Shortcut Trick

Check for a Pythagorean triplet: 20² + 21² = 400 + 441 = 841, and 29² = 841.

Since AB² + BC² = AC², ΔABC is a right-angled triangle.

The circumcentre of a right-angled triangle always lies exactly at the midpoint of its hypotenuse.

⇒ Circumradius = Hypotenuse ÷ 2

⇒ Circumradius = 29 ÷ 2 = 14.5 cm

∴ The correct answer is 14.5 cm.

Alternate Method

Given:

In ΔABC:

⇒ AB = 21 cm, BC = 20 cm, and AC = 29 cm

Formula Used:

⇒ Circumradius (R) = (a × b × c) ÷ (4 × Area)

⇒ Semi-perimeter (s) = (a + b + c) ÷ 2

⇒ Area = √[s(s − a)(s − b)(s − c)]

A C B O 21 cm 20 cm 29 cm R

Calculations:

First, find the semi-perimeter (s) of the triangle:

⇒ s = (21 + 20 + 29) ÷ 2 = 70 ÷ 2 = 35 cm

Calculate the area using Heron's formula:

⇒ Area = √[35 × (35 − 21) × (35 − 20) × (35 − 29)]

⇒ Area = √[35 × 14 × 15 × 6]

⇒ Area = √44100 = 210 cm²

Substitute the values into the circumradius formula:

⇒ R = (21 × 20 × 29) ÷ (4 × 210)

⇒ R = 12180 ÷ 840

⇒ R = 14.5 cm

∴ The correct answer is 14.5 cm.

Additional Information

Circumcentre of a Triangle

It is the point of intersection of the perpendicular bisectors of the sides. The distance from the circumcentre to all vertices is equal to the circumradius (R).

Location of Circumcentre

For an acute triangle, it lies inside. For an obtuse triangle, it lies outside. For a right-angled triangle, it lies exactly at the midpoint of the hypotenuse.

Inradius of a Right Triangle

The inradius (r) can be directly found using the formula: r = (a + b − c) ÷ 2, where a and b are perpendicular sides and c is the hypotenuse.

Shortcut Trick

Since they move in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 20 + 16 = 36 km/hr

Convert relative speed to m/s: 36 × (5 ÷ 18) = 10 m/s

Time = Distance ÷ Relative Speed = 2200 ÷ 10 = 220 seconds

220 seconds = 3 minutes 40 seconds

∴ The correct answer is 3 minutes 40 seconds.

Alternate Method

Given:

Track length (Distance) = 2200 m

Speed of Sarita = 20 km/hr

Speed of Kavita = 16 km/hr

Formula Used:

Relative Speed (when moving in opposite directions) = S₁ + S₂

Time = Distance ÷ Relative Speed

Calculation:

⇒ Relative Speed = 20 + 16 = 36 km/hr

⇒ Speed in m/s = 36 × (5 ÷ 18) = 10 m/s

⇒ Time taken to meet = 2200 ÷ 10

⇒ Time = 220 seconds

⇒ Convert seconds into minutes: 220 ÷ 60 = 3 minutes and 40 seconds

∴ The correct answer is 3 minutes 40 seconds.

Additional Information

Meeting in Same Direction

When moving in the same direction on a circular track, Relative Speed = |S₁ − S₂|. Time to meet = Distance ÷ |S₁ − S₂|.

Meeting at Starting Point

To find when multiple runners meet exactly at the starting point again, calculate the LCM of the times taken by each individual to complete one full round.

Conversion Factor

To convert km/hr to m/s, multiply by (5 ÷ 18). To convert m/s to km/hr, multiply by (18 ÷ 5).

Shortcut Trick

Let ∠H = x. In isosceles ΔHGK (HG = GK), ∠GKH = x, making exterior ∠JGK = 2x.

In isosceles ΔGKJ (GK = JK), ∠GJK = 2x. Since HJ = HK, ∠HKJ = ∠HJK = 2x.

Angle sum of ΔHJK: x + 2x + 2x = 180° ⇒ x = 36°.

∠HGK = 180° − 2x = 108° and ∠GKJ = 180° − 4x = 36°.

Required value = ²⁄₃ × (108° + 36°) = ²⁄₃ × 144° = 96°.

∴ The correct answer is 96°.

Alternate Method

Given:

⇒ HJ = HK

⇒ HG = GK = JK

Calculations:

⇒ Let ∠H = x.

⇒ HG = GK ⇒ ∠GKH = ∠H = x.

⇒ Exterior ∠JGK = ∠H + ∠GKH = 2x.

⇒ GK = JK ⇒ ∠GJK = ∠JGK = 2x ⇒ ∠HJK = 2x.

⇒ HJ = HK ⇒ ∠HKJ = ∠HJK = 2x.

⇒ ΔHJK angle sum: ∠H + ∠HJK + ∠HKJ = 180°

⇒ x + 2x + 2x = 180° ⇒ 5x = 180° ⇒ x = 36°.

⇒ ∠HGK = 180° − 2x = 180° − 72° = 108°.

⇒ ∠GKJ = 180° − 4x = 180° − 144° = 36°.

⇒ ²⁄₃ × (∠HGK + ∠GKJ) = ²⁄₃ × (108° + 36°)

⇒ ²⁄₃ × 144° = 96°.

∴ The correct answer is 96°.

Additional Information

Isosceles Triangle Property

In a triangle, angles opposite to equal sides are always equal in measure.

Exterior Angle Theorem

The measure of an exterior angle of a triangle equals the sum of the two opposite non-adjacent interior angles.

Angle Sum Property

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

Shortcut Trick

Total time from 14 Feb to 28 Apr = 14 + 31 + 28 = 73 days.

Convert days to years: 73 ÷ 365 = 1/5 years.

SI = P × R × T ÷ 100

SI = 2000 × 6.5 × (1/5) ÷ 100 = 20 × 1.3

SI = 26

∴ The correct answer is ₹26.

Alternate Method

Given:

Principal (P) = ₹2000

Rate of interest (R) = 6.5% per annum

Time period = 14 February 2023 to 28 April 2023

Formula Used:

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

where T is time in years.

Calculations:

⇒ Count the exact number of days from 14 Feb 2023 to 28 Apr 2023.

⇒ Since 2023 is not a leap year, February has 28 days.

⇒ Days in February (excluding the starting day) = 28 − 14 = 14 days

⇒ Days in March = 31 days

⇒ Days in April = 28 days

⇒ Total days = 14 + 31 + 28 = 73 days

⇒ Time in years (T) = 73 ÷ 365 = 1/5 years

⇒ Substitute values into the Simple Interest formula:

⇒ SI = (2000 × 6.5 × 1) ÷ (100 × 5)

⇒ SI = (20 × 6.5) ÷ 5

⇒ SI = 130 ÷ 5 = 26

∴ The correct answer is ₹26.

Additional Information

Simple Interest (SI) vs Compound Interest (CI)

Simple Interest is calculated only on the principal amount, whereas Compound Interest is calculated on the principal as well as the accumulated interest from previous periods.

Time Calculation Rule

When calculating simple interest between two exact dates, it is standard practice to exclude the first date and include the last date (or vice versa), such that the total number of days strictly equals the mathematical difference between dates.

Converting Days to Years

If the time period is given in days, T is obtained by dividing the total number of days by 365. Frequent standard test values include 73 days (1/5 years) and 146 days (2/5 years).

Shortcut Trick

Effective % increase for 2 years at 6% p.a. = 6 + 6 + (6 × 6)/100 = 12.36%

Multiplying factor for 2 years = 1.1236

Population in 2020 (B) = 50,000 × 1.1236 = 56,180

Population in 2016 (A) = 50,000 ÷ 1.1236 ≈ 44,500 (approximated to next integer)

Difference B − A = 56,180 − 44,500 = 11,680

∴ The correct answer is 11680.

Alternate Method

Given:

Population in 2018 = 50,000

Rate of increase = 6% per annum

A = Population in 2016, B = Population in 2020

Formula Used:

Population after n years = Initial Population × (1 + R/100)ⁿ

Calculation:

⇒ Time period from 2016 to 2018 = 2 years

⇒ 50,000 = A × (1 + 6/100)²

⇒ A = 50,000 ÷ 1.1236 = 44499.82 ≈ 44,500

⇒ Time period from 2018 to 2020 = 2 years

⇒ B = 50,000 × (1 + 6/100)²

⇒ B = 50,000 × 1.1236 = 56,180

⇒ Value of B − A = 56,180 − 44,500 = 11,680

∴ The correct answer is 11680.

Additional Information

Successive Percentage Change

When a value changes successively by a% and then b%, the effective percentage change is calculated as: a + b + (a × b) / 100.

Population Growth Formula

If the initial population is P and it grows at a compound rate of R% per annum, the population after n years is given by: P × (1 + R/100)ⁿ.

Depreciation Formula

If the value or population decreases at R% per annum, the value after n years is given by: P × (1 − R/100)ⁿ.

Shortcut Trick

The ratio of corresponding sides of similar triangles is exactly equal to the square root of the ratio of their areas.

⇒ Ratio of corresponding sides = √121 : √225

⇒ Ratio of corresponding sides = 11 : 15

∴ The correct answer is 11 : 15.

