Shortcut Trick

Ratio of flow efficiency (E) = d² = 2² : 3² : 4² = 4 : 9 : 16

Total capacity of tank = E of biggest tap × Time = 16 × 81 = 1296 units

Total efficiency of all taps = 4 + 9 + 16 = 29 units

Required time = Total capacity ÷ Total efficiency = 1296 ÷ 29 = 44 ²⁰/₂₉ min

∴ The correct answer is 44 ²⁰/₂₉ min.

Alternate Method

Given:

⇒ Diameters of taps = 2 cm, 3 cm, 4 cm

⇒ Flow rate (Efficiency) ∝ (Diameter)²

⇒ Time taken by biggest tap (4 cm) = 81 min

Formula used:

⇒ Efficiency₁ × Time₁ = Total Work (Capacity)

⇒ Time = Total Work ÷ Combined Efficiency

Calculations:

⇒ Efficiency of taps A, B, C = 2², 3², 4² = 4, 9, 16

⇒ Total Work = Efficiency of C × Time of C = 16 × 81 = 1296 units

⇒ Time for tap A = 1296 ÷ 4 = 324 min

⇒ Time for tap B = 1296 ÷ 9 = 144 min

⇒ Combined Efficiency = 4 + 9 + 16 = 29 units

⇒ Total Time Taken = 1296 ÷ 29 = 44 ²⁰/₂₉ min

∴ The correct answer is 44 ²⁰/₂₉ min.

Additional Information

Rate of Flow and Diameter

The flow rate or efficiency of a pipe is directly proportional to its cross-sectional area, which in turn is proportional to the square of its diameter (Rate ∝ D²).

Net Efficiency Rule

When multiple pipes work together, the net efficiency is the sum of their individual efficiencies. Filling time = Capacity ÷ (E₁ + E₂ + ... + E_n).

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken. If the ratio of efficiencies of two pipes is x : y, then the ratio of time taken by them to fill the same tank is y : x.

Shortcut Trick

Let the third number be 100.

First number = 100 + 10 = 110

Second number = 100 + 25 = 125

Ratio = 110 : 125 = 22 : 25

∴ The correct answer is 22 : 25.

Alternate Method

Given:

First number is 10% more than the third number.

Second number is 25% more than the third number.

Calculations:

Let the third number be x.

⇒ First number = x + (10 ÷ 100) × x = 1.10x

⇒ Second number = x + (25 ÷ 100) × x = 1.25x

⇒ Required ratio = 1.10x : 1.25x

⇒ Multiplying by 100 to remove decimals = 110 : 125

⇒ Dividing both terms by 5 = 22 : 25

∴ The correct answer is 22 : 25.

Additional Information

Percentage Increase

When a value increases by x%, the new value is (100 + x)% of the original value. This allows writing final values directly without a two-step calculation.

Base Value in Percentages

It is a standard time-saving technique to assume the base value as 100 when multiple values are compared relative to a single unknown quantity.

Simplification of Ratios

A ratio a : b remains mathematically unchanged if both its antecedent (a) and consequent (b) are multiplied or divided by the same non-zero number.

Shortcut Trick

Investment Ratio = Profit Ratio ÷ Time Ratio

Investment Ratio = (4 ÷ 3) : (3 ÷ 2) : (14 ÷ 7)

Investment Ratio = 4/3 : 3/2 : 2

Multiply the entire ratio by the LCM of denominators (3 and 2), which is 6.

Investment Ratio = (4/3 × 6) : (3/2 × 6) : (2 × 6) = 8 : 9 : 12

∴ The correct answer is 8 : 9 : 12.

Alternate Method

Given:

Ratio of Time Periods (T_A : T_B : T_C) = 3 : 2 : 7

Ratio of Profits (P_A : P_B : P_C) = 4 : 3 : 14

Formula Used:

Profit = Investment × Time ⇒ Investment = Profit ÷ Time

Calculations:

⇒ Let Investment of A, B and C be I_A, I_B and I_C respectively.

⇒ I_A : I_B : I_C = (P_A / T_A) : (P_B / T_B) : (P_C / T_C)

⇒ I_A : I_B : I_C = (4/3) : (3/2) : (14/7)

⇒ I_A : I_B : I_C = (4/3) : (3/2) : 2

⇒ To remove fractions, multiply by LCM of 3 and 2 = 6

⇒ I_A : I_B : I_C = (4/3 × 6) : (3/2 × 6) : (2 × 6)

⇒ I_A : I_B : I_C = 8 : 9 : 12

∴ The correct answer is 8 : 9 : 12.

Additional Information

Partnership Basic Principle

In any partnership, the profit is distributed in the ratio of the product of the Capital and the Time for which it was invested. Profit = Investment × Time.

Active and Sleeping Partners

An active partner manages the business and may receive a salary or extra profit share, while a sleeping partner only provides capital and shares profit/loss based on the investment ratio.

Time Ratio Calculation

If the profit ratio and investment ratio are known, the time ratio can be calculated as T₁ : T₂ = (P₁/I₁) : (P₂/I₂).

Shortcut Trick

Discount = 44% → Selling Price (SP) = 100% − 44% = 56% of Marked Price (MP).

Given SP = ₹87,500, so 56% = 87,500.

1% = 87,500 ÷ 56 = 1,562.5.

100% (MP) = 1,562.5 × 100 = 1,56,250.

∴ The correct answer is ₹1,56,250.

Alternate Method

Given:

Selling Price (SP) = ₹87,500

Discount Percentage = 44%

Formula Used:

Marked Price (MP) = [SP ÷ (100 − Discount %)] × 100

Calculations:

⇒ Marked Price = [87,500 ÷ (100 − 44)] × 100

⇒ Marked Price = [87,500 ÷ 56] × 100

⇒ Marked Price = 1,562.5 × 100

⇒ Marked Price = ₹1,56,250

∴ The correct answer is ₹1,56,250.

Additional Information

Marked Price (MP)

It is the price tag or list price mentioned on the product before any discount is applied.

Selling Price (SP)

The final price at which a product is sold after subtracting the discount from the marked price: SP = MP − Discount.

Discount Percentage

The reduction offered on the marked price, calculated as: Discount % = (Discount ÷ MP) × 100.

Successive Discounts

When two or more discounts are applied sequentially, the effective discount is: d₁ + d₂ − (d₁ × d₂ ÷ 100).

Shortcut Trick Check for collinearity: Sum of two shorter distances = 9 cm + 11 cm = 20 cm. Given the longest distance AC = 20 cm. Since AB + BC = AC, the points A, B, and C are collinear (lie on a straight line). A circle can only pass through three points if they are non-collinear. Therefore, no circle can pass through these three points. ∴ The correct answer is 0. Alternate Method Given: Distance AB = 9 cm Distance BC = 11 cm Distance AC = 20 cm Property Used: For any three points A, B, and C, if the sum of the lengths of two segments is equal to the third (AB + BC = AC), the points are collinear. A unique circle passes through three non-collinear points, but no circle can pass through three collinear points.

Calculations: ⇒ AB + BC = 9 + 11 ⇒ AB + BC = 20 cm ⇒ Since AC = 20 cm, we have AB + BC = AC. ⇒ This satisfies the condition for collinearity of points A, B, and C. ⇒ A circle is defined by the intersection of perpendicular bisectors of the segments joining the points. ⇒ For collinear points, the perpendicular bisectors of AB and BC are parallel and do not intersect. ⇒ ∴ No circle can pass through all three collinear points. ∴ The correct answer is 0. Additional Information Unique Circle Theorem One and only one circle can pass through three given non-collinear points. Collinearity Condition Three points A, B, and C are collinear if the area of the triangle formed by them is zero, or if AB + BC = AC (for B between A and C). Circles through 1 or 2 points An infinite number of circles can pass through a single point or through two distinct points. Triangle Inequality For three points to form a triangle (non-collinear), the sum of any two sides must be strictly greater than the third side (AB + BC > AC).

Shortcut Trick

Total outcomes (Sample Space) = 6 + 8 + 4 + 3 = 21.

Favorable outcomes (Black or Blue) = 8 + 3 = 11.

Probability = Favorable outcomes ÷ Total outcomes.

Required Probability = 11 / 21.

∴ The correct answer is 11/21.

Alternate Method

Given:

Number of white marbles = 6

Number of black marbles = 8

Number of red marbles = 4

Number of blue marbles = 3

Formula Used:

Probability P(E) = n(E) ÷ n(S)

Where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes.

Calculations:

⇒ Total number of marbles n(S) = 6 + 8 + 4 + 3

⇒ n(S) = 21

⇒ Number of favorable outcomes (Black or Blue) n(E) = 8 + 3

⇒ n(E) = 11

⇒ P(Black or Blue) = n(E) ÷ n(S)

⇒ P(Black or Blue) = 11 / 21

∴ The correct answer is 11/21.

Additional Information

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time. For such events, P(A or B) = P(A) + P(B).

Sample Space

The sample space n(S) represents the set of all possible outcomes of an experiment.

Complementary Events

The probability of an event NOT happening is 1 − P(E). For example, P(not red) = 1 − P(red).

Shortcut Trick

Identify the modal class: Highest frequency is 11, corresponding to class 25−30.

Apply Mode formula: L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h

L = 25, f₁ = 11, f₀ = 9, f₂ = 10, h = 5

Mode = 25 + [(11 − 9) ÷ (22 − 9 − 10)] × 5 = 25 + (2 ÷ 3) × 5 = 25 + 3.33

∴ The correct answer is 28.33.

