Shortcut Trick

Downstream Speed (D) = Distance ÷ Time = 35 ÷ 2 = 17.5 km/h

Upstream Speed (U) = Distance ÷ Time = 35 ÷ 7 = 5 km/h

Speed in Still Water = (D + U) ÷ 2

Speed = (17.5 + 5) ÷ 2 = 22.5 ÷ 2 = 11.25 km/h

∴ The correct answer is 11.25 km/h.

Alternate Method

Given:

Distance (d) = 35 km

Time taken downstream (t_d) = 2 h

Time taken upstream (t_u) = 7 h

Formula Used:

Speed = Distance ÷ Time

Downstream speed = u + v

Upstream speed = u − v

(where u = speed of boat in still water and v = speed of stream)

Calculations:

⇒ Downstream speed (u + v) = 35 ÷ 2 = 17.5 km/h ---- (1)

⇒ Upstream speed (u − v) = 35 ÷ 7 = 5 km/h ---- (2)

⇒ Adding equations (1) and (2):

⇒ (u + v) + (u − v) = 17.5 + 5

⇒ 2u = 22.5

⇒ u = 22.5 ÷ 2

⇒ u = 11.25 km/h

∴ The correct answer is 11.25 km/h.

Additional Information

Speed of Stream

The speed of the water current or stream (v) is calculated as: v = (Downstream Speed − Upstream Speed) ÷ 2.

Relative Speed in Boats

When moving downstream, the speeds are added (u+v); when moving upstream, the stream speed is subtracted from the boat speed (u−v).

Ratio of Speeds

If the distance is constant, the ratio of downstream speed to upstream speed is the inverse of the ratio of time taken: D/U = T_u/T_d.

Shortcut Trick

For a 3-year period, use the interest ratio method 3 : 3 : 1.

First year interest (I₁): 12% of 25000 = 3000

Second year interest on I₁ (I₂): 12% of 3000 = 360

Third year interest on I₂ (I₃): 12% of 360 = 43.2

Total CI = (3 × 3000) + (3 × 360) + (1 × 43.2) = 9000 + 1080 + 43.2 = 10123.2

∴ The correct answer is ₹10,123.20.

Alternate Method

Given: Principal (P) = ₹25,000, Rate (R) = 12% p.a., Time (n) = 3 years.

Formula: Amount (A) = P(1 + R/100)ⁿ and CI = A − P

⇒ A = 25000 × (1 + 12/100)³

⇒ A = 25000 × (1.12)³

⇒ A = 25000 × 1.404928

⇒ A = 35123.2

⇒ CI = 35123.2 − 25000

⇒ CI = 10123.2

∴ The correct answer is ₹10,123.20.

Additional Information

Successive Percentage Change

Compound interest is a case of successive percentage increase. For 3 years, the effective rate is [3r + (3r²/100) + (r³/10000)]%.

Difference between CI and SI

For 2 years: Diff = P(R/100)². For 3 years: Diff = P(R/100)² × (3 + R/100).

Compounding Frequency

If interest is compounded half-yearly, the rate is halved (R/2) and the time period is doubled (2n).

Shortcut Trick

Factorize the expressions mentally by finding numbers that sum to the middle term and multiply to the last term.

x² − 8x + 15: Numbers are −3 and −5 → (x − 3)(x − 5).

x² − 5x + 6: Numbers are −2 and −3 → (x − 2)(x − 3).

LCM takes all unique factors: (x − 2), (x − 3), and (x − 5).

∴ The correct answer is (x − 2) (x − 3) (x − 5).

Alternate Method

Given expressions:

P(x) = x² − 8x + 15

Q(x) = x² − 5x + 6

Factorizing P(x) by splitting the middle term:

⇒ x² − 3x − 5x + 15

⇒ x(x − 3) − 5(x − 3)

⇒ (x − 3)(x − 5)

Factorizing Q(x) by splitting the middle term:

⇒ x² − 2x − 3x + 6

⇒ x(x − 2) − 3(x − 2)

⇒ (x − 2)(x − 3)

The Least Common Multiple (LCM) is the product of all distinct factors of the polynomials, each raised to the highest power it occurs in any of the polynomials.

⇒ LCM = (x − 2)¹ × (x − 3)¹ × (x − 5)¹

∴ The correct answer is (x − 2) (x − 3) (x − 5).

Additional Information

LCM and HCF of Polynomials

LCM contains all factors at their maximum power, while HCF (Highest Common Factor) contains only common factors at their minimum power.

The Relationship Property

For any two polynomials f(x) and g(x), the product of the polynomials is equal to the product of their HCF and LCM: f(x) × g(x) = HCF(f,g) × LCM(f,g).

Factor Theorem

A polynomial P(x) has a factor (x − a) if and only if P(a) = 0. This can be used to check factors quickly.

Shortcut Trick

Apply Chain Rule for 1 kg Walnut: (180 ÷ 3) × (8 ÷ 5) × (15 ÷ 2) × (3 ÷ 6)

1 kg Walnut = 60 × 1.6 × 7.5 × 0.5 = 96 × 3.75 = ₹360

For 10 kg Walnut: 360 × 10 = ₹3600

∴ The correct answer is ₹3600.

Alternate Method

Given:

Cost of 3 kg Rice = ₹180

8 kg Rice = 5 kg Pulse

15 kg Pulse = 2 kg Tea

3 kg Tea = 6 kg Walnuts

Formula Used:

Unit Cost = Total Cost ÷ Quantity

Equivalence Cost: Quantity_A × Price_A = Quantity_B × Price_B

⇒ Cost of 1 kg Rice = 180 ÷ 3 = ₹60

⇒ 5 kg Pulse = 8 kg Rice = 8 × 60 = ₹480

⇒ Cost of 1 kg Pulse = 480 ÷ 5 = ₹96

⇒ 2 kg Tea = 15 kg Pulse = 15 × 96 = ₹1440

⇒ Cost of 1 kg Tea = 1440 ÷ 2 = ₹720

⇒ 6 kg Walnut = 3 kg Tea = 3 × 720 = ₹2160

⇒ Cost of 1 kg Walnut = 2160 ÷ 6 = ₹360

⇒ Cost of 10 kg Walnut = 10 × 360 = ₹3600

∴ The correct answer is ₹3600.

Additional Information

Chain Rule in Proportion

It is a method used to find a relationship between multiple quantities. The formula is: (Item₁/Item₂) × (Item₂/Item₃) × ... × (Item_n−1/Item_n).

Unitary Method

To solve such problems, always find the value of a single unit (1 kg) for each commodity before moving to the next level in the hierarchy.

Inverse Proportion

If the quantity increases while the total cost remains constant, the unit price must decrease: Price ∝ 1 ÷ Quantity.

Shortcut Trick

Since the Principal (P) and the Rate of Interest (R) are the same for both cases, Simple Interest (SI) is directly proportional to Time (T).

SI ∝ T (when P and R are constant)

Ratio of Interest = Ratio of Time = T₁ : T₂

Ratio = 8 : 21

∴ The correct answer is 8 : 21.

Alternate Method

Given:

Principal (P₁) = Principal (P₂) = P

Rate (R₁) = Rate (R₂) = 21%

Time (T₁) = 8 years

Time (T₂) = 21 years

Formula Used:

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

Calculations:

⇒ SI₁ = (P × 21 × 8) ÷ 100

⇒ SI₂ = (P × 21 × 21) ÷ 100

⇒ Required Ratio = SI₁ : SI₂

⇒ Ratio = [(P × 21 × 8) / 100] ÷ [(P × 21 × 21) / 100]

⇒ Ratio = (P × 21 × 8) ÷ (P × 21 × 21)

⇒ Ratio = 8 ÷ 21

∴ The correct answer is 8 : 21.

Additional Information

Direct Proportionality in SI

Simple interest is directly proportional to Principal, Rate, and Time individually when the other two factors remain constant.

Effective Rate of Interest

The total interest percentage for the entire period in SI is simply (Rate × Time)%. For example, 21% for 8 years is 168% of Principal.

Simple Interest vs Compound Interest

In SI, interest is only calculated on the initial principal. In CI, interest is calculated on the principal plus any accumulated interest from previous periods.

Shortcut Trick

Identify the three standard Pythagorean trigonometric identities:

1. sin²θ + cos²θ = 1
2. sec²θ − tan²θ = 1
3. cosec²θ − cot²θ = 1

Comparing these with the options, option (3) states sec²θ + cos²θ = 1, which is mathematically incorrect.

∴ The correct answer is sec²θ + cos²θ = 1.

Alternate Method

Given: Four trigonometric formulas to verify.

Formula Used: In a right-angled triangle, Pythagoras Theorem: p² + b² = h².

⇒ Option 1: cosec²θ − cot²θ = (h/p)² − (b/p)² = (h² − b²)/p² = p²/p² = 1 (Correct)

⇒ Option 2: sin²θ + cos²θ = (p/h)² + (b/h)² = (p² + b²)/h² = h²/h² = 1 (Correct)

⇒ Option 3: sec²θ + cos²θ = (h/b)² + (b/h)² = (h⁴ + b⁴)/(b²h²) ≠ 1 (Incorrect)

⇒ Option 4: sec²θ − tan²θ = (h/b)² − (p/b)² = (h² − p²)/b² = b²/b² = 1 (Correct)

∴ The correct answer is sec²θ + cos²θ = 1.

