, 4x+2y=6, into y-intercept form. y=-2x+3 is the result. Perpendicular equations will have the opposite reciprocal of the slope. 2’s opposite reciprocal is -1/2, so 3x-6y=5 is correct.

- Question 57 of 80
##### 57. Question

Mark is in debt of 6 dollars. In one year, his debt multiplies by the same amount. In the year after that, it again multiplies by the same amount he had to begin with. How much debt is Mark in now?

Correct$6 times $6 times $6 equals -$216. This could be represented by $6^3.

Incorrect$6 times $6 times $6 equals -$216. This could be represented by $6^3.

- Question 58 of 80
##### 58. Question

**In algebra, you learned that (A – B)^2 = A^2 – 2AB + B^2. Use this result to compute the exact value of the following expression:****23^2 – 2 * 23 * 13 + 13^2**Correct**In algebra you learn that (A – B)^2 = A^2 – 2AB + B^2. Put A = 23, B = 13:****(23 – 13)^2 = 23^2 – 2 * 23 * 13 + 13^2, hence 23^2 – 2 * 23 * 13 + 13^2 = 10^2 = 100.**Incorrect**In algebra you learn that (A – B)^2 = A^2 – 2AB + B^2. Put A = 23, B = 13:****(23 – 13)^2 = 23^2 – 2 * 23 * 13 + 13^2, hence 23^2 – 2 * 23 * 13 + 13^2 = 10^2 = 100.** - Question 59 of 80
##### 59. Question

**In algebra, you learn that (A + B)^2 = A^2 + 2AB + B^2. Use this result to compute the exact value of the following expression:****33^2 + 2 * 33 * 17 + 17^2**CorrectUse (A + B)^2 = A^2 + 2AB + B^2 with A = 33 and B = 17:

33^2 + 2 * 33 * 17 + 17^2 = (33 + 17)^2 = 50^2 = 2500.

IncorrectUse (A + B)^2 = A^2 + 2AB + B^2 with A = 33 and B = 17:

33^2 + 2 * 33 * 17 + 17^2 = (33 + 17)^2 = 50^2 = 2500.

- Question 60 of 80
##### 60. Question

**In algebra, you learn that (A + B)(C + D) = AC + AD + BC + BD. Use this result to compute the exact value of the following expression:****13 * 42 + 13 * 18 + 17 * 42 + 17 * 18**CorrectWe use (A + B)(C + D) = AC + AD + BC + BD with A = 13, B = 17, C = 42, and D = 18. Then 13 * 42 + 13 * 18 + 17 * 42 + 17 * 18 = (13 + 17)(42 + 18) = 30 * 60 = 1800.

IncorrectWe use (A + B)(C + D) = AC + AD + BC + BD with A = 13, B = 17, C = 42, and D = 18. Then 13 * 42 + 13 * 18 + 17 * 42 + 17 * 18 = (13 + 17)(42 + 18) = 30 * 60 = 1800.

- Question 61 of 80
##### 61. Question

**In algebra, you learn that a difference of squares factors as****A^2 – B^2 = (A – B)(A + B). Use this result to compute the exact value of the following expression:****250^2 – 150^2**Correct**We can use A^2 – B^2 = (A – B)(A + B) with A = 250 and B = 150.****250^2 – 150^2 = (250 – 150)*(250 + 150) = 100 * 400 = 40000.**Incorrect**We can use A^2 – B^2 = (A – B)(A + B) with A = 250 and B = 150.****250^2 – 150^2 = (250 – 150)*(250 + 150) = 100 * 400 = 40000.** - Question 62 of 80
##### 62. Question

**In algebra, you learn that (A – B)^3 = A^3 – 3A^2B + 3AB^2 – B^3. Use this result to compute the exact value of the following expression:****17^3 – 3 * 17^2 * 12 + 3 * 17 * 12^2 – 12^3**Correct**We can use (A – B)^3 = A^3 – 3A^2B + 3AB^2 – B^3. Put A = 17, B = 12:****17^3 – 3 * 17^2 * 12 + 3 * 17 * 12^2 – 12^3 = (17 – 12)^3 = 5^3 = 125.**Incorrect**We can use (A – B)^3 = A^3 – 3A^2B + 3AB^2 – B^3. Put A = 17, B = 12:****17^3 – 3 * 17^2 * 12 + 3 * 17 * 12^2 – 12^3 = (17 – 12)^3 = 5^3 = 125.** - Question 63 of 80
##### 63. Question

**You know from algebra how to factor a quadratic polynomial such as**

x^2 – 16x + 39. So factor this polynomial, and then let x be a certain number to obtain the exact value of the following expression without any hard work:**73^2 – 16 * 73 + 39**Correctx^2 – 16x + 39 factors as (x – 13)(x – 3). Now put x = 73:

73^2 – 16 * 73 + 39 = (73 – 13)(73 – 3) = 60 * 70 = 4200.

