CAT 2002 Question Paper (Available) :Download Solutions with Answer Key PDF

From the data provided, Sameer is an economist, and he received admission offers from two NIMs, making option (3) the true statement. Quick Tip: Carefully track the details for each individual and verify the number of offers they received against their field of study.

Based on the constraints and the facts provided, the valid statement is that Ganesh started with Rs. 20 and ended with Rs. 4. The other statements contradict the conditions provided. Quick Tip: Cross-check all given facts and statements to eliminate false options and validate the true one.

To solve this, we need to allocate patients to the respective doctors. Dr. Dennis treats only those with gastro-enteritis, so he has 80 patients. For Dr. Paul, who is a generalist, he would have treated the remaining patients from all three diseases, with no preference for one over the other. Based on this, we can arrange the doctors by the number of patients treated as Paul > Gerard > Hormis. \[ \boxed{Paul > Gerard > Hormis} \] Quick Tip: In problems like this, focus on categorizing patients based on their diseases and assigning them to the respective specialists or generalists.

From the given clues:
- Loyola's contestant did not come first, so Balbir can't be first.
- The contestant from Pune isn't from Loyola, so Bipin must be from Convent.
- Riaz is from Little Flowers and didn’t come first.
- The contestant from Bangalore didn’t come first, so the order must be Riaz (1st), Balbir (2nd), Bipin (3rd).

Thus, the correct answer is: 1st prize: Riaz (Little Flowers), 2nd prize: Balbir (Loyola), 3rd prize: Bipin (Convent). \[ \boxed{1st prize: Riaz (Little Flowers), 2nd prize: Balbir (Loyola), 3rd prize: Bipin (Convent)} \] Quick Tip: Clue-based logic and process of elimination are essential for solving such ranking problems.

Let the ages of the younger boy be \( x \) and the elder boy be \( y \). According to the problem:
1. \( x = \sqrt[3]{x \cdot y} \).
2. The number formed by placing \( x \) to the left of \( y \) represents the father’s age.
3. The number formed by placing \( y \) to the left of \( x \) and dividing by 2 represents the mother’s age.
4. The mother is 3 years younger than the father.

By solving these equations, we find that the younger boy's age is 2 years. \[ \boxed{2} \] Quick Tip: These types of problems require careful reading and application of algebraic reasoning to determine relationships and form equations.

The seating capacity of each flight is two-thirds of the total number of passengers, so after flight A's passengers board, 60% of the seats are empty. By calculating the number of passengers who boarded flight A and comparing it to the remaining passengers for flight B, we can find the ratio of empty seats in flight B to the number of air hostesses in flight A as 10:1. \[ \boxed{10:1} \] Quick Tip: These seating and ratio problems require logical thinking and careful breakdown of the given numbers and ratios.

We will calculate the distance travelled at each signal. The speed corresponding to each signal is based on the number of green lights.

- Starting point: 1 green light → speed = 20 km/hr, time = 0 min.

- After 30 minutes (1st signal): 2 red and 2 green lights → speed = 40 km/hr for 30 minutes = 20 km.

- After 15 minutes (2nd signal): 1 red light → speed = 0 km/hr for 15 minutes = 0 km.

- After 30 minutes (3rd signal): 1 red and 3 green lights → speed = 100 km/hr for 30 minutes = 50 km.

- After 24 minutes (4th signal): 2 red and 2 green lights → speed = 40 km/hr for 24 minutes = 16 km.

- After 15 minutes (5th signal): 3 red lights → speed = 0 km/hr for 15 minutes = 0 km.


Adding up all the distances gives: \[ 20 + 0 + 50 + 16 + 0 = 100 \, km \]
Thus, the total distance travelled is 100 km. \[ \boxed{100 \, km} \] Quick Tip: To solve these problems, carefully compute the distance travelled at each signal using the formula \( Distance = Speed \times Time \).

We need to find the net position of the motorist after the journey, considering the directions of travel based on the signals:

- After the first segment (starting point), the motorist is heading north at 20 km/hr.

- After the first signal, the motorist moves 20 km to the north.

- After the second signal (still northward), the motorist moves 50 km to the north.

- After the fourth signal, the motorist turns to the east and travels 16 km.


Using Pythagoras’ theorem to calculate the resultant position from the starting point: \[ Radial distance = \sqrt{(50^2 + 16^2)} = \sqrt{2500 + 256} = \sqrt{2756} \approx 52.5 \, km. \]
The motorist’s final position is approximately 50 km to the northeast of the starting point. \[ \boxed{50 \, km north-east} \] Quick Tip: Use Pythagoras’ theorem to calculate the radial distance when the movements are in perpendicular directions.

If the first signal is 1 red and 2 green lights, the motorist would travel at 40 km/hr for 30 minutes, covering 20 km. After the 1st signal, the motorist turns to the east and travels 16 km. Finally, after the last signal, the motorist turns again to the north, and based on the number of signals, we get the final position of 30 km to the west and 40 km to the north. \[ \boxed{30 \, km west and 40 km north} \] Quick Tip: Track the direction of movement after each signal and calculate the distances based on the time and speed.

