NUST NET cheat sheet

Stress = Force / Area (Pa); strain = ΔL / L (dimensionless). Young's modulus E = stress/strain only in the linear-elastic region.

For a simply supported beam with central point load W and span L: max bending moment = WL/4 at midspan; max deflection = WL³/(48EI).

Bernoulli along a streamline: P + ½ρv² + ρgh = constant for steady, incompressible, frictionless flow.

Carnot efficiency of a heat engine = 1 − T_cold/T_hot; temperatures must be in kelvin.

Ohm's law V = IR; resistors in series add (R₁+R₂), in parallel add as reciprocals (1/R = 1/R₁ + 1/R₂).

Projectile: horizontal range R = v²sin2θ/g, maximum at θ = 45°; time of flight = 2v sinθ/g.

Moment of inertia of a solid cylinder about its axis = ½MR²; of a thin hoop = MR².

Binary-to-decimal: each bit position i (from right, starting 0) contributes bit × 2^i; e.g. 1011₂ = 8+0+2+1 = 11.

Two's complement: negate a value by inverting all bits and adding 1; an n-bit signed range is −2^(n−1) to 2^(n−1)−1.

Stack is LIFO, queue is FIFO; push/pop are O(1), enqueue/dequeue are O(1) with a proper implementation.

Big-O of common algorithms: binary search O(log n), merge sort O(n log n), bubble sort O(n²), hash lookup average O(1).

SQL SELECT execution order: FROM → WHERE → GROUP BY → HAVING → SELECT → ORDER BY; WHERE filters rows before grouping.

OSI model layers (bottom up): Physical, Data Link, Network, Transport, Session, Presentation, Application.

In relational design, a primary key is unique and non-null; a foreign key must match an existing primary key value or be null.

Forward kinematics maps joint angles to end-effector pose via chained homogeneous transforms; inverse kinematics is generally multi-solution and nonlinear.

Degrees of freedom equal the number of independent joint variables; a 6-DOF arm can generally reach an arbitrary position and orientation in 3D.

PID control output = Kp·e(t) + Ki·∫e dt + Kd·de/dt; integral term removes steady-state error, derivative term damps overshoot.

Perceptron update rule: w ← w + η(y − ŷ)x, where η is the learning rate and (y − ŷ) is the error.

Backpropagation computes gradients via the chain rule; gradient descent step is θ ← θ − η∇L(θ).

Convolution output size = (W − F + 2P)/S + 1, with W input width, F filter size, P padding, S stride.

A* search uses f(n) = g(n) + h(n); it is optimal when h is admissible (never overestimates true cost).

Ideal gas law PV = nRT; at STP (273 K, 1 atm) one mole occupies about 22.4 L.

pH = −log₁₀[H⁺]; a change of one pH unit means a tenfold change in hydrogen-ion concentration.

First law of thermodynamics: ΔU = Q − W, with W the work done by the system.

Beer–Lambert law A = εcl; absorbance is proportional to concentration and path length.

Half-life relation: after n half-lives, remaining fraction = (1/2)^n; radioactive decay N = N₀e^(−λt).

Arrhenius equation k = A·e^(−Ea/RT); reaction rate roughly doubles for a 10 °C rise near room temperature.

Significant figures: multiplication/division keeps the least number of sig figs; addition/subtraction keeps the least decimal places.

Water activity (a_w) is the ratio of vapor pressure of water in food to that of pure water; most bacteria need a_w above about 0.91.

D-value is the time at a given temperature to reduce a microbial population by 90% (one log); z-value is the temperature change giving a tenfold change in D.

F₁₂₁ or 12-D concept: 12 decimal reductions of Clostridium botulinum spores at 121 °C for commercial sterility of low-acid canned foods.

Blanching inactivates enzymes (peroxidase, lipoxygenase) before freezing or drying to prevent off-flavors and browning.

Maillard browning is a non-enzymatic reaction between reducing sugars and amino groups; it is favored by heat and intermediate moisture.

Freezing concentrates solutes and lowers water activity; most microbial growth stops but enzymes and some cells survive.

HACCP has seven principles, starting with hazard analysis and ending with verification and record-keeping.

Scale 1:100 means 1 unit on the drawing equals 100 units in reality; a 5 cm drawing line represents 5 m.

Golden ratio φ ≈ 1.618; a rectangle with sides a and b is golden when a/b = (a+b)/a.

A standard door leaf is about 0.9 m wide by 2.1 m high; a typical storey height is 3.0–3.6 m.

Ergonomic desk height is roughly 72–75 cm; seat height about 42–45 cm for average adult users.

Load-bearing masonry resists compression but not tension, so openings need lintels or arches to span them.

Design brief components: problem statement, user, constraints, criteria, and deliverables; form follows function is a stated decision rule, not a slogan.

In technical drawing, first-angle projection places the top view below the front view; third-angle places it above.

Cell theory: all living things are made of cells, the cell is the basic unit of life, and cells arise from existing cells.

DNA base pairing: adenine pairs with thymine (two hydrogen bonds), guanine with cytosine (three).

Photosynthesis: 6CO₂ + 6H₂O + light → C₆H₁₂O₆ + 6O₂, occurring in chloroplasts.

Newton's second law F = ma; weight W = mg with g ≈ 9.8 m/s² near Earth's surface.

Speed of light in vacuum c ≈ 3 × 10⁸ m/s; wavelength × frequency = c.

Mendelian monohybrid cross Aa × Aa gives a 3:1 phenotype and 1:2:1 genotype ratio.

Molarity = moles of solute / liters of solution; dilution obeys M₁V₁ = M₂V₂.

Break-even quantity = fixed costs / (price − variable cost per unit); contribution margin per unit is price minus variable cost.

Gross profit = revenue − cost of goods sold; net profit further subtracts operating expenses, interest, and tax.

Current ratio = current assets / current liabilities; a ratio below 1 signals possible short-term liquidity strain.

Demand is elastic when |%Δquantity| / |%Δprice| > 1; total revenue falls when price rises under elastic demand.

GDP = C + I + G + (X − M); it counts final goods within a country's borders in a period.

Inflation erodes purchasing power; real interest rate ≈ nominal rate − inflation rate (Fisher approximation).

Maslow's hierarchy orders needs from physiological to safety, belonging, esteem, and self-actualization.

A valid contract needs offer, acceptance, consideration, intention to create legal relations, and capacity; absence of any element can defeat it.

Consideration must be sufficient but need not be adequate; past consideration is generally not good consideration.

An invitation to treat (e.g., displaying goods in a shop) is not an offer; the customer makes the offer.

Mens rea is the guilty mind; actus reus is the prohibited act; both are generally required for a criminal offence.

Burden of proof in criminal cases lies with the prosecution and must meet beyond reasonable doubt; in civil cases the standard is balance of probabilities.

Stare decisis: lower courts are bound by ratio decidendi of higher courts in the same hierarchy; obiter dicta are persuasive only.

A tort is a civil wrong giving a remedy in damages; negligence requires duty of care, breach, causation, and damage.