Alternate Method

Given:

Ratio of the areas of two similar triangles = 121 : 225

Formula Used:

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides: Area₁ / Area₂ = (Side₁ / Side₂)²

Area = 121 Side 1

Area = 225 Side 2

Calculations:

Let the areas of the two similar triangles be A₁ and A₂, and their corresponding sides be S₁ and S₂ respectively.

⇒ A₁ / A₂ = 121 / 225

⇒ (S₁ / S₂)² = 121 / 225

Taking the square root on both sides,

⇒ S₁ / S₂ = √(121 / 225)

⇒ S₁ / S₂ = 11 / 15

⇒ S₁ : S₂ = 11 : 15

∴ The correct answer is 11 : 15.

Additional Information

Ratio of Perimeters

The ratio of the perimeters of two similar triangles is directly equal to the ratio of their corresponding sides.

Ratio of Medians and Altitudes

The ratio of the corresponding medians, corresponding altitudes, and corresponding angle bisectors of two similar triangles is exactly equal to the ratio of their corresponding sides.

Volumes of Similar Solid Figures

If two solid geometric figures are similar, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (V₁ / V₂ = (S₁ / S₂)³).

Shortcut Trick

Multiply the second equation by 2 to get: 2X − 2Y + 2Z = 18.

Subtract the first equation from this result: (2X − 2Y + 2Z) − (X − 2Y + 2Z) = 18 − 16 ⇒ X = 2.

Substitute X = 2 into equations 2 and 3 to get: −Y + Z = 7 and −3Y − Z = 5.

Add these two new equations to eliminate Z: (−Y + Z) + (−3Y − Z) = 7 + 5 ⇒ −4Y = 12 ⇒ Y = −3.

∴ The correct answer is -3.

Alternate Method

Given:

X − 2Y + 2Z = 16   --- (1)

X − Y + Z = 9   --- (2)

2X − 3Y − Z = 9   --- (3)

Calculations:

⇒ From equation (2), express Z in terms of X and Y: Z = 9 − X + Y

⇒ Substitute the value of Z in equation (1):

⇒ X − 2Y + 2(9 − X + Y) = 16

⇒ X − 2Y + 18 − 2X + 2Y = 16

⇒ −X + 18 = 16

⇒ X = 2

⇒ Substitute X = 2 in equation (2):

⇒ 2 − Y + Z = 9

⇒ −Y + Z = 7   --- (4)

⇒ Substitute X = 2 in equation (3):

⇒ 2(2) − 3Y − Z = 9

⇒ 4 − 3Y − Z = 9

⇒ −3Y − Z = 5   --- (5)

⇒ Adding equations (4) and (5):

⇒ (−Y + Z) + (−3Y − Z) = 7 + 5

⇒ −4Y = 12

⇒ Y = −3

∴ The correct answer is -3.

Additional Information

System of Linear Equations

A set of two or more linear equations with the same variables. Finding the solution involves calculating the values for the variables that make all equations true simultaneously.

Elimination Method

This technique solves a system of equations by multiplying equations by constants and then adding or subtracting them to cancel out one of the variables, making the system easier to solve.

Substitution Method

A systematic method where one equation is algebraically solved for one variable, and that result is plugged into the other equations to reduce the number of unknown variables.

Shortcut Trick

Total work = LCM(12, 16) = 48 units.

Efficiencies: Rani = 48 ÷ 12 = 4 units/day, Adya = 48 ÷ 16 = 3 units/day.

Work done in 2 days (Rani + Adya) = 4 + 3 = 7 units.

Work in 12 days (6 cycles) = 6 × 7 = 42 units. Remaining work = 48 − 42 = 6 units.

Day 13: Rani does 4 units. Remaining = 2 units. Day 14: Adya takes 2/3 day to finish.

∴ The correct answer is 13 ²/₃ days.

Alternate Method

Given:

Time taken by Rani = 12 days

Time taken by Adya = 16 days

They work in stretches of one day alternately, starting with Rani.

Calculations:

⇒ Work done in 1^st day by Rani = 4 units

⇒ Work done in 2^nd day by Adya = 3 units

⇒ Work done in a 2-day cycle = 4 + 3 = 7 units

⇒ Number of complete 2-day cycles in 48 units = 48 ÷ 7 = 6 cycles

⇒ Work done in 12 days (6 cycles) = 6 × 7 = 42 units

⇒ Remaining work = 48 − 42 = 6 units

⇒ On the 13^th day, Rani does 4 units of work.

⇒ Work done in 13 days = 42 + 4 = 46 units

⇒ Remaining work = 48 − 46 = 2 units

⇒ On the 14^th day, Adya completes the remaining work.

⇒ Time taken by Adya = Remaining work ÷ Efficiency = 2/3 day

⇒ Total time taken = 13 days + 2/3 day = 13 ²/₃ days

∴ The correct answer is 13 ²/₃ days.

Additional Information

Alternate Days Work

When individuals work on alternate days, find the total work done in one complete cycle (usually 2 days) and use it to find the maximum possible full cycles before calculating the leftover work.

Work and Efficiency Ratio

Efficiency is inversely proportional to the time taken. If the time taken ratio is A : B, then the efficiency ratio is B : A.

Pipes and Cisterns

Similar to Time and Work, an inlet pipe has a positive efficiency (filling) while an outlet pipe has a negative efficiency (emptying).

Shortcut Trick

Direct formula for the exact occurrence of one event out of two independent events:

P(Exactly one) = P(A) + P(B) − 2 × P(A) × P(B)

⇒ P = 2/3 + 3/4 − 2 × (2/3 × 3/4)

⇒ P = 17/12 − 12/12 = 5/12

∴ The correct answer is 5/12.

Alternate Method

Given:

Probability that player A hits the target, P(A) = 2/3

Probability that player B hits the target, P(B) = 3/4

Formula Used:

P(Exactly one hits) = [P(A hits) × P(B misses)] + [P(A misses) × P(B hits)]

Start A Hits A Misses

2/3 1/3

3/4 1/4 3/4 1/4

B Hits (Both hit) B Misses → P = 2/12 B Hits → P = 3/12 B Misses (Both miss)

Calculations:

⇒ P(A misses) = P(A') = 1 − 2/3 = 1/3

⇒ P(B misses) = P(B') = 1 − 3/4 = 1/4

⇒ P(Exactly one hits) = [P(A) × P(B')] + [P(A') × P(B)]

⇒ P(Exactly one hits) = (2/3 × 1/4) + (1/3 × 3/4)

⇒ P(Exactly one hits) = 2/12 + 3/12

⇒ P(Exactly one hits) = 5/12

∴ The correct answer is 5/12.

Additional Information

Independent Events

Two events are considered independent if the occurrence of one does not affect the probability of the other. The key formula used is: P(A and B) = P(A) × P(B).

Mutually Exclusive Events

Events that cannot occur at the same time are mutually exclusive. For such events, the probability of both occurring is zero. The key formula used is: P(A or B) = P(A) + P(B).

Probability of 'At Least One' Event

The probability that at least one of the independent events occurs is easily calculated by subtracting the probability of none occurring from 1. Formula: P(At least one) = 1 − P(None).

Shortcut Trick

Savings = Discount % × Total Purchase Value

Savings = 26% of 2050

Savings = 26 × 20.5 = 533

∴ The correct answer is ₹533.

Alternate Method

Given:

Total purchase amount = ₹2,050

Discount percentage = 26%

Formula Used:

Discount Amount = (Discount % ÷ 100) × Total Value

Calculations:

⇒ Savings = (26 ÷ 100) × 2050

⇒ Savings = 0.26 × 2050

⇒ Savings = 26 × 20.5

⇒ Savings = 533

∴ The correct answer is ₹533.

Additional Information

Marked Price (MP) and Discount

Discount is always calculated on the Marked Price or the initial Total Value. Selling Price = Marked Price − Discount.

Successive Discounts

If two successive discounts of a% and b% are given, the effective net discount % = a + b − (a × b) ÷ 100.

Percentage Base Change

When calculating savings, the original purchase amount always acts as the 100% base value.

Shortcut Trick

Fraction of milk in A = 4/9, B = 2/3, and Final Mixture = 8/13.

By Alligation cross-multiplication on the fraction of milk:

Ratio of A to B = (2/3 − 8/13) : (8/13 − 4/9)

⇒ Ratio = 2/39 : 20/117

⇒ Ratio = (2 × 117) : (39 × 20) = 234 : 780

⇒ x : y = 3 : 10

∴ The correct answer is 3 : 10.

Alternate Method

Given:

Ratio of milk to water in vessel A = 4 : 5

Ratio of milk to water in vessel B = 2 : 1

Ratio of milk to water in the final mixture = 8 : 5

Formula Used:

The rule of alligation is applied on the fraction of milk to find the required ratio of mixtures.

Calculations:

⇒ Fraction of milk in Vessel A = 4 / (4 + 5) = 4/9

⇒ Fraction of milk in Vessel B = 2 / (2 + 1) = 2/3

⇒ Fraction of milk in the final mixture = 8 / (8 + 5) = 8/13

⇒ Ratio of x to y = (Fraction in B − Mean Fraction) : (Mean Fraction − Fraction in A)

⇒ x : y = (2/3 − 8/13) : (8/13 − 4/9)

⇒ x : y = [(26 − 24) / 39] : [(72 − 52) / 117]

⇒ x : y = (2 / 39) : (20 / 117)

⇒ x : y = (2 / 39) × (117 / 20)

⇒ x : y = 117 / (39 × 10)

⇒ x : y = 3 : 10

∴ The correct answer is 3 : 10.

