MCA CET Prep मराठी

Last-week revision

Revision sheet

One line of analogy and the key formulas for every topic. Read it the night before; if a line doesn't trigger the full idea, open that topic again.

Speed maths: know these cold

Every value below is a tool you should not have to compute in the hall. The numbers are generated by the build, not typed, so they are exact.

Squares

nn²nn²nn²
1²12²43²9
4²165²256²36
7²498²649²81
10²10011²12112²144
13²16914²19615²225
16²25617²28918²324
19²36120²40021²441
22²48423²52924²576
25²62526²67627²729
28²78429²84130²900

Cubes

nn³nn³nn³
1³12³83³27
4³645³1256³216
7³3438³5129³729
10³100011³133112³1728
13³219714³274415³3375

Powers of 2

2ⁿ2ⁿ2ⁿ
201212224
23824162532
26642712828256
2951221010242112048
2124096213819221416384
2153276821665536

Fraction ↔ percent

%%
1/250%1/333.33%
1/425%1/520%
1/616.67%1/714.29%
1/812.5%1/911.11%
1/1010%1/119.09%
1/128.33%1/166.25%
1/205%1/254%

Divisibility rules

ByRule
2last digit even
3digit sum divisible by 3
4last two digits divisible by 4
5ends in 0 or 5
6divisible by both 2 and 3
8last three digits divisible by 8
9digit sum divisible by 9
11(sum of odd-place digits) − (sum of even-place digits) is 0 or a multiple of 11

Constants and logs

√21.414
√31.732
√52.236
π3.1416 ≈ 22/7
e2.718
log10 20.3010
log10 30.4771
log10 70.8451
ln 102.3026

Trig values

0°30°45°60°90°
sin01/21/√2√3/21
cos1√3/21/√21/20
tan01/√31√3—

Mathematics & Statistics

Basic Algebra: Expansion, Factorisation, Indices

Making a square farm plot bigger: (a + b)² is a bigger square plot: the old plot, two new strips and one small corner piece.

  • (a ± b)² = a² ± 2ab + b²; a² − b² = (a − b)(a + b)
  • (a ± b)³ = a³ ± 3a²b + 3ab² ± b³ = a³ ± b³ ± 3ab(a ± b)
  • a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
  • (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
  • If a + b + c = 0, then a³ + b³ + c³ = 3abc
  • aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ
  • a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; ap/q = (q-th root of a)p
  • x + 1/x = k ⇒ x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k

Quadratic Equations

A cricket ball thrown upward: The roots are the two moments the ball is at ground level.

  • Roots: x = (−b ± √D) / 2a, where D = b² − 4ac
  • D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: complex conjugate roots
  • D a perfect square (with a, b, c rational): roots are rational
  • Sum of roots α + β = −b/a; product αβ = c/a
  • Equation from roots: x² − (sum)x + (product) = 0
  • α² + β² = (α + β)² − 2αβ

Logarithms

A savings scheme that doubles your money: A logarithm counts how many times you multiplied by the base, like counting how many doubling periods your savings needed.

  • loga x = y ⟺ aʸ = x (x > 0, a > 0, a ≠ 1)
  • loga 1 = 0; loga a = 1; aloga x = x
  • log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x
  • Change of base: loga x = logb x / logb a; so loga b × logb a = 1
  • logaᵐ xⁿ = (n/m) loga x
  • log₁₀ 2 ≈ 0.3010, log₁₀ 3 ≈ 0.4771, log₁₀ 5 = 1 − log₁₀ 2 ≈ 0.6990
  • Number of digits in N (N ≥ 1) = (integer part of log₁₀ N) + 1

Progressions: AP, GP, HP

Gokulashtami: the Dahi Handi pyramid and the WhatsApp invite: AP adds the same number each step (pyramid tiers); GP multiplies by the same number each step (forwarded invites).

  • AP: aₙ = a + (n − 1)d; Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l)
  • GP: aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1; infinite sum = a/(1 − r) when |r| < 1
  • HP: reciprocals are in AP; nth term of HP = 1/(nth term of that AP)
  • AM = (a + b)/2; GM = √(ab); HM = 2ab/(a + b)
  • AM ≥ GM ≥ HM for positive numbers, and GM² = AM × HM
  • Three terms in AP: a − d, a, a + d; three terms in GP: a/r, a, ar
  • aₙ = Sₙ − Sₙ₋₁ (find any term from the sum formula)

Binomial Theorem

Five friends ordering in the college canteen: Each bracket is one friend choosing poha or upma; a coefficient counts how many ways the same order slip can happen.

  • (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ (r = 0 to n); it has n + 1 terms
  • General term: Tr+1 = ⁿCᵣ aⁿ⁻ʳ bʳ
  • n even: one middle term Tn/2 + 1; n odd: two middle terms T(n+1)/2 and T(n+3)/2
  • ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ; sum of odd-position coefficients = sum of even-position = 2ⁿ⁻¹
  • ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
  • In (a − b)ⁿ the general term has (−1)ʳ: signs alternate
  • Term independent of x: put the power of x in Tr+1 equal to 0

Permutations and Combinations

Planning the sweets at a wedding: Choosing sweets for the menu card is a combination; deciding the order the servers bring them is a permutation.

  • ⁿPᵣ = n!/(n − r)!; ⁿCᵣ = n!/[r!(n − r)!]; ⁿPᵣ = r! × ⁿCᵣ
  • Arrangements of n objects with p alike, q alike, …: n!/(p! q! …)
  • Circular arrangements of n different objects: (n − 1)!; necklace or garland (can be flipped): (n − 1)!/2
  • ⁿCᵣ = ⁿCₙ₋ᵣ; if ⁿCₓ = ⁿCᵧ then x = y or x + y = n
  • Selections of at least one from n different things: 2ⁿ − 1
  • Diagonals of an n-sided polygon: ⁿC₂ − n; triangles from n points (no three collinear): ⁿC₃
  • “Together”: tie the group into one block, arrange, then multiply by the arrangements inside the block

Straight Lines

An ST bus climbing a ghat road: Slope is the ghat road’s gradient: how much you climb for every step forward.

  • Distance: √[(x₂ − x₁)² + (y₂ − y₁)²]
  • Section formula (internal, ratio m : n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); external: replace + by − in the ratio part
  • Centroid of a triangle: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
  • Slope m = tan θ = (y₂ − y₁)/(x₂ − x₁); for ax + by + c = 0, m = −a/b
  • Forms: y = mx + c; y − y₁ = m(x − x₁); x/a + y/b = 1
  • Parallel: m₁ = m₂; perpendicular: m₁m₂ = −1; angle: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
  • Distance of (x₁, y₁) from ax + by + c = 0: |ax₁ + by₁ + c| / √(a² + b²); between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0: |c₁ − c₂| / √(a² + b²)
  • Area of triangle: ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; area 0 means the points are collinear

Circles

A goat tied to a peg: The peg is the centre, the rope is the radius, and the edge of the grazed patch is the circle.

