11.1.1 Section A ordering logic

Section A runs Quant first for 22 minutes, Reasoning next for 18 minutes, English last for 10 minutes, with a 10 minute buffer held in reserve. Quant leads because its items are the most procedure dense, and procedures from the early chapters run fastest while working memory is fresh. Reasoning follows because elimination discipline needs sustained focus, and focus is still strong in the second block. English closes the section because each item is a single short read with low setup cost, so it fits the phase when mental speed starts to fall. The buffer is never assigned in advance. It exists only to absorb overruns and to fund returns to marked items. A buffer that is spent by plan is not a buffer.

11.1.2 Section B two-pass sweep logic

Section B runs as two passes over the 50 domain items. Pass one is a 60 minute formula sweep. Every item solvable in under a minute by direct substitution is solved at once. Every item that is solvable but lengthy is star marked and left for later. Every item with negative or near zero expected value for the solver is crossed and left for the day. Pass two is a 30 minute lengthy round spent only on star marked items, easiest star first. The logic is coverage before depth. A single slow march in serial order lets a small number of long items consume the minutes that would have paid for many short items. Two passes invert that loss by collecting all low cost marks first and then spending remaining time only where a known method already exists.

11.1.3 Time-box summary

The full allocation reads as one line. Section A is 22 plus 18 plus 10 plus 10 reserve. Section B is 60 sweep plus 30 star round. First contact with any item is capped at 90 seconds. The cap forces the three way decision of solve now, star for later, or cross for the day. Items still unsolved after pass two are left alone. Coverage is the scoring strategy, persistence on a crossed item is the scoring leak.

11.2.1 Derivation from the marking scheme

Each item pays plus one for a correct choice and minus zero point two five for a wrong choice. Let p denote the solver probability of selecting the correct choice on an attempt. The expected increment from attempting is the probability weighted sum of the two outcomes, stated as EV equals p times one minus the quantity one minus p times zero point two five. Algebra gives EV equals one point two five times p minus zero point two five. The break even point follows by setting EV to zero, which gives p equals zero point two. In pure arithmetic every attempt with p above zero point two has positive expected value. Time cost then tightens the practical rule, as developed below.

11.2.2 Threshold table

The table below evaluates the derived expression at the four operational values of p.

Solver state	p	EV computation	EV	Decision under time cost
Method known, computation direct	near one	near one minus near zero	near plus one	solve at once
Reduced to two options	zero point five	zero point five minus zero point one two five	plus zero point three seven five	attempt, the gain is large
Reduced to three options, one eliminated	one third	one third minus zero point two five times two thirds	plus zero point one six seven	attempt on first contact, still clearly positive
No elimination among four	zero point two five	zero point two five minus zero point one eight seven five	plus zero point zero six two five	leave for later, time earns more on a star marked item

The slogan form of the table is compact. Attempt when the method is known. Attempt when the item is reduced to two options. Attempt when one option is eliminated on first contact. Leave the no clue item while star marked items remain pending.

QUICK EXAMPLE — Fifty-fifty attempt value

Given $p = 0.5$, plus one for a correct choice and minus zero point two five for a wrong choice.

Step one writes $EV = p(1) - (1 - p)(0.25)$.

Step two substitutes $EV = 0.5 - 0.5(0.25) = 0.5 - 0.125$.

Result is $EV = +0.375$.

Insight is that a two-option reduction carries large positive value, so the attempt proceeds on first contact.

11.2.3 Time-cost reasoning

The plus zero point zero six two five at p equals zero point two five is arithmetically positive and practically weak. A no clue attempt still costs sixty to ninety seconds of reading and weighing. The same minute spent on a star marked item with a known method carries expected value near plus one. Rational allocation therefore sends minutes to the highest expected value per minute, not merely to any positive expected value. The no clue item is rational only when no star marked item remains and residual seconds cannot start a fresh solution. Even there, flow discipline favors leaving it alone, because breaking accounting rhythm near the end creates slips on items already banked. Time is part of value, and the threshold table is read together with the clock.

QUICK EXAMPLE — Single elimination attempt value

Given $p = 1/3$, with the same plus one and minus zero point two five marking.

Step one writes $EV = 1.25p - 0.25$.

Step two substitutes $EV = 1.25(1/3) - 0.25 = 0.4167 - 0.25$.

Result is $EV = +0.167$.

Insight is that one elimination still leaves clearly positive value, so the guess verdict is to attempt on first contact while no-clue items wait.

