RESONANCE SUMMER VACATION HOMEWOR OF MATH

 

RESONANCE SUMMER VACATION HOMEWOK  OF MATH

1. Uses of Mathematics in Real Life (100 words)

Mathematics is all around us! It's used in science, technology, engineering, and mathematics (STEM) fields, finance, economics, and even everyday activities like cooking and shopping. Math helps us understand patterns, make predictions, and solve problems. It's essential for coding, data analysis, and medical research. From calculating tips to understanding climate change, math is a vital tool for navigating the world.

 

2. Pythagoras

Pythagoras was an ancient Greek mathematician and philosopher who lived around 570-495 BCE. He's best known for the Pythagorean theorem, which describes the relationship between the sides of a right-angled triangle.

 

3. Rational Numbers between 3/5 and 2/3

To find rational numbers between 3/5 and 2/3, we can find a common denominator, which is 15. Then, 3/5 = 9/15 and 2/3 = 10/15. Some rational numbers between them are:

 

- 91/150

- 92/150

- 93/150

- 94/150

- 95/150

 

4. Representing √5 on the Number Line

To represent √5 on the number line, we can use the Pythagorean theorem. Draw a right triangle with legs of length 2 and 1. The hypotenuse will have a length of √5 (since 2^2 + 1^2 = (√5)^2). Use a compass to draw an arc from the origin with a radius equal to the length of the hypotenuse, and mark the point where it intersects the number line.

 

5. Rationalizing

a. 1/(2+√3) = (2-√3)/(2+√3)(2-√3) = (2-√3)/(4-3) = 2-√3

 

b. 3/(2+2√5) = (3(2-2√5))/((2+2√5)(2-2√5)) = (6-6√5)/(4-20) = (6√5-6)/16

 

6. Expressing 0.2353535... in p/q Form

Let x = 0.2353535...

 

100x = 23.535353...

 

10000x = 2353.535353...

 

Subtracting the first equation from the second:

 

9900x = 2330

 

x = 2330/9900 = 233/990

 

7. Verifying Zeroes of Polynomials

a. P(x) = 3x^2 - 1

 

P(-1/√3) = 3(-1/√3)^2 - 1 = 3(1/3) - 1 = 0

 

P(2/√3) ≠ 0 (so 2/√3 is not a zero)

 

b. P(x) = (x+1)(x+2)

 

P(-1) = (-1+1)(-1+2) = 0

 

P(2) ≠ 0 (so 2 is not a zero, but -2 is)

 

8. Spiral Square Root

To create a spiral square root, start with a right triangle with legs of length 1. The hypotenuse will have a length of √2. Then, draw another right triangle with one leg of length √2 and the other leg of length 1. The hypotenuse will have a length of √3. Continue this process to create a spiral pattern representing √2, √3, √4, √5, and so on.