Incorrectx^2 – 16x + 39 factors as (x – 13)(x – 3). Now put x = 73:

73^2 – 16 * 73 + 39 = (73 – 13)(73 – 3) = 60 * 70 = 4200.

- Question 64 of 80
##### 64. Question

**We can factor A^4 – 2A^2B^2 + B^4 as (A^2 – B^2)^2 = ((A – B)(A + B))^2. Use this result to compute the exact value of the following expression:****25^4 – 2 * 25^2 * 15^2 + 15^4**Correct**We can use A^4 – 2A^2B^2 + B^4 = ((A – B)(A + B))^2. Let A = 25, B = 15. Then****25^4 – 2 * 25^2 * 15^2 + 15^4 = ((25 – 15)(25 + 15))^2 = (10 * 40)^2 = 400^2 = 160000.**Incorrect**We can use A^4 – 2A^2B^2 + B^4 = ((A – B)(A + B))^2. Let A = 25, B = 15. Then****25^4 – 2 * 25^2 * 15^2 + 15^4 = ((25 – 15)(25 + 15))^2 = (10 * 40)^2 = 400^2 = 160000.** - Question 65 of 80
##### 65. Question

**By multiplying, you can show that (A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2AC + 2BC. Use this result to compute the exact value of the following expression:****19^2 + 20^2 + 21^2 + 2*19*20 + 2*19*21 + 2*20*21**Correct**We can use (A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2AC + 2BC. So put A = 19, B = 20, and C = 21:****19^2 + 20^2 + 21^2 + 2*19*20 + 2*19*21 + 2*20*21 = (19 + 20 + 21)^2 = 60^2 = 3600.**Incorrect**We can use (A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2AC + 2BC. So put A = 19, B = 20, and C = 21:****19^2 + 20^2 + 21^2 + 2*19*20 + 2*19*21 + 2*20*21 = (19 + 20 + 21)^2 = 60^2 = 3600.** - Question 66 of 80
##### 66. Question

**By multiplying, you can show that (A + B – C)^2 = A^2 + B^2 + C^2 + 2AB – 2AC – 2BC. Using this result, compute the exact value of the following expression:****15^2 + 17^2 + 22^2 + 2*15*17 – 2*15*22 – 2*17*22.**CorrectWe can use (A + B – C)^2 = A^2 + B^2 + C^2 + 2AB – 2AC – 2BC. So put A = 15, B = 17, and C = 22:

15^2 + 17^2 + 22^2 + 2*15*17 – 2*15*22 – 2*17*22 = (15 + 17 – 22)^2 = 10^2 = 100.

IncorrectWe can use (A + B – C)^2 = A^2 + B^2 + C^2 + 2AB – 2AC – 2BC. So put A = 15, B = 17, and C = 22:

15^2 + 17^2 + 22^2 + 2*15*17 – 2*15*22 – 2*17*22 = (15 + 17 – 22)^2 = 10^2 = 100.

- Question 67 of 80
##### 67. Question

**In algebra, you learned how to simplify rational expressions, such as**

(A^4 – B^4)/(A^3 + AB^2 + BA^2 + B^3). The method was to factor the numerator and denominator, then cancel common factors. Do this, and then use your result to compute the exact value of the following expression:**(85^4 – 75^4)/(85^3 + 85*75^2 + 75*85^2 + 75^3).**Correct**We want (A^4 – B^4)/(A^3 + AB^2 + BA^2 + B^3) where A = 85, B = 75. Let’s first factor the numerator and denominator:****A^4 – B^4 = (A^2 – B^2)(A^2 + B^2) = (A – B)(A + B)(A^2 + B^2)****A^3 + AB^2 + BA^2 + B^3 = A(A^2 + B^2) + B(A^2 + B^2) = (A + B)(A^2 + B^2).****Dividing gives (A^4 – B^4)/(A^3 + AB^2 + BA^2 + B^3) = A – B. So the answer is 85 – 75 = 10.**Incorrect**We want (A^4 – B^4)/(A^3 + AB^2 + BA^2 + B^3) where A = 85, B = 75. Let’s first factor the numerator and denominator:****A^4 – B^4 = (A^2 – B^2)(A^2 + B^2) = (A – B)(A + B)(A^2 + B^2)****A^3 + AB^2 + BA^2 + B^3 = A(A^2 + B^2) + B(A^2 + B^2) = (A + B)(A^2 + B^2).****Dividing gives (A^4 – B^4)/(A^3 + AB^2 + BA^2 + B^3) = A – B. So the answer is 85 – 75 = 10.** - Question 68 of 80
##### 68. Question

