Statistics

Algebra

Calculus

Geometry

Number System

100

For her calculus class, Marie has scored 88, 78, and 95 on three of her tests so far. What is the minimum score Marie needs to receive on her 4th test in order to have an average of 90?

99

100

3x+5=14

x=3

100

Find the derivative of the function f(x)= 3x^{2} + 5x - 2

f′(x)=6x+5

100

In a triangle ABC, angle A is 90 degrees, and the lengths of the sides AB and AC are 6 units and 8 units, respectively. Find the length of the hypotenuse BC and the area of the triangle.

24 units^{2 }and 10 units

100

2x+3 / 4 = 5

8.5

200

What is the central line theorem?

The central limit theorem (CLT) is a fundamental concept in statistics that describes how the distribution of sample means approaches a normal distribution as the sample size becomes large, regardless of the original distribution of the data.

200

What is a polynomial?

An expression made up of variables and coefficients, involving only addition, subtraction, and multiplication. For example, 2x^{2}+3x+4.

200

What is the meaning of indifinite integration? (Chat GPT)

Indefinite integration, also known as anti-differentiation, is the process of finding a function F(x)F(x)F(x) whose derivative is a given function f(x)f(x)f(x). The result is called the indefinite integral and includes a constant of integration (usually denoted as CCC) because the derivative of a constant is zero.

200

What is circumcircle?

The circle that passes through all the vertices of a polygon. For a triangle, it's the circle that passes through all three vertices.

200

What is Modular Arithmetic?

A system of arithmetic for integers, where numbers wrap around upon reaching a certain value (the modulus). For example, in modulo 12 arithmetic, after 11 comes 0.

300

In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state. What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

0.24

300

Solve: 2(x−3)=4x+5

x= -11/2

300

Find the indefinite integral of f(x). Given the function f(x)= x^{3} - 4x +1.

∫f(x)dx= f(x)=x^{4}/4-2x^{2}+x+C

300

In a triangle ABC, let D and E be the midpoints of AC and AB, respectively. If F is the midpoint of BC, prove that the line segments DE and AF are parallel and that DE= 1/2 AF

Therefore, DE ll AF and DF= 1/2 AF are proven.

300

Consider the polynomial P(x)= x^{4} - 10x^{2} + 9. Determine the number of distinct real roots of the polynomial P(x) and find them.

It has four distinct real roots: ±1 and ±3.

400

A mathematics competition uses the following scoring procedure to discourage students from guessing (choosing an answer randomly) on the multiple-choice questions. For each correct response, the score is 7, For each question left unanswered, the score is 2. For each incorrect response, the score is 0. If there are 5 choices for each question, what is the minimum number of choices that the student must eliminate before it is advan- tageous to guess among the rest?

0.922

400

Solve: x^{2}-4/x-2= x+2/x+1

x=0 and x=−2

400

Evaluate the following integral:∫^{1}_{0 }x^{2} e^{x} dx

∫^{1}_{0 }x^{2} e^{x} dx= 2-e

400

In triangle △ABC, ∠BAC=90 degrees, and the circumcircle of △ABC intersects BC again at point D. Let M be the midpoint of BC, and let E be the foot of the altitude from A to BC. Prove that AD is the angle bisector of ∠BAD and that AM=AE.

Therefore, AD is indeed the angle bisector of ∠BAD, and AM=AE has been proven.

400

Find all integers x such that: 7x ≡ 11(mod 23)

x=18+23k

500

A Pois() number of people vote in a certain election. Each voter votes for Candidate A with probability p and for Candidate B with probability q = 1 p, independently of all the other voters. Let V be the di↵erence in votes, defined as the number of votes for A minus the number of votes for B. Find E(V ) (simplify).

E(V)=λ(2p−1)

500

x^{3}-2x^{2}+3x-6/ x^{2}-2x=2

x=2, x=1+i(square root)2, and x=1−i (square root)2

500

Evaluate the following integral:∫^{1}_{0 }ln(1+ x^{2}) dx

∫^{1}_{0 }ln(1+ x^{2}) dx= ln(2)−2+ π/2

500

{ x^{2 }+ y^{2}= 25

{ x^{3}-y^{3}=14

(x,y)=(3,4) or (4,3)

500

What is my favorite number category in the number system

Irrational numbers