Probability, Statistics & Data Analytics — Master Guide (Placement-Ready)

Introduction — How to use this page

This file is a complete revision vault for quantitative aptitude and analytics interviews. It's arranged to move you from fundamentals to advanced analytics problems. Each section has quick formulas, real-world context, and practice problems with revealable answers. Use the Formula-Tweak section to unlock solving many problems by small substitutions.

Workflow I recommend: timed drills → concept check → formula-tweak practice → mock DI caselets → interview Q&A recap.

I. Number Systems & Fundamentals

Key Concepts

Number classification: natural, whole, integer, rational (p/q), irrational (π, √2), prime numbers (no divisors ≠ 1,n).
Primality checks: trial division up to √n; deterministic Miller-Rabin bases for 32/64-bit integers if needed.
Divisibility shortcuts: last digit (2,5), last two (4), digit sum (3,9), alternating sum (11).
Modular arithmetic: use congruences; for large exponents reduce via Euler φ(n) or Carmichael λ(n) when coprime.
LCM/HCF: gcd by Euclid; lcm(a,b) = a*b / gcd(a,b). Extend to multiple numbers iteratively.

Real-world uses

Remainder tricks help in hashing and sharding logic; gcd/lcm appear in scheduling and cyclic pattern detection.

Quick Example

Find remainder when 3^100 is divided by 5.
3 ≡ 3 (mod 5). Powers cycle: 3,4,2,1 (period 4). 100 mod 4 = 0 -> remainder 1.

II. Core Arithmetic

Percentages

Factor form: new = old × (1 ± p/100). Multiply successive factors for multi-step changes. Use multiplicative inverse 1/(1+p) to reverse a percentage increase.

Profit & Loss

Profit% = (SP - CP)/CP × 100. When discount applied on MP: SP = MP × (1 - discount%). Markup vs margin distinction is crucial in business problems.

Ratio & Proportion

If A:B = x:y and total T, A = T × x/(x+y). Ratios used heavily in allocation and revenue-share problems.

Averages

Weighted average: (Σ w_i x_i)/Σ w_i. Replacement problems: new_avg = (old_sum - removed + added)/n — memorize and apply fast.

Interest

Simple interest: SI = P × R × T / 100. Compound: A = P×(1 + r/n)^(n×t). Continuous: A = P×e^(r t). Financial approximations often use continuous compounding for limits.

III. Time, Motion & Work

Time & Work

Work = rate × time. If A takes a days, rate_A = 1/a. Combine rates by sum. For partial work or leaving mid-job apply time-weighted contributions.

Speed & Distance

Distance = speed × time. Average speed:

Equal distances: Harmonic mean = n / Σ(1/v_i).
Equal times: Arithmetic mean.

Trains & Boats

Relative speed (opposite) = v1 + v2. Crossing time = (L_train + L_object) / relative_speed. For boats: downstream = u + v, upstream = u - v (where u = boat speed in still water, v = current).

IV. Modern Mathematics

Permutations & Combinations

nPr = n!/(n-r)!, nCr = n!/(r!(n-r)!). Use stars-and-bars: combinations with repetition C(n + r - 1, r). Permutation with repeated items: n! / (k1! k2! ...).

Probability

Basic rules:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Conditional: P(A|B) = P(A ∩ B) / P(B).
Independence: P(A ∩ B) = P(A) P(B).
Complement: P(at least one) = 1 - P(none).

Set Theory & Venn

Inclusion-exclusion for n sets; visualize with Venn diagrams for interviews.

Real-world note

Probability underpins A/B testing, conversion rate estimation, and risk calculations — you'll see these in analytics interviews and product/data roles.

V. Geometry, Algebra & Mensuration

Key formulas

Area triangle = ½ × base × height.
Circle area = π r², circumference = 2π r.
Rect/Parallelogram area rules; volume cylinder = π r² h, sphere = 4/3 π r³.
Coordinate geometry: distance formula, slope, line eqn y = mx + c.

Algebra & Logs

Solve linear & quadratic equations. Log properties: log_b(x y) = log_b x + log_b y; log_b(x^k) = k log_b x. These appear in growth and doubling-time problems.

VI. Data Interpretation & Analytics Skills

Quick DI strategy

Read the question first — know which numbers you need.
Extract base values and transform to percentages for comparison.
Use ratios and differences; prefer approximate arithmetic to narrow answer quickly.
Double-check edge cases (zeros, missing data) and whether questions ask relative or absolute changes.

Charts & Common traps

Watch axes (absolute vs indexed), stacked charts (composition vs absolute), and moving averages that hide volatility. In caselets ask for contribution to change — compute absolute delta and divide by global delta.

Mini-caselet (practice)

Company X: Region A revenue grew from 200 → 260, Region B from 500 → 550. Which contributed more to company revenue increase?

Compute absolute deltas: A = +60, B = +50 → A contributed more despite B being larger base. For percent growth, A = 30%, B = 10%.

Formula-Tweak Patterns — One core formula, many problems

Memorize base formulas; then learn 2–3 small tweaks for families of problems. Below are canonical patterns with examples and 'tweak' variants you will use in interviews.

A — Work / Rate

Base: Work = Rate × Time. If task = 1 unit, time = 1 / (sum rates).

Tweak 1: Worker leaves at time t: work_done = t × (sum rates) + remaining by others.

Tweak 2: Different jobs/time-shared — convert to 'per unit' rates and aggregate.

Example: A=12 days (rate 1/12), B=20 days (1/20). If A works x days with B, then leaves; B finishes in 6 days:
1 - x*(1/12+1/20) = 6*(1/20) -> solve for x

B — Percentage Compound

Base: Net factor = Π(1 + p_i). For repeated increases/decreases, multiply properly. To reverse a p% increase use divide by (1+p/100).

C — Probability via Complement

Base: P(at least one) = 1 - Π (1 - p_i) for independent small events. For dependent events, use inclusion-exclusion.

D — Mean/Replacement

Base: new_avg = (old_sum - removed + added) / n. Tweak for sequence of replacements — apply iteratively or form direct algebraic expression.

E — Permutations with constraints

Use complementary counting (total - bad) or place constrained items first and arrange rest. For repeated letters use factorial division.

Placement-Style Problems — Beginner → Advanced (50+ included)

Attempt without looking. Problems labeled B / M / A. Use reveal buttons to check answers.

Total Questions: 200

1. (B) Number Theory — remainder powers

Find remainder when 7^222 is divided by 8.

7 ≡ -1 (mod 8). (-1)^222 = 1 → remainder 1.

2. (B) Percentage — successive change

Price increases by 10% then decreases by 10%. Net change?

Net factor = 1.1 × 0.9 = 0.99 → net -1%.

3. (M) Ratio & Partnership

A:B invest 3:2. A adds an amount equal to his investment after 6 months. Profit = 5000 after 12 months. Find A's share.

A time-weighted = 3*6 + 6*(3+3?) — correct calculation: A initial=3 for 12 months, but actually A adds 3 after 6 months, so A's capital-months = 3*6 + 6*(3+3?) Wait: simpler: months: first 6 -> A=3,B=2; next 6 -> A=6 (3+3),B=2. So A = 3*6 + 6*6 = 18+36=54; B = 2*12=24. Ratio 54:24=9:4 -> A's share = 9/13*5000 ≈ 3461.54.