  • Centre-radius form: (x − h)² + (y − k)² = r²
  • General form x² + y² + 2gx + 2fy + c = 0: centre (−g, −f), radius √(g² + f² − c)
  • Diameter form with ends (x₁, y₁), (x₂, y₂): (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
  • Tangent at (x₁, y₁) on x² + y² = a²: xx₁ + yy₁ = a²
  • y = mx + c touches x² + y² = a² if and only if c² = a²(1 + m²)
  • Length of tangent from (x₁, y₁): √S₁, where S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
  • Two circles touch externally if C₁C₂ = r₁ + r₂, internally if C₁C₂ = |r₁ − r₂|

Conics: Parabola, Ellipse, Hyperbola

A torch beam on the wall during a power cut: Tilt a torch against a wall and its patch of light goes circle → ellipse → parabola → hyperbola.

  • Parabola y² = 4ax: vertex (0, 0), focus (a, 0), directrix x = −a, latus rectum 4a
  • Parabola x² = 4ay: focus (0, a), directrix y = −a
  • Ellipse x²/a² + y²/b² = 1 (a > b): b² = a²(1 − e²), foci (±ae, 0), directrices x = ±a/e
  • Ellipse: latus rectum 2b²/a; PF₁ + PF₂ = 2a (major axis)
  • Hyperbola x²/a² − y²/b² = 1: b² = a²(e² − 1), foci (±ae, 0), latus rectum 2b²/a, |PF₁ − PF₂| = 2a
  • Rectangular hyperbola (a = b, e.g. x² − y² = a² or xy = c²): e = √2
  • Focus distance from centre: ellipse ae = √(a² − b²); hyperbola ae = √(a² + b²)

Pair of Straight Lines and Transformations

Drawing a Diwali rangoli with a stencil: Two crossing chalk lines are one pair-of-lines equation; sliding, turning and enlarging the stencil are translation, rotation and scaling.

  • ax² + 2hxy + by² = 0: two lines through the origin; real and distinct if h² > ab, coincident if h² = ab
  • Slopes m₁, m₂: m₁ + m₂ = −2h/b, m₁m₂ = a/b
  • Angle between them: tan θ = 2√(h² − ab) / |a + b|; perpendicular if a + b = 0
  • ax² + 2hxy + by² + 2gx + 2fy + c = 0 is a pair of lines if abc + 2fgh − af² − bg² − ch² = 0
  • Shift origin to (h, k) (axes moved, not turned): new coordinates X = x − h, Y = y − k
  • Rotate a point anticlockwise by θ about the origin: (x cos θ − y sin θ, x sin θ + y cos θ); by 90°: (x, y) → (−y, x)
  • Rotate the axes by θ: X = x cos θ + y sin θ, Y = −x sin θ + y cos θ
  • Scaling by sₓ and sᵧ: (x, y) → (sₓx, sᵧy); areas are multiplied by sₓsᵧ

Differential Equations

A glass of cutting chai cooling at the tapri: The chai cools faster when it is hotter; writing that rule down is a differential equation, and the cooling curve is its solution.

  • Order = highest derivative; degree = its power (after clearing radicals/fractions); number of arbitrary constants = order
  • Variables separable: f(y) dy = g(x) dx, integrate both sides
  • Homogeneous dy/dx = F(y/x): put y = vx, dy/dx = v + x dv/dx
  • Linear dy/dx + Py = Q: IF = e∫P dx; solution y × IF = ∫(Q × IF) dx + C
  • y″ + ay′ + by = 0: solve m² + am + b = 0
  • Distinct real roots m₁, m₂: y = c₁em₁x + c₂em₂x; equal roots m: y = (c₁ + c₂x)emx; roots α ± iβ: y = eαx(c₁ cos βx + c₂ sin βx)
  • Growth or decay dy/dt = ky gives y = y₀ekt

Trigonometric Identities and Equations

Riding the giant wheel at the village jatra: Your seat’s height is sin θ and its sideways distance is cos θ; you are always the same distance from the axle, so sin²θ + cos²θ = 1.

  • sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
  • sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
  • sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1; tan 2A = 2 tan A/(1 − tan²A)
  • sin 3A = 3 sin A − 4 sin³A; cos 3A = 4 cos³A − 3 cos A
  • sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2); cos C + cos D = 2 cos((C + D)/2) cos((C − D)/2)
  • General solutions: sin θ = sin α ⇒ θ = nπ + (−1)ⁿα; cos θ = cos α ⇒ θ = 2nπ ± α; tan θ = tan α ⇒ θ = nπ + α
  • sin θ = 0 ⇒ θ = nπ; cos θ = 0 ⇒ θ = (2n + 1)π/2; n is any integer

Solution of Triangles, Heights and Distances

Finding the height of the school flagpole on Republic Day: Stand at a known distance, measure how far you tilt your head, and tan gives the height without climbing.

  • Sine rule: a/sin A = b/sin B = c/sin C = 2R (R = circumradius)
  • Cosine rule: cos A = (b² + c² − a²)/(2bc), i.e. a² = b² + c² − 2bc cos A
  • Projection rule: a = b cos C + c cos B
  • Area Δ = ½ bc sin A = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2
  • R = abc/(4Δ); inradius r = Δ/s
  • Height h = d tan θ; angle of depression from the top = angle of elevation from the ground point
  • Elevations α and β (α > β) from two points d apart in line with the base: h = d tan α tan β/(tan α − tan β)

Inverse Trigonometric Functions

Guessing the time from the temperature on a Nagpur summer day: The same temperature comes twice a day, so to read time from temperature you agree to look only at the rising morning hours: that is the principal value.

  • sin⁻¹x: domain [−1, 1], range [−π/2, π/2]
  • cos⁻¹x: domain [−1, 1], range [0, π]; tan⁻¹x: domain all real x, range (−π/2, π/2)
  • sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + cosec⁻¹x = π/2
  • sin⁻¹(−x) = −sin⁻¹x; tan⁻¹(−x) = −tan⁻¹x; but cos⁻¹(−x) = π − cos⁻¹x
  • tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1
  • 2 tan⁻¹x = tan⁻¹[2x/(1 − x²)] for |x| < 1
  • sin⁻¹(sin θ) = θ only for θ in [−π/2, π/2]; cos⁻¹(cos θ) = θ only for θ in [0, π]

Probability: Basics, Dependent and Independent Events

The lucky draw at the Ganpati mandal: Drawing chits without putting them back makes the next draw dependent; putting them back keeps draws independent.

  • P(A) = n(A)/n(S); 0 ≤ P(A) ≤ 1; P(A′) = 1 − P(A)
  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive means P(A ∩ B) = 0
  • P(B | A) = P(A ∩ B)/P(A)
  • Independent: P(A ∩ B) = P(A) P(B); then A′ and B′ are independent too
  • P(at least one) = 1 − P(none)
  • Bayes: P(Eᵢ | A) = P(Eᵢ) P(A | Eᵢ) / Σ P(Eⱼ) P(A | Eⱼ)
  • Two dice: 36 outcomes; a sum of 7 has 6 ways (the most likely sum)
  • Odds in favour a : b means P = a/(a + b)

Averages, Dispersion, Skewness and Kurtosis

Evening milk collection at the village dairy: Mean, median and mode describe the typical farmer’s milk; standard deviation measures how much farmers differ; one big farmer creates skew.