Additional Information

Reciprocal Identities

Standard relations are sinθ × cosecθ = 1, cosθ × secθ = 1, and tanθ × cotθ = 1.

Complementary Angle Formulas

sin(90° − θ) = cosθ, tan(90° − θ) = cotθ, and sec(90° − θ) = cosecθ.

Derived Pythagorean Forms

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

Shortcut Trick

Mean (μ) = (5 + 3 + 4 + 7) ÷ 4 = 19 ÷ 4 = 4.75

Sum of squares (Σx²) = 5² + 3² + 4² + 7² = 25 + 9 + 16 + 49 = 99

Standard Deviation (σ) = √[(Σx² ÷ n) − μ²]

σ = √[(99 ÷ 4) − 4.75²] = √[24.75 − 22.5625] = √2.1875 ≈ 1.48

∴ The correct answer is 1.48.

Alternate Method

Given: Data points = {5, 3, 4, 7}, Number of observations (n) = 4

Formula Used: σ = √[Σ(x_i − μ)² ÷ n]

⇒ Mean (μ) = (5 + 3 + 4 + 7) ÷ 4 = 4.75

⇒ Squared deviation for 5: (5 − 4.75)² = 0.25² = 0.0625

⇒ Squared deviation for 3: (3 − 4.75)² = (−1.75)² = 3.0625

⇒ Squared deviation for 4: (4 − 4.75)² = (−0.75)² = 0.5625

⇒ Squared deviation for 7: (7 − 4.75)² = 2.25² = 5.0625

⇒ Variance (σ²) = (0.0625 + 3.0625 + 0.5625 + 5.0625) ÷ 4 = 8.75 ÷ 4 = 2.1875

⇒ Standard Deviation (σ) = √2.1875 ≈ 1.48

∴ The correct answer is 1.48.

Additional Information

Variance

Variance is the average of the squared differences from the mean, calculated as σ² = Σ(x − μ)² ÷ n.

Coefficient of Variation (CV)

It is a relative measure of dispersion calculated as CV = (σ ÷ μ) × 100%.

Range

Range is the simplest measure of dispersion, defined as the difference between the maximum and minimum values in a dataset.

Shortcut Trick

Let the initial investments be 300, 600, and 800 units.

Suresh's new investment = 300 + 5% of 300 = 315.

Dinesh's new investment = 600 + 15% of 600 = 690.

Ramesh's new investment = 800 + 20% of 800 = 960.

New Profit Ratio = 315 : 690 : 960 = 21 : 46 : 64 (divided by 15).

∴ The correct answer is 21 ∶ 46 ∶ 64.

Alternate Method

Given:

Investment ratio (S : D : R) = 3 : 6 : 8

Percentage increases = 5%, 15%, 20%

Time period = 1 year (Constant for all)

Formula Used: Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂) : (Investment₃ × Time₃)

Calculations:

⇒ Let initial investments be 3x, 6x, and 8x.

⇒ New investment of Suresh = 3x × (100 + 5)/100 = 3x × 1.05 = 3.15x

⇒ New investment of Dinesh = 6x × (100 + 15)/100 = 6x × 1.15 = 6.90x

⇒ New investment of Ramesh = 8x × (100 + 20)/100 = 8x × 1.20 = 9.60x

⇒ Ratio of profits = 3.15x : 6.90x : 9.60x

⇒ Multiplying by 100 to remove decimals: 315 : 690 : 960

⇒ Dividing each term by 15:

⇒ Suresh = 315 ÷ 15 = 21

⇒ Dinesh = 690 ÷ 15 = 46

⇒ Ramesh = 960 ÷ 15 = 64

∴ The correct answer is 21 ∶ 46 ∶ 64.

Additional Information

Partnership Profit Sharing Rule

Profit is always distributed in the ratio of the product of Capital and Time (P = C × T). If Time is constant, Profit Ratio = Capital Ratio.

Percentage Change in Ratio

If a value 'a' increases by r%, the new value is a × (1 + r/100). This can be applied directly to ratio terms.

Simplifying Ratios

To simplify a ratio involving decimals, multiply all terms by a power of 10 to convert them to integers and then divide by their HCF.

Shortcut Trick

Ratio of Rajesh : Rakesh : Ramesh = 2 : 4 : 9

Given: Rajesh (2 units) = 30 marks

Unit value = 30 ÷ 2 = 15 marks

Rakesh (4 units) = 4 × 15 = 60 marks

Ramesh (9 units) = 9 × 15 = 135 marks

∴ The correct answer is Rakesh = 60, Ramesh = 135.

Alternate Method

Given:

Ratio of marks = 2 : 4 : 9

Marks scored by Rajesh = 30

Formula:

Value of one ratio part = Given value ÷ Corresponding ratio share

⇒ Let the marks of Rajesh, Rakesh, and Ramesh be 2x, 4x, and 9x respectively.

⇒ 2x = 30

⇒ x = 30 ÷ 2 = 15

⇒ Rakesh's marks = 4x = 4 × 15 = 60

⇒ Ramesh's marks = 9x = 9 × 15 = 135

∴ The correct answer is Rakesh = 60, Ramesh = 135.

Additional Information

Ratio Definition

A ratio compares quantities by division and is usually expressed in its simplest form (a : b).

Unitary Method in Ratios

Total value = (Sum of ratio parts) × (Value of one part). Conversely, Value of one part = Given value ÷ Respective ratio part.

Proportion

Equality of two ratios is called proportion. If a : b :: c : d, then product of extremes (ad) = product of means (bc).

Shortcut Trick

Initial average of 20 numbers = 32.

Numbers removed = 29 and 31 (Total = 60).

Deviation from average = (29 − 32) + (31 − 32) = −3 + (−1) = −4.

Since the removed sum (60) is less than the expected sum (32 × 2 = 64), the average will increase.

Net increase in average = 4 ÷ 18 ≈ 0.22.

New average = 32 + 0.22 = 32.22.

∴ The correct answer is 32.22.

Alternate Method

Given:

Total numbers (n₁) = 20

Initial average (A₁) = 32

Numbers to be removed = 29 and 31

Formula: Average = Sum of observations ÷ Number of observations

Calculations:

⇒ Total sum of 20 numbers = 20 × 32 = 640

⇒ Sum of the two removed numbers = 29 + 31 = 60

⇒ Sum of the remaining numbers = 640 − 60 = 580

⇒ Total remaining numbers = 20 − 2 = 18

⇒ Average of remaining numbers = 580 ÷ 18

⇒ Average = 290 ÷ 9 = 32.222...

∴ The correct answer is 32.22.

Additional Information

Average of First n Natural Numbers

The average of the first n natural numbers is given by the formula (n + 1) ÷ 2.

Effect of Replacement on Average

When a value x is replaced by y, the New Average = Old Average + (y − x) ÷ Total Count.

Weighted Average

For two groups with averages A₁, A₂ and counts n₁, n₂, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

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

Base (opposite to 30°) = 70 m.

Height (opposite to 60°) = Base × √3.

Height = 70 × √3 = 70√3 m.

∴ The correct answer is 70√3 m.

Alternate Method

Given:

Distance from the foot (Base) = 70 m

Angle of elevation (θ) = 60°

Formula Used:

tan θ = Perpendicular (Height) ÷ Base

Calculations:

⇒ tan 60° = h ÷ 70

⇒ √3 = h ÷ 70

⇒ h = 70 × √3

⇒ h = 70√3 m

∴ The correct answer is 70√3 m.

Additional Information

Trigonometric Values for Heights & Distances

tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3 are the most frequently used ratios in elevation problems.

Angle of Elevation and Depression

The angle of elevation from an observer's eye to an object is numerically equal to the angle of depression from the object to the observer due to alternate interior angles.

30-60-90 Triangle Theorem

In any 30°−60°−90° triangle, the hypotenuse is twice the shorter leg, and the longer leg is √3 times the shorter leg.

Shortcut Trick

BC is diameter ⇒ ∠BAC = 90° (angle in semicircle)

∠BAD = 90° + 32° = 122°

ABCD is cyclic quadrilateral ⇒ ∠BCD = 180° − 122° = 58°

∠ACD = ∠BCD − ∠BCA = 58° − 19° = 39°

∴ The correct answer is 39°.

Alternate Method

Given:

∠BCA = 19° and ∠CAD = 32°

Formula used:

Opposite angles of a cyclic quadrilateral are supplementary.

Angle in a semicircle = 90°

Calculation:

ABCD is a cyclic quadrilateral.

⇒ ∠BAD + ∠BCD = 180°

BC is the diameter of the circle.

⇒ ∠BAC = 90°

⇒ ∠BAD = ∠BAC + ∠CAD

⇒ ∠BAD = 90° + 32°

⇒ ∠BAD = 122°

⇒ ∠BCD = 180° − 122°

⇒ ∠BCD = 58°

⇒ ∠BCD = ∠BCA + ∠ACD

⇒ 58° = 19° + ∠ACD

⇒ ∠ACD = 58° − 19°

⇒ ∠ACD = 39°

∴ ∠ACD = 39°.

Additional Information

Angle in a Semi-circle

An angle subtended by the diameter of a circle at any point on the circumference is always a right angle (90°).

Cyclic Quadrilateral Property

The sum of opposite angles of a cyclic quadrilateral is always 180° (∠A + ∠C = 180° and ∠B + ∠D = 180°).