4. (M) Work/Rate — leaving early

A finishes in 12 days, B in 20. They work together x days, A leaves, B finishes 6 days later. Find x.

Rates: 1/12 + 1/20 = 2/15. After x days, remaining = 1 - x*(2/15) = B's 6 days = 6*(1/20)=3/10. Solve: x*(2/15)=7/10 -> x=5.25 days.

5. (A) Permutation with repeated letters

Number of arrangements of \"ANALYTICS\" where A's are not together?

Total=9!/2!=181440; Treat A together: 8!*2=80640; Non-adjacent = 181440-80640=100800.

6. (A) Conditional Probability (Bayes basics)

Two boxes: B1 {2R,3B}, B2 {1R,4B}. Pick box randomly, draw ball. P(red)? P(box1 | red)?

P(red)=0.5*(2/5)+0.5*(1/5)=3/10. P(B1|red)=(0.5*2/5)/(3/10)= (1/5)/(3/10)=2/3.

7. (B) DI quick — quarter growth

Sales Q1..Q4: 100,120,150,130. Which quarter had highest % growth?

Q1->Q2:20%; Q2->Q3:25%; Q3->Q4:-13.33%. Highest = Q2->Q3 (25%).

8. (B) Mean/Median/Mode

Dataset [2,3,3,5,7,9,11]. Find mean, median, mode.

Mean = 40/7 ≈5.714; Median = 5; Mode = 3.

9. (M) Inclusion-exclusion

Class of 100: tea 70, coffee 60, juice 50; tea&coffee 40, tea&juice 30, coffee&juice 20; all three 10. How many at least one?

Sum singles 180 - pairs 90 + triple 10 = 100 → all 100.

10. (B) Replacement average

Average of 50 numbers = 60. Replace 70 with 40. New average?

Old sum=3000. New sum=2970 => avg=59.4.

11. (M) Bayes in practice

Disease prevalence 0.5%; sensitivity 99%; specificity 98%. P(disease|positive)?

P=0.005*0.99/(0.005*0.99+0.995*0.02)=0.00495/0.02485≈0.199 ≈19.9%.

12. (A) WMA — Analytics

3-day WMA with weights 0.5,0.3,0.2 for values [100,110,120] (most recent 120).

WMA = 0.5*120 + 0.3*110 + 0.2*100 = 113.

13. (M) DI Caselet — Contribution to growth

Given regional numbers (old & new), how to find region with highest % growth and top contributor to revenue increase? Outline steps.

Compute % change per region and absolute deltas; highest % = max((new-old)/old); top contributor = max(abs_delta); contribution% = abs_delta / total_delta.

14. (M) Geometric-series quick

Sum S = a(1 - r^n)/(1 - r) for r ≠ 1. If a=3, r=2, n=4, what's S?

S = 3(1 - 16)/(1 - 2) = 3(-15)/(-1) = 45.

15. (A) Probability — coupon collector (approx)

Expected draws to collect n coupons ≈ n * H_n (Harmonic number). For n=5, expected ≈ 5*(1+1/2+...+1/5)=5*2.2833=11.4165.

≈11.4165 draws.

16. (A) Chi-squared quick concept

Explain when to use chi-squared test vs t-test in analytics interviews.

Chi-sq: categorical variables independence/goodness-of-fit. t-test: compare means of continuous variables (small sample/normal approx).

17. (M) Sampling — Central Limit Theorem

State CLT and explain relevance in product analytics for sample means.

CLT: sample mean ~ Normal(μ, σ/√n) for sufficiently large n. Justifies normal-based CI and hypothesis tests for means.

18. (A) Hypothesis testing quick

Define p-value and Type I/II errors concisely.

p-value = prob of observing data at least as extreme assuming H0 true. Type I error = false positive (α); Type II = false negative (β).

19. (A) Correlation vs Causation

Short explanation and how to investigate causation in product data.

Correlation measures linear association; causation needs experiment (A/B test) or causal inference (instrumental variables, diff-in-diff).

20. (M) Expected value — multi-stage

Roll two dice. What is expected value of maximum face?

Compute P(max ≤ k)= (k/6)^2 -> P(max = k) = P(max ≤ k) - P(max ≤ k-1). EV=Σ k*P(max=k). Calculation yields EV ≈ 4.4722.

21. (M) Logarithm solving

Solve log_2(x) + log_2(x-2) = 3.

log_2(x(x-2))=3 -> x^2 - 2x = 8 -> x^2 -2x -8=0 -> (x-4)(x+2)=0 -> x=4 (x>2 only) -> 4.

22. (A) Linear equation practical

If mean of n numbers is m. Adding a new number x increases mean to m+δ. Find x in terms of m,n,δ.

Old sum = nm. New sum = (n+1)(m+δ) = nm + x -> x = (n+1)(m+δ) - nm = m + (n+1)δ.

23. (M) Confidence interval quick

Given sample mean 50, σ=10, n=100. 95% CI for mean?

SE=σ/√n=10/10=1. 95% CI = 50 ± 1.96*1 => [48.04,51.96].

24. (A) Regression basics

Explain R² and overfitting in simple terms for interviews.

R² = fraction of variance explained. Overfitting: model fits noise — test on unseen data and prefer simpler models if generalization suffers.

25. (M) Transformations — log rules in modeling

When to log-transform a target variable in regression?

When distribution is skewed, multiplicative effects present, or to stabilize variance. Interpret coefficients accordingly (elasticities).

26. (A) Sampling bias check

List three types of sampling bias and quick fixes.

Selection bias (use randomization), survivorship bias (include dropped/failed units), non-response bias (follow up or weight data).

27. (M) A/B test power

What is statistical power and how do you increase it?

Power = 1-β, probability to detect true effect. Increase by larger sample, larger effect size, lower variance, or higher α.

28. (A) Correlation — Spearman vs Pearson

When to use Spearman rank instead of Pearson?

Use Spearman for monotonic relationships or ordinal data (nonlinear). Pearson measures linear correlation.

29. (A) Poisson process quick

If average 5 events/hour, probability of 2 events in half hour?

λ_half = 5 × 0.5 = 2.5. P(2) = e^-2.5 * 2.5^2 / 2! ≈ 0.2565.

30. (M) Chi-square example

Given contingency table, outline chi-square test steps (no numeric calc needed).

Compute expected counts under independence, sum (obs-exp)^2/exp, compare to χ²(df) critical value, conclude reject/fail to reject.

31. (A) Bootstrapping quick

Explain bootstrap resampling and when you'd use it.

Bootstrap: sample with replacement to estimate sampling distribution empirically; use when analytic CI hard or small sample assumptions fail.

32. (M) Time series — stationarity

What is stationarity and why important for ARIMA?

Stationarity: mean/variance constant over time; ARIMA assumes stationarity (use differencing to induce).

33. (A) ROC / AUC intuition

What does ROC curve show and why AUC matters?

ROC: TPR vs FPR tradeoff across thresholds. AUC = probability classifier ranks random positive above random negative. Useful for unbalanced data.

