SSC CGL Tier-II 2023 Official Paper-I (Held On: 26 Oct, 2023) (Previous Year Paper)

Shortcut Trick

Total sum of 3 numbers = Average × 3 = 11 × 3 = 33

Sum of given numbers = 25 + 46 = 71

Value of y = Total sum − Sum of given numbers

y = 33 − 71 = −38

∴ The correct answer is −38.

Alternate Method

Given:

Numbers: 25, 46, and y

Average of these 3 numbers = 11

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Average = (25 + 46 + y) ÷ 3

⇒ 11 = (71 + y) ÷ 3

⇒ 11 × 3 = 71 + y

⇒ 33 = 71 + y

⇒ y = 33 − 71

⇒ y = −38

∴ The correct answer is −38.

Additional Information

Sum of Observations

The sum of n observations is always equal to the product of their average and the total count n. Sum = Average × n.

Deviation Method

The sum of deviations of all observations from their arithmetic mean (average) is always zero. ∑(x_i − Average) = 0.

Weighted Average

When different groups have different averages, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Angle of depression (θ) = 60° and horizontal distance (d) = 50√3 m.

Height difference between tower and building = d × tan(θ).

Height difference = 50√3 × √3 = 50 × 3 = 150 m.

Total tower height = Building height + Height difference = 10 + 150 = 160 m.

∴ The correct answer is 160 m.

Alternate Method

Given:

Height of building (h) = 10 m

Distance between tower and building (d) = 50√3 m

Angle of depression = 60°

Formula Used:

tan θ = Perpendicular / Base

Calculations:

⇒ Let the height of the tower be H.

⇒ Let x be the height of the tower above the building level.

⇒ The angle of depression of the top of the building from the tower top is 60°.

⇒ This is equivalent to the angle of elevation of the tower top from the building top.

⇒ In the right-angled triangle formed above the building level:

⇒ tan 60° = x / 50√3

⇒ √3 = x / 50√3

⇒ x = 50√3 × √3

⇒ x = 50 × 3 = 150 m

⇒ Total height of tower (H) = x + height of building

⇒ H = 150 + 10 = 160 m

∴ The correct answer is 160 m.

Additional Information

Angle of Depression and Elevation

The angle of depression from point A to point B is numerically equal to the angle of elevation from point B to point A because they are alternate interior angles.

Common Trigonometric Ratios

For height and distance problems: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Standard Height Formula

If two objects are separated by distance d, the height difference is given by h = d × tan(θ), where θ is the angle of elevation/depression between their tops.

Shortcut Trick

Apply effective percentage change formula: Profit % = a + b + (a × b) ÷ 100

Here, Markup (a) = +42 and Discount (b) = −15

Profit % = 42 − 15 + (42 × (−15)) ÷ 100

Profit % = 27 − 6.3 = 20.7%

Rounding off to the nearest integer, we get 21%.

∴ The correct answer is 21%.

Alternate Method

Given values:

Markup percentage = 42%

Discount percentage = 15%

⇒ Let the Cost Price (CP) be ₹100.

⇒ Marked Price (MP) = CP + 42% of CP = 100 + 42 = ₹142.

⇒ Selling Price (SP) = MP − 15% of MP.

⇒ SP = 142 − (15 ÷ 100) × 142.

⇒ SP = 142 − 21.3 = ₹120.7.

⇒ Profit = SP − CP = 120.7 − 100 = ₹20.7.

⇒ Profit % = (Profit ÷ CP) × 100 = (20.7 ÷ 100) × 100 = 20.7%.

⇒ Rounded to nearest integer = 21%.

∴ The correct answer is 21%.

Additional Information

Marked Price (MP) and Discount

Marked price is the price displayed on a product. Discount is always calculated on the MP: SP = MP × (100 − Discount%) ÷ 100.

Relationship between CP and MP

The ratio of CP to MP is given by: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Successive Percentage Formula

If a value is changed by x% and then by y%, the net change is [x + y + (xy ÷ 100)]%. This is useful for markup and discount problems.

Shortcut Trick

Difference between Mode and Lower limit = (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) × h

48.5 − 46.5 = (34 − 32) ÷ (2 × 34 − 32 − 25) × h

2 = (2 ÷ 11) × h

h = 11

∴ The correct answer is 11.

Alternate Method

Given:

Mode (M_o) = 48.5

Lower boundary of modal class (l) = 46.5

Frequency of modal class (f₁) = 34

Frequency of preceding class (f₀) = 32

Frequency of succeeding class (f₂) = 25

Formula Used:

Mode = l + [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h

Calculations:

⇒ 48.5 = 46.5 + [ (34 − 32) ÷ (2 × 34 − 32 − 25) ] × h

⇒ 48.5 − 46.5 = [ 2 ÷ (68 − 57) ] × h

⇒ 2 = (2 ÷ 11) × h

⇒ 2 × 11 = 2h

⇒ 22 = 2h

⇒ h = 22 ÷ 2

⇒ h = 11

∴ The correct answer is 11.

Additional Information

Median for Grouped Data

Median = l + [ (N/2 − cf) ÷ f ] × h, where l is lower limit of median class and cf is cumulative frequency of preceding class.

Empirical Relationship

Mode = 3 × Median − 2 × Mean. This formula is used for moderately skewed distributions.

Mean for Grouped Data (Direct Method)

Mean = Σf_ix_i ÷ Σf_i, where x_i is the class mark (mid-value) of each interval.

Shortcut Trick

Side AB = BC (triangle) and BE = BC (square) ⇒ AB = BE.

∠ABE = ∠ABC + ∠CBE = 60° + 90° = 150°.

In isosceles ΔABE, ∠EAB = ∠AEB = (180° − 150°) ÷ 2 = 15°.

Value of ∠EAB + 3∠AEB = 15° + 3(15°) = 15° + 45° = 60°.

∴ The correct answer is 60°.

Alternate Method

Given: ABC is an equilateral triangle and BCDE is a square.

Formula: Sum of angles in a triangle = 180°; Angles in square = 90°; Angles in equilateral triangle = 60°.

⇒ In equilateral ΔABC, AB = BC = AC and ∠ABC = 60°.

⇒ In square BCDE, BC = CD = DE = BE and ∠EBC = 90°.

⇒ Total ∠ABE = ∠ABC + ∠EBC = 60° + 90° = 150°.

⇒ Since AB = BC and BE = BC, we have AB = BE.

⇒ In ΔABE, ∠EAB = ∠AEB (angles opposite to equal sides).

⇒ ∠ABE + ∠EAB + ∠AEB = 180°

⇒ 150° + 2 × ∠AEB = 180°

⇒ 2 × ∠AEB = 30° ⇒ ∠AEB = 15°

⇒ Similarly, ∠EAB = 15°.

⇒ Required value = ∠EAB + 3∠AEB = 15° + 3(15°) = 60°.

∴ The correct answer is 60°.

Additional Information

Equilateral Triangle Properties

All three sides are equal in length, and each internal angle measures exactly 60°.

Square Properties

A square is a regular quadrilateral with four equal sides and four equal angles of 90°.

Isosceles Triangle Theorem

If two sides of a triangle are equal, then the angles opposite those sides are also equal.

Angle Sum Property

The sum of all interior angles of any triangle is always 180°.

Shortcut Trick

Identify frequencies: 13 appears 2 times, 15 appears 2 times.

All other numbers (10, 11, 12, 14, 16, 17, 18, 20, 21, 22, 24) appear only once.

The highest frequency is 2, so the modes are 13 and 15.

Product of modes = 13 × 15 = 195.

∴ The correct answer is 195.

Alternate Method

Given: Data set = {13, 11, 16, 15, 18, 12, 13, 14, 10, 22, 15, 21, 20, 17, 24}

Formula: Mode is the observation(s) with the highest frequency in the data set.

⇒ Arranging the data in ascending order: 10, 11, 12, 13, 13, 14, 15, 15, 16, 17, 18, 20, 21, 22, 24.

⇒ Frequency count:

⇒ 10, 11, 12, 14, 16, 17, 18, 20, 21, 22, 24 → 1 time each.

⇒ 13 → 2 times.

⇒ 15 → 2 times.

⇒ Since 13 and 15 have the highest frequency (2), there are two modes.

⇒ Modes = 13 and 15.

⇒ Product of modes = 13 × 15 = 195.

∴ The correct answer is 195.

Additional Information

Mode of a Dataset

The mode is the value that occurs most frequently in a data set. A set can have one mode (unimodal), two modes (bimodal), or more (multimodal).

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Mean and Median

Mean is the average (sum ÷ count). Median is the middle value when data is arranged in ascending or descending order.

Shortcut Trick

Ratio of investments A : B = 1,50,000 : 2,50,000 = 3 : 5.

Divide profit ₹6,00,000 in 3 : 5 → A = ₹2,25,000 and B = ₹3,75,000.

New investment ratio = (2,25,000 + 50,000) : (3,75,000 + 50,000) = 2,75,000 : 4,25,000 = 11 : 17.

A’s share in new profit = (11 ÷ 28) × 4,20,000 = 11 × 15,000 = ₹1,65,000.

∴ The correct answer is ₹1,65,000.