Additional Information

Rule of Alligation

When two ingredients of different prices or concentrations are mixed, the ratio of their quantities is given by: (Quantity of Cheaper) / (Quantity of Dearer) = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper).

Concentration in Mixtures

The fraction of any specific component in a mixture is calculated as the quantity of that component divided by the total volume of all components present in the mixture.

Replacement Method

If V is the total volume and x quantity is removed and replaced by water n times, the remaining quantity of pure liquid = V × (1 − x/V)ⁿ.

Shortcut Trick

The average of the first n even natural numbers is directly given by the formula: (n + 1).

Here, the total number of even numbers given is n = 91.

Average = 91 + 1 = 92

∴ The correct answer is 92.

Alternate Method

Given:

Total number of terms (n) = 91

The series of the first 91 even numbers is 2, 4, 6, 8, ..., 182

Formula Used:

Since the given series is in an Arithmetic Progression, Average = (First Term + Last Term) ÷ 2

2 1^st term

4 2^nd term

...

Average 92

...

180 90^th term

182 91^st term

Average = (2 + 182) ÷ 2

Calculations:

⇒ First term (a) = 2

⇒ Last term (l) = 91 × 2 = 182

⇒ Average = (2 + 182) ÷ 2

⇒ Average = 184 ÷ 2

⇒ Average = 92

∴ The correct answer is 92.

Additional Information

Average of first n natural numbers

The average of the first n natural numbers is calculated as: (n + 1) ÷ 2.

Average of first n odd numbers

The average of the first n consecutive odd numbers is always exactly equal to n.

Average of squares of first n natural numbers

The average of squares of first n natural numbers is: (n + 1)(2n + 1) ÷ 6.

Average of cubes of first n natural numbers

The average of cubes of first n natural numbers is: n(n + 1)² ÷ 4.

Shortcut Trick

Radius (r) = Diameter ÷ 2 = 12 ÷ 2 = 6 cm.

Volume of Cone = (1/3) × π × r² × h

⇒ Volume = (1/3) × (22/7) × 6² × 5

⇒ Volume = (22/7) × 12 × 5 = 1320 ÷ 7

⇒ Volume = 188.5714... ≈ 188.57 cm³

∴ The correct answer is 188.57 cm³.

Alternate Method

Given:

Diameter of the base (d) = 12 cm

Height of the cone (h) = 5 cm

Value of π = 22/7

Formula Used:

Radius (r) = d ÷ 2

Volume of a right circular cone = (1/3) × π × r² × h

Calculations:

⇒ Radius (r) = 12 ÷ 2 = 6 cm

⇒ Volume = (1/3) × (22/7) × (6)² × 5

⇒ Volume = (1/3) × (22/7) × 36 × 5

⇒ Volume = (22/7) × 12 × 5

⇒ Volume = (22/7) × 60

⇒ Volume = 1320 ÷ 7

⇒ Volume = 188.5714... cm³

⇒ Rounding off to 2 decimal places, Volume ≈ 188.57 cm³

∴ The correct answer is 188.57 cm³.

Additional Information

Slant Height of a Cone

The slant height (l) is calculated using Pythagoras theorem: l = √(r² + h²).

Curved Surface Area (CSA)

The area of the curved surface of a cone is given by: CSA = π × r × l.

Total Surface Area (TSA)

The sum of the curved surface area and the base area: TSA = π × r × (r + l).

Volume of a Cylinder

For a given radius and height, the volume of a cylinder is exactly three times the volume of a cone: V = π × r² × h.

Given:

$\frac{\sqrt[3]{6859}}{\sqrt[4]{1296}} \times \frac{3}{57} \times 42=?$

Formula used:

19³ = 6859

6⁴ = 1296

Calculation:

$\frac{\sqrt[3]{6859}}{\sqrt[4]{1296}} \times \frac{3}{57} \times 42 = ?$

⇒ $\frac{19}{6} \times \frac{3}{57} \times 42 = ?$

⇒ $\frac{19}{6} \times \frac{1}{19} \times 42 = ?$

⇒ $\frac{1}{6} \times 42 = ?$

⇒ ? = 7

∴ The correct answer is option (1).

Shortcut Trick

For conjugate surds of the form x = a + √b and y = a − √b, the sum of their squares is given directly by: x² + y² = 2(a² + b)

Here, a = 8 and b = 5.

⇒ x² + y² = 2(8² + 5)

⇒ x² + y² = 2(64 + 5) = 2 × 69 = 138

∴ The correct answer is 138.

Alternate Method

Given:

x = 8 + √5

y = 8 − √5

Formula Used:

(a + b)² = a² + b² + 2ab

(a − b)² = a² + b² − 2ab

Calculations:

⇒ x² = (8 + √5)² = 8² + (√5)² + 2 × 8 × √5

⇒ x² = 64 + 5 + 16√5 = 69 + 16√5

⇒ y² = (8 − √5)² = 8² + (√5)² − 2 × 8 × √5

⇒ y² = 64 + 5 − 16√5 = 69 − 16√5

Adding the expanded values of x² and y²:

⇒ x² + y² = (69 + 16√5) + (69 − 16√5)

⇒ x² + y² = 69 + 69 = 138

∴ The correct answer is 138.

Additional Information

Algebraic Expansion Identity

The sum of the squares of the sum and difference of two numbers directly simplifies to: (a + b)² + (a − b)² = 2(a² + b²).

Difference of Squares

The product of the sum and difference of two numbers yields the difference of their squares: (a + b)(a − b) = a² − b².

Properties of Surds

When multiplying a square root by itself, the root sign is removed: √x × √x = x. The conjugate of an irrational number a + √b is a − √b.

Shortcut Trick

Difference in ratio parts of P and R = 5 − 4 = 1 part.

Given difference = ₹1,500, so 1 part = ₹1,500.

Q's share is 3 parts.

Q's share = 3 × 1,500 = ₹4,500.

∴ The correct answer is ₹4,500.

Alternate Method

Given:

Ratio of shares of P, Q, and R = 5 : 3 : 4

P gets ₹1,500 more than R.

Calculations:

⇒ Let the individual shares of P, Q, and R be 5x, 3x, and 4x respectively.

⇒ According to the condition, P's share − R's share = 1500

⇒ 5x − 4x = 1500

⇒ x = 1500

⇒ Q's share = 3x

⇒ Q's share = 3 × 1500 = 4500

∴ The correct answer is ₹4,500.

Additional Information

Properties of Ratio

Multiplying or dividing all terms of a ratio by the same non-zero number does not change the ratio. It is universally used to scale down or parameterise given quantities.

Sum of Shares in Proportion

If the total sum is given and needs to be divided in the ratio A : B : C, an individual share (e.g., A's share) is calculated directly as: (A ÷ (A + B + C)) × Total Sum.

Difference Between Shares

When the difference between two parts is known, the numeric value of 1 ratio part is equal to the given difference amount divided by the difference in their ratio parts. This skips algebraic variables.

Shortcut Trick

If tan A × tan B = 1, then A + B = 90°.

Given: tan (x + y) × tan (x − y) = 1

⇒ (x + y) + (x − y) = 90°

⇒ 2x = 90° ⇒ x = 45°

⇒ tan x = tan 45° = 1

∴ The correct answer is 1.

Alternate Method

Given:

tan (x + y) tan (x − y) = 1

Formula Used:

tan (A + B) = (tan A + tan B) ÷ (1 − tan A tan B)

tan (A − B) = (tan A − tan B) ÷ (1 + tan A tan B)

Calculations:

⇒ [(tan x + tan y) ÷ (1 − tan x tan y)] × [(tan x − tan y) ÷ (1 + tan x tan y)] = 1

⇒ (tan²x − tan²y) ÷ (1 − tan²x tan²y) = 1

⇒ tan²x − tan²y = 1 − tan²x tan²y

⇒ tan²x + tan²x tan²y = 1 + tan²y

⇒ tan²x (1 + tan²y) = 1 + tan²y

⇒ tan²x = 1

⇒ tan x = 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Properties

If A + B = 90°, then sin A = cos B, tan A = cot B, and sec A = cosec B.

Important Trigonometric Rules

If tan A × tan B = 1, then A + B = 90°. Similarly, if sin A × sec B = 1, then A + B = 90°.

Double Angle Formulas

tan 2A = (2 tan A) ÷ (1 − tan²A)

Shortcut Trick Total terms (n) = 12 and Sum of data = 264 Mean = 264 ÷ 12 = 22 For 12 sorted items, Median = average of 6^th and 7^th term = (22 + 24) ÷ 2 = 23 Empirical formula: Mode = 3 × Median − 2 × Mean Mode = 3 × 23 − 2 × 22 = 69 − 44 = 25 ∴ The correct answer is 25.

Alternate Method Given: Data set: 17, 20, 21, 18, 25, 28, 24, 22, 16, 24, 25, 24 Formula Used: Mean = Sum of terms ÷ Total number of terms Median = Average of (n ÷ 2)th and ((n ÷ 2) + 1)th terms for even 'n' Mode = 3 × Median − 2 × Mean

Empirical Formula

Mode = 3 × Median − 2 × Mean

Mean 22

Median 23

Calculated Mode 25

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Calculations: ⇒ Arranging data in ascending order: 16, 17, 18, 20, 21, 22, 24, 24, 24, 25, 25, 28 ⇒ Total number of terms (n) = 12 ⇒ Sum of all terms = 16 + 17 + 18 + 20 + 21 + 22 + 24 + 24 + 24 + 25 + 25 + 28 = 264 ⇒ Mean = 264 ÷ 12 = 22 ⇒ The data has an even number of terms, so the median lies between the 6^th and 7^th terms. ⇒ Median = (22 + 24) ÷ 2 = 23 ⇒ Applying empirical formula: Mode = 3 × Median − 2 × Mean ⇒ Mode = 3 × 23 − 2 × 22 ⇒ Mode = 69 − 44 = 25 ∴ The correct answer is 25.