34. (A) Outliers handling quick

List 4 ways to handle outliers in analytics.

Remove if data error, winsorize, transform (log), or use robust models (median-based, tree-based).

35. (M) PCA intuition

Explain principal component analysis in two sentences.

PCA finds orthogonal directions capturing max variance; reduces dimensionality while preserving variance (linear projection).

36. (M) Rolling averages application

Why use rolling averages in dashboards? One downside?

Smooths noise and reveals trends; downside: it lags and can hide sudden changes.

37. (A) LTV / CAC quick definition

Quick formula for customer lifetime value (simple) and acquisition cost.

Simplest LTV = ARPU × Gross margin % × Avg customer lifetime. CAC = total marketing cost / # customers acquired.

38. (A) Transform check — heteroscedasticity

What is heteroscedasticity and quick fix?

Non-constant variance of residuals. Fix: transform target (log), use weighted least squares or robust SEs.

39. (M) Bootstrap CI example

Outline steps to create bootstrap CI for median from sample of 50.

Resample 10k bootstrap samples with replacement, compute median each time, take 2.5/97.5 percentiles for 95% CI.

40. (A) Business metric sanity check

You see daily active users jump 3× overnight — list immediate diagnostic steps.

Check instrumentation, logs, sampling, GA filters, timezone shifts, bot traffic, deployment changes, A/B tests, data pipeline lag or duplicates.

41. (M) A/B test sample size calc

High-level formula ingredients for sample size to detect δ difference with power 80% and α=5%.

n ≈ (Z_{1-α/2} √(2p(1-p)) + Z_{power} √(p1(1-p1)+p2(1-p2)))^2 / δ^2 (approx). Use online calculators in practice.

42. (A) Entropy quick

Define Shannon entropy and give one analytics use-case.

Entropy H = -Σ p_i log p_i. Use-case: measure uncertainty in user behavior or decision tree splits.

43. (M) Imputation strategies

List 4 imputation strategies and when to use each.

Mean/median for MCAR; model-based imputation (regression) for MAR; KNN for local similarity; indicator for missingness as feature.

44. (A) Log-odds interpretation

In logistic regression, interpret coefficient β of a predictor.

Exp(β) gives multiplicative change in odds for 1 unit increase of predictor (odds ratio).

45. (M) Funnel conversion check

If funnel conversion drops at step 3, list experimental checks and quick analyses.

Check instrumentation, segment by device/OS, A/B tests, deployments, UX changes, backend errors, load times; compute diagnostics and run cohort analysis.

46. (A) Weighted average vs median

When is median preferred to mean?

Median preferred for skewed distributions or when outliers distort mean (e.g., income data).

47. (A) Quick combinatorics

Number of ways to choose 3 people from 10 when order doesn't matter?

C(10,3)=120.

48. (M) Jensen's inequality intuition

Explain Jensen's inequality and give a small analytic example.

For convex φ, φ(E[X]) ≤ E[φ(X)]. Example: variance relation or log concavity use cases; shows transformation bias.

49. (A) Percentile quick

Definition: 90th percentile in a sorted list.

Value such that 90% of observations are <= that value (interpolate if needed).

50. (A) Final mock: end-to-end case

You have weekly active users drop 15% after a release. Outline investigative steps and metrics you'd present to PM in 30 minutes.

Immediate: segment by cohort/device/region, check releases/rollbacks, check instrumentation and error rates, A/B exposure, retention curves, funnel step drop, and suggest rollback or hotfix if impact large. Present absolute and relative changes, CI for effect, and mitigation plan.

50. (A) Final mock: end-to-end case

You have weekly active users drop 15% after a release. Outline investigative steps and metrics you'd present to PM in 30 minutes.

Immediate: segment by cohort/device/region, check releases/rollbacks, check instrumentation and error rates, A/B exposure, retention curves, funnel step drop, and suggest rollback or hotfix if impact large. Present absolute and relative changes, CI for effect, and mitigation plan.

Number Systems & Fundamentals

51. (B) HCF Application [TCS, Infosys]

Find the largest number that divides 615 and 963 leaving remainders 6 and 7 respectively.

Subtract remainders: 615-6=609, 963-7=956. HCF(609,956) = HCF(609,347) = HCF(262,347) = HCF(262,85) = HCF(7,85) = 1... Actually: 609=3×7×29, 956=4×239. Direct: HCF(609,956)=29. Answer: 29.

52. (M) Prime Factorization [Wipro, Accenture]

How many prime factors does 2520 have? What is the sum of distinct prime factors?

2520 = 2³×3²×5×7. Prime factors (with repetition): 8 total. Distinct primes: {2,3,5,7}. Sum = 17.

53. (B) Divisibility Rule [Cognizant, Capgemini]

Find the smallest 4-digit number divisible by 88.

1000 ÷ 88 = 11.36... Next integer: 12. 12×88 = 1056.

54. (M) LCM-HCF Relationship [Amazon, Microsoft]

Two numbers have HCF 16 and LCM 160. If one number is 32, find the other.

Product = HCF × LCM. 32 × other = 16 × 160 = 2560. Other = 2560/32 = 80.

55. (A) Modular Exponentiation [Google, Adobe]

Find the last two digits of 17^256.

Last 2 digits = 17^256 mod 100. φ(100)=40. 256 mod 40 = 16. Compute 17^16 mod 100 by successive squaring: 17²=289≡89, 17⁴≡89²=7921≡21, 17⁸≡21²=441≡41, 17^16≡41²=1681≡81. Answer: 81.

56. (B) Unit Digit Pattern [TCS Digital, Infosys]

What is the unit digit of 3^47 × 7^58 × 13^72?

3 cycles {3,9,7,1} period 4: 47 mod 4 = 3 → 7. 7 cycles {7,9,3,1}: 58 mod 4 = 2 → 9. 13≡3 cycles: 72 mod 4 = 0 → 1. Product: 7×9×1=63 → unit digit 3.

57. (M) Remainder Theorem [Wipro, Tech Mahindra]

Find remainder when (7^100 + 13^100) is divided by 10.

7^100 mod 10: cycle {7,9,3,1}, 100 mod 4 = 0 → 1. 13^100 mod 10: 3^100, same cycle → 1. Sum = 1+1 = 2.

58. (A) Euler's Totient [Adobe, Oracle]

Calculate φ(72) where φ is Euler's totient function.

72 = 2³×3². φ(72) = 72(1-1/2)(1-1/3) = 72×(1/2)×(2/3) = 24.

59. (B) Co-prime Numbers [Cognizant, Capgemini]

How many numbers between 1 and 50 are co-prime to 50?

50 = 2×5². φ(50) = 50(1-1/2)(1-1/5) = 50×(1/2)×(4/5) = 20.

60. (M) Perfect Numbers [TCS Ninja, Infosys]

Is 28 a perfect number? Verify by showing sum of its proper divisors.

Divisors of 28: 1,2,4,7,14. Sum = 1+2+4+7+14 = 28. Yes, it's perfect.

61. (A) Chinese Remainder Theorem [Amazon, Microsoft]

Find the smallest positive integer x such that x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7).