The diagram plots EV against p as a straight line from minus zero point two five at p equals zero to plus one at p equals one. The horizontal line marks EV equals zero. The break even dash stands at p equals zero point two. The leave zone covers the low p band. The middle band covers reduced option attempts. The right band covers direct solutions.

11.3.1 The three marks and their meaning

Every item receives exactly one mark at first reading. A tick means solving now, because the method is known and the path is under a minute. A star means solvable but lengthy, so the item is deferred to the designated second round. A cross means left for the day, because expected value for this solver is negative or too small to justify the minutes. The star must carry a three word method note in the margin, naming the tool that will restart the solution warm. Examples of such notes are torque formula, Venn check, pump energy split, Otto relation, and wear versus pressure. A cross carries no return. Re reading a crossed trap is the standard path by which minus zero point two five entries accumulate.

11.3.2 Return logic and finality

The buffer at the end of each section is spent only on stars, ordered from easiest star to hardest star. No item gets more than 90 seconds on first contact, so the first pass never sinks. Both decisions are final in the sense that matters for scoring. A star is a promise of a known method with a warm start, and a cross is a refusal to fund a low value attempt. Finality protects attention. Without it, attention leaks back to crossed items while starred marks wait unfunded.

11.4.1 Four cause classes

A lost mark belongs to one of four classes. A concept loss means the governing rule was unknown or misremembered. An arithmetic loss means the rule was known and the computation slipped, through unit conversion, term count, radius versus diameter, or fraction handling. A trap loss means the formula was known but an applicability condition was violated, for instance Bernoulli applied across a pump, uniform wear used where the item names uniform pressure, or isolation assumed below the root two threshold. A time loss means the method was correct and the minutes ran out, so the item was rushed, truncated, or never reached.

11.4.2 Why each class needs different treatment

Each class has a distinct repair because each class has a distinct defect. Concept losses need rule restudy from the compendium, with the condition stated aloud alongside the formula. Arithmetic losses need slower writing and unit first substitution, not more theory, because the defect is execution under speed. Trap losses need condition drills that pair each formula with its boundary cases, because the defect is recognition of when the tool fails. Time losses need protocol repair, through stricter first contact caps and earlier starring, because the defect is allocation rather than subject matter. Treating all losses as concept losses wastes study hours on theory when the fault lies in writing speed, condition checks, or clock control.

11.4.3 Diary method and review logic

The diary method records each lost mark in a four column row. Column one holds date and item tag. Column two holds the fault plus the correct rule in one line. Column three holds the cause class from the four classes above. Column four holds a single repair action tied to that class. Review logic follows the class. Before the next timed paper, only concept and trap rows are re solved, because those rows carry reusable rules. Arithmetic rows are repaired by rewriting the computation line by line at reduced speed. Time rows are repaired by rehearsing the starring decision on the same item type. A diary with ten honest rows outperforms further volume, because it converts each paid loss into exactly one targeted repair.

Copy this section by hand group by group. Cover each group, write it from memory, then check against the source. Every entry below pairs the formula with the condition that keeps it valid. A formula without its condition is a future trap row.

11.5.1 Aptitude procedures

Simple interest equals principal times rate times time divided by one hundred, valid for linear growth with no compounding. Compound amount equals principal times one plus rate divided by one hundred raised to n, valid for annual compounding at fixed rate. Two year difference of compound minus simple equals principal times rate divided by one hundred squared, valid only for n equals two. Successive changes of a percent then b percent give net equals a plus b plus a times b divided by one hundred, valid because the second change acts on the grown base. Equal distance average speed equals two times x times y divided by x plus y, valid for equal legs, never the plain mean. Work allocation fixes total work as the least common multiple of the given times, valid for constant rates. Product of two numbers equals highest common factor times least common multiple, valid for two numbers only. Arithmetic progression sum equals n divided by two times first plus last, with term count n equals last minus first divided by d plus one, valid for constant difference. At least one equals one minus none, valid for complementary counting. Dice pair outcomes total thirty six, valid for two fair dice. Train crossing time uses distance equals train length plus bridge or tunnel length, valid when the full length must clear. Downstream speed equals still water speed plus stream speed and upstream speed equals still water speed minus stream speed, valid for constant current. Sphere volume scales with the cube of radius, so doubling gives eight times and tripling gives twenty seven times. Quadratic sum of roots equals minus b divided by a, valid for a non zero. Net fill rate with a leak equals fill rate minus leak rate, valid for constant rates. Ratio shares divide the total in proportion to the stated ratio, valid for a fixed total.

