What is the value of the limit lim (x -> 0) [sin(4x) / (2x)]?
Mathematics
Question · Easy
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What is the value of the first derivative of f(x) = ln(3x^2 + 1) evaluated at x = 1?
Which French sentence has correct noun-adjective agreement?
French Language
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Which French sentence correctly distinguishes a from à?
Three identical sensor rows each contain x sensing elements, and two shared reference elements are mounted outside the rows. If the assembly has 29 elements in total, what is x?
Given three-dimensional vectors u = (2, -1, 3) and v = (4, 2, -1), what is their scalar (dot) product u . v?
What is the probability of rolling a sum of exactly 7 with two fair, independent six-sided dice?
A calibration model uses g(t) = 1/sqrt(t - 3). For which real values of t is the model defined?
What is the real solution x to the exponential equation 2^(x + 1) = 16?
What is the value of the limit lim (x -> +infinity) [sqrt(x^2 + 4x) - x]?
Question · Medium
An open-top box is made by cutting squares of side x from each corner of a 20 cm by 14 cm sheet and folding the sides. If V(x) = x(20 - 2x)(14 - 2x), which equation must an interior maximum satisfy?
What is the general solution y(x) to the first-order differential equation y' - 3y = 6?
Which completion gives the standard French construction after bien que?
What is the sum S of the infinite geometric series: S = sum_{n=0}^infinity 4 * (1/3)^n?
What is the cross product w = u x v of the vectors u = (1, 2, 0) and v = (0, 3, 1)?
The line r(t) = (1, -1, 2) + t(2, 1, -1) intersects the plane x + 2y + z = 7. What is the intersection point?
A discrete random variable X follows a binomial distribution B(n = 5, p = 0.4). What is the exact probability P(X = 2)?
Which French relative pronoun correctly completes this sentence: « Le projet ___ nous avons discuté sera révisé. »?
What is the unique solution to the initial value problem y'' + 4y = 0 with y(0) = 1 and y'(0) = 2?
What is the perpendicular distance from the point M(1, 2, 3) to the plane with equation 2x - 2y + z - 6 = 0?
What is the determinant of the 3x3 matrix A = [[2, 1, 0], [1, 3, 2], [0, 4, 1]]?
A continuous random variable X has probability density function f(x) = k*x for 0 <= x <= 2, and 0 elsewhere. What are the values of k and the expected value E(X)?
What is the value of the complex number (1 + i)^4?
Six distinct sensor positions include two corner positions. If three positions are selected uniformly at random, what is the probability that at least one corner position is selected?
What is the third-order Taylor polynomial (Maclaurin expansion) of f(x) = e^(2x) centered at x = 0?
Question · Hard
A closed cylindrical tank must hold 500pi cubic centimeters. If its radius r and height h are positive, which relation holds at the dimensions that minimize total surface area?
Solve the second-order linear differential equation y'' - 4y' + 4y = 0 subject to initial conditions y(0) = 2 and y'(0) = 7.
What is the volume under the surface z = 4 - x^2 - y^2 and above the xy-plane?
An urn contains 4 red balls and 6 blue balls. Three balls are drawn at random without replacement. What is the probability of drawing exactly 2 red balls and 1 blue ball?
Read the French passage: « Une solution technique n'est pas robuste parce qu'elle fonctionne dans des conditions idéales. Elle l'est si ses performances restent acceptables malgré les variations de charge, de température et de fabrication. Cependant, accroître toutes les marges sans distinction peut alourdir et renchérir le système. Le concepteur doit donc identifier les variations critiques et dimensionner les marges en conséquence. » Which statement best expresses its central claim?
In the French sentence « Une méthode fiable donne des résultats reproductibles », which phrase is closest in meaning to fiable?
What is the kinetic energy of a 1,000 kg automobile travelling at a constant velocity of 20 m/s?
Physics
A sound wave travels through air at 340 m/s with a wavelength of 0.85 m. What is the frequency of the sound wave?
What is the gravitational potential energy gained by a 5.0 kg mass lifted vertically by 4.0 m in a uniform gravitational field where g = 9.8 m/s^2?
A light ray in air (refractive index n_1 = 1.0) enters a glass prism (n_2 = 1.5) at an incidence angle of 30 degrees. According to Snell's law, what is the value of sin(r), where r is the angle of refraction?
What is the electric power dissipated as heat in a resistor of resistance R = 10 ohms carrying a direct current of I = 3.0 A?
A cart starts from rest and accelerates uniformly at 2.5 m/s^2 for 8.0 s. How far does it travel?
What is the magnitude of linear momentum of an object of mass 0.5 kg moving in a straight line at 12 m/s?
What is the equivalent capacitance of two 6.0 microfarad capacitors connected together in series?
A projectile is launched from ground level with an initial speed of 50 m/s at an angle of 30 degrees above the horizontal over flat ground. Neglecting air resistance and taking g = 10 m/s^2, what is the maximum height reached?
A mass m = 0.20 kg oscillates on an ideal horizontal spring of stiffness k = 80 N/m. What are the angular frequency omega and the oscillation period T?
Which French sentence correctly agrees the past participle with a direct object placed before a verb conjugated with avoir?
A 1,200 kg vehicle negotiates an unbanked circular curve of radius 50 m at a constant speed of 20 m/s. What minimum coefficient of static friction mu_s between the tires and the dry pavement is required to prevent skidding (g = 9.8 m/s^2)?
A train whistle emits sound at 600 Hz as the train approaches a stationary observer at 34 m/s in calm air where sound travels at 340 m/s. What apparent frequency does the observer hear?
A proton (charge q = 1.6 x 10^-19 C, mass m = 1.67 x 10^-27 kg) enters perpendicularly into a uniform 0.50 T magnetic field with speed v = 2.0 x 10^6 m/s. What is the radius of the resulting circular trajectory?
One mole of an ideal gas undergoes a reversible isothermal expansion at T = 300 K from initial volume V_1 = 0.010 m^3 to V_2 = 0.030 m^3. Given R = 8.314 J/(mol*K), how much work is performed by the gas?
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