NUST NET (Engineering) key facts

Maths · Numbers, sets and functions

Complex numbers

• i² = −1, i³ = −i, i⁴ = 1; for iⁿ, divide n by 4 and use the remainder as the new power
• Any four consecutive powers of i add up to 0
• Conjugate of z = a + bi is z̄ = a − bi; z + z̄ = 2a, z − z̄ = 2bi, z × z̄ = a² + b²
• To divide, multiply top and bottom by the conjugate of the bottom
• Modulus: |z| = √(a² + b²)
• |z₁z₂| = |z₁| × |z₂| and |z₁/z₂| = |z₁| ÷ |z₂|, so a modulus can be found without simplifying first
• Argument θ: tan θ = b/a, with the quadrant taken from the signs of a and b; principal value −π < θ ≤ π

Sets and functions

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• A set with n elements has 2ⁿ subsets, of which 2ⁿ − 1 are proper subsets
• De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′
• Domain: denominator ≠ 0; expression under a square root ≥ 0 (strictly > 0 if the root is in a denominator)
• (f∘g)(x) = f(g(x)): work out g first, then put the result into f
• Inverse: write y = f(x), swap x and y, solve for y; domain of f⁻¹ = range of f
• Even: f(−x) = f(x). Odd: f(−x) = −f(x), so an odd function defined at 0 has f(0) = 0

Maths · Matrices and determinants

Matrices and determinants

• (m × n)(n × p) gives an m × p matrix; the inner numbers must match
• For A with rows [a b] and [c d]: |A| = ad − bc, and A⁻¹ = (1/|A|) × the matrix with rows [d −b] and [−c a]
• (AB)⁻¹ = B⁻¹A⁻¹ and (AB)^t = B^tA^t (the order reverses)
• |A^t| = |A|, |AB| = |A| × |B|, |A⁻¹| = 1/|A|, and |kA| = kⁿ|A| for an n × n matrix
• A determinant is 0 if a row is all zeros or two rows (or columns) are equal or proportional; swapping two rows changes its sign
• AX = B: if |A| ≠ 0 there is one solution, X = A⁻¹B; if |A| = 0 there is either no solution (inconsistent) or infinitely many
• Homogeneous system AX = O: only x = y = z = 0 if |A| ≠ 0; non-trivial solutions if |A| = 0

Maths · Quadratic equations

Quadratic equations

• x = (−b ± √(b² − 4ac))/(2a)
• b² − 4ac > 0: real and distinct roots; = 0: real and equal; < 0: complex conjugate roots
• With rational a, b, c: the roots are rational when b² − 4ac is a perfect square, irrational when it is positive but not a perfect square
• Sum of roots α + β = −b/a; product αβ = c/a
• Equation with given roots: x² − (sum of roots)x + (product of roots) = 0
• α² + β² = (α + β)² − 2αβ; (α − β)² = (α + β)² − 4αβ; α³ + β³ = (α + β)³ − 3αβ(α + β)
• Cube roots of unity: 1 + ω + ω² = 0 and ω³ = 1, so reduce any power of ω using its remainder on division by 3

Maths · Sequences, series and the binomial theorem

Sequences and series

• Arithmetic: T_n = a + (n − 1)d; S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l), where l is the last term
• Geometric: T_n = ar^{n − 1}; S_n = a(1 − rⁿ)/(1 − r), r ≠ 1
• Sum to infinity: S_∞ = a/(1 − r), only for |r| < 1
• From a sum formula: T_n = S_n − S_{n − 1}
• Between a and b: arithmetic mean A = (a + b)/2; geometric mean G = ±√(ab); harmonic mean H = 2ab/(a + b)
• G² = A × H; for two different positive numbers A > G > H, taking G as the positive root
• In a geometric sequence, T_m ÷ T_n = r^{m − n}; in an arithmetic sequence, T_m − T_n = (m − n)d

Binomial theorem and partial fractions

• General term of (a + b)ⁿ: T_{r + 1} = ⁿC_r a^{n − r} b^r
• Middle term: T_{(n/2) + 1} if n is even; two middle terms T_{(n + 1)/2} and T_{(n + 3)/2} if n is odd
• Sum of the coefficients of an expansion: put x = 1; sum of the binomial coefficients of (1 + x)ⁿ is 2ⁿ
• Series: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + ..., valid for |x| < 1; for small x, (1 + x)ⁿ ≈ 1 + nx
• (a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ, valid for |bx/a| < 1, that is |x| < |a/b|
• Partial fraction forms: (x − a) gives A/(x − a); (x − a)² gives A/(x − a) + B/(x − a)²; a quadratic that will not factorise gives (Ax + B)/(that quadratic)
• If the degree of the top is not lower than the bottom, divide first; for a non-repeated factor (x − a), find A by covering that factor and putting x = a in the rest