11.5.2 Engineering mathematics

Two by two eigenvalues equal trace plus or minus square root of trace squared minus four times determinant, all divided by two, valid for a two by two matrix. Steady state of dy by dx plus a times y equals b is y equals b divided by a, valid for constant a non zero with the transient decayed. Laplace of e raised to minus a times t equals one divided by s plus a, valid for s greater than minus a. Simpson one third rule is exact up to cubic polynomials, valid for smooth integrands with fine uniform spacing. The integral from zero to one of x squared dx equals one third, used as a calibration value for numerical checks.

11.5.3 Mechanics and strength of materials

Belt contact angle must enter friction relations in radians, with one hundred eighty degrees equal to pi. Zero force joint rule one states that two non collinear members with no external load carry zero force, valid only with no load at the joint. Rule two states that with three members, two collinear and no load, the non collinear member carries zero force. Self locking needs mu greater than or equal to tan of alpha, valid for the wedge or screw contact considered. Projectile range equals u squared times sin of two theta divided by g, valid for level ground with no air drag. Drop velocity from height h equals square root of two times g times h, valid for free fall from rest. Axial elongation equals P times L divided by A times E, valid for linear elastic prismatic bars. Solid shaft shear equals sixteen times T divided by pi times d cubed, valid for solid circular sections with diameter d, not radius. Thin cylinder hoop stress equals p times d divided by two times t and longitudinal stress equals p times d divided by four times t, so hoop is double longitudinal, valid only for thin shells with t less than d divided by ten. Euler effective lengths are L for pinned pinned, two times L for fixed free, L divided by two for fixed fixed, and L divided by root two for fixed pinned, valid for ideal Euler columns. Uniaxial absolute maximum shear equals sigma divided by two, valid for one non zero principal stress. Ductile design uses von Mises and brittle design uses Rankine, valid as stated material pairing. Cantilever tip deflection under end load P equals P times L cubed divided by three times E times I, valid for linear elastic small deflection. Simply supported mid load deflection equals P times L cubed divided by forty eight times E times I, under the same linear conditions.

QUICK EXAMPLE — Hoop stress safety call

Given $p = 2$ MPa, $d = 500$ mm, $t = 10$ mm, allowable stress $100$ MPa, thin shell check $t < d/10$ holds since $10 < 50$.

Step one writes hoop stress $\sigma_h = pd/(2t)$.

Step two substitutes $\sigma_h = 2 \times 500/(2 \times 10) = 1000/20 = 50$ MPa, with longitudinal stress $25$ MPa, half the hoop value.

Result is safety factor $100/50 = 2.0$.

Insight is that hoop stress governs the call, and the section stays safe with margin two under the stated allowable.

11.5.4 Theory of machines, vibration, and design

Planar degree of freedom equals three times n minus one minus two times j, valid for planar links with lower pairs only. Bearing L one zero life in million revolutions equals C divided by P raised to p, with p equals three for ball bearings and ten divided by three for roller bearings, valid for the stated contact type. Flat clutch under uniform wear transmits T equals mu times W times capital R plus small r divided by two, valid for worn in service where pressure varies inversely with radius. Under uniform pressure the torque equals two thirds times mu times W times capital R cubed minus small r cubed divided by capital R squared minus small r squared, valid for new clutches with uniform pressure. The item wording decides, so the named theory must be matched exactly. Isolation needs frequency ratio r greater than root two, valid for the linear isolator model, and below that value transmitted force exceeds input. Grashof condition for at least one full rotation is s plus l less than or equal to p plus q, valid for four bar chains with s shortest and l longest. Log decrement delta equals one divided by n times natural log of x one divided by x n plus one, valid for viscous underdamped free decay. External gear train value is negative and equals minus driven teeth divided by driver teeth, valid for external mesh with direction reversal. Lewis bending stress equals tangential load divided by face width times module times form factor Y, valid for the stated bending model with Y set mainly by tooth count.

