Questions involving 3-D solids: cubes, cuboids, cylinders, cones, spheres, hemispheres, frustums, prisms, and pyramids. Focus on knowing exact formulae and applying them to problems involving melting, recasting, and composite solids.
LSA = 2h(l + b)
TSA = 2(lb + bh + lh)
Diagonal = √(l² + b² + h²)
LSA = 4a²
TSA = 6a²
Diagonal = a√3
CSA = 2πrh
TSA = 2πr(r + h)
Volume = ⅓πr²h
CSA = πrl
TSA = πr(r + l)
Surface Area = 4πr²
CSA = 2πr²
TSA = 3πr²
Slant l = √(h² + (R−r)²)
CSA = π(R + r)l
TSA = π[R² + r² + (R+r)l]
LSA = Perimeter of Base × Height
TSA = LSA + 2 × Base Area
- Cylinder : Cone : Sphere (same diameter and height)
- Volumes are in ratio 3 : 1 : 2
- Sphere vs Cube (equal surface areas)
- Ratio of volumes = √π : √6
- Largest sphere in a cube of side a
- Radius = a/2; Volume = (4/3)π(a/2)³
- Largest cube in a sphere of radius r
- Side = 2r/√3; Volume = (8r³)/(3√3)
- Volume becomes n³ times when edge is multiplied by n
- Surface area becomes n² times
Problems about comparing speeds, starts, and handicaps in races.
- Race
- A contest of speed in running, riding, driving, sailing or rowing.
- Start / Handicap
- If A gives B a start of x metres, B begins x metres ahead. In a 100 m race A runs 100 m while B runs (100 − x) m.
- Dead Heat
- All contestants reach the goal at exactly the same time.
- Games
- "A game of 100" means the first to score 100 wins. "A gives B 20 points" means A scores 100 while B scores 80.
Ex 1. In a km race, A beats B by 28 metres or 7 seconds. A's time over the course?
B covers 28 m in 7 s → B's time = (7/28) × 1000 = 250 sec. A's time = 250 − 7 = 243 sec (4 min 3 sec).
Ex 2. A runs 1¾ times as fast as B. A gives B a start of 84 m. Where must the winning post be?
A : B = 7 : 4. In a race of 7 m, A gains 3 m. So 84 m gained when race = (7/3) × 84 = 196 m.
Ex 3. A can run 1 km in 3 min 10 sec, B in 3 min 20 sec. By what distance can A beat B?
A beats by 10 sec. B's distance in 10 s = (1000/200) × 10 = 50 m.
Ex 6. A, B, C in a km race. A gives B a 40 m start and C a 64 m start. How many metres start can B give C?
A:B = 1000:960, A:C = 1000:936. B:C = (960/1000) × (1000/936) = 960/936 → B:C = 1000:975. B gives C a 25 m start.
Ex 8. In a game of 80 points, A gives B 5 points and C 15 points. How many points can B give C in a game of 60?
A:B = 80:75, A:C = 80:65. B:C = (75/80)×(80/65) = 75/65 = 60:52. B gives C 8 points.
11.(a) 12.(c) 13.(b) 14.(c) 15.(b) 16.(c) 17.(a) 18.(b) 19.(c) 20.(c)
21.(c) 22.(d) 23.(a) 24.(c) 25.(a)
Finding the day of the week for any given date using the concept of "odd days".
- Odd Days
- The number of days more than complete weeks in a given period.
- Ordinary Year
- 365 days = 52 weeks + 1 day → 1 odd day.
- Leap Year
- 366 days = 52 weeks + 2 days → 2 odd days. Leap year: divisible by 4 but not century unless also divisible by 400.
200 years → 3 odd days
300 years → 1 odd day
400 years → 0 odd days (and multiples: 800, 1200, 1600, 2000…)
Day mapping: 0=Sun, 1=Mon, 2=Tue, 3=Wed, 4=Thu, 5=Fri, 6=Sat
What day was 16th July 1776?
1600 yrs = 0 odd days; 100 yrs = 5; 75 yrs = 2 odd days. Total = 7 ≡ 0. Jan–July 16 in 1776 (leap): 31+29+31+30+31+30+16 = 198 days = 28w + 2 → 2 odd days. Grand total = 0+2 = 2 → Tuesday.
What day was 15th August 1947?
1600 = 0; 300 = 1; 46 yrs (11 leap, 35 ord) = 22+35 = 57 = 8w+1. Jan–Aug 15: 31+28+31+30+31+30+31+15 = 227 = 32w+3. Total = 0+1+1+3 = 5 → Friday.
