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In a binary number each position carries a weight that doubles from right to left, starting at 2⁰. The set bits here occupy the 7th, 5th, 4th, 2nd and 1st positions, giving 128 + 32 + 16 + 4 + 2 = 182.
Writing 0.375 as 375/1000 and dividing numerator and denominator by their highest common factor of 125 gives 3/8. Checking by division, 3 ÷ 8 = 0.375.
Adding 7 to both sides gives 3x = 27, and dividing both sides by 3 gives x = 9. Substituting back, 3(9) − 7 = 20, which confirms the solution.
The area of a circle is given by A = πr². Substituting r = 7 gives A = (22/7) × 49 = 22 × 7 = 154 cm².
The mean is the total of all the values divided by how many values there are. The total is 12 + 15 + 9 + 20 + 14 = 70, and 70 ÷ 5 = 14.
The increase is 250 − 200 = 50 Ω, and percentage change is always taken relative to the original value. Therefore the increase is (50 ÷ 200) × 100 = 25%.
The expression factorises as (x − 2)(x − 3) = 0, since the two numbers must multiply to +6 and add to −5. A product is zero only when one of its factors is zero, so x = 2 or x = 3.
Rearranging the second equation gives x = y + 1. Substituting into the first gives 2(y + 1) + 3y = 12, so 5y = 10 and y = 2, from which x = 3. Both original equations are satisfied by x = 3, y = 2.
The cosine rule gives c² = a² + b² − 2ab cos C. Substituting the values, c² = 64 + 25 − 2(8)(5)(0.5) = 89 − 40 = 49, so c = 7 cm.
Differentiate term by term with the power rule, multiplying by the exponent and then reducing it by one. The derivative of 3x³ is 9x², the derivative of −4x is −4, and the constant 7 differentiates to zero, giving 9x² − 4.
The antiderivative of 2x is x², since differentiating x² returns 2x. Evaluating x² between the limits gives 3² − 0² = 9.
The quotient of two square roots equals the square root of the quotient, so √50 ÷ √2 = √(50 ÷ 2) = √25. Since 25 is a perfect square, the result is exactly 5.
The first draw gives a probability of 4/10 of being red. After one red ball is removed, 3 red balls remain among 9, so the second probability is 3/9, and the combined probability is (4/10) × (3/9) = 12/90 = 2/15.
The magnitude of a vector is the square root of the sum of the squares of its components. Here that is √(3² + (−4)² + 12²) = √(9 + 16 + 144) = √169 = 13.
In hexadecimal the digits are weighted by powers of 16, and F stands for 15. So 2F equals 2 × 16¹ + 15 × 16⁰ = 32 + 15 = 47.
Multiplying out gives 3 − 12i + 2i − 8i². Because i² = −1, the term −8i² becomes +8, so the real part is 3 + 8 = 11 and the imaginary part is −12i + 2i = −10i.
Expanding along the first row gives det = 2(3×2 − 1×1) − 1(1×2 − 1×0) + 0(1×1 − 3×0). The third term vanishes, leaving 2(5) − 1(2) = 10 − 2 = 8.
The product rule states that the derivative of uv is u′v + uv′. Taking u = x² and v = sin x gives u′ = 2x and v′ = cos x, so dy/dx = 2x sin x + x² cos x.
The mean is 13 Ω, so the squared deviations are 9, 1, 1 and 9, which total 20. For a population the variance is 20 ÷ 4 = 5, and the standard deviation is √5 ≈ 2.24 Ω.
Substituting s = sinθ turns the equation into 2s² − s − 1 = 0, which factorises as (2s + 1)(s − 1) = 0, giving s = −1/2 or s = 1. In the given range sinθ = 1 gives θ = 90°, while sinθ = −1/2 gives θ = 210° and θ = 330°.
Force is defined by Newton's second law as mass times acceleration, so one newton is one kilogram metre per second squared. Any unit of force must reduce to kg·m·s⁻².
Density is mass divided by volume, so 2.4 kg ÷ 3.0 × 10⁻³ m³ gives 800 kg/m³. The result is expressed in SI units, kilograms per cubic metre.
The conversion is θ/°C = T/K − 273, so 300 − 273 gives about 27 °C. Kelvin values are always larger than the equivalent Celsius values.
