CUET 2026 General Aptitude Test May 24 Shift 1 Question Paper- Download Answer Key and Solution PDF

Concept:
The core concept involved in this question is the mathematical principle of ratios and proportions.
A ratio is a relationship between two or more quantities that indicates how many times the first number contains the second.
In business partnerships, total profits are usually distributed among partners based on their initial investment ratio or a pre-agreed sharing ratio.
When exact profit amounts are given, their relative ratio can be found by simplifying the respective amounts to their lowest possible terms.

Step 1: Analyzing the Given Data:
The problem provides the total profit earned by the three partners over a year, which is Rs. 12 lakh.
It also specifies the exact amount that each individual partner retained from this total profit pool.
Ankit retained a total of Rs. 6 lakh as his specific share of the annual profit.
Ashish retained a total of Rs. 3 lakh as his designated share of the profit.
Arun also retained a total of Rs. 3 lakh as his portion of the profit.
The sum of these shares (\(6 + 3 + 3\)) equals the total profit of 12 lakh, which confirms the data is consistent.

Step 2: Formulating the Ratio:
To find the respective ratio of their profits, we must place their individual profit shares side by side in the exact order of their names as presented in the question.
The ratio of profits for Ankit : Ashish : Arun will simply correspond to the actual amounts they retained.
Therefore, the unsimplified ratio is 6 lakh : 3 lakh : 3 lakh.
Since the unit "lakh" is common across all three terms, we can safely eliminate it for the calculation.
This gives us the numerical sequence of \( 6 : 3 : 3 \).

Step 3: Simplifying the Ratio:
To simplify a ratio to its lowest terms, we need to find the greatest common divisor (GCD) or highest common factor (HCF) for all the numbers involved.
The numbers in our unsimplified ratio are 6, 3, and 3.
The greatest common divisor for the numbers 6, 3, and 3 is exactly 3.
By dividing each individual term of the ratio by 3, we arrive at the simplified components.
Ankit’s simplified ratio component is \( \frac{6}{3} = 2 \).
Ashish’s simplified ratio component is \( \frac{3}{3} = 1 \).
Arun’s simplified ratio component is \( \frac{3}{3} = 1 \).
Thus, the final simplified ratio of their profits is \( 2 : 1 : 1 \).

Final Answer:
Therefore, the correct answer is (B) 2:1:1.

Concept:
The fundamental concepts required here belong to commercial mathematics, specifically involving Cost Price, Selling Price, and Profit Percentage.
Cost Price (CP) is the initial price at which an article is manufactured or purchased by the seller.
Selling Price (SP) is the final price at which the article is sold to the customer.
Profit is earned when the Selling Price is strictly greater than the Cost Price, and it is calculated as \( \text{Profit} = \text{SP} - \text{CP} \).
Profit Percentage is always calculated over the base Cost Price using the formula \( \text{Profit \%} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \).

Step 1: Finding the Cost Price (CP):
We are given the initial Selling Price (SP) as Rs. 2000 and the initial Profit Percentage as 25%.
We can establish the relationship between CP, SP, and Profit % using the standard mathematical formula.
\[ \text{SP} = \text{CP} \times \left(1 + \frac{\text{Profit \%}}{100}\right) \]
Substituting the given numerical values into our formula gives:
\[ 2000 = \text{CP} \times \left(1 + \frac{25}{100}\right) \]
\[ 2000 = \text{CP} \times \left(1 + 0.25\right) \]
\[ 2000 = \text{CP} \times 1.25 \]
To isolate and find the CP, we divide the SP by 1.25:
\[ \text{CP} = \frac{2000}{1.25} = 1600 \]
Thus, the base Cost Price of the article is definitely Rs. 1600.

Step 2: Calculating Profit under New Conditions:
The question asks for the new Profit Percentage if the article were instead sold for Rs. 1800.
Here, our New Selling Price (New SP) is Rs. 1800.
The Cost Price remains unchanged at Rs. 1600.
First, we find the new absolute Profit amount.
\[ \text{New Profit} = \text{New SP} - \text{CP} \]
\[ \text{New Profit} = 1800 - 1600 = 200 \]
The new absolute profit earned is Rs. 200.

Step 3: Calculating New Profit Percentage:
Now, we apply the Profit Percentage formula using the new profit amount and the original Cost Price.
\[ \text{New Profit \%} = \left( \frac{\text{New Profit}}{\text{CP}} \right) \times 100 \]
\[ \text{New Profit \%} = \left( \frac{200}{1600} \right) \times 100 \]
Simplifying the fraction inside the parentheses by cancelling out the zeros:
\[ \text{New Profit \%} = \left( \frac{2}{16} \right) \times 100 \]
\[ \text{New Profit \%} = \left( \frac{1}{8} \right) \times 100 = 12.5\% \]

Final Answer:
Therefore, the correct answer is (C) 12.5%.

Concept:
This question relies on the mathematical principles governing Compound Interest (CI).
Unlike Simple Interest, where interest is generated only on the principal, Compound Interest calculates interest on the principal as well as the accumulated interest of previous periods.
The fundamental formula determining the final Amount (A) under annual compound interest is \( A = P \left(1 + \frac{R}{100}\right)^n \).
In this specific formula, \(P\) stands for the Principal amount initially invested or borrowed.
\(R\) represents the annual Rate of interest expressed as a percentage.
\(n\) represents the total number of time periods (usually years) for which the sum is compounded.

Step 1: Identifying Given Variables:
From the problem description, we can clearly extract the given values to substitute into our formula.
The Principal sum (\(P\)) is given as Rs. 1800.
The final compounded Amount (\(A\)) is given as Rs. 2178.
The time period (\(n\)) is stated as exactly 2 years.
The Rate of interest (\(R\)) is the unknown variable we need to mathematically deduce.

Step 2: Applying the Formula and Simplifying:
We substitute the identified variables into the standard Compound Interest formula.
\[ 2178 = 1800 \left(1 + \frac{R}{100}\right)^2 \]
To isolate the variable part, we divide both sides of the equation by 1800.
\[ \frac{2178}{1800} = \left(1 + \frac{R}{100}\right)^2 \]
We must now simplify the fraction on the left side to its lowest terms to make the calculation manageable.
Dividing the numerator and the denominator by their common factor of 18, we get:
\[ \frac{2178 \div 18}{1800 \div 18} = \frac{121}{100} \]
This simplifies our main equation beautifully to:
\[ \frac{121}{100} = \left(1 + \frac{R}{100}\right)^2 \]

Step 3: Solving for the Interest Rate (R):
We recognize that the fraction \(\frac{121}{100}\) is a perfect square, specifically the square of \(\frac{11}{10}\).
Taking the square root on both sides of the equation:
\[ \sqrt{\frac{121}{100}} = 1 + \frac{R}{100} \]
\[ \frac{11}{10} = 1 + \frac{R}{100} \]
Subtracting 1 from both sides to isolate the term with R:
\[ \frac{R}{100} = \frac{11}{10} - 1 \]
\[ \frac{R}{100} = \frac{11 - 10}{10} = \frac{1}{10} \]
Finally, we solve for R by multiplying both sides by 100:
\[ R = \frac{100}{10} = 10 \]
The rate of compound interest per annum is therefore exactly 10%.

Final Answer:
Therefore, the correct answer is (D) 10%.

Concept:
This question rigorously tests the foundational knowledge of two-dimensional geometry, specifically focusing on the mathematical properties of a circle.
A circle is defined as a 2D shape consisting of all points in a plane that are at a given distance (called the radius) from a given center point.
Because a circle is strictly a two-dimensional geometric figure, it solely possesses properties related to length (circumference) and surface space (area).
Three-dimensional properties like volume simply do not exist for a two-dimensional shape like a circle.

Step 1: Evaluating Statement (A):
Statement (A) claims that the Volume of a circle is \( \frac{2}{3}\pi r^2 \).
As discussed in the underlying concept, a circle is flat and two-dimensional, meaning it has zero thickness or depth.
Consequently, calculating the "volume" of a circle is mathematically impossible and incorrect; volume only applies to 3D shapes like spheres or cylinders.
Therefore, statement (A) is completely incorrect.

Step 2: Evaluating Statement (B):
Statement (B) states that the Circumference of a circle is \( 2\pi r \).
The circumference is the total linear distance around the outside boundary of the circle.
The universally established mathematical formula for the perimeter or circumference of a circle is indeed \( 2\pi r \), where \(r\) is the radius.
Therefore, statement (B) is perfectly correct.

Step 3: Evaluating Statement (C):
Statement (C) claims that the Area of a circle is \( \pi r^2 \).
The area represents the total two-dimensional space enclosed within the boundary of the circle.
The standard geometric formula for the area of a full circle is universally accepted as \( \pi r^2 \).
Therefore, statement (C) is strictly correct.

Step 4: Evaluating Statement (D):
Statement (D) defines the Area of a sector as \( \frac{\pi r^2 \theta}{360} \).
A sector is a specific portion of a circle enclosed by two radii and an arc, determined by a central angle \(\theta\).
Since a full circle has a central angle of \(360^\circ\) and an area of \(\pi r^2\), the area of a fractional sector is proportionally \( \frac{\theta}{360} \times \pi r^2 \).
This precisely matches the given formula in the statement.
Therefore, statement (D) is absolutely correct.

Step 5: Synthesizing the Final Conclusion:
Upon detailed review, we have logically deduced that statements (B), (C), and (D) are geometrically factual and correct.
Statement (A) is flawed due to confusing 2D shapes with 3D properties.
Looking at the provided options, the combination representing this conclusion is Option (B) which includes (B), (C), and (D) only.

Final Answer:
Therefore, the correct answer is (B) (B), (C) and (D) only.

Concept:
This matching question is rooted deeply in the foundational concepts of Cartesian coordinate geometry.
The Cartesian coordinate system uses a two-dimensional plane consisting of a horizontal axis (x-axis) and a vertical axis (y-axis).
Every specific point in this plane is definitively identified by an ordered pair \((x, y)\) representing its spatial coordinates.
Understanding the specific properties of the origin, the axes, and fundamental distance principles is crucial to solve this accurately.

Step 1: Analyzing (A) Coordinates of origin:
The origin is mathematically defined as the central point where the x-axis and the y-axis physically intersect each other.
Because it is exactly at the starting zero point for both dimensions, its coordinates are always \((0, 0)\).
Looking at List-II, this matches perfectly with option (III).
Therefore, (A) must pair with (III).

Step 2: Analyzing (B) Distance of a point from itself:
The mathematical distance between any two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) is found using the formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
If we are measuring the distance from a point to itself, both sets of coordinates are completely identical (\(x_1 = x_2\) and \(y_1 = y_2\)).
Plugging this into the formula gives \(\sqrt{0^2 + 0^2}\), which ultimately equals exactly 0.
Looking at List-II, this matches perfectly with option (IV).
Therefore, (B) must pair with (IV).

Step 3: Analyzing (C) Any point on the y axis:
The y-axis is the vertical line that runs exactly through the zero mark of the horizontal x-axis.
Consequently, every single point located anywhere on the y-axis has an x-coordinate that is strictly zero.
Thus, the general coordinate format for such a point is always written as \((0, y)\), where y represents any real number.
Looking at List-II, this matches perfectly with option (I).
Therefore, (C) must pair with (I).

Step 4: Analyzing (D) Any point on the x axis:
Conversely, the x-axis is the horizontal line that runs exactly through the zero mark of the vertical y-axis.
This inherently means that every single point located on the x-axis has a vertical y-coordinate that is strictly zero.
Thus, the standard coordinate format for a point on the x-axis is \((x, 0)\), where x is any valid real number.
Looking at List-II, this matches perfectly with option (II).
Therefore, (D) must pair with (II).

Step 5: Finalizing the Matches:
Reviewing our logical deductions, the correct matching sequence is undeniably: (A)-(III), (B)-(IV), (C)-(I), and (D)-(II).
Scanning the given multiple-choice options, Option (C) reflects this exact specific combination.

Final Answer:
Therefore, the correct answer is (C) (A) - (III), (B) - (IV), (C) - (I), (D) - (II).

Concept:
The central concept in this question is the statistical measure known as the Arithmetic Mean, commonly referred to as the average.
The average provides a single central value that summarizes a larger set of varying numerical data.
In mathematics, the average is universally calculated by taking the total combined sum of all given observations and dividing it strictly by the total number of individual observations.
The fundamental formula used is: \( \text{Average} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} \).