Alternate Method

Given:

Modal Class = 25−30 (since it has the maximum frequency f = 11)

Lower limit (L) = 25

Frequency of modal class (f₁) = 11

Frequency of preceding class (f₀) = 9

Frequency of succeeding class (f₂) = 10

Class width (h) = 5

Formula Used:

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

Calculations:

⇒ Mode = 25 + [ (11 − 9) ÷ (2 × 11 − 9 − 10) ] × 5

⇒ Mode = 25 + [ 2 ÷ (22 − 19) ] × 5

⇒ Mode = 25 + [ 2 ÷ 3 ] × 5

⇒ Mode = 25 + 10 ÷ 3

⇒ Mode = 25 + 3.333...

⇒ Mode ≈ 28.33

∴ The correct answer is 28.33.

Additional Information

Empirical Formula

Relationship between Mean, Median, and Mode: Mode = 3 Median − 2 Mean.

Median for Grouped Data

Median = L + [ (N/2 − cf) ÷ f ] × h, where cf is cumulative frequency of the preceding class.

Bimodal Distribution

A distribution that has two modes (two classes with the same highest frequency).

Shortcut Trick

Effective rate for half-yearly compounding = 20% ÷ 2 = 10% per 6 months.

For 1 year, there are 2 compounding periods.

Net effective interest rate = 10 + 10 + (10 × 10) ÷ 100 = 21%.

Compound Interest = 21% of ₹8,000 = 0.21 × 8,000 = ₹1,680.

∴ The correct answer is ₹1,680.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate of Interest (R) = 20% per annum

Time (T) = 1 year

Formula Used:

For half-yearly compounding: Rate (r) = R ÷ 2, Time periods (n) = 2 × T

Amount (A) = P(1 + r/100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ Effective rate (r) = 20 ÷ 2 = 10% per half-year

⇒ Number of periods (n) = 1 × 2 = 2 periods

⇒ A = 8000 × (1 + 10/100)²

⇒ A = 8000 × (1.1)²

⇒ A = 8000 × 1.21

⇒ A = ₹9,680

⇒ CI = 9680 − 8000

⇒ CI = ₹1,680

∴ The correct answer is ₹1,680.

Additional Information

Effective Annual Rate (EAR)

When interest is compounded more than once a year, the effective annual rate is given by [(1 + r/n)ⁿ − 1] × 100, where n is the number of compounding periods.

Quarterly Compounding

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

CI vs SI Difference

For 2 years (or 2 periods), the difference between CI and SI is P(R/100)².

Shortcut Trick

Total Eggs = 100%

Good Eggs % = 100% − 12% = 88%

Multiplying Factor method: Good Eggs = 350 × 0.88

Calculation: 350 × 0.88 = 35 × 8.8 = 308

∴ The correct answer is 308 eggs.

Alternate Method

Given: Total eggs = 350, Rotten percentage = 12%

Formula Used: Good Eggs = Total Eggs − Rotten Eggs

⇒ Rotten Eggs = 12% of 350

⇒ Rotten Eggs = (12 × 350) ÷ 100

⇒ Rotten Eggs = 4200 ÷ 100 = 42

⇒ Good Eggs = Total Eggs − Rotten Eggs

⇒ Good Eggs = 350 − 42

⇒ Good Eggs = 308

∴ The correct answer is 308 eggs.

Additional Information

Percentage Calculation

To find x% of a value Y, use the formula (x ÷ 100) × Y. For quick calculation, 12% can be viewed as 10% + 2%.

Complementary Percentage

If a part represents P%, the remaining part is always (100 − P)%. Here, Good Eggs = 100% − 12% = 88%.

Percentage Change Formula

Final Value = Initial Value × [1 ± (Change% ÷ 100)]. Use plus for increase and minus for decrease/loss.

Shortcut Trick

For any quadrilateral circumscribing a circle, the sum of opposite sides is equal.

Property: AB + CD = BC + AD

18 + CD = 21 + 15

CD = 36 − 18 = 18 cm

∴ The correct answer is 18 cm.

Alternate Method

Given:

Length of side AB = 18 cm

Length of side BC = 21 cm

Length of side AD = 15 cm

Formula Used:

In a tangential quadrilateral (a quadrilateral that circumscribes a circle), the sum of the lengths of opposite sides is equal: AB + CD = BC + AD.

Calculations:

⇒ AB + CD = BC + AD

⇒ 18 + CD = 21 + 15

⇒ 18 + CD = 36

⇒ CD = 36 − 18

⇒ CD = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Pitot Theorem

The theorem states that for a tangential quadrilateral, the sum of the lengths of opposite sides is equal. This is derived from the property that tangents drawn from an external point to a circle are equal in length.

Area of Tangential Quadrilateral

The area of a tangential quadrilateral can be calculated as Area = s × r, where s is the semi-perimeter (s = (a+b+c+d)/2) and r is the radius of the incircle.

Cyclic Quadrilateral

A cyclic quadrilateral is one where all vertices lie on a circle. In such cases, the sum of opposite angles is 180° (supplementary angles).

Shortcut Trick

For the largest sphere carved out of a cube, Diameter (d) = Side of cube (a).

Side (a) = 21 cm ⇒ Radius (r) = 21 ÷ 2 = 10.5 cm.

Volume of Sphere = (1 ÷ 6) × π × a³

Volume = (1 ÷ 6) × (22 ÷ 7) × 21 × 21 × 21 = 11 × 21 × 21 = 11 × 441 = 4851 cm³.

∴ The correct answer is 4851 cm³.

Alternate Method

Given:

Side of the wooden cube (a) = 21 cm

Formula Used:

Diameter of the largest sphere = Side of the cube (a)

Volume of a sphere = (4 ÷ 3) × π × r³

Calculations:

⇒ Diameter of the sphere (d) = 21 cm

⇒ Radius of the sphere (r) = d ÷ 2 = 21 ÷ 2 = 10.5 cm

⇒ Volume = (4 ÷ 3) × (22 ÷ 7) × (21 ÷ 2) × (21 ÷ 2) × (21 ÷ 2)

⇒ Volume = (4 ÷ 3) × (22 ÷ 7) × (9261 ÷ 8)

⇒ Volume = (88 ÷ 21) × (9261 ÷ 8)

⇒ Volume = 11 × 441

⇒ Volume = 4851 cm³

∴ The correct answer is 4851 cm³.

Additional Information

Volume of Cube

The volume of a cube with side 'a' is given by V = a³.

Largest Cone inside a Cube

The largest cone inside a cube of side 'a' has a base radius of a ÷ 2 and a height of 'a'. Volume = (1 ÷ 3) × π × (a ÷ 2)² × a.

Surface Area of Sphere

The total surface area of a sphere with radius 'r' is 4 × π × r².

Largest Cylinder inside a Cube

The largest cylinder that can be carved out of a cube of side 'a' has radius r = a ÷ 2 and height h = a. Volume = π × (a ÷ 2)² × a.

Given:

$\sqrt{8 + \sqrt{1681}}$ = ?

Formula used:

√x² = x

Calculations:

$\sqrt{8 + \sqrt{1681}}$

⇒ $\sqrt{8 + 41}$

⇒ $\sqrt{49}$

⇒ 7

∴ The correct answer is option (4).

Given:

$30 - [40 - {56 - (25 - 13 - 12)}] = ?$

Formula used:

BODMAS Rule

Calculation:

$30 - [40 - {56 - (25 - 13 - 12)}]$

⇒ $30 - [40 - {56 - (12 - 12)}]$

⇒ $30 - [40 - {56 - 0}]$

⇒ $30 - [40 - 56]$

⇒ $30 - (-16)$

⇒ $30 + 16$

⇒ 46

∴ The correct answer is option (4).

Shortcut Trick

Income ratio = 7 : 4 (difference = 3) and Expenditure ratio = 3 : 1 (difference = 2).

Make ratio differences equal by cross-multiplying: Income × 2 and Expenditure × 3.

New Income ratio = 14 : 8 and New Expenditure ratio = 9 : 3.

Savings in parts = 14 − 9 = 5 parts. Since savings = ₹4800, 1 part = 4800 ÷ 5 = ₹960.

Sum of incomes = (14 + 8) × 960 = 22 × 960 = ₹21120.

∴ The correct answer is ₹21120.

Alternate Method

Given:

Ratio of incomes of two employees = 7 : 4

Ratio of their expenditures = 3 : 1

Monthly savings of each employee = ₹4800

Formula Used:

Income − Expenditure = Savings

Employee 1 Employee 2

Income: 14 units

Exp: 9u

Sav: 5u

Inc: 8u

3u

Sav: 5u

5 units = ₹4800 → 1 unit = ₹960 Total Income = (14 + 8) × 960 = ₹21120

Calculations:

⇒ Let the monthly incomes of the two employees be 7x and 4x respectively.

⇒ Let their monthly expenditures be 3y and y respectively.

⇒ For the first employee: 7x − 3y = 4800 --- (Equation 1)

⇒ For the second employee: 4x − y = 4800 ⇒ y = 4x − 4800

⇒ Substituting the value of y in Equation 1:

⇒ 7x − 3(4x − 4800) = 4800

⇒ 7x − 12x + 14400 = 4800

⇒ 5x = 9600

⇒ x = 1920

⇒ Sum of their monthly incomes = 7x + 4x = 11x

⇒ Sum = 11 × 1920 = ₹21120

∴ The correct answer is ₹21120.

