IPN HIGHER LEVEL Geometry and Trigonometry Simulator Exam 2024

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By definition the exponential and logarithmic form are expressed as

Complete the related exponential and logarithmic form shown below:

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Logarithm base 𝑎 of 𝑥 is equal to the exponent to which the base 𝑎 is raised to obtain 𝑥:

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1D. The power is multiplying the base by itself as many times as the exponent indicates.

2A. Logarithm is the exponent to which the base is raised to obtain the power.

3B. Reciprocal is exchanging numerator for denominator.

4C. Radication is calculating the number that, multiplied by itself, as many times as the index of the root indicates, results in the radical.

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Apply definition of logarithm:

Apply hierarchy of operations and law of subtraction:

Apply law of signs:

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By definition of logarithm base 10 we can express:

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Calculate prime factors:

Apply properties of exponents:

Apply laws of exponents and distributive property:

Apply laws of exponents and law of addition and law of subtraction, with similar terms:

Apply trichotomy law:

11x-26=2-2x

Apply inverse operations and the law of addition with like terms:

13x = 28

Apply reverse operation:

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By definition, the base 10 logarithmic form we have 𝑙𝑜𝑔 1 = 0 analogously 𝑙𝑜𝑔 10 = 1

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Apply laws of exponents and calculate prime factors:

Apply inverse operations and laws of exponents:

Apply properties of exponents:

Apply definition of logarithm:

Apply base change, logarithm properties and reflective property:

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The quotient is made between the functions:

Then:

Applying the logarithm base e, we have:

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By drawing the dotted line to vertex B, we can identify an equilateral triangle with angles
60° interiors.

Analogous to the previous one, an isosceles triangle is formed with an angle of 30° and the other two angles have to be equal, therefore they are . Finally, the angle of 180 must be the sum
next:

𝑥 + 60 +75 = 180

Then 𝑥 = 45°

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Apply laws of exponents:

Apply properties of logarithms and definition of logarithm:

Apply equality property, using the lcm:

Apply distributive property, law of signs and hierarchy of operations:

Apply law of addition and law of subtraction with similar terms:

Apply inverse operations and laws of signs:

Apply definition of logarithm:

Apply property of exponents and rationalize the denominator:

Apply laws of exponents:

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The units for measuring angles are sexagesimal degrees, radians, gradians and revolutions. However, here we convert from degrees to radians for which we remember that 𝜋 𝑟𝑎𝑑 = 180°, then:

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By complementary angles:

Solving equation 2:

Substituting into equation (1):

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The similarity relationship is Side Angle Side (LAL), then the ratio is fulfilled:

Substituting the lengths:

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The relationship to find the number of diagonals of a regular polygon is:

Where 𝑁𝐷 = 9 is the number of diagonals and 𝑛 the number of sides.

Solving to find 𝑛 we have:

For which 𝑛 = 6 is a regular 6-sided polygon (hexagon)

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1D. Height is the segment that has as its end a vertex of the triangle and is perpendicular to the opposite side of that vertex or its extension.

2 C. Median is the segment that joins a vertex of a triangle to the midpoint of the opposite side.

3A. Bisector is the ray that divides an angle into two congruent angles.

4B. Bisector is the line that passes perpendicularly through the midpoint of a segment.

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The circle has an inscribed hexagon, so the interior angle is calculated with the relationship:

This angle is equal to 𝛼 because they are opposite at the vertex.

The triangle that is formed is equilateral and has two equal angles, therefore:

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The diagonals of the square measure , so the segments measure

Now the triangle  It has the following area:

Entonces

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According to the rule of interior angles of a triangle:

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According to the properties of quadrilaterals, the sum of the interior angles is 360° as shown below:

In the inner triangle

Entonces

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1D. Incenter.- Point where the three bisectors intersect, it is always inside the triangle. It is the center of a circle that touches each side at a single point (circle inscribed in the triangle).

2A. Centroid or centroid.- It is the point where the three medians intersect, it is always inside the triangle. The center of gravity of a triangle-shaped body is considered.

3C. Orthocenter.- It is the point where the three heights intersect.

4B. Circumcenter.- Point where the three bisectors intersect. It is the center of a circle that passes through the three vertices of the triangle (circle circumscribed by the triangle).

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The triangles that are formed are similar under the criterion (LAL).

So it is fulfilled:

It has

If fulfilled, it provides for:

Then:

Where to get it by Pythagoras

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In the figure you can distinguish the angles in the circle:

1 C. The semiinscribed angle is one with a vertex on the circle, one side secant and the other tangent to the circle 𝛾.

2D. The exterior angle has a vertex on the outside of the circle and its sides are secant to the circle 𝛿.