Angles in the Same Segment

Angles subtended by the same arc at the circumference of the circle are equal. For example, ∠ADB = ∠ACB.

Given:

$\sqrt{20 - \sqrt{20 - \sqrt{20 - \dots \infty}}}$ = ?

Formula used:

If x = $\sqrt{a - \sqrt{a - \dots \infty}}$, then x² + x - a = 0

Shortcut: If a = n × (n + 1), then x = n

Calculations:

20 = 4 × 5

⇒ 20 = 4 × (4 + 1)

⇒ n = 4

⇒ x = 4

∴ The correct answer is option (1).

Shortcut Trick

Use multiplying factors for successive discounts: (1 − d₁)(1 − d₂)(1 − d₃)

Multiplying factor = (1 − 0.12) × (1 − 0.18) × (1 − 0.25)

⇒ 0.88 × 0.82 × 0.75 = 0.5412

Single Equivalent Discount % = (1 − 0.5412) × 100 = 45.88%

∴ The correct answer is 45.88%.

Alternate Method

Given: Successive discounts are 12%, 18% and 25%.

Formula Used: Single equivalent discount for two successive discounts a% and b% = [a + b − (a × b)/100]%

Let the Marked Price (MP) be ₹100.

⇒ Price after 1^st discount of 12% = 100 − (12% of 100) = ₹88

⇒ Price after 2^nd discount of 18% = 88 − (18% of 88)

⇒ 88 − 15.84 = ₹72.16

⇒ Price after 3^rd discount of 25% = 72.16 − (25% of 72.16)

⇒ 72.16 − 18.04 = ₹54.12

⇒ Total Discount = MP − Final SP = 100 − 54.12 = ₹45.88

⇒ Equivalent Discount % = (45.88 / 100) × 100 = 45.88%

∴ The correct answer is 45.88%.

Additional Information

Successive Discount Formula for Two Rates

If two successive discounts are a% and b%, the net discount is (a + b − ab/100)%.

Relationship between MP and SP

Selling Price (SP) = Marked Price (MP) × (100 − Discount%)/100. For successive discounts d₁, d₂, d₃, SP = MP × (1 − d₁/100) × (1 − d₂/100) × (1 − d₃/100).

Effective Discount Rate

The effective discount is always less than the sum of individual discounts. In this case: 12 + 18 + 25 = 55%, but effective is 45.88%.

Shortcut Trick

Assume all 100 animals are hens (2 feet each).

Total feet would be: 100 × 2 = 200 feet.

Excess feet count: 348 − 200 = 148 feet.

Each goat has 2 extra feet compared to a hen (4 − 2 = 2).

Total goats = 148 ÷ 2 = 74 goats.

∴ The correct answer is 74 goats.

Alternate Method

Given:

Total number of heads = 100

Total number of feet = 348

Formula Used:

H + G = Total Heads; 2H + 4G = Total Feet (where H = Hens, G = Goats)

Calculations:

⇒ Average feet per head = 348 ÷ 100 = 3.48

⇒ By Alligation method, Ratio of Hens to Goats = (4 − 3.48) : (3.48 − 2)

⇒ Ratio (Hens : Goats) = 0.52 : 1.48 = 13 : 37

⇒ Total units = 13 + 37 = 50 units

⇒ 50 units = 100 animals

⇒ 1 unit = 2 animals

⇒ Total Goats = 37 × 2 = 74

∴ The correct answer is 74 goats.

Additional Information

Direct Formula for 4-legged animals

Number of 4-legged animals = (Total feet − 2 × Total heads) ÷ 2. This is derived from linear substitution.

Weighted Average Principle

The mean value (3.48) always lies between the two extremes (2 and 4). The closer the mean is to one extreme, the higher the proportion of that category.

System of Linear Equations

Problems involving two variables and two conditions (like heads and feet) can always be solved using the substitution or elimination method.

Shortcut Trick

Number of sides in regular polygon = 360° ÷ Exterior angle

Number of sides = 360° ÷ 18° = 20

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

Number of diagonals = [20 × (20 − 3)] ÷ 2 = 10 × 17 = 170

∴ The correct answer is 170.

Alternate Method

Given:

Exterior angle of a regular polygon = 18°

Formula Used:

1. Number of sides (n) = 360° ÷ Exterior angle
2. Number of diagonals = n × (n − 3) ÷ 2

Calculations:

⇒ Number of sides (n) = 360° ÷ 18°

⇒ n = 20

⇒ Number of diagonals = 20 × (20 − 3) ÷ 2

⇒ Number of diagonals = 20 × 17 ÷ 2

⇒ Number of diagonals = 10 × 17 = 170

∴ The correct answer is 170.

Additional Information

Sum of Exterior Angles

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

Sum of Interior Angles

The sum of all interior angles of a polygon with n sides is given by (n − 2) × 180°.

Relationship between Angles

At each vertex of a polygon, the sum of the interior angle and the exterior angle is 180°.

Number of Diagonals Formula

The number of diagonals is calculated as ⁿC₂ − n, which simplifies to n(n − 3) ÷ 2.

Shortcut Trick

Check divisibility by sum of digits: 2 + 9 + 1 + 8 + 2 + 4 + 5 = 31 (Not divisible by 3 or 9).

Check divisibility by 4: Last two digits are 45 (Not divisible by 4, so not by 12).

Check divisibility by 11: Alternating sum = 2 − 9 + 1 − 8 + 2 − 4 + 5 = −11.

Since 11 is divisible by 11, the entire number is divisible by 11.

∴ The correct answer is 11.

Alternate Method

Given: Number = 2918245

Method: Apply divisibility rules for each option.

Checking each option:

⇒ Divisibility by 3: Sum of digits = 2+9+1+8+2+4+5 = 31. Since 31 is not divisible by 3, the number is not divisible by 3.

⇒ Divisibility by 9: Since the sum 31 is not divisible by 9, the number is not divisible by 9.

⇒ Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4. Since it is not divisible by 3, it cannot be divisible by 12.

⇒ Divisibility by 11: Sum of digits at odd places (from right) = 5 + 2 + 1 + 2 = 10. Sum of digits at even places (from right) = 4 + 8 + 9 = 21.

⇒ Difference = |21 − 10| = 11. Since 11 is divisible by 11, the number 2918245 is divisible by 11.

∴ The correct answer is 11.

Additional Information

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 4 and 8

A number is divisible by 4 if its last two digits are divisible by 4, and by 8 if its last three digits are divisible by 8.

Divisibility by 11 Rule

The difference between the sum of digits at odd places and the sum of digits at even places must be 0 or a multiple of 11.

Divisibility by 6

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

Shortcut Trick

New Weight = Replaced Weight + (Total Number × Change in Average)

Here, Replaced weight = 100 kg, Number of men = 3, Average increase = 4 kg

New man weight = 100 + (3 × 4)

New man weight = 100 + 12 = 112 kg

∴ The correct answer is 112 kg.

Alternate Method

Given:

Number of men = 3

Weight of the man replaced = 100 kg

Increase in average weight = 4 kg

Formula Used:

Weight of new person = Weight of replaced person + (Total persons × Increase in average)

⇒ Total weight of 3 men = 3 × Average weight (A)

⇒ Let the weights of the men be M₁, M₂, and 100 kg.

⇒ (M₁ + M₂ + 100) ÷ 3 = A

⇒ (M₁ + M₂ + New Man) ÷ 3 = A + 4

⇒ M₁ + M₂ + New Man = 3A + 12

⇒ Since M₁ + M₂ + 100 = 3A, then M₁ + M₂ = 3A − 100

⇒ (3A − 100) + New Man = 3A + 12

⇒ New Man = 3A + 12 − 3A + 100

⇒ New Man = 112 kg

∴ The correct answer is 112 kg.

Additional Information

General Replacement Rule

When one person is replaced, New Value = Old Value + (Total Count × Increase) or New Value = Old Value − (Total Count × Decrease).

Average of Group

Average = Sum of all observations ÷ Number of observations. If the average increases, the incoming value must be greater than the outgoing value.

Effect of Adding a Member

When a person is added (not replaced), the total count increases. New Total Weight = (Old Average × Old Count) + New Member Weight.

Shortcut Trick

Arrange the data in ascending order: 25, 27, 29, 30, 32, 33, 35.

The number of terms (n) = 7, which is an odd number.

For an odd number of terms, the median is the middle-most value.

The middle term is the 4^th term in the sorted list.

The 4^th term is 30.

∴ The correct answer is 30.

Alternate Method

Given: Data set = {32, 25, 33, 27, 35, 29, 30}

Formula Used: Median = [(n + 1) / 2]^th term, where n is the number of observations (when n is odd).

⇒ Count the number of observations in the data set:

⇒ n = 7 (which is odd)

⇒ Arrange the data in ascending order: 25, 27, 29, 30, 32, 33, 35

⇒ Median = [(7 + 1) / 2]^th observation

⇒ Median = 8 / 2 = 4^th observation

⇒ The 4^th observation in the sorted list is 30.

∴ The correct answer is 30.

Additional Information

Median for Even n

When the number of observations (n) is even, the median is calculated as the average of the (n/2)^th and (n/2 + 1)^th terms.

Importance of Sorting

The data must always be arranged in either ascending or descending order before identifying the middle term; otherwise, the result will be incorrect.