Step 1: Identifying the Given Information:
We must carefully extract the relevant data points provided in the question text.
The problem explicitly states that there is a class containing a total of 50 students.
Therefore, the Total number of observations (which is the total number of students) equals 50.
The problem also explicitly provides the total aggregate sum of the marks obtained by all these 50 students combined.
Therefore, the Sum of all observations (total marks) is precisely 2250.

Step 2: Applying the Average Formula:
We substitute the identified numerical values directly into our standard average formula.
\[ \text{Average Marks} = \frac{\text{Total sum of marks}}{\text{Total number of students}} \]
\[ \text{Average Marks} = \frac{2250}{50} \]

Step 3: Calculating the Final Value:
To make the division simpler, we can first cancel out the common zero at the end of both numbers.
\[ \text{Average Marks} = \frac{225}{5} \]
Now, we perform simple division: 225 divided by 5.
Since \( 5 \times 40 = 200 \) and \( 5 \times 5 = 25 \), then \( 200 + 25 = 225 \).
Thus, \( 225 \div 5 = 45 \).
The average marks of a student in the class is 45.

Final Answer:
Therefore, the correct answer is (B) 45.

Concept:
This question tests the foundational concepts of classical probability.
Probability is a mathematical measure quantifying the likelihood that a specific event will occur.
The theoretical probability of a single event occurring is calculated using a standard ratio.
The formula is: \( \text{Probability of an Event } P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \).
This formula assumes that all possible outcomes in the sample space are equally likely to happen.

Step 1: Determining the Total Number of Possible Outcomes:
A standard, fair die is a solid cube, meaning it has exactly six distinct flat faces.
Each face is marked with a different number of dots, representing the integers from 1 to 6.
When this die is thrown or rolled, it can land with any one of these six faces pointing upwards.
Therefore, the entire sample space of possible outcomes is \( \{1, 2, 3, 4, 5, 6\} \).
This means the Total Number of Possible Outcomes is exactly 6.

Step 2: Determining the Number of Favorable Outcomes:
The event we are interested in is specifically "getting a 6" on the top face of the die.
Looking at our sample space \( \{1, 2, 3, 4, 5, 6\} \), the number 6 appears exactly one time.
Therefore, there is only one specific outcome that satisfies our condition.
This means the Number of Favorable Outcomes is exactly 1.

Step 3: Calculating the Final Probability:
Now, we substitute these determined values into our classical probability formula.
\[ P(\text{getting a 6}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
\[ P(\text{getting a 6}) = \frac{1}{6} \]
The calculated probability is 1/6, which cannot be simplified any further.

Final Answer:
Therefore, the correct answer is (D) 1/6.

Concept:
This problem is a classic example from the field of combinatorics, specifically dealing with Combinations.
Combinations are used when we need to find the number of ways to select a subset of items from a larger set, where the order of selection absolutely does not matter.
Since a cricket team is just a group of players, selecting player A then player B is the same as selecting player B then player A.
The standard mathematical formula for combinations is denoted as \( ^nC_r \), which represents "n choose r".
The formula is \( ^nC_r = \frac{n!}{r! \times (n-r)!} \), where ’n’ is the total items and ’r’ is the number of items to choose.

Step 1: Identifying the Variables:
We need to carefully extract the values for ’n’ and ’r’ from the problem statement.
The total number of available cricket players from which we can choose is 14.
Therefore, our total set size \( n = 14 \).
The required size of the final cricket team we need to form is 11 players.
Therefore, the number of players to be selected is \( r = 11 \).

Step 2: Applying the Combination Formula:
We substitute \( n = 14 \) and \( r = 11 \) into the combination formula.
\[ ^{14}C_{11} = \frac{14!}{11! \times (14-11)!} \]
\[ ^{14}C_{11} = \frac{14!}{11! \times 3!} \]

Step 3: Utilizing Combination Properties for Simplification:
To make the calculation much easier, we can use the fundamental property of combinations: \( ^nC_r = ^nC_{n-r} \).
This property implies that choosing 11 players to be ON the team is mathematically identical to choosing 3 players to be LEFT OUT of the team (\(14 - 11 = 3\)).
So, \( ^{14}C_{11} \) is exactly equal to \( ^{14}C_3 \).
\[ ^{14}C_3 = \frac{14 \times 13 \times 12 \times 11!}{3! \times 11!} \]
The \( 11! \) terms in the numerator and denominator cancel each other out completely.
\[ ^{14}C_3 = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} \]

Step 4: Final Calculation:
Now we simplify the remaining fraction.
The denominator is \( 3 \times 2 \times 1 = 6 \).
We can divide the 12 in the numerator by the 6 in the denominator, which gives us exactly 2.
The expression simplifies down to:
\[ 14 \times 13 \times 2 \]
First, multiply 13 by 2 to get 26.
Next, multiply 14 by 26.
\( 14 \times 26 = 14 \times (20 + 6) = 280 + 84 = 364 \).
Therefore, there are 364 distinct ways to choose the cricket team.

Final Answer:
Therefore, the correct answer is (A) 364.

Concept:
This problem is fundamentally based on solving linear equations involving two variables.
A linear equation in two variables, such as x and y, represents a straight line when graphed on a coordinate plane.
When the specific numerical value of one of the variables is explicitly provided, we can easily find the value of the other unknown variable.
This is achieved by using the method of substitution, where we replace the known variable in the main equation with its given numerical value and then isolate the remaining unknown variable.

Step 1: Identifying the Given Equations:
We are provided with a primary linear equation that establishes a relationship between variables x and y.
The primary equation is: \( 2x + 3y = 18 \).
We are also directly provided with the exact numerical value for one of the variables.
The given value for the variable y is: \( y = 2 \).

Step 2: Substituting the Known Value:
Our objective is to find the corresponding value for the variable x.
To do this, we substitute the known value of \( y \) into our primary linear equation.
Wherever there is a ’y’ in the equation, we will replace it with the number 2.
Substituting \( y = 2 \) into \( 2x + 3y = 18 \):
\[ 2x + 3(2) = 18 \]

Step 3: Solving the Algebraic Equation for x:
Now, we must perform basic arithmetic operations to simplify and solve the equation for x.
First, multiply the constant term: \( 3 \times 2 = 6 \).
The equation now becomes:
\[ 2x + 6 = 18 \]
Next, we need to isolate the term containing x. We do this by subtracting 6 from both sides of the equation.
\[ 2x = 18 - 6 \]
\[ 2x = 12 \]
Finally, to find the single value of x, we divide both sides of the equation by 2.
\[ x = \frac{12}{2} \]
\[ x = 6 \]
The exact value of x that satisfies these conditions is 6.

Final Answer:
Therefore, the correct answer is (A) 6.

Concept:
This question involves the practical application of basic Trigonometry, specifically dealing with Heights and Distances.
When sunlight hits a vertical object like a tower, it casts a shadow on the horizontal ground, forming a right-angled triangle.
The tower represents the perpendicular height (opposite side), the shadow represents the base (adjacent side), and the line connecting the tip of the shadow to the top of the tower represents the hypotenuse.
The angle of elevation of the sun is the angle formed between the ground and the line of sight pointing towards the sun.
The trigonometric ratio tangent (\(\tan\)) is perfectly suited here, as it defines the relationship between the opposite side and the adjacent side: \( \tan(\theta) = \frac{\text{Perpendicular}}{\text{Base}} \).

Step 1: Setting up the Variables:
Let the angle of elevation of the sun, which we need to find, be represented by the Greek letter \(\theta\).
Let the vertical height of the tower be denoted by the variable \(h\).
According to the explicit conditions given in the problem, the length of the shadow is exactly \( \sqrt{3} \) times the height of the tower.
Therefore, we can express the length of the shadow mathematically as \( \sqrt{3} \times h \), or simply \( h\sqrt{3} \).

Step 2: Applying the Trigonometric Ratio:
We envision a right-angled triangle where the tower is the perpendicular and the shadow is the base.
The angle \(\theta\) is situated on the ground at the far tip of the shadow.
We use the tangent formula:
\[ \tan(\theta) = \frac{\text{Height of Tower (Perpendicular)}}{\text{Length of Shadow (Base)}} \]
Substitute the variable expressions we defined in Step 1 into this formula:
\[ \tan(\theta) = \frac{h}{h\sqrt{3}} \]

Step 3: Simplifying and Solving for the Angle:
Since the variable \(h\) represents height, it is a non-zero positive number.
Therefore, we can safely cancel out \(h\) from both the numerator and the denominator.
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]
Now, we must recall the standard trigonometric values for common angles.
We know from standard trigonometric tables that the tangent of \(30^\circ\) is exactly \( \frac{1}{\sqrt{3}} \).
Therefore, the angle \(\theta\) must be equal to \(30^\circ\).
The angle of elevation of the sun is \(30^\circ\).

Final Answer:
Therefore, the correct answer is (D) \(30^\circ\).

Concept:
This question assesses knowledge of solid geometry, specifically regarding the surface area of a right circular cylinder.
A cylinder is a 3D geometric shape with two identical flat circular ends (bases) connected by a curved surface.
The Total Surface Area (TSA) of a solid cylinder is the complete measure of all the exposed surfaces combined.
It includes the area of the curved middle part (Curved Surface Area = \( 2\pi rh \)) plus the area of the two flat circular bases (Top base = \( \pi r^2 \), Bottom base = \( \pi r^2 \)).
Adding these together gives the comprehensive standard formula: \( \text{TSA} = 2\pi rh + 2\pi r^2 \), which is more commonly factored and written as \( \text{TSA} = 2\pi r(r + h) \).

Step 1: Identifying the Given Parameters:
The problem clearly provides the necessary dimensions required to calculate the surface area.
The radius of the circular base of the cylinder (\(r\)) is given as exactly 7 cm.
The vertical height of the cylinder (\(h\)) is given as exactly 11 cm.
We will use the standard mathematical approximation for Pi (\(\pi\)) which is \( \frac{22}{7} \), as this value typically allows for easy cancellation of numbers.

Step 2: Applying the Formula for Total Surface Area:
We substitute our known values into the factored formula for the Total Surface Area of a cylinder.
\[ \text{TSA} = 2\pi r(r + h) \]
Substituting \( r = 7 \), \( h = 11 \), and \( \pi = \frac{22}{7} \):
\[ \text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 11) \]

Step 3: Executing the Mathematical Calculation:
First, evaluate the expression inside the parentheses: \( 7 + 11 = 18 \).
Next, look at the fraction part. We can cancel out the 7 in the numerator with the 7 in the denominator.
This simplifies our main equation considerably:
\[ \text{TSA} = 2 \times 22 \times 18 \]
Now, perform the multiplication step by step.
First, multiply 2 by 22, which gives us 44.
Now multiply 44 by 18:
\[ \text{TSA} = 44 \times 18 \]
To calculate this mentally or on paper: \( 44 \times 10 = 440 \), and \( 44 \times 8 = 352 \).
Add them together: \( 440 + 352 = 792 \).
The Total Surface Area is precisely \( 792 \text{ cm}^2 \).

Final Answer:
Therefore, the correct answer is (B) \(792 \text{ cm}^2\).

Concept:
This is a standard problem based on the concept of Time and Work.
The core principle in such problems is dealing with the rate of work, which is the fraction of the total work completed in a single unit of time (like one day).
If a person can complete a whole task in ’n’ days, then their rate of work or "one day’s work" is exactly \( \frac{1}{n} \) of the task.
When two or more individuals work simultaneously, their combined one day’s work is simply the sum of their individual one day’s work rates.
The total time taken to finish the task together is the reciprocal of their combined daily work rate.

Step 1: Calculating Individual Work Rates:
We determine the amount of work each person can accomplish in just one day.
The problem states Neha can complete the total work in 6 days.
Therefore, Neha’s 1 day’s work rate is \( \frac{1}{6} \).
The problem states Ishrat can complete the exact same work in 4 days.
Therefore, Ishrat’s 1 day’s work rate is \( \frac{1}{4} \).

