Simplify the algebraic expression: (6x³ - 18x² + 12x) / [6x(x - 2)], assuming x ≠ 0 and x ≠ 2.
Matematicas
Question · Easy
Solve the system of linear equations: 4x - 3y = 17 2x + 5y = -11
Determine the value of k such that the quadratic equation 3x² - 12x + 2k = 0 has exactly one real solution (a double root).
Question · Medium
Factor completely the difference of cubes: 8x³ - 27y³.
Simplify the product of rational expressions: [(x² - 9) / (2x² + 7x + 3)] · [(4x + 2) / (x - 3)].
Rationalize the denominator of the radical expression: 6 / (√5 - √2).
Solve for x in the logarithmic equation: log₂(x + 6) + log₂(x - 1) = 3.
Find the real solutions to the exponential equation: 9^(x + 1) - 28 · 3^x + 3 = 0.
The 3rd term of an arithmetic progression is 11 and the 8th term is 31. What is the sum of the first 15 terms of this progression?
What is the constant term (the term independent of x) in the binomial expansion of (2x² - 1/x)⁶?
Question · Hard
In triangle ABC, line segment DE is drawn parallel to base BC, with D on side AB and E on side AC. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, what is the length of segment EC?
In a circle with center O, inscribed angle ∠ACB subtends arc AB. If the central angle ∠AOB measures 76°, what is the measure of the inscribed angle ∠ACB?
Simplify the trigonometric identity: sin(2θ) / [1 + cos(2θ)].
In triangle ABC, side a = 7 cm, side b = 5 cm, and the included angle C = 60°. What is the exact length of side c?
An observer stands 30 meters from the base of a vertical antenna on level ground. The angle of elevation to the top of the antenna is 30°. Given tan(30°) = √3 / 3, what is the height of the antenna?
Calculate the perpendicular distance from point P(2, -3) to the straight line given by 3x - 4y + 7 = 0.
What is the general equation of the circle centered at (3, -2) that is tangent to the horizontal line y = 4?
A parabola has its vertex at the origin (0, 0) and its focus at (0, -3). What is the standard equation of the parabola and the equation of its directrix?
For the ellipse with equation x² / 25 + y² / 16 = 1, determine the coordinates of its foci and its eccentricity e.
Determine the equations of the asymptotes for the hyperbola x² / 16 - y² / 9 = 1.
Calculate the limit: lim (x → 3) [(x² - x - 6) / (x² - 9)].
Find the derivative with respect to x of the function f(x) = (3x² - 5)⁴.
Determine the equation of the line tangent to the parabola y = 2x² - 3x + 1 at the point where x = 2.
Evaluate the definite integral: ∫₀² (3x² - 4x + 5) dx.
Evaluate the indefinite integral: ∫ 2x √(x² + 4) dx.
In which of the following Spanish words is the orthographic accent (tilde) correctly placed according to standard RAE accentuation rules?
Competencia Escrita Y Lectora
Which of the following sentences exhibits correct punctuation according to the normative rules of the Real Academia Española (RAE)?
Identify the sentence that contains a spelling error involving homophones or parophones in Spanish:
Which of the following Spanish sentences displays correct grammatical agreement (concordancia nominal y verbal)?
Identify the syntactic classification of the sentence: 'Aunque el equipo dedicó semanas enteras a la investigación, los resultados preliminares resultaron inconclusos.'
Which of the following sentences incurs the linguistic error known as 'dequeísmo'?
According to Roman Jakobson's communication theory, what language function predominates in the statement: 'El punto de ebullición del etanol a una atmósfera de presión es de 78.37 °C'?
In the sentence: 'El decano pronunció un discurso conciso para evitar dilaciones innecesarias en el protocolo ceremonial', which word serves as the most precise contextual synonym for 'conciso'?
Select the word that functions as the exact antonym of 'efímero' in scientific and literary writing:
Choose the discourse marker that correctly establishes an adversative relationship to complete the statement: 'El candidato dominaba los fundamentos teóricos del área físico-matemática; _______, carecía de experiencia práctica en el manejo de instrumental de laboratorio.'
Which textual typology is primarily defined by formulating a controversial thesis and defending it through logical evidence, deductive warrants, and refutation of counterarguments?
Read the following passage: 'La automatización de procesos industriales no busca reemplazar la creatividad humana, sino liberar a los operarios de labores repetitivas y peligrosas. Cuando las máquinas asumen tareas mecánicas con precisión milimétrica, los profesionales técnicos pueden reenfocar su energía en la resolución de problemas complejos, la supervisión estratégica y la innovación de procesos.' What is the central thesis of the author?
Read the excerpt: 'Durante el siglo XX, el Instituto Politécnico Nacional formó a las generaciones de ingenieros que construyeron las presas hidroeléctricas, las redes de telecomunicaciones y el sistema de transporte colectivo de México. Sin esta infraestructura técnica, el proceso de urbanización e industrialización del país habría enfrentado cuellos de botella insalvables.' What can be directly inferred from the passage?
Analyze the formal structure of the following argument: Premise 1: Todos los metales alcalinos reaccionan vigorosamente con el agua liberando hidrógeno gaseoso. Premise 2: El cesio es un metal alcalino del grupo 1 de la tabla periódica. Conclusion: Por lo tanto, el cesio reacciona vigorosamente con el agua liberando hidrógeno gaseoso. Which mode of reasoning is exemplified here?
In the literary fragment: 'El tiempo devoró las murallas centenarias de la antigua fortaleza con sus implacables dientes de óxido y viento', which rhetorical figure is primarily employed?
Identify the informal logical fallacy committed in the statement: 'No debemos tomar en consideración el proyecto del investigador sobre energías renovables, porque en el pasado tuvo disputas personales con los miembros del consejo directivo.'
In the passage: 'Los investigadores publicaron la patente del nuevo polímero térmico; esta fue validada de inmediato por el organismo certificador', which word functions as an anaphoric referent?
Which discursive mode predominates in an engineering technical manual that systematically explains the operating principles, safety protocols, and calibration parameters of a spectrophotometer without expressing personal value judgments?
In the legal sentence: 'El juez de control dictó un auto de vinculación a proceso contra el imputado', what is the exact technical meaning of the polysemous word 'auto'?
Choose the most faithful paraphrase of the following premise: 'La divergencia epistemológica en una comunidad académica no fractura su solidez institucional, sino que estimula la rigurosidad analítica a través de la controversia fundamentada.'
A test automobile moves along a straight track at an initial velocity of v₀ = 25 m/s. The driver applies the brakes, causing a uniform deceleration of a = -5.0 m/s² until coming to a complete stop (v = 0 m/s). What is the total braking distance covered by the car?
Fisica
A crate of mass m = 20 kg rests on a horizontal concrete surface. The coefficient of kinetic friction between the crate and the surface is μ_k = 0.25. A constant horizontal pulling force of F = 99 N is applied to the crate. Taking g = 9.8 m/s², what is the resulting horizontal acceleration of the crate?
A pendulum bob is pulled back and released from rest at a height of h = 2.5 meters above its lowest equilibrium position. Assuming air resistance and friction are negligible and taking g = 9.8 m/s², what is the maximum speed attained by the bob as it passes through its lowest point?
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