Use CRT. M=3×5×7=105. M₁=35, M₂=21, M₃=15. Find inverses: 35⁻¹≡2(mod 3), 21⁻¹≡1(mod 5), 15⁻¹≡1(mod 7). x = (2×35×2 + 3×21×1 + 2×15×1) mod 105 = (140+63+30) mod 105 = 233 mod 105 = 23.

62. (B) Sum of Digits [Wipro, Accenture]

Find the sum of digits of 10^20 - 1.

10^20 - 1 = 99999...9 (20 nines). Sum = 20×9 = 180.

63. (M) Base Conversion [Google, Adobe]

Convert (ABC)₁₆ to decimal.

A=10, B=11, C=12. 10×16² + 11×16 + 12 = 2560 + 176 + 12 = 2748.

64. (A) Factorial Trailing Zeros [Amazon, Microsoft]

How many trailing zeros does 100! have?

Count factors of 5: ⌊100/5⌋ + ⌊100/25⌋ + ⌊100/125⌋ = 20 + 4 + 0 = 24.

65. (B) Arithmetic Progression [TCS, Cognizant]

Find the 50th term of AP: 3, 7, 11, 15, ...

a=3, d=4. a₅₀ = a + 49d = 3 + 49×4 = 199.

66. (M) Geometric Progression [Infosys, Wipro]

Sum of infinite GP: 4 + 2 + 1 + 0.5 + ...

a=4, r=0.5. S∞ = a/(1-r) = 4/(1-0.5) = 8.

67. (A) Diophantine Equation [Adobe, Oracle]

Find positive integer solutions for 7x + 12y = 100.

Try values: x=4, y=6 works (28+72=100). General: x=4-12t, y=6+7t. Positive: t=0 gives (4,6); t=-1 invalid. Only (4,6).

68. (B) Digital Root [Capgemini, Tech Mahindra]

Find the digital root of 9875.

9+8+7+5=29 → 2+9=11 → 1+1=2. Digital root = 2.

69. (M) Catalan Numbers [Google, Amazon]

Calculate the 4th Catalan number (C₃).

Cₙ = (2n)!/(n+1)!n!. C₃ = 6!/(4!×3!) = 720/(24×6) = 5.

70. (A) Wilson's Theorem [Microsoft, Adobe]

Verify Wilson's theorem for p=7 and use it to find 6! mod 7.

Wilson: (p-1)! ≡ -1 (mod p) for prime p. 6! ≡ -1 (mod 7) ≡ 6 (mod 7). Verify: 720 = 7×102 + 6.

Percentages & Core Arithmetic

71. (B) Basic Percentage [TCS, Infosys]

If 25% of a number is 60, find 40% of that number.

Number = 60/(0.25) = 240. 40% = 0.4×240 = 96.

72. (M) Successive Percentage Change [Wipro, Cognizant]

Population increases by 20% in year 1 and decreases by 20% in year 2. Net change?

Net = 1.2×0.8 = 0.96 → -4%.

73. (A) Reverse Percentage [Amazon, Google]

After increasing price by 25%, the new price is ₹500. What was the original price?

Original × 1.25 = 500. Original = 500/1.25 = ₹400.

74. (B) Profit & Loss Basic [Accenture, Capgemini]

CP = ₹800, SP = ₹1000. Find profit percentage.

Profit = 1000-800 = 200. Profit% = 200/800 × 100 = 25%.

75. (M) Discount Problem [TCS Digital, Infosys]

MP = ₹2000. After 20% discount and 10% tax, find final price.

After discount: 2000×0.8 = 1600. After tax: 1600×1.1 = ₹1760.

76. (A) Markup vs Margin [Microsoft, Adobe]

If profit margin is 25% on SP, what is the markup on CP?

Margin 25% on SP means profit/SP = 0.25. SP = CP/(1-0.25) = CP/0.75 = 1.333CP. Markup = 33.33%.

77. (B) False Weight [Wipro, Tech Mahindra]

Shopkeeper uses 800g weight instead of 1kg. Find profit%.

He gives 800g for price of 1000g. Profit = 200g on 800g = 25%.

78. (M) Two Successive Discounts [Cognizant, Capgemini]

Find single equivalent discount for 20% and 30% successive discounts.

Net = 0.8×0.7 = 0.56 = 44% discount.

79. (A) Break-even Analysis [Amazon, Oracle]

Fixed cost ₹50,000. Variable cost per unit ₹20. SP ₹50. Find break-even quantity.

Contribution per unit = 50-20 = 30. Break-even = 50000/30 ≈ 1667 units.

80. (B) Partnership Profit [TCS, Infosys]

A invests ₹3000 for 12 months, B invests ₹4000 for 8 months. Profit ₹5600. Find A's share.

Capital-months: A=36000, B=32000. Ratio 36:32 = 9:8. A's share = 9/17 × 5600 = ₹2964.71.

81. (M) Ratio Change [Wipro, Accenture]

Ratio of milk to water is 5:3. If 8L water is added, ratio becomes 5:5. Find initial milk quantity.

Let milk=5x, water=3x. After adding: 5x:(3x+8) = 5:5 → 5x = 3x+8 → x=4. Milk = 20L.

82. (A) Weighted Average Revenue [Google, Microsoft]

Product A: 1000 units at ₹50, Product B: 500 units at ₹80. Find weighted avg price.

WA = (1000×50 + 500×80)/(1000+500) = (50000+40000)/1500 = ₹60.

83. (B) Simple Interest [TCS, Cognizant]

Principal ₹5000, Rate 8% p.a., Time 3 years. Find SI.

SI = 5000×8×3/100 = ₹1200.

84. (M) Compound Interest [Infosys, Wipro]

₹10,000 at 10% CI compounded annually for 2 years. Find amount.

A = 10000(1.1)² = 10000×1.21 = ₹12,100.

85. (A) CI vs SI Difference [Amazon, Adobe]

For 2 years at 10% p.a., difference between CI and SI is ₹100. Find principal.

Diff = P(r/100)² = P×0.01 = 100 → P = ₹10,000.

86. (B) Mixture & Alligation [Capgemini, Tech Mahindra]

Mix milk at ₹50/L with water. To get ₹40/L mixture (water free), what ratio?

Alligation: milk(50) water(0) mean(40). Ratio = (40-0):(50-40) = 40:10 = 4:1.

87. (M) Salary Increment [TCS Digital, Infosys]

Salary increased by 15% to ₹69,000. What was original salary?

Original × 1.15 = 69000. Original = 69000/1.15 = ₹60,000.

88. (A) Tax Calculation [Oracle, Microsoft]

Income ₹10L. Tax: 0% up to ₹2.5L, 5% on next ₹2.5L, 10% above. Find total tax.

Tax = 0 + 0.05×250000 + 0.1×500000 = 0 + 12500 + 50000 = ₹62,500.

89. (B) Average Replacement [Wipro, Cognizant]

Average of 10 numbers is 15. Replace one number 20 with 35. New average?

Old sum = 150. New sum = 150-20+35 = 165. New avg = 16.5.

90. (M) Ratio Adjustment — Practical

A mixture contains milk and water in ratio 7:5. How much water must be added to 60 L mixture to make ratio 7:8?