Step 2: Calculating Combined Work Rate:
To find out how much work they accomplish together in a single day, we must add their individual rates.
Combined 1 day’s work = Neha’s rate + Ishrat’s rate
Combined 1 day’s work = \( \frac{1}{6} + \frac{1}{4} \)
To add these two fractions, we need to find their Least Common Multiple (LCM).
The LCM of the denominators 6 and 4 is precisely 12.
Convert both fractions to have 12 as the common denominator:
\[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \]
\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]
Now, add the converted fractions:
Combined 1 day’s work = \( \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \)
This indicates they complete \( \frac{5}{12} \) of the total task in a single day.

Step 3: Calculating Total Time Taken Together:
The total time taken to finish the work together is simply the reciprocal of their combined 1 day’s work.
Total time = \( \frac{1}{\text{Combined 1 day's work}} \)
Total time = \( \frac{1}{5/12} = \frac{12}{5} \text{ days} \)
To match the formatting of the given options, we must convert this improper fraction into a mixed number.
Dividing 12 by 5 gives a quotient of 2 and a remainder of 2.
Therefore, \( \frac{12}{5} = 2\frac{2}{5} \text{ days} \).

Final Answer:
Therefore, the correct answer is (C) \(2\frac{2}{5} \text{ days}\).

Concept:
The fundamental relationship between Speed, Distance, and Time is a core concept in basic kinematics and practical mathematics.
Speed is defined as the rate at which a specific object covers distance over a given period of time.
The primary mathematical formula linking these three variables together is: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
Assuming the person’s speed remains perfectly constant throughout both parts of the journey, time becomes directly proportional to the distance traveled.

Step 1: Calculating the Initial Speed:
First, we must determine the constant speed of the person using the initial data provided in the question.
The problem explicitly states that the person covers an initial distance of 72 km in a time span of exactly 3 hours.
Using our standard formula, we can calculate the constant speed of this person.
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
\[ \text{Speed} = \frac{72 \text{ km}}{3 \text{ hours}} \]
Performing the division gives us:
\[ \text{Speed} = 24 \text{ km/h} \]
The person is traveling at a constant rate of 24 kilometers per hour.

Step 2: Calculating the Required Time for the New Distance:
We now need to figure out how much time it will take this person to cover a completely new distance of 288 km, assuming they maintain the same constant speed of 24 km/h.
By rearranging our primary formula algebraically to solve for Time, we get:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Now, we substitute our new target distance and our previously calculated constant speed into this rearranged formula.
\[ \text{Time} = \frac{288 \text{ km}}{24 \text{ km/h}} \]

Step 3: Final Mathematical Calculation:
We need to divide 288 by 24.
Let’s simplify the fraction by dividing both numerator and denominator by 12 first.
\( 288 \div 12 = 24 \), and \( 24 \div 12 = 2 \).
So the fraction simplifies to \( \frac{24}{2} \).
Finally, \( 24 \div 2 = 12 \).
Therefore, it will undoubtedly take the person exactly 12 hours to complete the new journey.

Final Answer:
Therefore, the correct answer is (C) 12 hours.

Concept:
This question tests your theoretical understanding of the classification of real numbers, specifically distinguishing between rational and irrational numbers.
A rational number is any number that can be expressed as a simple fraction \( \frac{p}{q} \), where ’p’ and ’q’ are integers and ’q’ is not equal to zero. They have terminating or repeating decimal expansions.
An irrational number is a real number that cannot possibly be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating.
Common examples of irrational numbers include \( \pi \), Euler’s number ’e’, and the square roots of non-perfect squares.

Step 1: Evaluating Statement (A):
Statement (A) claims: "The sum of two rational numbers is always irrational."
Let’s test this with an example. Take the rational numbers 2 and 3.
Their sum is \( 2 + 3 = 5 \). The number 5 is clearly a rational number because it can be written as \( \frac{5}{1} \).
In mathematics, the sum of any two rational numbers is mathematically guaranteed to always be a rational number, never irrational.
Therefore, statement (A) is completely incorrect.

Step 2: Evaluating Statement (B):
Statement (B) claims: "\( \pi \) is an irrational number."
The mathematical constant \( \pi \) represents the ratio of a circle’s circumference to its diameter.
Its decimal representation begins as 3.14159265... and continues infinitely without ever falling into a repeating pattern.
Because it cannot be represented as an exact simple fraction, it perfectly fits the definition of an irrational number.
Therefore, statement (B) is absolutely correct.

Step 3: Evaluating Statement (C):
Statement (C) claims: "The value of \( \sqrt{49} \) is an irrational number."
Let’s evaluate the mathematical expression \( \sqrt{49} \).
The number 49 is a perfect square, as it is exactly equal to \( 7 \times 7 \).
Therefore, \( \sqrt{49} = 7 \).
The number 7 can be written as a fraction \( \frac{7}{1} \), making it a rational number, not irrational.
Therefore, statement (C) is strictly incorrect.

Step 4: Evaluating Statement (D):
Statement (D) claims: "The value of \( \sqrt{5} \) is an irrational number."
The number 5 is not a perfect square, meaning its square root cannot be simplified into an integer or a simple fraction.
The decimal expansion of \( \sqrt{5} \) is non-terminating and non-repeating (2.23606...).
The square root of any prime number, or any non-perfect square, is fundamentally irrational.
Therefore, statement (D) is perfectly correct.

Step 5: Synthesizing the Final Conclusion:
Based on our logical evaluation, only statements (B) and (D) are mathematically correct facts about irrational numbers.
Looking at the provided multiple-choice options, Option (D) corresponds to exactly "(B) and (D) only".

Final Answer:
Therefore, the correct answer is (D) (B) and (D) only.

Concept:
This question requires an understanding of basic numerical sequences, specifically an Arithmetic Progression (AP).
An Arithmetic Progression is a sequence of numbers in which the consecutive terms follow a very specific rule: the difference between any two successive terms is always a constant value.
This constant value is officially termed the "common difference" and is usually denoted by the mathematical variable ’d’.
To find the common difference ’d’, one must simply subtract a preceding term from its immediately succeeding term.
The formula is: \( d = a_{n} - a_{n-1} \), where \(a_n\) is any term and \(a_{n-1}\) is the term right before it.

Step 1: Identifying the Terms of the Sequence:
The problem explicitly provides a sequence of numbers forming an Arithmetic Progression.
Let’s clearly identify the first few terms of this AP for our calculation.
The first term, denoted as \( a_1 \), is \( \frac{3}{2} \).
The second term, denoted as \( a_2 \), is \( \frac{1}{2} \).
The third term, denoted as \( a_3 \), is \( -\frac{1}{2} \).
The sequence continues, but we only need two consecutive terms to find the difference.

Step 2: Calculating the Common Difference (d):
We will apply the formula by subtracting the first term (\(a_1\)) from the second term (\(a_2\)).
\[ d = a_2 - a_1 \]
Substitute the numerical values of the identified terms into the equation:
\[ d = \frac{1}{2} - \frac{3}{2} \]

Step 3: Performing the Subtraction:
Since both fractions already have exactly the same denominator (which is 2), we can directly subtract their numerators while keeping the common denominator.
\[ d = \frac{1 - 3}{2} \]
Calculate the result in the numerator: \( 1 - 3 = -2 \).
\[ d = \frac{-2}{2} \]
Finally, simplify the fraction by performing the division:
\[ d = -1 \]
To verify, let’s check the difference between the third and second terms: \( a_3 - a_2 = -\frac{1}{2} - \frac{1}{2} = \frac{-1-1}{2} = \frac{-2}{2} = -1 \).
The common difference is consistently -1.

Final Answer:
Therefore, the correct answer is (A) -1.

Concept:
This question requires fundamental knowledge of Ancient Indian History and the ability to arrange major historical events on a chronological timeline.
Chronological order means arranging events according to the exact sequence in time in which they occurred, starting from the oldest (earliest) event and moving towards the most recent ones.
When dealing with dates in "BC" or "BCE" (Before Christ / Before Common Era), the numbers count backwards. Therefore, a larger number in BC represents an older, earlier event in history.
Dates in "AD" or "CE" (Anno Domini / Common Era) count forwards normally.

Step 1: Analyzing the Historical Events and their Timelines:
We need to assign rough historical dates or time periods to each of the four listed events based on historical facts.
(A) Maurya Empire Founded: Chandragupta Maurya founded this massive empire in ancient India around 322 BC, shortly after the power vacuum left by Alexander’s departure.
(B) Birth of Gautam Buddha: Siddharth Gautam, who later became known as the Buddha, was born much earlier, historically accepted to be around 563 BC (in the 6th Century BCE).
(C) Invasion by Alexander: Alexander the Great, the Macedonian king, crossed the Indus River and invaded the northwestern part of the Indian subcontinent around 327 - 326 BC.
(D) Gupta Empire Founded: The Gupta Empire, often referred to as the Golden Age of India, was founded by Sri Gupta, but came to prominence under Chandragupta I much later, around 319 AD (or CE).

Step 2: Arranging in Chronological Order:
Now we sort these dates from oldest to most recent. Remember that larger BC numbers are older.
1st (Earliest): 563 BC \(\rightarrow\) Birth of Gautam Buddha (Event B)
2nd: 326 BC \(\rightarrow\) Invasion by Alexander (Event C)
3rd: 322 BC \(\rightarrow\) Maurya Empire Founded (Event A)
4th (Most Recent): 319 AD \(\rightarrow\) Gupta Empire Founded (Event D)
Therefore, the correct chronological sequence of these historical events is precisely: (B) \(\rightarrow\) (C) \(\rightarrow\) (A) \(\rightarrow\) (D).

Step 3: Matching with Options:
We look for the option that perfectly matches our deduced sequence (B), (C), (A), (D).
Option (D) exactly reflects this historically correct sequence.

Final Answer:
Therefore, the correct answer is (D) (B), (C), (A), (D).

Concept:
This question explores the cultural heritage of India, specifically focusing on the classical dance forms recognized by the Sangeet Natak Akademi.
Classical Indian dances are deeply rooted in the Natya Shastra, an ancient Sanskrit text on the performing arts.
Different dance forms have unique etymologies (origins of their names) that often describe their core characteristics or geographic origins.
Understanding the breakdown of the name of the dance can directly lead to the correct answer.

Step 1: Analyzing the Dance Forms and their Meanings:
Let’s evaluate the etymology and defining features of the options provided.
Kathak (A): Derived from the Sanskrit word ’Katha’ meaning "story". Kathakars are storytellers. Its name focuses on storytelling rather than a combination of four specific elements.
Kathakali (B): Also translates to "story-play" (’Katha’ = story, ’Kali’ = play/performance). It is a highly stylized dance-drama from Kerala, but the name itself doesn’t explicitly break down into the four elements mentioned.
Manipuri (C): This dance is simply named after its geographical region of origin, the northeastern state of Manipur. The name does not indicate its internal stylistic elements.
Bharatnatyam (D): This is one of the oldest classical dance traditions in India, originating in Tamil Nadu. The word "Bharatnatyam" is widely considered an acronym or a composite word derived from the essential elements of the dance.

Step 2: Deconstructing the word "Bharatnatyam":
The name "Bharata" is traditionally understood to be composed of three distinct syllables, each representing a crucial element of the performance:
1. Bha stands for Bhava, which translates to expression or emotion.
2. Ra stands for Raga, which translates to melody or musical framework.
3. Ta stands for Tala, which translates to rhythm or beat.
4. The suffix Natyam means dance or dramatic performance (drama).
When you combine these meanings, "Bha-ra-ta-natyam" literally translates into a dance (drama) that harmoniously combines expression, melody, and rhythm.
This perfectly aligns with the detailed description provided in the question.

Final Answer:
Therefore, the correct answer is (D) Bharatnatyam.

Concept:
This question relates to Medieval Indian History, specifically the establishment of the Delhi Sultanate.
The Delhi Sultanate was an Islamic empire based in Delhi that stretched over large parts of the Indian subcontinent for over 300 years (1206–1526).
Five unrelated dynasties ruled over the Delhi Sultanate sequentially: the Mamluk (Slave) dynasty, the Khalji dynasty, the Tughlaq dynasty, the Sayyid dynasty, and the Lodi dynasty.
Identifying the founder of the very first dynasty provides the answer to who the first ruler of the entire Sultanate was.

