Solve the quadratic equation 2x² - 8x + 6 = 0 for all real roots.
Matematica Naval
Question · Easy
Factor it
Short
In a right-angled triangle, if the opposite side to angle θ is 6 meters and the adjacent side is 8 meters, what is the value of cos(θ)?
Two naval patrol vessels depart from point O. Vessel A travels vector u = (3, 4) nautical miles and Vessel B travels vector v = (4, -3) nautical miles. What is the dot product u · v?
What is the domain of the real-valued function f(x) = √(2x - 10)?
In an arithmetic progression (AP), the first term a₁ = 4 and the common difference d = 3. What is the 15th term (a₁₅)?
What is the derivative of the function f(x) = 3x⁴ - 5x² + 7x - 9 with respect to x?
Evaluate the trigonometric expression: sin²(35°) + cos²(35°) + tan(45°).
Solve the logarithmic equation log₂(x) + log₂(x - 2) = 3.
Question · Medium
What is the distance between point A(2, -3) and point B(6, 0) in a Cartesian plane representing a maritime chart grid?
Evaluate the limit: lim (x → 3) [ (x² - 9) / (x - 3) ].
What is the sum of the first 8 terms of the geometric progression: 3, 6, 12, 24, ...?
A naval radar installation tracks a target whose bearing angle θ satisfies sin(θ) = 1/2 in the first quadrant. What is the value of sin(2θ)?
What is the equation of the line passing through the point P(1, 4) with a slope m = -2?
Find the critical point (stationary point) of the trajectory function f(x) = x³ - 3x² + 5.
A naval communications code uses 4 distinct digits selected from the set {1, 2, 3, 4, 5, 6, 7}. How many different 4-digit codes can be formed without repeating any digit?
What is the center and radius of the circle defined by the equation x² + y² - 6x + 8y = 0?
If the polynomial P(x) = x³ + kx² - 4x + 12 is divisible by (x - 2), what is the value of k?
What is the period of the trigonometric function f(t) = 4·cos(3t + π/4)?
Find the determinant of the 2×2 matrix M = [[5, -2], [3, 4]].
Calculate the definite integral: ∫ (from 0 to 2) (3x² - 2x + 1) dx.
Question · Hard
What is the solution set for the inequality |2x - 3| < 7?
Two vectors A and B have magnitudes |A| = 6 and |B| = 8, and the angle between them is 60°. What is the magnitude of their cross product |A × B|?
What is the vertical asymptote of the rational function f(x) = (2x + 1) / (x - 4)?
Find the sum of the infinite geometric series: 12 + 6 + 3 + 1.5 + ...
If matrix A is a 2×2 matrix such that A · A⁻¹ = I, what is the inverse of A = [[3, 1], [5, 2]]?
What is the second derivative f''(x) of the function f(x) = sin(2x)?
Simplify the algebraic expression: (x³ - 8) / (x - 2).
In a maritime triangle, side a = 7, side b = 5, and the included angle C = 60°. What is the length of side c using the Law of Cosines?
What is the solution to the system of linear equations: 3x + 2y = 12 and x - y = 4?
What is the modulus and argument of the complex number z = 1 + i√3?
A naval vessel accelerates uniformly from rest to a speed of 20 m/s in 10 seconds. What is the total distance traveled during this period?
Fisica E Hidrostatica
According to Archimedes' principle, what buoyant force acts on a submerged submarine hull displacing 500 m³ of seawater with a density of 1,025 kg/m³? (Assume g = 9.8 m/s²)
What hydrostatic pressure is exerted by seawater (density ρ = 1,025 kg/m³) at a depth of 40 meters on a diver? (Use g = 9.8 m/s²)
A naval crane lifts an anchor weighing 4,000 N vertically by 15 meters in 20 seconds. What is the average power developed by the crane?
A coastal artillery projectile is fired horizontally from a cliff 45 meters above sea level with an initial velocity of 200 m/s. How long does it take to hit the water? (Use g = 10 m/s²)
What is the net force required to accelerate a 50,000 kg naval patrol boat at 0.4 m/s² through water, overcoming a hydrodynamic drag resistance of 10,000 N?
A ship's diesel engine absorbs 500 kJ of heat from the combustion chamber and discharges 320 kJ of heat to the cooling system and exhaust per cycle. What is its thermal efficiency?
A naval radar pulse operating at a frequency of 3.0 GHz (3.0 × 10⁹ Hz) travels at the speed of light c = 3.0 × 10⁸ m/s. What is its wavelength in meters?
In a shipboard DC electrical circuit, three resistors of 6 Ω, 3 Ω, and 2 Ω are connected in parallel. What is their equivalent resistance?
A hydraulic steering actuator on a frigate has an input piston of area 0.02 m² and an output ram of area 0.50 m². If a force of 400 N is applied to the input piston, what force is exerted by the output ram?
A torpedo of mass 1,200 kg is propelled from a submarine tube at 25 m/s. What is its kinetic energy upon exiting the tube?
According to the ideal gas law, if an air tank for naval clearance divers has its absolute temperature doubled while volume remains constant, what happens to its internal pressure?
A light ray traveling through air (n₁ = 1.00) enters seawater (n₂ = 1.33) at an angle of incidence of 30°. What is the sine of the angle of refraction sin(θ₂)?
What is the center of buoyancy (centro de carena) of a surface vessel?
For a surface ship to have positive initial transverse stability (estabilidade transversal inicial positiva), where must the transverse metacenter (M) be located relative to the center of gravity (G)?
An active sonar transducer transmits an acoustic pulse into seawater (speed of sound c = 1,500 m/s). If the echo from the seabed is received 0.8 seconds later, what is the water depth?
A gun turret rotates at a constant angular velocity of 0.5 rad/s. What is the linear tangential speed of an observation camera mounted 4 meters from the rotation axis?
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