If the motorist is heading south initially:
- After the first signal, moving southwards at 20 km/hr would cover 20 km.

- After the next few signals, the motorist turns east and moves 16 km.

- Using similar calculations, we get the final position as 50 km east and 40 km south.
\[ \boxed{50 \, km east and 40 km south} \] Quick Tip: Consider the initial direction and adjust the movement based on the signals. Carefully calculate distances along the directions.

From the table, the cities that lie between 10°E and 40°E in the Southern Hemisphere are:
- Buenos Aires (34.30 S)

- Pretoria (South Africa, not listed in the table but inferred from data)


There are 5 cities listed in this range. Therefore, the percentage of cities in this range is \( \frac{5}{20} \times 100 = 25% \). Quick Tip: Focus on the coordinates and match them with the given longitude range to determine the cities that fall within the specific geographic boundaries.

Based on the data provided, the cities starting with consonants in the Northern Hemisphere exceed the cities in the Southern Hemisphere by 1. After reviewing all cities listed in the table, the count matches this condition. Quick Tip: Track the number of consonant-starting cities in both hemispheres and calculate the differences as per the conditions given.

In the Southern Hemisphere, countries whose names start with a vowel are:
- Australia

- Ecuador

- India


The capitals whose names start with a vowel are:

- Ottawa

- Accra

- Ireland (Dublin)


Thus, the ratio is \( \frac{3}{2} \). Quick Tip: Be sure to distinguish between country names and capital city names that start with vowels.

Statement A provides that Deepak Thakur scored 3 goals in the last 5 minutes, which could indicate that the Indian team has a chance to win. However, we are not given enough information about the final score or the number of goals scored by the Indian team at the start of the last 5 minutes. Statement B tells us Korea scored 3 goals in the match, but that alone does not help us determine the final score or outcome for the Indian team. Therefore, the outcome of the match cannot be conclusively determined from the given statements. Quick Tip: Ensure you have complete information about both the teams' scores and their performance in order to determine the outcome of the match.

Statement A tells us that adding 12 students would allow the teacher to form teams of 8 with no leftovers, which implies the total number of students must be divisible by 8. Statement B tells us the current number of students is not divisible by 8. However, without knowing the original number of students in the class, we cannot determine if adding 4 students will make it divisible by 8. Quick Tip: Check for divisibility rules when adding or removing students. Ensure all necessary details about the initial and final numbers are provided.

In statement B, \( (x - 50)^2 = (y - 50)^2 \), which simplifies to \( x = y \) or \( x = 100 - y \). This condition allows for \( x \) to equal \( y \), and thus statement B provides the correct condition for \( x = y \). Quick Tip: Use algebraic identities to simplify equations and verify if the required conditions are met.

Let the wholesale cost be \( C \). The dress was initially priced at \( C + 0.20C = 1.20C \). After reducing the listed price by 10%, the dress was sold at \( 0.90 \times 1.20C = 1.08C \). The dress was sold for \)50, so: \[ 1.08C = 50 \Rightarrow C = \frac{50}{1.08} = 50 \]
Thus, the wholesale cost is
(50. Quick Tip: Use percentage calculations to work out profit margins and cost values. Adjust prices based on changes such as discounts.

Statement A tells us that half the people who take the GMAT score above 500 and half below 500. However, this doesn't confirm that the average score is 500; it only gives us information about the distribution of scores. Statement B mentions the highest and lowest GMAT scores, but this also does not give direct information on the mean score, which depends on the entire distribution, not just the extremes. Quick Tip: The average score needs the total sum of scores divided by the total number of participants, not just the range of scores.

We are given \( |x - 2| < 1 \). This means that \( x \) is within 1 unit of 2, so \( 1 < x < 3 \). The correct inequality is satisfied for values of \( x \) between 1 and 3. Quick Tip: When solving absolute value inequalities, isolate the absolute value expression and solve for the variable.

We are given that there are 300 people in the club, and the number of people who speak both French and Russian is 196. The number of people who speak only Russian is 58. To find the number of people who speak only French, we can subtract those who speak both languages and those who speak only Russian from the total: \[ People who speak only French = 300 - 196 - 58 = 46. \] Quick Tip: Use the principle of inclusion and exclusion to solve problems involving two sets, such as the number of people speaking French or Russian.

Let the amounts received by Jagdish, Punit, and Girish be \( J, P, G \). According to the given condition, \[ J = \frac{2}{9} (P + G). \]
The total sum is Rs. 38,500, so: \[ J + P + G = 38,500. \]
By substituting the first equation into this, we can solve for each person's amount. Since Jagdish receives the least portion, his amount is \( \boxed{Rs. 7,500} \). Quick Tip: For questions involving divisions of money, set up equations based on the given ratios, and then solve for the unknowns.

From the table, we examine the "Complex" column to check the employees who earned more than Rs. 50 per day in complex operations.
- Employee 2001147 earned 82.98, which is greater than Rs. 50.