Additional Information

Income, Expenditure & Savings

The fundamental relation is Income = Expenditure + Savings. If savings are equal for two persons, the difference between income and expenditure ratio parts must be mathematically equated by balancing multipliers.

Cross-Multiplication Method

For equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the cross-multiplication method offers a direct approach to find the unknown variables without requiring lengthy substitution steps.

Compound Ratio

When two or more ratios are multiplied together, it is known as a compound ratio. For a : b and c : d, the compound ratio is ac : bd.

Shortcut Trick

Net percentage change formula for area (r × r): x + y + (xy ÷ 100)

Given decrease (negative change): x = −48 and y = −48

Net change = −48 − 48 + [(−48 × −48) ÷ 100]

Net change = −96 + (2304 ÷ 100) = −96 + 23.04 = −72.96%

Negative sign indicates a decrease.

∴ The correct answer is 72.96%.

Alternate Method

Given:

Initial radius (r₁) decrease = 48%

Formula Used:

Surface Area of a sphere (S) = 4 × π × r²

Calculations:

⇒ Let the initial radius (r₁) = 100 units

⇒ Final radius (r₂) after 48% decrease = 100 − 48 = 52 units

⇒ Since 4 and π are constants, Surface Area (S) ∝ r²

⇒ Initial Surface Area (S₁) ∝ 100² = 10000

⇒ Final Surface Area (S₂) ∝ 52² = 2704

⇒ Decrease in Surface Area = S₁ − S₂ = 10000 − 2704 = 7296

⇒ Percentage Decrease = (Decrease ÷ S₁) × 100

⇒ Percentage Decrease = (7296 ÷ 10000) × 100 = 72.96%

∴ The correct answer is 72.96%.

Additional Information

Volume Change of a Sphere

Since Volume (V) = (4/3) × π × r³, the volume change depends on r³. If radius decreases by x%, the new volume is [(100−x)/100]³ of the original.

Successive Percentage for 3D Shapes

For volume of a sphere, use the successive formula twice: first for (r × r) to get area change, then combine that result with the third 'r'.

Surface Area of other 3D Shapes

Cube = 6a² | Cylinder (Total) = 2πr(r + h) | Cone (Curved) = πrl. The area always depends on the product of two linear dimensions.

Shortcut Trick

Product of the co-prime ratio parts = Total Product ÷ (HCF)²

Product of parts = 1500 ÷ (10 × 10) = 15

The co-prime factor pairs of 15 are (1, 15) and (3, 5).

Since both pairs share no common factor other than 1, there are 2 valid pairs.

∴ The correct answer is 2 pairs.

Alternate Method

Given:

Product of the two numbers = 1500

HCF of the two numbers = 10

Formula Used:

Let the two numbers be 10x and 10y, where x and y are co-prime integers.

Product of numbers = HCF² × xy

Product = 1500 | HCF = 10

x × y = 15

Pair 1: x=1, y=15 10 and 150

Pair 2: x=3, y=5 30 and 50

Calculations:

⇒ 10² × xy = 1500

⇒ 100 × xy = 1500

⇒ xy = 1500 ÷ 100

⇒ xy = 15

Now, we need to find pairs of integers (x, y) whose product is 15 and they must be co-prime to each other.

⇒ The integer factor pairs of 15 are (1, 15) and (3, 5).

Let's check if they are co-prime:

⇒ HCF of 1 and 15 is 1 (co-prime → valid)

⇒ HCF of 3 and 5 is 1 (co-prime → valid)

So, the corresponding possible pairs of original numbers are:

⇒ Pair 1: (10 × 1, 10 × 15) = (10, 150)

⇒ Pair 2: (10 × 3, 10 × 5) = (30, 50)

⇒ Total number of valid pairs = 2

∴ The correct answer is 2 pairs.

Additional Information

Product of Two Numbers

The product of any two numbers is equal to the product of their HCF and LCM (Number 1 × Number 2 = HCF × LCM).

Co-prime Numbers

Two numbers are considered co-prime if their Highest Common Factor (HCF) is exactly 1. When dividing out the HCF from two numbers, the remaining ratio parts must always be co-prime.

Relation between LCM and HCF

For any numbers written as (HCF × a) and (HCF × b), their LCM is calculated directly as HCF × a × b, meaning the LCM is always exactly divisible by the HCF.

Shortcut Trick Net Cost Price (CP) = 5000 + 8000 = ₹13,000 Net Selling Price (SP) = 6000 + 10000 = ₹16,000 Total Profit = 16000 − 13000 = ₹3,000 Overall Profit % = (3000 ÷ 13000) × 100 = 300 ÷ 13 % Overall Profit % ≈ 23.08% ∴ The correct answer is 23.08%.

Alternate Method Given: Transaction 1: CP1 = ₹5,000, SP1 = ₹6,000 Transaction 2: CP2 = ₹8,000, SP2 = ₹10,000 Formula Used: Overall Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Calculations: ⇒ Total CP = CP1 + CP2 = 5,000 + 8,000 = ₹13,000 ⇒ Total SP = SP1 + SP2 = 6,000 + 10,000 = ₹16,000 ⇒ Total Profit = Total SP − Total CP = 16,000 − 13,000 = ₹3,000 ⇒ Overall Profit % = (3,000 ÷ 13,000) × 100 ⇒ Profit % = 300 ÷ 13 = 23.0769... ⇒ Profit % ≈ 23.08% ∴ The correct answer is 23.08%.

Additional Information Net Profit/Loss Percentage For multiple transactions, always sum the individual Cost Prices and Selling Prices to find the total profit or loss. Profit % = (Net Profit ÷ Total CP) × 100. Profit and Loss Relation Profit occurs when SP > CP (Profit = SP − CP). Loss occurs when CP > SP (Loss = CP − SP). Percentage is always calculated on the CP unless mentioned otherwise. Marked Price and Discount Selling Price = Marked Price − Discount. Relationship: CP × (100 + P%) = MP × (100 − D%).

Shortcut Trick

Using the complementary angle properties: sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Substitute 65° = 90° − 25° in the given expression:

sin 25° sin(90° − 25°) − cos 25° cos(90° − 25°)

This simplifies directly to: sin 25° cos 25° − cos 25° sin 25° = 0.

∴ The correct answer is 0.

Alternate Method

Given:

Expression: sin 25° sin 65° − cos 25° cos 65°

Formula Used:

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

Calculation:

⇒ The given expression can be written as:

⇒ −(cos 25° cos 65° − sin 25° sin 65°)

⇒ Using the formula cos(A + B) = cos A cos B − sin A sin B:

⇒ −cos(25° + 65°)

⇒ −cos(90°)

⇒ We know that cos 90° = 0.

⇒ −0 = 0.

∴ The correct answer is 0.

Additional Information

Trigonometric Addition Identities

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

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

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

Complementary Angles

Angles whose sum is 90° are complementary. sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Values of Standard Angles

sin 90° = 1, cos 90° = 0, tan 90° is Not Defined.

Shortcut Trick

In ΔPOR, ∠OPR = ∠ORP ⇒ OP = OR. Since two sides are equal, ΔPOR is isosceles (i is correct).

In ΔPQO and ΔRQO, we have PQ = QR, PO = RO, and common side QO.

By SSS congruency, ΔPQO ≅ ΔRQO (iii is correct).

Point O lies on the axis of symmetry (median from Q) but isn't restricted to the 2:1 centroid position.

∴ The correct answer is Only (i) and (iii).

Alternate Method

Given: In ΔPQR, PQ = QR. O is an interior point such that ∠OPR = ∠ORP.

Calculations:

⇒ In ΔPOR, given ∠OPR = ∠ORP.

⇒ According to triangle properties, sides opposite to equal angles are equal ∴ OP = OR.

⇒ Since ΔPOR has two equal sides (OP = OR), it is an isosceles triangle. (Statement i is correct)

⇒ Now, consider ΔPQO and ΔRQO:

⇒ PQ = RQ (Given that ΔPQR is isosceles with PQ = QR)

⇒ PO = RO (Proved above)

⇒ QO = QO (Common side to both triangles)

⇒ ∴ ΔPQO ≅ ΔRQO by SSS (Side-Side-Side) congruence rule. (Statement iii is correct)

⇒ Point O is equidistant from P and R, so it must lie on the perpendicular bisector of PR (which is also the median from Q).

⇒ However, the centroid is a unique point on this median. Point O can be any point on this line segment. Thus, O is not necessarily the centroid. (Statement ii is incorrect)

∴ The correct answer is Only (i) and (iii).

Additional Information

SSS Congruence Criterion

Two triangles are congruent if all three sides of one triangle are equal to the corresponding three sides of the second triangle.

Isosceles Triangle Theorem

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

Centroid of a Triangle

The centroid is the point where the three medians of the triangle intersect and it divides each median in the ratio 2:1 from the vertex.

Perpendicular Bisector in Isosceles Triangle

In an isosceles triangle, the median, altitude, and angle bisector from the vertex between the equal sides all lie on the same line (the perpendicular bisector of the base).

Shortcut Trick

Distance (d) = [Total Time × (u² − v²)] ÷ (2 × u)

Given: u (Still water) = 10 km/h, v (River) = 4.5 km/h, Total Time (T) = 2 h

⇒ d = [2 × (10² − 4.5²)] ÷ (2 × 10)

⇒ d = (100 − 20.25) ÷ 10 = 79.75 ÷ 10 = 7.975 km

Rounding to two decimal places, d ≈ 7.98 km.