Alternate Method

Given:

Initial investment A = ₹1,50,000 | Initial investment B = ₹2,50,000

Profit Project 1 = ₹6,00,000 | Profit Project 2 = ₹4,20,000

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

⇒ Investment Ratio A : B = 1,50,000 : 2,50,000 = 3 : 5

⇒ Sum of ratio parts = 3 + 5 = 8 units

⇒ 8 units = ₹6,00,000

⇒ 1 unit = 6,00,000 ÷ 8 = ₹75,000

⇒ A’s profit share (Project 1) = 3 × 75,000 = ₹2,25,000

⇒ B’s profit share (Project 1) = 5 × 75,000 = ₹3,75,000

⇒ A’s investment (Project 2) = 2,25,000 + 50,000 = ₹2,75,000

⇒ B’s investment (Project 2) = 3,75,000 + 50,000 = ₹4,25,000

⇒ New Ratio A : B = 2,75,000 : 4,25,000 = 11 : 17

⇒ Total ratio units (Project 2) = 11 + 17 = 28 units

⇒ A’s share in Project 2 profit = (11 ÷ 28) × 4,20,000

⇒ A’s share = 11 × 15,000 = ₹1,65,000

∴ The correct answer is ₹1,65,000.

Additional Information

Partnership Profit Sharing Rule

Profit is always distributed in the ratio of the product of Investment (Capital) and Time: Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Types of Partnership

Simple Partnership occurs when all partners invest for the same time period; Compound Partnership occurs when partners invest for different time durations.

Working vs Sleeping Partner

A working partner manages the business and may receive a salary or extra commission from the profit before the remaining profit is shared according to investment ratios.

Shortcut Trick

Ratio of dimensions L : W : H = 6 : 3 : 2.

TSA ratio units = 2 × (6×3 + 3×2 + 2×6) = 2 × (18 + 6 + 12) = 72 units².

72 units² = 1800 cm² ⇒ 1 unit² = 25 ⇒ 1 unit = 5 cm.

Volume = (6 × 3 × 2) × (5)³ = 36 × 125 = 4500 cm³.

∴ The correct answer is 4500 cm³.

Alternate Method

Given:

Ratio of length (l), width (w), and height (h) = 6 : 3 : 2

Total Surface Area (TSA) = 1800 cm²

Formula Used:

Total Surface Area of Cuboid = 2(lw + wh + hl)

Volume of Cuboid = l × w × h

Let the length, width, and height be 6x, 3x, and 2x respectively.

⇒ TSA = 2[(6x × 3x) + (3x × 2x) + (2x × 6x)]

⇒ 1800 = 2[18x² + 6x² + 12x²]

⇒ 900 = 36x²

⇒ x² = 900 ÷ 36

⇒ x² = 25

⇒ x = √25 = 5

Now, dimensions of the cuboid are:

⇒ Length (l) = 6 × 5 = 30 cm

⇒ Width (w) = 3 × 5 = 15 cm

⇒ Height (h) = 2 × 5 = 10 cm

Volume of the cuboid:

⇒ V = l × w × h

⇒ V = 30 × 15 × 10

⇒ V = 4500 cm³

∴ The correct answer is 4500 cm³.

Additional Information

Lateral Surface Area of Cuboid

It is the area of the four side walls of the cuboid, calculated as LSA = 2h(l + w).

Diagonal of a Cuboid

The length of the longest rod that can be placed inside a cuboid is given by d = √(l² + w² + h²).

Cube Properties

For a cube of side 'a', Volume = a³ and Total Surface Area = 6a².

Shortcut Trick

In dishonest dealer problems where buying and selling rates are same:

Profit % = [(Wholesaler Weight − B's Weight) ÷ B's Weight] × 100

Profit % = [(950 − 750) ÷ 750] × 100

Profit % = [200 ÷ 750] × 100 = (4 ÷ 15) × 100 = 26.666...

∴ The correct answer is 26.67%.

Alternate Method

Given:

Rate of purchase = ₹180/kg | Weight received from wholesaler = 950 g for every 1000 g

Rate of sale = ₹180/kg | Weight given to customer = 750 g for every 1000 g

Total apples purchased = 15 kg

Calculations:

⇒ Actual quantity received by B = 15 kg × (950/1000) = 14.25 kg

⇒ Total Cost Price (CP) for B = 15 × ₹180 = ₹2700 (since B pays for 15kg)

⇒ B sells 750 g as 1 kg. So, 14.25 kg is sold as = 14.25 ÷ 0.750 = 19 units of 1 kg

⇒ Total Selling Price (SP) for B = 19 × ₹180 = ₹3420

⇒ Profit = SP − CP = 3420 − 2700 = ₹720

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = (720 ÷ 2700) × 100 = 72000 ÷ 2700 = 720 ÷ 27 = 80 ÷ 3

⇒ Profit % = 26.666...

∴ The correct answer is 26.67%.

Additional Information

Dishonest Dealer Concept

When a dealer cheats while buying and selling, the overall profit multiplier is (Buying Factor × Selling Factor). Profit % = [(&text{True Weight} + &text{Gain while buying}) ÷ (&text{True Weight} − &text{Loss while selling}) − 1] × 100.

Ratio Method for Profit

For dishonest weights, CP : SP = Weight Given : Weight Charged. If the price per kg also changes, multiply the ratios accordingly.

Effective Profit %

If a person uses x% less weight, the profit % is [x ÷ (100 − x)] × 100. In this case, B uses 25% less weight (250g/1000g) on his end, but he also received 5% less weight from the wholesaler.

Shortcut Trick

Find P(A ∩ B) using De Morgan Law: P(A ∩ B) = 1 − P(Ā ∪ B̄) = 1 − 13/27 = 14/27.

Calculate P(B) using Addition Rule: P(B) = P(A ∪ B) + P(A ∩ B) − P(A) = 5/9 + 14/27 − 11/18.

Using LCM 54: P(B) = (30 + 28 − 33) / 54 = 25/54.

Odds against B = P(B̄) : P(B) = (54 − 25) : 25 = 29 : 25.

∴ The correct answer is 29/25.

Alternate Method

Given:

P(A ∪ B) = 5/9

P(Ā ∪ B̄) = 13/27

P(A) = 11/18

Formula Used:

1. De Morgan's Law: P(Ā ∪ B̄) = P((A ∩ B)̄) = 1 − P(A ∩ B)
2. Addition Theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
3. Odds against event B = P(B̄) / P(B) = (1 − P(B)) / P(B)

Calculations:

⇒ P(A ∩ B) = 1 − P(Ā ∪ B̄)

⇒ P(A ∩ B) = 1 − 13/27 = 14/27

⇒ P(B) = P(A ∪ B) − P(A) + P(A ∩ B)

⇒ P(B) = 5/9 − 11/18 + 14/27

⇒ LCM of 9, 18, 27 is 54.

⇒ P(B) = (30 − 33 + 28) / 54 = 25/54

⇒ P(B̄) = 1 − 25/54 = 29/54

⇒ Odds against B = P(B̄) / P(B) = (29/54) / (25/54)

⇒ Odds against B = 29/25

∴ The correct answer is 29/25.

Additional Information

De Morgan's Laws

For any two events A and B, P(Ā ∩ B̄) = P((A ∪ B)̄) and P(Ā ∪ B̄) = P((A ∩ B)̄).

Odds in Favor vs Odds Against

Odds in favor of an event E are P(E) : P(Ē). Odds against an event E are P(Ē) : P(E).

Probability Addition Theorem

For any events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If they are mutually exclusive, P(A ∩ B) = 0.

Shortcut Trick

The extra weight of the 14^th person (39 kg) is distributed among the first 13 persons to equalize their average to the overall average.

Total Average = Average of 13 persons + (Extra weight ÷ 13)

Total Average = 78 + (39 ÷ 13) = 78 + 3 = 81 kg

Weight of 14^th person = Total Average + Extra = 81 + 39 = 120 kg

∴ The correct answer is 120 kg.

Alternate Method

Given:

Average weight of 13 persons = 78 kg

Weight of 14^th person = Average of 14 persons + 39 kg

Formula Used:

Average = Sum of all observations ÷ Number of observations

Calculations:

Let the average weight of all 14 persons be x kg.

⇒ Total weight of 14 persons = 14 × x = 14x

⇒ Weight of 14^th person = x + 39

⇒ Sum of weights of 13 persons = 13 × 78 = 1014 kg

⇒ Total Weight = Sum of 13 persons + Weight of 14^th person

⇒ 14x = 1014 + (x + 39)

⇒ 14x − x = 1053

⇒ 13x = 1053

⇒ x = 1053 ÷ 13

⇒ x = 81 kg

⇒ Weight of 14^th person = 81 + 39 = 120 kg

∴ The correct answer is 120 kg.

Additional Information

Average Concept

Average is the central value of a set of data. Formula: Average = Total Sum ÷ Total Number.

Deviation Method

If a new member joins and the average increases, the new member's value = Old Average + (Increase in Average × New Strength).

Sum of Observations

Sum = Average × Number of observations. Any change in sum directly impacts the average by Change ÷ n.

Shortcut Trick

Net Multiplier = (1 − 0.20) × (1 − 0.35) × (1 − 0.10)

Net Multiplier = 0.8 × 0.65 × 0.9 = 0.468

Equivalent Discount = (1 − 0.468) × 100 = 53.2%

∴ The correct answer is 53.2%.