  • Mean x̄ = Σfx/Σf; grouped median = l + [(N/2 − cf)/f] × h; grouped mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
  • Empirical relation (moderately skewed data): Mode = 3 Median − 2 Mean
  • Variance σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; SD σ = √variance
  • Coefficient of variation CV = (σ/x̄) × 100; lower CV means more consistent
  • Range = max − min; quartile deviation = (Q₃ − Q₁)/2
  • y = ax + b ⇒ ȳ = ax̄ + b and σᵧ = |a|σₓ (adding b does not change SD)
  • Karl Pearson’s skewness Sk = (Mean − Mode)/σ = 3(Mean − Median)/σ
  • Kurtosis β₂ = μ₄/μ₂²: β₂ = 3 mesokurtic (normal), β₂ > 3 leptokurtic (peaked), β₂ < 3 platykurtic (flat)

Random Variables and Probability Distributions

Packing Devgad hapus mangoes: Spoilt mangoes per box is a random variable; it follows a Binomial law, rare pests follow Poisson, and mango weights follow a Normal bell curve.

  • Discrete RV: Σp(x) = 1; E(X) = Σx p(x); Var(X) = E(X²) − [E(X)]²
  • E(aX + b) = aE(X) + b; Var(aX + b) = a² Var(X)
  • Binomial B(n, p): P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, q = 1 − p; mean np, variance npq (variance < mean)
  • Poisson: P(X = r) = e−λ λʳ / r!; mean = variance = λ
  • Normal N(μ, σ²): symmetric; mean = median = mode; total area 1; P(X = a) = 0 for a single point
  • Standard normal Z = (X − μ)/σ has mean 0 and SD 1
  • About 68%, 95% and 99.7% of values lie within 1, 2 and 3 SDs of the mean
  • Continuous RV: f(x) ≥ 0 and the total area ∫f(x) dx = 1

Correlation, Regression and Curve Fitting

Study hours and marks in a Pune coaching class: Correlation measures how tightly the dots follow a line; regression is the best ruler line through them, used for prediction.

  • r = cov(x, y)/(σₓσᵧ), cov(x, y) = Σ(x − x̄)(y − ȳ)/n; −1 ≤ r ≤ 1
  • y on x: y − ȳ = byx(x − x̄), byx = r σᵧ/σₓ
  • x on y: x − x̄ = bxy(y − ȳ), bxy = r σₓ/σᵧ
  • byx × bxy = r²; r = ±√(byx bxy) with the same sign as the coefficients
  • Both regression lines pass through (x̄, ȳ); they coincide if r = ±1
  • Least squares line y = a + bx: Σy = na + bΣx and Σxy = aΣx + bΣx²
  • Spearman’s rank correlation: R = 1 − 6Σd²/[n(n² − 1)]

Ratio, Proportion and Percentage

A day at Ramesh Kaka’s kirana shop: Mixing two rice qualities is a ratio, the discount board is a percentage, and two offers one after another do not simply add.

  • a : b = c : d ⇒ ad = bc; mean proportional of a and b = √(ab); third proportional = b²/a
  • Divide N in ratio a : b : c → shares aN/(a + b + c), bN/(a + b + c), cN/(a + b + c)
  • x% of y = y% of x
  • Successive changes a% and b%: net = a + b + ab/100
  • Price up r% ⇒ cut consumption by r/(100 + r) × 100% to keep spending the same
  • If A is r% more than B, then B is r/(100 + r) × 100% less than A
  • Profit% = (SP − CP)/CP × 100; Discount% = (MP − SP)/MP × 100
  • Alligation: cheaper : dearer = (dearer price − mean price) : (mean price − cheaper price)

Time and Work (with Pipes and Cisterns)

Filling the housing society’s overhead tank: Think per hour: each motor fills a fraction of the tank every hour, fractions add, and a leak subtracts.

  • A finishes in a days ⇒ A’s one-day work = 1/a
  • Two together: time T = ab/(a + b); three together: 1/T = 1/a + 1/b + 1/c
  • M₁D₁H₁/W₁ = M₂D₂H₂/W₂ (men × days × hours per unit of work)
  • Pipes: filling pipe +, emptying pipe or leak −; net rate × time = 1 tank
  • If A is k times as efficient as B, A takes 1/k of B’s time
  • Wages are shared in the ratio of work done
  • LCM method: total work = LCM of the given days; each rate = total ÷ own days

Time, Speed and Distance

Watching Mumbai locals at Dadar station: A train must cover its own length to pass a pole, its length plus the platform to clear the platform, and two trains feel only their relative speed.

  • Distance = speed × time; 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h
  • Average speed = total distance ÷ total time; equal distances at u and v: 2uv/(u + v)
  • Train passing a pole or a person: time = train length ÷ speed
  • Train passing a platform, bridge or another train: distance = sum of the lengths
  • Relative speed: same direction u − v; opposite directions u + v
  • Boat speed b, stream s: downstream b + s, upstream b − s; b = (down + up)/2, s = (down − up)/2
  • For a fixed distance, time is inversely proportional to speed

Sets, Relations and Functions

The Aadhaar enrolment camp at the gram panchayat: Giving every villager exactly one Aadhaar number is a function; no two sharing a number makes it one-one; every number used makes it onto.

  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B); n(A ∪ B ∪ C) = Σn(A) − Σn(A ∩ B) + n(A ∩ B ∩ C)
  • De Morgan: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′
  • A set with n elements has 2ⁿ subsets (2ⁿ − 1 proper subsets)
  • n(A × B) = mn; number of relations from A to B = 2mn; relations on a set of n elements = 2n²
  • Equivalence relation = reflexive + symmetric + transitive
  • Functions from A (m elements) to B (n elements): nᵐ in all; one-one: ⁿPₘ (m ≤ n); bijections when m = n: n!
  • (f ∘ g)(x) = f(g(x)); in general f ∘ g ≠ g ∘ f; f⁻¹ exists only if f is a bijection

Mensuration 2D: Triangles, Quadrilaterals, Circles

Tiling the floor of a new flat: Tiles needed is the area, the skirting strip along the walls is the perimeter, and doubling the room’s size needs four times the tiles.

  • Triangle: ½ × base × height = ½ ab sin C; Heron: √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2
  • Equilateral triangle of side a: area (√3/4)a², height (√3/2)a
  • Rectangle: area lb, perimeter 2(l + b), diagonal √(l² + b²); square: area a² = d²/2, diagonal a√2
  • Parallelogram: base × height; rhombus: ½ d₁d₂, side = ½√(d₁² + d₂²); trapezium: ½ (sum of parallel sides) × height
  • Circle: area πr², circumference 2πr; semicircle perimeter πr + 2r; ring: π(R² − r²)
  • Sector of angle θ°: area (θ/360)πr², arc length (θ/360) × 2πr
  • Wheel: distance in one revolution = circumference; revolutions = distance ÷ 2πr
  • Each length × k ⇒ perimeter × k, area × k² (r up 10% ⇒ area up 21%)

Mensuration 3D: Cube, Cuboid, Sphere, Cylinder, Cone

An ice-cream parlour on a Sunday evening: The ice-cream inside is volume, the paper wrapper is surface area, and three cones fill one tub of the same base and height.