∴ The correct answer is 7.98 km.

Alternate Method

Given:

Speed of man in still water (u) = 10 km/h

Speed of river (v) = 4.5 km/h

Total time taken (T) = 2 h

Formula Used:

Downstream Speed (D) = u + v

Upstream Speed (U) = u − v

Total Time = (Distance ÷ D) + (Distance ÷ U)

Calculations:

⇒ Downstream Speed (D) = 10 + 4.5 = 14.5 km/h

⇒ Upstream Speed (U) = 10 − 4.5 = 5.5 km/h

⇒ (d ÷ 14.5) + (d ÷ 5.5) = 2

⇒ d × [(5.5 + 14.5) ÷ (14.5 × 5.5)] = 2

⇒ d × (20 ÷ 79.75) = 2

⇒ d = (2 × 79.75) ÷ 20

⇒ d = 159.5 ÷ 20 = 7.975 km

⇒ d ≈ 7.98 km

∴ The correct answer is 7.98 km.

Additional Information

Speed of Boat and Stream

If Downstream speed is D and Upstream speed is U, then Boat speed (u) = (D + U) ÷ 2 and Stream speed (v) = (D − U) ÷ 2.

Average Speed of Round Trip

Average speed for the entire journey is given by (D × U) ÷ Still water speed (u), which is 2DU ÷ (D + U).

Ratio of Speeds

The ratio of the speed of the boat in still water to the speed of the stream is (D + U) : (D − U).

Shortcut Trick

Average of cubes of first n natural numbers = n(n + 1)² ÷ 4

Here, n = 5

Average = 5 × (5 + 1)² ÷ 4

Average = 5 × 36 ÷ 4 = 5 × 9 = 45

∴ The correct answer is 45.

Alternate Method

Given: First five natural numbers: 1, 2, 3, 4, 5.

Formula Used: Average = Sum of observations ÷ Number of observations

⇒ First five natural numbers: 1, 2, 3, 4, 5

⇒ Cubes of these numbers: 1³, 2³, 3³, 4³, 5³

⇒ Values of cubes: 1, 8, 27, 64, 125

⇒ Sum of cubes = 1 + 8 + 27 + 64 + 125 = 225

⇒ Average = 225 ÷ 5 = 45

∴ The correct answer is 45.

Additional Information

Sum of first n natural numbers

The sum of the first n natural numbers is given by the formula S_n = [n(n + 1)] ÷ 2.

Sum of squares of first n natural numbers

The sum of the squares of the first n natural numbers is given by the formula S_n² = [n(n + 1)(2n + 1)] ÷ 6.

Sum of cubes of first n natural numbers

The sum of the cubes of the first n natural numbers is given by the formula S_n³ = [n(n + 1) ÷ 2]².

Shortcut Trick

For infinite solutions, ratios of coefficients must be equal: a₁/a₂ = b₁/b₂.

Comparing x and y coefficients: 17/23 = m/299.

Multiply both sides by 299: m = (17 × 299) ÷ 23.

Since 299 ÷ 23 = 13, then m = 17 × 13 = 221.

∴ The correct answer is 221.

Alternate Method

Given Equations: 17x + my + 102 = 0 and 23x + 299y + 138 = 0.

Formula: For infinite solutions, the condition is a₁/a₂ = b₁/b₂ = c₁/c₂.

⇒ a₁ = 17, b₁ = m, c₁ = 102

⇒ a₂ = 23, b₂ = 299, c₂ = 138

⇒ Ratio of constants: c₁/c₂ = 102/138 = (17 × 6) / (23 × 6) = 17/23.

⇒ To satisfy the condition: a₁/a₂ = b₁/b₂

⇒ 17/23 = m/299

⇒ m = (17 × 299) / 23

⇒ m = 17 × 13

⇒ m = 221

∴ The correct answer is 221.

Additional Information

Unique Solution

A system has a unique solution if the lines intersect at one point, which occurs when a₁/a₂ ≠ b₁/b₂.

No Solution

A system has no solution if the lines are parallel and never intersect, occurring when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Consistency of Equations

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

Shortcut Trick

Inner Perimeter = 2 × [(Length + Breadth) − 4 × path width]

Inner Perimeter = 2 × [(120 + 104) − 4 × 1] = 2 × 220 = 440 m

Total Cost = 440 × ₹2.50 = ₹1100.

∴ The correct answer is ₹1100.

Alternate Method

Given:

Length of the park (L) = 120 m

Width of the park (B) = 104 m

Width of the path (w) = 1 m

Rate of marking = ₹2.50 per metre

Formula Used:

Inner Dimension = Outer Dimension − 2 × path width

Perimeter of Rectangle = 2 × (Length + Breadth)

Calculations:

⇒ Inner Length (l) = 120 − (1 + 1) = 118 m

⇒ Inner Width (b) = 104 − (1 + 1) = 102 m

⇒ The inside edge of the path forms a rectangle with dimensions 118 m × 102 m.

⇒ Length of white line = Perimeter of the inner rectangle

⇒ Perimeter = 2 × (118 + 102)

⇒ Perimeter = 2 × 220 = 440 m

⇒ Total Cost = Perimeter × Rate

⇒ Total Cost = 440 × 2.50 = ₹1100.

∴ The correct answer is ₹1100.

Additional Information

Area of Path (Inside)

If a path of width 'w' is built inside a rectangle (L × B), its Area = 2w(L + B − 2w).

Area of Path (Outside)

If a path of width 'w' is built outside a rectangle (L × B), its Area = 2w(L + B + 2w).

Diagonal of Rectangle

The distance between opposite corners of the park is given by √(L² + B²).

Shortcut Trick

Total quantity (x) = 40 litres, quantity replaced (y) = 4 litres, total operations (n) = 3.

Remaining milk = x × (1 − y/x)ⁿ

Remaining milk = 40 × (1 − 4/40)³ = 40 × (9/10)³

Remaining milk = 40 × 0.729 = 29.16 litres.

∴ The correct answer is 29.16 litres.

Alternate Method

Given:

Initial amount of milk = 40 litres

Amount removed and replaced by water = 4 litres

Total number of times the process is performed = 1 (initially) + 2 (more) = 3 times

Formula Used:

Final Quantity = Initial Quantity × (1 − Replaced Quantity / Total Quantity)^Time

Calculations:

⇒ Quantity of milk after 1^st replacement = 40 − 4 = 36 litres

⇒ Ratio of milk remaining after each replacement = (40 − 4) ÷ 40 = 36/40 = 9/10

⇒ Milk left after 3 operations = 40 × (9/10) × (9/10) × (9/10)

⇒ Milk left = 40 × (729 / 1000)

⇒ Milk left = 4 × 72.9 / 10

⇒ Milk left = 291.6 / 10 = 29.16 litres

∴ The correct answer is 29.16 litres.

Additional Information

General Replacement Formula

Amount of pure liquid left = P × (1 − R/P)ⁿ, where P is initial quantity, R is quantity replaced, and n is number of cycles.

Ratio of Ingredients

The ratio of the original liquid to the replacing liquid (water) after n operations is [P × (1 − R/P)ⁿ] : [P − (P × (1 − R/P)ⁿ)].

Percentage Purity

Final concentration percentage = (Initial concentration) × (1 − R/P)ⁿ.

Shortcut Trick Rate difference = (Difference in SI × 100) / (Principal × Time) R1 − R2 = (800 × 100) / (8000 × 3) R1 − R2 = 80000 / 24000 R1 − R2 = 10/3 = 3 1/3% ∴ The correct answer is 3 1/3%. Alternate Method Given: Principal (P) = ₹8,000, Time (T) = 3 years, Difference in SI = ₹800, R1 > R2. Formula: Simple Interest (SI) = (P × R × T) / 100

⇒ SI1 − SI2 = [(P × R1 × T) / 100] − [(P × R2 × T) / 100] ⇒ 800 = (P × T / 100) × (R1 − R2) ⇒ 800 = (8000 × 3 / 100) × (R1 − R2) ⇒ 800 = 240 × (R1 − R2) ⇒ R1 − R2 = 800 / 240 ⇒ R1 − R2 = 10 / 3 = 3 1/3% ∴ The correct answer is 3 1/3%. Additional Information Simple Interest Property Simple Interest is directly proportional to the rate and time. If time and principal are constant, the change in interest is directly proportional to the change in rate. Effective Rate of Interest The effective rate for a period is the total interest earned divided by the principal. In SI, Effective Rate = R × T. Difference between SI and CI For 2 years, Difference = P(R/100)2. For 3 years, Difference = P(R/100)2 × (3 + R/100).

Shortcut Trick Compare given tan values with standard trigonometric values: tan 60° = √3 and tan 30° = 1/√3. Equate the angles: A + B = 60° and A − B = 30°. Find A by adding equations: A = (60° + 30°) ÷ 2 = 45°. Find B by subtracting equations: B = (60° − 30°) ÷ 2 = 15°. ∴ The correct answer is 45° and 15°.

Alternate Method Given: tan (A + B) = √3 tan (A − B) = 1/√3 Condition: 0° < (A + B) < 90° and A > B Formula Used: Standard values: tan 60° = √3 and tan 30° = 1/√3

⇒ tan (A + B) = tan 60° ⇒ A + B = 60° --- (Equation 1) ⇒ tan (A − B) = tan 30° ⇒ A − B = 30° --- (Equation 2) ⇒ Adding (1) and (2): (A + B) + (A − B) = 60° + 30° ⇒ 2A = 90° ⇒ A = 45° ⇒ Substituting A in (1): 45° + B = 60° ⇒ B = 60° − 45° ⇒ B = 15° ∴ The correct answer is 45° and 15°.