Additional Information Mean The mean represents the average value of a dataset. It is calculated by dividing the sum of all observations by the total number of observations. Median The median is the middle value of a sorted dataset. If the dataset has an odd number of observations, it is the exact middle value. For an even number of observations, it is the average of the two central values. Mode The mode is the most frequently occurring value in a dataset. In cases of moderate skewness, it can be approximated using the empirical relationship: Mode = 3 × Median − 2 × Mean. Skewness When Mean < Median < Mode, the data distribution is negatively skewed (left-skewed). When Mean > Median > Mode, it is positively skewed (right-skewed).

Shortcut Trick

Using Componendo and Dividendo on (secθ + tanθ) / (secθ − tanθ) = 9 / 1:

⇒ secθ / tanθ = (9 + 1) / (9 − 1) ⇒ 1 / sinθ = 10 / 8 = 5 / 4 ⇒ sinθ = 4/5

Since sinθ = 4/5, in a right-angled triangle, Opposite = 4, Hypotenuse = 5, Adjacent = 3. Thus, tanθ = 4/3.

⇒ Value = (sinθ − tan²θ) / (sinθ + tan²θ) = (4/5 − 16/9) / (4/5 + 16/9)

⇒ (−44 / 45) ÷ (116 / 45) = −44 / 116 = −11/29

∴ The correct answer is −11/29.

Alternate Method

Given:

(secθ − tanθ) / (secθ + tanθ) = 1/9

Formula Used:

secθ = 1 / cosθ, tanθ = sinθ / cosθ

Calculations:

⇒ [(1/cosθ) − (sinθ/cosθ)] / [(1/cosθ) + (sinθ/cosθ)] = 1/9

⇒ (1 − sinθ) / (1 + sinθ) = 1/9

⇒ 9 × (1 − sinθ) = 1 + sinθ

⇒ 9 − 9sinθ = 1 + sinθ

⇒ 10sinθ = 8

⇒ sinθ = 8/10 = 4/5

Since θ lies in the first quadrant, we can visualize this using a right-angled triangle:

⇒ Opposite = 4, Hypotenuse = 5

⇒ Adjacent = √(Hypotenuse² − Opposite²) = √(5² − 4²) = √(25 − 16) = √9 = 3

⇒ tanθ = Opposite / Adjacent = 4/3

⇒ tan²θ = (4/3)² = 16/9

Now, substituting the values into the required expression:

⇒ (sinθ − tan²θ) / (sinθ + tan²θ)

⇒ (4/5 − 16/9) / (4/5 + 16/9)

⇒ [(36 − 80) / 45] / [(36 + 80) / 45]

⇒ (−44 / 45) / (116 / 45)

⇒ −44 / 116

⇒ −11/29

∴ The correct answer is −11/29.

Additional Information

Componendo and Dividendo

If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). This algebraic property is highly effective for rapidly simplifying equations that consist of sum and difference pairs in the numerator and denominator.

Pythagorean Triplets

A set of integers satisfying a² + b² = c² is called a triplet. Remembering fundamental sets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) significantly improves speed in geometry and trigonometry problems.

Trigonometric Quadrant Rules (ASTC Rule)

In the 1^st Quadrant, All trigonometric functions are positive. In the 2^nd Quadrant, only Sine is positive. In the 3^rd Quadrant, only Tangent is positive. In the 4^th Quadrant, only Cosine is positive. Since θ lies in the first quadrant here, tanθ and sinθ are both positive.

Given:

$\frac{a^7 \times b^8 \times c^7}{a^9 \times b^5 \times c^4}$

Formula used:

$\frac{x^m}{x^n} = x^{m-n}$

Calculation:

$\frac{a^7 \times b^8 \times c^7}{a^9 \times b^5 \times c^4}$

⇒ a^7-9 × b^8-5 × c^7-4

⇒ a^-2 × b³ × c³

⇒ (a^-2) × (b³) × (c³)

∴ The correct answer is option (3).

Shortcut Trick

60% is equal to the fraction 3/5.

First value = 3/5 × 75 = 45

Second value = 1/5 × 25 = 5

Difference = 45 − 5 = 40

∴ The correct answer is 40.

Alternate Method

Given:

First expression = 60% of 75

Second expression = 1/5 of 25

Method Used:

Convert the percentage to a fraction and apply the BODMAS rule, where 'Of' is treated as multiplication.

Calculations:

⇒ First expression = (60 ÷ 100) × 75

⇒ First expression = (3 ÷ 5) × 75 = 3 × 15 = 45

⇒ Second expression = (1 ÷ 5) × 25

⇒ Second expression = 5

⇒ Difference = 45 − 5 = 40

∴ The correct answer is 40.

Additional Information

BODMAS Rule

In mathematical expressions, 'Of' signifies multiplication and is resolved before standard division and multiplication, according to the BODMAS sequence (Brackets, Of, Division, Multiplication, Addition, Subtraction).

Percentage to Fraction Conversion

Converting percentages to standard fractions speeds up calculations. Common fraction equivalents include 20% = 1/5, 40% = 2/5, 60% = 3/5, and 80% = 4/5.

Base Value in Percentages

The term 'Percentage' means 'per hundred'. Thus, an expression like x% of y is mathematically calculated as (x ÷ 100) × y.

Shortcut Trick

Average of ratio parts = (2 + 5 + 7) ÷ 3 = 14/3 parts

14/3 parts → 56 years

1 part → (56 × 3) ÷ 14 = 12 years

Youngest person's age = 2 parts = 2 × 12 = 24 years

∴ The correct answer is 24 years.

Alternate Method

Given:

Average age of 3 persons = 56 years

Ratio of their ages = 2 : 5 : 7

Formula Used:

Total Sum = Average × Number of observations

Calculations:

⇒ Total age of the three persons = 56 × 3 = 168 years

⇒ Let the ages of the persons be 2x, 5x, and 7x respectively.

⇒ Sum of ages = 2x + 5x + 7x = 14x

⇒ 14x = 168

⇒ x = 168 ÷ 14

⇒ x = 12

⇒ Age of the youngest person = 2x = 2 × 12 = 24 years

∴ The correct answer is 24 years.

Additional Information

Sum and Average Relation

The sum of all values in a dataset is always equal to the average multiplied by the total number of values. Sum = Average × N.

Proportional Distribution

When a total quantity is divided among multiple parts in a given ratio (a : b : c), the share of the first part is calculated as (a ÷ (a + b + c)) × Total Quantity.

Shortcut Trick

Total Surface Area (TSA) of the toy = CSA of cone + CSA of hemisphere = πr(l + 2r)

Slant height, l = √(r² + h²) = √(36² + 105²) = √(1296 + 11025) = √12321 = 111 cm

TSA = π × 36 × (111 + 2 × 36)

TSA = π × 36 × (111 + 72)

TSA = π × 36 × 183 = 6588π cm²

∴ The correct answer is 6588π cm².

Alternate Method

Given:

Radius of hemisphere and cone, r = 36 cm

Height of the cone, h = 105 cm

Formula Used:

Slant height of cone, l = √(r² + h²)

Curved Surface Area (CSA) of cone = πrl

Curved Surface Area (CSA) of hemisphere = 2πr²

Calculation:

⇒ First, let's find the slant height (l) of the conical portion.

⇒ l = √(36² + 105²)

⇒ l = √(1296 + 11025)

⇒ l = √12321 = 111 cm

⇒ CSA of the cone = π × 36 × 111 = 3996π cm²

⇒ CSA of the hemisphere = 2 × π × (36)²

⇒ CSA of the hemisphere = 2 × π × 1296 = 2592π cm²

⇒ Total Surface Area = CSA of cone + CSA of hemisphere

⇒ Total Surface Area = 3996π + 2592π

⇒ Total Surface Area = 6588π cm²

∴ The correct answer is 6588π cm².

Additional Information

Volume of Cone

The volume of a cone is calculated as one-third of the volume of a cylinder with the same base radius and height: V = (1 ÷ 3)πr²h.

Volume of Hemisphere

Since a hemisphere is exactly half of a full sphere, its volume is given by the formula V = (2 ÷ 3)πr³.

Total Surface Area of Solid Hemisphere

If the hemisphere stands alone as a solid body, its total surface area includes the curved surface and the flat circular base: TSA = 3πr².

Shortcut Trick

Let Anil's age be A.

Dinesh = 3A and Charu = (A + 5) ÷ 2.

Given difference: 3A − (A + 5) ÷ 2 = 45

⇒ 6A − A − 5 = 90 ⇒ A = 19

Dinesh = 57 years, Charu = 12 years ⇒ Sum = 69 years.

∴ The correct answer is 69 years.

Alternate Method

Given:

Difference between Beena and Anil's age = 5 years (Beena is older)

Dinesh's age = 3 × Anil's age

Beena's age = 2 × Charu's age

Difference between Dinesh and Charu's age = 45 years

Calculations:

Let Anil's age be A.

