GATE PI cheat sheet

Merchant orthogonal cutting: shear angle phi satisfies tan(phi) = (rcos(alpha))/(1 - rsin(alpha)), where r = t1/t2 = chip thickness ratio and alpha is rake angle; r = (chip thickness before cut)/(chip thickness after cut).

Merchant shear-angle relation: 2phi + beta - alpha = 90 degrees when minimising cutting force (beta = friction angle); friction coefficient mu = tan(beta) = (F_csin(alpha) + F_tcos(alpha))/(F_ccos(alpha) - F_t*sin(alpha)) using cutting force F_c and thrust force F_t.

Specific cutting energy u = F_c/(b*t1) with b = width of cut and t1 = uncut chip thickness; Taylor tool life VT^n = C, with n ~ 0.1-0.15 for HSS and ~0.2-0.4 for carbide.

Metal cutting temperatures rise with speed; the shear-plane model gives shear strain gamma = cos(alpha)/[sin(phi)cos(phi - alpha)], and shear stress on the shear plane tau = (F_s)/(bt1/sin(phi)).

Turning: material removal rate = piDNfd (D diameter in m, N rev/s, f feed per rev, d depth of cut); cutting speed V = piDN. Milling MRR = wdf_zzN with z teeth.

Casting: Chvorinov's rule solidification time t = B*(V/A)^2, where V/A is the modulus; riser must have a larger modulus than the casting it feeds. Shrinkage allowance for steel ~ 2% of linear dimension.

Welding: heat input per unit length H = (etaVI)/v, where eta is arc efficiency (~0.85 for SAW, ~0.7-0.8 for SMAW); resistance spot welding heat = I^2Rt.

Metal forming: true strain epsilon = ln(L/L0) = ln(A0/A); flow stress for a strain-hardening metal sigma = Kepsilon^n (Hollomon). For rolling, maximum draft = mu^2R (R = roll radius).

Machining time in shaping/planing: time = L/(fN) per stroke with L = length + approach; in drilling, machining time = (depth + 0.5D*tan(90 - half-point angle))/feed rate.

Forecasting: simple moving average of n periods = sum of last n demands / n; exponential smoothing F(t+1) = alpha*D(t) + (1 - alpha)*F(t), alpha in [0,1].

EOQ = sqrt(2DS/(H)), where D = annual demand, S = ordering cost per order, H = holding cost per unit per year. Total cost at EOQ = sqrt(2DS*H); reorder point = demand rate * lead time.

Inventory with shortages and quantity discounts: compare total annual cost (purchase + ordering + holding) at EOQ and at each price break; the feasible EOQ uses the holding cost of its own price band.

Line balancing: cycle time = available time per period / required output; theoretical minimum number of workstations = ceil(sum of task times / cycle time); balance delay = (nC - sum t)/ (nC).

PERT: expected time t_e = (a + 4m + b)/6, variance = ((b - a)/6)^2; project variance = sum of variances on the critical path; CPM uses deterministic durations and identifies the longest path with zero slack.

Work study: standard time = observed time * performance rating * (1 + allowance fraction); normal time = observed time * rating. Allowances cover personal, fatigue, and delay needs.

Queuing (M/M/1): utilisation rho = lambda/mu; average number in system L = rho/(1 - rho); average waiting time in queue W_q = lambda/[mu*(mu - lambda)].

Quality control: control limits for an x-bar chart = grand mean +/- A2R_bar (A2 depends on subgroup size); for an R chart, limits = D3R_bar and D4R_bar. Process capability Cp = (USL - LSL)/(6sigma); Cpk = min[(USL - mean), (mean - LSL)]/(3*sigma).

Project crashing: crash cost per unit time = (crash cost - normal cost)/(normal time - crash time); crash the cheapest activity on the critical path first, and recheck the critical path after each step.

Percentage and successive changes: a single equivalent change of +a% then +b% is a + b + ab/100 percent; a discount of d% on marked price gives selling price = MP*(1 - d/100).

Profit and loss: profit% = (SP - CP)/CP * 100; if two articles are sold at the same price with equal gain and loss percentages, there is always an overall loss of (x/10)^2 percent.

Time-speed-distance: distance = speed * time; for equal distances at speeds u and v, average speed = 2uv/(u + v). Trains crossing: time = (sum of lengths)/(relative speed).

Work and time: if A finishes a job in a days, A's one-day work = 1/a; combined rate of A and B is 1/a + 1/b, so together they take ab/(a + b) days.

Permutations and combinations: nPr = n!/(n - r)!; nCr = n!/[r!(n - r)!]; number of arrangements of a word with repeated letters = n!/(p!q!...).

Probability: P(A or B) = P(A) + P(B) - P(A and B); for independent events P(A and B) = P(A)*P(B); Bayes' rule P(A|B) = P(B|A)*P(A)/P(B).

Data interpretation: weighted average = sum(w_ix_i)/sum(w_i); for grouped data, mean = sum(f_ix_i)/sum(f_i), and median class is where cumulative frequency reaches N/2.

Series and progressions: arithmetic sum = (n/2)(2a + (n-1)d); geometric sum = a(r^n - 1)/(r - 1) for r not equal to 1; recognise differences and ratios to identify the pattern.

Verbal and numerical reasoning: in analogy and odd-one-out items, test the relationship (part-to-whole, cause-effect, synonym-antonym) before choosing; in sentence completion, use grammatical agreement and logical contrast markers (although, whereas).