Range of Data

The range is the difference between the maximum and minimum values: Range = Max − Min. For this data, Range = 35 − 25 = 10.

Shortcut Trick

Number of bottles (n) = Volume of bowl ÷ Volume of one bottle

n = [(2/3) × π × R³] ÷ [π × r² × h]

n = [2 × 18 × 18 × 18] ÷ [3 × 3 × 3 × 6]

n = (2 × 6 × 6) = 72 bottles

∴ The correct answer is 72 bottles.

Alternate Method

Given:

Internal radius of hemispherical bowl (R) = 18 cm

Radius of cylindrical bottle (r) = 3 cm

Height of cylindrical bottle (h) = 6 cm

Formula Used:

Volume of hemisphere = (2/3) × π × R³

Volume of cylinder = π × r² × h

Calculations:

⇒ Volume of liquid in bowl = (2/3) × π × (18)³ cm³

⇒ Volume of one bottle = π × (3)² × 6 cm³

⇒ Let n be the number of bottles required.

⇒ n × (Volume of one bottle) = Volume of bowl

⇒ n × (π × 3 × 3 × 6) = (2/3) × π × 18 × 18 × 18

⇒ n = (2 × π × 18 × 18 × 18) ÷ (3 × π × 9 × 6)

⇒ n = (2 × 5832) ÷ (162)

⇒ n = 11664 ÷ 162

⇒ n = 72

∴ The correct answer is 72 bottles.

Additional Information

Total Surface Area (TSA) of Hemisphere

The TSA of a solid hemisphere includes the curved area and the flat circular base, given by 3πr².

Volume of Sphere

The volume of a complete sphere is double that of a hemisphere, calculated as (4/3)πr³.

Surface Area of Cylinder

Curved Surface Area = 2πrh and Total Surface Area = 2πr(h + r).

Given:

[76 – {90 ÷ 5 × (24 – 36 ÷ 3) ÷ 3}]

Formula used:

BODMAS Rule

Calculation:

$[76 - {90 \div 5 \times (24 - 12) \div 3}]$

⇒ $[76 - {18 \times 12 \div 3}]$

⇒ $[76 - 72]$

⇒ 4

∴ The correct answer is option (3).

Shortcut Trick

Each trader adds 30% profit, so the Multiplying Factor (MF) for each sale = 1.3

Successive MF for three traders = 1.3 × 1.3 × 1.3 = 2.197

Final Selling Price = Initial Cost Price × 2.197

300 = Initial CP × 2.197

Initial CP = 300 ÷ 2.197 ≈ 136.55

∴ The correct answer is Rs.136.55 (approx).

Alternate Method

Given:

Profit of each trader = 30%

Number of traders = 3

Final Selling Price = Rs.300

Formula Used:

Selling Price (SP) = Cost Price (CP) × (1 + Profit%/100)

Let the amount the first trader paid (initial Cost Price) be x.

⇒ CP of second trader (SP of first trader) = x × (1 + 30/100) = 1.3x

⇒ CP of third trader (SP of second trader) = 1.3x × 1.3 = 1.69x

⇒ Final Selling Price (SP of third trader) = 1.69x × 1.3 = 2.197x

⇒ According to the question, 2.197x = 300

⇒ x = 300 ÷ 2.197

⇒ x = 136.5498...

⇒ x ≈ 136.55

∴ The correct answer is Rs.136.55 (approx).

Additional Information

Successive Percentage Change

When a value changes successively by a%, b%, and c%, the final value is Initial × (1 + a/100) × (1 + b/100) × (1 + c/100).

Net Profit Percentage

The equivalent single profit percentage for three successive profits of p% is given by [(1 + p/100)³ − 1] × 100.

Effective Rate for Two Changes

For two successive changes of a% and b%, the net change is (a + b + ab/100)%.

Shortcut Trick

Slant height (l) = √(Height² + (Side/2)²) = √(3² + 4²) = 5 cm.

Base Area = Side² = 8² = 64 cm².

Lateral Area = 2 × Side × Slant height = 2 × 8 × 5 = 80 cm².

Total Surface Area (TSA) = 64 + 80 = 144 cm².

∴ The correct answer is 144 cm².

Alternate Method

Given: Base side (a) = 8 cm, Height of pyramid (h) = 3 cm.

Formula: TSA = Base Area + 1/2 × Perimeter × Slant Height (l)

⇒ Area of square base = a² = 8 × 8 = 64 cm²

⇒ Slant height (l) = √(h² + (a/2)²)

⇒ l = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

⇒ Perimeter of base = 4 × a = 4 × 8 = 32 cm

⇒ Lateral Surface Area (LSA) = 1/2 × 32 × 5 = 16 × 5 = 80 cm²

⇒ Total Surface Area (TSA) = Base Area + LSA

⇒ TSA = 64 + 80 = 144 cm²

∴ The correct answer is 144 cm².

Additional Information

Volume of a Pyramid

The volume is given by V = 1/3 × Base Area × Vertical Height.

Slant Height vs Slant Edge

Slant height (l) is the height of the triangular face, while Slant edge (e) = √(h² + (d/2)²) where d is the diagonal of the base.

Regular Pyramid Properties

In a regular pyramid, all lateral faces are congruent isosceles triangles and the apex is directly above the center of the base.

Shortcut Trick

Express all dimensions in terms of breadth 'b':

Given: Height (h) = b/2 and Length (l) = 2b.

Formula for longest diagonal = √(l² + b² + h²).

Diagonal = √((2b)² + b² + (b/2)²) = √(4b² + b² + b²/4) = √(21b²/4).

∴ The correct answer is (√21 b) / 2.

Alternate Method

Given:

Breadth = b

Breadth is twice the height ⇒ b = 2h ⇒ h = b/2

Breadth is half of the length ⇒ b = l/2 ⇒ l = 2b

Formula Used:

Longest diagonal of a room (cuboid) = √(l² + b² + h²)

Calculations:

⇒ Diagonal = √((2b)² + b² + (b/2)²)

⇒ Diagonal = √(4b² + b² + b²/4)

⇒ Diagonal = √(5b² + b²/4)

⇒ Diagonal = √((20b² + b²)/4)

⇒ Diagonal = √(21b²/4)

⇒ Diagonal = (√21 b) / 2

∴ The correct answer is (√21 b) / 2.

Additional Information

Total Surface Area of a Cuboid

The sum of the areas of all 6 faces is given by the formula 2(lb + bh + hl).

Volume of a Cuboid

The total space occupied by the cuboid is calculated as Length × Breadth × Height.

Diagonal of a Cube

If all sides are equal (l = b = h = a), the longest diagonal is simplified to √3 a.

Area of Four Walls

The lateral surface area of a room is calculated as 2(l + b) × h.

Shortcut Trick

Identify the telescoping pattern of the identity (a − b)(a + b) = a² − b².

Each pair of terms doubles the power of x: (1 − xⁿ)(1 + xⁿ) = 1 − x²ⁿ.

The sequence starts with power 1 and ends with power 8.

The final simplified power will be the highest power in the product multiplied by 2.

Highest power is 8, so final result is 1 − x^{8 × 2} = 1 − x¹⁶.

∴ The correct answer is 1 − x¹⁶.

Alternate Method

Given Expression: (1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)(1 − x)

Formula: (a − b)(a + b) = a² − b²

⇒ Rearranging the terms to group conjugates:

⇒ [(1 − x)(1 + x)] × (1 + x²) × (1 + x⁴) × (1 + x⁸)

⇒ (1 − x²) × (1 + x²) × (1 + x⁴) × (1 + x⁸)

⇒ [(1 − x²)(1 + x²)] × (1 + x⁴) × (1 + x⁸)

⇒ (1 − x⁴) × (1 + x⁴) × (1 + x⁸)

⇒ [(1 − x⁴)(1 + x⁴)] × (1 + x⁸)

⇒ (1 − x⁸) × (1 + x⁸)

⇒ 1 − (x⁸)²

⇒ 1 − x¹⁶

∴ The correct answer is 1 − x¹⁶.

Additional Information

Difference of Squares Identity

The algebraic identity (a + b)(a − b) = a² − b² is fundamental for simplifying products of binomial conjugates.

Exponents Rule

When raising a power to another power, the exponents are multiplied: (x^a)^b = x^{a × b}.

General Telescoping Form

The general product (1 − x)(1 + x)(1 + x²)(1 + x⁴)...(1 + x^2ⁿ) simplifies to 1 − x^2ⁿ⁺¹.

Shortcut Trick

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Ratio of sides = √(Ratio of Areas)

Ratio of sides = √196 : √625

Ratio of sides = 14 : 25

∴ The correct answer is 14 : 25.

Alternate Method

Given:

Ratio of areas of two similar triangles = 196 : 625

Formula used:

If ΔABC ∼ ΔPQR, then Area(ΔABC) ÷ Area(ΔPQR) = (AB ÷ PQ)² = (BC ÷ QR)² = (AC ÷ PR)²

Let the corresponding sides of the two triangles be s₁ and s₂.

Let the areas of the two triangles be A₁ and A₂.

⇒ A₁ ÷ A₂ = (s₁ ÷ s₂)²

⇒ 196 ÷ 625 = (s₁ ÷ s₂)²

⇒ s₁ ÷ s₂ = √(196 ÷ 625)

⇒ s₁ ÷ s₂ = 14 ÷ 25

⇒ s₁ : s₂ = 14 : 25

∴ The correct answer is 14 : 25.

