Geometry
What is the geometric definition of the absolute value of a number on a number line? (or the complex plane)
Distance of the number from 0.
Simplify (1 + i)(3 - 2i)
5 + i
the nth root of xa is equal to?
xa/n
If h(x) = f(x)/g(x), then h is undefined whenever:
a.) g(x) = 0
b.) f(x) and g(x) are both = 0
c.) h(x) = 0
a and b
What is the fallacy you commit when you try to defend against an argument by attacking a person rather than their claims and reasoning?
Argumentum ad hominem (or just ad hominem)
In ΔABC, AB = 3, BC = x, and AC = 5. Find x.
x = 4.
The absolute value of √(-144) is?
Explanation: the absolute value function is defined for all complex numbers. In the complex plane, the distance between z = 12i and z = 0 is 12.
The inverse function of f(x) = a ⋅ bx is?
*{a≠0, b>0}
f-1(x) = logb(x/a)
The sum of the roots of a quadratic equation of the form ax2 + bx + c, in terms of a, b, and c is?
-b/a
Explain what is wrong with the following argument:
"Studies have shown that a higher temperature is connected with more crime. Therefore, people commit more crimes because the temperature is higher."
Correlation does not imply causation
The angle θ, in degrees, that a line connecting the origin with the point (3, 4) makes with the x axis is (to two decimal places)?
53.13
ix cycles between i, -1, -i, and 1 for integer values of x. What two trigonometric functions cycle between 0, -1, 0, and 1?
cos(x) and sin(x).
Find x such that 9x+1 + 9x+1 = 6
x = -1/2
If h(x) = (f ⋅ g)(x) and h has two distinct zeroes, then which of the following must be true?
a.) g has at least one zero
b.) f has at least one zero
c.) the sum of the number of zeroes of f and g is 2.
Name the law that justifies your answer.
c by the zero product law.
Consider the following statement:
"If it is raining and I am not using an umbrella, then I will get wet."
Are "it is not raining and I am not using an umbrella" and "I will get wet" necessarily logically equivalent statements? Justify or prove your answer.
Not necessarily.
Explanation: if they were, then "I will get wet" is true if and only if "it is not raining and I am not using an umbrella is true". It could be true that "I will get wet" if someone shoots a fire hose at you. So, by contradiction, they are not necessarily logically equivalent.
Let A(x) be the area of a rectangle. The rectangle has dimensions of L(x) = (x + 3), w(x) = (1 - x), and h(x) = (x2 -1). Express A(x) in expanded polynomial form.
-x4 - 2x3 + 4x2 + 2x - 3
What is i^2025?
i
Given x is an integer, solve xx = 224
x = 8
The number of non-real solutions to x4 + 1 = 0 is:
4 distinct solutions.
Suppose for propositions a and b: If a, then b.
Which must be true:
If not b, then not a.
If not a, then b.
If not b, then a.
If not b, then not a.
A circle is centered at the point C(-1, 1). A line tangent to the circle has the equation y = -x +14. What is the area of the circle?
*hint: the radius of the circle is perpendicular to the tangent line at the point of tangency.
98 pi
Let z = 4/(2+i). Suppose z = a + bi, where a, b are real number constants. Find a and b.
a = 8/5
b = -4/5
9x+1 ⋅ 32x-1 = 81x
No real solutions
Find all x such that √(x + 4) + √(x + 11) = 7.
x = 5 only
Consider the following logic:
"If every student in the class studies hard, then they will pass the test."
Suppose only one student studies hard. With this being the case, is it always true, sometimes true, or never true that at least one student will pass the test?
Justify your answer.
Sometimes true.
Explanation: the original statement only guarantees that everyone will pass if everyone studies hard. So if one student studies hard, then it is not guaranteed that everyone will pass the test, but it is also not guaranteed that everyone will fail the test. So, in all of the possibilities there exists a possibility in which one student studies hard and everybody still fails the test. By counterexample, the statement that "at least one student will pass the test" is not necessarily true and thus only sometimes true.