**What is the mean of the following series- 2 2 2 4 4 5 5 6 8 11 12 16 18 ?**CorrectThe mean is the average, determined by adding all numbers of the sequence and then dividing by the number of items in the sequence. The mean is sensitive to extreme values in the series- these can ‘distort’ the average. This is why ‘mean income’ would be a misleading statistic- you could have hundreds of poor people and just a few very rich people, and the mean income would seem to be rather high, as the large salaries of the rich would unbalance it.

IncorrectThe mean is the average, determined by adding all numbers of the sequence and then dividing by the number of items in the sequence. The mean is sensitive to extreme values in the series- these can ‘distort’ the average. This is why ‘mean income’ would be a misleading statistic- you could have hundreds of poor people and just a few very rich people, and the mean income would seem to be rather high, as the large salaries of the rich would unbalance it.

- Question 69 of 80
##### 69. Question

What is the median of the following series- 2 2 2 4 4 5 5 6 8 11 12 16 18 ?

Correct**The median is the middle score (or the average of the two scores closest to the middle if there is an even number of data points.) Medians are less influenced by extreme scores and thus most common statistics (income, house prices, etc.) use a median score.**Incorrect**The median is the middle score (or the average of the two scores closest to the middle if there is an even number of data points.) Medians are less influenced by extreme scores and thus most common statistics (income, house prices, etc.) use a median score.** - Question 70 of 80
##### 70. Question

**What is the mode of the following series- 2 2 2 4 4 5 5 6 8 11 12 16 18 ?**CorrectThe mode is the most common score.

IncorrectThe mode is the most common score.

- Question 71 of 80
##### 71. Question

**If you graph a normally distributed set of data (one without obvious irregularities or biases, such as the height of all people in your school), what will the shape of that graph be?**Correct**The graph will be bell-shaped, with a large bulge of ‘average’ heights and smaller ‘tails’ of people farther away from the mean- midgets and basketball players.**Incorrect**The graph will be bell-shaped, with a large bulge of ‘average’ heights and smaller ‘tails’ of people farther away from the mean- midgets and basketball players.** - Question 72 of 80
##### 72. Question

**What is the most commonly used statistical measure of spread in a normally-distributed population?**Correct**The standard deviation lets us know how a data sample is distributed around the sample mean. A data set with a small standard deviation has all data points relatively close to the mean, a sample with a large standard deviation is more spread out. In a normal distribution, about 68 percent of scores are within one standard deviation of the mean, about 95 percent are within two standard deviations of the mean, and 99.7 percent are within 3 standard deviations of the mean.**Incorrect**The standard deviation lets us know how a data sample is distributed around the sample mean. A data set with a small standard deviation has all data points relatively close to the mean, a sample with a large standard deviation is more spread out. In a normal distribution, about 68 percent of scores are within one standard deviation of the mean, about 95 percent are within two standard deviations of the mean, and 99.7 percent are within 3 standard deviations of the mean.** - Question 73 of 80
##### 73. Question

**Which of the following would be appropriate for a t-test?**Correct**The t-test determines whether two sets of scores are significantly different. It can be used with small samples and even if the standard deviation is not known. As sample sizes increase, the measurement becomes more accurate.**Incorrect**The t-test determines whether two sets of scores are significantly different. It can be used with small samples and even if the standard deviation is not known. As sample sizes increase, the measurement becomes more accurate.** - Question 74 of 80
##### 74. Question

**Which of the following would be appropriate for a z test?**Correct**The z-test also determines whether two sets of scores are significantly different. However, the z test can only be used when a population is known to have a normal distribution and the standard deviation is known. In this case, all weights are known, so the distribution and standard deviations can be checked. Weights of all members of a population is a normally-distributed statistic. The z test is also used when population samples are large.**Incorrect**The z-test also determines whether two sets of scores are significantly different. However, the z test can only be used when a population is known to have a normal distribution and the standard deviation is known. In this case, all weights are known, so the distribution and standard deviations can be checked. Weights of all members of a population is a normally-distributed statistic. The z test is also used when population samples are large.** - Question 75 of 80
##### 75. Question