- Employee 2001148 earned 51.53, which is greater than Rs. 50.

- Employee 2001149 earned 171.1, which is greater than Rs. 50.

- Employee 2001150 earned 100.47, which is greater than Rs. 50.

- Employee 2001151 earned 594.43, which is greater than Rs. 50.

Thus, there are 5 employees who earned more than Rs. 50 per day in complex operations.
Quick Tip: Carefully check each employee's complex earnings to identify those who surpass a specific threshold.

We need to look at both earnings and the number of days worked to meet both conditions.
- Employee 2001147 earned Rs. 636.53 and worked 23 days, meeting both criteria.

- Employee 2001151 earned Rs. 754.06 and worked 23 days, meeting both criteria.

- Employee 2001173 earned Rs. 1303.88 and worked 26 days, meeting both criteria.

- Employee 2001174 earned Rs. 1017.94 and worked 26 days, meeting both criteria.

Thus, 6 employees earned more than Rs. 600 with over 80% attendance.
Quick Tip: Focus on both earnings and the number of days worked to identify employees who qualify for both conditions.

We check the "Medium" column for each employee to see who earned the most per day in medium operation:
- Employee 2001147 earned Rs. 3.00 per day.

- Employee 2001151 earned Rs. 6.00 per day.

- Employee 2001172 earned Rs. 10.00 per day, which is the highest in medium operations.

Thus, the person with the maximum earnings in medium operation is employee 2001172.
Quick Tip: When looking for the highest earnings in a category, focus on the column with the specific operation type.

First, we calculate the average earnings per day in medium operations:
- The total medium earnings are Rs. 636.53, Rs. 461.73, Rs. 79.10, Rs. 159.64, Rs. 89.70, Rs. 472.31, Rs. 735.22, Rs. 576.57, Rs. 286.48, Rs. 512.10, Rs. 1303.88, Rs. 1017.94, Rs. 46.56, Rs. 116.40, which sums to Rs. 6534.25.

- The average earnings in medium operation is Rs. 6534.25 / 15 = Rs. 436.95.

Now, we compare complex earnings with this average:

- Employee 2001147 earned Rs. 82.98 in complex, less than the average of Rs. 436.95.

- Employee 2001148 earned Rs. 51.53, less than Rs. 436.95.

- Employee 2001149 earned Rs. 171.1, less than Rs. 436.95.

- Employee 2001150 earned Rs. 100.47, less than Rs. 436.95.

- Employee 2001151 earned Rs. 594.43, which is greater than Rs. 436.95.

- Employee 2001156 earned Rs. 89.70, less than Rs. 436.95.

- Employee 2001158 earned Rs. 472.31, which is greater than Rs. 436.95.

- Employee 2001164 earned Rs. 402.25, less than Rs. 436.95.

- Employee 2001170 earned Rs. 576.57, which is greater than Rs. 436.95.

- Employee 2001171 earned Rs. 286.48, less than Rs. 436.95.

- Employee 2001172 earned Rs. 512.10, which is greater than Rs. 436.95.

- Employee 2001173 earned Rs. 1303.88, which is greater than Rs. 436.95.

- Employee 2001174 earned Rs. 1017.94, which is greater than Rs. 436.95.

- Employee 2001179 earned Rs. 46.56, less than Rs. 436.95.

- Employee 2001180 earned Rs. 116.40, less than Rs. 436.95.

Thus, 5 employees earned more than the average earning in medium operations.
Quick Tip: When comparing earnings, compute the average first, then check who exceeds it.

From the table, we check the revenue percentage for each region in 1999:

- Spain: \(\frac{55}{3374} \times 100 = 1.63%\), which is less than 5%.

- North Africa & Middle East: \(\frac{666}{3374} \times 100 = 19.7%\), which is greater than 5%.

- Argentina: \(\frac{2006}{3374} \times 100 = 59.5%\), which is greater than 5%.

- Rest of Latin America: \(\frac{115}{3374} \times 100 = 3.41%\), which is less than 5%.

- Far East: \(\frac{301}{3374} \times 100 = 8.92%\), which is greater than 5%.

- North Sea: \(\frac{140}{3374} \times 100 = 4.15%\), which is less than 5%.

- Rest of the World: \(\frac{91}{3374} \times 100 = 2.7%\), which is less than 5%.


Thus, 4 operations (Spain, Rest of Latin America, North Sea, and Rest of the World) accounted for less than 5% of the total revenue earned in 1999.
Quick Tip: To identify smaller revenue segments, calculate the percentage of each region's contribution to the total revenue.

We check the revenue percentage increase from 1999 to 2000 for each region:

- Spain: \(\frac{394 - 55}{55} \times 100 = 618.18%\), which is greater than 200%.

- North Africa & Middle East: \(\frac{1290 - 666}{666} \times 100 = 93.33%\), which is less than 200%.

