Shortcut Trick

Base Area (Equilateral) = (√3/4) × 8² = 16√3 cm².

Slant height (l) of a face = √(Slant edge² − (Side/2)²) = √(24² − 4²) = 4√35 cm.

Lateral Surface Area (LSA) = 3 × (1/2 × 8 × 4√35) = 48√35 cm².

Whole Surface Area = Base Area + LSA = (16√3 + 48√35) cm².

∴ The correct answer is (16√3 + 48√35) cm².

Alternate Method

Given:

Side of equilateral base (a) = 8 cm

Slant edge (e) = 24 cm

Formula Used:

Area of equilateral triangle = (√3/4) × side²

Slant height (l) = √(slant edge² − (side/2)²)

Whole Surface Area (WSA) = Base Area + 3 × (Lateral Face Area)

⇒ Area of base = (√3 / 4) × 8 × 8 = 16√3 cm²

⇒ Half of base side = 8 ÷ 2 = 4 cm

⇒ Slant height (l) = √(24² − 4²) = √(576 − 16) = √560

⇒ l = √(16 × 35) = 4√35 cm

⇒ Area of one lateral face = 1/2 × base × l = 1/2 × 8 × 4√35 = 16√35 cm²

⇒ Total Lateral Surface Area = 3 × 16√35 = 48√35 cm²

⇒ Whole Surface Area = Base Area + LSA = (16√3 + 48√35) cm²

∴ The correct answer is (16√3 + 48√35) cm².

Additional Information

Volume of a Pyramid

The volume of any pyramid is given by V = (1/3) × Base Area × Vertical Height (h).

Slant Height vs Vertical Height

For a regular pyramid, l² = h² + r², where l is the slant height, h is the vertical height, and r is the inradius of the base.

Regular Tetrahedron

A regular tetrahedron is a special pyramid where all four faces are congruent equilateral triangles; its total surface area is √3 × a².

Shortcut Trick

Water percentage (Initial : Final) = 40% : 20% = 2 : 1

Total volume ratio (inverse of concentration) = 1 : 2

Original volume (1 unit) = 80 litres

Volume to be added = (2 − 1) = 1 unit

Added milk = 1 × 80 = 80 litres

∴ The correct answer is 80 litres.

Alternate Method

Given: Initial volume = 80 litres, Milk (Initial) = 60%, Milk (Final) = 80%

Formula: Alligation method or Quantity balance method.

Initial Milk % Added Milk %

60%

100%

80%

20 (100−80) 20 (80−60) Ratio = 1 : 1

⇒ Initial milk quantity = 60% of 80 = 48 litres

⇒ Initial water quantity = 80 − 48 = 32 litres

⇒ Let x litres of milk be added.

⇒ Final total volume = (80 + x) litres

⇒ New milk concentration = 80%, so new water concentration = 20%

⇒ 20% of (80 + x) = 32

⇒ 0.20 × (80 + x) = 32

⇒ 80 + x = 32 ÷ 0.20

⇒ 80 + x = 160

⇒ x = 160 − 80 = 80 litres

∴ The correct answer is 80 litres.

Additional Information

Constant Quantity Rule

In a mixture, if only one component is added or removed, the quantity of the other component remains constant. Original Volume × Old % = New Volume × New %.

Alligation Method

Ratio of quantities = (Higher Value − Mean Value) : (Mean Value − Lower Value). Pure milk is always taken as 100%.

Percentage to Ratio Conversion

60% milk means Milk : Water = 3 : 2. To get 80% milk (Milk : Water = 4 : 1), equalize the water parts to find the change in milk units.

Shortcut Trick

Let x = 15 and y = 15 as HCF(15, 15) = 15.

First term: 36(15²) − 81(15²) = 225 × (36 − 81) = 225 × (−45)

Second term: 81(15²) − 9(15²) = 225 × (81 − 9) = 225 × 72

HCF of terms = 225 × HCF(45, 72) = 225 × 9 = 2025.

Among options, 2025 ÷ 135 = 15 (Perfectly divisible).

∴ The correct answer is 135.

Alternate Method

Given: HCF(x, y) = 15

Formula: If HCF(x, y) = h, then x = ha and y = hb, where HCF(a, b) = 1.

⇒ Let x = 15a and y = 15b.

⇒ 36x² − 81y² = 36(15a)² − 81(15b)²

⇒ 36(225a²) − 81(225b²) = 225(36a² − 81b²) = 225 × 9(4a² − 9b²) = 2025(4a² − 9b²)

⇒ 81x² − 9y² = 81(15a)² − 9(15b)²

⇒ 81(225a²) − 9(225b²) = 225(81a² − 9b²) = 225 × 9(9a² − b²) = 2025(9a² − b²)

⇒ The HCF of the two expressions is a multiple of 2025.

⇒ We need to find which option divides 2025.

⇒ 2025 ÷ 135 = 15

⇒ 2025 ÷ 120 = 16.875

⇒ 2025 ÷ 180 = 11.25

⇒ 2025 ÷ 90 = 22.5

∴ The correct answer is 135.

Additional Information

HCF Scaling Property

If HCF(x, y) = k, then HCF(xⁿ, yⁿ) = kⁿ. Similarly, HCF(cx, cy) = c × HCF(x, y).

Algebraic Divisibility

For any expressions f(x, y) and g(x, y), their HCF will always be divisible by the HCF of the common numerical coefficients and the HCF of the base variables.

Divisibility Rule of 135

A number is divisible by 135 if it is divisible by both 5 and 27 (where 27 requires the sum of digits to be a multiple of 9 and satisfying higher power checks).

Shortcut Trick

Diagonal of an inscribed square = Diameter of the circle.

Diameter = 2 × Radius = 2 × 5 = 10 cm.

Perimeter of square = 2√2 × Diagonal.

Perimeter = 2√2 × 10 = 20√2 cm.

∴ The correct answer is 20√2 cm.

Alternate Method

Given:

Radius of the circle (r) = 5 cm

Formula Used:

Diameter of circle (d) = 2 × r

Diagonal of square (D) = Side × √2

Perimeter of square = 4 × Side

Calculations:

⇒ Diameter of the circle = 2 × 5 = 10 cm

⇒ Since the square is inscribed, its diagonal (D) = Diameter of the circle = 10 cm

⇒ Side × √2 = 10

⇒ Side = 10 ÷ √2 = 5√2 cm

⇒ Perimeter of square = 4 × 5√2

⇒ Perimeter = 20√2 cm

∴ The correct answer is 20√2 cm.

Additional Information

Area of Inscribed Square

The area of a square inscribed in a circle of radius r is given by 2r².

Square Circumscribing a Circle

If a square circumscribes a circle, its side is equal to the diameter (2r) and its perimeter is 8r.

Diagonal Property

The diagonal of a square with side 'a' is a√2, and it is the longest distance between any two points in the square.

Shortcut Trick

Radius (r) = Curved Surface Area ÷ (2 × π × height)

r = 126π ÷ (2 × π × 14) = 126 ÷ 28 = 4.5 cm

Volume of cylinder = Curved Surface Area × (r ÷ 2)

Volume = 126π × (4.5 ÷ 2) = 126π × 2.25 = 283.5π cm³

∴ The correct answer is 283½ π cm³.

Alternate Method

Given:

Curved Surface Area (CSA) = 126π cm²

Height (h) = 14 cm

Formula Used:

CSA of cylinder = 2πrh

Volume of cylinder = πr²h

Calculations:

⇒ 2 × π × r × 14 = 126 × π

⇒ 28 × r = 126

⇒ r = 126 ÷ 28 = 4.5 cm

⇒ Volume = π × (4.5)² × 14

⇒ Volume = π × 20.25 × 14

⇒ Volume = 283.5π cm³

⇒ Volume = 283½ π cm³

∴ The correct answer is 283½ π cm³.

Additional Information

Total Surface Area (TSA) of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πr(h + r).

Volume of a Cone

A cone with the same radius and height as a cylinder has exactly one-third of its volume: V = (1/3)πr²h.

Relation between Volume and CSA

For any cylinder, the ratio of its volume to its curved surface area is always r/2, simplified as V = CSA × (r/2).

Given:

$\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt{16+19.25 \times 4^2}}$

Formula used:

√x² = x

Calculation:

⇒ Numerator = $\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}}$

⇒ $\sqrt{10+\sqrt{25+\sqrt{108+13}}}$

⇒ $\sqrt{10+\sqrt{36}}$

⇒ $\sqrt{10+6} = 4$

⇒ Denominator = $\sqrt{16+19.25 \times 16}$

⇒ $\sqrt{16+308}$

⇒ $\sqrt{324} = 18$

⇒ Expression = $\frac{4}{18}$

⇒ $\frac{2}{9}$

∴ The correct answer is option (2).

Shortcut Trick

In partnership: Profit = Capital × Time ⇒ Time = Profit ÷ Capital.

Time ratio = (2 ÷ 3) : (3 ÷ 4) : (5 ÷ 8).

LCM of denominators (3, 4, 8) = 24.