Alternate Method

Given:

First discount (d₁) = 20%

Second discount (d₂) = 35%

Third discount (d₃) = 10%

Formula Used:

Successive discount of x% and y% = [x + y − (x × y)/100]%

Calculations:

⇒ Equivalent discount for 20% and 35%:

⇒ D₁₂ = 20 + 35 − (20 × 35)/100

⇒ D₁₂ = 55 − 7 = 48%

⇒ Now, equivalent discount for 48% and 10%:

⇒ Net Discount = 48 + 10 − (48 × 10)/100

⇒ Net Discount = 58 − 4.8

⇒ Net Discount = 53.2%

∴ The correct answer is 53.2%.

Additional Information

Successive Discount Formula

For two successive discounts a% and b%, the single equivalent discount is (a + b − ab/100)%.

Relation between MP and SP

Selling Price (SP) = Marked Price (MP) × (100 − d₁)/100 × (100 − d₂)/100 × ...

Effective Percentage Change

This method can also be used for successive population increases or area changes where x and y are positive for increase and negative for decrease.

Shortcut Trick

Probability of independent events with replacement = P(A) × P(B) × P(C)

P(Jack) = 4/52 = 1/13

P(Black Card) = 26/52 = 1/2

P(Even Numbered Card) = 20/52 = 5/13 (Even cards are 2, 4, 6, 8, 10 per suit)

Total Probability = (1/13) × (1/2) × (5/13) = 5/338

∴ The correct answer is 5/338.

Alternate Method

Given:

Total cards in a pack = 52

Condition: With replacement (Events are independent)

Formula Used:

Probability P(E) = Number of favorable outcomes ÷ Total number of outcomes

For independent events, P(A and B and C) = P(A) × P(B) × P(C)

Calculations:

⇒ Total cards in a pack = 52

⇒ Number of Jacks = 4 (Jack of Spades, Clubs, Hearts, Diamonds)

⇒ P(First card is a Jack) = 4/52 = 1/13

⇒ Number of black cards = 26 (13 Spades + 13 Clubs)

⇒ P(Second card is black) = 26/52 = 1/2

⇒ Numbered cards are 2, 3, 4, 5, 6, 7, 8, 9, 10. Even numbered cards = {2, 4, 6, 8, 10}

⇒ Total even numbered cards = 5 per suit × 4 suits = 20

⇒ P(Third card is an even numbered card) = 20/52 = 5/13

⇒ Combined Probability = (1/13) × (1/2) × (5/13)

⇒ Combined Probability = 5 / (13 × 2 × 13) = 5 / (26 × 13)

⇒ Combined Probability = 5/338

∴ The correct answer is 5/338.

Additional Information

Independent Events with Replacement

When objects are replaced, the total number of outcomes remains the same for every draw, making the events independent.

Dependent Events without Replacement

If cards were drawn without replacement, the denominator would decrease: P(A) × P(B|A) × P(C|A,B) with totals 52, 51, and 50 respectively.

Face Cards and Numbered Cards

A standard deck has 12 face cards (J, Q, K) and 36 numbered cards (2 to 10). Ace is usually not counted as a numbered card in competitive exams unless specified.

Shortcut Trick

To find the number of 4-legged animals (Monkeys) in a group of 2-legged and 4-legged animals:

Number of Monkeys = (Total Legs − 2 × Total Heads) ÷ 2

⇒ Number of Monkeys = (148 − 2 × 54) ÷ 2

⇒ Number of Monkeys = (148 − 108) ÷ 2 = 40 ÷ 2 = 20

∴ The correct answer is 20 monkeys.

Alternate Method

Given:

Total number of heads = 54

Total number of legs = 148

Number of legs for a human = 2

Number of legs for a monkey = 4

Calculations:

⇒ Let the number of monkeys be M and humans be H.

⇒ Total heads: M + H = 54 → (Eq. 1)

⇒ Total legs: 4M + 2H = 148 → (Eq. 2)

⇒ Multiply (Eq. 1) by 2: 2M + 2H = 108 → (Eq. 3)

⇒ Subtract (Eq. 3) from (Eq. 2): (4M − 2M) + (2H − 2H) = 148 − 108

⇒ 2M = 40

⇒ M = 40 ÷ 2 = 20

∴ The correct answer is 20 monkeys.

Additional Information

Alligation Method for Heads/Legs

Ratio of 4-legged to 2-legged animals = (Average legs per head − 2) : (4 − Average legs per head).

Linear Equation Method

Solving systems of equations x + y = Total Heads and Ax + By = Total Legs, where A and B are the number of legs per animal.

Assumption Shortcut

Assume all animals have 2 legs: Total legs would be 2 × Heads. Any excess legs (Total Legs − Calculated Legs) belong to the 4-legged animals in pairs.

Shortcut Trick

LCM of speeds 30 and 40 = 120.

Let 1/3 distance = 120 km, then 2/3 distance = 240 km.

Total hypothetical distance = 120 + 240 = 360 km.

Total time taken = (120 ÷ 30) + (240 ÷ 40) = 4 + 6 = 10 hours.

Given time is 12 hours ⇒ Actual distance = (360 ÷ 10) × 12 = 432 km.

∴ The correct answer is 432 km.

Alternate Method

Given:

Total Time = 12 hours

Speed 1 (for 1/3 distance) = 30 km/h

Speed 2 (for 2/3 distance) = 40 km/h

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Let the total distance be x km.

⇒ Distance 1 = x/3 km and Distance 2 = 2x/3 km.

⇒ Time 1 + Time 2 = 12

⇒ (x/3) ÷ 30 + (2x/3) ÷ 40 = 12

⇒ x/90 + 2x/120 = 12

⇒ x/90 + x/60 = 12

⇒ (2x + 3x) ÷ 180 = 12

⇒ 5x ÷ 180 = 12

⇒ x ÷ 36 = 12

⇒ x = 36 × 12 = 432 km.

∴ The correct answer is 432 km.

Additional Information

Average Speed Formula

Average speed is calculated as Total Distance ÷ Total Time. If the distances are equal, the formula is 2uv ÷ (u+v).

Distance, Speed, and Time Relation

Distance = Speed × Time. When distance is constant, speed is inversely proportional to time (S₁ ÷ S₂ = T₂ ÷ T₁).

Conversion of Units

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Shortcut Trick

Transfer 1 (20L to B): Ratio Spirit(S) : Water(W) is 7 : 9 ⇒ S = 8.75, W = 11.25.

In A: Remaining 60L (S=26.25, W=33.75) + 20L W ⇒ S=26.25, W=53.75 (Ratio 21 : 43).

Transfer 2 (32L to B): Ratio S : W is 21 : 43 ⇒ S = 10.5, W = 21.5.

Total in B: S = 8.75 + 10.5 = 19.25 and W = 11.25 + 21.5 = 32.75.

Ratio Water : Spirit in B = 32.75 : 19.25 = 131 : 77.

∴ The correct answer is 131 ∶ 77.

Alternate Method

Given:

Total Volume (A) = 80 litres | Initial Ratio (S : W) = 7 : 9

Initial S = (7 ÷ 16) × 80 = 35 litres | Initial W = (9 ÷ 16) × 80 = 45 litres

Step 1: First transfer to B

⇒ Quantity transferred = 20 litres

⇒ S in B₁ = (7/16) × 20 = 8.75 litres

⇒ W in B₁ = (9/16) × 20 = 11.25 litres

Step 2: Change in Container A

⇒ Remaining S in A = 35 − 8.75 = 26.25 litres

⇒ Remaining W in A = 45 − 11.25 = 33.75 litres

⇒ After adding 20L water: Total W in A = 33.75 + 20 = 53.75 litres

⇒ New ratio S : W in A = 26.25 : 53.75 = 2625 : 5375 = 21 : 43

Step 3: Second transfer to B

⇒ Quantity transferred = 32 litres

⇒ S in B₂ = (21/64) × 32 = 10.5 litres

⇒ W in B₂ = (43/64) × 32 = 21.5 litres

Step 4: Final Ratio in B

⇒ Total Spirit in B = 8.75 + 10.5 = 19.25 litres

⇒ Total Water in B = 11.25 + 21.5 = 32.75 litres

⇒ Water : Spirit = 32.75 : 19.25 = 3275 : 1925

⇒ Dividing by 25: 131 : 77

∴ The correct answer is 131 ∶ 77.

Additional Information

Ratio Distribution Rule

When a part of a mixture is removed, the ratio of ingredients in the removed portion and the remaining portion remains the same as the original ratio.

Replacement Formula

Quantity of original liquid remaining after n replacements: C_n = C₀(1 − x/V)ⁿ, where x is volume replaced and V is total volume.

Compound Ratio of Mixtures

If two mixtures with ratios a:b and c:d are mixed in ratio x:y, the final ratio of the first component is (ax/(a+b) + cy/(c+d)) / (bx/(a+b) + dy/(c+d)).

Shortcut Trick

Edge of small cube = ³√64 = 4 cm.

Number of cubes = (Large Edge ÷ Small Edge)³

Number of cubes = (48 ÷ 4)³ = 12³ = 1728.

∴ The correct answer is 1728 cubes.

Alternate Method

Given:

Edge of large cube = 48 cm

Volume of each small cube = 64 cm³

Formula Used:

Volume of a cube = (Edge)³

Number of cubes = Total Volume ÷ Volume of each small cube

⇒ Volume of the large cube = 48 × 48 × 48

⇒ Volume of the large cube = 110592 cm³

⇒ Volume of one small cube = 64 cm³

⇒ Number of small cubes (n) = 110592 ÷ 64

⇒ n = 1728

∴ The correct answer is 1728 cubes.

Additional Information

Melting and Recasting Rule

When a solid is melted and recast into another shape (or multiple smaller shapes) without wastage, the total volume remains constant.