11.(c) 12.(a) 13.(b) 14.(c) 15.(c) 16.(c) 17.(b) 18.(c)
Angle between clock hands, coincidence, right angles, opposite positions, fast/slow clocks.
- Minute hand speed
- 6° per minute
- Hour hand speed
- 0.5° per minute (360° in 12 hrs)
- Relative speed
- Minute hand gains 5.5° per minute over the hour hand
- Hands coincide
- Every 65 5/11 minutes (22 times per day)
- Right angle
- 44 times per day (hands 15 min spaces apart)
- Opposite direction
- 22 times per day (hands 30 min spaces apart)
- Same straight line
- 44 times per day (coincident + opposite)
(Take value ≤ 180°, else 360° − θ for reflex)
Minutes to gain x spaces = (60/55) × x = (12/11) × x min
Ex 1. Angle at 3:25?
Hour hand at 3 hrs 25 min = 3×30 + 25×0.5 = 90+12.5 = 102.5°. Minute hand = 25×6 = 150°. Angle = 150−102.5 = 47.5°.
Ex 2. When between 2 and 3 will hands coincide?
At 2 o'clock, hands are 10 min apart. Minute must gain 10 min. Time = (12/11)×10 = 10 10/11 min past 2.
Ex 6. Clock gains if minute hand overtakes hour hand every 65 min. Daily gain?
Correct interval = 65 5/11 min. Gain per 65 min = 5/11 min. Gain in 24 hrs = (5/11)×(60×24/65) = 10 10/43 min.
11.(b) 12.(c) 13.(d) 14.(a) 15.(b) 16.(c) 17.(c) 18.(b) 19.(d) 20.(c)
22.(c) 23.(b) 24.(c) 25.(b) 26.(c) 27.(d) 28.(d) 29.(c) 30.(d)
35.(a) 36.(d) 37.(c) 38.(d) 39.(d) 40.(b) 49.(b) 52.(b) 53.(c) 54.(a)
Investment in stocks, calculating income, yield, market value, brokerage, and comparing investments.
- Face Value (Nominal Value)
- Value printed on the share certificate (usually ₹100). Dividend is always paid on this.
- Market Value
- At premium: MV > FV. At par: MV = FV. At discount: MV < FV.
- Brokerage
- Added to cost when buying; subtracted from sale price when selling.
- Dividend
- Annual profit paid on Face Value.
- Rate of Return
- Annual income from an investment of ₹100 (using MV).
Income = Dividend% × FV of stock
Number of shares = Total Investment / MV per share
For "x% stock at y":
• FV = ₹100, MV = ₹y
• Dividend per share = x% of 100 = ₹x
• Yield = (x/y) × 100 %
Ex 5. Find income from ₹2500, 8% stock at 106.
Income from ₹100 stock = ₹8. Income from ₹2500 = (8/100)×2500 = ₹200.
Ex 6. Which is better: 7½% stock at 105 or 6½% stock at 94?
Invest ₹(105×94) in each. Case I income: (15/2)×(1/105)×(105×94) = 705. Case II: (13/2)×(1/94)×(105×94) = 682.5. 7½% at 105 is better.
Ex 11. A man buys ₹25 shares paying 9% dividend. He wants 10% on investment. At what price?
Dividend = 9% of 25 = ₹2.25. Need 10% return → investment = 2.25/10% = ₹22.50.
11.(c) 12.(d) 13.(a) 14.(a) 15.(b) 16.(b) 17.(b) 18.(c) 19.(b) 20.(b)
21.(c) 22.(a) 23.(b) 24.(b) 25.(d) 26.(b) 27.(b) 28.(c) 29.(b) 30.(b)
= n(n−1)(n−2)…(n−r+1)
All n things: ⁿPₙ = n!
With repeated items: n! / (p₁! p₂! … pₖ!)
ⁿCᵣ = ⁿC(n−r)
ⁿC₀ = ⁿCₙ = 1
0! = 1
Ex 4. Arrangements of letters in DAUGHTER so vowels always come together?
Treat AUE as one block → 6 letters arranged in 6! = 720 ways. AUE arrange in 3! = 6 ways. Total = 720×6 = 4320.
Ex 6. Arrangements of ENGINEERING?
11 letters: 3E, 3N, 2G, 2I, 1R. Total = 11! / (3!×3!×2!×2!×1!) = 277200.
Ex 10. Vowels always in odd positions in DETAIL (3 vowels, 3 consonants)?