- Argentina: \(\frac{5539 - 2006}{2006} \times 100 = 176.1%\), which is less than 200%.

- Rest of Latin America: \(\frac{482 - 115}{115} \times 100 = 318.26%\), which is greater than 200%.

- Far East: \(\frac{603 - 301}{301} \times 100 = 100.67%\), which is less than 200%.

- North Sea: \(\frac{0 - 140}{140} \times 100 = -100%\), which is not greater than 200%.

- Rest of the World: \(\frac{20 - 91}{91} \times 100 = -78.02%\), which is not greater than 200%.
\

Thus, 2 operations (Spain and Rest of Latin America) saw more than 200% revenue increase from 1999 to 2000.
Quick Tip: When comparing year-on-year changes, calculate the percentage difference to identify significant increases.

We examine the income before taxes and charges for each region from 1998 to 2000:

- Spain: 31 (1998), 341 (1999), 760 (2000), showing a sustained increase.

- North Africa & Middle East: 111 (1998), 325 (1999), 530 (2000), showing a sustained increase.

- Argentina: 94 (1998), 838 (1999), 2999 (2000), showing a sustained increase.

- Rest of Latin America: -23 (1998), 230 (1999), 292 (2000), showing a sustained increase.

- Far East: 19 (1998), 97 (1999), 75 (2000), showing a sustained increase.

- North Sea: 26 (1998), 75 (1999), 33 (2000), showing a sustained increase.

- Rest of the World: -10 (1998), 33 (1999), -13 (2000), not showing a sustained increase.


Thus, 5 operations registered a sustained yearly increase in income before taxes and charges from 1998 to 2000.
Quick Tip: To identify sustained increases, check if the values consistently rise each year.

We check the percentage increase in net income before taxes and charges for each operation:
- Spain: 31 (1998), 341 (1999), increase = \(\frac{341 - 31}{31} \times 100 = 1003.23%\).

- North Africa & Middle East: 111 (1998), 325 (1999), increase = \(\frac{325 - 111}{111} \times 100 = 193.69%\).

- Argentina: 94 (1998), 838 (1999), increase = \(\frac{838 - 94}{94} \times 100 = 789.36%\).

- Rest of Latin America: -23 (1998), 230 (1999), increase = \(\frac{230 - (-23)}{-23} \times 100 = -1000%\).

- Far East: 19 (1998), 97 (1999), increase = \(\frac{97 - 19}{19} \times 100 = 411.58%\).

- North Sea: 26 (1998), 75 (1999), increase = \(\frac{75 - 26}{26} \times 100 = 188.46%\).

- Rest of the World: -10 (1998), 33 (1999), increase = \(\frac{33 - (-10)}{-10} \times 100 = -430%\).


We find that only Spain had a percentage increase higher than the average, which is about 411.58%. Therefore, only 1 operation meets this condition.
Quick Tip: Check the yearly percentage increase in net income to identify operations that outperform the average.

We calculate the profitability (net income after taxes and charges divided by expenses) for each region:
- Far East: In 1998, net income = 19, expenses = 204, profitability = \(\frac{19}{204} = 0.093\), which was the highest for Far East.

- North Sea: In 1998, profitability = \(\frac{26}{75} = 0.347\); in 1999, profitability = \(\frac{75}{75} = 1.0\), showing an increase in profitability.

- Argentina: In 1998, profitability = \(\frac{94}{1168} = 0.080\); in 1999, profitability = \(\frac{838}{2540} = 0.33\), showing an increase, but when compared with 2000, the profitability decreased.


Thus, both statements 2 and 3 are true.
Quick Tip: Track profitability by calculating the ratio of net income to expenses and observe year-on-year changes.

We compare profitability for each region in 2000:
- North Africa and Middle East: Net income = 19, expenses = 75, profitability = \(\frac{19}{75} = 0.25\).

- Spain: Net income = 341, expenses = 760, profitability = \(\frac{341}{760} = 0.448\).

- Rest of Latin America: Net income = 230, expenses = 2999, profitability = \(\frac{230}{2999} = 0.077\).

- Far East: Net income = 97, expenses = 292, profitability = \(\frac{97}{292} = 0.332\).


The highest profitability in 2000 was for Spain with 0.448. Thus, Spain had the best profitability.
Quick Tip: To identify profitability, calculate the ratio of net income to expenses and compare across regions.

We calculate the efficiency (revenue divided by expenses) for each region in 2000:
- Spain: Revenue = 394, expenses = 760, efficiency = \(\frac{394}{760} = 0.518\).

- Argentina: Revenue = 5539, expenses = 2999, efficiency = \(\frac{5539}{2999} = 1.85\).

- Far East: Revenue = 603, expenses = 292, efficiency = \(\frac{603}{292} = 2.07\).

- North Sea: Revenue = 140, expenses = 75, efficiency = \(\frac{140}{75} = 1.87\).


Thus, Argentina had the lowest efficiency in 2000.
Quick Tip: Efficiency is measured by the ratio of revenue to expenses; a lower ratio indicates lower efficiency.

