🌿 • Module 6 : Surds, Quadratics and Analysis

Question 1

What is a surd?

Answer 1

A surd is an irrational root of a rational number that cannot be simplified into a whole number or fraction.

Example : √2, √3, √5.

Question 2

How do you simplify √50?

Answer 2

Find factors that include a perfect square:
√50 = √(25x2)= √25 x √2= 5√2

Question 3

How do you add and subtract surds?

Answer 3

You can only add or subtract like surds (with the same root).

Example : 3√2 + 5√2= 8√2,
But 3√2 + 4√3 cannot be simplified.

Question 4

How do you multiply surds?

Answer 4

Multiply the numbers inside the square roots.

Example : √3 x √12 = √(3x12) = √36 = 6

Question 5

How do you rationalize the denominator of 5/√3?

Answer 5

Multiply the numerator and denominator by √3 to remove the surd from the denominator:

(5/√3)x(√3/√3)=5 √3/3

Question 6

What is the difference of two squares?

Answer 6

It’s a special factorization rule:
a²-b²=(a-b)(a+b)

Example : 𝑥²-9 = (𝑥-3)(𝑥+3)

Question 7

What is a perfect square expansion?

Answer 7

It follows the formula:
(a+b)²=a²+2ab+b²

Example : (𝑥+4)² = 𝑥² + 8𝑥 + 16

Question 8

How do you solve a quadratic equation by factorizing?

Answer 8

Set the equation to zero, then factorize and solve for 𝑥.

Example : 𝑥²-5𝑥+6=0
(𝑥-2)(𝑥-3)=0
So, 𝑥=2 or 𝑥=3.

Question 9

What is the modal class in grouped data?

Answer 9

The modal class is the range with the highest frequency in a grouped frequency table.

Question 10

What is the conjugate of a surd?

Answer 10

The conjugate of a surd expression…
a+√b is a-√b

It’s useful for rationalizing denominators, as multiplying by the conjugate removes the surd from the denominator.

Question 11

How do you factorize a quadratic expression of the form a𝑥²+b𝑥+c?

Answer 11

Find two numbers that multiply to ac and add to b. Split the middle term, group, and factor out common terms.

Example : 𝑥²+7𝑥+10
1 — Find the two numbers that multiply to 10 and add to 7. These are 5 and 2.
2 — Factor (𝑥+5)(𝑥+2)
That’s the final answer.

Question 12

How do you find the area scale factor for similar shapes?

Answer 12

Square the length scale factor. If the length scale factor is k, then the area scale factor is k².

Question 13

What is the formula for the volume scale factor in similar solids?

Answer 13

Cube the length scale factor. If the length scale factor is k, then the volume scale factor is k³.

Question 14

What are the four conditions for triangle congruence?

Answer 14

SSS : All three sides are equal.
SAS : Two sides and the included angle are equal.
ASA : Two angles and the included side are equal.
RHS : Right angle, hypotenuse, and one other side are equal.

Question 15

How do you find the interquartile range (IQR)?

Answer 15

Order the data from smallest to largest, then find the median, which is the middle value (Q2).
Next, find Q1 (the lower half of the data), and Q3 (the upper half).
The IQR is Q3-Q1.

Question 16

What is the quadratic formula?

Answer 16

The quadratic formula is used to solve any quadratic equation of the form a𝑥²+b𝑥+c=0 :

𝑥 = (-b ± √b² - 4ac)/2a

Question 17

What does it mean for two triangles to be congruent?

Answer 17

Two triangles are congruent if they have the same size and shape.
This means their corresponding sides and angles are equal.

Question 18

What is the mean of grouped data?

Answer 18

To find the mean of grouped data, use the midpoint of each class, multiply by the frequency, sum them all up, and divide by the total frequency.

Mean = ∑(midpoint x frequency)/∑frequency

Question 19

What is the Venn diagram representation of the union of two sets A and B?

Answer 19

The union of two sets is represented by all elements in either set A or set B or both.

It is denoted as A ∪ B

Question 20

What is the Venn diagram representation of the intersection of two sets A and B?

Answer 20

The intersection of two sets is represented by the area where both sets overlap in a Venn diagram. It includes only the elements common to both sets.

It is denoted as A ∩ B

Question 21

What is the complement of a set A in Venn diagram notation?

Answer 21

The complement of a set A includes all the elements that are not in set A. In a Venn diagram, this is represented by the area outside of set A.

It is denoted as A’

Question 22

What does A∩B’ represent in set notation?

Answer 22

A∩B’ represents the elements that are in A but not in B.
It is the intersection of A with the complement of B.

Question 23

How do you represent “A is a subset of B” in set notation?

Answer 23

“A is a subset of B” is written as… A⊆B, meaning that every element of A is also an element of B.

Question 24

How do you represent “A is not equal to B” in set notation?

Answer 24

To represent “A is not equal to B,” you write A ≠ B, meaning that the two sets do not contain exactly the same elements.

Question 25

How do you find the elements in either A or B, but not both?

Answer 25

The set notation for “A or B but not both” is the symmetric difference, represented as A Δ B.
It includes elements that are in either A or B, but not in both.