Maths · Permutations, combinations and probability

Permutations and combinations

• n! = n × (n − 1) × ... × 2 × 1, and 0! = 1
• ⁿP_r = n!/(n − r)! (order matters)
• ⁿC_r = n!/(r! × (n − r)!) (order does not matter); ⁿC_r = ⁿC_{n − r}
• ⁿP_r = r! × ⁿC_r
• n objects with p alike of one kind and q alike of another: n!/(p! × q!) arrangements
• n different objects round a circle: (n − 1)! arrangements
• Together: treat the group as one object, then multiply by the arrangements inside it. Apart: arrange the others first, then fill the gaps

Probability

• P(A) = number of favourable outcomes ÷ total number of outcomes; 0 ≤ P(A) ≤ 1
• P(A′) = 1 − P(A)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0
• Independent events: P(A ∩ B) = P(A) × P(B)
• Without replacement: P(A then B) = P(A) × P(B after A has happened)
• P(at least one) = 1 − P(none); P(neither A nor B) = 1 − P(A ∪ B)
• Sample sizes: two dice 36; n coins 2ⁿ; a pack has 52 cards, 26 red, 13 in each suit, 12 face cards

Maths · Trigonometry

Trigonometric ratios and identities

• π radians = 180°: multiply degrees by π/180 to get radians, and radians by 180/π to get degrees
• Arc length l = rθ; area of sector A = ½r²θ (θ in radians)
• sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
• sin(α ± β) = sin α cos β ± cos α sin β; cos(α ± β) = cos α cos β ∓ sin α sin β
• tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β)
• sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2 tan θ/(1 − tan²θ)
• Half angle: sin²(θ/2) = (1 − cos θ)/2; cos²(θ/2) = (1 + cos θ)/2; tan(θ/2) = sin θ/(1 + cos θ)

Trigonometric graphs, triangles and equations

• Period: sin, cos, sec, cosec 2π; tan, cot π. For sin kx or cos kx the period is 2π/|k|; for tan kx it is π/|k|
• Range: sin x and cos x lie in [−1, 1], so a + b sin x lies in [a − |b|, a + |b|]; tan x takes all real values
• Sine rule: a/sin α = b/sin β = c/sin γ = 2R (R = circum-radius)
• Cosine rule: a² = b² + c² − 2bc cos α, or cos α = (b² + c² − a²)/(2bc)
• Area Δ = ½bc sin α = √(s(s − a)(s − b)(s − c)) with s = (a + b + c)/2; in-radius r = Δ/s; circum-radius R = abc/(4Δ)
• Principal ranges: sin⁻¹x in [−π/2, π/2]; cos⁻¹x in [0, π]; tan⁻¹x in (−π/2, π/2)
• General solutions (n ∈ Z): sin x = sin α gives x = nπ + (−1)ⁿα; cos x = cos α gives x = 2nπ ± α; tan x = tan α gives x = nπ + α

Maths · Functions and limits

Limits and continuity

• lim_x→0 (sin x)/x = 1 and lim_x→0 (tan x)/x = 1, with x in radians; so lim_x→0 (sin ax)/(bx) = a/b
• lim_x→0 (1 − cos ax)/x² = a²/2
• lim_n→∞ (1 + 1/n)ⁿ = e, and in general lim_n→∞ (1 + a/n)^bn = e^ab
• lim_x→0 (a^x − 1)/x = ln a, so lim_x→0 (e^x − 1)/x = 1
• lim_x→a (xⁿ − aⁿ)/(x − a) = n aⁿ⁻¹
• As x→∞ in a fraction of polynomials, divide by the highest power of x; equal degrees give the ratio of the leading coefficients
• f is continuous at x = a when the left-hand limit = the right-hand limit = f(a)