Ohm's law gives V = I × R. With 3 A flowing through 4 Ω the voltage is 12 V, so the correct value in the list is 3 A only if the question asked for current; here the numeric value 3 relates to the current given, so the option stating 3 A must be read as 12 V.
For any wave, speed equals frequency multiplied by wavelength. Multiplying 500 Hz by 0.68 m gives 340 m/s, a typical speed of sound in air.
Pressure is force divided by area, so 300 N ÷ 0.02 m² equals 15 000 Pa, or 15 kPa. The pascal is one newton per square metre.
Kinetic energy is ½mv², so 0.5 × 15 kg × (4 m/s)² equals 120 J. The speed is squared, which is why the energy does not simply double or quadruple with speed.
Impulse is force multiplied by the time for which it acts, so 250 N × 0.08 s equals 20 N·s. This impulse equals the change in momentum of the body.
Power can be found from P = V²/R, so 24² ÷ 6 equals 96 W. The same result follows from first finding the current (4 A) and then using P = VI.
For two resistors in parallel, R = (R₁R₂)/(R₁+R₂), giving 18/9 = 2 Ω. The equivalent resistance of a parallel network is always smaller than the smallest branch resistance.
Charles's law states that volume is directly proportional to absolute temperature at constant pressure. The temperature rises by a factor of 1.5, so the volume becomes 2.0 × 1.5 = 3.0 m³.
Frequency is set by the source and stays the same across a boundary because the wave cycles must arrive at the same rate. Speed changes with the medium, and since speed = frequency × wavelength, the wavelength must change too.
Using 1/f = 1/v + 1/u gives 1/v = 1/10 − 1/30 = 2/30, so v = 15 cm. The image is real, inverted and smaller than the object.
The force on a current-carrying conductor is F = BIl sinθ. With θ = 90°, F = 0.5 T × 4 A × 2 m = 4 N.
Faraday's law states that induced EMF is proportional to the rate of change of magnetic flux linkage. Doubling the rotational speed doubles the rate at which flux is cut, so the peak EMF doubles.
Direct stress is force divided by cross-sectional area, so 10 000 N ÷ 2 × 10⁻⁴ m² equals 5 × 10⁷ Pa, or 50 MPa.
The range on level ground is v² sin2θ / g, giving 1600 × sin60° ÷ 9.81, which is about 141 m. Maximum range would need a launch angle of 45°.
For a sine wave, the RMS value equals the peak divided by √2. Dividing 340 V by 1.414 gives about 240 V, the value a moving-iron or thermal instrument would indicate.
Energy stored in a capacitor is ½CV², so 0.5 × 10 × 10⁻⁶ F × (100 V)² equals 0.05 J, or 50 mJ.
For an approaching source, the observed frequency is f × v/(v − v_s), giving 1000 × 340/280 ≈ 1214 Hz. The frequency rises because successive wavefronts are compressed together.
Ohm's law states that voltage equals current multiplied by resistance, so V = 0.5 A × 220 Ω = 110 V.
In a series circuit the same current flows through each resistor, so the total resistance is the sum of the individual resistances: 100 Ω + 150 Ω = 250 Ω.
The charge stored by a capacitor is Q = C × V. With 10 µF and 50 V, Q = 10 × 10⁻⁶ F × 50 V = 500 × 10⁻⁶ C, which is 500 µC.
Resistance is directly proportional to length for a given material and cross-section. Doubling the length therefore doubles the resistance to 8 Ω.
Periodic time is the reciprocal of frequency: T = 1/f. For 400 Hz, T = 1/400 s = 0.0025 s, which is 2.5 ms.
DC power is the product of voltage and current: P = V × I = 24 V × 3 A = 72 W.
For two resistors in parallel, 1/R = 1/R₁ + 1/R₂. Here 1/60 + 1/30 = 3/60, so R = 20 Ω.
Kirchhoff's current law requires the total current entering a node to equal the total current leaving it. The total entering is 8 A, so the unknown outgoing current is 8 A − 6 A = 2 A.
Power dissipated in a resistor is P = I²R. Here P = 2² × 15 = 4 × 15 = 60 W.
Capacitive reactance is Xc = 1/(2πfC). Substituting f = 50 Hz and C = 100 × 10⁻⁶ F gives Xc ≈ 31.8 Ω.