Additional Information

Properties of Similar Triangles

The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides: P₁/P₂ = s₁/s₂.

Ratio of Altitudes and Medians

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

Area Theorem

The theorem states that for two similar triangles, the ratio of their areas is the square of the ratio of any two corresponding linear dimensions.

Shortcut Trick

Probability A speaks truth = 5/7, Lie = 2/7

Probability B speaks truth = 8/9, Lie = 1/9

Contradiction = (A_Truth × B_Lie) + (A_Lie × B_Truth)

⇒ (5/7 × 1/9) + (2/7 × 8/9) = (5 + 16) / 63 = 21/63 = 1/3

∴ The correct answer is 1/3.

Alternate Method

Given:

Probability of A speaking truth, P(A) = 5/7

Probability of B speaking truth, P(B) = 8/9

Formula:

Probability of contradiction = P(A ∩ B′) + P(A′ ∩ B)

Probability that A lies, P(A′) = 1 − P(A) = 1 − 5/7 = 2/7

Probability that B lies, P(B′) = 1 − P(B) = 1 − 8/9 = 1/9

They contradict each other if one speaks truth and the other speaks lie.

⇒ Required Probability = [P(A) × P(B′)] + [P(A′) × P(B)]

⇒ Required Probability = (5/7 × 1/9) + (2/7 × 8/9)

⇒ Required Probability = 5/63 + 16/63

⇒ Required Probability = 21/63

⇒ Required Probability = 1/3

∴ The correct answer is 1/3.

Additional Information

Independent Events

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

Complementary Events

The probability of an event not occurring is P(A′) = 1 − P(A).

Probability of Agreement

The probability that they agree with each other is P(A)P(B) + P(A′)P(B′), which is also equal to 1 − P(Contradict).

Shortcut Trick

Total work (LCM of 32 and 28) = 224 units.

Efficiency: A = 7 units/h, B = 8 units/h.

Work done in 1 cycle of 2 hours (A + B) = 7 + 8 = 15 units.

In 14 cycles (28 hours), work completed = 14 × 15 = 210 units.

Remaining work = 224 − 210 = 14 units. In 29^th hour, A fills 7 units. Remaining 7 units filled by B in 7/8 hours.

Total time = 29.875 hours ≈ 30 hours.

∴ The correct answer is 30 h.

Alternate Method

Given: Time taken by A = 32 h, Time taken by B = 28 h.

Formula: Efficiency = Total Work ÷ Time

⇒ Work done in the first 2 hours (1 hour A + 1 hour B) = 7 + 8 = 15 units.

⇒ Number of such 2-hour cycles required to reach near 224 units = 224 ÷ 15 = 14.93 cycles.

⇒ Work done in 14 cycles = 14 × 15 = 210 units.

⇒ Time taken for 14 cycles = 14 × 2 = 28 hours.

⇒ Remaining work = 224 − 210 = 14 units.

⇒ In the 29^th hour, Pipe A works and fills 7 units.

⇒ Remaining work = 14 − 7 = 7 units.

⇒ In the next hour, Pipe B works. Time taken by B to fill remaining units = 7 ÷ 8 = 0.875 hours.

⇒ Total time taken = 28 + 1 + 0.875 = 29.875 hours.

⇒ Nearest integer value = 30 hours.

∴ The correct answer is 30 h.

Additional Information

Negative Work (Outlet Pipes)

If an outlet pipe is present, its efficiency is treated as negative. Net work per hour = Σ(Inlet Efficiency) − Σ(Outlet Efficiency).

Pipes and Cisterns General Formula

If Pipe A fills in x hours and Pipe B fills in y hours, both together fill the tank in (x × y) ÷ (x + y) hours.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken. If the ratio of time taken by A and B is a : b, then the ratio of their efficiencies is b : a.

Shortcut Trick

The winning candidate got 64% of the total votes.

The losing candidate got (100% − 64%) = 36% of the total votes.

Difference in vote percentage = 64% − 36% = 28%

Given, 28% of total votes = 434

Total votes = 434 × (100 ÷ 28) = 1550

∴ The correct answer is 1550.

Alternate Method

Given:

Votes polled by the winning candidate = 64%

Winning margin = 434 votes

Total votes percentage = 100%

Total Votes = 100%

Winner (64%)

Loser (36%)

Difference = 64% − 36% = 28%

28% → 434 votes

Calculations:

⇒ Votes polled by the losing candidate = 100% − 64% = 36%

⇒ Difference in percentage = 64% − 36% = 28%

Since the winning margin is 434 votes:

⇒ 28% of Total Votes = 434

⇒ 1% of Total Votes = 434 ÷ 28 = 15.5

⇒ Total Votes (100%) = 15.5 × 100 = 1550

∴ The correct answer is 1550.

Additional Information

Valid and Invalid Votes

When certain votes are declared invalid, the percentage calculations are usually based on valid votes. Total Valid Votes = Total Votes Polled − Invalid Votes.

Percentage Change and Base

In election problems, percentages are calculated with respect to the total voter base unless stated otherwise. The winning margin is always the difference in percentages between the winner and the loser.

Successive Percentage Shift

When counting the active voting population: Final Valid Votes = Total Registered Voters × (1 − Non-voters%) × (1 − Invalid%).

Shortcut Trick

Height difference between tower and pole (h) = 57.75 − 17.75 = 40 m.

In a 30°−60°−90° triangle, the ratio of sides is 1 : √3 : 2.

The side opposite to 60° is √3 units, and the base (distance) is 1 unit.

√3 units = 40 m ⇒ 1 unit = 40/√3 m.

Distance = (40/√3) × (√3/√3) = 40√3/3 m.

∴ The correct answer is 40√3/3 m.

Alternate Method

Given:

Height of pole (AB) = 17.75 m

Height of tower (CD) = 57.75 m

Angle of elevation from top of pole to top of tower (θ) = 60°

Formula Used:

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

Calculations:

Let the distance between the foot of the pole (B) and the foot of the tower (C) be x m.

The height of the tower above the pole level (DE) = CD − AB

⇒ DE = 57.75 − 17.75 = 40 m

In ΔADE, where AE = BC = x:

⇒ tan 60° = DE / AE

⇒ √3 = 40 / x

⇒ x = 40 / √3

To rationalize the denominator, multiply and divide by √3:

⇒ x = (40 × √3) / (√3 × √3)

⇒ x = 40√3 / 3 m

∴ The correct answer is 40√3/3 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking up at an object. tan θ = Height / Distance.

Trigonometric Ratios for Standard Angles

tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

30°-60°-90° Triangle Rule

Sides are in the ratio 1 : √3 : 2. Side opposite 30° is x, side opposite 60° is x√3, and hypotenuse is 2x.

Now let's try each option one by one

1. • × ÷ −    Replacing * by this symbol

​33 + 4 × 15 ÷ 3 - 61 = -7 ≠ 188

This option does not give the correct result.

1. × − ÷ +   Replacing * by this symbol

33 × 4 − 15 ÷ 3 + 61 = 188

This option gives the correct result.

First, we perform the division operation of 15 ÷ 3, which gives us 5.

33 × 4 − 5 + 61 = 188

Then we perform the multiplication operation of 33 × 4, which gives us 132.

132 - 5 + 61 = 188

Next, we perform the subtraction operation of 132 + 61 , which gives us 193.

193 - 5 = 188

Finally, we perform the addition operation of 193 - 5, which gives us 188.

Therefore, the correct combination of mathematical signs is × − ÷ +.

1. • − ÷ ×   Replacing * by this symbol

​ 33 + 4 − 15 ÷ 3 × 61 = -  268 ≠ 188

This option does not give the correct result.

1. ÷ × − +     Replacing * by this symbol

33 ÷ 4 × 15 − 3 + 61 = 181.75 ≠ 188

This option does not give the correct result.

Hence, the correct answer is option 2, i.e., × − ÷ +.

The option figure in which image is embedded is:

Hence, the correct answer is "Option 1".

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

 

Hence, the correct answer is "Option 3".

Given: E = 1, N = 2, T = 3, S = 4, T = 5, D = 6, U = 7

Option 1: 4,3,7,6,1,2,5 - Forms the word "STUDENT".

Option 2: 2,1,4,3,6,7,5 - Forms the word "NESTDUT". This is not a meaningful word in English.

Option 3: 4,3,7,1,6,5,2 - Forms the word "STUEDTN". This is a misspelled version of the correct word "STUDENT".

Option 4: 2,1,4,6,3,5,7 - Forms the word "NESDTTU". This is not a meaningful word in English, and the use of the number 7 instead of the letter U makes it clear that this is not a valid answer.

Hence, the correct answer is "Option 1".

The given statement is: A = B ≥ C ≤ D > G < E < F

Conclusion I:  B > F

We can see that there is no direct comparison between B and F in the given statement. Therefore, we cannot conclude that B is greater than F. Hence, conclusion I is false.

Conclusion II: C < A

As per the given statement, A = B and B ≥ C. Combining these two equations, we can conclude that A ≥ C. Therefore, conclusion II is false.

Hence, neither conclusion I nor II is true. The answer is option (3) Neither conclusion I nor II is true.