Milk = 7/12 × 60 = 35 L. Let water added = x. New water = 25 + x. Ratio: 35:(25+x) = 7:8 → 35×8 = 7(25+x) → 280 = 175 + 7x → x = 15 L.

91. (B) Simple Algebra — Reverse Operation

If 4x + 12 = 36, find x.

4x = 24 → x = 6.

92. (M) Boats & Streams

A boat takes 3 hours upstream and 2 hours downstream to travel the same distance. Speed of stream is 2 km/h. Find ratio of boat speed to stream speed.

Let distance = D. Upstream speed = D/3; downstream = D/2. Let boat speed = b. Then b-2 = D/3, b+2 = D/2. Divide: (b+2)/(b-2)=3/2 → 2b+4=3b-6 → b=10. Ratio = 10:2 = 5:1.

93. (A) Probability — Defective Items

A batch has 5 defective and 45 good items. Two items picked without replacement. Probability both are defective?

P = (5/50) × (4/49) = 20/2450 = 2/245.

94. (M) Mixture Increase Problem

A 40 L solution has 25% acid. How much pure acid must be added to make it 40% acid?

Acid initially = 10 L. Let x be added. Final %: (10+x)/(40+x)=0.4 → 10+x=16+0.4x → 0.6x=6 → x=10 L.

95. (B) Speed-Time

A man walks at 6 km/h. How much time to cover 3 km?

Time = 3/6 = 0.5 h = 30 min.

96. (A) Conditional Expectation — Interview Level

A fair die is rolled. Find E[X | X is even].

Even outcomes: {2,4,6}. Mean = (2+4+6)/3 = 4.

97. (M) Averages — Weighted Increase

Class A avg = 60 (40 students), Class B avg = 70 (20 students). Combined avg?

Total = 40×60 + 20×70 = 2400 + 1400 = 3800. Combined avg = 3800/60 = 63.33.

98. (A) Prime Gaps — Analytical

Find the number of primes between 100 and 150.

Primes: 101,103,107,109,113,127,131,137,139,149 → 10 primes.

99. (M) Pipes & Cisterns

Pipe A fills tank in 12 h, Pipe B in 18 h. Both work for 4 h, then A is closed. Time to finish?

Rate AB = 1/12 + 1/18 = 5/36. Work done in 4h = 20/36 = 5/9. Remaining = 4/9. B alone: (4/9)/(1/18) = 8 h.

100. (A) Statistics — Variance Scaling

If random variable X has variance 9, what is Var(3X + 4)?

Var(3X+4) = 3² Var(X) = 9 × 9 = 81.

101. (M) Percentage Error

A value is measured as 198 instead of 200. Find % error.

% error = |200 - 198| / 200 × 100 = 1%.

102. (A) Probability — Drawing Cards

From a deck of 52, find probability that two cards drawn are both Kings.

P = (4/52) × (3/51) = 12/2652 = 1/221.

103. (M) Simple Interest Relationship

SI on ₹2000 for 2 years at r% is ₹400. Find r.

SI = PRT/100 → 400 = 2000×r×2/100 → r = 10%.

104. (A) Expected Value — Non-uniform Die

A biased die shows 6 with probability 0.4; remaining faces equally share remaining probability. Find expected value.

Other faces probability = 0.6/5 = 0.12. EV = 1*0.12 + 2*0.12 + 3*0.12 + 4*0.12 + 5*0.12 + 6*0.4 = 0.12(15) + 2.4 = 1.8 + 2.4 = 4.2.

105. (B) Linear Equation Simple

Solve 3x - 9 = 0.

3x = 9 → x = 3.

106. (A) Standard Deviation — Scaling

If SD of X is 5, find SD of 4X + 3.

SD = |4| × 5 = 20.

107. (M) Time & Work — Sharing Work

A does a job in 10 days, B in 15 days. They work 3 days together, then A leaves. Days for B to finish?

Work done in 3 days = 3(1/10 + 1/15)=3(1/6)=1/2. Remaining = 1/2. B alone = (1/2)/(1/15)=7.5 days.

108. (A) Combinatorics — Team Forming

Choose a team of 4 from 8 men and 5 women if at least 1 woman must be included.

Total C(13,4)=715. All-men teams = C(8,4)=70. Valid = 715 - 70 = 645.

109. (M) Ratio Scaling

The ratio of boys:girls is 3:4. If 20 more boys join, ratio becomes 4:5. Find initial girls.

Let boys=3x, girls=4x. New: (3x+20)/4x = 4/5 → 5(3x+20)=16x → 15x+100=16x → x=100. Girls=400.

110. (A) Probability — Coin + Die Hybrid

A coin is tossed, then a die rolled. Find probability of (Head AND even number).

P = (1/2) × (3/6) = 1/4.

111. (M) Alligation — Two Milk Solutions

Mix 30% milk solution with 70% solution to get 50% solution. Ratio?

Alligation: (70-50):(50-30)=20:20=1:1.

112. (A) Expected Value — Random Tip Model

A restaurant tip is 5% with prob 0.6, 10% with prob 0.3, 20% with prob 0.1. Expected tip%?

EV = 0.6(5)+0.3(10)+0.1(20)=3+3+2=8%.

113. (B) Simple Geometry

Perimeter of a square is 40 cm. Find area.

Side = 40/4 = 10 → area = 100.

114. (A) Probability — Bayes Simple

Machine A produces 30% of items, defect rate 2%. Machine B produces 70%, defect rate 1%. If item is defective, probability it came from A?

P(A|D)=0.3×0.02/(0.3×0.02+0.7×0.01)=0.006/(0.006+0.007)=0.462.

115. (M) Train Crossing

A 200 m train crosses a pole in 10 sec. Find speed km/h.

Speed = 200/10 = 20 m/s = 72 km/h.

116. (A) Regression — Interpretation

If regression slope is 0.7, interpret it in simple terms.

For each 1-unit increase in X, Y is expected to increase by 0.7 units on average.

117. (M) Time Zones — Analytics

User logs from Region A (UTC+2) and Region B (UTC−3). If events are recorded at 15:00 local in both regions, what is the UTC difference in timestamps?

UTC times: A: 13:00; B: 18:00. Difference = 5 hours.

118. (A) Entropy — Binary Variable

Compute entropy for p=0.8, q=0.2.

H = -0.8 log2(0.8) - 0.2 log2(0.2) ≈ 0.721.

119. (M) Quadratic Equation Roots

Find roots of x² - 7x + 12 = 0.

(x-3)(x-4)=0 → x=3,4.

120. (A) Decision Trees — Gini

For a node with class split (40 positive, 60 negative), compute Gini impurity.

p=0.4 → G=1-(p²+(1-p)²)=1-(0.16+0.36)=0.48.

121. (B) Series — Identify Pattern

Series: 3, 6, 12, 24, 48, ?. Find next.

Multiply by 2 → 96.

122. (A) Optimization — Cost Function

If cost C = x² + 10x, find x that minimizes C.

Vertex at -b/2a = -10/2 = -5.

123. (M) Profit & Loss Advanced

An item sold at 20% loss. If sold for ₹240 more, it becomes 10% profit. Find CP.

Let CP=x. 0.8x +240 =1.1x → 240=0.3x → x=800.