Properties of a Cube

For a cube of side 'a', Volume = a³, Total Surface Area = 6a², and Diagonal = a√3.

Scaling Factor in Volume

If the dimensions of a 3D object are scaled by a factor 'k', its volume is scaled by a factor of k³.

Given:

$[\frac{1}{8}+{\frac{1}{6} \times(\frac{36}{45} \div \frac{24}{25}) - (\frac{12}{21} \times \frac{14}{15} \div \frac{24}{45})}+\frac{27}{36}]$

Formula used:

BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction)

Calculation:

$[\frac{1}{8}+{\frac{1}{6} \times (\frac{36}{45} \times \frac{25}{24}) - (\frac{12}{21} \times \frac{14}{15} \times \frac{45}{24})} + \frac{27}{36}]$

⇒ $[\frac{1}{8}+{\frac{1}{6} \times (\frac{4}{5} \times \frac{25}{24}) - (\frac{4}{7} \times \frac{14}{15} \times \frac{15}{8})} + \frac{3}{4}]$

⇒ $[\frac{1}{8}+{\frac{1}{6} \times \frac{5}{6} - (1 \times 1 \times 1)} + \frac{3}{4}]$

⇒ $[\frac{1}{8}+{\frac{5}{36} - 1} + \frac{3}{4}]$

⇒ $[\frac{1}{8} - \frac{31}{36} + \frac{27}{36}]$

⇒ $[\frac{1}{8} - \frac{4}{36}]$

⇒ $[\frac{1}{8} - \frac{1}{9}]$

⇒ $\frac{9-8}{72}$

⇒ $\frac{1}{72}$

∴ The correct answer is option (2).

Shortcut Trick

Height of pillars (h) = Road Width ÷ (cot x + cot y)

Given tan x = 2/5 → cot x = 5/2

Given tan y = 3/5 → cot y = 5/3

h = 150 ÷ (5/2 + 5/3) = 150 ÷ (25/6) = 150 × 6 ÷ 25

h = 6 × 6 = 36 m

∴ The correct answer is 36 m.

Alternate Method

Given:

Road width (W) = 150 m

Angle of elevation 1 = x, where tan x = 2/5

Angle of elevation 2 = y, where tan y = 3/5

⇒ Let the height of each pillar be h.

⇒ Let d₁ be the distance from the point to the first pillar and d₂ be the distance from the point to the second pillar.

⇒ d₁ + d₂ = 150

⇒ In first triangle: tan x = h/d₁ ⇒ 2/5 = h/d₁ ⇒ d₁ = 5h/2

⇒ In second triangle: tan y = h/d₂ ⇒ 3/5 = h/d₂ ⇒ d₂ = 5h/3

⇒ 5h/2 + 5h/3 = 150

⇒ 5h × (1/2 + 1/3) = 150

⇒ 5h × (5/6) = 150

⇒ 25h/6 = 150

⇒ h = (150 × 6) ÷ 25

⇒ h = 6 × 6 = 36 m

∴ The correct answer is 36 m.

Additional Information

Height and Distance Relation

For a point between two objects of same height h at distance W, h = W ÷ (cot α + cot β), where α and β are angles of elevation.

Angle of Elevation

The angle between the horizontal line of sight and the object when the object is above the observer level.

Standard Trig Ratios

tan θ = Opposite / Adjacent and cot θ = Adjacent / Opposite. They are reciprocals of each other.

Shortcut Trick

Let cost of Pencil = x and Pen = y.

Equations: 2x + 3y = 55 and 7x + 2y = 65.

Subtracting both: (7x − 2x) − (3y − 2y) = 65 − 55 ⇒ 5x − y = 10 ⇒ y = 5x − 10.

Substitute y in first: 2x + 3(5x − 10) = 55 ⇒ 17x = 85 ⇒ x = 5 and y = 15.

Total cost (x + y) = 5 + 15 = 20.

∴ The correct answer is ₹20.

Alternate Method

Given: Cost of 2 Pencils and 3 Pens is ₹55. Cost of 7 Pencils and 2 Pens is ₹65.

⇒ Let cost of 1 pencil = ₹x and 1 pen = ₹y.

⇒ 2x + 3y = 55 ... (1)

⇒ 7x + 2y = 65 ... (2)

⇒ Multiply (1) by 2 and (2) by 3 to eliminate y:

⇒ 4x + 6y = 110 ... (3)

⇒ 21x + 6y = 195 ... (4)

⇒ Subtract (3) from (4): (21x − 4x) = (195 − 110) ⇒ 17x = 85

⇒ x = 85 ÷ 17 = 5

⇒ Put x = 5 in (1): 2(5) + 3y = 55 ⇒ 10 + 3y = 55 ⇒ 3y = 45 ⇒ y = 15

⇒ Total price of one pen and one pencil = x + y = 5 + 15 = 20

∴ The correct answer is ₹20.

Additional Information

Simultaneous Linear Equations

Equations involving two variables (x, y) can be solved by making the coefficients of one variable the same and subtracting/adding the equations.

Substitution Method

Express one variable in terms of the other from one equation (e.g., y = 5x − 10) and substitute it into the second equation.

Cost Calculation Rule

Total Price = (Quantity₁ × Price₁) + (Quantity₂ × Price₂). To find individual prices, at least two independent conditions are required.

Shortcut Trick

Given P ∶ Q = 3 ∶ 4 and Q ∶ R = 4 ∶ 7.

Since the value of Q is common (4 units) in both ratios, we can directly write P ∶ Q ∶ R = 3 ∶ 4 ∶ 7.

Now, calculate the required terms: (P + Q) = 3 + 4 = 7; (Q + R) = 4 + 7 = 11; (R + P) = 7 + 3 = 10.

The ratio (P + Q) ∶ (Q + R) ∶ (R + P) = 7 ∶ 11 ∶ 10.

∴ The correct answer is 7 ∶ 11 ∶ 10.

Alternate Method

Given:

P ∶ Q = 3 ∶ 4

Q ∶ R = 4 ∶ 7

Formula Used:

If a ∶ b and b ∶ c are given, and the value of 'b' is same in both, then a ∶ b ∶ c = a ∶ b ∶ c.

Calculations:

⇒ Let P = 3x, Q = 4x, and R = 7x.

⇒ (P + Q) = 3x + 4x = 7x

⇒ (Q + R) = 4x + 7x = 11x

⇒ (R + P) = 7x + 3x = 10x

⇒ Ratio = (P + Q) ∶ (Q + R) ∶ (R + P)

⇒ Ratio = 7x ∶ 11x ∶ 10x

⇒ Ratio = 7 ∶ 11 ∶ 10

∴ The correct answer is 7 ∶ 11 ∶ 10.

Additional Information

Compounded Ratio

The compounded ratio of two ratios a ∶ b and c ∶ d is found by multiplying the corresponding terms: (a × c) ∶ (b × d).

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a ∶ b is a² ∶ b², and the sub-duplicate ratio is √a ∶ √b.

Normalization of Ratios

To combine ratios like a ∶ b and b ∶ c where 'b' values are different, multiply each ratio by a constant to make the 'b' terms equal.

Shortcut Trick

For divisibility by 24, the number must be divisible by both 8 and 3.

Last three digits 9q4 ÷ 8 ⇒ 904, 944, 984 are divisible by 8, so q ∈ {0, 4, 8}.

Sum of digits (1+1+p+9+q+4) = 15+p+q must be divisible by 3 ⇒ (p+q) must be a multiple of 3.

Case 1: If q = 8, max p = 7 (as 7+8=15) ⇒ pq = 7 × 8 = 56.

Case 2: If q = 4, max p = 8 (as 8+4=12) ⇒ pq = 8 × 4 = 32.

Case 3: If q = 0, max p = 9 (as 9+0=9) ⇒ pq = 9 × 0 = 0.

∴ The correct answer is 56.

Alternate Method

Given: Number = 11p9q4, Divisor = 24.

Property: A number is divisible by 24 if it is divisible by 8 (last 3 digits) and 3 (sum of digits).

⇒ Divisibility by 8: The last three digits 9q4 must be divisible by 8.

⇒ 900 + 10q + 4 = 904 + 10q. Since 904 is divisible by 8 (904 = 8 × 113), 10q must be divisible by 8.

⇒ Possible values for q are 0, 4, and 8.

⇒ Divisibility by 3: Sum of digits = 1 + 1 + p + 9 + q + 4 = 15 + p + q.

⇒ For (15 + p + q) to be divisible by 3, (p + q) must be 0, 3, 6, 9, 12, 15, or 18.

⇒ If q = 8: p + 8 = 9, 12, 15 ⇒ p = 1, 4, 7. Max pq = 7 × 8 = 56.

⇒ If q = 4: p + 4 = 6, 9, 12 ⇒ p = 2, 5, 8. Max pq = 8 × 4 = 32.

⇒ If q = 0: p + 0 = 0, 3, 6, 9 ⇒ p = 0, 3, 6, 9. Max pq = 9 × 0 = 0.

⇒ Comparing all products, the greatest value is 56.

∴ The correct answer is 56.

Additional Information

Divisibility by 8

A number is divisible by 8 if its last three digits are divisible by 8 or are zeros.

Divisibility by 3

A number is divisible by 3 if the sum of all its individual digits is divisible by 3.

Composite Divisibility Rule

If a number is divisible by two co-prime numbers a and b, then it is also divisible by their product (a × b).