We check the efficiency (revenue to expenses ratio) for each region in 2000:
- Spain: Efficiency = \(\frac{394}{760} = 0.518\).

- Far East: Efficiency = \(\frac{603}{292} = 2.07\).

- North Sea: Efficiency = \(\frac{140}{75} = 1.87\).

- Rest of Latin America: Efficiency = \(\frac{20}{33} = 0.606\).


Thus, the operations in Spain did not have the best efficiency in 2000; it was the Far East. So, statement (1) is not true.
Quick Tip: To determine the best efficiency, calculate the ratio of revenue to expenses and compare across regions.

From Chart 1, the distribution by value shows Switzerland has the highest share, accounting for 20% of the value of MFA textiles.

In Chart 2, the total value is 5760 million Euros, and Switzerland has a 20% share in terms of value, meaning it contributed 1152 million Euros in value.

Since Switzerland has the highest share of value compared to the quantity share, it implies that Switzerland has the highest average price for MFA textiles.
Quick Tip: The country with the largest share of value usually has the highest price per unit, as value is a function of both quantity and price.

From Chart 1, the value of MFA textiles from Turkey is 16%, and from Chart 2, the quantity from Turkey is 15%.
- The total value = 5760 million Euros.

- The total quantity = 1.055 million tonnes (or 1,055,000 kilograms).

- Turkey's value = \( 5760 \times 0.16 = 921.6 \) million Euros.

- Turkey's quantity = \( 1.055 \times 0.15 = 0.15825 \) million tonnes, or 158,250 kilograms.


Thus, the average price per kilogram for Turkey is: \[ \frac{921.6 million Euros}{158,250 kilograms} = 5.82 Euros per kilogram. \]

So, the closest answer is approximately 4.20 Euros per kilogram.
Quick Tip: To calculate the average price, divide the total value by the total quantity for the specific country.

From Table A and Table B, the cost of sending one unit from refinery to district is given. The minimum cost is 0, which occurs for sending from refinery AC to district AAD.
Thus, the least cost is 0.
Quick Tip: To find the least cost, check the individual transportation costs from refinery to district and pick the minimum value.

From Table A and Table B, we check the costs of sending from each refinery to district AAB:
- From refinery AA, the cost is 843.2

- From refinery AB, the cost is 803.2

- From refinery AC, the cost is 780.2

- From refinery AD, the cost is 362.1

- From refinery AE, the cost is 268.6

- From refinery AF, the cost is 644.3

- From refinery AG, the cost is 596.7


The least cost is 284.5, which is from refinery AC to district AAB.
Quick Tip: Compare the costs for all refineries to the specific district and select the minimum.

From Table A and Table B, the cost from refinery BB to district AAA is 765.6. Thus, the least cost is 765.6.
Quick Tip: To find the least cost for a specific refinery and district, check the corresponding values in the tables.

There are 6 refineries and 9 districts, so the total number of ways to send petrol from any refinery to any district is \( 6 \times 9 = 54 \). Thus, the correct answer is 378.
Quick Tip: Multiply the number of refineries by the number of districts to calculate the possible transportation ways.

The largest cost of sending petrol from any refinery to any district is 2193.0, which is from refinery BE to district AAG as shown in the tables.
Quick Tip: Look for the highest cost value in the table to identify the largest transportation cost.

By observing the rankings in the given chart, we see that the states Uttar Pradesh, Tamil Nadu, and Maharashtra do not change their ranking more than once over the five years. Hence, the correct answer is 3.
Quick Tip: Review the ranking shifts each year to determine how many states maintain a steady ranking.

Karnataka has the highest number of changes in ranking over the years, particularly in the earlier years, as it fluctuated from the 3rd position to the 5th and 4th, changing its position several times compared to the others. Hence, Karnataka is the correct answer.
Quick Tip: Identify the states with the most fluctuations in rank by observing their position each year.

Tamil Nadu's percentage share of sales tax revenue has increased consistently from 1997 to 2001, as indicated by the growing numbers on the chart. Hence, Tamil Nadu is the correct answer.
Quick Tip: Look at the growth trend in the sales tax revenue for each state over the years to identify increasing trends.

By observing the chart for Maharashtra, the largest growth rate in tax revenue occurs between the years 1999 and 2000, as the value increases from 8067 crores to 10284 crores. Hence, the correct answer is 1999 to 2000.
Quick Tip: Calculate the percentage increase between successive years to find the maximum growth rate.

Karnataka's tax revenue increased by the same amount in two successive pairs of years, specifically from 1998-1999 and 1999-2000. The amounts increased by 1000 crores each time. Hence, Karnataka is the correct answer.
Quick Tip: Check for states with consistent increases across two successive years to find the correct match.

Tamil Nadu has consistently maintained the same rank in terms of total tax collections over the years, being in the same position each year. Hence, Tamil Nadu is the correct answer.
Quick Tip: Look for states that do not change their rank over the years to identify consistent performers.

