BEL Probationary Engineer cheat sheet

Ohm's law: V = IR; power P = VI = I²R = V²/R. In AC, impedance Z = R + jX, with |Z| = √(R² + X²) and phase angle θ = arctan(X/R).

Thevenin's theorem: any linear two-terminal network can be replaced by V_th (open-circuit voltage) in series with R_th (resistance looking back with sources dead).

Boolean algebra: A + A'B = A + B; De Morgan's laws: (A·B)' = A' + B' and (A + B)' = A'·B'. A NAND gate is functionally complete.

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

Stress σ = F/A; strain ε = ΔL/L; Young's modulus E = σ/ε. Thermal stress in a fully constrained bar: σ = EαΔT (α = coefficient of thermal expansion).

In a BJT, β = I_C/I_B and α = I_C/I_E with α = β/(1+β). For a silicon diode, forward cut-in voltage ≈ 0.7 V; germanium ≈ 0.3 V.

Time complexity: binary search O(log n); merge sort O(n log n); quicksort average O(n log n), worst O(n²); hash table average O(1) lookup.

Soil bearing capacity: Terzaghi's equation q_u = cN_c + qN_q + 0.5γBN_γ for a strip footing; factor of safety against bearing failure typically 2.5–3.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x; product rule (uv)' = u'v + uv'; chain rule d/dx f(g(x)) = f'(g(x))·g'(x).

Integration by parts: ∫u dv = uv − ∫v du. Standard results: ∫eˣ dx = eˣ + C; ∫(1/x) dx = ln|x| + C; ∫sin x dx = −cos x + C.

Laplace transforms: L{1} = 1/s; L{e^{at}} = 1/(s−a); L{sin ωt} = ω/(s²+ω²); L{cos ωt} = s/(s²+ω²); L{tⁿ} = n!/s^{n+1}.

Matrix: determinant of 2×2 [[a,b],[c,d]] = ad − bc. A square matrix is invertible iff det(A) ≠ 0; eigenvalues solve det(A − λI) = 0.

Probability: P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B). Bayes: P(A|B) = P(B|A)P(A)/P(B).

Mean μ = Σx/n; variance σ² = Σ(x−μ)²/n; standard deviation σ = √σ². For binomial distribution mean = np, variance = np(1−p).

Newton's second law F = ma; work W = F·d cos θ; kinetic energy KE = ½mv²; potential energy PE = mgh. Hooke's law F = kx.

Ideal gas law PV = nRT; R = 8.314 J/(mol·K). First law of thermodynamics ΔU = Q − W (work done by the system).

Percentage change: new value = original × (1 ± r/100). Successive changes of a% and b% give net change a + b + ab/100 percent.

Simple interest SI = PRT/100; compound interest CI = P[(1 + R/100)^T − 1]. For two years, CI − SI = P(R/100)².

Speed–time–distance: distance = speed × time. Average speed for equal distances at speeds u and v is 2uv/(u+v), not (u+v)/2.

Ratio and proportion: if a:b = c:d then ad = bc. Partnership profit shares are proportional to capital × time invested.

Time and work: if A alone takes x days and B alone takes y days, together they take xy/(x+y) days. Work rate = 1/time.

Profit and loss: profit% = (SP − CP)/CP × 100. If an article is sold at a loss of x%, SP = CP(1 − x/100).

Permutations and combinations: nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]. Number of diagonals of an n-sided polygon = n(n−3)/2.

Mensuration: area of circle = πr²; circumference = 2πr; volume of cylinder = πr²h; surface area of sphere = 4πr²; volume of cone = ⅓πr²h.

In a number/letter series, check differences between consecutive terms first; if differences are constant it is arithmetic; if ratios are constant it is geometric.

Coding–decoding: map each letter to its position (A=1 … Z=26) and test shifts (+1, −1, +2, −2) or reversals (A↔Z, B↔Y) systematically.

Blood relations: draw a family tree; vertical lines for parent–child, horizontal for siblings/spouse. 'Maternal' refers to mother's side, 'paternal' to father's side.

Direction sense: use a compass sketch. Net displacement after moves is the straight-line distance from start; use Pythagoras for right-angle turns.

Syllogism: with 'All A are B' and 'All B are C', conclude 'All A are C'. 'Some' statements are not reversible; 'No A is B' converts to 'No B is A'.

Seating arrangement: fix the reference person, then place others clockwise/counter-clockwise as stated. For circular arrangements, n people can be seated in (n−1)! ways.

Analogy: identify the relationship type (part–whole, cause–effect, synonym, worker–tool) before choosing the matching pair.

Clocks: angle between hands = |30H − 5.5M| degrees, where H is hour and M is minutes. Hands coincide 11 times in 12 hours.

Subject–verb agreement: a singular subject takes a singular verb; with 'either…or' / 'neither…nor', the verb agrees with the nearer subject.

Articles: 'a' before consonant sounds, 'an' before vowel sounds (an hour, a university). 'The' is used for unique nouns and superlatives.

Tenses: present perfect (has/have + past participle) for past actions with present relevance; past perfect (had + past participle) for the earlier of two past actions.

Active to passive: object becomes subject; verb becomes be + past participle; subject moves to 'by' phrase. Example: 'She wrote a letter' → 'A letter was written by her'.

Common error: 'one of the' takes a plural noun but a singular verb - 'One of the boys is absent.' 'Each' and 'every' take singular verbs.

Prepositions: 'different from', 'similar to', 'capable of', 'composed of', 'superior to', 'married to'. Do not use 'than' after 'superior'.

Reading comprehension: locate the topic sentence (usually first or last in a paragraph); answer must be supported by the passage, not by outside knowledge.

Idioms: 'to bite the bullet' = to endure a difficult situation; 'once in a blue moon' = very rarely; 'a blessing in disguise' = an apparent misfortune that turns out well.

Bharat Electronics Limited (BEL) is a Government of India enterprise under the Ministry of Defence, established in 1954, headquartered in Bengaluru.

Indian Constitution: adopted 26 November 1949, came into force 26 January 1950. Fundamental Rights are in Part III (Articles 12–35); Fundamental Duties in Article 51A.

Indian Parliament: Lok Sabha maximum strength 552; Rajya Sabha maximum 250. Money bills can be introduced only in the Lok Sabha.

Space: ISRO launched Chandrayaan-3 in July 2023; Vikram lander touched down near the lunar south pole on 23 August 2023.

Defence: DRDO is India's defence research agency; BrahMos is a supersonic cruise missile developed jointly by India and Russia.

Economics: RBI is India's central bank, established 1935, nationalised 1949. Repo rate is the rate at which RBI lends to commercial banks.

Geography: Tropic of Cancer passes through eight Indian states, including Gujarat, Rajasthan, Madhya Pradesh, and West Bengal.

Awards: the Bharat Ratna is India's highest civilian award; the Param Vir Chakra is the highest wartime military decoration.