Step 1: Identifying the Start of the Sultanate:
The foundation of Muslim rule in northern India was laid by Muhammad Ghori in the late 12th century.
However, Ghori did not establish a capital in India; he returned to his base in Ghur, leaving his conquests under the administration of his trusted generals.
Upon Muhammad Ghori’s assassination in 1206, his vast empire fragmented.
One of his most prominent generals and former slaves, Qutb-ud-din Aibak, took control of Ghori’s Indian territories and declared himself the independent ruler.

Step 2: Evaluating the Historical Figures in the Options:
Let’s look at the roles of the individuals listed in the options.
Iltutmish (A): He was the son-in-law of Aibak and the third ruler of the Slave Dynasty. While considered the real consolidator of the empire, he was not the first ruler.
Shahjahan (B): He was the fifth emperor of the Mughal Empire, reigning much later in history (1628–1658). The Mughals overthrew the Delhi Sultanate.
Qutub-U-Din-Aibak (C): As established, he assumed power in 1206 immediately after Ghori’s death, establishing the Mamluk (Slave) Dynasty and effectively becoming the first official Sultan of Delhi.
Dara Shikhoh (D): He was the eldest son and heir-apparent of the Mughal emperor Shah Jahan, and was defeated by his brother Aurangzeb. He was never a Sultan of Delhi.

Step 3: Conclusion:
Qutb-ud-din Aibak is historically recognized universally as the founder of the Mamluk dynasty and the very first ruler of the Delhi Sultanate.
He is also famous for beginning the construction of the Qutub Minar in Delhi.

Final Answer:
Therefore, the correct answer is (C) Qutub-U-Din-Aibak.

Concept:
This question tests your knowledge of the Indian National Movement, specifically connecting prominent freedom fighters with the significant political organizations they founded or major national movements they led.
Identifying the key contributions of these historical figures is essential for understanding India’s struggle for independence.

Step 1: Analyzing (A) Bal Gangadhar Tilak:
Bal Gangadhar Tilak, one of the famous "Lal-Bal-Pal" triumvirate, was a staunch nationalist.
He is best known for starting the Indian Home Rule League in April 1916 (along with Annie Besant who started her own league later that year) to demand self-government for India within the British Empire.
Therefore, (A) accurately pairs with (III) Home Rule Movement.

Step 2: Analyzing (B) Subhash Chandra Bose:
Subhash Chandra Bose, affectionately known as Netaji, had ideological differences with Mahatma Gandhi regarding the path to independence.
After resigning from the presidency of the Indian National Congress in 1939, he founded a new political faction within the Congress to consolidate radical and anti-imperialist forces.
This faction was named the All India Forward Bloc.
Therefore, (B) accurately pairs with (II) Forward Block.

Step 3: Analyzing (C) Mahatma Gandhi:
Mahatma Gandhi was the preeminent leader of the Indian independence movement, known for his philosophy of non-violent resistance.
He launched several massive nationwide campaigns, the most prominent being the Non-Cooperation Movement, the Civil Disobedience Movement (which started with the Dandi March in 1930), and the Quit India Movement.
Among the given options, the Civil Disobedience Movement is the clear match.
Therefore, (C) accurately pairs with (IV) Civil Disobedience Movement.

Step 4: Analyzing (D) Lala Lajpat Rai:
Lala Lajpat Rai, another crucial member of the "Lal-Bal-Pal" trio, was a prominent leader from the Punjab region.
He was popularly known as "Punjab Kesari" (Lion of Punjab) and played a massive role in mobilizing nationalist sentiment in Punjab against British rule.
He was heavily involved in political activism and nationalist movements centered in that region.
Therefore, (D) accurately pairs with (I) Punjab Nationalist Movement.

Step 5: Finalizing the Matches:
Reviewing our historical deductions, the correct matching sequence is undeniably: (A)-(III), (B)-(II), (C)-(IV), and (D)-(I).
Looking at the provided multiple-choice options, Option (A) reflects this exact specific combination.

Final Answer:
Therefore, the correct answer is (A) (A) - (III), (B) - (II), (C) - (IV), (D) - (I).

Concept:
This question tests basic knowledge of physics and chemistry regarding the electrical conductivity of various liquids.
For a liquid to conduct electricity, it must contain mobile, charged particles called ions. These ions carry the electric charge through the fluid.
Liquids that contain dissolved salts, acids, or bases dissociate into positively and negatively charged ions, thereby becoming excellent conductors (electrolytes).
Liquids that lack these free-moving ions are classified as insulators or poor conductors.

Step 1: Evaluating Option (A) - Distilled Water:
Distilled water is pure \( H_2O \) that has been boiled into vapor and condensed back into liquid in a separate container, effectively removing all impurities, natural minerals, and dissolved salts.
Because it is purely covalent and lacks free ions, it is an extremely poor conductor of electricity, essentially acting as an insulator.
Thus, (A) is incorrect.

Step 2: Evaluating Option (B) - Lemon Juice:
Lemon juice is highly acidic. It contains a significant concentration of citric acid, along with ascorbic acid (Vitamin C).
When acids are in an aqueous solution, they readily dissociate to release hydrogen ions (\(H^+\)) and respective anions.
The abundance of these free-flowing charged ions makes lemon juice a very good conductor of electricity.
Thus, (B) is a strong candidate for the correct answer.

Step 3: Evaluating Option (C) - Sugar Solution:
A sugar solution is formed by dissolving sugar (sucrose) in water.
Sugar is a covalent molecular compound. When it dissolves in water, it breaks apart into individual neutral molecules, but it does not dissociate into charged ions.
Without charged ions to carry the current, the sugar solution remains a poor conductor of electricity.
Thus, (C) is incorrect.

Step 4: Evaluating Option (D) - Kerosene:
Kerosene is a liquid hydrocarbon, fundamentally a non-polar organic compound composed entirely of carbon and hydrogen atoms sharing covalent bonds.
It absolutely does not ionize or break down into charged particles. Therefore, like most oils, it is a complete insulator.
Thus, (D) is incorrect.

Step 5: Final Conclusion:
Comparing all options based on scientific principles, Lemon juice is the only liquid listed that contains the necessary ions to conduct an electric current effectively.

Final Answer:
Therefore, the correct answer is (B) Lemon Juice.

Concept:
This question tests basic biological knowledge regarding the practical application of specific microorganisms in everyday food production processes.
Fermentation is a metabolic process that produces chemical changes in organic substrates through the action of enzymes produced by microorganisms.
The transformation of milk into curd (or yogurt) is a classic example of bacterial fermentation.
Knowing which specific biological agent is responsible for this transformation is a standard general science fact.

Step 1: Evaluating the process of Curd formation:
Milk naturally contains a specific type of sugar called lactose, as well as a protein called casein.
To make curd, a small amount of pre-existing curd (a ’starter’) is added to warm milk. This starter contains millions of specific lactic acid bacteria.
These bacteria multiply in the warm milk and consume the lactose sugar, converting it into lactic acid through fermentation.
This increased acidity causes the milk proteins (casein) to coagulate and solidify, giving curd its characteristic thick texture and slightly sour taste.

Step 2: Evaluating the Microorganisms in the Options:
Let’s look at the specific microorganisms provided in the options.
Yeast (A): Yeast is a fungus commonly used in baking (to make bread rise) and brewing (to produce alcohol). It is not the primary agent for making curd.
Rhizobium (B): Rhizobium is a soil bacterium that fixes atmospheric nitrogen in the root nodules of leguminous plants. It has no role in dairy fermentation.
Lactobacillus (C): As the prefix "Lacto-" suggests (relating to milk), Lactobacillus is a genus of lactic acid-producing bacteria. It is specifically the primary microorganism responsible for converting milk into curd.
Penicillium (D): Penicillium is a genus of mold (fungus) famous for producing the antibiotic penicillin, and used in making some cheeses (like blue cheese), but it is not the primary bacterium for regular curd.

Step 3: Conclusion:
Based on biological facts, Lactobacillus is undeniably the correct microorganism that drives the fermentation process to turn milk into curd.

Final Answer:
Therefore, the correct answer is (C) Lactobacillus.

Concept:
This question tests a fundamental fact from basic chemistry regarding the physical states of elements on the periodic table.
Metals are generally known for their specific physical properties: they are solid, lustrous, malleable, ductile, and excellent conductors of heat and electricity.
At standard room temperature (typically defined as around \(20^\circ\)C to \(25^\circ\)C), almost every single metal in the periodic table exists in a solid state.
However, there is one very famous exception to this rule—a single metal that remains entirely liquid under these normal temperature conditions.

Step 1: Evaluating the given options:
We need to identify the physical state of each listed metal at room temperature based on chemical facts.
Iron (B): Iron is a heavy, structural transition metal. Its melting point is extremely high, at \(1,538^\circ\)C. It is definitely a hard solid at room temperature.
Copper (C): Copper is a reddish-gold transition metal widely used for electrical wiring. Its melting point is \(1,085^\circ\)C. It is undoubtedly a solid at room temperature.
Aluminium (D): Aluminium is a lightweight, silvery-white metal used in airplanes and cans. Its melting point is \(660^\circ\)C. It is completely solid at room temperature.
Mercury (A): Mercury, known historically as quicksilver, is a unique transition metal. It has an exceptionally low melting point of \(-38.83^\circ\)C.

Step 2: Conclusion based on facts:
Because Mercury melts at nearly \(-39^\circ\)C, it has already transitioned into its liquid phase long before reaching normal room temperatures (like \(25^\circ\)C).
It remains a dense, silvery liquid under standard environmental conditions, which is why it was historically used extensively in thermometers and barometers.
It is the only stable metal on the periodic table with this property.

Final Answer:
Therefore, the correct answer is (A) Mercury.

Concept:
This question pertains to the history of physics, specifically acknowledging major scientific contributions made by prominent Indian scientists.
Scattering of light is a physical phenomenon that occurs when a beam of light interacts with particles (like molecules in the air or a liquid) and is forced to deviate from its straight trajectory, scattering in multiple different directions.
Identifying the scientist who made a ground-breaking, Nobel Prize-winning discovery related to this phenomenon is key.

Step 1: Analyzing the Contributions of the Scientists Listed:
Let’s briefly review the major scientific fields of the individuals provided in the options.
J.C. Bose (A): Sir Jagadish Chandra Bose was a polymath who made significant contributions to plant biology (inventing the crescograph to measure plant growth) and was a pioneer in the investigation of radio and microwave optics. He is not primarily known for the scattering of light.
Homi Jahangir Bhabha (C): Dr. Homi J. Bhabha was a prominent nuclear physicist. He is widely recognized as the "father of the Indian nuclear program" and founded the Tata Institute of Fundamental Research (TIFR). His work was focused on quantum theory and cosmic rays.
A.P.J. Abdul Kalam (D): Dr. Kalam was an aerospace scientist who served as the 11th President of India. He was intimately involved in India’s civilian space program and military missile development efforts, earning him the title "Missile Man of India."
C.V. Raman (B): Sir Chandrasekhara Venkata Raman was an exceptional Indian physicist. In 1928, he and his student K. S. Krishnan discovered a remarkable physical phenomenon related to light.

Step 2: Identifying the Specific Discovery:
C.V. Raman discovered that when light traverses a transparent material, a small fraction of the scattered light changes its wavelength (and thus its color).
This specific inelastic scattering of photons by molecules is now globally known as the "Raman Effect" or Raman scattering.
For this ground-breaking discovery concerning the scattering of light, he was awarded the 1930 Nobel Prize in Physics, becoming the first Asian to receive a Nobel Prize in any branch of science.
National Science Day is celebrated in India on February 28th every year to commemorate this exact discovery.

Final Answer:
Therefore, the correct answer is (B) C.V. Raman.

Concept:
This is a Current Affairs and General Knowledge question that connects a contemporary milestone year (2025) with the birth year of a prominent historical Indian leader.
To find whose 150th birth anniversary falls in the year 2025, we must perform a simple mathematical subtraction.
The target birth year is calculated as: \( 2025 - 150 = 1875 \).
Therefore, the question is essentially asking which of these famous leaders was born in the year 1875.

Step 1: Identifying the Birth Years of the Given Leaders:
We need to rely on basic historical knowledge regarding the lives of the prominent figures listed in the options.
Mahatma Gandhi (C): He was born on October 2, 1869. His 150th birth anniversary was celebrated nationwide in the year 2019 (\(1869 + 150 = 2019\)).
Netaji Subhash Chandra Bose (B): He was born on January 23, 1897. His 125th birth anniversary was celebrated in the year 2022 (\(1897 + 125 = 2022\)).
Lal Bahadur Shastri (D): He shared a birthday with Gandhi but was born much later, on October 2, 1904.
Sardar Vallabhbhai Patel (A): Known as the "Iron Man of India", he was born on October 31, 1875.

