What is the limit of the function f(x) = sqrt(x^2 + 3x + 1) - x as x approaches positive infinity?
Mathematics
Question · Medium
What is the first derivative of the function f(x) = ln(x + sqrt(x^2 + 1)) with respect to x?
What is the value of the definite integral of x * e^(2x) dx evaluated from x = 0 to x = 1?
Question · Hard
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Which function solves the first-order differential equation y' - 3y = 6 with the initial condition y(0) = 1?
What is the solution to the initial value problem y'' + 4y = 0 with y(0) = 2 and y'(0) = 6?
Question · Easy
For the cubic function f(x) = x^3 - 3x + 2, which equation represents the tangent to its graph at x = 2?
Evaluate L = lim[x -> 0] sin(3x)/tan(5x). What is L?
What is the value of the definite integral of ln(x)/x dx from x = 1 to x = e?
What are the coordinates of the inflection point of the cubic curve f(x) = 2x^3 - 9x^2 + 12x - 5?
What is the equation of the oblique asymptote of the rational function f(x) = (2x^2 + 5x - 1) / (x + 1) as x approaches infinity?
Write z = -1 + i*sqrt(3) in polar form. Which pair gives its modulus and principal argument?
What is the exponential form of the quotient z = (1 - i*sqrt(3)) / (1 + i)?
What are the complex roots of the quadratic equation z^2 - 2z + 4 = 0 in C?
In the complex plane, what geometric locus is described by the equation |z - 2 + i| = 3?
Let R be the plane rotation centered at Omega with affix omega = 1 + 2i and angle theta = pi/2. What is the affix z' of the image of point A with affix z_A = 3 + i under rotation R?
What is the cross product u x v of the vectors u = (2, -1, 3) and v = (1, 4, -2) in R^3?
A plane contains A(1, 2, -1) and is perpendicular to n = (3, -2, 4). Which Cartesian equation represents that plane?
What is the distance from the point P(2, -3, 1) to the plane with equation 2x - y + 2z - 3 = 0?
What is the determinant of the 3x3 matrix M = [[2, 1, 3], [0, -1, 4], [1, 2, 1]]?
What is the angle theta between the vectors u = (1, 1, 0) and v = (0, 1, 1) in R^3?
In how many distinct ways can an engineering project committee of 3 students be selected from a class of 12 students?
In a manufacturing facility, Machine A produces 60% of all components with a defect rate of 2%, while Machine B produces 40% with a defect rate of 5%. What is the total probability that a randomly selected component is defective?
Continuing from the previous factory problem (Machine A produces 60% with 2% defects, Machine B produces 40% with 5% defects; overall defect rate is 3.2%), if a randomly selected component is found to be defective, what is the probability that it was produced by Machine B?
For a binomial random variable X following B(n = 100, p = 0.36), what are its variance Var(X) and standard deviation sigma(X)?
A projectile is launched from ground level with initial speed v_0 at an angle alpha above the horizontal in a uniform gravitational field of magnitude g. Neglecting air resistance, what is the maximum height H reached by the projectile?
Physics
For a projectile launched from level ground with given initial speed v_0, at what launch angle alpha is the horizontal range maximized, and what is that maximum range?
A block of mass m = 5 kg is sliding on a rough horizontal surface with coefficient of kinetic friction mu = 0.2. A constant horizontal force F = 20 N acts in the direction of motion. Taking g = 10 m/s^2, what is the acceleration of the block?
A vehicle of mass m = 1,000 kg accelerates uniformly on a straight road from an initial speed v_1 = 10 m/s to a final speed v_2 = 20 m/s. According to the work-energy theorem, what is the net work done by the resultant force on the vehicle?
A horizontal spring-mass system consists of a block of mass m = 0.2 kg connected to an ideal spring of stiffness k = 80 N/m on a frictionless surface. What is the natural period of oscillation T_0?
A simple pendulum of length L = 1.0 m and bob mass m = 0.5 kg is released from rest at an angle theta_0 = 60 degrees to the vertical. Taking g = 10 m/s^2 and neglecting air resistance, what is the speed of the bob as it passes through its lowest equilibrium position?
A satellite of mass m moves in a circular orbit around Earth (mass M, radius R) at an altitude h above the surface. If G is the universal gravitational constant, what is the orbital speed v of the satellite?
According to Kepler's third law for circular planetary orbits around a central body of mass M, what is the exact expression for the ratio T^2 / r^3, where T is the orbital period and r is the orbital radius?
A cart of mass m_1 = 2 kg moving at speed v_1 = 6 m/s on a frictionless horizontal track undergoes a perfectly inelastic collision with a stationary cart of mass m_2 = 4 kg, sticking together. What is their common speed V after the collision, and what fraction of the initial kinetic energy is lost?
A solid uniform disk of mass M = 4 kg and radius R = 0.5 m is free to rotate about its central perpendicular axis. Its moment of inertia is I = (1/2)*M*R^2. If a constant tangential torque tau = 10 N*m is applied, what is the resulting angular acceleration alpha of the disk?
A resistor R = 40 kOhm and capacitor C = 50 microF are connected in series. What is the time constant of the RC circuit?
An initially uncharged capacitor is connected to a constant voltage source through a resistor. Approximately what percentage of its final voltage has the capacitor reached after one time constant?
An inductor L = 0.20 H and a resistor R = 50 Ohm are connected in series to a DC source. What is the time constant of the RL circuit?
An inductor of inductance L = 0.50 H carries a current I = 4.0 A. How much magnetic energy is stored in it?
An ideal LC circuit has L = 40 mH and C = 25 microF. What is its natural angular frequency?
A proton of charge magnitude 1.6 * 10^-19 C moves at 2.0 * 10^6 m/s perpendicular to a uniform magnetic field of magnitude 0.50 T. What is the magnetic-force magnitude?
A charged particle moves perpendicular to a uniform magnetic field in a circular path of radius r. If its speed doubles while its mass, charge magnitude, and the field remain unchanged, what is the new radius?
A 200-turn coil experiences a uniform magnetic flux per turn that decreases from 0.050 Wb to 0.010 Wb in 0.10 s. What is the magnitude of the average induced emf?
A straight wire segment of length 0.20 m carries a current of 5.0 A perpendicular to a uniform magnetic field of 0.40 T. What magnetic-force magnitude acts on the segment?
Which statement correctly describes resonance in an ideal series RLC circuit?
An ideal gas expands at constant pressure P = 1.0 * 10^5 Pa from 0.010 m^3 to 0.030 m^3 while receiving heat Q = 5000 J. Using the convention Delta U = Q + W, where W is work done on the gas, what is Delta U?
Light travels from air with refractive index 1.0 into glass with refractive index 1.5 at an incidence angle of 30 degrees. What is the sine of the refraction angle?
An object is placed 60 cm in front of a thin converging lens of focal length 20 cm. Where is the image, and what is its transverse magnification?
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