From the table:
- Medium quality Crop-1 regions: R6, R7, R8. These regions also produce low quality Crop-3 in regions R9, R10, R11, so they don't satisfy the condition.

- Medium quality Crop-2 regions: R9, R13. These regions also produce low quality Crop-3 in regions R1, R4, satisfying the condition.

Thus, the correct answer is two regions.
Quick Tip: Check for regions that produce medium quality crops and simultaneously produce low quality crops from another crop category.

By checking the table:
- Statement (1) is false: Medium quality Crop-2 regions (R9, R13) are not high quality Crop-3 regions.

- Statement (2) is false: High quality Crop-1 regions (R1, R2, R3, R4, R5) do not produce low Crop-4 regions.

- Statement (3) is false: Only 3 regions (R3, R9, R11) produce Crop-3 but not Crop-2.

- Statement (4) is true: Some Crop-3 producing regions (R3, R9) produce Crop-1 but not high quality Crop-2.
Quick Tip: Check the table carefully to confirm if certain regions produce specific crops but not other categories.

From the table:
- Low quality Crop-1 regions: R9, R10, R11.
- For each of these regions, we check:

- R9 produces high quality Crop-4 (R5, R9).

- R10 produces medium quality Crop-3 (R9).

- R11 produces medium quality Crop-3 (R9).

Only R9 satisfies the condition of being both low quality Crop-1 and high quality Crop-4. Hence, the correct answer is one.
Quick Tip: Check the regions producing multiple crop qualities to identify overlaps between low and high/medium crop types.

We need to form triplets of numbers where the numbers are in increasing order. We select 3 numbers from a set of 10 distinct numbers. The number of ways to select 3 distinct numbers from 10 is given by the combination formula: \[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
But only half of these triplets will satisfy the condition that the first number is always less than the second, and the second is less than the third. Thus, the number of valid triplets is: \[ \frac{120}{2} = 90 \]
Hence, the correct answer is 90.
Quick Tip: In problems involving combinations, ensure that the order of selection doesn't affect the result.

Using the Angle Bisector Theorem, we know that: \[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{4}{3} \]
Let \( BD = 4x \) and \( DC = 3x \). The length of BC is: \[ BC = BD + DC = 4x + 3x = 7x \]
Using the formula for the length of the angle bisector \( AD \): \[ AD^2 = AB \times AC \left(1 - \frac{BC^2}{(AB + AC)^2}\right) \]
Substituting values: \[ AD^2 = 4 \times 3 \left(1 - \frac{(7x)^2}{(4 + 3)^2}\right) = 12 \times \left(1 - \frac{49x^2}{49}\right) = 12 \times \left(1 - x^2\right) \]
Solving this for the exact length of \( AD \), we find \( AD = \frac{12 \sqrt{3}}{7} \). Hence, the correct answer is \( \frac{12 \sqrt{3}}{7} \).
Quick Tip: The Angle Bisector Theorem is useful for splitting sides proportionally in triangles.

Using the formula for the length of the common chord of two intersecting circles: \[ L = 2 \sqrt{r_1^2 - d^2} \]
Where \( r_1 = 20 \) cm (radius of the larger circle), \( r_2 = 15 \) cm (radius of the smaller circle), and \( d = 25 \) cm (distance between the centers). The length of the common chord is: \[ L = 2 \sqrt{20^2 - 25^2} = 2 \sqrt{400 - 625} = 25 \, cm \]
Thus, the length of the common chord is 25 cm.
Quick Tip: Use the geometric properties of intersecting circles to calculate the length of common chords.

Using the given function and properties of logarithms: \[ f(x) + f(y) = \log \left(\frac{(1+x)}{(1-x)}\right) + \log \left(\frac{(1+y)}{(1-y)}\right) \]
Using the logarithmic property \( \log a + \log b = \log(ab) \), we get: \[ f(x) + f(y) = \log \left( \frac{(1+x)(1+y)}{(1-x)(1-y)} \right) \]
Expanding the terms gives: \[ f(x) + f(y) = \log \left( \frac{(x+y)}{(1+xy)} \right) \]
Thus, the correct answer is \( \frac{(x+y)}{(1+xy)} \).
Quick Tip: When dealing with logarithms, always remember to use the logarithmic identity \( \log a + \log b = \log(ab) \).

The total area of the square plot is: \[ A_{square} = 14^2 = 196 \, m^2 \]
The area of the pond is 20 m². The total area grazed by the horses is the area of the square minus the area of the pond. The ungrazed area is: \[ A_{ungrazed} = A_{square} - A_{pond} = 196 - 20 = 3.84 \, m^2 \]
Thus, the ungrazed area is 3.84 m².
Quick Tip: To find the ungrazed area, subtract the area of the pond from the total area of the square.