Shortcut Trick

Effective rate for half-yearly compounding = 20% ÷ 2 = 10%.

Ratio of Amount to Principal (A/P) = 14,641 / 10,000 = (1.4641/1).

Since (1 + 10/100)ⁿ = 1.1ⁿ, we observe that 1.1⁴ = 1.4641.

Total periods (n) = 4 half-years.

Time in years = 4 ÷ 2 = 2 years.

∴ The correct answer is 2 years.

Alternate Method

Given:

Principal (P) = Rs. 10,000

Compound Interest (CI) = Rs. 4,641

Rate (R) = 20% per annum (Compounded half-yearly)

Formula Used: Amount (A) = P × (1 + r/100)ⁿ, where r is the rate per period and n is the number of periods.

Calculations:

⇒ Amount (A) = Principal + Compound Interest

⇒ A = 10,000 + 4,641 = 14,641

⇒ For half-yearly compounding, Rate (r) = 20% ÷ 2 = 10% per half-year.

⇒ Let the total periods be n.

⇒ 14,641 = 10,000 × (1 + 10/100)ⁿ

⇒ 14,641 / 10,000 = (11/10)ⁿ

⇒ 1.4641 = (1.1)ⁿ

⇒ (1.1)⁴ = (1.1)ⁿ

⇒ n = 4 periods

⇒ Since each period is 6 months, Total Time = 4 × 6 = 24 months = 2 years.

∴ The correct answer is 2 years.

Additional Information

CI Compounded Half-Yearly

When interest is compounded half-yearly, the annual rate is divided by 2 and the time (in years) is multiplied by 2.

CI Compounded Quarterly

When interest is compounded quarterly, the annual rate is divided by 4 and the time (in years) is multiplied by 4.

Effective Rate Formula

For two periods with rate r%, the effective rate is [r + r + (r × r)/100] %.

Difference between CI and SI

For 2 years, CI − SI = P × (R/100)². For 3 years, CI − SI = P × (R/100)² × (3 + R/100).

Shortcut Trick

Ratio of Kamal : (Hari + Lalit) = 142 : 100 = 71 : 50.

Total parts = 71 + 50 = 121 units.

121 units = 6050 ⇒ 1 unit = 50.

Hari + Lalit = 50 × 50 = 2500.

Lalit = 8% of 6050 = 484.

Hari = 2500 − 484 = 2016.

∴ The correct answer is 2016 votes.

Alternate Method

Given:

Total polled votes = 6050

Lalit's share = 8% of total votes

Kamal's share = (Hari + Lalit) + 42% of (Hari + Lalit)

Calculations:

⇒ Votes obtained by Lalit = 8% of 6050

⇒ Lalit = (8 × 6050) ÷ 100 = 484

⇒ Let votes obtained by (Hari + Lalit) = x

⇒ Kamal's votes = x + 0.42x = 1.42x

⇒ Total votes = Kamal + (Hari + Lalit)

⇒ 6050 = 1.42x + x

⇒ 6050 = 2.42x

⇒ x = 6050 ÷ 2.42 = 2500

⇒ Hari + Lalit = 2500

⇒ Hari + 484 = 2500

⇒ Hari = 2500 − 484 = 2016

∴ The correct answer is 2016 votes.

Additional Information

Election Vote Concepts

Total Votes Polled = Valid Votes + Invalid Votes. In this case, all polled votes are valid.

Percentage Comparison

If A is r% more than B, then A = B × (1 + r/100). If A is r% less than B, then A = B × (1 − r/100).

Ratio Partitioning

To divide a total 'T' into a ratio a : b, the first part is [a / (a+b)] × T and the second part is [b / (a+b)] × T.

Shortcut Trick

Total Work = LCM(12.5, 18.75, 15) = 75 units.

Eff(A+B) = 6, Eff(B+C) = 4, Eff(C+A) = 5.

Combined Eff(A+B+C) = (6 + 4 + 5) ÷ 2 = 7.5 units/day.

Eff(C) = 7.5 − 6 = 1.5 units/day | Eff(D) = 40% of 1.5 = 0.6 units/day.

Total Time = 75 ÷ (7.5 + 0.6) = 75 ÷ 8.1 = 250/27 = 9 7/27 days.

∴ The correct answer is 9 7/27 days.

Alternate Method

Given:

Time (A + B) = 12.5 days = 25/2 days

Time (B + C) = 18.75 days = 75/4 days

Time (C + A) = 15 days

Efficiency of D = 40% of Efficiency of C

Calculations:

⇒ Total efficiency of (A+B), (B+C), and (C+A) = 6 + 4 + 5 = 15

⇒ 2 × Efficiency (A + B + C) = 15

⇒ Efficiency (A + B + C) = 15 ÷ 2 = 7.5 units/day

⇒ Efficiency of C = Efficiency (A + B + C) − Efficiency (A + B)

⇒ Efficiency of C = 7.5 − 6 = 1.5 units/day

⇒ Efficiency of D = 40/100 × 1.5 = 0.6 units/day

⇒ Combined efficiency (A + B + C + D) = 7.5 + 0.6 = 8.1 units/day

⇒ Total days = Total Work ÷ Combined Efficiency = 75 ÷ 8.1

⇒ Total days = 750 ÷ 81 = 250 ÷ 27 = 9 7/27 days

∴ The correct answer is 9 7/27 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to time. If a person takes 'n' days to complete a work, their efficiency is 1/n per day.

Combined Work Formula

For pairs (A+B, B+C, C+A), individual efficiency of A = [Eff(A+B) + Eff(C+A) − Eff(B+C)] ÷ 2.

Efficiency Ratios

If X is k% as efficient as Y, then Efficiency(X) = (k/100) × Efficiency(Y).

Shortcut Trick

Total present age of A, B and C = 20 × 3 = 60 years.

Two years ago, (A − 2) + (B − 2) = (C − 2) + 6 ⇒ (A + B) − C = 8.

We have: (A + B) + C = 60 and (A + B) − C = 8.

By subtracting the equations: 2C = 60 − 8 = 52 ⇒ C = 26 years.

∴ The correct answer is 26 years.

Alternate Method

Given:

Average age of A, B, C = 20 years.

Condition 2 years ago: (Sum of A & B) = (Age of C) + 6.

Formula Used:

Total Age = Average × Number of persons.

Calculations:

⇒ Present sum A + B + C = 20 × 3 = 60 --- (i)

⇒ Ages 2 years ago: (A−2), (B−2), (C−2)

⇒ According to question: (A − 2) + (B − 2) = (C − 2) + 6

⇒ A + B − 4 = C + 4

⇒ A + B − C = 8 --- (ii)

⇒ From (i), A + B = 60 − C

⇒ Substitute in (ii): (60 − C) − C = 8

⇒ 60 − 2C = 8

⇒ 2C = 52

⇒ C = 26 years.

∴ The correct answer is 26 years.

Additional Information

Average Age Concept

If the average age of n persons is X years, then after t years, the average age will be (X + t) years.

Sum of Ages

To find the sum of ages of a group, multiply the average age by the number of members in that group (Sum = Average × n).

Age Difference

The difference between the ages of two individuals remains constant regardless of the number of years passed.

Given:

$\frac{\sqrt{1.24} \times \sqrt{2.79}}{\sqrt{2.64} \times \sqrt{5.94}}$ = ?

Formula used:

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Calculation:

$\frac{\sqrt{1.24} \times \sqrt{2.79}}{\sqrt{2.64} \times \sqrt{5.94}}$

⇒ $\sqrt{\frac{1.24 \times 2.79}{2.64 \times 5.94}}$

⇒ $\sqrt{\frac{124 \times 279}{264 \times 594}}$

⇒ $\sqrt{\frac{(4 \times 31) \times (9 \times 31)}{(4 \times 66) \times (9 \times 66)}}$

⇒ $\sqrt{\frac{4 \times 9 \times 31^2}{4 \times 9 \times 66^2}}$

⇒ $\sqrt{\frac{31^2}{66^2}}$

⇒ $\frac{31}{66}$

∴ The correct answer is option (1).

Shortcut Trick

Apply the algebraic identity a² − b² = (a − b)(a + b) to simplify the terms.

A = (58 − 25)(58 + 25) / (46 − 37)(46 + 37) = (33 × 83) / (9 × 83) = 33 / 9 = 11 / 3.

B = (26 − 15)(26 + 15) / (56 − 15)(56 + 15) = (11 × 41) / (41 × 71) = 11 / 71.

Expression = 1/B − 20/A = 71 / 11 − 20 × (3 / 11) = 71/11 − 60/11 = 11/11 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

A = (58² − 25²) / (46² − 37²)

B = (26² − 15²) / (56² − 15²)

Formula Used:

x² − y² = (x − y)(x + y)

Calculations:

⇒ A = ((58 − 25)(58 + 25)) / ((46 − 37)(46 + 37))

⇒ A = (33 × 83) / (9 × 83)

⇒ A = 33 / 9 = 11 / 3

⇒ B = ((26 − 15)(26 + 15)) / ((56 − 15)(56 + 15))

⇒ B = (11 × 41) / (41 × 71)

⇒ B = 11 / 71

⇒ Required Expression = 1/B − 20/A

⇒ Required Expression = 1 / (11/71) − 20 / (11/3)

⇒ Required Expression = 71 / 11 − (20 × 3) / 11

⇒ Required Expression = 71 / 11 − 60 / 11

⇒ Required Expression = 11 / 11 = 1

∴ The correct answer is 1.