124. (A) Normal Distribution

Z ~ N(0,1). Find P(Z > 1.64).

≈ 0.05.

125. (M) Pipes & Cisterns 2

A fills in 8 h, B in 10 h, and C empties in 20 h. All operate together. Time to fill?

Rate = 1/8+1/10−1/20=1/5. Time=5 h.

126. (A) Correlation Significance

If correlation r = 0.9, what does it imply?

Strong positive linear relationship.

127. (M) Inequalities

Solve 5x - 3 < 12.

5x < 15 → x < 3.

128. (A) Sample Size for Proportion

Estimate sample size for margin ±5% with 95% confidence, p≈0.5.

n ≈ (1.96² × 0.25)/0.05² ≈ 384.

129. (M) Distance-Speed

A car travels 60 km at 40 km/h and returns 60 km at 60 km/h. Average speed?

Avg = total distance / total time =120/(1.5+1)=120/2.5=48 km/h.

130. (A) Probability — Continuous

X uniform on [0,10]. Find P(3 < X < 7).

Length = 4/10 = 0.4.

131. (M) Logarithm Simplification

Simplify log(a²b³) - log(ab).

= log(a²b³/(ab)) = log(ab²).

132. (A) Gradient Descent Intuition

What happens if learning rate is too high?

Model overshoots minima and may diverge.

133. (B) Simple Triangles

Find area of triangle with base 10, height 6.

Area = 1/2 × 10 × 6 = 30.

134. (A) Confidence Intervals — Interpretation

What does a 95% confidence interval for mean imply?

If repeated many samples, 95% of constructed CIs contain true mean.

135. (M) Percentage Change

Value rises from 400 to 500. % increase?

100/400 ×100 = 25%.

136. (A) PCA — Variance Capture

If first 2 PCs explain 85% of variance, what does this mean?

The 2D projection retains 85% of dataset variability.

137. (M) Venn Diagram Basic

If |A|=40, |B|=30, |A∩B|=10, find |A∪B|.

40+30−10=60.

138. (A) Random Variables — Transformation

If X~N(10,4), find distribution of Y = 3X + 2.

Y ~ N(3*10+2, (3²)*4) = N(32,36).

139. (M) Work Efficiency

A completes work in 6 days, B in 9. How many days if B works twice as efficiently as before?

Rates: 1/6 + 2*(1/9)=1/6+2/9= (3+4)/18 =7/18. Time=18/7.

140. (A) Optimization — L1 vs L2

Why does L1 create sparsity while L2 doesn’t?

L1 penalty has sharp corner at zero → pushes weights exactly to zero.

141. (B) Simple Interest

Find SI on ₹3000 at 4% for 5 years.

SI = 3000×4×5/100=600.

142. (A) Linear Algebra — Rank

What does rank of a matrix represent?

Number of linearly independent rows or columns.

143. (M) Compound Interest

₹4000 at 5% compounded annually for 3 years. Find amount.

A=4000(1.05)³=4000×1.157625=4630.5.

144. (A) Gradient Meaning

What does gradient represent in optimization?

Direction of steepest ascent of function.

145. (M) Functional Equation

Solve f(x)=2x+3 for f(7).

2*7+3=17.

146. (A) Logistic Regression Limit

Why logistic regression cannot model non-linear boundaries?

Decision boundary is linear unless features are transformed.

147. (M) Trains — Relative Speed

Two trains 120m and 180m long pass each other in 12 sec at speeds 40 and 50 km/h. Check correctness.

Relative speed=90 km/h=25 m/s. Time=300/25=12s ✔️.

148. (A) Bias-Variance

What happens when model complexity increases?

Bias decreases, variance increases.

149. (B) Mensuration

Area of circle radius 7.

= π*49.

150. (A) t-test Use

When is a t-test preferable to z-test?

Small samples, unknown population SD.

151. (M) Discount & Marked Price

After 10% discount, selling price is 900. Find MP.

MP = 900/0.9 = 1000.

152. (A) KL Divergence Meaning

What does KL divergence measure?

Information loss when approximating P by Q.

153. (M) Ratio Difference

Divide 600 in ratio 5:7.

Sum=12 → parts=600/12=50 → shares 250 & 350.

154. (A) Confidence Level

Higher confidence → CI becomes wider or narrower?

Wider.

155. (M) Clocks Problem

At what time between 3 and 4 do hour and minute hands coincide?

At (3*5.5)/5.5+5.5? Easier: (3*30)/(5.5)= approx 16.36 minutes → 3:16:21.

156. (A) Lasso vs Ridge

Which technique is used for feature selection?

Lasso (L1).

157. (M) Profit Split

A:B invest in ratio 2:3. Profit 5000. Find A’s share.

=2/5×5000=2000.

158. (A) ROC Curve Meaning

What is a ROC curve used for?

Visualizing classifier TPR vs FPR tradeoffs.

159. (M) Linear Equation Pair

Solve: x + y = 10, x - y = 4.

Add: 2x=14 → x=7, y=3.

160. (A) Type I Error

What is Type I error?

Rejecting true null.

161. (M) Boats & Streams

Boat speed 12, stream 3. Find upstream speed.

12−3=9.

162. (A) Correlation Interpretation

If r = −0.5, what does it imply?

Moderate negative linear relationship.

163. (M) Percentage Mixture

A solution is 20% salt. If 10 L water added to 40 L solution, final %?

Salt=8 L. New vol=50 → 8/50=16%.

164. (A) Entropy Property

Entropy is maximum when?

Probabilities are uniform.

165. (M) Discount Successive

Two discounts of 10% and 20%. Equivalent?

=0.9×0.8=0.72 → 28% discount.

166. (A) p-value Interpretation

What does small p-value imply?

Data unlikely under H0.

167. (M) Ratio Increase

Ratio becomes 4:5 after adding 5 to both numbers. Original ratio was 3:4. Find numbers.

Let 3x+5 : 4x+5 = 4:5 → 5(3x+5)=4(4x+5) → 15x+25=16x+20 → x=5 → numbers 15,20.

168. (A) Overfitting Sign

Large train accuracy, low test accuracy means?

Overfitting.

169. (M) Clock Angle

Angle between hands at 2:20?

Hour=70°, minute=120° → diff=50°.

170. (A) Decision Trees — Overfitting

How to reduce overfitting in trees?

Pruning, depth limit, min samples per leaf.

171. (M) Age Problem

Sum of ages of A and B is 60. A is twice B. Find ages.

A=40,B=20.

172. (A) Feature Scaling

Why is scaling important for gradient descent?

Prevents slow convergence from uneven feature scales.

173. (M) Area — Composite Shapes

Find area of rectangle 10×6 plus semicircle radius 3.

Rect=60; semi=½π9=4.5π → total=60+4.5π.

174. (A) Bias Meaning

What is bias in ML?

Error from simplifying assumptions.

175. (M) Simplify Expression

Simplify (x² − 9)/(x − 3).

= x+3 (for x≠3).

176. (A) Variance of Bernoulli

Find Var(X) for Bernoulli(p).

p(1-p).

177. (M) Average Speed

Bike travels 50 km at 25 km/h and 50 km at 50 km/h. Average?