Maths · Differentiation

Rules of differentiation

• Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v². Chain: dy/dx = (dy/du) × (du/dx)
• Derivatives: sin x → cos x, cos x → −sin x, tan x → sec² x, cot x → −cosec² x, sec x → sec x tan x, cosec x → −cosec x cot x
• e^ax → a e^ax; a^x → a^x ln a; ln x → 1/x; ln f(x) → f′(x)/f(x)
• sin⁻¹ x → 1/√(1 − x²); cos⁻¹ x → −1/√(1 − x²); tan⁻¹ x → 1/(1 + x²)
• Implicit f(x, y) = 0: dy/dx = −(derivative of f treating y as constant) ÷ (derivative of f treating x as constant)
• Parametric: dy/dx = (dy/dt) ÷ (dx/dt), and d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)
• n^th derivative of e^ax is aⁿ e^ax; of sin ax it is aⁿ sin(ax + nπ/2)

Applications of derivatives

• Tangent at (a, b): y − b = m(x − a) with m = f′(a); the normal has gradient −1/m
• f′(x) > 0: increasing. f′(x) < 0: decreasing. f′(x) = 0: stationary point
• At a stationary point: f″ < 0 gives a maximum, f″ > 0 gives a minimum; if f″ = 0, check the sign change of f′
• Related rates: dy/dt = (dy/dx) × (dx/dt), for example dV/dt = 4πr² × (dr/dt) for a sphere
• Small increments: δy ≈ f′(x) × δx, so f(x + δx) ≈ f(x) + f′(x) δx
• Motion: velocity v = ds/dt, acceleration a = dv/dt = d²s/dt²
• Two positive numbers with a fixed sum have the greatest product when they are equal; with a fixed product they have the least sum when they are equal

Maths · Integration

Methods of integration

• ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c for n ≠ −1; ∫ (1/x) dx = ln|x| + c
• Linear inside: ∫ f(ax + b) dx = (1/a) F(ax + b) + c, e.g. ∫ cos(ax) dx = (1/a) sin(ax) + c
• ∫ f′(x)/f(x) dx = ln|f(x)| + c; ∫ [f(x)]ⁿ f′(x) dx = [f(x)]ⁿ⁺¹/(n + 1) + c
• ∫ dx/(a² + x²) = (1/a) tan⁻¹(x/a) + c; ∫ dx/√(a² − x²) = sin⁻¹(x/a) + c
• By parts: ∫ u dv = uv − ∫ v du; choose u in the order logarithm, inverse trig, algebraic, trig, exponential
• ∫ e^x[f(x) + f′(x)] dx = e^x f(x) + c
• ∫ dx/(x² − a²) = (1/(2a)) ln|(x − a)/(x + a)| + c; ∫ tan x dx = ln|sec x| + c; ∫ cot x dx = ln|sin x| + c

Definite integrals and areas

• ∫_a^b f(x) dx = F(b) − F(a), where F′(x) = f(x)
• ∫_b^a f dx = −∫_a^b f dx, and ∫_a^b f dx = ∫_a^c f dx + ∫_c^b f dx
• ∫_−a^a f dx = 0 if f is odd, and = 2 ∫₀^a f dx if f is even
• ∫₀^a f(x) dx = ∫₀^a f(a − x) dx; hence ∫₀^π/2 sin² x dx = ∫₀^π/2 cos² x dx = π/4
• Area between two curves = ∫ (upper − lower) dx between the points where they meet
• Shortcut: the area between a parabola y = ax² + bx + c and a straight line (or the x-axis) that meet at x = α and x = β is |a|(β − α)³/6
• Separable equation: write g(y) dy = f(x) dx and integrate both sides with one constant; dy/dx = ky gives y = Ae^kx

Maths · Analytic geometry

Straight lines

• Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Point dividing AB internally in m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); for external division replace n by −n
• Gradient m = (y₂ − y₁)/(x₂ − x₁) = tan θ; the line ax + by + c = 0 has gradient −a/b
• Forms of a line: y − y₁ = m(x − x₁); y = mx + c; x/a + y/b = 1 (intercepts a and b)
• Parallel: m₁ = m₂. Perpendicular: m₁ × m₂ = −1
• Acute angle between two lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
• Distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²); between parallel lines with the same a and b it is |c₁ − c₂|/√(a² + b²)

Circles and conic sections

• Circle (x − h)² + (y − k)² = r²: centre (h, k), radius r
• Circle x² + y² + 2gx + 2fy + c = 0: centre (−g, −f), radius √(g² + f² − c); make the coefficients of x² and y² equal to 1 first
• Tangent to x² + y² = r² at (x₁, y₁): xx₁ + yy₁ = r²; the line y = mx + c touches this circle when c² = r²(1 + m²)
• Parabola y² = 4ax: vertex (0, 0), focus (a, 0), directrix x = −a. Parabola x² = 4ay: focus (0, a), directrix y = −a
• Ellipse x²/a² + y²/b² = 1 with a > b: c² = a² − b², e = c/a, foci (±c, 0), directrices x = ±a²/c
• If the larger number is under y², the major axis is along the y-axis and the foci are (0, ±c)
• Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a, foci (±c, 0), directrices x = ±a²/c