⇒ Beena's age (B) = A + 5

⇒ Dinesh's age (D) = 3 × A = 3A

⇒ Charu's age (C) = B ÷ 2 = (A + 5) ÷ 2

Since Dinesh is Anil's father, his age is evidently greater than Charu's age.

⇒ D − C = 45

⇒ 3A − (A + 5) ÷ 2 = 45

Multiply the entire equation by 2 to remove the fraction:

⇒ 6A − (A + 5) = 90

⇒ 5A − 5 = 90

⇒ 5A = 95

⇒ A = 19 years

Now, substitute the value of A back to find Dinesh and Charu's ages:

⇒ Dinesh's age (D) = 3 × 19 = 57 years

⇒ Charu's age (C) = (19 + 5) ÷ 2 = 24 ÷ 2 = 12 years

⇒ Sum of their ages = 57 + 12 = 69 years

∴ The correct answer is 69 years.

Additional Information

Constant Age Difference

The age difference between two individuals is always constant throughout their lives. If person A is 5 years older than person B today, A will always remain 5 years older than B in the future.

Linear Equations in Ages

When a question states relationships like "X is three times as old as Y", it translates directly into a linear equation (X = 3Y). Solving simultaneous linear equations is the primary method to find exact current ages.

Handling Fractional Ages

Statements like "A is twice as old as B" can be written as A = 2B or B = A ÷ 2. It is often mathematically easier to multiply the entire equation by the denominator to avoid working with fractions, as done in the solution step: 6A − (A + 5) = 90.

Shortcut Trick

Given CSA = 5 × Base Area ⇒ πrl = 5πr² ⇒ l = 5r.

Using l² = r² + h² ⇒ 25r² = r² + h² ⇒ 24r² = h².

Since h = 63, r² = 63² ÷ 24 = 3969 ÷ 24 = 1323/8.

Volume = (1/3)πr²h = (1/3) × (22/7) × (1323/8) × 63

Volume = 22 × 441 × 9 ÷ 8 = 43659 ÷ 4 = 10914.75 cm³.

∴ The correct answer is 10914.75 cm³.

Alternate Method

Given:

Height of the cone (h) = 63 cm

Curved Surface Area = 5 × Base Area

Formula Used:

Curved Surface Area of a cone = πrl

Base Area of a cone = πr²

Volume of a cone = (1/3)πr²h

l² = r² + h² (where l is slant height)

Calculations:

⇒ πrl = 5 × πr²

⇒ l = 5r

⇒ l² = 25r²

We know that, l² = r² + h²

⇒ 25r² = r² + 63²

⇒ 24r² = 3969

⇒ r² = 3969 ÷ 24 = 1323 ÷ 8 cm²

Now, Volume = (1/3) × (22/7) × r² × h

⇒ Volume = (1/3) × (22/7) × (1323/8) × 63

⇒ Volume = 22 × 441 × 9 ÷ 8

⇒ Volume = 11 × 3969 ÷ 4

⇒ Volume = 43659 ÷ 4 = 10914.75 cm³

∴ The correct answer is 10914.75 cm³.

Additional Information

Total Surface Area of a Cone

The total surface area includes the curved surface area and the flat base area: TSA = πr(l + r).

Frustum of a Cone

When a right circular cone is cut by a plane parallel to its base, the lower part is called a frustum. Volume = (1/3)πh(R² + r² + Rr).

Slant Height Relationship

The slant height (l), radius (r), and vertical height (h) always form a right-angled triangle where l² = r² + h².

Shortcut Trick

Cost Price (CP) of 1 article = 15 ÷ 10 = &rupee;1.5

Selling Price (SP) of 1 article = &rupee;1.7

Profit = 1.7 − 1.5 = &rupee;0.2

Profit % = (0.2 ÷ 1.5) × 100 = 13.33%

∴ The correct answer is 13.33%.

Alternate Method

Given:

Total Cost Price (CP) of 10 articles = &rupee;15

Selling Price (SP) of 1 article = &rupee;1.7

Formula Used:

Total SP = SP of 1 article × Total articles

Profit = Total SP − Total CP

Profit % = (Profit ÷ Total CP) × 100

Calculations:

⇒ Total SP of 10 articles = 1.7 × 10 = &rupee;17

⇒ Total CP of 10 articles = &rupee;15

⇒ Profit = 17 − 15 = &rupee;2

⇒ Profit % = (2 ÷ 15) × 100

⇒ Profit % = 200 ÷ 15 = 13.33%

∴ The correct answer is 13.33%.

Additional Information

Loss Percentage

When Cost Price is greater than Selling Price, a loss occurs. Loss % = (Loss ÷ CP) × 100.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Percentage Change

If there are successive changes of a% and b%, the net percentage change is given by: a + b + (a × b ÷ 100).

Shortcut Trick Total frequency N = 32 → N/2 = 16. Cumulative frequency just > 16 is 21, so median class is 20-25. L = 20, CF = 14, f = 7, h = 5. Median = 20 + [(16 − 14) ÷ 7] × 5 = 20 + 10/7 ≈ 21.43. ∴ The correct answer is 21.43. Alternate Method Given: Total frequency (N) = 3 + 5 + 4 + 2 + 7 + 6 + 5 = 32 Cumulative frequencies (CF) for classes: 3, 8, 12, 14, 21, 27, 32 Formula Used: Median = L + [(N/2 − CF) ÷ f] × h Calculations: ⇒ N/2 = 32 ÷ 2 = 16 ⇒ The CF just greater than 16 is 21, which corresponds to the class interval 20-25. ⇒ Lower limit of median class (L) = 20 ⇒ CF of preceding class (CF) = 14 ⇒ Frequency of median class (f) = 7 ⇒ Class size (h) = 5 ⇒ Median = 20 + [(16 − 14) ÷ 7] × 5 ⇒ Median = 20 + (2 ÷ 7) × 5 ⇒ Median = 20 + 10 ÷ 7 ⇒ Median = 20 + 1.428... ⇒ Median ≈ 21.43 ∴ The correct answer is 21.43. Additional Information Mode of Grouped Data Mode = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h, where f1 is frequency of modal class, f0 and f2 are frequencies of preceding and succeeding classes. Mean of Grouped Data Direct method formula: Mean = Σfixi ÷ Σfi, where xi represents the class mark (midpoint) of each class interval. Empirical Relationship The empirical relationship between Mean, Median, and Mode is given by: Mode = 3 × Median − 2 × Mean.

Shortcut Trick

In right-angled ΔABC with ∠B = 90°, the acute angles A and C must sum to 90° (∠A + ∠C = 90°).

We can directly apply compound angle formulas without needing the exact angle values:

⇒ sin A cos C + cos A sin C = sin(A + C) = sin 90° = 1

⇒ cos A cos C − sin A sin C = cos(A + C) = cos 90° = 0

∴ The correct answer is 1, 0.

Alternate Method

Given: In right-angled ΔABC, ∠B = 90° and tan A = √3.

⇒ Since tan 60° = √3, we have ∠A = 60°.

⇒ The sum of angles in a triangle is 180°, so ∠C = 180° − (90° + 60°) = 30°.

Calculating the first expression (sin A cos C + cos A sin C):

⇒ sin 60° cos 30° + cos 60° sin 30°

⇒ (√3/2) × (√3/2) + (1/2) × (1/2)

⇒ 3/4 + 1/4 = 4/4 = 1

Calculating the second expression (cos A cos C − sin A sin C):

⇒ cos 60° cos 30° − sin 60° sin 30°

⇒ (1/2) × (√3/2) − (√3/2) × (1/2)

⇒ √3/4 − √3/4 = 0

∴ The correct answer is 1, 0.

Additional Information

Compound Angle Formulas

sin(A + B) = sin A cos B + cos A sin B

cos(A + B) = cos A cos B − sin A sin B

Complementary Angles

In a right-angled triangle, the two acute angles are complementary, meaning their sum is always equal to 90°.

Standard Trigonometric Values

sin 30° = cos 60° = 1/2

sin 60° = cos 30° = √3/2

Shortcut Trick

Use the direct algebraic identity: (ax − by)² + (bx + ay)² = (a² + b²)(x² + y²)

Here, a = 5, b = 12, x = z, and y = y.

⇒ (5z − 12y)² + (12z + 5y)² = (5² + 12²)(z² + y²)

⇒ (25 + 144)(z² + y²) = 169z² + 169y²

Now, simply subtract 144z²: 169z² + 169y² − 144z² = 25z² + 169y²

∴ The correct answer is 25z² + 169y².

Alternate Method

Given:

(5z − 12y)² + (12z + 5y)² − 144z²

Formula Used:

(a − b)² = a² − 2ab + b²

(a + b)² = a² + 2ab + b²

Calculations:

⇒ (5z − 12y)² = (5z)² − 2 × 5z × 12y + (12y)² = 25z² − 120yz + 144y²

⇒ (12z + 5y)² = (12z)² + 2 × 12z × 5y + (5y)² = 144z² + 120yz + 25y²

Adding both squared expressions together:

⇒ (25z² − 120yz + 144y²) + (144z² + 120yz + 25y²)

⇒ 25z² + 144z² + 144y² + 25y² − 120yz + 120yz

⇒ 169z² + 169y²

Now, substitute this result back into the complete original expression by subtracting 144z²:

⇒ 169z² + 169y² − 144z²

⇒ 25z² + 169y²

∴ The correct answer is 25z² + 169y².