  • Cuboid: V = lbh; TSA = 2(lb + bh + hl); diagonal √(l² + b² + h²)
  • Cube: V = a³; TSA = 6a²; diagonal a√3
  • Cylinder: V = πr²h; CSA = 2πrh; TSA = 2πr(r + h)
  • Cone: V = ⅓πr²h; CSA = πrl; TSA = πr(l + r); l = √(r² + h²)
  • Sphere: V = (4/3)πr³; surface area 4πr²
  • Hemisphere: V = (2/3)πr³; CSA = 2πr²; TSA = 3πr²
  • Recasting: volume before = volume after; number of small pieces = big volume ÷ one small volume
  • Lengths × k ⇒ surface area × k², volume × k³

Logical / Abstract Reasoning

Number Series

Your gullak (piggy bank) savings: Do not stare at the jar total; look at how much was added each day.

  • Step order: differences → second differences → ratios → squares/cubes → alternate positions
  • Equal differences: arithmetic series (a, a + d, a + 2d …)
  • Differences 2, 4, 6, 8 … or 5, 7, 9, 11 …: second difference is constant
  • Fast growth: test ×k, ×k + c, ×k − c (e.g. 5, 11, 23, 47 is ×2 + 1)
  • Know squares to 30² and cubes to 15³ by heart
  • n(n + 1): 2, 6, 12, 20, 30 …; n² − 1: 3, 8, 15, 24 …; n³ + n: 2, 10, 30, 68 …
  • Long series (7–8 terms) with a zig-zag: check odd and even positions separately
  • Wrong-number questions: the wrong term usually makes two consecutive differences look odd

Letter & Alphanumeric Series

House numbers on a circular society lane: Every letter is a house number from 1 to 26 on a lane that loops back to the start.

  • EJOTY: E = 5, J = 10, O = 15, T = 20, Y = 25
  • Position from the right end = 27 − position from the left
  • Opposite pairs add to 27: A–Z, B–Y, C–X, D–W, E–V, F–U, G–T, H–S, I–R, J–Q, K–P, L–O, M–N
  • Moving forward beyond Z: subtract 26; moving back beyond A: add 26
  • “k-th to the left of X” = position of X − k; “k-th to the right” = position + k
  • Clusters: solve the 1st letters, 2nd letters (and digits) as separate series
  • Vowels A, E, I, O, U are at 1, 5, 9, 15, 21

Coding–Decoding

The Mumbai dabbawala tiffin code: Every tiffin carries a short code; the same rule is applied to every tiffin, so one decoded tiffin teaches you all the rest.

  • Always write positions above and below: compare letter by letter
  • Check for a constant shift first, then reversal, then opposite letters (sum 27)
  • Opposite-letter coding: GIRL → TRIO because G↔T, I↔R, R↔I, L↔O
  • Number codes: try sum of positions, then product, then number of letters
  • Sentence coding: common word ↔ common code; use pairs of sentences
  • Substitution: answer the real question first, then give its code name
  • If the code is shorter or longer than the word, letters are being dropped, added or grouped

Blood Relations

The family photo at a wedding: Arrange everyone like the wedding photographer does: grandparents on top, parents in the middle, children below, couples side by side.

  • Symbols: + male, − female, = married couple, | parent to child, — siblings
  • Same generation on the same horizontal level
  • Father’s or mother’s brother = uncle; their sister = aunt; their children = cousins
  • Son’s wife = daughter-in-law; daughter’s husband = son-in-law
  • Wife’s or husband’s brother = brother-in-law; sister’s husband = brother-in-law
  • “Only son of my father’s father” = my father; “only daughter of my mother” = me (if I am female)
  • Gender is unknown unless stated — “A is the child of B” says nothing about A’s gender
  • Decode long “pointing” sentences from the end backwards

Direction Sense

Directions from the ST stand to your house in the village: Mark the ST stand as the origin, draw every road on paper, and turn the way the walker turns — not the way you are sitting.

  • Right turn = 90° clockwise; left turn = 90° anticlockwise (from current facing)
  • Clockwise order: N → E → S → W → N; 45° turns: N → NE → E → SE → S …
  • Net x = East − West; net y = North − South; distance = √(x² + y²)
  • Triples: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 7-24-25
  • Morning: sun in east, shadow to the west. Evening: sun in west, shadow to the east
  • “Direction of A from B” means stand at B and look towards A
  • A U-turn is 180°; two right turns or two left turns also reverse the direction

Ranking & Ordering

The queue at the ration shop: Your place from the front plus your place from the back counts you twice, so subtract one.

  • Total = L + R − 1
  • Position from right = Total − L + 1
  • Between two people (same end positions a and b): |a − b| − 1
  • Interchange: A’s new position = B’s old position (from the same end)
  • Ordering: build one chain with > and read the k-th from the top
  • Minimum total in a row when A is p-th from left and B is q-th from right (B to the right of A, no overlap) = p + q

Seating Arrangement — Linear & Circular

Seats in a Mumbai local train compartment: Fix one person first, and remember that “your right” depends on which way your bench faces.

  • Facing north: their right = page right. Facing south: their right = page left
  • Circle facing centre: right = anticlockwise, left = clockwise
  • Circle facing outward: right = clockwise, left = anticlockwise
  • “Immediate right” = the very next seat; “second to the right” = skip one seat
  • n people in a circle: opposite person is n/2 seats away (n even)
  • Place definite clues first, then relative clues, then negative clues (“not adjacent”)
  • Two possible cases? Draw both side by side and eliminate

Puzzles — Scheduling, Floors & Multi-attribute Grids

The Ganpati mandal volunteer roster: Make a table of volunteers × days, tick the sure things first, cross out the impossible ones, and the gaps fill themselves.

  • Draw the table before reading clue 2
  • Order: definite → linked → negative clues
  • “Immediately above/after” = next one; “above/after” = any one higher/later
  • Each person–value pair is used exactly once (one tick per row and column)
  • Two cases? Solve both; the survivor is the answer
  • Floor 1 is the bottom unless the question says “ground floor = 0”
  • Re-check every clue against the final table before answering

Syllogisms

WhatsApp groups and their members: Each category is a WhatsApp group; a conclusion follows only if it is true for every possible way the groups could overlap.

  • All A are B → A inside B; also Some A are B and Some B are A
  • Some A are B → Some B are A (at least one common)
  • No A is B → No B is A
  • All A–B + All B–C → All A are C; All A–B + No B–C → No A is C; Some A–B + All B–C → Some A are C; Some A–B + No B–C → Some A are not C
  • Some + Some, or No + No, or All (A in B) + Some (B with C): no definite A–C conclusion
  • “All B are A” does NOT follow from “All A are B” — but it is a possibility
  • Possibility fails only when a definite statement forbids it
  • Either–or: a complementary pair on the same two terms (Some/No, or All/Some-not) where neither is definite alone

Statements — Assumptions, Conclusions & Inferences

The gram panchayat’s drum announcement (dawandi): The announcement is the statement; the unspoken belief behind it is the assumption; what villagers can safely take away is the conclusion.

  • Assumption = unstated belief the statement depends on
  • Negation test: if the assumption is false, does the statement lose its point?
  • Conclusion = must follow from the given words only
  • Extreme words (only, all, never, always, best) usually make a choice wrong
  • Advice/appeal assumes the advice is useful and people can follow it
  • Inference questions: no new facts, no future predictions, no generalising
  • Real-world truth is irrelevant — only the statement matters

Analogy & Classification (Odd One Out)

The shelves of a kirana shop: Find the shelf label (the common property); the item that does not belong on that shelf is the odd one out.