Step 2: Verifying the Calculation:
We established that the leader we are looking for must have been born in 1875 to have their 150th anniversary in 2025.
Sardar Vallabhbhai Patel was born in 1875.
Calculating the anniversary: \( 1875 \text{ (birth year)} + 150 \text{ years} = 2025 \).
The Government of India celebrates his birthday, October 31st, as Rashtriya Ekta Diwas (National Unity Day). The year 2025 marks his sesquicentennial (150th) anniversary.

Final Answer:
Therefore, the correct answer is (A) Sardar Ballabh Bhai Patel.

Concept:
This question tests your awareness of major global current affairs and international relations, specifically regarding the ongoing geopolitical conflict in Eastern Europe.
The invasion of Ukraine by Russia, which escalated significantly in February 2022, is one of the most impactful global events of the decade.
Understanding the stated primary geopolitical motivations behind the conflict is essential for answering this question correctly.

Step 1: Analyzing the Geopolitical Context:
Following the collapse of the Soviet Union in 1991, NATO (the North Atlantic Treaty Organization, a Western military alliance) began expanding eastward, admitting several former Soviet bloc countries.
Russia, under the leadership of Vladimir Putin, has consistently viewed NATO’s eastward expansion as a severe, direct threat to its own national security.
Ukraine, a large nation sharing a massive border with Russia, had increasingly expressed strong aspirations to join the European Union and eventually become a member state of NATO.

Step 2: Evaluating the Given Options:
Let’s review the options to identify the primary catalyst cited for the invasion.
(D) Change of leadership in Russia: This is incorrect. Vladimir Putin has been the dominant political leader in Russia (as either President or Prime Minister) continuously since 1999. There was no recent change in leadership.
(C) Ukraine buying oil from UAE: This is a minor economic action and completely irrelevant to sparking a massive, full-scale military invasion.
(B) Friendship with USA: While Ukraine’s growing alignment with the West and the USA frustrated Russia, "friendship" alone is too broad and was not the specific military trigger.
(A) Intention of Ukraine to join NATO: This was explicitly cited by the Russian government as a red line. Russia demanded strict legal guarantees that Ukraine would never be allowed to join NATO, fearing that Western military infrastructure would be placed directly on its borders. When these demands were not met, Russia launched its "special military operation."

Step 3: Conclusion:
While the conflict has deep historical, cultural, and political layers, the immediate and primary geopolitical reason consistently cited for initiating the war was Ukraine’s intention and progress toward joining the NATO military alliance.

Final Answer:
Therefore, the correct answer is (A) The intention of Ukraine to join NATO.

Concept:
This question tests your geographical knowledge regarding the mineral wealth and distribution across various states in India.
Rajasthan is a highly mineral-rich state and holds a near-monopoly in the production of certain critical non-ferrous metals in India.
Understanding which states are the leading producers of specific minerals is a key part of Indian Geography.

Step 1: Analyzing the Distribution of Zinc and Lead:
Zinc and Lead are typically found together in the same ore bodies.
Rajasthan is overwhelmingly the largest producer of both Lead and Zinc in India, accounting for almost the entirety of the country’s production.
The Zawar mines in the Udaipur district and the Rampura Agucha mines in the Bhilwara district are world-famous for their massive deposits of Lead and Zinc.
Therefore, minerals (A) Zinc and (D) Lead are definitely mainstays of Rajasthan.

Step 2: Analyzing the Distribution of Copper:
Rajasthan is also one of the leading producers of Copper in India, second only to Madhya Pradesh.
The Khetri Copper Belt located in the Jhunjhunu district of Rajasthan has been a major historical and contemporary source of copper mining.
Therefore, mineral (B) Copper is also mainly found in Rajasthan.

Step 3: Analyzing the Distribution of Iron:
While Rajasthan has some minor deposits of iron ore, it is not considered a primary or major producer of Iron on a national scale.
The vast majority of India’s Iron ore is heavily concentrated in the eastern and central states such as Odisha, Jharkhand, Chhattisgarh, and Karnataka.
Therefore, mineral (C) Iron is not a correct fit for this specific combination.

Step 4: Final Conclusion:
Based on geographical facts, Zinc, Copper, and Lead are heavily mined in Rajasthan, while Iron is not a major mineral of the state.
Thus, the correct combination is (A), (B), and (D).
Looking at the options, Option (A) represents this exact combination. (Note: The provided PDF has a typo where option 1 and option 4 are identical, both represent the correct answer).

Final Answer:
Therefore, the correct answer is (A) (A), (B) and (D) only.

Concept:
This question addresses the fundamental astronomical mechanics that dictate Earth’s climate and seasonal variations.
The Earth experiences two primary types of motion: Rotation (spinning on its own axis) and Revolution (orbiting around the Sun).
While the revolution dictates the length of our year, it alone does not cause the drastic temperature and weather shifts we call seasons.
The true cause lies in the specific geometry of Earth’s orientation as it travels around the Sun.

Step 1: Evaluating the Role of Rotation:
Earth rotates on its axis once every 24 hours.
This spinning motion is solely responsible for the daily cycle of day and night, as different parts of the planet face toward or away from the Sun.
Rotation does not cause the long-term cycle of seasons. Thus, option (B) is incorrect.

Step 2: Evaluating Gravitational Pull:
The gravitational pull of the Moon interacts with Earth’s oceans to create the daily rising and falling of tides.
While it subtly affects the Earth’s rotation over millions of years, it has absolutely no bearing on the annual changing of the seasons.
Thus, option (C) is strictly incorrect.

Step 3: Analyzing Axial Tilt and Revolution:
Earth’s axis of rotation is not perfectly upright; it is tilted at an angle of approximately \(23.5^\circ\) relative to its orbital plane around the Sun.
As the Earth revolves around the Sun over the course of a year, this fixed axial tilt causes the Northern and Southern Hemispheres to alternately point towards or lean away from the Sun.
When a hemisphere is tilted towards the Sun, it receives more direct, intense sunlight and experiences longer days (Summer).
Conversely, when it is tilted away, the sunlight hits at a shallower angle, spreading the energy over a larger area, resulting in colder temperatures and shorter days (Winter).
Without the tilt, even with revolution, both hemispheres would receive consistent sunlight year-round, and seasons would not exist.
Therefore, it is the combination of the axial tilt AND the annual revolution that creates the seasons.

Final Answer:
Therefore, the correct answer is (D) Tilt of Earth’s axis combined with revolution of the Earth.

Concept:
This question tests geographical awareness regarding the major river systems of India, specifically tracing the path of the river Ganga.
The Ganga is the longest and most sacred river in India, originating in the Himalayas in Uttarakhand and flowing southeast through the Gangetic plains before emptying into the Bay of Bengal.
Knowing the major cities situated along its banks is a standard requirement for Indian Geography.

Step 1: Tracing the Path of the Ganga:
The river originates as the Bhagirathi from the Gangotri glacier and officially becomes the Ganga at Devprayag where it meets the Alaknanda.
It then flows down into the plains. The very first major towns it passes through in the foothills of the Himalayas are Rishikesh and then Haridwar.
Therefore, cities (B) and (C) are definitely situated directly on the banks of the Ganga.

Step 2: Evaluating Prayagraj:
As the Ganga flows further into Uttar Pradesh, it reaches the city of Prayagraj (formerly Allahabad).
Prayagraj is highly famous for the ’Triveni Sangam’, which is the sacred confluence of the Ganga, the Yamuna, and the mythical Saraswati rivers.
Because the Ganga flows right through this confluence point, Prayagraj is undeniably situated on the Ganga.
Therefore, city (D) is correct.

Step 3: Evaluating Delhi:
The capital city of Delhi is historically and geographically situated on the banks of the river Yamuna, not the Ganga.
The Yamuna is a major tributary of the Ganga, running parallel to it for hundreds of kilometers before they finally meet at Prayagraj.
Therefore, city (A) is incorrect as the main Ganga river does not pass through Delhi.

Step 4: Final Conclusion:
Based on our geographical tracing, the river Ganga passes through Rishikesh, Haridwar, and Prayagraj. It does not pass through Delhi.
The correct combination is (B), (C), and (D).
Looking at the options, Option (C) matches this precisely.

Final Answer:
Therefore, the correct answer is (C) (B), (C) and (D) only.

Concept:
This question assesses basic knowledge of India’s political geography and its international boundaries.
India shares its vast land borders with several sovereign nations: Pakistan and Afghanistan to the northwest, China, Nepal, and Bhutan to the north, and Myanmar and Bangladesh to the east.
Knowing the relative lengths of these international borders is a frequent topic in competitive exams.

Step 1: Evaluating the given Options:
Let’s analyze the borders of the countries provided in the options.
Sri Lanka (C): Sri Lanka is an island nation located in the Indian Ocean. It shares a maritime boundary with India (separated by the Palk Strait), but it does not share any physical land border at all. Thus, it is immediately eliminated.
Pakistan (B): India shares a highly sensitive and heavily guarded border with Pakistan on its western front, running through Gujarat, Rajasthan, Punjab, and Jammu & Kashmir. The length is approximately 3,323 kilometers.
China (D): India shares a long, high-altitude border with China across the Himalayas (the LAC). It spans approximately 3,488 kilometers, making it the second-longest border.
Bangladesh (A): Bangladesh is almost entirely enclaved by Indian territory on three sides (West Bengal, Assam, Meghalaya, Tripura, and Mizoram).

Step 2: Identifying the Longest Border:
While the border with China spans across the entire northern length of the subcontinent, the border with Bangladesh is incredibly convoluted, twisting and turning intricately around rivers and local jurisdictions.
Because of this highly meandering and zigzagging geography, the total measured length of the India-Bangladesh border is effectively the longest.
It measures approximately 4,096 kilometers.
This makes it the fifth-longest land border in the world and the longest one India shares with any neighbor.

Final Answer:
Therefore, the correct answer is (A) Bangladesh.

Concept:
This question tests core knowledge of the Indian Polity, specifically focusing on the evolution and significant amendments of the Indian Constitution.
When the Constitution was originally adopted in 1950, it laid out Fundamental Rights for citizens but did not contain a specific list of Fundamental Duties.
The concept of explicitly defining the duties of citizens was inspired by the Constitution of the USSR.
These duties were added later during a period of national crisis to remind citizens that enjoying rights comes with corresponding civic responsibilities.

Step 1: Identifying the Context of the Addition:
During the Internal Emergency (1975–1977), the ruling government formed the Sardar Swaran Singh Committee to study and recommend the inclusion of fundamental duties.
Based on the strong recommendations of this committee, the government moved to amend the Constitution to explicitly list these civic obligations.

Step 2: Identifying the Correct Constitutional Amendment:
The recommendations were officially enacted into law through the highly comprehensive 42nd Constitutional Amendment Act in the year 1976.
This specific amendment is often referred to as a "Mini-Constitution" because it made such extensive and widespread changes to various parts of the document.
It inserted a brand new part, Part IVA, which contained a single new article, Article 51A, explicitly detailing 10 Fundamental Duties for all Indian citizens (an 11th duty was added much later by the 86th Amendment).

Step 3: Evaluating the Other Options:
The 41st Amendment changed the retirement age of State Public Service Commission members.
The 43rd and 44th Amendments (passed post-Emergency in 1977 and 1978) were primarily aimed at undoing many of the controversial and restrictive changes that had just been introduced by the 42nd Amendment.
None of these other amendments introduced the Fundamental Duties.

Final Answer:
Therefore, the correct answer is (B) 42nd.

Concept:
This question requires an understanding of India’s political administration, specifically the unique geographical structuring of its Union Territories (UTs).
While states are large, contiguous landmasses with their own elected governments, UTs are generally smaller, federally administered regions.
Most UTs are localized to a single, continuous geographic area or a specific archipelago.
However, due to specific historical colonial legacies, one UT in India is a geographical anomaly, composed of unconnected enclaves located in entirely different states.

