Algebra and Geometry: Circle Properties

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Questions and Answers

What is the formula to calculate the roots of a quadratic equation represented as $ax^2 + bx + c = 0$?

$x = \frac{-b \pm 2\sqrt{b^2 - 4ac}}{2a}$
$x = \frac{-2b \pm \sqrt{b^2 - 4ac}}{a}$
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (correct)
$x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a}$

Which of the following expansions correctly represents $(a + b)^2$?

$(a + b)(a + b)$
$a^2 + 2ab + b^2$ (correct)
$a^2 - 2ab + b^2$
$a^2 + ab + b^2$

Which property applies to the angle at the center of a circle compared to the angle at the circumference?

It is equal to the angle at the circumference.
It is independent of the circumference angle.
It is twice the angle at the circumference. (correct)
It is half the angle at the circumference.

What is the expression for $a^2 - b^2$ in terms of its factors?

$(a + b)(a - b)$ (B) Signup and view all the answers

What result comes from applying the distributive property to $a(b + c)$?

$ab + ac$ (B) Signup and view all the answers

How is the mode of a series of numbers determined?

It is the number that occurs most frequently in the series. (B) Signup and view all the answers

What is the relationship between exclusive events A and B?

p(A or B) = p(A) + p(B) (D) Signup and view all the answers

What is the correct way to calculate the probability of independent events A and B occurring at the same time?

p(A and B) = p(A) × p(B) (A) Signup and view all the answers

To find the median of a series of numbers, which step is necessary?

Arrange the series in ascending order and select the middle number. (A) Signup and view all the answers

What is the formula for calculating the probability of an event?

p(Event) = total occurrences / total outcomes (D) Signup and view all the answers

What is the formula for converting speed from km/hr to m/sec?

$rac{5}{18} imes ext{km/hr}$ (B) Signup and view all the answers

What is the sum of the interior angles of a polygon with n sides?

$(n − 2) × 180°$ (B) Signup and view all the answers

If the exterior angles of a polygon are summed, what total do they equal to?

$360°$ (A) Signup and view all the answers

What is the relationship between tangent and radius at the point of contact on a circle?

They form a right angle (D) Signup and view all the answers

Using the Pythagorean theorem, what does c represent?

The hypotenuse of a right triangle (C) Signup and view all the answers

What does TOA in trigonometric ratios stand for?

Tangent equals opposite divided by adjacent (D) Signup and view all the answers

For a polygon with 5 sides, what is the sum of the interior angles?

$540°$ (C) Signup and view all the answers

Which statement is true about the types of polygons?

A square is a polygon (C) Signup and view all the answers

What is the value of $a^0$ for any non-zero number $a$?

1 (B) Signup and view all the answers

How is the multiplication of two square roots expressed mathematically?

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ (D) Signup and view all the answers

According to the triangular law of addition, which of the following expresses how two vectors can be added?

$\vec{OA} + \vec{OB} = \vec{OC}$ (C) Signup and view all the answers

What is the equation for simple interest?

$i = \frac{Prt}{100}$ (D) Signup and view all the answers

What does the formula for compound interest represent?

$A = P(1 + \frac{r}{100})^n$ (D) Signup and view all the answers

Which of the following expressions represents a vector addition scenario?

$\vec{OA} + \vec{OB} = \vec{OC}$ (C) Signup and view all the answers

What is the result of $(a^m)^{-n}$?

$\frac{1}{a^{mn}}$ (B) Signup and view all the answers

Which statement about opposite angles in a quadrilateral is correct?

They add up to 180 degrees. (A) Signup and view all the answers

For the equation $y$ is inversely proportional to $x$, which of the following is true?

$y = k \div x$ (D) Signup and view all the answers

What does the formula for the exterior angle of a quadrilateral state?

It is equal to the sum of two interior opposite angles. (D) Signup and view all the answers

What is the relationship expressed by the variation formula $y = kx$?

y is directly proportional to x. (A) Signup and view all the answers

Which of the following is a correct interpretation of the equation $(\sqrt{a})^2 = a$?

The square root of a number squared equals the number. (B) Signup and view all the answers

Which law states that when two sides of a triangle are added together, the result is equal to the third side?

Triangle law of addition (C) Signup and view all the answers

How is the expression $(a imes b)^m$ simplified?

$a^mb^m$ (C) Signup and view all the answers

Which equation represents a direct proportional relationship where the slope is positive?

y = mx + c (C) Signup and view all the answers

Which of the following properties is not a criterion for triangle congruence?

HL property (B) Signup and view all the answers

What can be concluded if two triangles satisfy the AAS property?

They are congruent. (A) Signup and view all the answers

What is a characteristic of similar solid figures?

Their corresponding linear dimensions are proportional. (C) Signup and view all the answers

In the SAS property, which elements are compared between the two triangles?

Two sides and one angle (D) Signup and view all the answers

Which equation represents a quadratic relationship with a negative leading coefficient?

y = -ax^2 + bx + c (C) Signup and view all the answers

Which property confirms that two right triangles are congruent based on the hypotenuse and one other side?

RHS property (C) Signup and view all the answers

For two triangles to be considered congruent under the SSS property, what must be true?

Their corresponding sides must be equal in length. (B) Signup and view all the answers

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Study Notes

Algebra

Distributive property: ( a(b + c) = ab + ac )
Expansion of squares:
- ( (a + b)^2 = a^2 + 2ab + b^2 )
- ( (a - b)^2 = a^2 - 2ab + b^2 )
Difference of squares: ( a^2 - b^2 = (a + b)(a - b) )

Quadratic Equations

Standard form: ( ax^2 + bx + c = 0 )
Solutions using the quadratic formula:
- ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
Completing the square for solving quadratics.

Properties of Circles

Angle at the center is twice the angle at the circumference.
The geometrical relationship between angles in a quadrilateral:
- Opposite angles sum to ( 180^\circ ) (e.g., ( \angle A + \angle C = 180^\circ )).

Operations with Exponents

Rules of exponents:
- ( (a^m)^n = a^{mn} )
- ( a^0 = 1 )
- ( a^{-n} = \frac{1}{a^n} )

Simple and Compound Interest

Simple Interest calculation: ( i = \frac{P \times r \times t}{100} )
Compound Interest formula:
- ( A = P(1 + \frac{r}{100})^n )

Measurement Conversions

Speed conversions:
- ( \text{km/hr} \times \frac{5}{18} = \text{m/sec} )
- ( \text{m/sec} \times \frac{18}{5} = \text{km/hr} )

Geometry

Pythagorean theorem: ( c^2 = a^2 + b^2 )
Sum of exterior angles in polygons: equal to ( 360^\circ )
Interior angles sum: ( (n - 2) \times 180^\circ ) for n-sided polygons.

Trigonometry

Mnemonic for trigonometric functions: TOA (Tangent = Opposite/Adjacent), CAH (Cosine = Adjacent/Hypotenuse), SOH (Sine = Opposite/Hypotenuse).

Vectors

Triangular law of vector addition:
- Combined vectors can be expressed using their components.

Statistical Measures

Mode: the most frequently occurring number in a data set.
Median: the middle value when numbers are arranged in ascending order.

Probability Concepts

Exclusive events: cannot occur simultaneously; use ( p(A \text{ or } B) = p(A) + p(B) ).
Independent events: can occur simultaneously; use ( p(A \text{ and } B) = p(A) \times p(B) ).

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