Multiply each part by 24: (2/3 × 24) : (3/4 × 24) : (5/8 × 24).

Time ratio = 16 : 18 : 15.

∴ The correct answer is 16 : 18 : 15.

Alternate Method

Given:

Ratio of Capitals (C_A : C_B : C_C) = 3 : 4 : 8

Ratio of Profits (P_A : P_B : P_C) = 2 : 3 : 5

Formula Used:

Profit = Capital × Investment Time

Time = Profit ÷ Capital

Calculations:

⇒ Time of A (T_A) = P_A ÷ C_A = 2 ÷ 3

⇒ Time of B (T_B) = P_B ÷ C_B = 3 ÷ 4

⇒ Time of C (T_C) = P_C ÷ C_C = 5 ÷ 8

⇒ T_A : T_B : T_C = (2/3) : (3/4) : (5/8)

⇒ To simplify, multiply by LCM(3, 4, 8) = 24

⇒ T_A = 2/3 × 24 = 16

⇒ T_B = 3/4 × 24 = 18

⇒ T_C = 5/8 × 24 = 15

⇒ Ratio = 16 : 18 : 15

∴ The correct answer is 16 : 18 : 15.

Additional Information

Capital Calculation

If profit ratio and time ratio are given, the ratio of capital can be found as Capital = Profit ÷ Time.

Active vs Sleeping Partner

An active partner manages the business and may receive a fixed percentage of total profit as salary before the remaining profit is distributed.

Compound Partnership

When partners invest different amounts for different periods of time, the profit is shared according to the ratio of their (Capital × Time) products.

Shortcut Trick

Probability (Solved) = 1 − Probability (None solve it)

Failure rates: A = 4/7, B = 4/9, C = 4/5

P(None solve) = 4/7 × 4/9 × 4/5 = 64/315

P(Solved) = 1 − 64/315 = 251/315

∴ The correct answer is 251/315.

Alternate Method

Given:

P(A) = 3/7, P(B) = 5/9, P(C) = 1/5

Formula Used:

Probability that the problem is solved = 1 − P(A') × P(B') × P(C')

Calculations:

⇒ P(A') = 1 − 3/7 = 4/7

⇒ P(B') = 1 − 5/9 = 4/9

⇒ P(C') = 1 − 1/5 = 4/5

⇒ Probability that none of them solve the problem = P(A') × P(B') × P(C')

⇒ P(none) = (4/7) × (4/9) × (4/5)

⇒ P(none) = 64 / 315

⇒ Probability that the problem is solved = 1 − P(none)

⇒ P(solved) = 1 − 64/315

⇒ P(solved) = (315 − 64) / 315 = 251 / 315

∴ The correct answer is 251/315.

Additional Information

Independent Events

Two events are independent if the occurrence of one does not affect the other. For such events, P(A ∩ B) = P(A) × P(B).

At Least One Event

The probability of at least one event occurring among independent events is calculated as 1 − P(none of the events occur).

Complementary Probability

The sum of the probability of an event and its complement is always 1: P(E) + P(E') = 1.

Shortcut Trick

Total Distance = 45 + 80 + 6 = 131 km

Total Time = (45 ÷ 15) + (80 ÷ 40) + (6 ÷ 2) = 3 + 2 + 3 = 8 hours

Average Speed = Total Distance ÷ Total Time = 131 ÷ 8 = 16.375 km/h

Rounding to 2 decimal places ≈ 16.38 km/h

∴ The correct answer is 16.38 km/h.

Alternate Method

Given:

Distance₁ (d₁) = 45 km, Speed₁ (s₁) = 15 km/h

Distance₂ (d₂) = 80 km, Speed₂ (s₂) = 40 km/h

Distance₃ (d₃) = 6 km, Speed₃ (s₃) = 2 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = d₁ + d₂ + d₃

⇒ Total Distance = 45 + 80 + 6 = 131 km

⇒ Time t₁ = 45 ÷ 15 = 3 hours

⇒ Time t₂ = 80 ÷ 40 = 2 hours

⇒ Time t₃ = 6 ÷ 2 = 3 hours

⇒ Total Time = t₁ + t₂ + t₃ = 3 + 2 + 3 = 8 hours

⇒ Average Speed = 131 ÷ 8

⇒ Average Speed = 16.375 km/h

∴ The correct answer is 16.38 km/h.

Additional Information

Average Speed for Two Equal Distances

If a journey is divided into two equal parts with speeds x and y, then Average Speed = 2xy ÷ (x + y).

Unit Conversion

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

Average Speed General Case

Average speed is always the total displacement (or total distance for scalar cases) divided by the total time taken, regardless of intermediate stops.

Shortcut Trick

Net Multiplier = (Markup Factor) × (Discount Factor) × (Weight Factor)

Markup 50% → 3/2 | Discount 20% → 4/5 | Faulty weight (reads 1000 for 900) → 10/9

Net Ratio = (3/2) × (4/5) × (10/9) = (120 / 90) = 4/3

Profit % = (4 − 3)/3 × 100 = 33.33% ≈ 33%

∴ The correct answer is 33%.

Alternate Method

Given:

Markup % = 50%

Discount % = 20%

Faulty Balance = 1 kg (1000 gm) instead of 900 gm

Let the CP of 1000 gm be ₹1000.

⇒ CP of 900 gm = ₹900 (This is the dealer's actual cost for what he gives).

⇒ Marked Price (MP) for 1000 gm = ₹1000 + 50% of 1000 = ₹1500.

⇒ Selling Price (SP) after 20% discount on MP = ₹1500 − 20% of 1500

⇒ SP = ₹1500 − ₹300 = ₹1200.

⇒ The dealer gives only 900 gm but charges for 1000 gm at SP.

⇒ Profit = SP − Actual CP of 900 gm = ₹1200 − ₹900 = ₹300.

⇒ Profit % = (Profit / Actual CP) × 100

⇒ Profit % = (300 / 900) × 100 = 1/3 × 100 = 33.33%

⇒ Rounded to nearest integer = 33%.

∴ The correct answer is 33%.

Additional Information

Dishonest Dealer Formula

Profit % = [(True Weight − False Weight) / False Weight] × 100. This is the profit due to weight manipulation only.

Successive Percentage Rule

When two changes x% and y% are applied, net change = [x + y + (xy/100)]%. For markup (positive) and discount (negative), x = markup and y = −discount.

Combined Effective Profit

Net SP/CP Ratio = (1 + markup/100) × (1 − discount/100) × (True Weight / False Weight). Profit % = (Ratio − 1) × 100.

Shortcut Trick

In a right-angled triangle with angles 30°, 60°, and 90°, the ratio of sides opposite to these angles is 1 : √3 : 2.

Here, ∠B = 90°, ∠A = 60°, therefore ∠C = 30°.

The side AB is opposite to ∠C (30°), so AB = 1 unit = 12 cm.

The side BC is opposite to ∠A (60°), so BC = √3 units = 12 × √3 cm.

∴ The correct answer is 12√3 cm.

Alternate Method

Given:

Triangle ABC is right-angled at B (∠ABC = 90°).

Length of AB = 12 cm.

Angle ∠CAB = 60°.

Formula Used:

In a right-angled triangle, tan θ = (Side Opposite to θ) ÷ (Side Adjacent to θ)

Calculations:

In ΔABC, for angle A = 60°:

⇒ tan(∠CAB) = BC ÷ AB

⇒ tan 60° = BC ÷ 12

⇒ √3 = BC ÷ 12

⇒ BC = 12 × √3

⇒ BC = 12√3 cm

∴ The correct answer is 12√3 cm.

Additional Information

Trigonometric Ratios

For any angle θ, sin θ = Opp/Hyp, cos θ = Adj/Hyp, and tan θ = Opp/Adj.

Special Values

Common values include tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Pythagoras Theorem

In a right triangle, Hypotenuse² = Base² + Perpendicular² (AC² = AB² + BC²).

Sum of Angles

The interior angles of any triangle always add up to 180° (∠A + ∠B + ∠C = 180°).

Shortcut Trick

Number of sides (n) = (Sum of interior angles ÷ 180) + 2

n = (2700 ÷ 180) + 2 = 15 + 2 = 17

Number of diagonals = n × (n − 3) ÷ 2

Number of diagonals = 17 × 14 ÷ 2 = 119

∴ The correct answer is 119.

Alternate Method

Given:

Sum of interior angles = 2700°

Formula Used:

1. Sum of interior angles of a polygon = (n − 2) × 180°
2. Number of diagonals = [n × (n − 3)] ÷ 2

where n = number of sides.

Calculations:

⇒ (n − 2) × 180° = 2700°

⇒ n − 2 = 2700 ÷ 180

⇒ n − 2 = 15

⇒ n = 15 + 2 = 17

⇒ Number of diagonals = [17 × (17 − 3)] ÷ 2

⇒ Number of diagonals = 17 × 7 = 119

∴ The correct answer is 119.

Additional Information

Sum of Exterior Angles

For any convex polygon, the sum of all exterior angles is always 360°.