Step 1: Evaluating the given Union Territories:
Let’s look at the geography of the options provided.
Jammu and Kashmir (A): This is a large, contiguous territory located entirely in the extreme northern part of the Indian subcontinent. It does not exist in pieces across other states.
Andaman and Nicobar Islands (B): This is an archipelago (a continuous chain of islands) located together in the Bay of Bengal. It operates as a single geographic block.
Lakshadweep (C): Similar to the Andamans, this is a distinct group of islands located together in the Arabian Sea off the coast of Kerala. It is geographically unified.

Step 2: Analyzing the Geography of Puducherry:
Puducherry (formerly Pondicherry) was historically a French colonial settlement. When it was integrated into the Indian Union, its original territorial boundaries were strictly maintained to preserve its distinct French cultural heritage.
Because the French had multiple, separated trading posts along the Indian coastline, the modern UT of Puducherry remains scattered across four distinct, non-contiguous districts located inside three different Indian states:
1. Puducherry district is an enclave situated within the state of Tamil Nadu.
2. Karaikal district is also an enclave situated further south within Tamil Nadu.
3. Yanam district is a small enclave situated within the state of Andhra Pradesh.
4. Mahe district is a small enclave situated on the western coast within the state of Kerala.
Thus, Puducherry is the only UT physically spread out over more than one state.

Final Answer:
Therefore, the correct answer is (D) Puducherry.

Concept:
This question tests a foundational concept in macroeconomics and government budgeting: the definition of deficits.
A government creates a budget estimating its total expected earnings (Revenue/Receipts) and its total planned spending (Expenditure) for a financial year.
A "deficit" fundamentally occurs when money going out is greater than money coming in.
The "Fiscal Deficit" is a very specific and crucial economic indicator that strictly defines the total borrowing requirements of the government.

Step 1: Understanding Government Revenue and Expenditure:
When a government’s total expenditure (spending on infrastructure, salaries, subsidies, etc.) is exactly equal to its total revenue (taxes, dividends, etc.), it is called a Balanced Budget (Option C).
When a government’s total revenue is greater than its total expenditure, it generates a Budget Surplus (Option B).
When total expenditure exceeds total revenue, a deficit is created.

Step 2: Defining Fiscal Deficit Correctly:
If a government spends more money than it earns, it must borrow money from the market, the public, or foreign institutions to make up the exact difference.
Therefore, the Fiscal Deficit represents the total amount of borrowed funds needed to sustain the government’s spending.
The exact economic formula for Fiscal Deficit is:
\( \text{Fiscal Deficit} = \text{Total Government Expenditure} - \text{Total Government Receipts (excluding borrowings)} \).
We must explicitly exclude borrowings from the "Revenue/Receipts" side of this equation; otherwise, the budget would always artificially balance to zero (since borrowing fills the gap).

Step 3: Evaluating the Options:
Option (A) accurately states that Fiscal Deficit occurs when "Government Expenditure exceeds Government Revenue (Excluding Borrowings)." This aligns perfectly with the standard economic definition.
Option (D) comparing expenditure directly to the total GDP is incorrect; while deficit is often expressed as a percentage of GDP to show scale, it is not defined by exceeding the GDP.

Final Answer:
Therefore, the correct answer is (A) Government Expenditure exceeds Government Revenue (Excluding Borrowings).

Concept:
This question tests general knowledge regarding famous Indian literature and the prominent authors or historical figures who wrote them.
Matching landmark books with their correct authors is a staple in aptitude and general awareness tests.

Step 1: Analyzing (A) Godaan:
"Godaan" (The Gift of a Cow) is a classic Hindi novel published in 1936.
It is widely considered one of the greatest novels of modern Indian literature, brilliantly highlighting the struggles of Indian peasantry under colonial rule.
It was famously written by the legendary Hindi-Urdu writer Dhanpat Rai Shrivastava, universally known by his pen name, Munshi Premchand.
Therefore, (A) matches with (II).

Step 2: Analyzing (B) Letters to my daughter:
"Letters from a Father to His Daughter" is a renowned collection of 30 letters written in the summer of 1928.
These letters contained lessons on natural history and the story of civilizations.
They were written by Jawaharlal Nehru, the first Prime Minister of India, to his young daughter, Indira Gandhi (who was 10 years old at the time).
Therefore, (B) matches with (IV).

Step 3: Analyzing (C) My Experiments with Truth:
"The Story of My Experiments with Truth" is a highly famous autobiographical work.
It covers the life of the author from early childhood through to 1921, detailing his spiritual and political journey, heavily focusing on non-violence.
It is the official autobiography of Mohandas Karamchand Gandhi (Mahatma Gandhi).
Therefore, (C) matches with (I).

Step 4: Analyzing (D) Train to Pakistan:
"Train to Pakistan" is a historical novel published in 1956.
It is one of the most powerful and harrowing literary accounts of the Partition of India in 1947, detailing the brutal communal violence that tore a village apart.
It was written by the prominent Indian author and journalist Khushwant Singh.
Therefore, (D) matches with (III).

Step 5: Finalizing the Matches:
Based on our literary knowledge, the correct mapping is unequivocally: (A)-(II), (B)-(IV), (C)-(I), and (D)-(III).
Looking closely at the provided multiple-choice options, Option (C) reflects this exact combination.

Final Answer:
Therefore, the correct answer is (C) (A) - (II), (B) - (IV), (C) - (I), (D) - (III).

Concept:
This question tests your knowledge of the history of the Indian Space Research Organisation (ISRO) and major milestones in India’s space program.
Chronological order requires arranging the satellite launches sequentially from the earliest historical date to the most recent.
Knowing the general decade or order of India’s major space missions is crucial here.

Step 1: Analyzing the Satellite Missions:
Let’s assign historical launch years to each of the space missions listed.
(D) Aryabhatta: This was India’s very first satellite, named after the famous ancient Indian astronomer. It was launched successfully by the Soviet Union on April 19, 1975.
(B) INSAT 1A: The Indian National Satellite System (INSAT) series revolutionized telecommunications in India. INSAT-1A, the first of this series, was launched on April 10, 1982.
(C) Chandrayaan-1: This was India’s historic first lunar probe, marking a massive leap in ISRO’s deep space exploration capabilities. It was launched successfully on October 22, 2008.
(A) Mangalyaan: Officially known as the Mars Orbiter Mission (MOM), this made India the first Asian nation to reach Martian orbit and the first in the world to do so in its maiden attempt. It was launched on November 5, 2013.

Step 2: Arranging in Chronological Order:
Now we sort these launch dates from the earliest to the most recent.
1st (Earliest): 1975 \(\rightarrow\) Aryabhatta (Event D)
2nd: 1982 \(\rightarrow\) INSAT 1A (Event B)
3rd: 2008 \(\rightarrow\) Chandrayaan-1 (Event C)
4th (Most Recent): 2013 \(\rightarrow\) Mangalyaan (Event A)
Therefore, the correct chronological sequence of these satellite launches is: (D) \(\rightarrow\) (B) \(\rightarrow\) (C) \(\rightarrow\) (A).

Step 3: Matching with Options:
We scan the given options for this exact sequence.
Option (B) perfectly matches our logically deduced sequence (D), (B), (C), (A).

Final Answer:
Therefore, the correct answer is (B) (D), (B), (C), (A).

Concept:
This question tests knowledge of major government policies, schemes, and missions launched in India regarding agriculture and food security.
To ensure the growing population was adequately fed, the government periodically introduces targeted missions to boost the production yields of staple crops.
Knowing the specific names and launch years of these flagship programs is important for general awareness.

Step 1: Identifying the Context of 2007:
In the mid-2000s, India faced stagnating agricultural yields in essential staple crops, leading to concerns about future food scarcity.
To combat this, a resolution was passed by the National Development Council (NDC) in May 2007 during the 11th Five Year Plan.
The resolution called for a massively funded, targeted scheme to increase the annual production of rice, wheat, and pulses by 10 million, 8 million, and 2 million tonnes, respectively.

Step 2: Identifying the Specific Mission:
In October 2007, a Centrally Sponsored Scheme was officially launched by the Government of India to fulfill this resolution and heavily sustain productivity gains in the agricultural sector.
The official name of this highly successful, large-scale initiative is the National Food Security Mission (NFSM).
Its primary objective was to increase food grain production through area expansion and productivity enhancement in targeted districts across the country.

Step 3: Evaluating the Options:
Options (A), (B), and (C) are fabricated or incorrect names that sound similar to the actual program but are not official government schemes launched in that year for that purpose.
Option (D) correctly identifies the exact flagship program launched in 2007.

Final Answer:
Therefore, the correct answer is (D) National Food Security Mission.

Concept:
This question involves decoding an Alpha-Numeric series, where a logical pattern dictates the progression of both letters (alphabets) and numbers simultaneously.
To solve such problems, we must isolate the alphabetical components from the numerical components and analyze their individual underlying patterns.
Often, the numbers are directly related to the standard positional value of the corresponding letter in the English alphabet (A=1, B=2, C=3, etc.).

Step 1: Analyzing the Alphabetical Series:
Let’s extract just the letters from the given sequence:
D, F, H, J, L...
We need to find the pattern linking these letters. Let’s look at their positions in the standard English alphabet:
D is the 4th letter.
F is the 6th letter. (Skip E)
H is the 8th letter. (Skip G)
J is the 10th letter. (Skip I)
L is the 12th letter. (Skip K)
The pattern is clearly adding 2 to the position value of each successive letter (alternating letters).
To find the next letter, we add 2 to the position of L (12).
\(12 + 2 = 14\).
The 14th letter of the English alphabet is N.
So, the next letter in our series must definitely be N.

Step 2: Analyzing the Numerical Series:
Let’s extract just the numbers from the given sequence:
4, 6, 8, 10, 12...
This is a very simple arithmetic progression consisting of consecutive even numbers, increasing by exactly 2 each time.
To find the next number, we simply add 2 to the last number in the series (12).
\(12 + 2 = 14\).
So, the next number must definitely be 14.
(Notice that in this specific series, the number always perfectly matches the alphabetical position of its paired letter).

Step 3: Combining the Results:
We take the deduced letter (N) and combine it with the deduced number (14).
The final term that completes the series is N14.
Looking at the options, Option (D) is exactly N14.

Final Answer:
Therefore, the correct answer is (D) N14.

Concept:
This is a standard logical reasoning question based on Blood Relations.
Solving these requires carefully breaking down a complex relational statement into smaller, understandable familial links, usually by tracing the relationships backwards from the speaker or the subject.
Drawing a quick mental or physical family tree is the most reliable method to visualize how the two individuals connect to a common ancestor.

Step 1: Deconstructing the Statement:
The speaker is Vishnu, and he is directly addressing Radha.
The core statement is: "Your mother (Radha’s mother) is the daughter of my grandmother (Vishnu’s grandmother)."
Let’s break this down into two distinct parts centered around the common figure (the grandmother).

Step 2: Analyzing Vishnu’s side of the family tree:
"My grandmother" refers to the mother of Vishnu’s parent (either maternal or paternal).
The statement mentions the "daughter" of this grandmother.
The daughter of Vishnu’s grandmother can logically be one of two people:
Scenario 1: The daughter is Vishnu’s own mother.
Scenario 2: The daughter is Vishnu’s aunt (his father’s sister or mother’s sister).

Step 3: Connecting Radha to the family tree:
The statement explicitly says that Radha’s mother IS this specific daughter.
Let’s apply this to our two scenarios:
If Scenario 1 is true (the daughter is Vishnu’s mother): Then Radha’s mother is Vishnu’s mother. This means Vishnu and Radha share the exact same mother, making them brother and sister (Siblings).
If Scenario 2 is true (the daughter is Vishnu’s aunt): Then Radha’s mother is Vishnu’s aunt. This means Vishnu’s parent and Radha’s mother are siblings. The children of siblings are cousins.

Step 4: Evaluating the Options:
We must check our deduced relationships (Siblings or Cousins) against the provided multiple-choice options.
(B) Uncle-Niece is incorrect.
(C) Father-Daughter is incorrect.
(D) Grandfather-Granddaughter is incorrect.
(A) Cousins is presented as an option. Since "Siblings" is not an option, the puzzle dictates that Scenario 2 is the intended logical path.
Therefore, Radha’s mother is Vishnu’s aunt, making Vishnu and Radha first cousins.

Final Answer:
Therefore, the correct answer is (A) Cousins.