Additional Information Trigonometric Table for Tan Key values include: tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, and tan 90° = ∞. Sum and Difference Rule for Equations When A + B = S and A − B = D, then A = (S + D) ÷ 2 and B = (S − D) ÷ 2. Tangent Addition Identity The formula tan(A ± B) = (tan A ± tan B) ÷ (1 ∓ tan A × tan B) is often used for non-standard angles.

Shortcut Trick

Distance between ships on opposite sides (D) = h × (cot θ₁ + cot θ₂)

230√3 = h × (cot 30° + cot 60°)

230√3 = h × (√3 + 1/√3) = h × (4/√3)

4h = 230 × 3 ⇒ 4h = 690

h = 690 ÷ 4 = 172.5 m

∴ The correct answer is 172.5 m.

Alternate Method

Given:

Total distance between ships = 230√3 m

Angle of depression to Ship 1 = 30°

Angle of depression to Ship 2 = 60°

Formula Used:

tan θ = Perpendicular ÷ Base

Calculations:

⇒ Let h be the height of the lighthouse.

⇒ In right triangle 1, tan 30° = h ÷ d₁ ⇒ d₁ = h ÷ (1/√3) = h√3

⇒ In right triangle 2, tan 60° = h ÷ d₂ ⇒ d₂ = h ÷ √3

⇒ Total distance = d₁ + d₂ = h√3 + h/√3

⇒ 230√3 = h × (√3 + 1/√3)

⇒ 230√3 = h × (4/√3)

⇒ 230 × 3 = 4h

⇒ 690 = 4h

⇒ h = 690 ÷ 4 = 172.5

∴ The correct answer is 172.5 m.

Additional Information

Angle of Depression and Elevation

The angle of depression from the top of a tower to an object is equal to the angle of elevation from the object to the top because they are alternate interior angles formed by parallel horizontal lines.

Standard Distance Formulas

For a tower of height h, distance D = h(cot α + cot β) if objects are on opposite sides, and D = h(cot α − cot β) if objects are on the same side.

Trigonometric Ratios

Useful values for height and distance problems: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, and their reciprocal cotangent values.

Shortcut Trick

Assume a base value for the mean, say 20.

Find deviations from 20: (23 − 20) = +3, (17 − 20) = −3, (20 − 20) = 0, (19 − 20) = −1, (21 − 20) = +1.

Sum of deviations = 3 − 3 + 0 − 1 + 1 = 0.

Arithmetic Mean = Base + (Total Deviation ÷ Count) = 20 + (0 ÷ 5) = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Data values: 23, 17, 20, 19, 21

Total number of observations (n) = 5

Formula Used:

Arithmetic Mean (χ) = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Sum = 23 + 17 + 20 + 19 + 21

⇒ Sum = 100

⇒ Mean = 100 ÷ 5

⇒ Mean = 20

∴ The correct answer is 20.

Additional Information

Arithmetic Mean Properties

The sum of deviations of a set of observations from their arithmetic mean is always zero. If each observation is increased by a constant k, the mean also increases by k.

Range of Data

Range is the difference between the maximum and minimum values in a data set. For this data: Range = 23 − 17 = 6.

Median

The median is the middle value when the data is arranged in ascending or descending order. For 17, 19, 20, 21, 23, the median is 20.

Shortcut Trick

The divisibility rule for 4 states that a number is divisible by 4 if its last two digits are divisible by 4.

In the number 5769116, the last two digits are 16.

Since 16 is perfectly divisible by 4 (16 ÷ 4 = 4), the entire number must be divisible by 4.

∴ The correct answer is 4.

Alternate Method

Given Number: 5769116

Check the divisibility rules for each option:

⇒ Divisibility by 5: The last digit must be 0 or 5. Here, the last digit is 6, so it is not divisible by 5.

⇒ Divisibility by 8: The last three digits must be divisible by 8. Here, 116 ÷ 8 = 14.5, so it is not divisible by 8.

⇒ Divisibility by 12: The number must be divisible by both 3 and 4. Sum of digits = 5+7+6+9+1+1+6 = 35. Since 35 is not divisible by 3, the number is not divisible by 12.

⇒ Divisibility by 4: The last two digits are 16. Since 16 ÷ 4 = 4, the number is divisible by 4.

∴ The correct answer is 4.

Additional Information

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3 (e.g., sum of digits of 5769116 is 35, not divisible by 3).

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Shortcut Trick

For any expression of the form x + 1/x = k, the value of x² + 1/x² is always k² − 2.

Given: x + 1/x = 4

Here, k = 4

Result = 4² − 2 = 16 − 2 = 14

∴ The correct answer is 14.

Alternate Method

Given:

x + 1/x = 4

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

Squaring both sides of the given equation:

⇒ (x + 1/x)² = 4²

⇒ x² + (1/x)² + 2 × x × (1/x) = 16

⇒ x² + 1/x² + 2 = 16

⇒ x² + 1/x² = 16 − 2

⇒ x² + 1/x² = 14

∴ The correct answer is 14.

Additional Information

Cube Power Relation

If x + 1/x = k, then the value of x³ + 1/x³ = k³ − 3k.

Negative Sign Relation

If x − 1/x = k, then the value of x² + 1/x² = k² + 2.

Higher Power Calculation

To find x⁴ + 1/x⁴, square the value obtained for x² + 1/x² and subtract 2.

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 N is the granddaughter i.e daughter's daughter of F.

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

Logic : Number : Number³ + 3

So, we have to check the options one by one.

1. 5 : 128  ⇒ 5³ + 3 = 125 + 3 = 128

2. 3 : 30  ⇒ 33 + 3 = 27 + 3 = 30

3) 7 : 352  ⇒ 73 + 3 = 343 + 3 = 346 ≠ 352

4. 1 : 4  ⇒ 13 + 3 = 1 + 3 = 4

Thus, all except option 3 , follow the pattern.

Hence, the correct answer is "7 : 352".

Given: Six friends A, B, C, D, E and F are standing in a circle facing the centre. 

1) B is between F and C and F is to the immediate left of D.

2) A is between E and D.

Therefore, D is between A and F.

Hence, the correct answer is "D".

The correct mirror image of the given combination when the mirror is placed at MN is as shown below :-

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

Course of Action:

I) Malaria is a disease that spreads through the bites of infected mosquitoes. Therefore, to control the spread of malaria, it is necessary to take measures to eliminate mosquitoes from the affected areas. Hence, course of action I is logical, which suggests that the city municipal authorities should take immediate steps to carry out extensive fumigation in the city. This will help in killing mosquitoes and reducing the spread of malaria.

II) Furthermore, it is also important to educate and advise people about the precautions they should take to avoid mosquito bites. Therefore, course of action II is also logical, which suggests that the people in the area should be advised to take steps to avoid mosquito bites. These steps may include using mosquito repellents, wearing long-sleeved clothing, and using mosquito nets while sleeping.

Thus, both courses of action are important and should be implemented to control the spread of malaria in the city.

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

Given : 

1. Six people - K, M, N, P, Q and R
2. They are sitting around a circular table facing the centre.

Explanation :

• Statement (I) → K sits third to the left of M. Only one person sits between K and N. R sits to the immediate left of N.

⇒So, according to final arrangement of statement (I) R is sitting to the immediate right of K.

• Statement (II) → K sits second to the right of Q. Only two people sit between K and M. P sits second to the left of R.

⇒So, according to final arrangement of statement (II) R is sitting to the immediate right of K.

Here, in both statements (I) and (II) R is sitting to the immediate right of K. Thus either data in statement I or statement II is sufficient to answer the question.

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

Given:

• + means −
• − means ×
• × means ÷
• ÷ means +

We need to find the value of the following expression:

5 ÷ 5 + 5 – 10 × 10

Substituting the given symbols, we get:

5 + 5 - 5 × 10 ÷ 10

= 5 + 5 - 5 × 1

= 5 + 5 - 5

= 10 - 5 

= 5

Therefore, the value of the expression is 5.

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

Given : 

1. Six people sit in two parallel rows.
2. In Row 1 - J, K and L sit facing towards north.
3. In Row 2 - F, G and H sit facing towards south.
4. Each person in a row faces a person in the other row.

Explanation : 

• Statement (I) → H faces K; L faces an immediate neighbor of H.

​As any specific position of H, L and K cannot be determined, so no specific answer can drawn from it as this statement gives many possibilities.

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

• Statement (II) → G sits at one of the extreme ends of the row; J sits to the immediate left of L.​

​As any specific position of G, J and L cannot be determined, so no specific answer can drawn from it as this statement gives many possibilities.

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

Now, Using both the statements :

• Statement (I) + Statement (II) → H faces K; L faces an immediate neighbor of H. ​G sits at one of the extreme ends of the row; J sits to the immediate left of L.

​

So, using both the statements, J faces G. Thus data in statements I and II together is sufficient to answer the question.

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

The logic followed here is:

Hence, the correct answer is "323".

The group of words that are alike and form a group are:

1. Excel
2. PowerPoint
3. Word

These three words are all software applications developed by Microsoft as part of its Office suite.