Time=2+1=3 → avg=100/3=33.33.

178. (A) Regression Residual

What is a residual?

Observed minus predicted value.

179. (M) Pipes — Percentage Work

A completes 40% in 4 days. How long for full work?

40% in 4 → 10%/day → 100% in 10 days.

180. (A) Logistic Curve

Why logistic curve is S-shaped?

Derivative increases then decreases due to saturation.

181. (M) Partnership Time

A invests 3000 for whole year; B invests 4000 for 6 months. Profit 3600. Find B’s share.

A:3000*12=36000; B:4000*6=24000 → ratio 3:2 → B=2/5×3600=1440.

182. (A) Central Limit Theorem Importance

Why does CLT matter in A/B tests?

Allows normal approximation of sample means for inference.

183. (M) Quadratic Maxima

Find max of f(x)=−x²+6x.

Vertex at x = −b/(2a)= −6/(−2)=3 → f(3)=9.

184. (A) Overfitting Fix

Name 3 ways to reduce overfitting.

Regularization, dropout, more data.

185. (M) Ratio Internal Division

Divide 70 into two parts in ratio 2:5.

Sum=7 → 70/7=10 → parts 20,50.

186. (A) Significance Level Meaning

What does α=0.05 mean?

5% chance of Type I error.

187. (M) Equation Roots

Solve x² = 49.

x=±7.

188. (A) Sampling Bias

Name one cause of sampling bias.

Non-random sample selection.

189. (M) Speed Conversion

Convert 54 km/h to m/s.

54×(1000/3600)=15 m/s.

190. (A) Variance Interpretation

High variance in model means?

Model is sensitive to training data.

191. (M) Linear Expression

Find x if 7x+14=49.

7x=35 → x=5.

192. (A) t-statistic Meaning

What does a larger t-statistic indicate?

More evidence against null.

193. (M) Profit % Increase

CP=500, SP=650. Profit%?

150/500×100=30%.

194. (A) Z-score Meaning

Z-score of +2 means?

Value is 2 SD above mean.

195. (M) Time & Distance Reverse

If trip takes 2 hours at 60 km/h, find distance.

=120 km.

196. (A) A/B Test Lift

If control CTR=4% and variant CTR=5%, lift?

=(5−4)/4=25% lift.

197. (M) Reverse Average

Average of 20 numbers is 15. Sum?

=300.

198. (A) Regularization Purpose

Why use regularization?

To prevent overfitting.

199. (M) Mixture Ratio

Mix 2 solutions: 30% and 50%, equal amounts. Final %?

=(30+50)/2=40%.

200. (A) Hypothesis Testing

Why do we never “accept” H0?

We can only fail to reject; insufficient evidence is not proof.

201. (B) Simple Ratio

Divide 90 in ratio 2:3.

Sum=5 → shares=36,54.

202. (M) Compound Percentage

Value increases 10% one year, 20% next. Net change?

1.1×1.2=1.32 → 32% increase.

203. (A) Time–Work Efficiency

A is 50% more efficient than B. Together finish in 12 days. Find A alone.

Let B=1, A=1.5. Combined=2.5 units/day. Work=2.5×12=30. A alone=30/1.5=20 days.

204. (M) Mixture Adjustment

A 40 L mixture is 30% alcohol. How much pure alcohol to add to make it 40%?

Alcohol=12 L. (12+x)/(40+x)=0.4 → x=6.67 L.

205. (B) Simple Interest

Find SI on ₹4000 for 2 years at 5%.

SI=4000×5×2/100=400.

206. (A) Permutation Count

How many permutations of 8 letters if 3 are identical?

8!/3!=6720.

207. (M) Profit–Loss Marked Price

MP=1200, SP after discount=960. Find discount%.

Discount=240 → 20%.

208. (A) Quadratic Problem

Find roots of x² − 5x + 6.

= (x−2)(x−3) → roots 2,3.

209. (M) Speed Difference

Car travels 120 km at 60 km/h. Then at what speed must it return to average 80 km/h overall?

Impossible: return time would need to be negative.

210. (B) Number System

Find unit digit of 7⁸.

Cycle 7,9,3,1 → 8 mod 4=0 → 1.

211. (A) Logarithmic Equation

Solve log₂(x) + log₂(4) = 5.

log₂(4x)=5 → 4x=32 → x=8.

212. (M) Pipes & Cisterns

A fills in 12 h, B empties in 24 h. Both together, time?

Net=1/12−1/24=1/24 → time=24 h.

213. (A) AP Sum

AP: a=5, d=3, n=20. Find Sₙ.

S=20/2(2*5+19*3)=10(10+57)=670.

214. (B) Geometry

Area of right triangle legs 6 & 8.

Area=24.

215. (M) Percentage Decrease

Value drops from 500 to 350. % decrease?

150/500=30%.

216. (A) Divisibility

Largest 5-digit number divisible by 72?

99999/72≈1388 → 1388×72=99936.

217. (M) Profit Ratio

A & B invest 6000 and 8000. Profit=7000. A’s share?

Ratio=6:8=3:4 → A=3/7×7000=3000.

218. (A) Time Zones

Call logged at 16:00 UTC; IST offset +5:30. Local time?

21:30 IST.

219. (M) Divisor Count

How many divisors does 360 have?

360=2³×3²×5¹ → divisors=(3+1)(2+1)(1+1)=4×3×2=24.

220. (B) Temperature Conversion

Convert 0°C to Fahrenheit.

32°F.

221. (A) Combinatorics

Number of subsets of a set of size 7?

2⁷=128.

222. (M) Train Speed

Train 150m passes pole in 6 sec. Speed?

150/6=25 m/s=90 km/h.

223. (A) GP Sum

GP: a=3, r=2, n=6. Find sum.

S=3(2⁶−1)/(2−1)=3(63)=189.

224. (M) Pipes

Two pipes fill in 10 & 15 h. Time together?

1/10+1/15=1/6 → 6 h.

225. (B) Basic Algebra

Solve x/5=7.

x=35.

226. (A) Coordinate Geometry

Distance between (3,4) and (7,1)?

√((4)²+(−3)²)=5.

227. (M) Profit %

CP=400, SP=480. Profit%?

80/400=20%.

228. (A) Number Theory

Find remainder 5¹⁰⁰ mod 7.

φ(7)=6 → 100 mod 6=4 → 5⁴=625≡2.

229. (M) Time–Distance

Travel 150 km at 50. Time?

3 hours.

230. (A) AP General

Find 25th term of AP: a=4, d=6.

a₂₅=4+24*6=148.

231. (B) Simple Percentage

Find 15% of 360.

54.

232. (A) Modulo Operation

Find 123456 mod 9.

Sum digits=1+2+3+4+5+6=21 → 21 mod 9=3.

233. (M) DI — Average

Values: 20, 30, 50, 100. Find average.

200/4=50.

234. (A) Boats & Streams

Downstream=15, upstream=5. Stream speed?

s=(15−5)/2=5.

235. (M) Ratio Adjustment

Ratio 4:9 becomes 1:2 when adding x to both. Find x.

4+x : 9+x = 1:2 → 2(4+x)=9+x → 8+2x=9+x → x=1.