  • Say the A–B relationship as a sentence, then reuse the sentence for C
  • Common word relations: tool–use, worker–workplace, item–unit, part–whole, synonym/antonym
  • Number analogies: test n², n³, n² ± 1, n³ ± n, n × (n + 1)
  • Number odd-one-out: prime, square, cube, even/odd, digit sum
  • Letter groups: convert to positions and compare gaps
  • A good rule leaves exactly one item out
  • Keep the order: A : B :: C : D, not B : A

Calendars & Clocks

The weekly village bazaar: Full weeks bring you back to the same bazaar day; only the leftover days move the weekday.

  • Odd day code: 0 Sun, 1 Mon, 2 Tue, 3 Wed, 4 Thu, 5 Fri, 6 Sat
  • Ordinary year = 1 odd day; leap year = 2 odd days
  • 100 yrs = 5, 200 yrs = 3, 300 yrs = 1, 400 yrs = 0 odd days
  • Leap year: ÷ 4, but century years must be ÷ 400
  • Angle = |30H − 5.5M| (use 360 − angle if > 180°)
  • Hands coincide at 60H/11 min past H; 22 times a day
  • Hands at right angles 44 times a day; in a straight line (0° or 180°) 44 times a day

Venn Diagrams & Data Sufficiency

Sports-day team lists: kabaddi and kho-kho: Students in both teams sit in the overlap and must be counted once; and the coach only asks whether the note is enough, not for the full answer.

  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
  • Only A = n(A) − n(A ∩ B); Neither = Total − n(A ∪ B)
  • Three sets: n(A ∪ B ∪ C) = A + B + C − (AB + BC + CA) + ABC
  • Subset (mothers ⊂ women): circle inside; possible overlap: intersecting circles; impossible overlap: separate circles
  • DS order: I alone → II alone → together
  • Sufficient = exactly one definite answer (for yes/no: a definite yes or a definite no)
  • Do not compute the final value in DS — just decide if you could

Non-verbal — Mirror/Water Images, Paper Folding, Dice & Figure Series

The barber shop mirror and the village pond: A wall mirror flips left and right; a pond flips top and bottom; nothing else changes.

  • Mirror (vertical): left ↔ right; water (horizontal): top ↔ bottom
  • Mirror clock time = 11:60 − actual time
  • Vertical-mirror-symmetric letters: A H I M O T U V W X Y
  • Water-symmetric letters: B C D E H I K O X
  • n folds → 2ⁿ layers → 2ⁿ holes from one punch (not on a fold)
  • Faces seen together on a dice are adjacent, never opposite
  • Standard dice: opposite faces sum to 7
  • Figure series: check rotation angle and direction first

English Comprehension & Verbal Ability

Reading Comprehension

Aai’s phone call from the village: Answer only from what Aai actually said and hinted, never from gossip you heard elsewhere.

  • Main idea = the one statement that covers the whole passage; reject too-narrow (one example) and too-broad (beyond the passage) options.
  • Detail question: find the keyword in the passage and read the sentence before and after it. The right option usually paraphrases; an option that copies exact words is often a trap.
  • Inference = must be true from the passage, one step beyond it. “Could be true” is not enough.
  • Out of scope: true in the real world but not supported by the passage = wrong.
  • Extreme words (always, never, only, all, completely) are red flags; moderate words (may, often, some, largely) are safer.
  • Tone: judge from adjectives and contrast words. Sarcasm says the opposite of what it means, or mocks with exaggeration.
  • Word in context: put each option into the sentence. The common dictionary meaning is often not the meaning used.
  • Best title: short, covers the whole passage, and matches the author’s focus and tone.

Synonyms and Antonyms

The shade card at the paint shop: A synonym is the nearest shade on the same strip; an antonym is the far end of that strip.

  • Synonym = closest meaning and same part of speech: abundant (adj) → plentiful (adj).
  • Antonym = the opposite on the same scale: verbose ↔ concise (not “silent”).
  • Decide the charge (+/−) first: frugal (+, thrifty) vs stingy (−). The right answer keeps the same charge for a synonym and flips it for an antonym.
  • High-frequency pairs (word – synonym / antonym): candid – frank / evasive; diligent – industrious / lazy; obsolete – outdated / modern; lethargic – sluggish / energetic; ephemeral – short-lived / permanent.
  • More pairs: benevolent – kind / malevolent; garrulous – talkative / reticent; meticulous – careful / careless; ameliorate – improve / worsen; adversity – hardship / prosperity; abundant – plentiful / scarce.
  • Prefixes in-, im-, un-, dis- often form antonyms, but not always: inflammable = easily set on fire (same as flammable); invaluable = priceless.
  • Look-alikes: eminent (famous) / imminent (about to happen); ephemeral (short-lived) / ethereal (delicate, heavenly); adverse (unfavourable) / averse (unwilling).

Vocabulary: One-word Substitution and Confused Words

The postman and the PIN code: One-word substitution is a PIN code for a long description; confused words are two houses with nearly the same name.

  • -cide (killing): homicide (a human), regicide (a king), patricide (father), matricide (mother), fratricide (brother), infanticide (infant), genocide (a whole people).
  • -cracy (rule by): autocracy (one person, absolute power), bureaucracy (officials), theocracy (religious authority), plutocracy (the rich), aristocracy (nobles).
  • omni- (all): omnipotent (all-powerful), omniscient (all-knowing), omnipresent (present everywhere), omnivorous (eats both plants and animals).
  • People: atheist (does not believe in God), agnostic (holds that God’s existence cannot be known), philanthropist (helps mankind through charity), misanthrope (hates mankind), misogynist (hates women), polyglot (speaks many languages), teetotaller (never drinks alcohol), novice (beginner), connoisseur (expert judge of art, food or taste).
  • Cannot be…: illegible (read), inaudible (heard), incorrigible (corrected/reformed), infallible (wrong), invincible (defeated), inevitable (avoided), irreparable (repaired). Also: posthumous (after death), anonymous (of unknown name), panacea (remedy for all ills).
  • Confused pairs (1): affect (v, to influence) / effect (n, result); principal (head; main) / principle (rule, belief); stationary (not moving) / stationery (pens, paper); complement (that which completes) / compliment (praise).
  • Confused pairs (2): advice (n) / advise (v); practice (n) / practise (v) in British spelling; loose (not tight) / lose (misplace); elicit (draw out) / illicit (illegal); council (a body) / counsel (advice; a lawyer); emigrate (leave one’s country) / immigrate (come into a country); beside (next to) / besides (in addition to).

Idioms and Phrases

The code on a dabbawala’s tiffin: An idiom is a tiffin code: learnt as a whole, never read letter by letter, and one changed letter sends it elsewhere.