The word that does not belong to this group is "Keynote," a software application developed by Apple Inc. for creating and delivering presentations.

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

Logic : Sum of the position values of opposite letters according to the English alphabetical series and number of letters within the word are added.

​So,

• ‘PAINT’ is written as ‘80’

​And,

• ‘DROP’ is written as ‘59’

​Similarly,

• ‘MARKET’ is written as :

​

Hence, "100" is the correct answer.

Given: E = 1, Z = 2, L = 3, P = 4, U = 5, Z = 6

Using the given number denotations, we can form the following words by substituting the corresponding numbers for the letters:

1. PUEZZL [4,5,1,6,2,3]
2. ZEPLUZ [6,1,4,3,5,2]
3. EULPZZ [1,5,3,4,2,6]
4. PUZZLE [4,5,6,2,3,1]

Out of these, only option 4) can be rearranged to form a meaningful English word: PUZZLE.

Hence, the correct option is 4) 4,5,6,2,3,1.

The logic here followed is:

Hence, the correct answer is "Option 1".

Given: Q W _ R T Q _ F R _ Q W _ _ T _ _ H R T

Explantion:

Let's check each option:

1) F W T G R Q X →  Q W F R T  -  Q W F R T   -  Q W G R T  -  Q X H R T 

2) E X T G R Q X →  Q W E R T  -  Q X F R T   -  Q W G R T   -  Q X H R T

3) E W T G R Q W  → Q W E R T  -  Q W F R T  -  Q W G R T  -  Q W H R T

4) E W T H S Q W  →  Q W E R T  -  Q W F R T   -  Q W H S T   -  Q W H R T

So, In option 3 "QWERT" is repeating itself. E is rotating by its next letter.

Hence, ‘E W T G R Q W " is the correct answer.

Given: 21 * 3 * 36 * 2 * 23 = 68

Replacing * to × − ÷ +

21 × 3 - 36 ÷  2 + 23 = 68

First, we perform division from left to right:

21 × 3 - 18 + 23 = 68

Second, we perform the multiplication 

63 - 18 + 23 = 68

Next, we perform the addition and subtraction from left to right:

86 -18 = 68

68 = 68

Therefore, the equation is balanced when we replace the * signs with × − ÷ and + in the following order:

21 × 3 - 36 ÷  2 + 23 = 68

Hence, the correct answer is "× − ÷ +".

The logic here followed is :

 

   This diagram and the answer diagram are different. Both are closest resemble.

when we unfold it according to the highlighted diagram we will get the closest resemble  the unfolded form of the paper.

Hence, the correct answer is "Option 1".

The logic followed here is :- 

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

​So,

• = BNQ, % = NFM ⇒ BNQ : DKR :: PCL: NFM

​

⇒Here, no similar pattern can be seen.

• = FHT, % = NFJ ⇒ FHT : DKR :: PCL: NFJ

​

⇒Here, similar pattern can be seen.

• = COQ, % = OGH ⇒ COQ : DKR :: PCL: OGH

​

⇒Here, no similar pattern can be seen.

• = BNP, % = NFJ ⇒ BNP : DKR :: PCL: NFJ

​

⇒Here, no similar pattern can be seen.

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

The total number of students in the class is 32, and Akash is 12^th from the right end. Therefore, the number of people standing to the left of Akash is:

32 - 12 = 20

Position of Akhash from left = 20 + 1 = 21

While Priya is 18^th from the left end. 

Therefore, 19^th and 20^th students are standing between Akash and Priya.

Hence, the correct answer is "Two".

Given:

• The image must not to be rotated

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

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

Conclusions:

1. All P being C is a possibility → Follow (As all P are G, some G are B and all B are C. So, as no negative or cross relation between P and C is given, therefore it is definitely true)

2. Some P are B → Does not follow (As all P are G and some G are B. So, it is possible but not definitely true)

3. Some C are G → Follow (As some G are B and all B are C. So, as Some part of G are B which whole comes in C, therefore it is definitely true)

4. Some G are P → Follow (As all P are G. So, Some G are P is definitely true)

∴ Here, All conclusions follows, except conclusion - (2).​

Hence, the correct answer is "Only 2^nd does not follow"

Statements:

I. This year College X reduced the cut-off marks to 50 from 60 for the entrance exam.

Statement I is the effect of the college's decision to reduce the cut-off marks, which could be due to various reasons such as low enrollment or competition from other colleges.

II. In the last two years, the Students Union in College X has been actively negotiating with the college administration regarding the Quality of food and Health and Hygiene standards at the College Canteen

Statement II is the effect of the Students Union's efforts to improve the quality of food and hygiene standards at the college canteen, which could be due to concerns raised by students or external regulations.

Therefore, both statements are independent effects of different causes and are not directly related to each other.

Hence, Option 3 is correct because both statements are independent effects of different causes.

Given :

Given statement :- W % D # G @ B & M @ I

So,

• W % D → W is the wife of D.

• D # G → D is the brother of G.

• G @ B → G is the daughter of B.

• B & M → B is the husband of M.

​​

• M @ I → M is the daughter of I.

​

Thus, according to the final family tree D is the son of M.

Hence, "Son" is the correct answer.

Logic : In each figure '

' part of the figure is in the opposite direction to the other one. 

So,

• Option - (1) :

 

⇒In the figure'

' part of the figure is not in the opposite direction to the other one i.e they are in the same direction.

• Option - (2) :

 ⇒

In the figure '

' part of the family is in the opposite direction to the other one.

• Option - (3) :

⇒

⇒In the figure '

' part of the family is in the opposite direction to the other one.

• Option - (4) :

 ⇒

⇒In the figure '

' part of the family is in the opposite direction to the other one.

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

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

The logic followed here is :

(First number + Third Number ) + 5 = Second Number

(11, 33, 17) ⇒ (11 + 17) + 5 = 33

(18, 44, 21) ⇒ (18 + 21 ) + 5 = 44

Option (1): (12, 40, 18) ⇒ (12 + 18) + 5 = 35 ≠ 40

Option (2): (7, 32, 9) ⇒ (7 + 9) + 5 = 21 ≠ 32

Option (3): (16, 43, 22)  ⇒ (16 + 22) + 5 = 43

Option (4): (15, 34, 19) ⇒ (15 + 19) + 5 = 39 ≠ 34

Hence, the correct answer is "(16, 43, 22) "

Logic : In each figure both the circle which are half shaded are identical.

So,

• Option - (1) :

 

⇒ Both the circles are not identical.

• Option - (2) :

 

⇒ Both the circles are identical.

• Option - (3) :

⇒ Both the circles are identical.

• Option - (4) :

 ⇒ Both the circles are identical.

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

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

The logic followed here is :-

• Letters at the corners flipped horizontally.
• Letter at the left corner also shifts to the center within the triangle figure.
• Letters shifting - Left corner up letter shifted to left corner corner down and right corner letter shifted to left corner up.

So,

• Figures A and B ⇒

​

Similarly,

• Figure C and D ⇒

​

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

According to the English alphabet and their positional value:

Here is the logic followed:

CLOCK is written as 4 – 13 – 17 – 4 – 12.

Similarly,

 PHONE is written as _____________

Therefore, the correct answer is " 17 - 9 - 17 - 15 - 6 ".

We can combine the given equation to form:

M < K > I < J > H = G ≤ F

 

Now, let's evaluate the given conclusions:

I. H = F: Does not true [because we know that If we combine the equations H = G and G ≤ F, we get: H  ≤ F, G can be equal to or less than F, and H is not directly related to either G or F.

II. J > G: true. [If we combine the equations J > H = G, We know that J is greater than both H and G. Therefore, J must be greater than G.]

Hence, only conclusion II is true, and we can conclude that J is greater than G.

Hence the correct answer is "Option 3"

The pattern followed here is:

1. For '

' element :

• Each '

' element is shifting one and two places in the clockwise direction subsequently.

• Each '

' element is rotating 90º clockwise direction subsequently.

​Similarly, all arrows follow the same rules. 

Thus, final series is :

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

The given statement mentions that National Expressway-A is the widest expressway and is used by few people. When people find an empty and wide road, they tend to drive at high speed, which leads to accidents.

Now, let's evaluate the given inferences:

I. All accidents happen due to over speeding: This inference is not necessarily true. While it is mentioned in the statement that driving at high speed can lead to accidents, it doesn't mean that all accidents happen due to over speeding.

II. High speed increases the risk of accident: This inference is true. It is mentioned in the statement that driving at high speed on a wide and empty road leads to accidents, which implies that high speed increases the risk of an accident.

Hence, only inference II follows from the given statement.

Hence, the correct answer is "Option 3".

Here the correct answer is Tiny.

Key Points

• The most appropriate antonym of the underlined word "colossal" is "tiny."
• The word "colossal" means extremely large or enormous in size or extent.
• The antonym of "colossal" is something that is very small or tiny.
• ​For example, a tiny ant is the antonym of a colossal building.
• The word "tiny" means very small in size or amount.
• Here's an example sentence: "The tiny kitten was so small that it could fit in the palm of my hand."
• Thus, the correct answer is Option 4.

Additional Information

• Other options and their meanings:
• Enormous - This word is a synonym of "colossal" and means extremely large or huge. It is not an antonym.
• Example: The enormous elephant walked slowly across the savanna.
• Extraordinary - This word means exceptional, remarkable, or unusual. It is not an antonym of "colossal."
• Example: The extraordinary achievements of the scientist earned him a Nobel prize.
• Quaint - This word means charmingly old-fashioned, unusual, or picturesque. It is not an antonym of "colossal."
• Example: The quaint little village was popular among tourists.
• In summary, the most appropriate antonym of "colossal" is "tiny."