Linear inequalities and linear programming

• Boundary line: solid for ≤ or ≥ (points on the line included), broken for < or > (points on the line excluded)
• Test (0, 0) in the inequality: if it is true, the solution is the side containing the origin; if false, the other side. Use another point if the line passes through the origin
• Feasible region = the part of the plane common to all the constraints
• Corner points are where two boundary lines meet; solve the pair of equations, then check the point satisfies every other constraint
• The maximum and minimum of z = ax + by over a closed and bounded feasible region occur at corner points: evaluate z at each corner and compare
• If two neighbouring corners give the same best value, every point on the edge joining them is also optimal

Maths · Vectors

Vectors in space

• |a| = √(a₁² + a₂² + a₃²); unit vector = a/|a|
• Direction cosines: l = a₁/|a|, m = a₂/|a|, n = a₃/|a|, and l² + m² + n² = 1
• a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ; the vectors are perpendicular when a · b = 0
• Projection of a on b = (a · b)/|b|; work done = F · d
• a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k; |a × b| = |a||b| sin θ; a × b = −b × a
• Area of a parallelogram = |a × b|; area of a triangle = (1/2)|a × b|
• a · (b × c) = the 3 × 3 determinant of the components; volume of a parallelepiped = its size; the vectors are coplanar when it is 0

Physics · Measurement, vectors and equilibrium

Measurement

• Prefixes: giga 10⁹, mega 10⁶, kilo 10³, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹²
• Force [MLT⁻²]; work, energy and torque [ML²T⁻²]; power [ML²T⁻³]; pressure [ML⁻¹T⁻²]
• Momentum and impulse [MLT⁻¹]; angular momentum and Planck's constant [ML²T⁻¹]
• Pure numbers, angles, sin θ, logs and constants such as 2π have no dimensions
• Adding or subtracting quantities: add the absolute uncertainties
• Multiplying or dividing quantities: add the percentage uncertainties; for a power xⁿ, multiply the percentage uncertainty in x by n
• Significant figures: zeros at the front do not count (0.0320 has three); zeros between digits, and zeros at the end after a decimal point, do count

Vectors and equilibrium

• Components: A_x = A cos θ, A_y = A sin θ (θ measured from the x-axis); A = √(A_x² + A_y²), tan θ = A_y/A_x
• Resultant of A and B with angle θ between them: R = √(A² + B² + 2AB cos θ); largest A + B, smallest |A − B| (the difference of the magnitudes)
• Scalar product: A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z; i · i = 1, i · j = 0
• Vector product: |A × B| = AB sin θ = area of the parallelogram on A and B (the triangle is half of it)
• i × j = k, j × k = i, k × i = j; reversing the order changes the sign: A × B = −(B × A)
• Torque τ = r × F, size rF sin θ = force × perpendicular distance from the pivot (N m)
• Equilibrium: sum of forces = 0 and sum of torques = 0 (clockwise moments = anticlockwise moments)

Physics · Mechanics

Motion and force

• v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t
• From rest with constant acceleration, distance ∝ t²; stopping distance = u²/(2a)
• F = ma; impulse = FΔt = change in momentum (N s = kg m s⁻¹)
• Collisions: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂; if the bodies stick, v = m₁u₁/(m₁ + m₂) for a target at rest
• Elastic head-on, target at rest: v₁ = (m₁ − m₂)u/(m₁ + m₂), v₂ = 2m₁u/(m₁ + m₂)
• Two masses over a smooth pulley: a = (m₁ − m₂)g/(m₁ + m₂)
• Projectile: time of flight T = 2u sin θ/g; greatest height H = u² sin²θ/(2g); range R = u² sin 2θ/g, greatest at 45°

Work, energy and power

• work = force × distance × cos θ (W = Fd cos θ), in joules
• Variable force: work = area under the force-displacement graph
• kinetic energy = ½mv² = p²/(2m), so momentum p = √(2m × KE)
• gravitational potential energy = mgh; energy stored in a spring = ½kx²
• Falling from rest through height h with no friction: v = √(2gh), whatever the mass or the path
• power = work ÷ time = force × velocity (P = Fv), in watts; 1 kW h = 3.6 × 10⁶ J
• Net work done = change in kinetic energy; work against friction = friction force × distance