Additional Information

Sum and Difference of Squares

(a + b)² + (a − b)² = 2(a² + b²). This is useful for quickly simplifying expressions with identical terms.

Brahmagupta-Fibonacci Identity

(a² + b²)(x² + y²) = (ax − by)² + (bx + ay)². This identity shows that the product of two sums of two squares is itself a sum of two squares.

Difference of Squares

a² − b² = (a + b)(a − b). This is extensively used in algebraic factorisation and rapid calculation.

Given:

4³ - 0² + $\left(\frac{22}{2}\right)^2$ - 8 + 7 × 6 = ?

Formula used:

BODMAS

Calculation:

4³ - 0² + $\left(\frac{22}{2}\right)^2$ - 8 + 7 × 6

⇒ 64 - 0 + 11² - 8 + 42

⇒ 64 - 0 + 121 - 8 + 42

⇒ 185 - 8 + 42

⇒ 219

∴ The correct answer is option (3).

Statements:

A < B ≥ C > D

C > E ≥ F = G

Condition needs to be checked, 

Option 1) A < E

A < B ≥ C > D

C > E ≥ F = G

No direct relation between A and E can be determined.

Option 2) D < F

A < B ≥ C > D

C > E ≥ F = G

No direct relation between D and F can be determined.

Option 3) G > C

A < B ≥ C > D

C > E ≥ F = G

The given statement shows C > E ≥ F = G which means G ≤ C, not G > C.

Option 4) B > G

A < B ≥ C > D

C > E ≥ F = G

Given B ≥ C and C > E ≥ F = G, so B ≥ C > G. Therefore, B > G is definitely true.

Hence, the correct answer is "Option 4".

Question:

In a certain code language, 'fire in mountain' is coded as 'si ph bo'. What is code for 'fire'?

Statements:

(I) 'mountain' is coded as 'bo'.

(II) 'ice and fire' is coded as 'si yu ca'.

Solution:

Statement I:

'mountain' is coded as 'bo'.

This statement alone tells us the code for 'mountain' but does not provide enough information to determine the code for 'fire'.

Statement II:

'ice and fire' is coded as 'si yu ca'.

From the original statement, 'fire in mountain' is coded as 'si ph bo'.

Comparing the two statements, 'fire' is common in both. 'si' is common in both.

Therefore, 'fire' is coded as 'si'.

Conclusion:

Statement II alone is sufficient to answer the question, while Statement I is not.

Hence, the correct answer is 4) Data in Statement II alone is sufficient to answer the question while data in Statement I is not.

The figure embedded will be,

Hence the correct answer is "Option 3"

Cause:

Many people in City X have reported receiving two electricity bills this month, instead of one.

Effect I:

The Electricity Board has published the details of areas where power shutdowns are expected tomorrow in City X.

Effect I is not a direct effect of the cause. Power shutdowns are often planned maintenance or due to other issues, not necessarily related to billing errors.

Effect II:

The municipal commissioner of City X has clarified that the technical glitch which led to multiple electricity bills being generated would be resolved within 48 hours.

Effect II is a direct effect of the cause. The clarification from the municipal commissioner directly addresses the issue of multiple bills and assures a resolution.

Effect III:

Long queues were seen outside the electricity board offices in many localities.

Effect III is too general as it does not specify the localities belong to City X, which is a standard requirement for such logical reasoning questions in SSC exams.

Conclusion:

Only Effect II is a possible effect of the given cause.

Hence, the answer is 'Only II is a possible effect'.

Logic followed here is,

S is common in both languages and 3 is common in both codes.

So, the code for S is 3.

Hence the correct answer is "Option 3"

The correct mirror image is:

Hence the correct answer is "Option 1"

Logic followed here is:

Logic: The first image is rotated 90° in the clockwise direction to make the second figure.

Hence the correct answer is "Option 2"

The completion of the incomplete figure will be,

Hence the correct answer is "Option 2"

The logic followed here is:

Logic: First number - 30 = Second number and Second number - 50 = Third number.

Here,

For (215, 185, 135):

215 - 30 = 185

185 - 50 = 135

For (276, 246, 196):

276 - 30 = 246

246 - 50 = 196

Similarly, for Option 4: (238, 208, 158)

238 - 30 = 208

208 - 50 = 158

Hence, the correct answer is Option 4.

Given Letters:

The word is RECOGNIZE.

First letter: R

Third letter: C

Sixth letter: N

Seventh letter: I

The letters are R, C, N, I.

Meaningful English Words:

CRIN: It means a horsehair or stiff fabric, though not widely used in everyday language.

From the above letters, the English word 'CRIN" can be formed.

So, the number of meaningful English words is 1.

Hence, the correct answer is "Option 1".

Given:

Word pairs: Park - Swim, Restaurant - Dinner, Temple - Pray, Theatre - Movies

We need to find the word pair that does NOT belong to the group.

1. Park - Swim:

Swim is an activity that can not be done in a park.

2. Restaurant - Dinner:

Dinner is an activity that can be done in a restaurant.

3. Temple - Pray:

Pray is an activity that can be done in a temple.

4. Theatre - Movies:

Movies are shown in a theatre.

The relationship in options 2, 3, and 4 is that the second word is an action performed at the first location.

But in option 1, the second word is an action not performed at the first location.

∴ The word pair that does NOT belong to the group is Park-Swim.

Given:

The statement mentions that Solver S employs systematic problem-solving approaches, similar to expert puzzle solvers, ensuring efficiency and accuracy in resolving complex challenges.

Inference:

Statement 1: "The efficiency and accuracy of Solver S in resolving complex challenges may be attributed to their systematic problem-solving methods."

⇒ This is a valid inference because the given statement suggests that Solver S ensures efficiency and accuracy through systematic problem-solving.

Statement 2: "Solver S utilizes systematic problem-solving approaches, similar to expert puzzle solvers."

⇒ This is directly stated in the given information, so it is not an inference but a restatement.

∴ The correct inference is Statement I only.

Family Chart:

Given:

U is the father of N.

N is the sister of T.

T is the wife of I.

I is son of L.

Family Tree:

So, U is the 'Son's wife's father' of L

Hence, the relation is 'Option 1'.

Given:

Words: Design, Test, Conceptualize, Develop, Implement

We need to arrange the given words in a logical and meaningful order.

3. Conceptualize - This is the first step, where the idea is formed.
4. Design - After conceptualization, the plan is designed.
5. Develop - Based on the design, the product is developed.
6. Test - The developed product is then tested.
7. Implement - Finally, the product is implemented.

So, the order would be: 3, 1, 4, 2, 5.

Hence, the correct answer is the "Option 3"

Given:

Words: Delivery, Finished Paper, Dry, Pulp, Press

We need to arrange the given words in a logical and meaningful order.

4. Pulp - This is the raw material used to make paper.
5. Press - The pulp is pressed to form a sheet.
6. Dry - The pressed sheet is dried.
7. Finished Paper - The dried sheet becomes finished paper.
8. Delivery - The finished paper is then delivered.

So, the order would be: 4, 5, 3, 2, 1.

Hence, the correct answer is the "option 1"

Logic followed here is,

Hence the correct answer is "Option 3"

Given: Each of P, Q, R, S, T, U and V has an exam on a different day of a week, starting from Monday and ending on Sunday of the same week.

1. Q has the exam on Saturday.

2. Only two people have exams between S and Q.

3. R has the exam before S, but after P.

4. U has the exam immediately after T.

V's exam is on the remaining day.

Therefore, V's exam is on Sunday.

So, V's exam is on 'Sunday'.

Hence the correct answer is "Option 3"

Family Chart:

Decoding the Symbols:

Given: U + V ÷ W - Y × Z

It means:

U is the mother of V.

V is the father of W.

W is the brother of Y.

Y is the wife of Z.

Family Tree:

So, U is the grandmother of W and Y. Therefore, U is the Wife's Father's Mother of Z.

Hence, the correct answer is "Option 2".

Logic followed here is,

Odd one is Figure C.

Hence the correct answer is "Option 1"

Logic followed here is 

Hence the correct answer is "Option 1"

Logic followed here is:

First number × 11 = Second number

Check each option,

Option 1: 13-144

13 × 11 = 143

⇒ 143 ≠ 144

Option 2: 31-341

31 × 11 = 341

⇒ 341 = 341

Option 3: 18-198

18 × 11 = 198

⇒ 198 = 198

Option 4: 22-242

22 × 11 = 242

⇒ 242 = 242

∴ The number pair that does NOT belong to the group is 13-144.

Hence the correct answer is "Option 1"

Given:

Arvind has blue, green, and white marbles.

Question: What is the percentage of white marbles out of all the marbles he has?

Statement I: The sum of blue and green marbles he has is 7.

Statement II: He has 3 white marbles out of 10 marbles.

Statement I alone:

Blue marbles + Green marbles = 7

We do not know the number of white marbles or the total number of marbles.

Therefore, Statement I alone is not sufficient.

Statement II alone:

White marbles = 3

Total marbles = 10

Percentage of white marbles = (3 / 10) × 100

⇒ Percentage = 30%

Therefore, Statement II alone is sufficient.

∴ Data in Statement II alone is sufficient to answer the question while data in Statement I is not.