3A. The inscribed angle is the one with a vertex on the circle and its sides are chords 𝛼.

4B. The central angle has a vertex at the center of the circle, its sides are radii 𝛽.

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Considering the right triangle formed by a radius 𝑟 = 2.5 and the apothem 𝑎 = 2

The base of the right triangle

Then one of the sides 𝐿 = 3

Considering the 5 sides we will have the perimeter 𝑃 = 5𝐿 = 15

The area will be A

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Of geometry

It can be seen that the length

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Applying the trigonometric identities we have the following:

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Using the right triangle you can see that

Then, calculating the hypotenuse, since 𝑡𝑎𝑛 𝛽 = then

you have

Thus

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Replace the quotient identity:

Add rational numbers:

Divide rational numbers:

Factor and apply partial fractions:

Substitute quotient identity and reciprocal identity:

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We use a sum identity:

where 𝑡𝑎𝑛 60 =

Rationalizing, we have:

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Factoring 𝑠𝑒𝑛 𝛼 of all the terms of the numerator and taking from the parentheses:

Developing the squared binomials:

Using the identity:

We have:

Finally, we used:

To get to the result

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The triangle is not right angled so we use the law of cosines to calculate side A in this way:

Substituting the sides 𝐵 = 5 and 𝐶 = 3, as well as the angle 𝛼 = 120

Now, to calculate the angle we use the law of sines:

Substituting the values ​​of 𝐴 = 7, 𝐶 = 3, 𝛼 = 120

Then 𝛽 = 38.2°

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[1] The correct thing is

[3] The interior angles of a triangle cannot add up to more than 180°

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The law of sines is never used on obtuse angles because the graph of the sine has symmetry between the angles 0 to 90 and 90 to 180.

The law of cosines can be used with obtuse angles because the cosine graph does not have symmetry.

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Apply equality property:

Simplify the fraction, multiply rational numbers and divide:

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It can be seen that the trigonometric function 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 has period 𝜋 as does its inverse function.

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Apply the law of sines to the triangle on the left:

Replace , perform inverse operation and divide:

Apply law of sines for the major triangle:

Substitute sin 90=1 and apply inverse operation:

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The dodecagon is made up of 12 isosceles triangles:

Calculate the measure of the vertex angle:

Share:

Calculate the measure of congruent angles:

Apply law of subtraction and divide:

Apply law of sines:

Replace , perform inverse operation and divide:

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The pentagon (regular polygon) does not have a center of symmetry since point A does not have a diametrically opposite symmetry like B, C, D, E

While the axes of symmetry are 5:

The answer is 0 centers of symmetry and 5 axes of symmetry

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Apply trigonometric function:

Apply inverse operation, simplify the fraction and reflective property:

Substitute into the speed=distance/time formula:

Factor by GCD, divide rational numbers and simplify the fraction:

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Apply law of cosines:

Apply hierarchy of operations, law of addition, inverse operation, inverse function and property
reflective:

Apply law of signs and simplify the fraction:

Apply law of sines:

Substitute sin 90=1 and apply inverse operation:

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When graphing the function, you can see the points where it crosses the x axis (zeroes of the function).

In the function 𝑓(𝑥) = 𝐴𝑐𝑜𝑠 (𝑥) the zeros are obtained if the argument takes the values

for 𝑛 = ⋯ ,−2,−1, 0, 1, 2, 3, …

However, for the function

the argument is

for 𝑛 = ⋯ ,−2,−1, 0, 1, 2, 3, …

Plays the zeros of the function

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Apply Pythagorean theorem:

Equate, apply definition of power and squared binomial:

Apply inverse operation, law of addition and law of subtraction:

16 20 a =

Apply inverse operation and simplify the fraction:

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Apply trigonometric function:

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The function has shifted by 𝜋/2 to the right with respect to the original , for the
that the function that has been advanced is

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As can be seen from the geometry

There are the following reasons:

Performing the quotient:

Adjusting the equation we have:

Solving the equation

Entonces

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is the distance from the lamp to the person.

Apply Thales Theorem:

Apply inverse operation and divide:

So the distance

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1. The graph of the cosecant (reciprocal function of the sine) is in part B.

2. The graph of the secant (inverse cosine function) in part D.

3. The graph of the cotangent (inverse function of the tangent) in part A.

4. The graph of the tangent is found in part C.