  • Secrets: spill the beans / let the cat out of the bag = reveal a secret; smell a rat = suspect something is wrong.
  • Effort and time: burn the midnight oil = study/work late at night; at the eleventh hour = at the last moment; call it a day = stop working; once in a blue moon = very rarely.
  • Trouble and surprise: in hot water = in trouble; a bolt from the blue = a complete, sudden surprise; in the same boat = in the same difficult situation.
  • People and talk: beat about the bush = avoid the main point; hit the nail on the head = say exactly the right thing; pull someone’s leg = tease; a wet blanket = one who spoils others’ fun; turn a deaf ear = deliberately ignore.
  • Things: a white elephant = costly but useless possession; Achilles’ heel = a weak point; a red-letter day = a memorable, happy day; break the ice = start a conversation in a stiff situation; cry over spilt milk = regret what cannot be undone.
  • Phrasal verbs (look): look after = take care of; look into = investigate; look up to = respect; look down on = despise; look forward to = wait eagerly for.
  • Phrasal verbs (others): call off = cancel; put off = postpone; put up with = tolerate; give in = surrender; give up = stop / abandon; turn down = reject; carry out = perform; run out of = use up completely; make up for = compensate; bring up = raise a child / mention.

Parts of Speech and Tenses

A live cricket scorecard: A word’s part of speech is the role it plays this over; tense is where the action sits on the scorecard’s timeline.

  • Adjective describes a noun; adverb describes a verb/adjective/adverb. Linking verbs (be, seem, look, feel, taste, smell, sound, become) take an adjective.
  • Simple present: habits, universal truths, fixed timetables. “The sun rises in the east.” “The train leaves at 7:15.”
  • Present perfect (has/have + V3): past action whose result matters now, with no fixed past time. It never goes with yesterday, ago, last year, in 2019.
  • Since + point of time (since 2019, since Monday); for + length of time (for three years). With these use the perfect or perfect continuous: “I have been living here since 2019.”
  • Past perfect (had + V3) marks the earlier of two past actions: “The train had left before we reached the station.”
  • In time and condition clauses (when, as soon as, before, after, until, if, unless) about the future, use the simple present, not will: “I will call you when he comes.”
  • Conditionals: If + present, will + V1; If + past (were), would + V1; If + had + V3, would have + V3. “If I had known, I would have come.”
  • Stative verbs (know, believe, own, have = possess, belong, consist, prefer) are not used in continuous forms.

Subject–Verb Agreement

Aaji’s steel almirah and its key: The verb is a key that must fit the real lock (the subject), not whichever lock is hanging nearest.

  • Find the true subject first; ignore “of…” phrases: “The quality of the mangoes is good.”
  • “As well as, along with, together with, with, in addition to, accompanied by, besides” do not make the subject plural: “The teacher, along with her students, was present.”
  • Each, every, either, neither, everyone, someone, nobody, each of + plural noun → singular verb: “Each of the boys has a bicycle.”
  • Either…or, neither…nor, not only…but also, or, nor → the verb agrees with the nearer subject: “Neither the coach nor the players were happy.”
  • Two subjects joined by and → plural, unless they name one person or one idea: “The secretary and treasurer is here” (one person); “Bread and butter is my breakfast.”
  • “The number of” → singular; “A number of” → plural. “Many a” + singular noun → singular verb: “Many a student has tried.”
  • Collective nouns: one unit → singular (“The jury has given its verdict”); members acting separately → plural (“The jury were divided in their opinions”).
  • Plural-looking but singular: news, mathematics, physics, economics, measles; a sum of money/distance as one amount (“Ten thousand rupees is a large sum”). Always plural: police, cattle, people, scissors, trousers (but “a pair of scissors is”).

Articles and Prepositions

The railway platform announcement: “A train” is any train, “the train” is the one we both know; at, on and in zoom from a clock point to a day to a whole month.

  • a/an by sound: an hour, an honour, an MBA, an FIR, an X-ray; a university, a uniform, a European, a one-way street.
  • The: specific/known things, unique things (the sun, the earth), superlatives and ordinals (the tallest, the first), rivers, seas, oceans, mountain ranges, island groups (the Godavari, the Arabian Sea, the Western Ghats, the Andaman Islands), holy books (the Gita), musical instruments (plays the tabla), “the + adjective” for a class (the poor, the rich).
  • No article: people’s names, cities, most countries (but the USA, the UK), languages, games, meals in general, and school/hospital/bed for their main purpose.
  • Time: at 7:15, at noon, at night; on Monday, on 15 August; in June, in 2026, in the morning, in the monsoon.
  • Place: at a point (at the bus stop), on a surface or line (on the table, on the road), in an enclosed space or area (in the room, in Pune).
  • Fixed partners: superior / inferior / senior / junior / prior / preferable to (never “than”); prefer A to B; comply with; abide by; consist of; depend on; different from; congratulate on; accused of; charged with; angry with a person, at a thing.
  • No preposition after: discuss, reach, enter, attack, resemble, marry, request, order, answer. “We discussed the plan”, “She reached Pune”.

Sentence Correction and Error Spotting

The tailor’s trial fitting: Check the sentence the way a tailor checks a shirt: part by part with a tape, not by a glance from the door.

  • Pronoun case: after a preposition use the object form (“between you and me”); after than/as, complete the clause (“He is taller than I [am]”). With one, use one’s: “One should keep one’s promise.”
  • Parallelism: items joined by and, or, both…and, not only…but also take the same form. “She likes reading, writing and to dance” ✗ → “dancing”.
  • Paired words: no sooner…than; hardly / scarcely…when; not only…but also; both…and; either…or; neither…nor; lest…should. Although is not followed by but.
  • Comparison: no double comparatives (more better, most cleverest); compare like with like (“The climate of Pune is better than that of Mumbai”); superior/senior/prefer take to.
  • Uncountable nouns have no plural and take no a/an: advice, information, furniture, luggage, equipment, scenery, machinery, news. Say “a piece of advice”, “much information”.
  • Redundancy: “return back”, “revert back”, “repeat again”, “cousin brother” → return, revert, repeat, cousin.
  • Adverb vs adjective: “He drives carefully”; but after linking verbs use an adjective: “The milk smells sour.”
  • Dangling modifier: “Walking to college, the rain started” ✗ (the rain was not walking) → “As I was walking to college, it started to rain.”

Fill in the Blanks and Cloze Test

Serving a festival thali: Each blank is an empty katori: the dish must fit the bowl, suit the meal, and agree with the dishes beside it.

  • Predict a word for the blank before reading the options.
  • Contrast signals (but, although, though, however, yet, despite, while) → the blank opposes the other part.
  • Continuation / cause signals (and, also, moreover, because, since, so, therefore) → the blank agrees with the other part.
  • Grammar clues: “an ___” needs a vowel-sound word; “to ___” usually needs a base verb; “the most ___” needs an adjective; “has ___” needs a past participle.
  • Collocations: heavy rain, strong tea, commit a crime, pay attention, make an effort, make a mistake, keep a promise, catch a cold.
  • Look-alikes appear as options: flout (openly disobey) / flaunt (show off); adapt (adjust) / adopt (take up); persecute (harass) / prosecute (take to court).
  • Two blanks: solve one, eliminate, then check the other. Cloze: read the whole passage first; keep the same tense through a narrative.

Para Jumbles (Jumbled Paragraphs)

A farmer’s sowing calendar: Farm work has an order you cannot break: ploughing first, sowing after the rain, weeding only once seedlings exist, harvest last.