Hence the correct answer is "Option 1"

For find the opposite surface of 'T', we need to compare the common surface in the given dices,

So, From Dice 1 and Dice 2,

From both dices, T is common and on the same surface. So, all remaining surface will be opposite to each other.

S is opposite to V.

U is opposite to Q.

So, P will be opposite to T.

Hence, the correct answer is "P".

Given Equation: 55 B 5 D 13 B 2 C 1 4 A 7 D 15 B 5 C 102 A 2 C 15 = ?

BODMAS Rule:

Decoding the given information:

Now replacing the signs from left to right using BODMAS Rule, we get:

⇒ 55 × 5 - 13 × 2 + 14 ÷ 7 - 15 × 5 + 102 ÷ 2 + 15 

⇒ 275 - 26 + 2 - 75 + 51 + 15 

⇒ 249 + 2 - 75 + 51 + 15 

⇒ 251 - 75 + 51 + 15 

⇒ 176 + 51 + 15 

⇒ 227 + 15 = 242 

Hence, the answer is "Option 2"

Given number: '5238641'

After arranged in ascending order:

Now,

There is two digit "2, 3" will remain unchanged if each of the digits in the number '5238641' is arranged in ascending order.

Hence, the correct answer is "Option 2".

Given:

Statement :

Hydroelectricity is a cleaner and greener energy than other forms of power generation.

Now,

Arguments :

I. Hydroelectric plants almost have nil CO₂ emissions - True (This supports the statement by highlighting a cleaner aspect of hydroelectricity).

II. Hydroelectric power projects submerge huge amounts under water thereby affecting local flora and fauna - True (This weakens the statement by pointing out a negative environmental impact).

So, II weakens while I strengthens the statement.

Hence, the correct answer is "Option 2".

Logic followed here is,

First code is decreased by 4 to make the second code.

Calculations:

Let's check each option:

Option 1: #BEJ % = QRW

Does not follow the same logic so, this option is incorrect.

Option 2: #BEJ % = ORW

⇒ Follow the same logic so, this option is correct.

Option 3: #BEJ % = ORP

⇒ Does not follow the same logic so, this option is incorrect.

Option 4: #BBJ % = ORW

⇒ Does not follow the same logic so, this option is incorrect.

Hence the correct answer is "Option 2"

Given, O, P, Q, R, S and T are sitting around a circular table facing the centre (but not necessarily in the same order).

S sits third to the right of Q.

O sits second to the right of S.

Q is not an immediate neighbour of P.

P sits second to the left of T.

R sits at the remaining place.

So, the position of O is Second to the left of R.

Hence the correct answer is "Option 2"

Given Statements:

All sheeps are cows.

Some cows are horses.

All horses are jackals.

The least possible Venn diagram for the given statements is:

Conclusions:

I. Some jackals are cows - True (Since all horses are jackals and some cows are horses, it logically follows that some jackals are cows).

II. Some horses are sheeps - False (There is no direct relationship or overlap shown between sheep and horses in the given statements or the Venn diagram).

So, Only conclusion I follows.

Hence, the correct answer is "Option 4".

Logic followed here is,

Common word: she

Common code: mp

⇒ she = mp

∴ The code for 'she' is 'mp'.

The correct answer is: Option 4.

Key Points

• The word "gauge" means to measure the size, amount, or degree of something. (मापना)
• Example: The company uses surveys to gauge customer satisfaction.
• The word "guage" is incorrectly spelled and should be corrected to "gauge". (गौज)
• Example: The trainer tried to gauge the learners' understanding of the subject by asking several new questions.

Therefore, the correct answer is: Option 4.

Additional Information

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

• Gaage (गैज): This is an incorrect spelling and does not have a meaning.
• Example: N/A
• Gaaug (गाउग): This is an incorrect spelling and does not have a meaning.
• Example: N/A
• Gaugg (गॉग): This is an incorrect spelling and does not have a meaning.
• Example: N/A

The correct answer is: Option 1.

Key Points

• The word "leisurely" means without haste, in a relaxed and unhurried manner. (बिना जल्दी के)
• Example: They enjoyed a leisurely walk in the park on Sunday afternoon.
• "Without haste" indicates doing something slowly and calmly, taking one's time. (बिना जल्दी के)
• Example: She completed the task without haste, ensuring everything was perfect.
• Hence, we can infer that the meaning of 'leisurely' is 'without haste'.

Therefore, the correct answer is: Option 1.

Additional Information

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

• Reassuringly (आश्वस्त करते हुए): In a way that removes someone’s doubts and fears.
• Example: He spoke reassuringly to calm her nerves before the interview.
• With a kickstart (एकदम से): To start something energetically or suddenly.
• Example: The team began their project with a kickstart, full of enthusiasm.
• Hastily (जल्दी में): Doing something quickly, often with little thought or planning.
• Example: She packed her bags hastily, eager to catch the early morning train.

The correct answer is Option 4.

Key Points

• The paragraph describes the discovery and details of Shi Huangdi's mausoleum.
• Sentence C is the first sentence as it introduces Shi Huangdi and gives historical context:
• "Shi Huangdi was the founder of Qin Dynasty and was the first ruler of unified China in 221 BC."
• Sentence D follows as it talks about the recent discovery of his mausoleum:
• "His mausoleum was recently discovered in 1974 by Chinese peasants who were drilling a well."
• Sentence A is the third as it describes the contents of the mausoleum:
• "It consists of a terracotta army of approximately 8000 life-sized men and horses."
• Sentence B is the fourth as it provides more details about the individual sculptures:
• "Each of the horses and men is individually sculpted by a 3-inch-thick terracotta clay and looks unique."
• The correct sequence is CDAB.

Coherent paragraph after rearrangement:

"Shi Huangdi was the founder of Qin Dynasty and was the first ruler of unified China in 221 BC. His mausoleum was recently discovered in 1974 by Chinese peasants who were drilling a well. It consists of a terracotta army of approximately 8000 life-sized men and horses. Each of the horses and men is individually sculpted by a 3-inch-thick terracotta clay and looks unique."

The correct answer is Option 2.

Key Points

• Sentence B comes first as it sets up the context of making life liveable:
• "In order for life to be liveable,"
• Sentence A follows as it adds the requirement of constant sustenance:
• "it requires constant sustenance"
• Sentence D follows next as it specifies the need for favourable circumstances:
• "and favourable circumstances"
• Sentence C comes last as it explains what these circumstances enable:
• "that enable it to grow and develop"

Coherent sentence after rearrangement:

"In order for life to be liveable, it requires constant sustenance and favourable circumstances that enable it to grow and develop."

The correct answer is 'much'.

Key Points

• The phrase "as much as" is used to compare the extent or degree of something between two objects or events.
• In this context, the sentence is comparing how captivating the final installment was in comparison to other books in the series.
• Options 1, 2, and 3 are incorrect as they do not fit the structure of the comparative phrase "as much as".

Therefore, the correct answer is Option 4.

Complete Sentence: "No other book in the series captivated readers as much as the final installment."

Additional Information

• Option 1: "many" is incorrect because "many" is used with countable nouns and does not fit the structure "as much as".
• Option 2: "more" is incorrect because "more" would be used in the structure "more than" rather than "as much as".
• Option 3: "most" is incorrect because "most" would imply the superlative form and does not fit the comparative "as much as".

The correct answer is Option 2.

Key Points

• The paragraph discusses the idea of India being a land of learning and the esteem given to learned men.
• Segment Q follows the first part as it logically continues describing India as a land of learning through the ages:
• "India has been a land of learning through the ages."
• Segment S is next because it describes that this learning is not because education has been universal:
• "Not indeed in the sense that education has been universal,"
• Segment R follows as it further explains this point:
• "But in the sense that learning has always been very highly valued,"
• Segment P is the fourth as it concludes by stating the esteem held for learned men:
• "And the learned man has been held in higher esteem,"
• The correct sequence is QSRP.

Coherent sentence after rearrangement:

"India has been a land of learning through the ages, not indeed in the sense that education has been universal, but in the sense that learning has always been very highly valued, and the learned man has been held in higher esteem than the warrior or the administrator."

The correct answer is Option 4.

Key Points

• Indirect Speech: The scientist confidently asserted that she was determined to publish her findings.
• Direct to Indirect Speech:
• Pronoun Change:
• "I" is changed to "she" to match the subject of the reporting verb.
• "my" is changed to "her" to align with the perspective of the scientist.
• Tense Change:
• "am" is changed to "was" to reflect the past tense in reported speech.
• Assertion Description:
• The phrase "asserted confidently" is kept intact to preserve the manner of the assertion.
• Punctuation and Format:
• No quotation marks are used in indirect speech.
• The reporting clause "The scientist confidently asserted that..." starts the sentence.

Therefore, the correct answer is Option 4.

Correct Sentence: "The scientist confidently asserted that she was determined to publish her findings."

Additional Information

• Option 1: "The scientist confidently asserted that she is determined to publish her findings."
• Fails to change "is" to "was," which is necessary to reflect the past tense in reported speech.
• Option 2: "The scientist confidently asserted that she had determined to publish her findings."
• Changes "am determined" to "had determined," which alters the meaning of the original sentence.
• Option 3: "The scientist confidently asserted that she would be determined to publish her findings."
• Changes "am determined" to "would be determined," which incorrectly suggests a future determination rather than an ongoing or past resolved intention.

The correct answer is: Option 4.