236. (A) Divisor Sum

Find σ(n) for n=28.

1+2+4+7+14+28=56.

237. (M) Mixture

2 solutions: 20% and 60%. Make 40%. Ratio?

(40−20):(60−40)=20:20=1:1.

238. (A) Time

Convert 2.75 hours into minutes.

2 hr 45 min → 165 minutes.

239. (M) Cylinder Volume

Cylinder r=4, h=10. Volume?

πr²h=16π×10=160π.

240. (A) Inequality

Solve 3x − 7 ≥ 2.

3x≥9 → x≥3.

241. (M) Reverse Profit

Profit=20%, SP=720. Find CP.

CP=720/1.2=600.

242. (A) Factorial Mod

Find 10! mod 9.

10! contains 9 as factor → 0 mod 9.

243. (M) Simple Age

Father 3× son's age. Sum=60. Find ages.

3s + s =60 → s=15, father=45.

244. (A) Roots

Solve 2x − 3 = 17.

2x=20 → x=10.

245. (M) Work Completion

A does 30% in 3 days. Full job time?

10 days.

246. (A) Simple Logic

If all A are B, and all B are C, then all A are?

C.

247. (M) Time–Distance

Speed 72 km/h. Time to travel 50 km?

50/72≈0.694 hr.

248. (A) Coordinate Midpoint

Midpoint of (2,5) and (6,9)?

(4,7).

249. (M) Square Root Estimate

Approximate √50.

≈7.07.

250. (A) Triangle

Sides 5,12,13. Type?

Right triangle.

251. (A) Standardization

Formula for z-score?

(x−μ)/σ.

252. (M) Probability Conditional

P(A)=0.3, P(B)=0.5, P(A∩B)=0.2. Find P(A|B).

=0.2/0.5=0.4.

253. (A) Central Limit

CLT applies to what?

Means of large samples → approx normal.

254. (B) Mean–Median–Mode

For symmetric distribution, relation?

Mean=Median=Mode.

255. (A) Type II Error

Define β error.

Failing to reject false null.

256. (M) A/B Test

If difference is statistically significant, what does it imply?

Observed effect unlikely due to random chance.

257. (A) Variance Formula

Population variance formula?

Σ(x−μ)²/N.

258. (M) Expected Value

Flip a coin: win ₹5 on Head, lose ₹2 on Tail. EV?

0.5(5)+0.5(−2)=1.5.

259. (A) Confidence Interval Width

What decreases CI width?

Larger sample size.

260. (M) Covariance Meaning

If cov(X,Y) > 0 → ?

Variables move together.

261. (A) Logistic Function

Range of logistic output?

0 to 1.

262. (M) Correlation

r=0 means?

No linear relation.

263. (A) Regression Coefficient

Interpret β in linear regression.

Change in Y for unit change in X.

264. (M) Poisson

λ=3. Find P(X=0).

e⁻³.

265. (A) Overfitting Sign

Train high accuracy, test low accuracy?

Overfitting.

266. (M) Sampling

Simple random sampling ensures?

Equal chance for all units.

267. (A) Hypothesis Terms

Null hypothesis symbolizes?

No effect / no difference.

268. (M) Entropy Zero

When is entropy zero?

When probability is 1 for one class.

269. (A) KL Divergence

KL(P‖Q) equals zero when?

P = Q.

270. (M) Mean of Means

Two groups: mean 50 (n=10), mean 70 (n=20). Combined?

=(500+1400)/30=63.33.

271. (A) L1 vs L2

Which creates sparsity?

L1.

272. (M) Probability Deck

Probability of drawing Ace?

4/52=1/13.

273. (A) z-value

Find P(Z<0).< /p>

0.5.

274. (M) Variance Scaling

Var(5X)=?

25 Var(X).

275. (A) Business Metrics

DAU/MAU ratio measures?

Stickiness.

276. (M) Histogram

Histogram represents?

Frequency distribution.

277. (A) Multicollinearity

High multicollinearity effect?

Unstable coefficients.

278. (M) Permutation Prob

Probability first card is red?

26/52=1/2.

279. (A) Logistic Regression

Output represents?

Probability of class 1.

280. (M) Dataset Scaling

Min–max scaling formula?

(x−min)/(max−min).

281. (A) Random Variable

Define continuous RV.

Takes values over interval.

282. (M) Expected Value

EV of uniform [0,1]?

0.5.

283. (A) Overfitting Example

Using too many trees in decision tree leads to?

Overfitting.

284. (M) Confidence

For 95% CI, z-score?

1.96.

285. (A) SVM Kernel

Why use kernel?

Map data to higher dim for separation.

286. (M) Variance

If variance=0, what is true?

All values equal.

287. (A) Bernoulli EV

EV of Bernoulli(p)?

p.

288. (M) p-value Meaning

Small p-value (<0.05) suggests?

Reject H0.

289. (A) Chi-square Usage

Used for?

Independence test for categorical data.

290. (M) Probability Dice

P(sum=7) on two dice?

6/36=1/6.

291. (A) ML Evaluation

F1-score useful when?

Class imbalance.

292. (M) Random Sampling

Sampling with equal probability is?

Simple random sampling.

293. (A) Standard Error Meaning

SE measures?

Variability of sample mean.

294. (M) Normal Distribution

Mean=50, SD=10. Z for X=60?

(60−50)/10=1.

295. (A) PCA Goal

PCA reduces?

Dimensionality.

296. (M) Independence

If A & B independent: P(A∩B)=?

P(A)P(B).

297. (A) Regression Residual Sum

OLS ensures mean residual?

Zero.

298. (M) Random Variable

Uniform RV variance on [0,1]?

1/12≈0.0833.

299. (A) Gradient Descent

Too small learning rate causes?

Slow convergence.

300. (A) Lift in A/B Test

CTR jumps from 5% → 6%. What is lift?

=(6−5)/5=20% lift.

Cheat Sheet & Quick Tricks

# GCD (Euclid)
def gcd(a,b): while b: a,b = b, a%b; return a

# LCM = a*b/gcd(a,b)

# Compound interest: A = P(1 + r/n)^(n*t)
# Continuous: A = P * exp(r*t)

# Bayes: P(A|B) = P(B|A)P(A)/P(B)

# Work/Rate: T = 1 / (Σ rates)
# Harmonic mean for avg speed (equal distances): HM = n / Σ(1/v_i)

# AMAT (cache-like mentality): AMAT = HitTime + MissRate * MissPenalty

# CLT: sample mean ~ Normal(μ, σ/√n)

# Quick z-scores: z = (x - μ) / σ
# CI for mean (known σ): μ ± z * σ/√n

Study Plan & Resources

30-day study plan:

Week 1: Number systems, LCM/GCD, divisibility, modular arithmetic.
Week 2: Percent, ratio, averages, work-rate, trains, DI practice.
Week 3: Probability, P&C, Bayes, sampling, hypothesis tests.
Week 4: Statistics, A/B testing, time-series basics, mocks + DI cases.

Suggested reading: Khan Academy (basics), MIT OCW probability/statistics, Hands-on: Kaggle micro-projects and LeetCode/Learn for math puzzles.