Interior and Exterior Angle Relation

At each vertex of a regular polygon, Interior Angle + Exterior Angle = 180°.

Each Interior Angle of Regular Polygon

The measure of each interior angle is calculated as [(n − 2) × 180°] ÷ n.

Each Exterior Angle of Regular Polygon

The measure of each exterior angle is calculated as 360° ÷ n.

Shortcut Trick

Interest = 2/5 of Principal = (2/5 × 100)% = 40% of Principal.

Total Interest % = Rate × Time

40% = Rate × 5 years

Rate = 40% ÷ 5 = 8%

∴ The correct answer is 8%.

Alternate Method

Given:

Principal (P) = Rs. X

Time (T) = 5 years

Simple Interest (SI) = 2/5 of X = 0.4X

Formula Used:

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

Calculations:

⇒ SI = (P × R × T) ÷ 100

⇒ (2/5) × X = (X × R × 5) ÷ 100

⇒ 2/5 = 5R ÷ 100

⇒ 0.4 = R ÷ 20

⇒ R = 0.4 × 20

⇒ R = 8%

∴ The correct answer is 8%.

Additional Information

Simple Interest Components

SI = (P × R × T) ÷ 100. If any three variables are known, the fourth can be calculated directly.

Effective Rate of Interest

The total percentage change in the principal over T years at R% simple interest per year is (R × T)%.

Total Amount Formula

Amount (A) = Principal (P) + Simple Interest (SI) = P × (1 + RT/100).

Given:

$3 - (- 6){- 2 - 9 - 3} \div 7{1 + (- 2)(- 1)}$ = ?

Formula used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

$3 - (- 6){- 14} \div 7{1 + 2}$

⇒ $3 - 84 \div (7 \times 3)$

⇒ $3 - 84 \div 21$

⇒ $3 - 4$

⇒ -1

∴ The correct answer is option (1).

Shortcut Trick

Sum of parallel sides = (2 × Area) ÷ Height

⇒ Sum = (2 × 1380) ÷ 60 = 46 cm.

Given Difference = 12 cm.

Sides = (Sum ± Difference) ÷ 2 = (46 ± 12) ÷ 2 = 29 cm and 17 cm.

∴ The correct answer is 29, 17 cm.

Alternate Method

Given:

Area of trapezium = 1380 cm²

Height (h) = 60 cm

Difference between parallel sides (a − b) = 12 cm

Formula used: Area of trapezium = 1/2 × (a + b) × h

⇒ 1380 = 1/2 × (a + b) × 60

⇒ 1380 = 30 × (a + b)

⇒ a + b = 1380 ÷ 30

⇒ a + b = 46 --- (Equation 1)

⇒ a − b = 12 --- (Equation 2)

⇒ Adding (1) and (2): 2a = 58 ⇒ a = 29 cm

⇒ Substituting in (1): 29 + b = 46 ⇒ b = 17 cm

∴ The correct answer is 29, 17 cm.

Additional Information

Area of Trapezium

The area is calculated as the product of the average of the parallel sides and the perpendicular distance between them: Area = 1/2 × (Sum of parallel sides) × Height.

Median of a Trapezium

The median (mid-segment) is parallel to the bases and its length is the average of the bases: Median = (a + b) ÷ 2.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal in length and the base angles are also equal.

Shortcut Trick Direct Empirical Formula: Mode = 3 × Median − 2 × Mean Substitute: Mode = 3(48) − 2(60) Calculate: 144 − 120 = 24 ∴ The correct answer is 24.

Alternate Method Given: Mean = 60 Median = 48 Formula Used: For a moderately skewed distribution, the empirical relation is: Mode = 3 × Median − 2 × Mean

Calculations: ⇒ Mode = 3 × 48 − 2 × 60 ⇒ Mode = 144 − 120 ⇒ Mode = 24 ∴ The correct answer is 24.

Additional Information Empirical Relation (Karl Pearson) The relationship Mean − Mode = 3 × (Mean − Median) is used for moderately skewed asymmetrical distributions. Symmetric Distribution In a perfectly symmetrical distribution, the Mean, Median, and Mode are all equal (Mean = Median = Mode). Skewness Identification When Mean > Median > Mode, the distribution is positively skewed; if Mode > Median > Mean, it is negatively skewed.

Shortcut Trick

In an equilateral triangle, the distance from any vertex to the centroid (L) is equal to its circumradius (R).

Shortcut formula: PL = Side / √3

Given side PQ = 6 cm

PL = 6 / √3 = (2 × 3) / √3 = 2√3 cm

∴ The correct answer is 2√3 cm.

Alternate Method

Given:

Side of equilateral triangle PQR (a) = 6 cm

L is the centroid of the triangle.

Formula used:

1. Altitude (h) of an equilateral triangle = (√3 / 2) × a
2. Centroid divides the median (altitude in this case) in the ratio 2 : 1 from the vertex.

Calculations:

⇒ Altitude (PM) = (√3 / 2) × 6

⇒ PM = 3√3 cm

⇒ PL = (2 / 3) × PM

⇒ PL = (2 / 3) × 3√3

⇒ PL = 2√3 cm

∴ The correct answer is 2√3 cm.

Additional Information

Centroid of a Triangle

The point where all three medians of a triangle intersect. It divides each median in the ratio 2 : 1 from the vertex to the midpoint of the opposite side.

Equilateral Triangle Properties

For a side length a, the circumradius R = a / √3, and the inradius r = a / (2√3). The centroid, orthocenter, circumcenter, and incenter all coincide at the same point.

Area and Height

Height h = (√3 / 2) × a and Area = (√3 / 4) × a².

Shortcut Trick

120° lies in the Second Quadrant (90° to 180°).

In the Second Quadrant, the Cosine function is always negative.

Reference Angle = 180° − 120° = 60°.

cos 120° = −cos 60° = −0.5.

∴ The correct answer is −0.5.

Alternate Method

Given: Angle = 120°

Formula Used: cos(180° − θ) = −cos θ

⇒ cos 120° can be written as cos(180° − 60°).

⇒ We know that cos(180° − θ) = −cos θ.

⇒ cos 120° = −cos 60°.

⇒ Since cos 60° = 0.5 (or 1/2).

⇒ cos 120° = −0.5.

∴ The correct answer is −0.5.

Additional Information

ASTC Rule (Quadrant System)

All (Q-I), Silver (Q-II: Sin/Cosec), Tea (Q-III: Tan/Cot), Cups (Q-IV: Cos/Sec) identifies where functions are positive.

Standard Cosine Values

cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 0.5, cos 90° = 0.

Angle Reduction Formulas

cos(90° + θ) = −sin θ | cos(180° − θ) = −cos θ | cos(180° + θ) = −cos θ | cos(360° − θ) = cos θ.

Shortcut Trick

Note the identity: (2x + y)² = 4x² + y² + 4xy.

Substitute the known values 4x² + y² = 40 and xy = 6.

(2x + y)² = 40 + 4 × 6 = 40 + 24 = 64.

Taking the square root on both sides: 2x + y = √64 = 8.

∴ The correct answer is 8.

Alternate Method

Given:

4x² + y² = 40

xy = 6

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ Consider the expression (2x + y)².

⇒ By applying the algebraic identity, (2x + y)² = (2x)² + (y)² + 2 × (2x) × (y)

⇒ (2x + y)² = 4x² + y² + 4xy

⇒ Substituting the given values: (2x + y)² = 40 + 4 × 6

⇒ (2x + y)² = 40 + 24

⇒ (2x + y)² = 64

⇒ 2x + y = √64

⇒ 2x + y = 8

∴ The correct answer is 8.

Additional Information

Square of a Binomial Sum

The identity (a + b)² = a² + 2ab + b² is used to relate the sum of squares and the product of two terms to the square of their sum.

Square of a Binomial Difference

The identity (a − b)² = a² − 2ab + b² helps in finding the difference between terms when squares and products are known.

Difference of Two Squares

The identity a² − b² = (a + b)(a − b) is essential for factorising expressions involving the subtraction of two perfect squares.

Shortcut Trick

Value ratio = (5 × 0.25) : (4 × 0.50) : (3 × 2) : (1 × 5) = 1.25 : 2 : 6 : 5

Sum of ratio parts = 1.25 + 2 + 6 + 5 = 14.25

Value of 1 unit = 285 ÷ 14.25 = 20

Difference in count between 25 paise and ₹5 coins = (5 − 1) × 20 = 4 × 20 = 80

∴ The correct answer is 80 coins.

Alternate Method

Given:

Ratio of number of coins (25p : 50p : ₹2 : ₹5) = 5 : 4 : 3 : 1

Total amount = ₹285

Formula Used:

Total Value = ∑ (Denomination × Number of coins)

Let the number of coins be 5x, 4x, 3x, and 1x.