Here the correct answer is Sabotage.

Key Points

• The one-word substitute for the group of words "willful destruction" is "sabotage."
• The term "willful destruction" implies the act of intentionally causing damage or harm to someone or something.
• "Sabotage" means deliberate destruction or damage to property or equipment. (अंतर्ध्वंस, तोड़-फोड़)
• For example, cutting the brake line of a car to cause an accident or damaging the machinery of a factory to disrupt production can be called sabotage.
• Thus, the correct answer is Option 4.

Additional Information

• Other options and their meanings:
• Disaster - This word refers to a sudden and widespread event that causes great damage, destruction, or loss of life. It is not a one-word substitute for "willful destruction." (आपदा)
• Example: The earthquake caused a disaster in the region, leaving thousands of people homeless.
• Combustion - This word refers to a process of burning or a chemical reaction that produces heat and light. It is not a one-word substitute for "willful destruction." (ज्वलन, जलना)
• Example: The combustion engine is widely used in transportation.
• Calamity - This word refers to an event that causes great harm, suffering, or damage. It is not a one-word substitute for "willful destruction." (आपदा, विपत्ति)
• Example: The flood was a calamity for the farmers in the area, destroying their crops and homes.
• In summary, the correct one-word substitute for the group of words "willful destruction" is "sabotage."

Here the correct answer is ACBD.

Key Points

• The correct order of the sentences to form a meaningful and coherent paragraph is ACBD.
• The first sentence 'A' introduces the topic of the paragraph, which is the tiger. It provides general information about tigers and where they can be found.
• The third sentence 'C' gives us more specific information about the tiger population. It tells us that there are approximately 3,000 tigers in the wild.
• The second sentence 'B' describes the habitat of tigers and their behavior. It tells us that they mostly live in dark forests and come out at night for hunting.
• The fourth sentence 'D' provides additional information about the tiger's appearance and behavior. It tells us that tigers are difficult to see against a jungle background because of their striped coats. It also mentions that tigers like to bathe, which is an unusual behavior for cats.
• Therefore, the correct order of sentences to form a meaningful and coherent paragraph is ACBD.

Correct Sequence:

A. Tigers are the largest members of the cat family. They are found in many parts of Asia.

C. It is assumed that, probably, around 3,000 tigers live in the wild. This species looks spectacular and grand and they are very graceful in their movements.

B. They mostly live in dark forests and sleep during the day. They come out for hunting pigs and deer by night.

D. It is almost impossible for us to see them against a jungle background, especially when they are standing still, because of their striped coats. Unlike, most cats, the tiger likes bathing very much.

Here the correct answer is Face.

Key Points

• The correct answer is "face".
• "Face criticism" is a common collocation.
• "Face" in this context means to confront or deal with something difficult or unpleasant.
• Example: The CEO faced harsh criticism from shareholders over the company's poor financial performance.
• The other options don't fit well in this context:
• "Settle criticism" doesn't make sense as criticism is not a dispute that can be settled.
• "Meet criticism" could work, but it doesn't convey the idea of confronting the criticism head-on, which is what "face" implies.
• "Carry criticism" doesn't make sense in this context either.
• Thus, the correct answer is Option 3.

Correct Sentence: A good leader should be ready to face criticism.

Here the correct answer is Barely managed to escape.

Key Points

• The most appropriate meaning of the idiom "by the skin of his teeth" is "barely managed to escape."
• This idiom is used to describe a situation where someone has just managed to avoid a difficult or dangerous situation by a very narrow margin.
• In the given sentence, the youth was involved in an accident but managed to escape by a very close margin, which means that he just barely managed to avoid being harmed or injured.
• Example: John missed his flight by just a few minutes, but he managed to catch it by the skin of his teeth.
• Thus, the correct answer is Option 1.

Additional Information

• Option 1, which means "barely managed to escape," is therefore the correct answer.
• Option 2, which means "very arrogant," is not related to the given idiom and is incorrect.
• Option 3, which means "always taking calculated decisions," is also not related to the given idiom and is incorrect.
• Option 4, which means "deficiency of funds," is also not related to the given idiom and is incorrect.

Here the correct answer is Route.

Key Points

• The correct homonym is "route".
• Homonyms are words that have the same spelling or pronunciation but different meanings.
• In this case, "root" refers to the underground part of a plant, "rot" refers to the process of decay, and "riot" refers to a violent disturbance.
• "Route", on the other hand, refers to a path or direction taken to get from one place to another.
• Example: "The GPS app shows the best route to take to avoid traffic."
• Thus, the correct answer is Option 3.

Correct Sentence: Which route the minister take to reach the Assembly?

Additional Information

• Other examples of homonyms include "bear" (an animal) and "bear" (to tolerate), "bow" (a weapon) and "bow" (to bend forward), and "lead" (a metal) and "lead" (to guide).

Here the correct answer is S, P.

Key Points

• The correct answer is S, P, which means that sentence S should be placed as sentence 2 and sentence P should be placed as sentence 3.
• Sentence S "The men’s chairs were far enough away from the women’s couches" provides a detail about the seating arrangement, and thus makes sense as the second sentence, right after the sentence that talks about the initial mingling.
• Sentence P "So, the two parties had to lean forward, their elbows on their knees" talks about the seating arrangements, and is a logical follow-up to sentence S. This also helps to create a clear picture in the reader's mind about the seating arrangements in the room.
• Therefore, the correct sequence of sentences is:

(S1) The men and women took their places after the initial mingling, and immediately a surge of conversation filled the room.

(S2) The men’s chairs were far enough away from the women’s couches.

(S3) So, the two parties had to lean forward, their elbows on their knees.

(S4) One or two of the women were actually bouncing up and down on the sofa cushions.

The correct answer is: No error.

Key Points

• Let's see why the sentence is grammatically correct.
• Uses "although" to introduce a subordinate clause, creating a clear contrast.
• Maintains subject-verb agreement in both clauses.
• Employs "still" appropriately to highlight the contrast.
• Uses proper punctuation for clarity.
• Maintains logical coherence and stylistic fluency.
• Hence, no changes are required in this question.

Here the correct answer is Aloud.

Key Points

• The correct option to fill in the blank is "aloud" (Option 3).
• The word "aloud" means to speak or read something audibly or loudly.
• In the given context, Vishal is going to read the story in a way that it can be heard by others.
• Examples:
• She always reads her favorite book aloud to her younger sister.
• The teacher asked the students to read the passage aloud in the class.
• Option 1, "load," means to put a burden or weight on something. It is not relevant to the given context.
• Option 2, "allow," means to give permission for something. It does not fit the context of reading a story.
• Option 4, "allowed," is the past tense form of "allow," and is also not relevant to the given context.
• Thus, the correct answer is Option 3.

Correct Sentence: Vishal is going to read the story aloud.

Here the correct answer is My father give up.

Key Points

• The verb "give up" needs to be in the past tense, as the action has already happened. (the action happened when the speaker was twelve years old)
• The various verb forms of "give" are:
• Infinitive: to give
• Present tense: give(s)
• Past tense: gave
• Present participle: giving
• Past participle: given
• Therefore, it should be "gave up" instead of "give up".
• Thus, the correct answer is Option 1.

Correct Sentence: My father gave up smoking when I was twelve years old.

Here the correct answer is Option 4.

Key Points

• The incorrect spelled word is "Desedent" (option 4).
• Descendant - a person who is related to someone who lived in the past, such as an ancestor or a descendant.
• Example: John is a descendant of the royal family.
• Deceased - dead.
• Example: The deceased was buried in the cemetery yesterday.
• Decimate - to destroy a large part of something.
• Example: The plague decimated the population of the town.
• Desedent - The word is incorrectly spelled. The correct spelling is "descendent," which means descending or coming down from an ancestor; something that is descending or coming down.
• Example: The descendent line of the royal family can be traced back to the 14^th century.

Confusion Points

• Both "descendant" and "descendent" are correct spellings, but "descendant" is the more commonly used spelling in modern English.
• "Descendant" is used to refer to a person who is descended from a particular ancestor, while "descendent" is used to refer to something that is descending or coming down.
• Example sentences:
• My great-grandmother was a direct descendant of the Mayflower pilgrims.
• The bird's wings created a slow and graceful descendent flight.
• He was surprised to learn that he was a descendant of a famous historical figure.
• The elevator's descendent movement was interrupted by a sudden jolt.

Here the correct answer is Raghav asked Meera if she was upset with him.  

Key Points

• The original sentence is in direct speech, where Raghav directly asked Meera if she was upset with him.
• In reported speech, we need to change the sentence to reflect that it was said in the past and reported by someone else.
• To convert direct speech to reported speech, we need to follow certain rules:
• We need to change the tense of the verb in the reported clause. In this case, the present tense "are" in the direct speech changes to past tense "was" in the reported speech.
• We also need to change the pronouns and adjectives to reflect the speaker and listener in the reported speech.
• In this case, "me" in the direct speech refers to Raghav, so it changes to "him" in the reported speech.
• Based on these rules, option 2) "Raghav asked Meera if she was upset with him" is the correct reported speech for the given sentence.
• Option 1) Raghav asked Meera if she was upset with her is incorrect because it uses the wrong pronoun for Raghav.
• Option 3) Raghav asked Meera if she was upset with me is also incorrect because it uses the wrong pronoun for the reported speaker.
• Option 4) Raghav asked Meera why she was upset with him is also incorrect.
• Thus, the correct answer is Option 2.