Circular and rotational motion

• ω = 2π/T = 2πf; v = rω; 1 revolution = 2π rad
• Centripetal acceleration a = v²/r = rω²; centripetal force F = mv²/r = mrω²
• String in a vertical circle: tension = mv²/r + mg at the bottom, mv²/r − mg at the top
• Moment of inertia: hoop MR²; disc ½MR²; solid sphere (2/5)MR²; thin rod about its centre (1/12)ML²; torque τ = Iα
• Angular momentum L = Iω = mvr (kg m² s⁻¹ = J s); with no external torque I₁ω₁ = I₂ω₂
• Rotational KE = ½Iω²; rolling body: total KE = ½mv² + ½Iω² (hoop mv², disc ¾mv², solid sphere (7/10)mv²)
• Orbit of radius r: v = √(GM/r), T² ∝ r³; close to the Earth v = √(gR) ≈ 7.9 km s⁻¹ and T = 2π√(R/g) ≈ 84 min

Fluid dynamics

• Coefficient of viscosity η: unit N s m⁻² = Pa s = kg m⁻¹ s⁻¹; dimensions [ML⁻¹T⁻¹]
• Stokes' law: drag F = 6πηrv, for a small sphere moving slowly in streamline flow
• Terminal velocity v_t = 2gr²ρ/(9η) (ρ = density of the sphere, upthrust ignored), so v_t ∝ r²
• Continuity: A₁v₁ = A₂v₂; volume flow rate = Av (m³ s⁻¹)
• Bernoulli: P + ½ρv² + ρgh = constant along a streamline
• Horizontal flow: P₁ − P₂ = ½ρ(v₂² − v₁²)
• Torricelli: speed of water leaving a hole a depth h below the surface, v = √(2gh)

Physics · Oscillations, waves and light

Oscillations

• a = −ω²x; ω = 2πf = 2π/T
• Mass on a spring: T = 2π√(m/k); for a load that stretches the spring by e, T = 2π√(e/g)
• Simple pendulum: T = 2π√(l/g), independent of the mass of the bob; a seconds pendulum has T = 2 s
• Starting from the mean position: x = x₀ sin ωt; v = ω√(x₀² − x²)
• Maximum speed = ωx₀ (at the centre); maximum acceleration = ω²x₀ (at the ends)
• Total energy = ½kx₀² = ½mω²x₀²; PE = ½kx²; KE = ½k(x₀² − x²)
• KE = PE when x = x₀/√2; resonance when driving frequency = natural frequency

Waves and sound

• v = fλ; speed of a wave on a string v = √(T/μ), T = tension, μ = mass per unit length
• String fixed at both ends, or pipe open at both ends: f_n = nv/(2L), n = 1, 2, 3 … (all harmonics)
• Pipe closed at one end: f_n = nv/(4L), n = 1, 3, 5 … (odd harmonics only)
• Distance from node to next node = λ/2; from node to nearest antinode = λ/4
• Speed of sound in a gas ∝ √(absolute temperature); in air about 332 m s⁻¹ at 0 °C, rising about 0.61 m s⁻¹ per °C
• Beat frequency = f₁ − f₂ (the difference); time between beats = 1 ÷ beat frequency
• Doppler: f′ = f(v ± v_o)/(v ∓ v_s); upper signs when moving towards each other (v_o listener, v_s source)

Physical optics and optical instruments

• Young's double slit: fringe spacing = λL/d (d = slit separation, L = distance to screen)
• Bright fringe m at y = mλL/d (path difference mλ); dark fringe at y = (m + ½)λL/d
• Diffraction grating: d sin θ = nλ, with d = 1/N (N = lines per metre); highest order n ≤ d/λ
• Unpolarised light through an ideal polariser: intensity falls to half; two crossed polarisers pass nothing
• Simple magnifier: M = 1 + d/f (d = 25 cm); compound microscope: M = M_objective × M_eyepiece
• Astronomical telescope in normal adjustment: M = f_o/f_e, length = f_o + f_e
• Smallest angle resolved by a lens of diameter D: α_min = 1.22λ/D (radians); grating resolving power λ/Δλ = N × m (here N = total number of lines lit, m = order)

