Mathematics Quiz: Various Topics

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Questions and Answers

The cost of a suit and a pair of shoes is 720 manats. The cost of the suit is reduced by 20% and the cost of the shoes is reduced by 30%. The total cost of the suit and shoes after the discounts is 564 manats. What is the original cost of the suit?

400

The population of a city was 70000 in 2023. The population has grown to 74200 in 2024. What is the percentage increase in the population?

6

Given that the complex numbers z1 = -1 + 2i and z2 = 1/(1- i) , find the modulus of the complex number z = 2z1 - 7z2 + 6

$2 \sqrt{10}$

Find the length of segment AD in the given figure.

<p>$4 \sqrt{13}$</p> Signup and view all the answers

Calculate the value of 0.2(6) + 0.2

<p>1.4</p> Signup and view all the answers

A straight line segment with a length of 1 cm is divided into two pieces with lengths of 0.1 cm and 0.5 cm. Find the projection of the line segment onto a line perpendicular to the original line segment.

<p>0.48 cm</p> Signup and view all the answers

A group of 50 people are randomly assigned numbers. What is the probability that a randomly selected person will have a number that is divisible by 7 and is even?

<p>1 / 25</p> Signup and view all the answers

Solve the inequality: (7 - 3k) / 2 - (8k + 1) / 3 < 12 - 10k

<p>k &gt; 17 / 23</p> Signup and view all the answers

A square is divided into 25 equal parts. If the angle inside the square is alpha and beta, find the value of tan(alpha) + cos(beta).

<p>34/15</p> Signup and view all the answers

Given that (x^4 + 3x)^8 = 1 and (x^6 + 2x + 3)^7 = 1, find the value of (x^4 + 3x)^5 * (x^6 + 2x + 3)^11.

<p>1</p> Signup and view all the answers

Flashcards

Discount problem with two items

The combined price after discounts: 564 manats Price of suit after discount: 564 - (Price of shoes after discount) Price of shoes after discount = 0.7 x (Original price of shoes) Price of suit after discount = 0.8 x (Original price of suit) Therefore: 0.8 x (Original price of suit) + 0.7 x (Original price of shoes) = 564 Original price of suit + Original price of shoes = 720 Solve the system of equations to get the original price of the suit.

Population Growth

Percentage increase = ((New population - Old population) / Old population) x 100% Percentage increase = ((74200 - 70000)/70000) x 100%

Modulus of a complex number

Modulus of a complex number: |z| = √(Real part² + Imaginary part²) Find z_2 by simplifying the given expression. Multiply z_1 and z_2 by their respective coefficients. Add all terms to get the complex number z. Find the modulus of z using the formula.

Finding length using Pythagorean theorem

Length of AD can be determined using the Pythagorean theorem. AC and CD are radii of the circle, so AC = CD = 8. Apply the Pythagorean theorem to triangle ACD: AD² = AC² + CD² AD = √(AC² + CD²)

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Simple calculation

0.2(6) + 0.2 = (0.2 x 6) + 0.2 = 1.2 + 0.2 = 1.4

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Projection of a line segment

The projection of a line segment onto a plane is the line segment formed by drawing perpendicular lines from each endpoint of the line segment to the plane. Length of the projection: √(1² - (0.5 - 0.1)²) = √(1² - 0.4²) = √(0.84) cm

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Probability of specific outcome

Probability = (Favorable outcomes) / (Total possible outcomes) Favorable outcomes: Even numbers divisible by 7 in the range of 1-50 are 14, 28, 42. Total possible outcomes: 50 Probability = 3/50

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Solving linear inequality

Multiply both sides of the inequality by 6 to eliminate fractions. Simplify and solve for k: 21 - 9k - 16k - 2 < 72 - 60k -25k + 19 < 72 - 60k 35k < 53 k < 53/35

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Finding trigonometric values in a square

The angle $

alpha$ is formed in the square, and it becomes the acute angle of a right triangle formed by dividing the square into 25 equal parts. The angle $

alpha$ is formed by the line segment between two adjacent points on the grid, so it represents the ratio of the side of the square to one part (i.e., 1/5).
The tangent of $

alpha$ can be found by dividing the opposite side of the triangle by the adjacent side (i.e., 1/4). The angle $

beta$ is also formed by the line segment between two points on the grid, and it represents the ratio of the side of the square to one part (i.e., 1/5). The cosine of $

beta$ can be found by dividing the adjacent side of the triangle by the hypotenuse (i.e., 4/√32). Therefore, the final answer is 1/4 + 4/√32

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Degree of a polynomial product

Degree of a polynomial is the highest power of its variable. Degree of a polynomial raised to a power is the degree of the polynomial multiplied by the exponent. Degree of $M_1 = 8 * 4 = 32$. Degree of $M_2 = 7 * 6 = 42$. Degree of $M_5 = 5 * 4 = 20$. Degree of $M_{11} = 11 * 4 = 44$. Degree of $M_{22} = 22 * 4 = 88$. Degree of $M_5

⋅ M_{11}

⋅ M_{22}$: 20 + 44 + 88 = 152

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Study Notes

Cost of Items

  • A suit and shoes cost a total of 720 manats
  • The suit's price decreased by 20%
  • The shoes' price decreased by 30%
  • The total price after discount is 564 manats
  • Determine the original price of the suit

Population Growth

  • A city's population was 70000
  • The population grew to 74200 in one year
  • Calculate the percentage increase in population

Complex Numbers

  • z₁ = -1 + 2i and z₂ = (1+i)/(1-i)
  • z = 2z₁ - 7z₂ + 6
  • Find the modulus of the complex number z

Geometry/Trigonometry

  • Find the length of AD in a diagram
  • A line segment from a point to a circle touches the circle at a point C
  • The circle's radius is 12
  • AC = 8
  • Find the length of the segment AD

Geometry/Trigonometry (2)

  • Length of a line segment is 1 cm
  • The ends of the segment are at distances of 0.1 cm and 0.5 cm from a plane
  • Calculate the projection length of the segment

Inequalities

  • Solve the inequality: (7-3k)/2 - (8k+1)/3 < 12 - 10k

Probability

  • A game with 50 participants
  • Participants' numbers drawn at random
  • Calculate the probability that a selected participant's number is an even number divisible by 7

Trigonometry

  • 25 is divided into 25 equal parts
  • Find the value of tan(a) + cos(β) in a diagram showing a triangle

Polynomials

  • M1(x⁴ + 3x)⁸ degree is 39
  • M2(x⁶ + 2x + 3)⁷ degree is 46
  • Calculate the degree of M₅ • M₁₁ • M₂₂

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