Total value in ₹ = (Number of 25p coins × 0.25) + (Number of 50p coins × 0.50) + (Number of ₹2 coins × 2) + (Number of ₹5 coins × 5)

⇒ (5x × 0.25) + (4x × 0.50) + (3x × 2) + (1x × 5) = 285

⇒ 1.25x + 2x + 6x + 5x = 285

⇒ 14.25x = 285

⇒ x = 285 ÷ 14.25

⇒ x = 20

Number of 25 paise coins = 5x = 5 × 20 = 100

Number of ₹5 coins = 1x = 1 × 20 = 20

Difference = 100 − 20 = 80

∴ The correct answer is 80 coins.

Additional Information

Unit Conversion

Always ensure all denominations are in the same unit (either all in paise or all in rupees) before calculation. 1 Rupee = 100 Paise.

Value vs Count Ratio

If the ratio of values is given as V₁ : V₂, the ratio of the number of coins is (V₁/D₁) : (V₂/D₂), where D is the denomination.

Combined Ratio

For three or more items, the total amount is ∑(Ratio Part × Denomination) × Constant = Total Money.

Shortcut Trick

Base Average of 4 papers = 51 marks

Marks in the 5th paper = 56 marks

Deviation (Extra marks) = 56 − 51 = +5 marks

Distribute deviation across all 5 papers = 5 ÷ 5 = +1 mark

New Average = 51 + 1 = 52 marks

∴ The correct answer is 52 marks.

Alternate Method

Given:

Average of 4 papers = 51

Marks in the 5th paper = 56

Formula Used:

Average = (Total Sum of Observations) ÷ (Number of Observations)

⇒ Total marks in 4 papers = 4 × 51 = 204

⇒ Total marks in 5 papers = 204 + 56 = 260

⇒ New average = 260 ÷ 5

⇒ New average = 52 marks

∴ The correct answer is 52 marks.

Additional Information

Weighted Average

When two groups with different averages are combined, the resulting average is (n₁×a₁ + n₂×a₂) ÷ (n₁ + n₂).

Effect of Constant Change

If every observation in a set is increased or decreased by a constant value 'k', the average also increases or decreases by 'k'.

Net Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always equal to zero.

Shortcut Trick

Efficiency Ratio (A : B) = 4 : 1

Total Efficiency (A + B) = 4 + 1 = 5 units/day

Total Work = Total Efficiency × Total Days = 5 × 48 = 240 units

B alone = Total Work ÷ Efficiency of B = 240 ÷ 1 = 240 days

∴ The correct answer is 240 days.

Alternate Method

Given:

Time taken by (A + B) = 48 days

Efficiency relation: A = 4 × B

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Let the efficiency of B be 1 unit/day.

⇒ Since A is 4 times as efficient as B, Efficiency of A = 4 × 1 = 4 units/day.

⇒ Combined efficiency of (A + B) = 4 + 1 = 5 units/day.

⇒ Total Work = (Combined Efficiency) × (Combined Time) = 5 × 48 = 240 units.

⇒ Time taken by B alone = Total Work ÷ Efficiency of B = 240 ÷ 1 = 240 days.

∴ The correct answer is 240 days.

Additional Information

Time and Efficiency Relation

Time is inversely proportional to efficiency. If the ratio of efficiency of A and B is x : y, then the ratio of time taken by them to complete the same work is y : x.

Combined Work Formula

If A can do a work in 'a' days and B can do it in 'b' days, they together can complete the work in (a × b) ÷ (a + b) days.

Work and Wages

Wages are generally distributed in the ratio of the work done by each person, which is the product of their efficiency and the time they worked.

Shortcut Trick

For a system of equations to have no solution, the lines must be parallel, where the ratio of x and y coefficients are equal.

Ratio of x-coefficients: 18 ÷ 7

Ratio of y-coefficients: −72 ÷ (−m) = 72/m

Equating ratios: 18/7 = 72/m

m = (72 × 7) ÷ 18 ⇒ 4 × 7 = 28

∴ The correct answer is 28.

Alternate Method

Given equations: 18x − 72y + 13 = 0 and 7x − my − 17 = 0.

Comparing with the standard form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

a₁ = 18, b₁ = −72, c₁ = 13

a₂ = 7, b₂ = −m, c₂ = −17

Formula for No Solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

⇒ 18/7 = −72/−m

⇒ 18/7 = 72/m

⇒ 18 × m = 72 × 7

⇒ m = (72 × 7) ÷ 18

⇒ m = 4 × 7

⇒ m = 28

Check: 18/7 ≠ 13/−17, which holds true.

∴ The correct answer is 28.

Additional Information

Unique Solution

A system of linear equations has a unique solution if the lines intersect: a₁/a₂ ≠ b₁/b₂.

Infinite Solutions

A system has infinite solutions if the lines are coincident: a₁/a₂ = b₁/b₂ = c₁/c₂.

Consistency

A system is consistent if it has at least one solution (unique or infinite) and inconsistent if it has no solution.

Shortcut Trick

For two successive discounts of x% and y%, use the net discount formula:

Net Discount = (x + y − (x × y)/100)%

Substitute x = 10 and y = 10:

Net Discount = (10 + 10 − (10 × 10)/100)% = (20 − 1)% = 19%

∴ The correct answer is 19%.

Alternate Method

Given: First discount (D₁) = 10%, Second discount (D₂) = 10%

Formula Used: Final Price = MP × (1 − d₁/100) × (1 − d₂/100)

⇒ Let the Marked Price (MP) be ₹100.

⇒ Price after 1st discount = 100 − (10% of 100) = 100 − 10 = ₹90

⇒ Price after 2nd discount = 90 − (10% of 90) = 90 − 9 = ₹81

⇒ Total Discount amount = 100 − 81 = ₹19

⇒ Net Discount Percentage = (19/100) × 100 = 19%

∴ The correct answer is 19%.

Additional Information

Successive Discounts for Three Terms

For three successive discounts x%, y%, and z%, first find the net of x and y (let it be d), then find the net of d and z using the same formula.

Marked Price and Profit Relation

The relation between CP and MP when a discount d% is given and a profit p% is made is MP/CP = (100 + p)/(100 − d).

Effective Percentage Change

Successive discounts are a form of successive percentage decreases. If values increase, the formula becomes (x + y + (xy/100))%.

∴ The correct answer is 19%.

Shortcut Trick

For elevation angle θ from a height h to another height H over distance d:

H = h + d × tan(θ)

Here, h = 24√3 feet, d = 60 feet, and θ = 60°

H = 24√3 + 60 × tan(60°) = 24√3 + 60√3 = 84√3 feet

∴ The correct answer is 84√3 feet.

Alternate Method

Given:

Height of pole (AB) = 24√3 feet

Distance between pole and tower (BD) = 60 feet

Angle of elevation (θ) = 60°

Formula Used:

In a right-angled triangle, tan θ = Perpendicular ÷ Base

Let CD be the height of the tower and AB be the height of the pole.

From the diagram, BD = AE = 60 feet and AB = DE = 24√3 feet.

In ▵CAE:

⇒ tan 60° = CE ÷ AE

⇒ √3 = CE ÷ 60

⇒ CE = 60√3 feet

Total height of the tower (CD) = CE + ED

⇒ CD = 60√3 + 24√3

⇒ CD = 84√3 feet

∴ The correct answer is 84√3 feet.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking upwards at an object.

Trigonometric Ratios for Standard Angles

tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3. These are frequently used in height and distance problems.

Relation between Elevation and Depression

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

Shortcut Trick

Mode is the value that appears most frequently in a given data set.

Frequency of 27 → 3 times (Max occurrence)

Frequency of 13 → 2 times

All other numbers (12, 15, 16, 28, 30, 31) → 1 time each

Since 27 has the highest frequency, it is the mode.

∴ The correct answer is 27.

Alternate Method

Given: Data set = 13, 15, 31, 12, 27, 13, 27, 30, 27, 28, 16

Formula: Mode = Observation with maximum frequency.

To find the mode, we count the frequency of each unique number:

⇒ Frequency of 12 = 1

⇒ Frequency of 13 = 2

⇒ Frequency of 15 = 1

⇒ Frequency of 16 = 1

⇒ Frequency of 27 = 3

⇒ Frequency of 28 = 1

⇒ Frequency of 30 = 1

⇒ Frequency of 31 = 1

⇒ Max Frequency = 3

⇒ Corresponding value = 27

∴ The correct answer is 27.

Additional Information

Arithmetic Mean

The average of all observations, calculated as Mean = (Sum of all observations) ÷ (Total number of observations).

Median

The middle value of a data set when arranged in ascending or descending order. For n observations, if n is odd, Median = ((n+1)/2)^th term.

Empirical Formula

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 Median − 2 Mean.

Types of Mode

Data sets can be Unimodal (one mode), Bimodal (two modes), or Multimodal (more than two modes).

Shortcut Trick

Ratio 6 years ago = 7 : 5 | Ratio 4 years later = 11 : 9

Difference in ratio parts: A (11 − 7) = 4 units, B (9 − 5) = 4 units.

Total time gap = 6 + 4 = 10 years.

4 units = 10 years ⇒ 1 unit = 2.5 years.