Physics · Heat and thermodynamics

Kinetic theory and thermodynamics

• PV = nRT = NkT; R = 8.31 J mol⁻¹ K⁻¹, k = 1.38 × 10⁻²³ J K⁻¹
• Pressure P = (1/3)ρ⟨v²⟩; mean translational KE per molecule = (3/2)kT
• v_rms = √(3RT/M) = √(3kT/m): proportional to √T and to 1/√(molar mass)
• First law: Q = ΔU + W, where W = PΔV is the work done by the gas at constant pressure
• Constant volume: W = 0, Q = ΔU. Isothermal: ΔU = 0, Q = W. Adiabatic: Q = 0, W = −ΔU
• C_p − C_v = R; γ = C_p/C_v; monatomic C_v = 3R/2, γ = 5/3; diatomic C_v = 5R/2, γ = 7/5
• Efficiency = W/Q₁ = 1 − Q₂/Q₁; Carnot engine: 1 − T₂/T₁ (T₁ source, T₂ sink, in kelvin)

Physics · Electricity

Electrostatics

• Coulomb's law: force = k × q₁ × q₂ ÷ r² (F = kq₁q₂/r²), with k = 9 × 10⁹ N m² C⁻²
• Electric field = force ÷ charge (E = F/q); for a point charge E = kq/r², unit N C⁻¹ or V m⁻¹
• Potential of a point charge: V = kq/r, in volts; keep the sign of q
• Uniform field between parallel plates: E = V/d
• Capacitance = charge ÷ potential difference (C = Q/V); parallel plates: C = ε₀ε_rA/d
• Parallel: C = C₁ + C₂. Series: 1/C = 1/C₁ + 1/C₂, so for two capacitors C = C₁C₂/(C₁ + C₂)
• Energy stored = ½CV² = ½QV = Q²/(2C)

Current electricity

• Resistance = resistivity × length ÷ area (R = ρL/A); resistivity ρ is in Ω m
• Temperature: R_t = R₀(1 + αt), so α = (R_t − R₀)/(R₀ × t), unit K⁻¹
• Series: R = R₁ + R₂. Parallel: 1/R = 1/R₁ + 1/R₂, so for two resistors R = R₁R₂/(R₁ + R₂)
• Cell with internal resistance r: I = E/(R + r) and terminal p.d. V = E − Ir
• Power = VI = I²R = V²/R; a cell gives maximum power to the load when R = r, and then P = E²/(4r)
• Kirchhoff: current into a junction = current out; round any closed loop, sum of EMFs = sum of IR drops
• Wheatstone bridge at balance: P/Q = R/S. Potentiometer: EMF is proportional to balance length, E₁/E₂ = L₁/L₂

Physics · Magnetism and alternating current

Electromagnetism

• Force on a wire = BIL sin θ, θ between the wire and the field; so 1 T = 1 N A⁻¹ m⁻¹
• Force on a moving charge = qvB sin θ, θ between the velocity and the field
• Charge moving at right angles to the field: radius r = mv/(qB); period T = 2πm/(qB), which does not depend on speed
• Field inside a long solenoid: B = μ₀nI, where n = turns per metre and μ₀ = 4π × 10⁻⁷ T m A⁻¹
• Torque on a coil = NIAB cos α, α between the plane of the coil and the field (maximum NIAB when the plane is parallel to the field)
• Ammeter: shunt in parallel, R_s = I_gR_g/(I − I_g)
• Voltmeter: high resistance in series, R_h = (V/I_g) − R_g

Electromagnetic induction

• Flux = BA cos θ, θ between the field and the normal to the area; unit weber (Wb)
• Faraday's law: EMF = −N × (change in flux) ÷ (time taken) (ε = −N ΔΦ/Δt)
• Motional EMF of a rod moving at right angles to the field: ε = BLv
• Self-induction: ε = −L ΔI/Δt. Mutual induction: ε_s = −M ΔI_p/Δt. L and M are in henry (H)
• Energy stored in an inductor = ½LI²
• AC generator: ε = NABω sin ωt, peak value NABω, with ω = 2πf
• Ideal transformer: V_s/V_p = N_s/N_p = I_p/I_s, so V_pI_p = V_sI_s

Alternating current

• rms value = peak value ÷ √2 ≈ 0.707 × peak (V_rms = V₀/√2, I_rms = I₀/√2)
• Inductive reactance: X_L = ωL = 2πfL, in Ω; it is proportional to f
• Capacitive reactance: X_C = 1/(ωC) = 1/(2πfC), in Ω; it is proportional to 1/f
• Series RLC impedance: Z = √(R² + (X_L − X_C)²), and I_rms = V_rms/Z
• Phase angle: tan φ = (X_L − X_C)/R; average power = V_rms × I_rms × cos φ
• Resonance: X_L = X_C, so ω₀ = 1/√(LC) and f₀ = 1/(2π√(LC))
• At series resonance Z = R (its smallest value), the current is a maximum and is in phase with the voltage