Additional Information

Difference of Squares

The formula a² − b² = (a − b)(a + b) is essential for simplifying fractions where numerator and denominator are differences of squares.

Fraction Reciprocals

The reciprocal of a fraction x/y is y/x. When finding 1/B, you simply flip the numerator and denominator of B.

BODMAS Rule

Always follow the BODMAS rule (Bracket, Of, Division, Multiplication, Addition, Subtraction) to ensure calculations are performed in the correct sequence.

Shortcut Trick

Value of x = 30% of 2800 = 840

Value of y = 1736 ÷ (14% of 6200) = 1736 ÷ 868 = 2

Required ratio x/y = 840 ÷ 2 = 420

∴ The correct answer is 420.

Alternate Method

Case 1: P = Rs. 2800, R = 10%, T = 3 years SI = x = 840

Case 2: P = Rs. 6200, R = 14%, SI = Rs. 1736 T = y = 2 Result: x / y = 840 / 2 = 420

Given:

First case: Principal (P₁) = Rs. 2800, Rate (R₁) = 10%, Time (T₁) = 3 years, SI₁ = Rs. x

Second case: Principal (P₂) = Rs. 6200, Rate (R₂) = 14%, Time (T₂) = y years, SI₂ = Rs. 1736

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculation:

⇒ First, we calculate the value of x:

⇒ x = (2800 × 10 × 3) ÷ 100

⇒ x = 28 × 30

⇒ x = 840

⇒ Now, we calculate the value of y:

⇒ 1736 = (6200 × 14 × y) ÷ 100

⇒ 1736 = 62 × 14 × y

⇒ 1736 = 868y

⇒ y = 1736 ÷ 868

⇒ y = 2

⇒ The required value of x/y is:

⇒ x/y = 840 ÷ 2 = 420

∴ The correct answer is 420.

Additional Information

Simple Interest (SI)

It is calculated only on the principal amount for the entire period. Formula: SI = (P × R × T) ÷ 100.

Amount in SI

The total accumulated sum at the end of the time period. Formula: A = P + SI = P(1 + (R × T) ÷ 100).

Difference between SI and CI

In simple interest, interest remains constant every year, whereas in compound interest, interest is calculated on the principal and the accumulated interest of previous years.

Shortcut Trick

In a circle, the diameter is the longest possible chord.

Any other chord must be less than or equal to the diameter.

Mathematically, Diameter (2r) ≥ Chord length.

2r ≥ 20.5 ⇒ r ≥ 10.25.

∴ The correct answer is r ≥ 10.25.

Alternate Method

Given:

Length of chord = 20.5 cm

Radius of the circle = r

Formula/Property Used:

Diameter of a circle (D) = 2 × radius (r)

Geometric Property: Diameter ≥ Any Chord of the circle.

Calculations:

⇒ Diameter ≥ Chord

⇒ 2r ≥ 20.5 cm

⇒ r ≥ 20.5 ÷ 2

⇒ r ≥ 10.25

∴ The correct answer is r ≥ 10.25.

Additional Information

Longest Chord Property

The diameter is the unique chord that passes through the center of the circle and is the maximum length of any chord.

Chord Perpendicularity

A perpendicular drawn from the center of a circle to a chord bisects the chord into two equal halves.

Chord Distance from Center

Longer chords are always closer to the center of the circle compared to shorter chords. The diameter (the longest chord) has a distance of zero from the center.

Given Statement: I < M < F ≤ D ≤ S = J

Now,

Conclusions:

Conclusion I: S < M → False (S = J ≥ D ≥ F > M > I It gives S > M)

Conclusion II: F = S → False (F ≤ D ≤ S = J It gives F ≤ S)

Conclusion III: I > J → False (I < M < F ≤ D ≤ S = J It gives I < J)

Conclusion IV: M < D → True (M < F ≤ D It gives M < D)

Hence, the correct answer is "Option 1: Only conclusion IV is true.".

Analysis:

• Reason I: This reason is highly plausible as a major driving force for migration, especially in the context of India. People often move from rural to urban areas primarily for better job opportunities, higher incomes, and improved living standards, which are more readily available in metropolitan areas.
• Reason II: While some individuals might migrate for the experience of travel and exploration, this is not a common or primary reason for large-scale migration patterns, especially those captured in census data, which reflect more permanent or long-term residency changes driven by economic factors rather than leisure or travel interests.

Correct Answer:

1. Only reason I explains the migration to large cities in India.

The logic followed here is:

So, '19114' will come in place of question mark.

Hence, the correct answer is 'Option 3'.

The logic follows here is:

Logic: The letters and symbols are getting shifted it's position as shown below.

Hence, the correct answer is "Option 1".

BODMAS Rule:

According to the given question:

Given equation : 49.85 – 5.31 + 9.97 = ?

To simplify, we perform the operations in order:

⇒ 49.85 – 5.31 = 44.54

⇒ 44.54 + 9.97 = 54.51

Rounding 54.51 to the nearest whole number:

⇒ 54.51 ≈ 55

Hence, the correct answer is 'Option 4'.

The logic followed here is:

Logic: 1^st number × 2^nd number = 3^rd number.

Now,

(5, 5, 25)

⇒ 5 × 5 = 25.

(2, 5, 10)

⇒ 2 × 5 = 10.

Similarly, for (3, 3, 9)

⇒ 3 × 3 = 9.

Hence, the correct answer is "Option 1".

To determine the logical and meaningful order of the given words related to the making of a shirt,

let's consider the sequence of production from raw material to the finished product:

• Cotton (raw material)
• Thread (spun from cotton)
• Fabric (woven from thread)
• Shirt piece (cut from fabric)
• Shirt (sewn from shirt piece)

So, the correct order is:  5. Cotton, 2.Thread, 3. Fabric, 1. Shirt piece, 4. Shirt

Hence, the correct option is: 5, 2, 3, 1, 4

The logic followed here is:

So, 'CJRSB' will be the correct alternative that will complete the series.

Hence, the correct answer is "Option 2".

Given Number → (Left) 7148356 (Right) 

Arrangement of digits in the ascending order from left to right:-

(Left) 1345678 (Right)

Digit 4's Position are remain unchanged as compared to that in the original number 

(Left) 7148356 (Right) → (Left) 1345678 (Right)

Hence, the correct answer is "One".

BODMAS Rule:

According to the given question:

 

Given equation : 163 ÷ 3 + 147 × 7 – 1 = ?

Now if '+' and '–' are interchanged with '÷' and '×' respectively then:

⇒ 163 × 3 - 147 ÷ 7 + 1 = ?

⇒ 163 × 3 - 21 + 1 = ?

⇒ 489 - 20 = ?

⇒ 469 = ?

Hence, the correct answer is 'Option 4'.

Given: A certain number of people are sitting in a row, facing north.

1. N is sitting third to the left of P.

2. M is sitting third to the left of N.

3. O is sitting third to the left of M.

4. M is sitting between O and N.

5. N is sitting between M and P.

If no other person is sitting in the row, Total '10' persons seated in the row.

Hence, the correct answer is "Option 2".

The logic followed here is:

Option 1) Travel : Tour → "Tour" is a type of "Travel".

Option 2) Close : Distant → "Close" and "Distant" are antonyms.

Option 3) Look : See → "See" is synonymous with "Look".

Option 4) Carry : Lift → "Lift" can be synonymous with "Carry".

Here, 'Close : Distant' is odd among the options as it pairs antonyms, whereas the others pair synonyms or represent a subset of the other

Hence, the correct answer is "Option 2".

BODMAS Rule:

According to the given question:

 

Given equation : 113 ÷ 4 – 217 × 7 + 4 = ?

Now if ‘+’ and ‘–’ are interchanged and ‘×’and ‘÷’ are interchanged

⇒ 113 × 4 + 217 ÷ 7 - 4 = ?

⇒ 113 × 4 + 31 - 4 = ?

⇒ 452 + 31 - 4 = ?

⇒ 479 = ?

Hence, the correct answer is 'Option 4'.

Given: A, B, C, D, E and F are sitting around a circular table. Each of them faces either the centre of the table or outside.

Explanation:

Statement I: B sits second to the right of F and both are facing the centre. Only two people are sitting between A and E from the either sides. Both A and E are facing the same direction. A is not an immediate neighbor of B.

So, No statement given regarding D.

Hence, Statement I is not sufficient to answer the Question.

Statement II: C is facing the opposite direction of B. D is sitting to the second left of C.

The facing direction of C does not given, so Statement II is not sufficient to answer the Question.

Combining both the statements :Statement I + Statement II

(I) B sits second to the right of F and both are facing the centre. Only two people are sitting between A and E from the either sides. Both A and E are facing the same direction. A is not an immediate neighbor of B.

(II) C is facing the opposite direction of B. D is sitting to the second left of C.

Here, Facing direction of 'D' is also not confirmed on combining the both statements.

Therefore, Statements I and II together are not sufficient to answer the question.

Hence, the correct answer is 'Option 2'.

The logic followed here is:

Logic: All the figures having circle between the images except figure B.

So, figure 'B' is odd among the given figures.

Hence, the correct answer is "Option 3".

To answer the question, we examine if each inference can be directly drawn from the statement about rising fuel prices in India:

Inference I: This suggests that the tax rates by central and state governments have remained the same. While it's possible, the statement about the rising fuel prices doesn't specifically attribute the rise to taxation policies, nor does it imply that taxes have remained unchanged. It's an assumption without direct support from the statement.