3 vowels in 3 odd positions: 3P₃ = 6 ways. 3 consonants in remaining 3 positions: 3P₃ = 6 ways. Total = 6×6 = 36.
Ex 13. Committee of 6 from 7 men and 5 ladies with 4 men and 2 ladies?
⁷C₄ × ⁵C₂ = 35 × 10 = 350.
| Word | Length/Repeats | Arrangements | Ans |
|---|---|---|---|
| DISPLAY | 7 distinct | 7! = 5040 | (d) 5040 |
| SMART | 5 distinct | 5! = 120 | (e) 120 |
| FORMULATE | 9 distinct | 9! = 362880 | (d) 362880 |
| RIDDLED | 7; D×3 | 7!/3! = 840 | (a) 840 |
| CREATE | 6; E×2 | 6!/2! = 360 | (c) 360 |
| TOTAL | 5; T×2 | 5!/2! = 60 | (b) 60 |
| OFFICES | 7; F×2 | 7!/2! = 2520 | (a) 2520 |
| BANANA | 6; A×3, N×2 | 6!/(3!×2!) = 60 | (a) 60 |
| ENGINEERING | 11; E×3,N×3,G×2,I×2 | 11!/(3!3!2!2!) = 277200 | (a) 277200 |
| ALLAHABAD | 9; A×4, L×2 | 9!/(4!×2!) = 7560 | (c) 7560 |
| RUMOUR | 6; R×2, U×2 | 6!/(2!×2!) = 180 | (c) 180 |
11.(b) 12.(c) 13.(b) 14.(b) 15.(c) 16.(c) 17.(d) 18.(c) 19.(b) 20.(d)
21.(b) 22.(e) 23.(c) 24.(b) 25.(d) 26.(c) 27.(c) 28.(c) 29.(a) 30.(c)
31.(b) 32.(a) 33.(c) 34.(a) 35.(d) 36.(e) 37.(b) 38.(e) 39.(b) 40.(b)
41.(d) 42.(b) 43.(c) 44.(d) 45.(b) 46.(c) 47.(b) 48.(a)
- Sample Space (S)
- Set of all possible outcomes of a random experiment.
- Event (E)
- Any subset of the sample space.
- Probability
- P(E) = n(E) / n(S). Always: 0 ≤ P(E) ≤ 1.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(not A) = P(Ā) = 1 − P(A)
P(A ∩ B) = P(A) × P(B) [independent events]
Deck of cards: 52 total, 13 per suit, 26 red, 26 black
Face cards = 12 (4 Jacks, 4 Queens, 4 Kings)
One coin toss: S = {H, T}, n(S) = 2
Two coin toss: S = {HH, HT, TH, TT}, n(S) = 4
Three coin toss: n(S) = 8
One die: S = {1,2,3,4,5,6}, n(S) = 6
Two dice: n(S) = 36
Cards: n(S) for 2 drawn = ⁵²C₂ = 1326
11.(d) 12.(c) 13.(b) 14.(c) 15.(c) 16.(a) 17.(b) 18.(d) 19.(d) 20.(d)
21.(b) 22.(d) 23.(b) 24.(c) 25.(b) 26.(d) 27.(c) 28.(b) 29.(d) 30.(d)
31.(b) 32.(b) 33.(a) 34.(b) 35.(a) 36.(b) 37.(a) 38.(d) 39.(c) 40.(a)
41.(b) 42.(c) 43.(a) 44.(d) 45.(b) 46.(a) 47.(a) 48.(b) 49.(b) 50.(e)
The present worth of a future sum, and the interest saved by paying now rather than later.
T.D. = (P.W. × R × T) / 100
T.D. = (Amount × R × T) / (100 + R×T)
Sum due = (S.I. × T.D.) / (S.I. − T.D.)
S.I. − T.D. = S.I. on T.D.
[Compound Interest case]
P.W. = Amount / (1 + R/100)ᵀ
Ex 1. Present worth of ₹930 due 3 years hence at 8% p.a.?
P.W. = (100×930) / (100+8×3) = 93000/124 = ₹750. T.D. = 930−750 = ₹180.
Ex 3. T.D. = ₹250, S.I. = ₹375 (same sum, time, rate). Find sum.
Sum = (375×250)/(375−250) = 93750/125 = ₹750.
11.(a) 12.(b) 13.(b) 14.(a) 15.(d) 16.(b) 17.(b) 18.(c) 19.(b) 20.(d) 21.(d)
- Banker's Discount (B.D.)