A’s age 6 years ago = 7 × 2.5 = 17.5 years.

A’s present age = 17.5 + 6 = 23.5 years.

∴ The correct answer is 23½ years.

Alternate Method

Given:

Ratio 6 years ago = 7 : 5

Ratio 4 years hence = 11 : 9

Calculations:

Let the ages of A and B six years ago be 7x and 5x respectively.

⇒ Present age of A = 7x + 6

⇒ Present age of B = 5x + 6

⇒ Ages after 4 years: A = 7x + 6 + 4 = 7x + 10, B = 5x + 6 + 4 = 5x + 10

⇒ (7x + 10) ÷ (5x + 10) = 11 ÷ 9

⇒ 9 × (7x + 10) = 11 × (5x + 10)

⇒ 63x + 90 = 55x + 110

⇒ 63x − 55x = 110 − 90

⇒ 8x = 20

⇒ x = 20 ÷ 8 = 2.5

⇒ A’s present age = 7 × (2.5) + 6

⇒ A’s present age = 17.5 + 6 = 23.5 years

∴ The correct answer is 23½ years.

Additional Information

Ratio Difference Rule

If the age difference between two people is constant, the difference between their ratio parts must be adjusted to be equal across different time periods.

Cross-Multiplication Method

For ages A and B with ratios a:b and c:d over time t, the value of one unit is calculated as |t × (c−d)| ÷ |ad − bc|.

Age Difference Invariance

The difference between the ages of two individuals remains the same regardless of how many years pass or have passed.

Shortcut Trick

Savings = 30% of Salary

Expenditure = (100 − 30)% = 70% of Salary

Monthly Expenditure = 70% of 42,000

Monthly Expenditure = 0.7 × 42,000 = 29,400

∴ The correct answer is ₹29,400.

Alternate Method

Given:

Total Salary = ₹42,000

Savings Percentage = 30%

Formula Used:

Expenditure = Total Salary − Savings

Savings = (Savings Percentage / 100) × Total Salary

Calculations:

⇒ Savings = (30 / 100) × 42,000

⇒ Savings = 3 × 4,200 = ₹12,600

⇒ Expenditure = Total Salary − Savings

⇒ Expenditure = 42,000 − 12,600

⇒ Expenditure = ₹29,400

∴ The correct answer is ₹29,400.

Additional Information

Percentage Breakdown

Total income is always considered as 100%. If any part (like savings) is known as x%, the remaining part (expenditure) is calculated as (100 − x)%.

Ratio Method

A 30% saving implies a ratio of Savings : Expenditure = 30 : 70, which simplifies to 3 : 7. Total units (3+7) = 10, where 10 units = ₹42,000.

Percentage to Decimal

To find a percentage of a number quickly, multiply the number by its decimal equivalent (e.g., 70% = 0.70 or 70/100).

Shortcut Trick

Rate per half-year = 10% ÷ 2 = 5%

Effective CI rate for 4 cycles of 5% ≈ 21.55%

Amount = Principal × (100% + Effective Rate)

Amount = 6500 × 121.55% = 6500 × 1.2155 ≈ 7900.75

The nearest integral value is ₹7,900.

∴ The correct answer is ₹7,900.

Alternate Method

Given:

Principal (P) = ₹6,500

Rate of Interest (R) = 10% per annum

Time (T) = 2 years

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Where r is the rate per period and n is the number of periods.

Half-Yearly Compounding Timeline

0 ₹6,500

1

2

3

4 Amount = ?

Half-Years (n = 4)

+5%

+5%

+5%

+5%

Total CI Rate ≈ 21.55%

Calculations:

⇒ Since interest is compounded half-yearly, r = 10% ÷ 2 = 5%.

⇒ Number of half-years (n) = 2 × 2 = 4 cycles.

⇒ A = 6500 × (1 + 5/100)⁴

⇒ A = 6500 × (105/100)⁴

⇒ A = 6500 × (21/20)⁴

⇒ A = 6500 × (194481 ÷ 160000)

⇒ A = 65 × (194481 ÷ 1600) = 12641265 ÷ 1600

⇒ A ≈ 7900.79

⇒ Taking the required integral value, the amount becomes ₹7,900.

∴ The correct answer is ₹7,900.

Additional Information

Quarterly Compounding

When interest is compounded quarterly, the given annual rate is divided by 4 (R/4) and the time periods are multiplied by 4 (4T).

Effective Rate Formula

The effective compound interest rate for 2 continuous cycles at a rate R% is given by R + R + (R × R)/100.

Simple vs Compound Interest

Simple Interest (SI) remains constant for each period, whereas Compound Interest (CI) increases exponentially as interest is calculated on both principal and accumulated interest.

Shortcut Trick

To check divisibility by 8, only the last three digits of the number need to be checked.

The last three digits of 1254216 are 216.

216 ÷ 8 = 27

Since 216 is perfectly divisible by 8, the entire number 1254216 is divisible by 8.

∴ The correct answer is 8.

Alternate Method

Given: Number = 1254216

Formula/Rules Used:

1. Divisibility by 5: Last digit must be 0 or 5.
2. Divisibility by 11: (Sum of digits at odd places) − (Sum of digits at even places) must be 0 or a multiple of 11.
3. Divisibility by 16: Last four digits must be divisible by 16.
4. Divisibility by 8: Last three digits must be divisible by 8.

Calculations:

⇒ Check for 5: Last digit is 6. Not divisible by 5.

⇒ Check for 11: (6 + 2 + 5 + 1) − (1 + 4 + 2) = 14 − 7 = 7. Not divisible by 11.

⇒ Check for 16: Last four digits are 4216. 4216 ÷ 16 = 263.5. Not divisible by 16.

⇒ Check for 8: Last three digits are 216. 216 ÷ 8 = 27. Divisible by 8.

∴ The correct answer is 8.

Additional Information

Divisibility by powers of 2

A number is divisible by 2ⁿ if its last n digits are divisible by 2ⁿ. For 2 (2¹), check last 1 digit; for 4 (2²), check last 2 digits; for 8 (2³), check last 3 digits.

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively.

Divisibility by 6

A number is divisible by 6 if it is simultaneously divisible by both 2 (even number) and 3 (sum of digits divisible by 3).

The logic here followed is

Hence, the correct answer is "Option 2".

Given :

Given statement : 77 ÷ 7 × 17 – 49 + 7 = ? 

After replacing the signs according to the question :

77 - 7 + 17 × 49 ÷ 7 = ?

77 - 7 + 17 × 7 = ?

77 + 119 - 7 = ?

196 - 7 = ?

189 = ?

Hence, "189" is the correct answer.

Given : A certain number of people are standing in the row facing in the north direction.

Explanation : 

• Statement (I) → M is fourth from the right end; only two people Stand between M and B. B is to the right of M.

<br>

​

​As it cannot be determined whether M is standing at the right end or there are any other people standing to the right of M, therefore total number of people are standing in the row where all people are facing north cannot be answered. 

So, statement (I) is not sufficient to answer the question.

• Statement (II) → M is fourth from the right end; only two people Stand between M and B.
• 
  

​As any specific position of B cannot be determined with respect to M, so there are two possible positions of B i.e B1 and B2 shown above.

So, statement (II) is not sufficient to answer the question.

Now, Using both the statements :

• Statement (I) + Statement (II) → A is fourth from the left end; M is second to the right of A; and R is second from the left end. M is fourth from the right end; only two people Stand between M and B.

​

So, using both statements there are a total of 9 people are standing in the row where all people are facing north. Thus data in statements I and II together is sufficient to answer the question.

Hence, "Option - (2)" is the correct answer.

All the creatures given in the options except chickens are Mammals while the chickens are considered as birds.

Therefore, Chickens are the odd ones out in this group.

Hence, the correct answer is "Option 1".

The logic followed here is :- Sum of the position value of the opposite letters according to English alphabetic series is added to the number of letters within the word.

​So,

• ‘MADRAS’ is written as ‘112’

<br>

And,

• ‘JAMMU’ is written as ‘82’

<br>

Similarly,

• ‘PUNJAB’ is written as :

<br>

Hence, the correct answer is "104".

Given :

Given statement : D @ N @ H & Y @ F % V

So,

• D @ N → D is the daughter of N.

• N @ H → N is the daughter of H.

• H & Y → H is the husband of Y.

• Y @ F → Y is the daughter of F.

• F % V → F is the wife of V.

​

Thus, according to the final family tree Y is the maternal grandmother i.e mother's mother of D.

Hence, "Mother’s mother" is the correct answer.

Logic : In each figure there Star, circle, parallel lines and hash elements are in this order in the clockwise direction.

So,

• Option - (1) :

 

⇒ Here the order of elements is star, circle, parallel lines and hash in the clockwise direction.

• Option - (2) :

 

⇒ Here the order of elements is star, circle, parallel lines and hash in the clockwise direction.

• Option - (3) :

⇒ Here the order of elements is star, parallel lines, circle and hash in the clockwise direction.