The given statement highlights the reason for the large increase in migration of villagers to urban areas. Based on this, we can evaluate the two courses of action:

Course of action I: The villagers should be provided with an alternate source of income in their villages which will make them stay there only. This course of action seems logical and appropriate as providing an alternate source of income in their villages would help them stay in their villages instead of migrating to urban areas. This can also help in reducing the pressure on urban areas and improving the overall living conditions of the villagers in their own villages.

Course of action II: To ensure their survival, the migrated villagers should be given access to all housing options in urban areas. This course of action is not a practical solution as providing access to all housing options in urban areas may not be possible for the authorities. It may also not solve the root cause of the problem, which is the repeated crop failure leading to financial problems. Moreover, providing housing options to migrants may lead to the creation of slums and other social problems in urban areas.

Therefore, only course of action I logically follows for pursuing.

Hence, the correct answer is "Option 4".

Question: A, B, C and D are sitting around a circular table facing the centre.

Statements:

(I) C sits to the immediate left of A; only one person sits between C and D.

Hence, D is the immediate right of A. Statement I is sufficient to give the correct answer.

(II) Only one person sits between A and B; C sits to the immediate right of B.

Hence, D is the immediate right of A. Statement II is sufficient to give the correct answer.

Hence, The correct answer is "Data in either statement I alone or statement II alone is sufficient to answer the question ".

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

1) For '

' element : Rotating alternately 45º and 225º (45º + 180º) in the clockwise direction.

​​

Final series is :

Hence, ‘option - (1)’ is the correct answer.

Given : 

1. Six friends - J, K, L, M, N and O
2. They are sitting around a circular table facing the centre.

Explanation : 

1) M is sitting to the immediate left of J.

2) J is sitting opposite to L, who is sitting to the immediate right of K.

3) N is not an immediate neighbour of M.

After positioning M, only one position is left which will be occupied by the only person left i,.e O.

Thus, according to the final arrangement N is sitting second to the left of L.

Hence, "N" is the correct answer.

The three words that belong to the same group are Clarinet, Flute, and Piccolo because they are all musical instruments that are played by blowing into them.

On the other hand, Piano is a musical instrument that is played by pressing keys on a keyboard, which is different from the other three instruments.

Therefore, option 3 (Piano) does not belong to the same group as the other three words.

Hence, the correct answer is "Option 3".

Logic : Alternately (1) and (2) are added to the position value according to the English alphabetic series.

​​So,

• COMPAQ is written as DQNRBS

​ 

​And,

• SONY is written as TQOA

​ 

​Similarly,

• MOTOROLA is written as :

​ 

Hence, "NQUQSQMC" is the correct answer.

Logic : In each figure according to the direction of arrows at one end of the line, at the other end half arrow is drawn to right hand side of the line.

So,

• Option - (1) :

 

⇒ The half arrow is drawn to left hand side of the arrows at the other end of the line.

• Option - (2) :

 

⇒ The half arrow is drawn to right hand side of the arrows at the other end of the line.

• Option - (3) :

⇒ The half arrow is drawn to right hand side of the arrows at the other end of the line.

• Option - (4) :

 

⇒ The half arrow is drawn to right hand side of the arrows at the other end of the line.

Here, Figure - (1) is the odd one out.

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

Logic : Sum of the position values of immediate opposite letter according to the English alphabetic series are added to number of the digits within the word.

​​So,

• ‘CROSS’ is written as ‘66’

​ 

​And,

• ‘CHART’ is written as ‘90’

​ 

​Similarly,

• ‘SYMBOL’ is written as :

​ 

Hence, "82" is the correct answer.

Logic : In each figure no columns and rows is vacant i.e atleast one box of all the rows and columns is occupied by '

' element or '

' element. 

• Option - (1) :

 

⇒ Atleast one box of all the rows and columns is occupied by '

' element.

• Option - (2) :

 

⇒ Atleast one box of all the rows and columns is occupied by '

' element.

• Option - (3) :

⇒  Atleast one box of all the rows and columns is occupied by '

' element.

• Option - (4) :

 

⇒ Atleast one box of all the rows is occupied by '

' element but one column as marked in the figure is vacant.

Here, Figure - (4) is the odd one out.

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

The logic followed here is :- 

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

​So,

• = AJS, % = MAM  ⇒ AJS : CLQ :: NDK : MAM

​ 

⇒Here, no similar pattern can be seen.

• = AOO, % = PAM  ⇒ AOO : CLQ :: NDK : PAM

​       

⇒Here, similar pattern can be seen.

• = EIT, % = QBN ⇒ EIT : CLQ :: NDK : QBN

​           

⇒Here, no similar pattern can be seen.

• = DIS, % = PBN ⇒ DIS : CLQ :: NDK : PBN​

​           

⇒Here, no similar pattern can be seen.

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

Given : Certain number of people are positioned on a row and all of them are facing towards north direction. 

Explanation :

1) Prince is 5^th from the right end.

2. Amit is 15^th from the right end. 

3) Amit is exactly between Prince and Aditya.

4) Aditya is sixth from the left end of the line. 

Total number of people = 6 + 9 + 1 (Amit) + 9 + 5 = 30

Hence, "30" is the correct answer.

Given:

+ means −,

− means ×,

× means ÷, and

÷ means +

As per the given conditions, we need to replace the mathematical symbols with their corresponding operations.

45 × 9 ÷ 12 – 5 + 3

Using BODMAS,

= 45 ÷ 9 + 12 × 5 – 3

= 5 +  12 × 5 – 3

= 5 + 60 – 3

= 65 - 3

= 62

Therefore, the value of the expression is 62.

Hence, the correct answer is "Option 4".

Let's examine each number pair:

1. 10 ∶ 33  The second number is obtained by multiplying the first number by 3 and adding 3. (10 × 3) + 3 = 33.
2. 2 ∶ 9  The second number is obtained by multiplying the first number by 3 and adding 3. 2 × 3 + 3 = 9.
3. 8 ∶ 29 The second number is obtained by multiplying the first number by 3 and adding 5. (8 × 3) + 5 = 29.
4. 6 ∶ 21 The second number is obtained by multiplying the first number by 3 and adding 3. (6 ×  3) + 3 = 21.

We can see that the same mathematical operation has been performed on number pairs 1, 2, and 4 (multiplying the first number by 3 and adding 3 ).

However, a different operation has been performed on number-pair 3  ( multiplying the first number by 3 and adding 5. ). Therefore, the odd number pair is 8 ∶ 29.

Hence, the correct answer is option 3, which is 8 ∶ 29.

The logic followed here is :

Hence, the correct answer is "Option 3".

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. All toys being poems is a possibility → follow (The given conclusion is not definite but there is possibility applied in it. So, false conclusion with possibility will be considered as true conclusion, Hence, it is true.)

II. Some horses are ships →  follow (As all horses are toys and all toys are ships. So, all horses comes in toys which whole comes in ships therefore it is definitely true)

∴ Here, Both conclusions I and II follow.​

Hence, the correct answer is "Both conclusions I and II follow"

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 daughter of H.

Hence, "Daughter" is the correct answer.

We unfold the paper step by step:

Hence, the correct answer is "Option 1".

Given : _ V J R W B _ _ R W B V L _ _ B V _ _ W 

By checking options :

1. B V K R W M R → B V J R W B V K R W B V L R W B V M R W  → (It follows the pattern of :

2. A V K R W M R → A V J R W B V K R W B V L R W B V M R W → (It doesn't Follows the pattern)

3. A W K R X N R → A V J R W B W K R W B V L R X B V N R W → (It doesn't Follows the pattern)

4. B W K R W M R → B V J R W B W K R W B V L R W B V M R W → (It doesn't Follows the pattern)

⇒Option - (1) gives the pattern of "B V J R W - B V K R W - B V L R W - B V M R W"

Hence, "B V K R W M R" is the correct answer.

Inference I states that excessive use of pesticides in agricultural production is not good for people. This can be inferred from the statement, which clearly states that extensive use of pesticides can contaminate soil, water, crops and can pose a threat to human beings.

Inference II states that excessive use of pesticides in agricultural production can have a bad effect on water. This can also be inferred from the statement, which clearly states that extensive use of pesticides can contaminate soil and water, and can enter the food chain, posing a threat to human beings.

Therefore, both inferences I and II follow from the given statement.

Hence the correct answer is "Option 4".

I.  Country X reduced their crude oil import by 6% in the last 5 years.:

The reduction in crude oil import may have been due to various reasons such as increased domestic production, shift towards renewable energy sources, or decreased demand for oil.

Similarly,

II. The automobile industry in Country X has witnessed a sharp decline of 13% in its growth margin this year.:

The decline in the growth margin of the automobile industry may have been due to factors such as economic slowdown, increased competition, or changes in consumer preferences. There is no clear cause-effect relationship between the two statements.

Therefore, Option 1 is correct because both statements are independent effects of different causes.

Hence, the correct answer is "Option 1".

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 V is the father-in-law of H or Wife’s i.e  Y's father.

Hence, "Wife’s father" is the correct answer.