- Simple interest on the face value (not P.W.) for the unexpired time. B.D. = S.I. on bill for unexpired time.
- Banker's Gain (B.G.)
- B.G. = B.D. − T.D. = S.I. on T.D. = (T.D.)² / P.W.
- True Discount
- T.D. = √(P.W. × B.G.)
T.D. = Amount × R × T / (100 + R×T)
Amount = B.D. × T.D. / (B.D. − T.D.)
T.D. = B.G. × 100 / (Rate × Time)
Ex 1. Bill for ₹6000 drawn July 14, 5 months. Discounted Oct 5 at 10%. Find B.D., T.D., B.G.
Legally due: Dec 17. Unexpired: Oct 5 to Dec 17 = 73 days = 1/5 year.
B.D. = 6000 × 10 × (1/5) / 100 = ₹120.
T.D. = (6000 × 10 × 1/5) / (100 + 10/5) = 12000/102 = ₹117.64.
B.G. = 120 − 117.64 = ₹2.36.
cos θ = Base / Hypotenuse
tan θ = Perpendicular / Base
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
sin: 0 1/2 1/√2 √3/2 1
cos: 1 √3/2 1/√2 1/2 0
tan: 0 1/√3 1 √3 ∞
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
- Angle of Elevation
- When you look up at an object. The angle the line of sight makes with the horizontal.
- Angle of Depression
- When you look down at an object. The angle the line of sight makes with the horizontal.
Ex 3. Angle of elevation from C is 30°, from D (24 m closer) is 60°. Height of tower?
AB/AD = tan 60° → AD = h/√3. AB/AC = tan 30° → AC = h√3. CD = AC − AD = h√3 − h/√3 = 2h/√3 = 24 → h = 12√3 ≈ 20.76 m.
Ex 5. Man on top of tower. Boat takes 10 min for angle of depression to change from 30° to 60°. Time to reach shore from the 60° position?
If h is tower height, at 60°: y = h/√3. At 30°: x+y = h√3. x = 2h/√3. x covered in 10 min. Remaining y = h/√3 covered in (10×(h/√3))/(2h/√3) = 5 minutes.
11.(c) 12.(b) 13.(b) 14.(a) 15.(d) 16.(a) 17.(c) 18.(a)
Identifying the element that doesn't fit, or the wrong/missing number in a sequence. Common patterns: arithmetic, geometric, prime, square, cube, alternating, Fibonacci-type.
- Odd Man Out — Types
- • All primes except one composite (or vice versa)
- • All even except one odd
- • All perfect squares/cubes except one
- • All multiples of n except one
- • Digital pattern (sum/product of digits follows a rule)
- Series — Common Patterns
- • x² + 1, x² − 1, x² + constant
- • Alternating: ×2 and +3; or two interleaved sequences
- • ×n + c (multiply and add)
- • Differences themselves form an AP or GP
- • Numbers are n³ − 1, n³ + 1, etc.
11.(b) 12.(b) 13.(a) 14.(b) 15.(d) 16.(b) 17.(b) 18.(d) 19.(c) 20.(c)
21.(b) 22.(a) 23.(c) 24.(d) 25.(a) 26.(d) 27.(d) 28.(a) 29.(a) 30.(b)
43.(c) 44.(b) 45.(a) 46.(a) 47.(c) 48.(b) 49.(c) 50.(a) 52.(b) 53.(b)
60.(c) 61.(e) 62.(d) 63.(c) 64.(d) 65.(b) 66.(c) 67.(e) 68.(d) 69.(c)
70.(b) 71.(b) 72.(c) 73.(c) 74.(e) 75.(e) 76.(c) 77.(d) 78.(e) 79.(b)
80.(a) 81.(a) 82.(e) 83.(b) 84.(d) 85.(c) 86.(b) 87.(d) 88.(e) 89.(a)
90.(a) 91.(a) 92.(c) 93.(c) 94.(e) 95.(a) 96.(d)
Reading tables with multiple variables (rows and columns) and answering questions about percentages, ratios, averages, and comparisons. Key skill: be careful with row vs column headers and units.
- Step 1: Read the table header carefully
- Identify what each row and column represents, and the units involved.
- Step 2: Understand the question type
- Percentage increase/decrease, ratio, average, difference, or approximate value.
- Step 3: Locate the exact cells needed
- Don't read more data than required. Focus only on relevant rows/columns.
- Step 4: Approximate when possible
- For % questions, round sensibly before computing to save time.