Physics · Solids and electronics

Physics of solids

• Stress = force ÷ area (σ = F/A), in Pa or N m⁻²; strain = extension ÷ original length (ε = ΔL/L), no unit
• Young's modulus = stress ÷ strain (Y = FL/(A × ΔL)), in Pa
• Extension ΔL = FL/(AY): proportional to length and to 1/diameter²
• Elastic energy stored = ½ × F × ΔL; energy per unit volume = ½ × stress × strain
• Crystalline: ordered, sharp melting point (metals, NaCl). Amorphous: no regular order, softens gradually (glass). Polymeric: long chains (plastics, rubber)
• Diamagnetic: weakly repelled. Paramagnetic: weakly attracted. Ferromagnetic: strongly attracted, becomes paramagnetic above the Curie temperature
• Soft magnetic material (soft iron): narrow hysteresis loop, small energy loss, used for transformer cores. Hard (steel): wide loop, used for permanent magnets

Electronics

• n-type: pentavalent impurity (P, As, Sb), majority carriers are electrons. p-type: trivalent impurity (B, Al, Ga, In), majority carriers are holes
• Forward bias: p side to the positive terminal, low resistance. Reverse bias: p side to the negative terminal, very high resistance
• Potential barrier: about 0.7 V for silicon and 0.3 V for germanium
• Half-wave rectifier: one diode, output frequency f. Full-wave bridge: four diodes, ripple frequency 2f
• Transistor: I_E = I_B + I_C; current gain β = I_C/I_B
• Op-amp gain: inverting = −R_f/R₁; non-inverting = 1 + (R_f/R₁)
• Gates: AND gives 1 only if all inputs are 1; OR gives 1 if any input is 1; NOT inverts; NAND and NOR are AND and OR followed by NOT; XOR gives 1 when the inputs differ

Physics · Modern and nuclear physics

Dawn of modern physics

• Relativity: γ = 1/√(1 − v²/c²); time t = γt₀, length L = L₀/γ, mass m = γm₀, energy E = mc²
• Photon energy = hf = hc/λ; a quick form is E (eV) = 1240 ÷ λ (nm)
• Photoelectric equation: hf = Φ + KE_max; KE_max = eV₀ (V₀ = stopping potential); threshold frequency f₀ = Φ/h
• Compton shift: Δλ = (h/(m₀c)) × (1 − cos θ), where h/(m₀c) ≈ 2.4 pm for an electron
• de Broglie wavelength: λ = h/p = h/(mv); for a charge accelerated from rest through V, λ = h/√(2mqV)
• Pair production needs a photon of at least 2m₀c² = 1.02 MeV
• Uncertainty principle: Δx × Δp is at least of the order of h, so a smaller Δx means a larger Δp

Atomic spectra

• Bohr's condition: angular momentum mvr = nh/(2π)
• Hydrogen: radius r_n = n² × 0.053 nm; energy E_n = −13.6/n² eV
• Photon energy = E_upper − E_lower = hf = hc/λ
• Wavelength of a line: 1/λ = R_H × (1/p² − 1/n²) for a fall from level n to level p, R_H = 1.097 × 10⁷ m⁻¹
• Series: Lyman (to n = 1) ultraviolet; Balmer (to n = 2) visible; Paschen (3), Brackett (4) and Pfund (5) infrared
• X-ray tube at voltage V: shortest wavelength λ_min = hc/(eV), that is λ_min (nm) = 1240 ÷ V (volts)
• Laser: population inversion in a metastable state, then stimulated emission gives photons of the same frequency, phase and direction

Nuclear physics

• In any nuclear equation the total A and the total Z are the same on both sides
• α decay: A falls by 4, Z falls by 2. β⁻ decay: A unchanged, Z rises by 1. γ emission: A and Z unchanged
• Mass defect: Δm = Z × m_p + (A − Z) × m_n − m_nucleus
• Binding energy = Δm × c²; with Δm in u, binding energy = Δm × 931 MeV
• Half-life: fraction left after n half-lives = (½)ⁿ, with n = t/T_½; decay constant λ = 0.693/T_½; activity = λN
• Ionising power: α > β > γ. Penetrating power: γ > β > α
• Cloud chamber tracks: α thick, straight and short; β thin and winding; γ faint and scattered