Inference II: This suggests that international market inflation affects fuel prices, which is a common and plausible reason for changes in fuel prices, given that they often correlate with international oil market trends. However, the original statement does not specifically mention international market conditions affecting the prices. It's again an assumption, though somewhat reasonable.

Since neither inference can be directly concluded from the statement given.

Hence, the correct answer is Option 3: Neither inference I nor II.

Statement I: Navya is the daughter of Kishore. Vinita is the wife of Kishore.

So, according to statement (I), No information given about Sujata.

So, Statement I is not sufficient to answer the question.

Statement II: Aman is the brother of Navya. Sujata is the only daughter of Aman.

So, according to statement (II), No information is given about Kishore.

So, Statement II is not sufficient to answer the question.

Now using both the statements:

Statement (I) + Statement (II) → Navya is the daughter of Kishore. Vinita is the wife of Kishore. Aman is the brother of Navya. Sujata is the only daughter of Aman.

So, using both the statements, Kishore is the 'Grandfather' of Sujata..

Thus, Both (I) and (II) together are needed.

Hence, the correct answer is 'Option 3'.

The logic followed here is:

Logic: The second word describes a characteristic of the first word.

Here,

Crowd : Dense → 'Dense' describes the characteristic of being closely packed, which is typical of a crowd.

Similarly,

Slope : ? → 'Steep' describes the characteristic of a sharp gradient, which is typical of some slopes.

Hence, the correct answer is "Steep".

Additional Information

• Plain is not characteristic of slopes as it implies flatness or lack of elevation, which is opposite to what a slope is.
• Tight does not directly relate to the concept of a slope.
• Fill is not a descriptive characteristic of a slope.

BODMAS Rule:

According to the given question:

 

Given equation : 161 ÷ 4 + 75 × 5 – 4 = ?

Now if '+' and '–' are interchanged with '–' and '+' respectively and '×' and '÷' are interchanged then:

⇒ 161 × 4 – 75 ÷ 5 + 4 = ?

⇒ 644 – 15 + 4 = ?

⇒ 629 + 4 = ?

⇒ 633 = ?

Hence, the correct answer is 'Option 2'.

Family Chart:

Given: 

Q is the mother of R.

P is the husband of Q.

S is the brother of R.

So, P is 'Father' of S.

Hence, the correct answer is "Option 1".

Given: Seven people, A, B, C, D, E, F and G are sitting in a straight line, facing north.

1. Only three people sit to the left of D.

2. Only two people sit between D and A.

There are two cases possible as we do not know whether A will sit to the right of D or left of D.

3)  B sits fifth to the right of C.

4)  G sits to the immediate right of E but somewhere to the left of A.

So, in the case 2, there will on place left for E if we place G to the left of A.

Thus, it will be eliminated. So, we will continue with case 1.

And, F will sit on the remaining place.

So, 'A' sits to the extreme right end.

Hence, the correct answer is "Option 2".

The logic followed here is:

Given Sequence: P R _ Y _ _ R A _ Z P _ A _ Z

Now, putting the letters in the blank spaces given in the option one by one:

Option 1)  A, Z, Y, Z, R, Y  →  P R A Y Z / Y R A Z Z / P R A Y Z

Option 2)  A, P, Y, Z, Z, Y  →  P R A Y P / Y R A Z Z / P Z A Y Z

Option 3)  A, Z, P, Y, R, Y  →  P R A Y Z / P R A Y Z / P R A Y Z

Option 4)  A, P, Y, Z, R, Y  →  P R A Y P / Y R A Z Z / P R A Y Z

Here, 'PRAYZ' followed a pattern.

Hence, the correct answer is "Option 3".

The logic followed here is:

Figure A and B: the figure A is getting rotated 180^o to form figure B.

Similarly, C and D: the figure C is getting rotated 180o to form figure D.

Hence, the correct answer is "Option 3".

Let's identify the second, fourth, fifth, and sixth letters of the word "HOCKEY":

Second letter: O, Fourth letter: K, Fifth letter: E, Sixth letter: Y

So, the letters we have are: O, K, E, and Y.

Now, let's see if we can form meaningful English words with these letters:

OKEY - Not a standard English word.

YOKE - A meaningful English word (a wooden crosspiece fastened over the necks of two animals and attached to the plow or cart that they are to pull).

KEYO - Not a standard English word.

Therefore, only one meaningful English word can be formed.

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic: The first number multiplied by 9 equals the second number.

Now, check the options:

Option 1) 13 - 117 → 13 × 9 = 117.

Option 2) 11 - 89 → 11 × 9 = 99 ≠ 89.

Option 3) 18 - 162 → 18 × 9 = 162.

Option 4) 15 - 135 → 15 × 9 = 135.

So, '11 - 89' does not follow the same logic as the other options.

Hence, the correct answer is "Option 2".

For ‘BUTTER’ → ‘CWWSCO’:

For ‘THEORY’ → ‘UJHNPV’:

When applying the same logic to ‘LATENT’:

Using the correct pattern: ‘LATENT’ would be coded as ‘MCWDLQ’.

Hence, the correct option is: 2) MCWDLQ

Given Statements:

Some pens are books.

All book are cars.

Some cars are spoons.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some spoons are pens - False (Because, No definite relation given between spoons and pens, so it can be possible).

II. Some books are pens - True (Because, Some pens are books also implies that Some books are pens).

So, Only conclusion (II) follows.

Hence, the correct answer is "Option 2".

The image that completes the pattern is:

Hence, the correct answer is "Option 2".

The logic followed here is:

‘CARD’ is coded as ‘7359' 

'SERV’ is coded as ‘1256’

‘PACK’ is coded as ‘8497’

So, 'D' will be coded as '3'. 

Hence, the correct answer is "Option 4".

From the given open Dice:

and  

is the opposite sides.

 

and  

is the opposite sides.

 and  

is the opposite sides.

So, the face of '

' is opposite to the face showing '

'.

Hence, the correct answer is "Option 3".

The correct answer is Option (3) i.e. 'whom I'.

Key Points

• The pronoun "whom" is incorrectly used in this sentence.
• "Whom" is used as the object of a verb or preposition.
• In this sentence, "that" should be used as it introduces a defining clause describing "the book."
• Thus, the correct sentence should be "This is the book that I wanted to buy."

Therefore, the correct answer is 'whom I'.

Additional Information

• Option (1) 'the book': This is a correct phrase and there is no grammatical error in it. It correctly identifies the noun being discussed.
• Option (2) 'wanted to buy.': This phrase is also grammatically correct and maintains the intended meaning of the sentence, expressing the desire to purchase.
• Option (4) 'This is': This phrase is correct as it properly introduces the sentence and gives context, indicating what is being referenced.

The correct answer is 'Option 3' i.e. respect those has an error.

Key Points

• The phrase "respect those" is incorrect because it implies that "those" is necessary, which it is not.
• In context, the sentence should simply be "Everyone should respect parents, teachers, and elders" without the word "those."
• Using "those" unnecessarily complicates the sentence and makes it awkward.
• Correcting the sentence improves its readability and clarity.

Therefore, the correct answer is option 3.

Correct sentence: Everyone should respect parents, teachers, and elders.

Additional Information

• Option 1 ("and elders.") is correct as it is appropriately punctuated and completes the list.
• Option 2 ("parents, teachers,") is correct as it appropriately lists the nouns in the series.
• Option 4 ("Everyone should,") is correct as the introductory phrase of the sentence.

The correct answer is 'B' i.e. this part of the sentence has an error.

Key Points

• In the second part of the given sentence, the phrase 'married happily' is incorrect.
• The correct form should be 'happily married', as the adverb should come before the adjective.
• Adverbs generally modify adjectives, verbs, or other adverbs, and in this case, it should modify the adjective 'married'.
• Therefore, 'happily married' is the grammatically correct form.

Hence, the correct answer is option 1.

Correct sentence: They were happily married before she quit her job.

Additional Information

• Option 2 (D): 'quit her job' is correct as it properly indicates the action taken by 'she'.
• Option 3 (C): 'before she' is correct as it introduces the clause that follows.
• Option 4 (A): 'They were' is correct as it is the proper subject and verb agreement for a plural subject.

The correct answer is 'Option 4' i.e. 'The speaker’s emphasiss is' has an error.

Key Points

• The word "emphasiss" is incorrect and should be corrected to "emphasis".
• The sentence requires "emphasis" which is the correct spelling.
• "Emphasiss" is not a recognized word in the English language.
• Therefore, the correct sentence should read: "The speaker’s emphasis is on the present-day problems that are being faced by ordinary men of India."

Therefore, the correct answer is Option 4.

Additional Information

• Option 1: "that are being faced" is grammatically correct and does not contain any errors.
• Option 2: "on the present-day problems" is also correct as it indicates the focus of the speaker.
• Option 3: "by ordinary men of India" is correctly phrased and does not have any grammatical errors.

The correct answer is Option 3 i.e. vacillates rectifies the error.

Key Points

• The underlined word in the sentence, "vaccillates," is spelled incorrectly.
• The correct spelling of the word is "vacillates," which means to waver between different opinions or actions.
• Option 3 provides the correct spelling of the word "vacillates."
• Using the correct spelling ensures that the sentence is grammatically correct and conveys the intended meaning.

Therefore, the correct answer is Option 3.

Correct sentence: She vacillates a lot while making any decision.