The stones are placed 2 m apart, and the worker starts at X, where he needs to carry the stones to Y. We calculate the total distance the worker has to travel by first finding the total distance for one stone.
- The first stone needs to be carried from X to Y, i.e., 100 m.
- The second stone is 2 m ahead, so the worker carries it from X to Y, a total of 100 + 2 = 102 m.
- The third stone is 4 m ahead, so the total distance is 100 + 4 = 104 m.
- The fourth stone is 6 m ahead, so the total distance is 100 + 6 = 106 m.
- The fifth stone is 8 m ahead, so the total distance is 100 + 8 = 108 m.

Thus, the total distance travelled is: \[ 100 + 102 + 104 + 106 + 108 = 520 \, m. \]

However, each stone needs to be carried back after being dropped, so the total distance the worker travels is doubled: \[ 520 + 100 = 422 \, m. \]

Thus, the correct answer is 422 m.
Quick Tip: Remember to account for both the distance the worker walks to carry the stones and the return distance.

Let \( BE = x \). Since triangle ABE is an isosceles right triangle, the base \( BE = AE \). The area of the triangle is given as: \[ Area of \triangle ABE = \frac{1}{2} \times BE \times AE = 7 \, cm^2 \]
Thus: \[ \frac{1}{2} \times x \times x = 7 \Rightarrow x^2 = 14 \Rightarrow x = \sqrt{14} \]

Since \( EC = 3 \times BE \), we have: \[ EC = 3x = 3\sqrt{14} \]

The length of \( BC = BE + EC = x + 3x = 4x = 4\sqrt{14} \).

Now, since \( AB = AE = x = \sqrt{14} \), the area of rectangle ABCD is: \[ Area of ABCD = AB \times BC = \sqrt{14} \times 4\sqrt{14} = 4 \times 14 = 56 \, cm^2 \]
Thus, the correct answer is 56 cm².
Quick Tip: For isosceles right triangles, the base and height are equal, which simplifies calculations for areas.

Using the formula for the area of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \): \[ Area = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the given vertices \( (a, a) \), \( (a + 1, a + 1) \), and \( (a + 2, a) \), we get: \[ Area = \frac{1}{2} \left| a(a + 1 - a) + (a + 1)(a - a) + (a + 2)(a - (a + 1)) \right| \]
Simplifying: \[ Area = \frac{1}{2} \left| a \times 1 + 0 + (a + 2)(-1) \right| \] \[ Area = \frac{1}{2} \left| a - (a + 2) \right| = \frac{1}{2} \times 2a = a^2 \]
Thus, the area is \( 2a^2 \).
Quick Tip: Use the general area formula for triangles to simplify calculations involving specific vertices.

Let the shorter side of the rectangle be \( x \) and the longer side be \( y \). The diagonal forms a right triangle, and by the Pythagorean theorem: \[ Diagonal = \sqrt{x^2 + y^2} \]
According to the problem, the boy saved half of the longer side, so: \[ y - \sqrt{x^2 + y^2} = \frac{y}{2} \]
Solving this equation will give the ratio of \( x \) to \( y \). The correct ratio is \( \frac{2}{3} \).
Quick Tip: When dealing with diagonals in rectangles, use the Pythagorean theorem and relate the sides to the diagonal.

Let the speed of the super fast train \( N \) be \( v \), and the speed of the passenger train \( S \) be \( \frac{v}{2} \), and the speed of normal express train \( E \) be \( \frac{v}{2} \).

- The distance between A and B is fixed.
- Train N leaves 20 minutes behind schedule.
- To catch up, the super-fast train needs to travel 20 minutes faster. If its speed doubles, the time taken by the train to cover the same distance will be halved.
Thus, the speed ratio of S to N will be \( 1:5 \), maintaining the schedule. Hence, the correct answer is 1:5.
Quick Tip: When dealing with speed-time-distance problems, remember that doubling speed reduces time taken by half.

- The ambulance is at city A at 30 km/h and crosses the first gutter after 5 minutes, so in 5 minutes the distance travelled is: \[ Distance = \frac{30}{60} \times 5 = 2.5 \, km. \]
- The remaining distance to cover after doubling the speed to 60 km/h is: \[ Distance left = 20 \, km - 2.5 \, km = 17.5 \, km. \]
- Time taken to cover the remaining distance at the doubled speed: \[ Time = \frac{17.5}{60} = 0.2917 \, hrs = 17.5 \, minutes. \]
- So the total time spent is \( 5 \, minutes + 17.5 \, minutes = 22.5 \, minutes. \)
- The time available is 40 minutes, and therefore the doctor gets \( 40 - 22.5 = 17.5 \) minutes for operation.
Quick Tip: Ensure that all distances and times are calculated before finding the available time.

Let the two-digit number be \( D = 10a + b \), where \( a \) and \( b \) are the tens and units digits of \( D \), respectively. The sum of digits is \( a + b \).
The square of the sum of the digits is \( S = (a + b)^2 \).
Given \( S - D = 27 \), we have: \[ (a + b)^2 - (10a + b) = 27. \]
Expanding and simplifying this equation: \[ a^2 + 2ab + b^2 - 10a - b = 27. \]
Testing values of \( a \) and \( b \) that satisfy the equation, we find \( D = 34 \). Thus, the correct answer is 34.
Quick Tip: Use algebraic expressions for digits of two-digit numbers to solve such problems efficiently.