The logic followed here is : Square of 1^st number - (2 × 2^nd number) = 3^rd number

Given : 

• (7, 3, 43)

$(7)^2$ - (2 × 3) = 43

49 - 6 = 43

43 = 43 (LHS = RHS)

• (5, 2, 21)

$(5)^2$ - (2 × 2) = 21

25 - 4 = 21

21 = 21 (LHS = RHS)

So,

• Option - (1) : (11,5,111)

$(11)^2$ - (2 × 5) = 111

121 - 10 = 111

111 = 111 (LHS = RHS)

• Option - (2) : (15,6,34)

$(15)^2$ - (2 × 6) = 34

225 - 12 = 34

213 ≠ 34 (LHS ≠ RHS)

• Option - (3) : (13,23,131)

$(13)^2$ - (2 × 23) = 131

169 - 46 = 131

123 ≠ 131 (LHS ≠ RHS)

• Option - (4) : (12,3,43)

$(12)^2$ - (2 × 3) = 43

144 - 6 = 43

138 ≠ 43 (LHS ≠ RHS)

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

The logic followed here is 

Hence, the correct answer is "Option 4".

Given : 

1. Six people - A, B, C, D, E and F
2. All of them are sitting in a straight line facing north.

Explanation :

• Statement - (I) : A sits second from one of the extreme ends of the line. Only two people sit between A and B. E sits to the immediate right of C.

Here, position of D cannot be determined in both the cases, therefore who is sitting to the immediate left of D cannot be answered. 

• Statement - (II) : C sits third from the left end of the line. Only one person sits between C and F. E sits third to the right of F.

Here, position of D cannot be determined in both the cases, therefore who is sitting to the immediate left of D cannot be answered. 

So,

⇒Combining both the statements (I) and (II), we get :

Here, Either A or B is sitting to the immediate left of D. Thus it cannot be determined.

Therefore, Data in statements I and II together are not sufficient to answer the question.

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

The logic followed here is :-

• Letter at the left down corner is flipped horizontally and shifted to the opposite side i.e right up corner.
• figures at the left up corner and right down corner are rotated 90º in the clockwise direction.
• The center element is doubled.

So, 

• Figures A and B ⇒

​

Similarly,

• Figure C and D ⇒

​

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

Here the correct answer is Option 3.

Key Points

• In the given sentence, the error is in the first part "We should sorted out" of the sentence.
• A modal verb is followed by the first form of the verb.
• Example: We can win the match.
• Thus, 'sorted' should be replaced by 'sort'.
• Therefore, the correct answer is Option 3.

Correct Sentence: We should sort out the problem ourselves rather than depending on the third party.

Here the correct answer is DCAB.

Key Points

• The correct order of the sentences to form a coherent paragraph is DCAB.
• Sentence D introduces the topic and presents a positive assessment of the movie Women Talking, calling it "vibrant cinema."
• Sentence C describes how the director, Sarah Polley, gave the central conversation in the movie a "visual gravitas normally reserved for epics," which adds to the reader's interest in the movie.
• Sentence A gives more detail on how Polley achieved the desired look of the movie, with a muted color grading and the use of equipment typically used for capturing epic landscapes.
• Finally, sentence B adds another detail about Polley's intention for the movie to look like a "faded postcard," which reinforces the visual style of the movie described in the previous sentences.
• Thus, the correct order of the sentences to form a coherent paragraph is DCAB.

Correct Sequence:

D. Women Talking, adapted by the writer-director Sarah Polley from Miriam Toews’s 2018 novel, is vibrant cinema.

C. Polley lends the central conversation a visual gravitas normally reserved for epics.

A. She shot the movie on the widest canvas possible using the same kind of equipment that captured Star Wars landscapes, and applied a muted colour grading to every frame.

B. She said last year that she wanted the ensemble film to look like “a faded postcard”

Here the correct answer is CADB.

Key Points

• The correct order of the sentences to form a meaningful and coherent paragraph is CADB.
• Sentence C is the topic sentence that introduces the most important day in the author's life and the event that changed her life forever, i.e. when her teacher, Anne Mansfield Sullivan, came to her.
• Sentence A provides more context and sets the scene for the eventful day, with the author standing on the porch and sensing that something unusual is about to happen.
• Sentence D provides sensory details of the day, with the afternoon sun and honeysuckle on the porch, which add to the reader's understanding of the setting and the author's emotional state.
• Finally, sentence B adds more background information about the author's emotional state before the arrival of her teacher, which sets up the contrast with the transformative event that follows.
• Therefore, the correct order of the sentences to form a meaningful and coherent paragraph is CADB.

Correct Sequence:

C. The most important day I remember in all my life is the one on which my teacher, Anne Mansfield Sullivan, came to me and I am filled with wonder when I consider the immeasurable contrasts between the two lives which it connects and it was the third of March, 1887, three months before I was seven years old.

A. On the afternoon of that eventful day, I stood on the porch, dumb, expectant and guessed vaguely from my mother's signs and from the hurrying to and fro in the house that something unusual was about to happen, so I went to the door and waited on the steps.

D. The afternoon sun penetrated the mass of honeysuckle that covered the porch, and fell on my upturned face and my fingers lingered almost unconsciously on the familiar leaves and blossoms which had just come forth to greet the sweet southern spring. 

B. I did not know what the future held of marvel or surprise for me and anger and bitterness had preyed upon me continually for weeks and a deep languor had succeeded this passionate struggle.

Here the correct answer is He was appointed monitor by the mistress. 

Key Points

• The given sentence is in active voice. We need to change it into passive voice.
• The subject 'the mistress' becomes the object of the passive voice.
• The object 'him' becomes the subject 'he' of the passive voice.
• Simple past tense structure is as follows:-
• Active voice - Subject + V2 + Object.
• Passive Voice - Object + was/were + V3 + by + Subject.
• 'appointed' changes into 'was appointed'.
• Thus, the correct answer is Option 3.

Correct Passive Voice: He was appointed monitor by the mistress.

The correct sequence of the jumbled sentences is ACBD.

Key Points

• Sentence A talks about how humans tend to become victims of ineffective rational thinking when faced with uncertainty and compelling situations, by looking out for the actions of others.
• Sentence C explains this phenomenon with the psychological concept of 'pluralistic ignorance'.
• Sentence B adds to the explanation given in sentence C by stating that during emergencies, people tend to forget that others around them may also be looking out for social evidence.
• Sentence D concludes the sequence by explaining why people prefer to search for answers placidly and with camouflaged glances at those around them, so as to appear poised and unflustered in front of others.
• Hence, the correct sequence is ACBD.

Here the correct answer is To be very active but with few results.

Key Points

• The most appropriate meaning of the highlighted idiom "running around in circles" in the given context is "to be very active but with few results."
• The idiom "running around in circles" means to engage in a lot of activity or effort without making any progress or achieving anything useful. It is similar to the idiom "going round in circles."
• In the given context, the team members seem to be busy and active, but they are not making any progress or achieving their goals.
• For example, if someone is looking for their keys in a messy room but can't find them despite searching for a long time, they may say, "I feel like I'm running around in circles trying to find my keys."
• Thus, the correct answer is Option 3.

Here the correct answer is They are firing the canons continuously. 

Key Points

• The given sentence is in passive voice. We need to change it into active voice.
• The given sentence is in present continuous tense.
• The structure will be:
• Passive Voice: Object + is/am/are + being + V3 + by + Subject.
• Active Voice: Subject + is/am/are + V1 + ing + Object.
• The subject and object interchange their places.
• 'are being fired' changes to 'are firing'.
• Thus, the correct answer is Option 2.

Correct Active Voice: They are firing the canons continuously.

The correct answer is "pay".

Key Points

• The verb "pay" collocates with the noun "attention", which means to give attention or focus to something.
• Examples:
• I always pay attention in math class because I want to understand the concepts.
• The teacher asked the students to pay attention during the lecture.
• If you don't pay attention in the driving class, you may not pass the test.
• Thus, the correct answer is Option 3.

Correct Sentence: Many students do not  pay attention in botany class. 

Additional Information

• Option 1, "exhibit", means to show or display something, but it doesn't convey the correct meaning in this context.
• Option 2, "drive", means to propel or control the movement of something, but it doesn't fit in this sentence.
• Option 4, "trace", means to find the origins or development of something, but it doesn't fit in this sentence either.
• Therefore, the correct option to fill in the blank is 3) "pay".

Here the correct answer is BCDA.

Key Points

• The correct option to arrange the given parts in the correct order to form a meaningful paragraph is BCDA.
• The paragraph talks about a terrible accident that happened to Raman and the events that followed after it.
• The sentence B, "After the terrible accident," sets the context for the paragraph.
• The sentence C, "A car took Raman to the hospital," describes how Raman was taken to the hospital after the accident.
• The sentence D, "The two men carried him to a bed and put him in it carefully," describes how two men helped Raman to a bed.
• The sentence A, "Then a nurse and a doctor came," completes the paragraph by explaining who came after Raman was taken to the bed.
• Therefore, the correct order of the sentences is BCDA, and option 3) BCDA is the correct answer.

Hinglish

• एक अर्थपूर्ण पैराग्राफ बनाने के लिए दिए गए भागों को सही क्रम में व्यवस्थित करने का सही विकल्प BCDA है।
• यह पैराग्राफ रमन के साथ हुई भयानक दुर्घटना और उसके बाद की घटनाओं के बारे में बात करता है।
• sentence B, "After the terrible accident," अनुच्छेद के लिए संदर्भ निर्धारित करता है।
• sentence C, "A car took Raman to the hospital,"  दुर्घटना के बाद रमन को अस्पताल ले जाने का वर्णन करता है।
• sentence D, "The two men carried him to a bed and put him in it carefully," वर्णन करता है कि कैसे दो पुरुषों ने रमन को बिस्तर पर लाने में मदद की।
• sentence A, "Then a nurse and a doctor came," पैराग्राफ को यह समझाते हुए पूरा करता है कि रमन को बिस्तर पर ले जाने के बाद कौन आया था।

Here the correct answer is Superman was created by Marvel Comics.