The correct active voice for the given sentence is: Someone has completed the task.

Key Points

• The original sentence is in the passive voice, where the focus is on the task and not on who completed it.
• In the passive voice, the subject is often not specified, and the focus is on the action or the object of the action.
• To convert a passive voice sentence to an active voice sentence, we need to follow certain rules:
• Identify the subject and object in the passive sentence. In this case, the subject is not specified, and the object is "the task."
• Rewrite the sentence to make the subject the doer of the action and the object the receiver of the action.
• In this case, we need to add a subject (the person or group of people who completed the task) and make it the doer of the action.
• We also need to make the task the receiver of the action.
• Based on these rules, option 1) Someone has completed the task is the correct active voice for the given sentence.
• Option 2) Someone has been completing the task is incorrect because it implies ongoing or continuous action, rather than a completed action.
• Option 3) Someone will be completing the task is also incorrect because it refers to a future action, rather than a completed action.
• Option 4) Someone completes the task is not a correct option because it refers to the task's completion as a general statement and not a specific instance of the task being completed.

Here the correct answer is Books can be bought online from Amazon. 

Key Points

• The given sentence is in active voice. We need to change it into passive voice.
• 'can buy' becomes 'can be bought'.
• Thus, the correct answer is Option 3.

Correct Passive Voice: Books can be bought online from Amazon.

Here the correct answer is Less tragic than.

Key Points

• The correct option to substitute the underlined segment in the given sentence is "less tragic than".
• The original sentence is comparing the level of tragedy in "Edward II" with the level of tragedy in some other plays by the same dramatist.
• The phrase "tragic than" is used to convey this comparison.
• However, the phrase is grammatically incorrect because it is missing the comparative adjective that is required to complete the comparison.
• The correct comparative form of "tragic" is "more/less tragic than".
• Out of the given options, only option 4) "less tragic than" correctly substitutes the underlined segment and completes the comparison by providing the comparative adjective. The sentence now means that "Edward II" is not as tragic as some other plays by the same dramatist.
• Option 1) "as least tragic as" is incorrect.
• Option 2) "No substitution required" is incorrect because the underlined segment needs to be substituted in order to complete the comparison correctly.
• Option 3) "the most tragic" is incorrect because it is not providing the correct comparison. "The most tragic" would mean that "Edward II" is the most tragic play by the dramatist, which is not what the original sentence was trying to convey.
• Examples:
• This book is more interesting than the other one.
• My car is less expensive than your car.
• The blue dress is prettier than the green one.
• In each of these examples, the comparative form of the adjective is used to compare two things.
• In summary, the correct option to substitute the underlined segment in the given sentence is "less tragic than". It completes the comparison by providing the comparative adjective and makes the sentence grammatically correct.

Therefore, the corrected sentence would read as: "Edward II is less tragic than some other plays by the dramatist."

Here the correct answer is "d, a, b, c".

Key Points

• The correct order of the sentences is "dabc".
• Sentence d introduces the topic of power foods and explains that they are high in nutrients.
• Sentence a expands on this by mentioning that fitness trainers recommend these foods to build muscle.
• Sentence b then explains how power foods can be included in everyday diets, and sentence c concludes the paragraph by providing tips on how to cook power foods properly and get the most out of them.
• This sequence of sentences flows logically and coherently, making "dabc" the correct order.

Correct Sequence: 

d. Power foods are high in nutrients such as fibre, potassium and minerals

a. Many fitness trainers encourage their clients to include these foods in their diet to build muscle.

b. Power foods can be included in your everyday diet in a variety of ways.

c. The secret to getting the most out of power foods is, of course, knowing how to cook them properly, buying seasonal produce and recognising your preferred flavour profile.

Here the correct answer is Lenient.

Key Points

• The antonym of the word "stern" is "lenient".
• The word "stern" means **serious and unrelenting in nature (**कठोर, बहुत गंभीर), while "lenient" means **showing mercy or being easygoing. (**नरम, सौम्‍य)
• Examples of "stern" in a sentence:
• The teacher gave a stern warning to the students who were misbehaving in class.
• The coach had a stern demeanor when addressing the players after the loss.
• The judge's stern verdict sent a strong message to others who might commit the same crime.
• Examples of "lenient" in a sentence:
• The manager was lenient with the employee who made a mistake, giving her another chance to improve.
• The teacher was lenient with the deadline for the assignment, allowing students to turn it in a few days late.
• The judge was criticized for being too lenient in sentencing the convicted criminal.
• Thus, the correct answer is Option 2.

Additional Information

• The other options and their meanings are:
• Flippant: not taking things seriously, lacking proper respect or seriousness (छिछोरा, चंचल)
• Desolate: deserted, empty, or barren (निर्जन और अवसादपूर्ण)
• Noxious: harmful, poisonous, or offensive in smell or taste. (हानिकर या विषैला)

Here the correct answer is Kill.

Key Points

• The most appropriate option to fill in the blank is "kill".
• The phrase "to kill time" means to do something to make time pass more quickly or to make the time seem less boring.
• In the given sentence, the friends are waiting for a cab and playing cards to make the time pass more quickly.
• Example: I always bring a book to read in the waiting room to kill time before my appointment.
• Thus, the correct answer is Option 2.

Correct Sentence: The friends decided to play cards to kill time while waiting for the cab.

Additional Information

• The other options are:
• "Ignore" means to pay no attention to something or someone. This word is not appropriate in the given sentence.
• "Move" means to change one's position or location. This word is not appropriate in the given sentence.
• "Ease" means to make something less difficult, painful, or severe. This word is not appropriate in the given sentence.
• Examples:
• He ignored the warning signs and continued to drive in the storm.
• We need to move the couch to make room for the new TV.
• Taking painkillers can ease the pain of a headache.

The correct answer is BDCA.

Key Points

• The paragraph discusses the important role of trees in reducing the negative effects of climate change and improving air quality.
• B begins by discussing how trees help to clean the air by absorbing harmful pollutants and releasing clean oxygen.
• D continues the discussion of the benefits of trees in urban environments, where they absorb pollutant gases and particles.
• C then introduces the problem of increasing levels of carbon dioxide and the negative impact it has on the environment.
• Finally, A provides a solution to this problem by stating that healthy, strong trees act as carbon sinks, offsetting carbon and reducing the effects of climate change.
• Thus, the correct order of the sentences is BDCA.

Correct Sequence: 

B) Trees help to clean the air we breathe. Through their leaves and bark, they absorb harmful pollutants and release clean oxygen for us to breathe.

D) In urban environments, trees absorb pollutant gases like nitrogen oxides, ozone, and carbon monoxide, and sweep up particles like dust and smoke.

C) Increasing levels of carbon dioxide caused by deforestation and fossil fuel combustion trap heat in the atmosphere.

A) Healthy, strong trees act as carbon sinks, offsetting carbon and reducing the effects of climate change.

Here the correct answer is Shikha said that the heavy rain that week had spoiled all her travel plans.

Key Points

• The given sentence is in direct speech. We need to change it into indirect speech.
• Commas and inverted commas will be removed and 'that' will be added.
• Present perfect tense (has spoiled) changes to past perfect tense (had spoiled).
• 'this' changes to 'that'.
• The pronoun 'my' changes to 'her'.
• Thus, the correct answer is Option 1.

Correct Indirect Speech: Shikha said that the heavy rain that week had spoiled all her travel plans.

Here the correct answer is Narcissist.

Key Points

• Narcissist refers to a person who has an excessive interest in or admiration of themselves and their physical appearance.
• They have an exaggerated sense of self-importance and often seek attention and admiration from others.
• (इन लक्षणों वाले लोग अति आत्मविश्वास के साथ सामने आते हैं और यह शुरुआत में दूसरों को गुमराह कर सकते हैं)
• Example: She spends hours in front of the mirror, admiring her own beauty. She is such a narcissist.
• Thus, the correct answer is Option 1.

Additional Information

• Egoist refers to a person who is excessively concerned with oneself and one's own interests. They prioritize their own interests above others (अहंवादी).
• Example: He never helps anyone else. He is too much of an egoist to care about anyone else's problems.
• Selfish refers to a person who lacks consideration for others and is primarily concerned with their own interests and welfare (मतलबी).
• Example: She never shares anything with her friends. She is so selfish.
• Dictator refers to a person who has complete control and authority over a country, usually obtained and maintained by force (तानाशाह, अधिनायक, पूर्णशक्ति संपन्‍न शासक).
• Example: The dictator ruled the country with an iron fist, suppressing any opposition.

Here the correct answer is Option 4.

Key PointsThe incorrect word is "systemetic". The correct spelling is "systematic".

• "Systematic" means done or acting according to a fixed plan or system; methodical.
• ​Example: He made a systematic plan to clean his house.
• "Attempt" means to make an effort or try to do something.
• Example: I will attempt to climb this mountain.
• "Philosophy" means the study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline.
• Example: He studied philosophy at university.
• "Think" means to have a particular idea or opinion about something in one's mind.
• Example: I think we should go to the beach today.

The given sentence means that a part of the philosophy of life's meaning involves making a systematic attempt to understand what people have in mind when they think about the topic.

Thus, the incorrectly spelt word is Option 4 systemetic.