• Option - (4) :

 ⇒ Here the order of elements is star, circle, parallel lines and hash in the clockwise direction.

Here, Figure - (3) is the odd one out.

Hence, "Option - (3)" is the correct answer.

The logic here followed is:

Hence, the correct answer is " 147".

Logic : Square of (1st number - 2nd number) - 3 = 3rd number

Given : 

• (8, 1, 46)

$(8 - 1)^2$ - 3 = 46

$(7)^2$ - 3 = 46

49 - 3 = 46

46 = 46 (LHS = RHS)

• (9, 4, 22)

$(9 - 4)^2$ - 3 = 22

$(5)^2$ - 3 = 22

25 - 3 = 22

22 = 22 (LHS = RHS)

So,

• Option - (1) : (4, 2, 5)

​$(4 - 2)^2$ - 3 = 5

$(2)^2$ - 3 = 5

4 - 3 = 5

1 ≠ 5 (LHS ≠ RHS)

• Option - (2) : (8, 5, 43)

​$(8 - 5)^2$ - 3 = 43

$(3)^2$ - 3 = 43

9 - 3 = 43

6 ≠ 43 (LHS ≠ RHS)

• Option - (3) : (7, 3, 13)

​$(7 - 3)^2$ - 3 = 13

$(4)^2$ - 3 = 13

16 - 3 = 13

13 = 13 (LHS = RHS)

• Option - (4) :  (7, 6, 13)

​$(7 - 6)^2$ - 3 = 13

$(1)^2$ - 3 = 13

1 - 3 = 13

-2 ≠ 13 (LHS ≠ RHS)

Hence, "Option - (3)" is the correct answer.

Given : 

1. Six people - P, Q, R, S, T, U and V
2. All of them are standing in a row all facing north.

Explanation :

• Statement - (I) : U is standing 3rd from the right end and 2nd to the left of R.

Here, Only position of U and R is given, therefore positions of the left five people cannot be determined and who is standing second to the right of T cannot be answered.

Thus, data in statement (I) alone is not sufficient to answer the question.

• Statement - (II) : Q is standing to the immediate right of T. Q is standing 2nd to the left of P

Here, position of P cannot be determined, therefore position of Q and T cannot be determined. Also position of the left four people cannot be determined therefore who is standing second to the right of T cannot be answered.

Thus, data in statement (II) alone is not sufficient to answer the question.

• Combining both the statements - U is standing 3rd from the right end and 2nd to the left of R. Q is standing to the immediate right of T. Q is standing 2nd to the left of P.

​

​Here, position of P cannot be determined, therefore position of Q and T cannot be determined. Also position of the left out four people cannot be determined therefore who is standing second to the right of T cannot be answered.

Therefore, The data in both the Statements (I) and (II) together are NOT sufficient to answer the question.

Hence, "Option - (4)" is the correct answer.

Given :

Given statement : S % D # F @ G & H @ J

So,

• S % D → S is the wife of D.

• D # F → D is the brother of F.

• F @ G → F is the daughter of G.

• G & H → G is the husband of H.

• H @ J → H is the daughter of J.

​

Thus, according to the final family tree F is the granddaughter i.e daughter's daughter of J.

Hence, "Daughter’s daughter" is the correct answer.

Explanation:

I. Even after declining employment, Country K has enough potters to continue contributing equally to the international market.: From the situation given, it can be inferred that despite the declining employment in the pottery industry in Country K, the country continues to contribute significantly (75%) to the world's pottery market. This implies that even after the decline in employment, Country K has enough potters to continue contributing equally to the international market. Therefore, the conclusion I can be concluded.

II. The local demand of pottery in Country K has substantially decreased leading to less interest among potters: On the other hand, there is no information provided in the situation about the local demand of pottery in Country K. We cannot assume or infer that the decline in employment is due to a decreased local demand for pottery, as the reduction in employment could be due to other factors, such as increased automation, outsourcing, or a change in manufacturing techniques. Therefore, conclusion II cannot be concluded.

Hence, the correct option here is "3) Only I can be concluded".

The logic here followed is 

Hence, the correct answer is "Option 3".

Given : 268 * 4 * 8 * 5 * 14 = 41

According to the question, by replacing the *signs :

So,

• Option - (1) : + × − ÷

268 + 4 × 8 − 5 ÷ 14 = 41

268 + 4 × 8 − 0.35 = 41

268 + 32 - 0.35 = 41

300 - 0.35 = 41

299.65 ≠ 41 (LHS ≠ RHS) 

• Option - (2) : × ÷ + −

268 × 4 ÷ 8 + 5 − 14 = 41

268 × 0.5 + 5 - 14 = 41

134 + 5 - 14 = 41

139 - 14 = 41

125 ≠ 41 (LHS ≠ RHS) 

• Option - (3) : ÷ × + −

268 ÷ 4 × 8 + 5 − 14 = 41

67 × 8 + 5 − 14 = 41

536 + 5 - 14 = 41

541 - 14 = 41

527 ≠ 41 (LHS ≠ RHS)

• Option - (4) : ÷ − × +

268 ÷ 4 − 8 × 5 + 14 = 41

67 − 8 × 5 + 14 = 41

67 + 14 - 40 = 41

81 - 40 = 41

41 = 41 (LHS = RHS)

Hence, "Option - (4)" is the correct answer.

The logic followed here is :- (2) is added to the position value according to English alphabetic diagonally.

​So,

• 'PLUMAGE' is written as ‘GICOWNR’

<br>

And,

• 'CRUSH' is written as 'ETWUJ'

<br>

Similarly,

• 'TRIBAL' is written as :

<br>

 

Hence, the correct answer is "NCDKTV".

Given : Each of A, B, C, D, E, F and G has an exam on a different day of a week starting from Monday and ending on Sunday of the same week.

Explanation : 

1) A has an exam on Wednesday.

2. Only one person has an exam between A and B.

As, Position of B with respect to A is not clear, therefore there are two possibilities for the position of B.

3) D has an exam on the day immediately before B.

4. Only one person has an exam between F and B. 

As D has an exam on the day immediately before B. Therefore Case- 1 is eliminated as there is no position for D.

5) G’s exam is on one of the days before A but on one of the days after C. 

After positioning C and G, only one position is left which will be occupied by the only person left i;e E.

Thus, according to the final arrangement G has exam on Tuesday.

Hence, "Tuesday" is the correct answer.

The logic followed here is:

When the folder paper is opened, the cuts will appear as follows

<br>

Hence, the correct answer is "Option 1".

Explanation:

Statement I indicates a decline in the agricultural produce in State M. Statement II indicates that there has been an increase in average rainfall in State M.

It's generally expected that increased rainfall would benefit agricultural produce, assuming all other factors are constant, as it would provide more water for crops. However, this is not always the case, too much rain can also lead to issues such as flooding or waterlogged soil that can negatively impact crops. But there is not enough information given to establish this correlation.

It's more reasonable to suggest that both these statements are effects, but they are of independent causes. The decline in agricultural produce could be due to factors like pests, disease, changes in farming practices, or changes in market demand. The increase in rainfall, on the other hand, could be due to climate change, changes in weather patterns, or other environmental factors.

Hence, the correct option is "4) Both I and II are effects of independent causes".

Logic : In each figure there is only one element or one circle is enclosed in the given figure.

So,

• Option - (1) :

 

⇒ Only one element or circle is enclosed in the given figure.

• Option - (2) :

 

⇒ Here two elements or circles are enclosed in the given figure.

• Option - (3) :

⇒ Only one element or circle is enclosed in the given figure.

• Option - (4) :

 ⇒ Only one element or circle is enclosed in the given figure.

Here, Figure - (2) is the odd one out.

Hence, "Option - (2)" is the correct answer.

Explanation:

I. Many Students are not paying attention to studies due to mobile games: From the statement given, it can be inferred that many students' addiction to mobile games is affecting their academic performance negatively. This suggests that students are likely not paying enough attention to their studies because they are playing mobile games, so inference I follows.

II. It is only because of mobile games that students fail in the examination: However, the statement does not imply that mobile games are the only reason students fail in examinations. There could be multiple other factors contributing to a student's failure, such as lack of understanding, insufficient study, personal issues, health problems, etc. Therefore, inference II does not follow from the statement given.

Hence, the correct answer is option 4, i.e., only inference I follow.

Given:

• The image must not to be rotated

=

Hence, the correct answer is "Option - (1)".

Given : F I R _ D F _ _ E D F _ T _ D F I _ E _

By checking options :

1. E I S I E U D → F I R E D -  F I S E D  - F I T E D -  F I U E D  → (It Follows the pattern of : )
3. E G S J E U E → F I R E D -  F G S E D  - F J T E D -  F I U E E → (It doesn't Follows the pattern)
4. E H S I E U D → F I R E D  - F H S E D  - F I T E D  - F I U E D → (It doesn't Follows the pattern)
5. ​E I S I E U E → F I R E D  - F I S E D -  F I T E D  -  F I U E E → (It doesn't Follows the pattern)

⇒Option - (1) gives the pattern of "F I R E D - F I S E D - F I T E D - F I U E D"

Hence, "E I S I E U D" is the correct answer.

