A ship travels 12 km due North and then 5 km due East. What is the magnitude of the ship's net displacement from its starting point?
Physics
Question · Easy
A force of 15 N is applied to a mass of 3 kg. What is the acceleration produced in the mass, assuming a frictionless surface?
An engine does 6000 J of work in 2 minutes. What is the average power developed by the engine?
If the distance between two point charges is doubled, what happens to the electrostatic force between them?
Which of the following thermal processes occurs at a constant volume?
A convex lens has a focal length of 20 cm. What is its power in diopters?
What is the maximum wavelength of electromagnetic radiation required to eject electrons from a metal surface having a work function of 2.0 eV? (Take h = 6.63 x 10⁻³⁴ J·s, c = 3.0 x 10⁸ m/s, and 1 eV = 1.6 x 10⁻¹⁹ J)
A projectile is launched from the ground at an angle of 30° with an initial velocity of 40 m/s. What is the horizontal range of the projectile? (Take g = 10 m/s²)
Question · Medium
A body of mass 2 kg is rotated in a horizontal circle of radius 0.5 m at a constant speed. If the tension in the string is 16 N, what is the angular velocity of the body?
A bullet of mass 20 g moving with a speed of 150 m/s penetrates a sandbag and comes to rest in 0.03 seconds. What is the average retarding force exerted by the sand on the bullet?
A block of mass 4 kg slides down an inclined plane of length 10 m and height 5 m. If the work done against friction is 40 J, what is the kinetic energy of the block at the bottom of the incline? (Take g = 10 m/s²)
A solid sphere of mass 2 kg and radius 0.2 m is rolling without slipping on a flat surface with a velocity of 5 m/s. What is its total kinetic energy?
At what height above the Earth's surface does the acceleration due to gravity become 36% of its value at the surface? (Take radius of Earth R = 6400 km)
A steel wire of length 2 m and cross-sectional area 1 mm² is stretched by a force of 100 N. If Young's modulus of steel is 2 x 10¹¹ N/m², what is the elongation produced in the wire?
A Carnot engine operates between two reservoirs at temperatures of 500 K and 300 K. What is the efficiency of the engine?
Three capacitors of capacitances 3 μF, 6 μF, and 9 μF are connected in series. What is the equivalent capacitance of the combination?
A proton enters a uniform magnetic field of 0.5 T with a velocity of 4 x 10⁶ m/s perpendicular to the field lines. What is the magnetic force acting on the proton? (Take charge of a proton = 1.6 x 10⁻¹⁹ C)
A square coil of side 10 cm consisting of 100 turns is placed perpendicular to a magnetic field. If the magnetic field changes from 0.1 T to 0.5 T in 0.2 seconds, what is the magnitude of the induced EMF in the coil?
In a Young's double-slit experiment, the slit separation is 0.2 mm and the screen is placed 1.2 m away. If the wavelength of the light used is 500 nm, what is the fringe width?
A radioactive sample has a half-life of 4 days. If the initial activity is 64 μCi, what will be its activity after 16 days?
A particle moves along a straight line such that its displacement x at time t is given by x = t³ - 6t² + 9t + 4 (where x is in meters and t is in seconds). What is the acceleration of the particle at the instant when its velocity is zero?
Question · Hard
A block of mass 5 kg is placed on a rough horizontal surface with coefficients of static and kinetic friction as 0.4 and 0.3 respectively. If a horizontal force of 15 N is applied to the block, what is the force of friction acting on the block? (Take g = 10 m/s²)
An infinite line charge produces an electric field of 9 x 10⁴ N/C at a distance of 2 cm. What is the linear charge density of the line charge? (Take 1 / (4πε₀) = 9 x 10⁹ N·m²/C²)
In a series LCR circuit, R = 30 Ω, L = 0.4 H, and C = 100 μF. The circuit is connected to an AC source of 220 V, 50 Hz. What is the impedance of the circuit?
A pipe open at both ends has a fundamental frequency of 300 Hz. If one end of the pipe is closed, what will be the frequency of its third harmonic?
If a matrix A is of order 3x4 and matrix B is of order 4x2, what is the order of the product matrix AB?
Mathematics
Find the distance between the parallel lines 3x + 4y - 9 = 0 and 3x + 4y + 6 = 0.
What is the principal value of sin⁻¹(-1/2)?
Two cards are drawn at random from a pack of 52 cards without replacement. What is the probability that both are aces?
What is the value of the limit as x approaches 0 of sin(5x) / x?
Find the projection of vector a = 2i + 3j + 2k on vector b = i + 2j + k.
If A is a square matrix of order 3 such that |A| = 4, what is the value of |2A|?
Find the derivative of y = x^(sin x) with respect to x.
Evaluate the definite integral of x * e^x dx from 0 to 1.
If z = (1 + i) / (1 - i), what is the value of z¹⁰⁰?
Find the equation of the circle which passes through the origin and cuts off intercepts of 4 and 6 on the positive x and y axes respectively.
In a class, 60% of students pass in Mathematics, 50% pass in Physics, and 30% pass in both. If a student is selected at random, what is the probability that they pass in Mathematics given that they have passed in Physics?
Find the area of a parallelogram whose diagonals are represented by the vectors d₁ = 3i + j - 2k and d₂ = i - 3j + 4k.
If the roots of the quadratic equation x² - px + q = 0 differ by unity, which of the following relations is correct?
What is the eccentricity of the ellipse 9x² + 25y² = 225?
Find the general solution of the differential equation dy/dx = (1 + y²) / (1 + x²).
If tan A = 1/2 and tan B = 1/3, what is the value of A + B?
Find the angle between the two planes 2x - y + z = 6 and x + y + 2z = 3.
What is the number of terms in the expansion of (x + y + z)¹⁰?
Find the maximum value of the function f(x) = x * (1 - x) on the interval [0, 1].
Evaluate the integral of (3x + 1) / (x² - x - 6) dx.
An urn contains 5 red and 7 black balls. A ball is drawn, its color is noted, and it is returned to the urn along with 3 additional balls of the same color. If a second ball is now drawn, what is the probability that it is red?
If a, b, and c are three unit vectors such that a + b + c = 0, what is the value of the scalar product a · b + b · c + c · a?
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