11.5.5 Fluid mechanics and machinery

Bernoulli needs steady, incompressible, inviscid flow along a streamline with no energy addition or extraction, so it never applies across a pump or turbine. Darcy friction head loss varies inversely with diameter to the fifth power at fixed discharge, so halving diameter multiplies loss by thirty two. Torricelli exit velocity equals square root of two times g times h, valid for ideal discharge under head h. Pump affinity at fixed impeller gives discharge proportional to N, head proportional to N squared, and power proportional to N cubed, valid for the same impeller and similar conditions. Orifice discharge coefficient equals contraction coefficient times velocity coefficient, near zero point six one in the standard case. Turbine choice follows head plus discharge. High head with low discharge calls for Pelton, medium head with medium discharge calls for Francis, low head with large discharge calls for Kaplan or propeller. Suction side throttling worsens cavitation, so the correct valve action is to throttle the delivery side. Net positive suction head margin rises by lifting sump level or cooling the liquid, valid because both raise available suction head or suppress vapour. Froude scaling gives velocity proportional to square root of length ratio, valid for free surface gravity dominated flows. Static head is read by a piezometer and total head needs a stagnation tap, with each tap matched to its head type.

11.5.6 Thermodynamics, heat transfer, and power

Otto air standard efficiency equals one minus one divided by r raised to gamma minus one, valid for the air standard Otto cycle at fixed gamma. At the same compression ratio Otto exceeds Diesel, and added cut off lowers Diesel efficiency further. At the same peak pressure and heat rejection Diesel exceeds Otto. Refrigerator coefficient of performance equals T sub L divided by T sub H minus T sub L, valid for the reversed Carnot bound between those reservoirs. One radiation shield of equal emissivity between parallel plates halves transfer, and two such shields cut it to one third, valid only for equal emissivity. Clearance volume increase lowers volumetric efficiency, valid for reciprocating compressors with fixed pressure ratio. Subcooling raises refrigerating effect by c sub p times delta T, valid before the expansion valve with no extra work. Spark ignition knock is resisted by high octane fuel, lower compression, and retarded spark, valid for premixed flame fronts. Compression ignition knock is resisted by high cetane fuel with short delay, valid for auto ignition timing. Critical insulation radius equals k divided by h, valid for cylindrical bodies, and adding insulation below this radius increases loss until the radius is reached.

11.5.7 Manufacturing and industrial engineering

Eutectoid steel at zero point seven seven percent carbon and seven hundred twenty seven degrees Celsius forms pearlite, valid at the eutectoid point. Eutectic cast iron at four point three percent carbon and eleven hundred forty seven degrees Celsius forms ledeburite, valid at the eutectic point. Chvorinov solidification time varies with volume divided by area squared, valid for the same mould conditions. Taylor tool life follows V times T raised to n equals C, valid for the fitted speed range. Economic order quantity equals square root of two times D times ordering cost divided by carrying cost, valid for constant deterministic demand with no stockouts. Program evaluation review expected time equals optimistic plus four times most likely plus pessimistic, all divided by six, with standard deviation equals pessimistic minus optimistic divided by six, valid for the beta distribution model. Tungsten inert gas welding uses a non consumable tungsten electrode, while metal inert gas welding uses a continuously fed consumable wire under gas shield. Surface cracks call for dye penetrant or magnetic particle inspection, while internal flaw depth and size call for ultrasonic inspection, with each method matched to defect location. ABC analysis ranks items by annual consumption value, with Class A under tight frequent control, valid for value based control policy.

QUICK EXAMPLE — EOQ bracket check

Given $D = 900$ units per year, ordering cost $S = 100$ per order, carrying cost $H = 50$ per unit per year, valid for constant deterministic demand with no stockouts.

Step one writes $EOQ = \sqrt{2DS/H}$.

Step two substitutes $EOQ = \sqrt{2 \times 900 \times 100/50} = \sqrt{3600} = 60$ units.

Result is reinforced by total cost $3000$ at $Q = 60$ versus $3250$ at both $Q = 40$ and $Q = 90$.

Insight is that the square root minimum sits inside the checked bracket, so $60$ units is the cost minimum order size.

The method closes the book in five lines. Section A follows Quant 22 plus Reasoning 18 plus English 10 plus a 10 minute reserve, because procedure density, focus demand, and read cost fall in that order. Section B follows a 60 minute substitution sweep plus a 30 minute star round, because coverage pays before depth. Every uncertain item is decided by EV equals one point two five times p minus zero point two five, read together with time cost, so direct solutions and reduced option attempts proceed while no clue items wait behind starred marks. The three marks stay final, with stars carrying a named method and crosses never revisited. Every lost mark enters a four column diary row under concept, arithmetic, trap, or time, and each class receives its own repair, with concept and trap rows re solved first. The compendium above is the hand copy source, and each formula travels only with the condition that keeps it valid.