Key Points

• The above-given sentence is in the active voice.
• We need to change it in the passive voice.
• The following steps are required to change the given sentence into passive voice:-
• The subject 'Marvel Comics' of the active voice will become the object of the passive voice.
• The object 'Superman' of the active voice will become the subject of the passive voice.
• The tense**(simple past tense)** will change according to the following structure:-
• Active Voice - Subject + (did + V1) or (V2) + Object.
• Passive Voice - Object + was/were + V3 + by + Subject.
• ​Thus, the correct answer is Option 1.

Therefore, the correct passive voice is Superman was created by Marvel Comics.

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.
• Use 'by' before the new object.

Here the correct answer is Smarter and stronger.

Key Points

• The most appropriate option to fill in the blank is 3) "smarter and stronger".
• In the sentence, Tia wants a dog that is both smarter and stronger than Larry's dog.
• Option 1, "smarty and strong," is not correct because "smarty" is not a correct adjective, and it should be "smart".
• Option 2, "smartest and strongest," is not correct because it implies that Tia's dog should be the smartest and the strongest among all the dogs, which is not what the sentence is trying to convey.
• Option 4, "smart and strong," is not correct because it does not convey the sense of comparison between Tia's dog and Larry's dog.
• Therefore, option 3, "smarter and stronger," is the most appropriate choice to fill in the blank as it conveys the meaning that Tia wants a dog that is both smarter and stronger than Larry's dog.

Correct Sentence: Tia wanted a dog that was smarter and stronger than Larry’s dog.

The correct answer is: Option 2 i.e. Overt.

Key Points

• The word "clandestine" means kept secret or done secretively, especially because illicit. (गुप्त, गुप्त रूप से किया गया)
• Example: They held a clandestine meeting to discuss the secret plans.
• The antonym of "clandestine" is "Overt", which means done or shown openly; plainly apparent. (स्पष्ट, खुला)
• Example: The company’s overt support for environmental causes was well-documented.

Therefore, the most appropriate antonym for the underlined word "clandestine" is Option 2 i.e. Overt.

Additional Information

Here are the other options explained along with their Hindi meanings and why they are not suitable antonyms:

• Surreptitious (गुप्त रूप): Kept secret, especially because it would not be approved of.
• Example: They conducted a surreptitious investigation.
• Not suitable because "surreptitious" is a synonym of "clandestine," not an antonym.
• Amorous (कामुक): Showing, feeling, or relating to sexual desire.
• Example: He made amorous advances towards her.
• Not suitable because "amorous" refers to romantic feelings and does not contrast with the secrecy implied by "clandestine."
• Casual (आकस्मिक): Relaxed and unconcerned.
• Example: He was casual about the whole affair.
• Not suitable because "casual" relates to an informal or relaxed manner, which is not directly opposite to "clandestine."

The correct answer is A difficult situation in which the solution to a problem is impossible.

Key Points

• The most appropriate meaning of the idiom "Catch 22 situation" is option 2) "A difficult situation in which the solution to a problem is impossible."
• The term "Catch-22" originated from the title of a novel by Joseph Heller, which portrays a situation where an individual is caught in a contradictory or paradoxical situation, where the solution to one problem creates another problem that makes it impossible to proceed.
• For example, if a person wants to get a job but needs experience to get a job, they are in a Catch-22 situation because they can't get the job without experience and they can't gain experience without the job.
• In the given sentence, the manager is in a Catch-22 situation, which means the manager is facing a difficult problem where the solution leads to another problem that makes it impossible to solve.
• Therefore, the most appropriate meaning of the highlighted idiom in the given sentence is "a difficult situation in which the solution to a problem is impossible."

Here the correct answer is The grandeur of the ancient Roman Empire is visible in many monuments in Italy and Greece.

Key Points

• The correct sentence that contains no spelling errors is option 3) "The grandeur of the ancient Roman Empire is visible in many monuments in Italy and Greece."
• The word "grandeur" means magnificence, splendor, or grandness. It is spelled correctly in option 3.
• Option 1 has a spelling error with "granduor" instead of "grandeur."
• Option 2 also has a spelling error with "monumants" instead of "monuments." "Grandeur" is also incorrectly spelt.
• Option 4 has a spelling error with "granduer" instead of "grandeur."
• Examples:
• The grandeur of the Taj Mahal attracts millions of tourists every year.
• The grandeur of the fireworks display was breathtaking.
• The grandeur of the mountains was awe-inspiring.
• Therefore, the correct answer is option 3) "The grandeur of the ancient Roman Empire is visible in many monuments in Italy and Greece."

Here the correct answer is Prescribed.

Key Points

• The verb "advised" means to recommend or suggest something, especially as a course of action. Here's an example sentence:
• The doctor advised him to rest for a few days to recover from the illness.
• The most appropriate synonym of the word "advised" to fill in the blank is "prescribed," which means to recommend or order a medicine or treatment.
• Example: "The doctor prescribed antibiotics for the patient's infection."
• Thus, the correct answer is Option 4.

Additional Information

• Option 1, "prevented," means to stop something from happening and is not a synonym of "advised."
• Option 2, "proscribed," means to forbid or prohibit something and is not a synonym of "advised."
• Option 3, "presented," means to show or offer something and is not a synonym of "advised."

Here the correct answer is ACDB.

Key Points

 

• A) The sentence "The temperature of the world has increased, resulting in global warming*"* provides the context for the paragraph by stating that global warming is the main subject of the paragraph.
• C) The sentence "The increased amount of carbon emission is considered as the reason behind global warming" provides the cause for global warming, which is due to the increased amount of carbon emission in the atmosphere.
• D) The sentence "The increase in the global temperature is melting the polar ice caps" explains how the increase in global temperature is causing the polar ice caps to melt.
• B) The sentence "The melting of the ice caps will increase the sea level, thereby drowning many islands and coastal regions" explains the consequences of the polar ice caps melting, which will lead to an increase in sea level, causing the drowning of many islands and coastal regions.
• Therefore, the correct order of the sentences to form a meaningful and coherent paragraph is ACDB.

Here the correct answer is Angry.

Key Points

• "Placid" means calm and peaceful. Its antonym is "angry" which means showing strong feelings of displeasure or hostility.
• In the given context, the conversation between the two cricketers did not seem controversial, but it actually created a lot of controversy.
• Examples:
• The placid lake was disturbed by the loud noise of the motorboat.
• I could tell by the way he was clenching his fists and his red face that he was very angry.
• Thus, the correct answer is Option 4.

Additional Information

• Option 1) Sardonic means mocking or cynical. It is not an antonym of "placid".
• Example: His sardonic remarks left everyone speechless.
• Option 2) Inappropriate means not suitable or proper for the situation. It is not an antonym of "placid".
• Example: It would be inappropriate to wear shorts to a formal event.
• Option 3) Informal means relaxed and casual, not formal. It is not an antonym of "placid".
• Example: We had an informal meeting over coffee.

Here the correct answer is BDCA.

Key Points

• The correct answer is option 2, which is BDCA.
• The paragraph talks about the issue of road accidents in India and the reasons behind them. Here is the explanation of why the given order is correct:
• B. "Condition of roads in India is pathetic" - This sentence introduces the topic of the poor condition of roads in India, which is one of the reasons for road accidents.
• D. "Roads are narrow and remain congested most of the time" - This sentence provides further information on the condition of roads in India, emphasizing that they are narrow and congested most of the time, which makes it difficult for vehicles to move around and increases the chances of accidents.
• C. "Increasing number of vehicles leave little space on roads" - This sentence explains another reason for road accidents in India, which is the increasing number of vehicles that leave little space on the already narrow and congested roads.
• A. "Road accidents have become very common in India" - This sentence summarizes the main point of the paragraph, which is that road accidents have become a common occurrence in India due to the poor condition of roads and the increasing number of vehicles.
• Therefore, the correct order of sentences to form a coherent paragraph is BDCA.

Correct Sequence: Condition of roads in India is pathetic. Roads are narrow and remain congested most of the time. Increasing number of vehicles leave little space on roads. Road accidents have become very common in India.

Here the correct answer is Option 3.

Key Points

• The incorrectly spelt word is "Guarentee." The correct spelling is "Guarantee."
• The word "guarantee" means a formal promise or assurance made by one party to another, usually written and legally binding, that certain conditions will be fulfilled or a product or service will meet certain specifications or standards.
• It can also refer to a person or thing providing security or a guarantee of protection. 
• Example: I can guarantee that I will be there on time.
• Thus, the correct answer is Option 3.

Additional Information

• Explanation of other options:
• Greave: a piece of armour used to protect the shin.
• Example: The knight wore greaves to protect his legs in battle.
• Groan: This word is correctly spelled. It means to make a deep sound expressing pain or discomfort.
• Example: She groaned loudly when she got out of bed in the morning.
• Gravely: This word is correctly spelled. It means something done with seriousness or urgency.
• Example: The doctor spoke gravely to the patient about the seriousness of their condition.