Two aircraft flight paths in a planar radar sector are modeled by the linear equations 2x - 3y = 6 and x + 2y = 10. At what coordinate point (x, y) do the two flight paths intersect?
Mathematics
Question · Easy
Factor it
Short
What are the real roots of the quadratic equation 2x² - 5x + 2 = 0?
According to the Remainder Theorem, what is the remainder when the polynomial P(x) = 3x³ - 4x² + 2x - 5 is divided by (x - 2)?
Question · Medium
What is the real solution to the logarithmic equation log₂(x) + log₂(x - 2) = 3?
Atmospheric pressure P at altitude h (in kilometers) decreases exponentially according to P(h) = P₀ · e^(-k·h). If the pressure at sea level is P₀ = 100 kPa and at altitude h = 6 km the pressure is 50 kPa, what is the exact value of the decay constant k?
Question · Hard
What is the exact value of cos(120°) + sin(150°)?
Which of the following simplifies the trigonometric expression sin(2θ) / (1 + cos(2θ)) for all valid θ?
A radar station tracks two aircraft at distances of 60 km and 80 km from the station. If the angle subtended between the two aircraft at the radar station is 60°, what is the straight-line distance between the two aircraft?
How many distinct solutions exist for the equation 2 sin²(x) - sin(x) - 1 = 0 in the interval [0, 2π)?
What is the exact value of cos(arcsin(3/5) + arccos(12/13))?
What is the straight-line distance between two tactical navigation beacons located at coordinates P(3, -4) and Q(-1, 2)?
What are the center coordinates and radius of the circular radar coverage area defined by x² + y² - 6x + 8y - 11 = 0?
What is the equation of the straight line perpendicular to 3x - 4y + 12 = 0 that passes through the point (2, -1)?
For a parabolic radar reflector modeled by the standard equation y² = 12x, what are the coordinates of the focus and the equation of the directrix?
An elliptical surveillance flight orbit is modeled by x²/25 + y²/16 = 1. What is the eccentricity e of this ellipse?
What is the magnitude and unit vector of the 2D velocity vector v = 6i - 8j (in m/s)?
What is the scalar dot product of 3D spatial vectors a = (2, 3, -1) and b = (1, -2, 2)?
What is the vector cross product u × v of vectors u = (1, 2, 0) and v = (0, 3, 4)?
A force vector F = (4i + 3j - 2k) N moves an aircraft mechanical actuator along a displacement vector d = (3i - 2j + 5k) m. What is the total mechanical work done by this force?
Given matrices A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], what is the resulting matrix 2A - B?
What is the product matrix AB of A = [[2, 1], [-1, 3]] and B = [[4, -2], [0, 5]]?
What is the inverse matrix M⁻¹ of M = [[3, 5], [2, 4]]?
What is the determinant of the 3×3 transformation matrix T = [[2, 0, 1], [3, 1, 2], [1, 2, 0]]?
An aircraft engine inspection schedule follows an arithmetic progression. If the first inspection occurs at hour a₁ = 50 and subsequent inspections occur every d = 75 operating hours, at what operating hour does the 15th inspection occur?
What is the sum of the infinite geometric series S = 12 + 6 + 3 + 1.5 + ...?
What is the value of the limit lim (x → 3) [(x² - 9) / (x - 3)]?
What is the value of the trigonometric limit lim (x → 0) [sin(6x) / (2x)]?
What is the first derivative f'(x) of the polynomial function f(x) = 4x³ - 5x² + 7x - 12?
What is the derivative of the function g(x) = e^(2x) · cos(3x) with respect to x?
An unmanned aerial vehicle moves along a straight trajectory with position function s(t) = 2t³ - 9t² + 12t (where s is in meters and t is in seconds). What are the vehicle's velocity and acceleration at time t = 2 seconds?
A test rocket's altitude h (in meters) after launch is given by h(t) = -5t² + 60t + 20 (where t is in seconds). What is the maximum altitude achieved by the rocket?
What is the exact value of the definite integral ∫₀³ (3x² + 4x - 1) dx?
What is the indefinite integral ∫ 2x √(x² + 5) dx?
A primary surveillance radar has a probability of 0.85 of detecting an incoming target on any single scan, while a secondary radar has a detection probability of 0.80. Assuming the two radar systems operate independently, what is the probability that at least one of the two systems detects the target?
From a squadron of 8 qualified pilot officers, how many different 3-officer mission crews can be formed if the order of selection does not matter?
An aircraft starts from rest at the beginning of a runway and accelerates uniformly to its liftoff speed of 70 m/s over a takeoff ground roll distance of 700 meters. What is the magnitude of the aircraft's constant acceleration?
Physics
A defensive projectile is launched from ground level with an initial velocity of 100 m/s at an angle of 30° above the horizontal. Taking g = 9.8 m/s² and neglecting air resistance, what is the maximum height achieved by the projectile?
An aircraft flies with a True Airspeed of 240 km/h heading due North. A crosswind is blowing from the West towards the East at a constant speed of 70 km/h. What is the aircraft's resulting ground speed and drift angle?
A military jet aircraft of mass 12,000 kg accelerates uniformly along a level runway at 4.5 m/s². If total aerodynamic and rolling drag opposing motion is 6,000 N, what is the total forward engine thrust required?
An equipment crate of mass 50 kg rests on a ramp inclined at 30° to the horizontal. If the coefficient of static friction between the crate and the ramp is μ_s = 0.40 and g = 9.8 m/s², what is the maximum static friction force available, and does the crate slide?
A fighter jet with a mass of 8,000 kg executes a horizontal coordinated turn at a constant airspeed of 200 m/s. If the turn radius is 2,000 meters, what is the magnitude of the centripetal force acting on the aircraft?
For an aircraft in a coordinated level turn at true airspeed v and radius r, which formula gives the required bank angle θ, and what is the load factor n in terms of θ?
What is the kinetic energy of a 2,500 kg reconnaissance drone flying at a steady cruise speed of 40 m/s?
A training glider of mass 600 kg descends without engine power from an altitude of 1,500 m to 1,000 m. Assuming air resistance is negligible, what is the increase in the glider's kinetic energy and its velocity gain if it started the descent at 30 m/s? (Take g = 9.8 m/s²)
A solid-fuel booster rocket expels exhaust gases at a constant exit velocity of v_e = 2,400 m/s relative to the nozzle. If the fuel burn rate is dm/dt = 15 kg/s, what is the thrust force produced by the rocket motor?
A jet engine turbine rotor has a moment of inertia I = 2.5 kg·m². A net torque of 150 N·m is applied during engine startup. Starting from rest, what is the rotor's angular velocity after 4.0 seconds?
What is the absolute hydrostatic pressure at the bottom of a hydraulic reservoir filled with fluid of density ρ = 850 kg/m³ to a depth of 4.0 meters, given atmospheric pressure P_atm = 101.3 kPa and g = 9.8 m/s²?
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