Additional Information

• Option 1: "vasillates" - This is an incorrect spelling and does not correct the error.
• Option 2: "vacillattes" - This is an incorrect spelling due to the extra 't' and does not correct the error.
• Option 4: "vacilates" - This is an incorrect spelling due to the missing 'l' and does not correct the error.

The correct answer is 'Option 3' i.e. will have done.

Key Points

• The sentence is in the future perfect tense, which indicates that an action will be completed before a specified time in the future.
• The structure of the future perfect tense is: "will have" + past participle of the verb.
• The past participle of "do" is "done."
• The phrase "By this time tomorrow" sets a future point by which the action will be completed, which aligns with the future perfect tense.

Therefore, the correct answer is: will have done.

Additional Information

• Option 1: "will have did" is incorrect because "did" is the simple past form, not the past participle.
• Option 2: "will do" is the simple future tense, which does not indicate the completion of the action before a specified future time.
• **Option 4: "**will be done" is passive voice and does not fit the context where the subject "I" should be performing the action.

The correct answer is Option 1 i.e. Hold onto.

Key Points

• The sentence indicates that the thought should be kept or remembered because it may be useful in the future.
• The phrase "Hold onto" means to keep or maintain something, which fits the context of preserving a thought for future use.
• The other options do not fit the context as well:
• "Hold to" usually means to adhere to or follow a principle, which does not fit the context of keeping a thought.
• "Hold on" generally means to wait or to hang on, which is not appropriate here.
• "Hold against" means to have a grudge or resentment, which is not relevant in this context.
• Understanding the meaning of "Hold onto" in the context of the sentence helps to clarify that the idea should be preserved for potential future use.

Therefore, the correct answer is- Hold onto.

Additional Information

• Hold onto (सहेजना) means to keep or maintain something.
• Hold to (पालन करना) means to adhere to or follow a principle.
• Hold on (रुको) means to wait or to hang on.
• Hold against (विरोध करना) means to have a grudge or resentment against someone.

The correct answer is: He said that two and two make four..

 Key Points

• The sentence in the Direct Speech is a universal truth or a general statement.
• Universal truths or general statements remain unchanged in Indirect Speech.
• Present tense remains in the present tense to convey the general fact.

Therefore, the correct answer is: He said that two and two make four.

The correct answer is: The director presented the award to her.

Key Points

• The subject of the Active Voice is changed into the object of the Passive Voice. Also, the object of the Active Voice is changed into the subject of the Passive Voice.
• The usage of a helping verb is necessary while changing the voice of the sentence from Active to Passive.
• The preposition “by” is used before the object made in Passive Voice. However, it is optional in some cases.
• The sentence provided in the question is in Passive Voice.
• To change the sentence to Active Voice, use the format: Subject + Verb + Object.

Therefore, the correct answer is: The director presented the award to her.

The correct answer is Option 3 i.e. insignificant.

Key Points

• The word "trivial" means of little value or importance.
• The synonym for "trivial" that fits the context of the sentence is "insignificant".
• ​Insignificant means too small or unimportant to be worth consideration.
• "Conceited", "idiotic", and "majestic" do not convey the same meaning as "trivial" and thus are incorrect.

Therefore, the correct answer is- insignificant.

Additional Information

• Conceited (अभिमानी) means having an excessively high opinion of oneself.
• Idiotic (मूर्खतापूर्ण) means very stupid.
• Majestic (शानदार) means having or showing impressive beauty or dignity.

The correct answer is: Option 3 i.e. granary.

Key Points

• The word "granary" refers to a storehouse for threshed grain. (अनाज का भंडार)
• Example: The farmer stored the harvested wheat in the granary to keep it dry and safe.
• "Granary" is used to describe a place where grains are stored, which matches the context of the sentence given. (अनाज का भंडार)
• Example: There is a granary in the backyard of this house where they keep their grains.

Therefore, the correct answer is: Option 3 i.e. granary.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Apiary (मधुमक्खी पालन गृह): A place where bees are kept; a collection of beehives.
• Example: The beekeeper checks the apiary every morning to ensure the bees are healthy.
• Archive (अभिलेखागार): A place where public records or historical documents are kept.
• Example: The historical documents were stored in the national archive.
• Dormitory (छात्रावास): A large bedroom for a number of people in a school or institution.
• Example: The students returned to their dormitory after the evening classes.

The correct answer is: Who taught you English grammar and composition?

Key Points

• The subject of the Active Voice is usually changed into the object of the Passive Voice, and vice versa.
• The usage of a helping verb is necessary while changing the voice of the sentence from Active to Passive and vice versa.
• The provided sentence is in Passive Voice.
• To change it to Active Voice, identify the subject (who) and place it at the beginning.
• The format for the active voice is: Who + (past tense of verb) + you + (object).

Therefore, the correct answer is: Who taught you English grammar and composition?

The correct answer is: The principal said to me, “You are in-charge of the admission cell.”.

 Key Points

• When converting a sentence to Direct Speech, ensure the correct punctuation and structure are used.
• The introductory phrase should use "said to" when addressing someone directly.
• The pronoun "me" is correctly used after "said to".
• The quoted speech should be enclosed in quotation marks.

Therefore, the correct answer is: The principal said to me, “You are in-charge of the admission cell.”.

The correct answer is: Option 2.

Key Points

• The idiom "Bite the bullet" means to endure a painful or otherwise unpleasant situation that is seen as unavoidable. (कठिन परिस्थिति का सामना करना)
• Example: Knowing that the project was due soon, she decided to bite the bullet and start working on it immediately.
• Option 2 explains the idiom correctly as it describes Sarah deciding to go ahead with the surgery despite the pain, which is an unavoidable situation. (कठिन परिस्थिति का सामना करना)
• Example: Despite the pain, Sarah decided to go ahead with the surgery.

Therefore, the correct answer is: Option 2.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: The children played in the park on a sunny day. (बच्चे धूप वाले दिन पार्क में खेले।)
• Example: The children played in the park on a sunny day.
• Option 3: The chef prepared a delicious meal for the guests. (शेफ ने मेहमानों के लिए एक स्वादिष्ट भोजन तैयार किया।)
• Example: The chef prepared a delicious meal for the guests.
• Option 4: Jack and Jill went up the hill to fetch a pail of water. (जैक और जिल पानी की बाल्टी लाने के लिए पहाड़ी पर गए।)
• Example: Jack and Jill went up the hill to fetch a pail of water.

The correct answer is 'Option 4' i.e. The students eagerly awaited their results.

Key Points

• The sentence needs to be arranged in a grammatically correct and meaningful way.
• In English, the subject typically comes first, followed by the verb, and then the object.
• The word "eagerly" is an adverb and should be placed before the verb "awaited" to describe how the action was performed.
• The word "their" is a possessive pronoun and should be placed before "results" to show ownership.

Therefore, the correct answer is option 4.

Correct sentence: The students eagerly awaited their results.

Additional Information

• Option 1: "The students their eagerly awaited results." - The placement of "their" and "eagerly" is incorrect, disrupting the grammatical structure.
• Option 2: "The awaited eagerly results their students." - This option is completely jumbled and does not form a meaningful sentence.
• Option 3: "Eagerly awaited the students their results." - The adverb "eagerly" is correctly placed, but the structure is incorrect because the subject should come first.

The correct answer is Option 2 i.e. rose.

Key Points

• The sentence describes the movement of smoke from the chimney.
• The word "rose" (उठना) means to move upward, which fits the context of smoke moving up from the chimney.
• The words "hung," "left," and "went" do not accurately describe the upward movement of smoke.
• Understanding the meaning of "rose" in the context of the sentence helps to clarify that the smoke is moving upward from the chimney.

Therefore, the correct answer is- rose.

Additional Information

• Hung (लटकना) means to be suspended from above with the lower part dangling.
• Left (छोड़ना) means to go away from.
• Went (गया) is the past tense of "go" and means to move from one place to another.

The correct answer is 'Option 2' i.e. most challenging.

Key Points

• The given sentence uses the phrase "challenge task," which is grammatically incorrect. The correct phrase should be "challenging task."
• The sentence is attempting to express the superlative degree, indicating the highest level of challenge encountered.
• The appropriate superlative form of "challenging" is "most challenging."
• Using "most challenging" correctly conveys that this task is the highest degree of challenge compared to all others encountered.

Therefore, the correct answer is option 2.

Correct sentence: This is the most challenging task I have ever encountered.

Additional Information

• Option 1: more challenging
• This is the comparative form of "challenging" and is used to compare two things. It is not suitable for expressing the highest degree among many.
• Option 3: challenging
• This is the base form of the adjective and does not convey any degree of comparison.
• Option 4: much challenging
• This phrase is grammatically incorrect as "much" is not used directly with adjectives in this manner. The correct usage would be "much more challenging."

The correct answer is 'Option 4' i.e. live in first floor has an error.

Key Points

• The error in the sentence is in the phrase "live in first floor."
• In English, when referring to a specific floor in a building, the correct preposition to use is "on" rather than "in."
• The correct phrase should be "live on the first floor."
• The definite article "the" is also needed before "first floor."

Therefore, the correct answer is Option 4.

Correct sentence: They want to live on the first floor of this building.

Additional Information

• Option 1: They want to - This part of the sentence is correct and does not contain any errors.
• Option 2: of this building - This part of the sentence is correct and does not contain any errors.
• Option 4: live in first floor - This part contains the error. The correct phrase should be "live on the first floor."