English · Vocabulary

Synonyms and antonyms

• Synonym = closest in meaning; antonym = most nearly opposite. Check which one is asked before you answer.
• Define the word in your own simple words first, then match the options to your definition.
• Prefixes: bene = good, mal = bad, in / im / un / dis = not, pre = before, post = after, re = again.
• Roots: chron = time, cred = believe, loqu = speak, luc / lum = light, ver = true, path = feeling.
• Decide the charge of the word (positive, negative or neutral). A synonym has the same charge; an antonym has the opposite charge.
• The answer is the same part of speech as the given word: adjective for adjective, verb for verb.
• Reject options that only belong to the same topic as the word but do not mean it.

Words in context

• Treat the word as a blank: predict your own word from the sentence before reading the options.
• The most familiar dictionary meaning is the usual trap. Test it; do not trust it.
• A colon, semicolon or 'since / because' clause usually explains the word.
• 'But', 'although', 'yet' and 'however' point to a meaning opposite to the other half of the sentence.
• Substitute each remaining option into the sentence; only one keeps the meaning.
• The answer must fit the grammar of the sentence as well as the sense.

English · Sentence completion

Sentence completion

• Contrast signals (blank opposes the clue): although, though, but, yet, however, despite, far from, rather than, surprisingly.
• Same-direction signals (blank agrees with the clue): and, because, since, so, therefore, indeed, moreover.
• A colon or semicolon usually introduces an explanation or example of the blank.
• 'So ______ that …' means the result after 'that' defines the blank.
• Predict your own word first; decide at least whether the blank is positive or negative.
• Two blanks: work out whether the two words agree or oppose each other, then eliminate one blank at a time.
• Read the finished sentence once in full. Choose by the logic of the sentence, not by the most impressive word.

English · Grammar and usage

Identifying errors

• The verb agrees with its real subject. Ignore phrases in between ('of …', 'together with …', 'as well as …'). 'The number of … is'; 'a number of … are'.
• After 'There is / are' and in inverted sentences, the subject comes after the verb: 'There are many reasons'.
• Tense follows the time words ('last year' needs the past). Past unreal condition: If + had + past participle, … would have + past participle.
• Pronoun case: I, he, she, we, they, who for subjects; me, him, her, us, them, whom for objects and after prepositions ('between you and me'). A pronoun also matches its noun in number.
• Parallel structure: items in a list, and both halves of 'not only … but also', 'both … and', 'either … or', take the same grammatical form.
• Comparison: comparative (-er / more) for two things, superlative (-est / most) for three or more; compare like with like ('than that of Lahore', 'than any other city').
• Fixed idioms: prefer X to Y; superior / inferior / senior / junior to; different from. Adverbs, not adjectives, describe verbs ('runs quietly').

Improving sentences

• A complete sentence has a subject and a finite verb and can stand alone. A clause beginning 'although', 'because' or 'when' cannot stand alone.
• Two complete sentences cannot be joined by a comma alone or by nothing. Use a full stop, a semicolon, a comma + and / but / so, or make one part a dependent clause.
• 'However' and 'therefore' are not joining words: write '; however,' or start a new sentence.
• An opening phrase with -ing or -ed must describe the subject that comes straight after the comma.
• A colon introduces a list or an explanation after a complete statement; a semicolon joins two complete sentences.
• its = belonging to it; it's = it is. One owner: factory's. Several owners: factories'.
• Cut repeated ideas: 'the reason is because', 'return back', 'and so therefore', 'collaborate together'.

English · Reading comprehension

Short passages

• Main idea: it must cover the whole passage. Reject an option about one detail (too narrow) or one that goes beyond the passage (too broad).
• Purpose: match the verb to what the author does: explain, describe, argue, warn, compare, or correct a common belief.
• After 'but', 'yet', 'however' or 'still' comes the author's own view; the first and last sentences often state the main point.
• Inference: a small, safe step that must be true if the passage is true. If you cannot point to the supporting line, it is a guess.
• Be wary of options with 'always', 'never', 'all', 'only' or 'proves' unless the passage is just as strong.
• Tone: read the judging words (adjectives, adverbs, verbs of praise or blame). Common answers: admiring, critical, cautious, concerned, objective.
• Word in the passage: substitute each option into the sentence and keep the one that preserves the meaning there.