Given:

 

1. 3, 4, 1, 5, 2, 6: If we arrange the letters according to this combination of numbers, we get "CELIPN," which is not a meaningful word.
2. 6, 5, 3, 4, 1, 2: If we arrange the letters according to this combination of numbers, we get "NICELP," which is not a meaningful word.
3. 2, 4, 6, 3, 5, 1: If we arrange the letters according to this combination of numbers, we get "PENCIL," which is a meaningful word.
4. 2, 4, 6, 3, 1, 5: If we arrange the letters according to this combination of numbers, we get "PENCLI," which is not a meaningful word.

Hence, option 3 is the correct answer.

The position of the English alphabet is given below:

The logic followed here is :- 

1. First letter → (2) is subtracted from the position value according to English alphabetic series.
2. Second letter → (2) is added to the position value according to English alphabetic series.
3. Third letter → (2) is subtracted from the position value according to English alphabetic series.

​So,

Option 1) # = IPD, % = OET ⇒ IPD : KNF :: QCV : OET

Does not follow the same pattern = Incorrect

Option 2) # = HPD, % = NFS ⇒ HPD : KNF :: QCV : NFS

Does not follow the same pattern = Incorrect

Option 3) # = MLH, % = OET ⇒ MLH : KNF :: QCV : OET

Follow the same pattern = Correct

Option 4) # = HQC, % = NFS ⇒ HQC : KNF :: QCV : NFS

Does not follow the same pattern = Incorrect

Hence, the correct answer is "# = MLH, % = OET".

The given statement is: Z > F ≥ A = B = G > S < E

From this, we can infer the following relationships:

• Z > F
• F ≥ A
• A = B
• B = G
• G > S
• S < E

Conclusion:

I) Z < A. ⇒Does not follow.[However, from the given statement, it's clear that Z > F ≥ A ⇒ Z > A. Therefore, Z cannot be less than A. So, conclusion I is not true.]

II) G < F. ⇒Does not follow [ However, from the given statement, it's clear that F ≥ A = B = G   ⇒ G ≤ F, which means G could be equal to or less than F. We cannot definitively conclude that G < F. So, conclusion II is not true either.]

Hence,, the correct answer is that neither conclusion I nor II is true.

The logic here followed is:

First number * 2 + 4 = Second Number

• In option 1, 18 * 2 + 4 = 36 + 4 = 40 ≠ 42
• In option 2, 1 * 2 + 4 = 2 + 4 = 6
• In option 3, 15 * 2 + 4 = 30 + 4 = 34
• In option 4, 7 * 2 + 4 = 18

Except option 1, all option follow same logic.

Hence, the correct answer is "Option 1".

The least possible Venn diagram is given below according to the given information.

Conclusion:

I. All pens are phones → Does not follow → (As all pen-drive are phones and all pen-drives are pens. So it is possible but not definitely true)

II. All laptops are pens → Does not follow → (As all phones are laptop and all pen-drives are pens. So it is possible but not definitely true)

III. All pen-drives are laptops → Follow → (As all phones are laptop and all pen-drive are phones. So as all pen-drives comes in phones which whole comes in laptops therefore it is definitely true)

Here, Only conclusion III follows

Hence, the correct answer is "Option - (1)".

The logic followed here is :-

• The circle and the arc elements exchanged positions.
• Whole figure is rotated 45º anti-clockwise.

​So,

• Figures A and B ⇒

​

Similarly,

• Figure C and D ⇒

​

Thus, final pattern is :

Hence, "Option - (4)" is the correct answer.

Given : There are 4 friends A, B, C and D standing in a straight line.

Explanation : 

1) B stands between C and D.

2. D does not stand last.

3) A and B are ahead of at least 1 person.

As A is ahead of at least 1 person, therefore case - 1 and case - 2 are eliminated as there is no position for A.

Thus, according to the final arrangement A stands first.

Hence, "A" is the correct answer.

The figure that will replace the question mark (?) in the following figure series is shown below:

1) For '

' and '

' elements :

• Number of the elements in each box are increasing subsequently.
• Bothe '

' and '

' elements are shown alternately.

​​

Final series is :

Hence, ‘option - (4)’ is the correct answer.

Here the correct answer is Through hand.

Key Points

• The segment that contains a grammatical error is "through hand" (option 4).
• The correct phrase would be "by hand" instead of "through hand".
• "By hand" means to do something manually, without the use of any equipment or machines.
• Whereas "through hand" doesn't make grammatical sense in the context of the sentence.
• Therefore, option 4 is the correct answer.
  

So the corrected sentence would be: "The postman delivers the letters by hand, door to door."

Additional Information

• "Through" is a preposition that can be used in various contexts. Here are some examples:
• Movement or direction: We walked through the park and saw a lot of animals.
• Passage from one side to another: You can enter the building through the front door.
• A means of accomplishing something: I learned English through watching movies and TV shows.
• From beginning to end: We read the book through the night and finished it before dawn.
• Because of: She got the job through her connections in the industry.
• In each of these examples**, "through"** is used to indicate something different. It's important to understand the context in which it is being used to determine its meaning.
• "By" is a versatile preposition with several meanings. Here are some examples of how "by" can be used:
• Means or method: I came by bus to the city.
• Agent or doer of an action: The book was written by Jane Austen.
• Time or deadline: I need to finish this project by Friday.
• Close proximity: He was sitting by the window.
• Quantity or measurement: I need five books by tomorrow.
• Indication of authorship or ownership: This painting is by Picasso.
• In each of these examples, "by" is used to indicate something different. It's essential to understand the context in which it is being used to determine its meaning.

Here the correct answer is Give in.

Key Points

• The most appropriate collocating word to fill in the blank is "give in" (option 2).
• The sentence means that mentally strong people don't give up or commit suicide over trivial issues.
• "Give in" is a phrasal verb that means to surrender, yield or submit to something. In this context, it means that mentally strong people do not give up on life, even in the face of difficulties.
• Example: Despite the numerous setbacks in his career, he never gave in to despair.
• "Make up" (option 1) means to invent or create something, and it doesn't make sense in this context.
• Example: She made up a story to explain her absence from work.
• "Conceive on" (option 3) is not a phrasal verb that makes sense in this context. "Conceive" means to form or devise a plan, and it is not appropriate to use it in this sentence.
• Example: He conceived a brilliant plan to win the competition.
• "Fall in" (option 4) means to conform or comply with something, and it doesn't fit in the context of the sentence.
• Example: He fell in line with the company policy to avoid any conflicts.
• Therefore, the correct answer is option 2, "give in".

<br>

Correct Sentence: Mentally strong people do not give in to suicide over trivial issues.

Here the correct answer is PQRS.

Key Points

• The paragraph begins by stating that Abraham Lincoln's definition of democracy is popular and widely accepted.
• Sentence Q defines democracy as "a government of the people, by the people and for the people".
• Then, the sentence R asserts that democracy is the most representative form of government.
• Finally, the last sentence (sentence S) concludes by stating that democracy is the most suitable form of government for India to take the nation forward on the path of progress.
• Therefore, the correct answer is option 3: PQRS.

The correct order of the sentences to form a meaningful and coherent paragraph is PQRS.

P. Perhaps no definition of democracy is more popular and universally acceptable than what is given by Abraham Lincoln.

Q. He defines democracy as "a government of the people, by the people and for the people."

R. Certainly, democracy is the most representative form of government.

S. To bind the people and take the nation ahead on the path of progress, democracy is the most suitable form of government for India.

Here the correct answer is E, D, C, B.

Key Points

• The sentence begins by stating the subject and verb: "I love to go".
• The next part of the sentence tells us where the speaker likes to go: "for a walk".
• The following part of the sentence indicates the frequency of this activity: "every morning".
• The next part of the sentence provides the reason for doing this: "so that".
• Finally, the last part of the sentence concludes with the purpose of the activity: "I can keep myself fit and healthy".
• Therefore, the correct answer is option 4: E, D, C, B.
  

The correct order of the parts to form a meaningful sentence is A, E, D, C, B, F.

A. I love

E. to go

D. for a walk

C. every morning

B. so that

F. I can keep myself fit and healthy.

Here, the correct answer is Steal the purse.

Key Points

• The segment that contains a grammatical error is segment 3: "steal the purse."
• The sentence is in the past tense, but this segment uses the present tense form of the verb "steal" instead of the past tense form "stole".
• Option 3: "steal the purse" is the correct answer as it contains the grammatical error.

Therefore, the correct sentence should be "She watched the thief who stole the purse in the bus."

Here the correct answer is Affluent.

Key Points

• The most appropriate antonym of the underlined word "insolvent" is "affluent".
• Insolvent means unable to pay debts owed, bankrupt or in financial distress.
• Example: John's expenses exceeded his income, and he was unable to pay his credit card bills, resulting in his being declared insolvent.
• Affluent, on the other hand, means wealthy or having a great deal of money.
• ​Example: The furniture company was declared bankrupt after it failed to meet its financial obligations to its creditors.
• Option 3: "affluent" is the correct answer as it is the opposite of "insolvent".