Concept:
This question falls under the logical reasoning topic of Calendars.
The core concept involves the calculation of "Odd Days."
A standard non-leap year consists of exactly 365 days.
A standard week consists of exactly 7 days.
When we divide 365 by 7, we get a quotient of 52 (meaning 52 full weeks) and a remainder of exactly 1 day.
This 1 remaining day is called an "Odd Day."
Because of this single odd day, moving forward exactly one normal year shifts the day of the week forward by exactly one day. Moving backward one normal year shifts it backward by one day.
A leap year has 366 days (adding Feb 29), resulting in 2 odd days.

Step 1: Identifying the Timeframe:
We are asked to find the day of the week for \(11^\text{th}\) January 2023, given that \(11^\text{th}\) January 2024 is a Thursday.
We are moving backward in time by exactly one full calendar year.

Step 2: Checking for Leap Year Impact:
We must meticulously check if the extra day of a leap year (February 29th) falls between our two target dates.
The year 2024 is a leap year (because 2024 is perfectly divisible by 4).
However, our relevant time period is from \(11^\text{th}\) January 2023 up to \(11^\text{th}\) January 2024.
The extra day, February 29th, 2024, occurs after \(11^\text{th}\) January 2024.
Therefore, the leap day is absolutely not included in the period we are calculating.
The duration between \(11^\text{th}\) Jan 2023 and \(11^\text{th}\) Jan 2024 consists of exactly 365 normal days.

Step 3: Calculating the Day:
Since the period is exactly 365 days, there is only 1 odd day involved.
Because we are navigating backward in time (from 2024 to 2023), we must subtract 1 day from the given day of the week.
The given day for 2024 is Thursday.
Thursday minus 1 day = Wednesday.
Therefore, \(11^\text{th}\) January 2023 undoubtedly fell on a Wednesday.

Final Answer:
Therefore, the correct answer is (A) Wednesday.

Concept:
This question is a classic "Missing Number in a Grid" puzzle, a common component of logical reasoning tests.
These puzzles require identifying a hidden mathematical operation (addition, subtraction, multiplication, division, or a combination) that consistently links the numbers within the rows or columns.
The key is to test simple arithmetic relationships horizontally across rows or vertically down columns until a rule applies universally to the fully populated lines.

Step 1: Analyzing the First Row:
Let’s look at the numbers horizontally in the first row: 7, 13, and 6.
We need to find an equation that connects these three specific numbers.
Notice that the number in the middle (13) is significantly larger than the numbers on the outside.
Let’s test simple addition of the outer numbers:
First column number + Third column number = \(7 + 6\).
\(7 + 6\) equals exactly 13, which is the number sitting in the middle column.
So, our working hypothesis rule is: Column 1 + Column 3 = Column 2.

Step 2: Verifying the Rule with the Second Row:
We must strictly verify if this identical rule applies to the second row to confirm it’s not a coincidence.
The numbers in the second row are: 4, 22, and 18.
Applying our hypothesized rule:
First column number + Third column number = \(4 + 18\).
\(4 + 18\) equals exactly 22.
This perfectly matches the number in the middle column of the second row.
The rule (Col 1 + Col 3 = Col 2) is now confirmed.

Step 3: Calculating the Missing Number:
Now, we confidently apply this verified mathematical rule to the final row to uncover the missing number.
The known numbers in the third row are: 17 (Column 1) and 7 (Column 3).
The middle number (Column 2) is represented by the question mark (?).
According to our rule: \(17 + 7 = \text{?}\)
\(17 + 7 = 24\).
Therefore, the missing number in the grid is definitively 24.

Final Answer:
Therefore, the correct answer is (D) 24.

Concept:
This is a standard Coding and Decoding problem involving the conversion of alphabetical letters into numerical strings.
The most common logic behind such codes is directly replacing each letter with its corresponding positional value in the standard English alphabet sequence.
(i.e., A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8, I=9, J=10, and so on).
Our task is to decode the rule from the first given example and meticulously apply it to the second word.

Step 1: Decoding the Rule from the First Word:
The problem states that the word "ACEGI" translates to the code "13579".
Let’s analyze the standard alphabetical position of each letter in the word ACEGI:
A is the \(1^\text{st}\) letter of the alphabet.
C is the \(3^\text{rd}\) letter.
E is the \(5^\text{th}\) letter.
G is the \(7^\text{th}\) letter.
I is the \(9^\text{th}\) letter.
When we string these specific positional numbers together in order, we get exactly "13579".
The rule is confirmed: Each letter is simply replaced by its numerical rank in the alphabet.

Step 2: Applying the Rule to the Target Word:
Now, we must apply this exact same translation rule to the target string "BDFHJ".
Let’s find the alphabetical position for each of these letters:
B is the \(2^\text{nd}\) letter of the alphabet.
D is the \(4^\text{th}\) letter.
F is the \(6^\text{th}\) letter.
H is the \(8^\text{th}\) letter.
J is the \(10^\text{th}\) letter.

Step 3: Formulating the Final Code:
Stringing these extracted positional numbers together exactly in the order they appear:
2, 4, 6, 8, 10 becomes the continuous string "246810".
This represents the coded version of the word BDFHJ.
Comparing this to our given options, Option (B) matches perfectly.

Final Answer:
Therefore, the correct answer is (B) 246810.

Concept:
This is a standard Direction Sense test question.
These questions require carefully tracking a subject’s movement across a mental compass grid (North, South, East, West).
The key is to accurately orient "left" and "right" turns based strictly on the direction the person is currently facing at that specific moment, not based on the top/bottom of your paper.
Visualizing or drawing a quick path diagram is the most foolproof way to solve these.

Step 1: The Initial Movement:
Let’s mark a specific dot on a conceptual grid and call it Vijay’s Starting Point (S).
"Vijay travels a distance of 5 km in the south."
From point S, draw a straight line downwards (representing South). Let’s call the end of this line Point A.
The distance from S to A is 5 km. Vijay is now standing at Point A, physically facing South.

Step 2: The First Turn:
"Then he turns to his right. After walking 5 km..."
Vijay is currently facing South. If you are facing South, your right hand points toward the West.
So, he turns 90 degrees to face West.
He walks straight for another 5 km. Let’s draw a line from Point A extending to the left (West) and call the end Point B.
The distance from A to B is 5 km. Vijay is now at Point B, physically facing West.

Step 3: The Second Turn:
"...he again turns to his right and walks 5 km."
Vijay is currently facing West. If you are facing West, your right hand points toward the North.
So, he turns 90 degrees to face North.
He walks straight for another 5 km. Let’s draw a line from Point B extending straight upwards (North) and call the final destination Point C.
The distance from B to C is 5 km.

Step 4: Determining the Final Position relative to Start:
We now analyze Vijay’s path. He formed three sides of a perfect square.
He went 5 km South, 5 km West, and then exactly 5 km North.
Because he walked the exact same distance North (5 km) as he initially walked South (5 km), he is now sitting perfectly level with his original Starting Point (S) on the horizontal axis.
The only difference is he is 5 km to the left (West) of where he began due to his second movement.
Therefore, drawing a straight line from his Starting Point (S) to his current position (Point C) points directly West.

Final Answer:
Therefore, the correct answer is (A) West.

Concept:
This question belongs to the logical reasoning topic involving Faulty Clocks.
A clock that "becomes fast" is essentially running faster than true, objective time. It covers more time on its face than the actual elapsed time.
The problem provides a specific error rate (gaining a certain number of seconds per a certain number of hours).
The solution requires calculating the total exact real-time elapsed between the initial setting and the final check, and then calculating how much total error the clock accumulated over that specific duration.

Step 1: Calculating Total Real Time Elapsed:
First, we must meticulously calculate the total number of hours that have passed in the real world.
Start Time: Sunday at 5:00 P.M.
Target Time: Monday at 9:00 A.M.
Let’s break this down to avoid errors:
From Sunday 5:00 P.M. to Sunday midnight (12:00 A.M.) is exactly 7 hours.
From Sunday midnight to Monday 9:00 A.M. is exactly 9 hours.
Total elapsed real time = \(7 \text{ hours} + 9 \text{ hours} = 16 \text{ hours}\).

Step 2: Calculating Total Accumulated Error:
The problem explicitly states the clock’s error rate: it gains exactly 15 seconds for every 2 hours that pass.
We need to find out how many 2-hour intervals exist within our total elapsed time of 16 hours.
Number of intervals = \(\frac{\text{Total hours}}{\text{Interval duration}} = \frac{16}{2} = 8 \text{ intervals}\).
The clock gains 15 seconds during each of these 8 intervals.
Total time gained = \(8 \text{ intervals} \times 15 \text{ seconds/interval} = 120 \text{ seconds}\).

Step 3: Converting Error and Finding Final Time:
We now convert the accumulated error from seconds into minutes to make it useful.
We know there are 60 seconds in one minute.
Total time gained = \(\frac{120 \text{ seconds}}{60 \text{ seconds/minute}} = 2 \text{ full minutes}\).
Because the clock is running "fast", it will display a time that is ahead of the true time.
The true time is Monday 9:00 A.M.
The clock will display: \(9:00 \text{ A.M.} + 2 \text{ minutes} = 9:02 \text{ A.M.}\)

Final Answer:
Therefore, the correct answer is (A) 9 : 02 A.M.

Concept:
This question tests non-verbal reasoning, specifically focusing on "Embedded Figures".
An embedded figure problem presents a simple, distinct target shape (the problem figure) and asks you to find which of the complex, busy option figures contains that exact target shape hidden within its lines.
The embedded shape must exactly match the original target in geometry and proportion, and typically in orientation (unless rotation is explicitly permitted).

Step 1: Analyzing the Target Figure:
Let’s carefully examine the specific geometry of the problem figure provided.
The figure resembles a wide, open triangular shape or an arrowhead pointing horizontally to the left.
It consists of two lines meeting at a sharp vertex on the left. The top line extends slightly further backward (to the right) than the bottom line.
Our goal is to trace this exact set of connected lines inside one of the complex boxes.

Step 2: Evaluating the Option Figures:
We visually scan the internal line structure of each option to find the hidden shape.
Option (B), (C), and (D): While these squares contain various internal horizontal, vertical, and diagonal lines, carefully tracing them does not reveal the specific, wide, asymmetric left-pointing angle required without breaking lines or changing proportions.
Option (A): This option shows a square box containing two full corner-to-corner diagonal lines, forming an ’X’ in the center, along with a top horizontal line.
If we trace the path starting from the center intersection of the ’X’, moving down and left along the diagonal, then sharply pivoting to trace back along the other diagonal line upwards and right, we perfectly recreate the left-pointing acute angle of the target figure.
The lines forming the diagonals inside Option (A) provide the exact geometrical framework needed to seamlessly embed the target shape.

Final Answer:
Therefore, the correct answer is (A) 1.

Concept:
This question evaluates pattern recognition skills through a "Figure Matrix" puzzle.
A figure matrix is a grid (in this case, 2x2) containing distinct shapes or symbols that change according to a specific, hidden logical rule.
The rule usually governs how the figures change as you move horizontally from left to right across a row, or vertically from top to bottom down a column.
Once the underlying rule is identified in the complete rows/columns, it is applied to the incomplete section to find the missing figure.

Step 1: Analyzing the Matrix Horizontally (Rows):
Let’s look at the first row (top row) of the matrix.
Top Left Box: Contains exactly 1 circle.
Top Right Box: Contains exactly 3 circles, arranged in a horizontal line.
The mathematical relationship here is an increase: \(1 \rightarrow 3\) (an addition of 2 circles).

Now let’s look at the second row (bottom row).
Bottom Left Box: Contains exactly 2 circles.
Bottom Right Box: This is the missing piece (?).
If we apply the same horizontal additive rule (+2) from the first row to the second row:
\(2 \text{ circles} + 2 \text{ circles} = 4 \text{ circles}\).
Our hypothesis is that the missing box must contain exactly 4 circles.

Step 2: Analyzing the Matrix Vertically (Columns) to Confirm:
Let’s double-check our hypothesis by looking at the columns vertically.
First Column (Left): Top box has 1 circle, Bottom box has 2 circles. Pattern is \(+1\).
Second Column (Right): Top box has 3 circles. If the \(+1\) vertical pattern holds, the bottom right box should have \(3 + 1 = 4\) circles.
Both horizontal and vertical logical analyses independently point to the missing box containing exactly 4 circles.