**Which of the following is used for statistic analysis?**Correct**The best known chi-square test was developed by Karl Pearson in the early twentieth century and is used for testing statistical hypotheses. Truth tables are used in logic and logarithms are associated with mathematics. Slide rules, superseded by calculators, were developed from logarithms as a calculation method, particularly used by engineers.**Incorrect**The best known chi-square test was developed by Karl Pearson in the early twentieth century and is used for testing statistical hypotheses. Truth tables are used in logic and logarithms are associated with mathematics. Slide rules, superseded by calculators, were developed from logarithms as a calculation method, particularly used by engineers.** - Question 76 of 80
##### 76. Question

**Which of the following would be appropriate for ANOVA?**Correct**ANOVA -analysis of variance- is used to determine whether significant differences exist between multiple sets of data.**Incorrect**ANOVA -analysis of variance- is used to determine whether significant differences exist between multiple sets of data.** - Question 77 of 80
##### 77. Question

**What is the probability of rolling 3 dice, and them all landing on a 6?**Correct**Because there are 216 different combinations that 3 dice could land on, the probability is 1/216. There are 216 possible combinations because there are 6 combinations for 1 die (‘die’ is the singular name for ‘dice’) – 1 times 6, 36 combinations for 2 dice – 6 times 6, which means that the combinations for 3 dice are 6 times 6 times 6, or 6 cubed. 6 cubed is 216.**Incorrect**Because there are 216 different combinations that 3 dice could land on, the probability is 1/216. There are 216 possible combinations because there are 6 combinations for 1 die (‘die’ is the singular name for ‘dice’) – 1 times 6, 36 combinations for 2 dice – 6 times 6, which means that the combinations for 3 dice are 6 times 6 times 6, or 6 cubed. 6 cubed is 216.** - Question 78 of 80
##### 78. Question

**I have 18 blue marbles, 16 green marbles and 22 red marbles in a bag. What is the chance I will pick a green marble if I pick a marble out at random (simplify the answer)?**Correct**You add up 18, 16 and 22, which gives you a total of 56. 16 of those 56 marbles are green, so we write it like this: 16/56. To simplify this, we could find the highest common factor of 16 and 56, which is 8, and divide 16 and 56 by 8 to get 2/7. We could also just keep halving 16 and 56 until we get to 2/7.**Incorrect**You add up 18, 16 and 22, which gives you a total of 56. 16 of those 56 marbles are green, so we write it like this: 16/56. To simplify this, we could find the highest common factor of 16 and 56, which is 8, and divide 16 and 56 by 8 to get 2/7. We could also just keep halving 16 and 56 until we get to 2/7.** - Question 79 of 80
##### 79. Question

**What is the probability you will get all 10 questions in this quiz correct if you narrow the options down so you have 2 answers to choose between, and 1 of them is correct?**Correct**The probability of you getting 1 question correct when there are 2 options to choose from 1/2, assuming 1 of those options is correct. . For 2 questions, getting them both correct is 1/4 or 1/2^2 so 10 questions would be 1/2^10 which is 1/1024**Incorrect**The probability of you getting 1 question correct when there are 2 options to choose from 1/2, assuming 1 of those options is correct. . For 2 questions, getting them both correct is 1/4 or 1/2^2 so 10 questions would be 1/2^10 which is 1/1024** - Question 80 of 80
##### 80. Question

**Which of these sentences accurately define the word ‘probability’?**Correct**According to thefreedictionary, probability is defined as ‘The quality or condition of being probable; likelihood.’. ‘Statistics’ refers to a branch of mathematics concerned with data, but not necessarily numerical data.**Incorrect**According to thefreedictionary, probability is defined as ‘The quality or condition of being probable; likelihood.’. ‘Statistics’ refers to a branch of mathematics concerned with data, but not necessarily numerical data.**

## Rusty Millabas

i want more elaborate explanation to this question. i think the correct answer is “cannot be determined”

## Jessie N. Toreros

Hope this will help me to pass the LET

## Stephen Cajigas

Thank you for this reviewer, Teacher Mark.

However, this reviewer seems to contain quite a number of items touching topics on Topology, Abstract Algebra and Complex Analysis. I checked the TOS for Math and it seems that these topics will not be covered in the LET. In the previous years, did the LET cover these topics as well?

## TeacherPH

Welcome po!