Additional Information

• Option 1: "independent" is not a suitable antonym for "insolvent" as it does not have any relation to the financial status of the bank or the manager.
• Option 2: "commercial" is also not a suitable antonym for "insolvent" as it pertains to commerce and trade, and does not have an opposite meaning to "insolvent".
• Option 4: "bankrupt" is a synonym for "insolvent" and not its antonym.

Here the correct answer is Seldom.

Key Points

• The most appropriate antonym for the word "often" in the given sentence is "seldom."
• "Often" means frequently or many times. Its antonym "seldom" means rarely or infrequently.
• Example sentence with the word "often":
• She often goes for a walk in the park after work. (meaning: she goes for a walk in the park frequently or regularly)
• Example sentence with the word "seldom":
• I seldom eat fast food because it's not good for my health. (meaning: I rarely or infrequently eat fast food because it's unhealthy)
• Option 1: "seldom" is the correct answer as it is the opposite of "often" and means rarely or infrequently.
• Option 2: "cherished" is not an antonym for "often" as it means beloved or valued.
• Option 3: "lovely" is not an antonym for "often" as it means beautiful or attractive.
• Option 4: "usually" is not an antonym for "often" as it means typically or generally.

Here the correct answer is The publishers will release a new book on games next week.

Key Points

• The above-given sentence is in the passive voice.
• We need to change it into active voice.
• The following steps need to be taken for converting the sentence into passive voice:-
• The subject 'A new book on games​' of the passive voice will become the object of the active voice.
• The object 'The publishers' of the passive voice will become the subject of the active voice.
• Since modal verb has been used in the active voice, therefore, the passive voice will follow the given structure:-
• Passive Voice - Object + modal verb*(will)* + be + V3 + Subject.
• Active Voice - Subject + modal verb*(will)* + V1 + Object.

Therefore, the correct passive voice is The publishers will release a new book on games next week.

Additional Information

• When we convert active voice to passive voice, three things need to be kept in mind:-
• Change object into subject.
• Change subject into object.
• Always use the third form of the verb in the passive voice.

The correct answer is option 1) later than. 

Key Points

• The correct phrase to fill in the blank is 'later than'.
• This is because the sentence is comparing the actual release date of the movie to the expected release date.
• "Later than" is the correct phrase to use in this context to indicate that the movie was released after the expected date.
• Options 2, 3, and 4 are incorrect because they use incorrect grammar and phrasing.
• "Most later than" and "more later than" are not proper phrases in English.
• Option 3 is also incorrect because "then" is not the correct word to use in this context.

<br>

Correct Sentence: The movie was released later than expected. 

Additional Information

• Than is used in comparisons as a conjunction (as in "she is younger than I am") and as a preposition ("he is taller than me").
• Then indicates time. It is used as an adverb ("I lived in Idaho then"), noun ("we'll have to wait until then"), and adjective ("the then-governor").

Here the correct answer is Experienced in.

Key Points

• The correct collocation is "experienced in", so the completed sentence is: "Rahul is experienced in coding."
• "Skilled on" is not a correct collocation. Instead, we can use "skilled in" or "skilled at".
• For example: "She is skilled in playing the piano" or "He is skilled at solving math problems".
• "Strong of" is also not a correct collocation. Instead, we can use "strong in" or "strong at".
• For example: "She is strong in mathematics" or "He is strong at playing basketball".
• "Experienced of" is not a correct collocation. Instead, we use "experienced in" or "experienced at".
• For example: "She is experienced in teaching English" or "He is experienced at using computer software".
• "Experienced in" means having knowledge or skill in a particular field or activity through practice or involvement. In the given sentence, it means that Rahul has expertise in coding.
• Overall, the correct collocation is "experienced in".

<br>

Correct Sentence: Rahul is experienced in Coding.

The correct answer is 2) Horticulture.

Key Points

• Horticulture refers to the science and art of growing fruits, vegetables, and ornamental plants. It includes plant propagation, cultivation, breeding, genetic engineering, and crop management.(फल, सब्जियां और सजावटी पौधे उगाने का विज्ञान और कला। इसमें पौधों का प्रसार, खेती, प्रजनन, जेनेटिक इंजीनियरिंग और फसल प्रबंधन शामिल हैं।)
• Horticulturists work in gardens, nurseries, greenhouses, research institutions, and academic institutions.
• Example sentence: The horticulture department of the university is conducting research on developing new varieties of tomatoes that are resistant to pests and diseases.
• Thus, the correct answer is Option 2.

Additional Information

• Other options:
• Herpetology: The study of reptiles and amphibians(सरीसृप और उभयचरों का अध्ययन).
• Example sentence: The herpetology club organized a field trip to the national park to observe different species of snakes and frogs.
• Haplology: The linguistic phenomenon of omitting one of two identical or similar syllables in a word(एक शब्द में दो समान या समान शब्दांशों में से एक को छोड़ने की भाषाई घटना).
• Example sentence: The word "probably" is an example of haplology, as the two syllables "ob" are omitted in pronunciation.
• Herbing: This is not a recognized word in English.(यह अंग्रेजी का कोई शब्द नहीं है।)

The correct answer is option 3, "Sycophant."

Key Points

• Gold-digger: someone who only associates with others for financial gain.
• Example: She only married him for his money; she's a gold-digger.
• Gregarious: someone who enjoys the company of others; sociable.
• Example: He's a gregarious person, always making new friends.
• Sycophant: someone who flatters others to gain advantage; a brown-noser.
• Example: He's always sucking up to the boss; he's a sycophant.
• Manipulative: someone who uses deceit or trickery to control others.
• Example: She's a very manipulative person, always getting what she wants through lies and manipulation.
• Therefore, option 3 is the correct one-word substitute for the given group of words.

Additional Information

• The main difference between a "gold-digger" and a "sycophant" is their motivation and behavior.
• A gold-digger is someone who is interested in forming a romantic relationship with someone primarily because of their wealth or financial status. They may try to manipulate or use the person for their own financial gain.
• On the other hand, a sycophant is someone who flatters and compliments others excessively in order to gain favor or advancement. They may be insincere and only interested in personal gain, but their focus is not necessarily on financial gain.
• In short, a gold-digger is motivated by money, while a sycophant is motivated by social or professional advancement.

The correct answer is option 3: "I told Mohan that he had a green book."

Key Points

• The given sentence is in direct speech. We need to change it into indirect speech.
• When converting direct speech to indirect speech, we need to make some changes to the original sentence. Here are the rules:
• The reporting verb (in this case, "said to") is changed to a suitable reporting verb (in this case, "told").
• The pronouns are changed to reflect the new subject of the sentence. In this case, "I" and "he" remains unchanged.
• The tense of the verb is changed to reflect the time of the reported speech.
• In this case, "has" becomes "had" because the original sentence was in the present tense, but the reported speech is in the past tense.
• Thus, the correct answer is Option 3.
  

Correct Sentence: "I told Mohan that he had a green book."

The correct answer is 'Option 4' i.e. 'maize'.

<br>

Key Points

• The sentence refers to a crop grown by farmers in a field, requiring a word that identifies a specific type of cereal grain.
• The word "maze" refers to a complex network of paths or passages; a labyrinth. (भूलभुलैया)
• Example: The children got lost in the garden maze.
• The word "maize" refers to a cereal plant that yields large grains set in rows on a cob; also known as corn. (मक्का)
• Example: The farmers harvested the maize just before the heavy rains started.
• Hence, we can infer that the correct word to fill the blank is 'maize'.

Therefore, the correct answer is- maize.

Additional Information

Here are the other options explained along with their meanings:

• Mays (ज़िया मेज़ - मक्का का वैज्ञानिक नाम का हिस्सा): While "Mays" is part of the botanical name for corn (Zea mays), it is not used in common English as a standalone noun for the crop.
• Example: Zea mays is the biological classification for corn.
• Mase (कोई मानक अर्थ नहीं): This is not a standard English word and is likely a misspelling or a distractor.

Here the correct answer is Option 3.

Key Points

• The INCORRECTLY spelled word is option 3) Acede, which should be spelled as "accede."
• Abhorrent - Correctly spelled word, meaning causing or deserving strong dislike or hatred.
• ​Example: The abhorrent behavior of the bullies was not tolerated in the school.
• Abseil - Correctly spelled word, meaning to descend a steep slope or vertical drop using a rope.
• Example: The mountaineers abseiled down the cliff to reach the base camp.
• Acede - INCORRECTLY spelled word, the correct spelling is "accede," meaning to agree to a request or demand.
• ​Example: The manager finally acceded to the employee's request for a day off.
• Abscess - Correctly spelled word, meaning a swollen area within body tissue, containing an accumulation of pus.
• Example: The doctor drained the abscess in the patient's leg to relieve the pain.
• Thus, the correct answer is Option 3.