- Step 5: Verify units
- If the table is "in hundreds" or "in lakhs", account for this in your answer.
11.(a) 12.(c) 13.(d) 14.(b) 15.(d) 16.(c) 17.(b) 18.(a) 19.(d) 20.(a)
21.(c) 22.(a) 23.(b) 24.(d) 25.(e)
11.(a) 12.(c) 13.(b) 14.(b) 15.(d) 16.(b) 17.(c) 18.(a) 19.(c) 20.(a)
21.(d) 22.(d) 23.(b) 24.(d) 25.(a) 26.(d) 27.(e) 28.(c) 29.(b) 30.(b)
31.(b) 32.(b) 33.(c) 34.(d) 35.(b)
11.(a) 12.(b) 13.(e) 14.(a) 15.(c) 16.(a) 17.(b) 18.(c) 19.(b) 20.(d)
21.(c) 22.(b) 23.(c) 24.(b) 25.(d)
Reading bar charts (vertical/horizontal), comparing bars, computing percentage change, and combining bar + table data.
- Percentage increase/decrease
- = (New − Old) / Old × 100%
- Average
- Sum of all bar values / Number of bars
- Ratio questions
- Read off the two values carefully and simplify the fraction
- Combining with tables
- Look up the ratio in the table, then use bar values to compute absolute numbers. e.g., if bar = total, and table gives A:B ratio, then A = bar × (ratio_A / sum_of_ratio).
11.(d) 12.(c) 13.(e) 14.(e) 15.(b) 16.(b) 17.(e) 18.(e) 19.(d) 20.(a)
21.(b) 22.(c) 23.(d) 24.(e) 25.(a) 27.(b) 28.(b) 29.(d) 30.(d) 31.(c)
11.(a) 12.(e) 13.(c) 14.(b) 15.(c) 16.(b) 17.(b) 18.(c) 19.(b) 20.(a)
21.(c) 22.(b) 23.(a) 24.(b) 25.(c)
11.(b) 12.(d) 13.(b) 14.(a) 15.(a) 16.(e) 17.(d) 18.(e) 19.(a) 20.(b)
21.(c) 22.(b) 23.(a) 24.(b) 25.(a)
Interpreting circle graphs showing percentage or degree distribution. Often combined with a second pie chart or table.
Value of a sector = (central angle / 360°) × total
Percentage = (central angle / 360°) × 100%
If two pie charts: combine proportions carefully.
e.g., Males from A = (% of A in pie 1) × total × (male % within A)
11.(c) 12.(c) 13.(a) 14.(a) 15.(c) 16.(a) 17.(b) 18.(b) 19.(b) 20.(b)
21.(c) 22.(d) 23.(d) 24.(d) 25.(d) 26.(c) 27.(b) 28.(b) 29.(c)
11.(c) 12.(c) 13.(b) 14.(b) 15.(c) 16.(b) 17.(c) 18.(d) 19.(a) 20.(e)
21.(a) 22.(d) 23.(b) 24.(e)
11.(a) 12.(b) 13.(a) 14.(a) 15.(a) 16.(d) 17.(b) 18.(b) 19.(c) 20.(a)
21.(b) 22.(a) 23.(d)
Line graphs show trends over time. Common tasks: find % change, ratio, average, and combine with ratio tables to compute absolute values for individual components.
- Reading a combined line graph + table
- The line gives the TOTAL (e.g., Company A's production). The table gives A:B ratio. So Company B = Total_A × (B_ratio / A_ratio).
- Profit % formula
- Profit% = [(Income − Expenditure) / Expenditure] × 100
- Import/Export ratio graph
- When ratio < 1, Exports > Imports (favourable). When ratio > 1, Imports > Exports.
- Be careful with data adequacy
- If only a ratio is given (not absolute value), percentage change in absolute terms cannot be found.
11.(b) 12.(c) 13.(b) 14.(a) 15.(e) 16.(b) 17.(e) 18.(e) 19.(c) 20.(d)
21.(a) 22.(b) 23.(d) 24.(d) 25.(c) 26.(c) 27.(c) 28.(d)
11.(d) 12.(c) 13.(b) 14.(e) 15.(d)
11.(b) 12.(e) 13.(d) 14.(c) 15.(a) 16.(e) 17.(d) 18.(e) 19.(a) 20.(b)
21.(a) 22.(a) 23.(b) 24.(c) 25.(d)
Chapters 25–39 · Volume & SA · Races · Calendar · Clocks · Stocks · Permutations · Probability · Discount · Heights · Series · Data Interpretation