  • Opener: introduces the subject with a full noun or a general statement; no pronoun pointing back, no “however / therefore / also”.
  • A sentence that begins with He, She, It, They, This, These, Such, However, Thus, Therefore or But cannot usually open the paragraph.
  • Noun before pronoun: “Sanjay…” must come before “He…”.
  • Article order: “a farmer” (new) comes before “the farmer” (already known).
  • Mandatory pairs: question → answer; claim → “For example…”; cause → “So / As a result…”; a point → “However…” that contrasts it.
  • Time order: dates and words like first, then, later, finally; “A decade ago” comes before “Today”.
  • The conclusion often gives the result or the present situation, or begins with Thus / Hence / In short / Today.

Computer Concepts

Computer Organization and the CPU

A kirana shopkeeper billing a grocery chit: Finger on the next line (PC), read it (fetch), understand it (decode), weigh and add (execute), repeat.

  • CPU = ALU + Control Unit + registers; the CU directs, the ALU computes
  • PC: address of the next instruction; IR: the instruction being executed now
  • MAR: address sent to memory; MDR/MBR: data coming from or going to memory
  • Accumulator holds ALU results; flag register holds Zero, Sign, Carry, Overflow bits
  • Fetch: MAR ← PC → MDR ← M[MAR] → IR ← MDR, PC ← PC + 1
  • Clock period = 1 ÷ frequency; instruction time = cycles × period
  • Von Neumann: one memory for program and data (the “von Neumann bottleneck”); Harvard: separate instruction and data memories
  • Control unit design: hardwired (fast, fixed logic) or microprogrammed (control memory, easy to change)

Instruction Structure and Addressing Modes

The Mumbai dabbawala’s lid code: The job is fixed (opcode); the lid code says how to find the place (addressing mode).

  • Instruction = opcode + address field(s); n opcode bits → 2n operations
  • Bits for address = instruction length − opcode bits − register-field bits
  • Immediate: operand = value in the instruction (a constant, no memory access)
  • Direct: EA = A; Indirect: EA = M[A] (one extra memory reference)
  • Register indirect: EA = (R); Indexed: EA = A + (XR); Base: EA = (BR) + A
  • PC-relative: EA = PC + offset, where PC already points to the next instruction
  • 0-address instructions work on a stack (PUSH, POP, ADD) — postfix form

Input / Output and Backup Devices

The talathi office and the farmer’s 7/12 photocopies: Form in = input, printed extract out = output, photocopy at home = backup.

  • MICR → bank cheques; OMR → bubble answer sheets; OCR → printed text to editable text; barcode reader → shop billing
  • Impact: dot-matrix (cps), line printer (lpm), daisy wheel; non-impact: inkjet, laser (ppm), thermal
  • Touch screen = input + output; plotter = output for large drawings
  • Magnetic tape is sequential access; hard disk, SSD and RAM are direct (random) access
  • CD ≈ 700 MB, DVD ≈ 4.7 GB (single layer), Blu-ray ≈ 25 GB (single layer)
  • Restore with incremental backups: last full + every incremental after it, in order
  • Restore with differential backups: last full + only the latest differential
  • 3-2-1 rule: 3 copies, on 2 types of media, 1 kept off-site

Memory Hierarchy and Organization

Studying at your desk for CET: Hand → desk → shelf → library: each step bigger, slower and cheaper.

  • Speed: registers > cache > RAM > SSD > HDD > tape; cost per bit falls in the same order
  • Cache = SRAM (no refresh); main memory = DRAM (needs refresh)
  • PROM: write once; EPROM: UV erase; EEPROM: electrical, byte-wise; Flash: electrical, block-wise
  • Capacity = 2address lines × word size; address lines = log₂(number of locations)
  • Chips needed = (total words ÷ chip words) × (total word size ÷ chip word size)
  • Hit ratio h = hits ÷ total accesses
  • EAT if a miss checks cache first, then memory: h·Tc + (1 − h)(Tc + Tm); if both are accessed together: h·Tc + (1 − h)·Tm
  • 1 KB = 210 B, 1 MB = 220 B, 1 GB = 230 B, 1 TB = 240 B

Number Systems and Conversions

Paying with one coin of each power of 2: Binary is paying with coins of ₹1, 2, 4, 8 … using at most one of each.

  • Value = Σ digit × rposition (positions …, 2, 1, 0, −1, −2 …)
  • Integer → base r: divide by r, read remainders bottom-up
  • Fraction → base r: multiply by r, read integer parts top-down
  • Binary → octal: groups of 3; binary → hex: groups of 4, from the point outwards
  • Hex digits: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
  • Useful powers: 2¹⁰ = 1024, 8² = 64, 8³ = 512, 16² = 256, 16³ = 4096
  • Largest n-digit number in base r = rn − 1 (8 bits → 255 = FF16)

Character and Data Representation (ASCII, Unicode, BCD, Parity)

Indian Railways train numbers: Every character has one agreed number everywhere, like every train has one train number.

  • ASCII: 7 bits, 128 characters; extended ASCII: 8 bits, 256 characters
  • ‘0’ = 48, ‘A’ = 65, ‘a’ = 97, space = 32; lowercase = uppercase + 32 (only bit 5 differs)
  • Unicode: one code point for every character of every script; the first 128 are ASCII
  • UTF-8: 1 byte for ASCII, 3 bytes for Devanagari (e.g. अ = U+0905)
  • BCD: 4 bits per decimal digit; 1010–1111 are invalid
  • Excess-3 = BCD of (digit + 3); Gray code: one bit changes between neighbours
  • Parity detects an odd number of bit errors (e.g. 1), misses 2, and cannot correct
  • n bits can represent 2n different characters

Binary Arithmetic, Signed Numbers and Two’s Complement

A scooter odometer rolling back below 0000: Wind 0000 back by one and it shows the top value — that is how −1 becomes all 1s.

  • n-bit two’s complement range: −2n−1 to +2n−1 − 1 (8 bits: −128 to +127)
  • n-bit sign-magnitude and one’s complement range: −(2n−1 − 1) to +(2n−1 − 1), with two zeros
  • Two’s complement = one’s complement + 1 = 2n − x
  • Quick method: copy bits from the right up to and including the first 1, then invert the rest
  • A − B = A + (two’s complement of B); discard the end carry
  • Overflow: (+) + (+) gives (−), or (−) + (−) gives (+); carry into MSB ≠ carry out of MSB
  • Sign extension: repeat the MSB when widening

Floating Point and Normalization (IEEE 754)

Property prices written in lakh and crore: A few significant digits (mantissa) plus a unit word like “crore” (exponent).

  • Single: 1 sign + 8 exponent + 23 fraction = 32 bits, bias 127
  • Double: 1 sign + 11 exponent + 52 fraction = 64 bits, bias 1023
  • Value = (−1)S × 1.F × 2E − 127 (the hidden 1 is not stored)
  • Stored exponent = actual exponent + 127 (so 1.0 has E = 127 = 01111111)
  • E all 0s and F = 0 → zero; E all 1s and F = 0 → infinity; E all 1s and F ≠ 0 → NaN
  • Normalizing: move the point until a single 1 is left of it; moving the point left raises the exponent
  • Only fractions with a power-of-2 denominator are exact (0.5, 0.25, 0.625 yes; 0.1, 0.2 no)

Boolean Algebra, Logic Gates, Truth Tables and K-maps

The switches and bulbs in your house: Switches in series = AND, in parallel = OR, the staircase two-way switch = XOR.