Key Points

• The word "Infallible" means incapable of making mistakes or being wrong. (अचूक)
• Example: The scientist’s method was considered infallible until new data disproved it.
• "Infallible" refers to someone or something that is flawless and cannot fail. (अचूक)
• Example: The CEO’s infallible judgment led the company to great success.

Therefore, the correct answer is: Option 4.

Additional Information

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

• Impregnable (अजेय): Incapable of being taken by assault; unconquerable.
• Example: The fortress was considered impregnable due to its strong walls and defenses.
• Inevitable (अपरिहार्य): Certain to happen; unavoidable.
• Example: Death is an inevitable part of life.
• Inimitable (अद्वितीय): So good or unusual as to be impossible to copy; unique.
• Example: Her inimitable style of writing has won her many fans.

The correct answer is: Option 3.

Key Points

• The word "cast" means to throw something forcefully or to discard something. (फेंकना या हटाना)
• Example: They cast their nets into the sea to catch fish.
• "Cast" in the context of the sentence means to discard or get rid of prejudices. (हटाना)
• Example: We should cast away all prejudices to live in harmony.

Therefore, the correct answer is: Option 3.

Additional Information

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

• Caste (जाति): A hereditary social class in Hindu society.
• Example: The caste system has been a part of Indian society for centuries.
• Coast (तट): The part of the land near the sea; the edge of the land.
• Example: They took a trip to the coast to enjoy the beach.
• Cost (लागत): The amount of money required to buy, do, or make something.
• Example: The cost of living in the city is quite high.

The correct answer is Option 1.

Key Points

• Sentence B should be the first as it introduces the context of Indian garment exporters catering to large MNCs in Europe and America.
• Sentence C should follow as it explains the nature of these large MNCs looking for cheap goods to maximize their profits:
• "These large MNCs with worldwide network look for the cheapest goods in order to maximize their profits."
• Sentence D comes next as it describes how Indian garment exporters attempt to fulfill large orders from MNCs:
• "To get these large orders, Indian garment exporters try hard to cut their own costs."
• Sentence A follows explaining the means which exporters use to reduce costs:
• "As cost of raw materials cannot be reduced, exporters try to cut labour costs."
• Sentence E concludes the paragraph describing the change in employment practices to cut costs:
• "Where earlier a factory used to employ workers on a permanent basis, now they employ workers only on a temporary basis so that they do not have to pay workers for the whole year."
• The correct order is BCDAE.

Coherent paragraph after rearrangement:

"Large MNCs in the garment industry in Europe and America order their products from Indian exporters. These large MNCs with worldwide network look for the cheapest goods in order to maximize their profits. To get these large orders, Indian garment exporters try hard to cut their own costs. As cost of raw materials cannot be reduced, exporters try to cut labour costs. Where earlier a factory used to employ workers on a permanent basis, now they employ workers only on a temporary basis so that they do not have to pay workers for the whole year."

The correct answer is: Option 2.

Key Points

• The word "niped" is incorrectly spelt. It should be spelt as "nipped". (गलत वर्तनी)
• "Nipped" means to pinch, squeeze, or bite sharply. (चुटकी लेना, काटना)
• Example: The cold wind nipped at their faces as they walked through the snow.

Therefore, the correct answer is: Option 2.

Additional Information

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

• Niiped (गलत वर्तनी): This is a misspelling and not a correct word.
• Nipeed (गलत वर्तनी): This is a misspelling and not a correct word.
• Nipedd (गलत वर्तनी): This is a misspelling and not a correct word.

The correct answer is Option 4.

Key Points

• Direct Speech: The teacher informed, "The exam has been postponed."
• Indirect to Direct Speech:
• Tense Change:
• "had been postponed" in indirect speech is often derived from "has been postponed" in direct speech. Therefore, it should be changed back to "has been postponed."
• Quotation Marks:
• Direct speech requires the use of quotation marks to enclose the exact words spoken.
• Reporting Verb:
• "informed" remains unchanged as it correctly signifies the reporting of information.

Therefore, the correct answer is Option 4.

Correct Sentence: "The teacher informed, 'The exam has been postponed.'"

Additional Information

• Option 1: "The teacher informs, 'The exam had been postponed.'"
• Incorrectly uses "informs" instead of "informed," and "had been postponed" instead of "has been postponed."
• Option 2: "The teacher informed, 'The exam was being postponed.'"
• Incorrectly changes the tense to "was being postponed," which alters the meaning of the original sentence.
• Option 3: "The teacher informs, 'The exam has been postponed.'"
• Incorrectly uses "informs" instead of "informed."

The correct answer is Option 3.

Key Points

• Sentence A should be the first as it introduces the core concept of psychotherapy and defines the relationship between the client and the therapist:
• "Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats or the therapist."
• Sentence C follows as it explains the purpose of this relationship:
• "The purpose of the relationship is to help the client to solve the psychological problems being faced by her or him."
• Sentence B comes next as it describes how the relationship builds trust, which is essential for discussing problems freely:
• "The relationship is conducive for building the trust of the client so that problems may be freely discussed."
• Sentence D follows as it elaborates on the goals of psychotherapies:
• "Psychotherapies aim at changing the maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment."
• Sentence E concludes by pointing out that changes might be required in the individual's environment due to inadequate adjustments:
• "Inadequate marital, occupational and social adjustment also requires that major changes be made in an individual's personal environment."
• The correct sequence is ACBDE.

Coherent paragraph after rearrangement:

"Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats or the therapist. The purpose of the relationship is to help the client to solve the psychological problems being faced by her or him. The relationship is conducive for building the trust of the client so that problems may be freely discussed. Psychotherapies aim at changing the maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment. Inadequate marital, occupational and social adjustment also requires that major changes be made in an individual's personal environment."

The correct answer is 'bear'.

Key Points

• The verb "to bear" means to endure or tolerate something difficult or unpleasant.
• In this context, Rashid is trying to endure the pain after the accident.
• Options 1, 2, and 4 are incorrect in this context as they do not convey the correct meaning.

Therefore, the correct answer is Option 3.

Complete Sentence: "Rashid was not able to bear the pain after the accident."

Additional Information

• Option 1: "beer" is incorrect because "beer" refers to an alcoholic beverage and is unrelated to enduring pain.
• Option 2: "buyer" is incorrect because "buyer" refers to a person who purchases something and is unrelated to enduring pain.
• Option 4: "bare" is incorrect because "bare" means uncovered or exposed and is unrelated to enduring pain.

The correct answer is Option 1.

Key Points

• In the active voice, the sentence is: "Did you hear that noise?"
• To convert this to the passive voice, we use the structure "was/were + object + past participle + by + subject."
• In this case, the object is "that noise".
• The verb "hear" in the passive voice becomes "heard".

Therefore, the correct answer is Option 1.

Correct Sentence: "Was that noise heard by you?"

Additional Information

• Option 2: "Is that noise heard by you?" – Incorrect tense, the sentence should be in the past simple tense.
• Option 3: "Wasn't that noise heard by you?" – Incorrect for the given sentence in the affirmative form, this one is negative and changes the meaning.
• Option 4: "Has that noise been heard by you?" – Incorrect tense, the sentence should be in the past simple tense, not present perfect.

The correct answer is Option 2.

Key Points

• Direct Speech: The District Magistrate told the police, "Work all night and put an end to these riots."
• Indirect to Direct Speech:
• Reporting Verb:
• "ordered" is converted to "told" to frame the direct speech appropriately.
• Imperative Mood:
• The sentence is in imperative mood which is used for commands, instructions, or requests.
• The command "work all night and put an end to those riots" is appropriately framed in direct speech.
• Pronoun Change:
• "those" changes to "these" to reflect the point of view of the speaker.
• Quotation Marks:
• Direct speech requires the use of quotation marks to enclose the exact words spoken.

Therefore, the correct answer is Option 2.

Correct Sentence: "The District Magistrate told the police, 'Work all night and put an end to these riots.'"

Additional Information

• Option 1: "The District Magistrate told the police, 'Work all night and put to end these riots.'"
• Contains the phrase "put to end," which is grammatically incorrect.
• Option 3: "The District Magistrate told the police, 'Work all night and put an end to their riots.'"
• Changes "these riots" to "their riots," which alters the intended meaning.
• Option 4: "The District Magistrate told the police, 'Work all night to put an end these riots.'"
• Incorrectly omits "to" after "put an end."

The correct answer is: Option 4.

Key Points

• The idiom "pull out all the stops" means to do everything you can to make something successful. (हर संभव प्रयास करना)
• Example: The team decided to pull out all the stops for their upcoming presentation.
• Option 4 implies doing everything possible to ensure success. (हर संभव प्रयास करना)
• Example: The organizers pulled out all the stops to make the event memorable.

Therefore, the correct answer is: Option 4.

Additional Information

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

• Option 1: Keep things simple and straightforward (साधारण और सीधा रखना): To not complicate things.
• Example: For the initial phase, we decided to keep things simple and straightforward.
• Option 2: Wait patiently for a more favourable time (एक और अनुकूल समय के लिए धैर्यपूर्वक प्रतीक्षा करना): To delay action until a better opportunity arises.
• Example: They decided to wait patiently for a more favourable time before making their move.
• Option 3: Give up easily and without thinking much (आसानी से हार मान लेना और ज्यादा नहीं सोचना): To stop trying without putting in much effort.
• Example: It's unlike him to give up easily and without thinking much.