Step 3: Evaluating the Options:
We need to find an option containing exactly 4 circles.
Option (A): Shows 3 circles in a vertical line. Incorrect.
Option (B): Shows exactly 4 circles arranged in a neat 2x2 square grid. This matches our logical deduction perfectly.
Option (C): Shows 5 circles (arranged like the face of a dice). Incorrect.
Option (D): Shows 6 circles (arranged in a 3x2 grid). Incorrect.

Final Answer:
Therefore, the correct answer is (B) 2.

Concept:
This matching question is an exercise in non-verbal reasoning, specifically dealing with Mirror Images.
When an object or word is placed in front of a standard vertical mirror, its reflection undergoes a phenomenon called "lateral inversion."
This means the left side of the original object becomes the right side of the image, and the right side becomes the left.
For a string of letters, this has two effects:
1. The sequence of the letters is entirely reversed (the last letter appears first, the first appears last).
2. Each individual letter is laterally inverted (flipped horizontally). (Note: Symmetrical letters like ’A’ look the same when flipped).

Step 1: Finding the Mirror Image for (A) A B C:
Original string: A, then B, then C.
In the mirror, the sequence reverses. The image will show the mirrored ’C’ first on the left, then the mirrored ’B’ in the middle, and finally the mirrored ’A’ on the far right.
Mirror of C is Ɔ. Mirror of B is ᙠ. Mirror of A remains A.
Therefore, the full mirror image is "Ɔ ᙠ A".
Looking at List-II, this matches exactly with (I). So, (A) pairs with (I).

Step 2: Finding the Mirror Image for (B) A C B:
Original string: A, then C, then B.
Sequence reverses: Mirrored ’B’ first, then mirrored ’C’, then mirrored ’A’.
Resulting image: "ᙠ Ɔ A".
Looking at List-II, this matches exactly with (II). So, (B) pairs with (II).

Step 3: Finding the Mirror Image for (C) B A C:
Original string: B, then A, then C.
Sequence reverses: Mirrored ’C’ first, then mirrored ’A’, then mirrored ’B’.
Resulting image: "Ɔ A ᙠ".
Looking at List-II, this matches exactly with (IV). So, (C) pairs with (IV).

Step 4: Finding the Mirror Image for (D) B C A:
Original string: B, then C, then A.
Sequence reverses: Mirrored ’A’ first, then mirrored ’C’, then mirrored ’B’.
Resulting image: "A Ɔ ᙠ".
Looking at List-II, this matches exactly with (III). So, (D) pairs with (III).

Step 5: Finalizing the Matches:
Reviewing our logical deductions, the correct matching sequence is: (A)-(I), (B)-(II), (C)-(IV), and (D)-(III).
Scanning the given multiple-choice options, Option (C) reflects this exact specific combination.

Final Answer:
Therefore, the correct answer is (C) (A) - (I), (B) - (II), (C) - (IV), (D) - (III).

Concept:
This question tests basic number classification skills through an "Odd One Out" format.
To find the odd number, we must analyze the mathematical properties of all four given numbers and identify a specific rule or characteristic that applies to three of them, but fails to apply to the fourth.
Common numerical properties tested include even/odd, perfect squares/cubes, divisibility rules, and prime/composite classifications.

Step 1: Analyzing the Set of Numbers:
The numbers provided are 23, 43, 73, and 93.
At a quick glance, they are all two-digit numbers, they are all odd numbers, and they all end in the digit 3.
Since these superficial traits are shared by all options, the distinguishing factor must lie deeper in their mathematical factors. We need to check for prime vs. composite numbers.
A prime number is a number strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself. A composite number has more than two divisors.

Step 2: Evaluating Each Option Individually:
Let’s check the divisibility of each number.
(A) 23: It is not divisible by 2, 3, 5, or 7. Its only divisors are 1 and 23. Therefore, 23 is a prime number.
(B) 43: It is not divisible by 2, 3, 5, or 7. Its only divisors are 1 and 43. Therefore, 43 is a prime number.
(C) 73: It is not divisible by 2, 3, 5, 7, or 8. Its only divisors are 1 and 73. Therefore, 73 is a prime number.
(D) 93: Let’s apply a basic divisibility rule. The sum of its digits is \(9 + 3 = 12\). Since 12 is cleanly divisible by 3, the number 93 is also completely divisible by 3.
Specifically, \(93 \div 3 = 31\).
This means 93 has multiple divisors: 1, 3, 31, and 93. Because it has more than two divisors, it is a composite number.

Step 3: Conclusion:
Options A, B, and C form a distinct group of prime numbers. Option D is the only composite number in the list.
This difference mathematically isolates it from the rest of the group.

Final Answer:
Therefore, the correct answer is (D) 93.

Concept:
This problem is based on the logical reasoning concept of Ranking and Ordering.
When objects or people are arranged in a linear row, a specific individual’s position can be counted from either the extreme left end or the extreme right end.
The total number of people in the row is mathematically linked to an individual’s left and right positions by a strict, standard formula.
The fundamental formula is: \(\text{Total} = \text{Position from Left} + \text{Position from Right} - 1\).
We subtract 1 at the end because adding the left and right positions intrinsically counts that specific individual twice.

Step 1: Identifying the Given Variables:
Let’s extract the known numbers from the problem statement to use in our formula.
The total number of boys standing in the row is explicitly given as 45.
So, \(\text{Total (T)} = 45\).
Vijay’s specific position counted from the right end of the row is given as 13.
So, \(\text{Position from Right (R)} = 13\).
We need to calculate Vijay’s position counting from the left end. Let’s call this variable \(L\).

Step 2: Applying the Ranking Formula:
We take our standard formula and rearrange it algebraically to solve directly for the left position (\(L\)).
\(\text{Total} = L + R - 1\)
To isolate \(L\), we move the other variables to the other side of the equation:
\(L = \text{Total} - R + 1\)

Step 3: Executing the Calculation:
Substitute the identified numerical values into the rearranged formula:
\(L = 45 - 13 + 1\)
First, perform the subtraction:
\(45 - 13 = 32\)
Now, add the 1 back in:
\(L = 32 + 1 = 33\)
Therefore, counting from the left end, Vijay occupies the \(33^\text{rd}\) position.

Final Answer:
Therefore, the correct answer is (B) \(33^\text{rd}\).

Concept:
This question tests logical reasoning through an arithmetic Number Series.
A number series is a sequence of numbers that follow a specific, hidden mathematical pattern or rule.
Often, complex-looking series are actually formed by grouping numbers together in pairs, or by interlacing two entirely separate independent series into one sequence.
Identifying how the numbers cluster or alternate is the key to cracking the code.

Step 1: Analyzing the Progression of the Series:
Let’s look at the full sequence provided: 5, 15, 9, ?, 11, 33, 13, 39.
The numbers bounce up and down dramatically: \(5 \text{ (up to)} 15 \text{ (down to)} 9 \dots\)
This erratic behavior strongly suggests the series is not a single, continuous progression but rather a combination.
Let’s try grouping the numbers into consecutive pairs to see if an internal rule emerges.
Pair 1: (5, 15)
Pair 2: (9, ?)
Pair 3: (11, 33)
Pair 4: (13, 39)

Step 2: Identifying the Pattern within Pairs:
Let’s examine the mathematical relationship strictly between the two numbers inside each confirmed pair.
Looking at Pair 1: How do we get from 5 to 15? We multiply by 3. (\(5 \times 3 = 15\)).
Looking at Pair 3: How do we get from 11 to 33? We multiply by 3. (\(11 \times 3 = 33\)).
Looking at Pair 4: How do we get from 13 to 39? We multiply by 3. (\(13 \times 3 = 39\)).
The underlying rule governing this series is extremely clear: Every odd-positioned number is multiplied by exactly 3 to generate the immediately succeeding even-positioned number.

Step 3: Applying the Pattern to Find the Missing Number:
We apply this solid, confirmed rule to the incomplete second pair.
Pair 2 is (9, ?).
The first number of the pair is 9.
Applying the rule, we multiply this number by 3.
\(9 \times 3 = 27\).
Therefore, the missing number required to complete the pattern is exactly 27.

Final Answer:
Therefore, the correct answer is (C) 27.

Concept:
This problem falls under the logical reasoning category of Linear Seating Arrangements.
These puzzles provide a set of specific positional clues (constraints) regarding a group of people or objects arranged in a straight line.
The objective is to synthesize all these distinct clues to construct the one unique, correct arrangement that satisfies every single condition.
It is highly recommended to draw a visual diagram of empty slots and carefully fill them in step-by-step.

Step 1: Setting up the Bench:
The problem states there are 6 distinct individuals: A, B, C, D, E, and F.
They are all sitting on a single bench, meaning they form a straight line of 6 consecutive positions.
Let’s create a visual representation of 6 empty slots, numbered from left to right:
[ 1 ] - [ 2 ] - [ 3 ] - [ 4 ] - [ 5 ] - [ 6 ]

Step 2: Placing the Central and Corner Positions:
Clue 1: "E and F are sitting in the centre."
In a 6-person row, the absolute center consists of slots 3 and 4.
So, E and F occupy slots 3 and 4 (in either order: EF or FE).
Clue 2: "A and B are sitting on the two corners."
The corners are the extreme ends: slot 1 (extreme left) and slot 6 (extreme right).
So, A and B occupy slots 1 and 6 (in either order: A...B or B...A).

Step 3: Using the Directional Clue to Lock Positions:
Clue 3: "C is sitting to the left of A."
This is the most critical clue. It tells us exactly which corner A must be sitting in.
If A were sitting on the extreme left corner (slot 1), it would be physically impossible for anyone (including C) to sit to his left, as there are no seats further left.
Therefore, A absolutely must be sitting on the extreme right corner (slot 6).
This forces B to occupy the remaining corner, the extreme left (slot 1).
Since C must sit immediately to the left of A, C must be placed in slot 5.
Current arrangement: [ B ] - [ ? ] - [ E/F ] - [ F/E ] - [ C ] - [ A ]

Step 4: Identifying the Final Position:
We have placed 5 out of the 6 individuals. The only person left without a designated seat is D.
The only remaining empty slot on our bench is slot 2.
Therefore, D must sit in slot 2.
Final complete arrangement: [ B ] - [ D ] - [ E/F ] - [ F/E ] - [ C ] - [ A ]

Step 5: Answering the Question:
The question asks: "Who is sitting to the right of B?"
Looking at our final, completed arrangement, B is sitting in slot 1.
The person sitting immediately to the right of B (in slot 2) is D.

Final Answer:
Therefore, the correct answer is (A) D.

Concept:
This question tests spatial non-verbal reasoning, specifically focusing on "Figure Construction" or shape synthesis.
You are provided with a "problem figure" containing several loose, disconnected shapes or puzzle pieces.
The task is to mentally manipulate (move, rotate, and combine) all these individual loose pieces without altering their fundamental size or shape to seamlessly construct one of the consolidated, complex figures presented in the options.

Step 1: Analyzing the Pieces in the Problem Figure:
Carefully examine the problem box to inventory the available puzzle pieces.
The problem figure displays exactly four distinct, separate pieces.
Each individual piece is a perfect, identical, small empty square with bold black outlines.
Our objective is to find which option can be built using exactly four small identical squares.

Step 2: Evaluating the Option Figures:
We must visually check if the shapes in each option can be broken down into exactly four identical small squares.
Option (A): This figure consists of a large square divided diagonally by two criss-crossing lines forming a large ’X’, creating four large triangles, not squares. This cannot be built from our pieces.
Option (B): This figure consists of a large square divided by three horizontal lines, creating four long, thin horizontal rectangles. This cannot be built from our small square pieces.
Option (C): This figure consists of a large square containing intersecting vertical, horizontal, and diagonal lines (resembling a Union Jack or an eight-point star framework). It creates a multitude of tiny triangles. This cannot be built from our pieces.
Option (D): This figure consists of a single large square divided perfectly by one central vertical line and one central horizontal line. This internal cross explicitly divides the large box into exactly a 2x2 grid.
This grid is composed of exactly four small, identical squares perfectly joined together at their edges.

Step 3: Conclusion:
If you take the four loose small squares from the problem figure and physically slide them together so their borders touch (two on top, two on the bottom), you construct the exact 2x2 larger grid shown perfectly in Option (D).

Final Answer:
Therefore, the correct answer is (D) 4.