  • AND: 1 only if all inputs are 1; OR: 0 only if all inputs are 0
  • NAND: 0 only if all inputs are 1; NOR: 1 only if all inputs are 0 — both are universal
  • XOR = A′B + AB′ (1 when inputs differ); XNOR = AB + A′B′
  • De Morgan: (A + B)′ = A′B′; (AB)′ = A′ + B′
  • Absorption: A + AB = A; A(A + B) = A; A + A′B = A + B
  • n variables: 2n truth-table rows and 22n possible functions
  • K-map order 00, 01, 11, 10; a group of 2k cells removes k variables
  • Venn: AND = intersection, OR = union, XOR = symmetric difference

Buses, Processor–I/O Communication and Interrupts

A teacher handling doubts in a classroom: Asking every student again and again = polling; a raised hand = interrupt; the monitor carrying notebooks = DMA.

  • Address bus: one-way (from CPU); data bus: two-way; control bus: individual signal lines
  • n address lines → 2n locations; data bus width = bits per transfer
  • Programmed I/O: CPU busy-waits; interrupt I/O: CPU free until the device signals; DMA: block moved without the CPU
  • DMA uses bus request/grant (HOLD/HLDA); modes: cycle stealing and burst
  • The CPU checks for interrupts at the end of each instruction cycle
  • On interrupt: finish instruction → save PC + flags on stack → run ISR → restore → resume
  • NMI cannot be disabled (power failure); divide by zero is an internal trap/exception
  • Daisy chain: priority by position — the device nearest the CPU is highest

Assembly vs High-Level Languages; Compiler, Interpreter, Assembler, Linker, Loader

Translating a Marathi novel vs a live interpreter at a gram sabha: Compiler = translate the whole book once; interpreter = translate sentence by sentence every time.

  • 1GL machine, 2GL assembly, 3GL high-level (C, Java), 4GL (SQL), 5GL (Prolog, AI)
  • Assembly: mnemonics, one-to-one with machine code, machine-dependent, needs an assembler
  • Compiler: whole program at once, all errors listed, produces object code, faster execution
  • Interpreter: line by line, stops at the first error, no object code, slower execution
  • C build chain: preprocessor → compiler → assembler → linker → loader
  • Linker resolves external references; loader allocates memory and relocates
  • Syntax error → compile time; missing function body → link time; divide by zero → run time

C Basics: Data Types, Operators, Precedence, Type Conversion, printf

Milk cans at the village dairy collection centre: Each data type is a container of fixed size; pouring into a smaller one loses milk.

  • Assume char 1, short 2, int 4, float 4, double 8 bytes unless told otherwise
  • 7 / 2 = 3; 7 % 2 = 1; -7 / 2 = −3; -7 % 2 = −1 (sign of the left operand)
  • Precedence: unary > * / % > + − > << >> > relational > == != > & > ^ > | > && > || > ?: > = > ,
  • Right-to-left grouping: unary operators, ?:, assignment (a = b = 5)
  • Short-circuit: in a && b, b is skipped if a is 0; in a || b, b is skipped if a is non-zero
  • x << n = x × 2n; x >> n = x ÷ 2n (for non-negative x)
  • Promotion: char → int → long → float → double; a cast applies only to the operand right after it
  • printf: %d %u %f %c %s %x %o %e %p %%; %.2f = 2 decimals; %5d = width 5

C Control Flow and Loops (Output Prediction)

A PMPML city bus on its route: Each stop is one loop pass; an empty stop is continue; a breakdown is break; a passenger riding on is switch fall-through.

  • Non-zero = true, 0 = false; if (x = 0) assigns and is always false
  • else pairs with the nearest unmatched if
  • switch: integer/char only; falls through until break; case labels must be constants
  • for/while: test first (0 or more passes); do-while: test last (at least 1 pass)
  • continue → next pass (in a for loop the update runs first); break → leave the innermost loop/switch
  • Nested loops with i = 1..n and j = i..n run n(n + 1)/2 times
  • for (i = 0; i < 5; i++); — the semicolon is an empty body; i ends at 5

C Arrays, Strings, Pointers and Functions (Recursion, Call by Value/Reference)

Rooms in a Mumbai chawl: An array is a row of equal rooms; a pointer is a chit with a room number on it.

  • a[i] ≡ *(a + i) ≡ i[a]; address of a[i] = base + i × sizeof(element)
  • 2-D array (row-major): address of a[i][j] = base + (i × columns + j) × size
  • char s[] = "CET"; → sizeof is 4 (includes '\0'), strlen is 3
  • & = address of; * = value at; p + 1 moves one element
  • C passes by value; pass addresses (pointers) to change the caller’s variables
  • Arrays are passed as pointers, so changes made inside the function are visible outside
  • Recursion: base case + smaller sub-problem; code after the recursive call runs in reverse order
  • A static local variable is initialized once and keeps its value between calls

OS Basics: Processes, CPU Scheduling, Paging, Deadlock, File Systems

The one-machine xerox shop near college: One xerox machine, many students: who goes next is CPU scheduling.

  • TAT = completion time − arrival time; WT = TAT − burst time
  • FCFS: non-preemptive, convoy effect; SJF: minimum average WT, may starve long jobs
  • SRTF = preemptive SJF; starvation in priority scheduling is fixed by aging
  • Round Robin: preemptive with a time quantum; a very large quantum makes it behave like FCFS
  • Process states: new, ready, running, waiting, terminated; the PCB stores each process’s details
  • Paging: page size = frame size; page-table entries = 2logical address bits ÷ page size
  • Deadlock conditions: mutual exclusion, hold and wait, no preemption, circular wait (all four needed)
  • FIFO page replacement can show Belady’s anomaly (more frames, yet more faults)

Computer Awareness: Internet, Email, Web and Software

Posting a wedding patrika from Pune to the village: Roads are the internet, the letter is email, name@village is username@domain, and the postman is DNS.

  • Internet = network of networks (TCP/IP); WWW = web pages on it, invented by Tim Berners-Lee (1989); email and FTP are other internet services
  • URL = protocol://domain/path, e.g. https://www.mahacet.org/exams/mca.html; HTTP port 80, HTTPS = HTTP + SSL/TLS, port 443
  • HTML is a markup language of tags, not a programming language; any browser on any OS can display an HTML page
  • Email address = user name @ domain name; SMTP sends, POP3 downloads to one device, IMAP keeps mail on the server; CC visible, BCC hidden
  • Software: system (OS, drivers, translators) · application (Word, Excel, browser) · utility (antivirus, compression, backup); firmware sits in ROM
  • PAN < LAN < MAN < WAN; IPv4 = 32 bits (four numbers 0–255), IPv6 = 128 bits; DNS: name → IP; modem: digital ↔ analog; router: joins networks
  • Virus needs a host file, worm spreads by itself, Trojan hides in useful-looking software; firewall filters traffic; phishing = fake message fishing for passwords/OTP
  • Size order micro < mini < mainframe < super; generations: vacuum tube → transistor → IC → microprocessor (VLSI) → AI (ULSI)