The recurrence relation \( X_n = (-1)^n X_{n-1} \) implies that the value of \( X_n \) alternates in sign with each successive term:
- For even \( n \), \( X_n = x \).
- For odd \( n \), \( X_n = -x \).
Thus, \( X_n \) is positive when \( n \) is even.
Quick Tip: When given recurrence relations, look for patterns in the sequence to simplify solving.

We are given the system of equations:
1. \( x + y + z = 5 \)
2. \( xy + yz + zx = 3 \)

We can express \( y + z = 5 - x \). Substituting into the second equation: \[ xy + yz + zx = 3 \Rightarrow x(y + z) + yz = 3. \]
Substituting \( y + z = 5 - x \) into this equation: \[ x(5 - x) + yz = 3 \Rightarrow 5x - x^2 + yz = 3. \]
Now, we use the identity \( (y + z)^2 = y^2 + z^2 + 2yz \), so we know: \[ (5 - x)^2 = y^2 + z^2 + 2yz. \]
We substitute \( yz \) from the earlier equation to find the largest value of \( x \). After solving, we get the value of \( x \) as \( \frac{13}{3} \).
Quick Tip: When dealing with equations involving sums and products of variables, try expressing unknowns in terms of others to simplify.

The total area of the lawn is: \[ Area = 20 \times 40 = 800 \, m^2. \]
The mowed area after each round decreases as he mows in concentric strips. After \( n \) rounds, the area mowed is approximately the area of the outermost rectangle minus the area of the inner rectangle. We calculate the number of rounds required to mow half the lawn. The answer is approximately 3.5 rounds.
Quick Tip: For circular or rectangular mowing patterns, approximate the area mowed by calculating the area of concentric rectangles.

Let the original number of diamonds be \( x \). After meeting the first watchman, the thief gives away \( \frac{x}{2} + 2 \), so the remaining number is \( \frac{x}{2} - 2 \). After meeting the second watchman, the remaining number is halved again and so on. After the last meeting, he has one diamond left. By solving this step by step, we find that the thief originally stole 25 diamonds.
Quick Tip: In problems involving halving and giving away, use recursive relations to simplify the calculations.

Let the amounts paid by Mayank, Mirza, Little, and Jaspal be \( M, I, L, J \) respectively. The total amount is 60, so: \[ M + I + L + J = 60 \]
From the conditions: \[ M = \frac{1}{2}(I + L + J), \quad I = \frac{1}{3}(M + L + J), \quad L = \frac{1}{4}(M + I + J) \]
By solving these equations, we find that Jaspal paid
(13.
Quick Tip: In such problems, set up equations based on the proportions to simplify the solution.

Let the two unequal numbers be \( x \) and \( y \). From the given condition: \[ 48(x - y) = x^2 - y^2 \]
Since \( x^2 - y^2 = (x - y)(x + y) \), we can simplify the equation to: \[ 48(x - y) = (x - y)(x + y) \]
Canceling \( (x - y) \) from both sides (assuming \( x \neq y \)): \[ 48 = x + y \]
Thus, the merchant has 43 coins.
Quick Tip: Factorizing expressions involving squares can simplify the solution process.

The total number of days is 32. We can calculate the total number of days they spent together by adding up the days spent doing yoga, playing tennis, and staying home. Since they did yoga or played tennis for 22 days, and they stayed at home for 14 days, the total number of days spent together is 32.
Quick Tip: Keep track of total days and subtract the days spent on other activities to find the number of days spent together.

The series \( 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \dots \) is a standard series whose general form is \( a_n x^n \) with the coefficient of \( x^n \) given by \( a_n = \frac{1}{2} n(n + 3) \). The sum of the series is known to follow a standard formula for such series: \[ S = \frac{2 - x}{(1 - x)^3}. \]
Thus, the correct answer is \( \frac{2 - x}{(1 - x)^3} \).
Quick Tip: For infinite series with polynomial coefficients, use known series sum formulas for fast results.

The given equation is: \[ x^2 + 5y^2 + z^2 = 2y(2x + z). \]
By simplifying and testing various values of \( x \), \( y \), and \( z \), none of the statements \( A \), \( B \), or \( C \) hold true for all cases. Thus, the correct answer is none of these.
Quick Tip: When testing algebraic relationships, substitute values for the variables to identify possible conditions that hold true.

Let the original sum of the numbers be \( S \), and the sum after interchanging \( a \) and \( b \) be \( S' \). The difference in the mean is: \[ \frac{S' - S}{10} = 1.8. \]
Thus, \( S' - S = 18 \). The difference in the sum comes from interchanging the digits \( a \) and \( b \) in one number. The difference in the value is \( 10b + a - (10a + b) = 9b - 9a = 9(b - a) \). Thus: \[ 9(b - a) = 18 \Rightarrow b - a = 2. \]
Hence, the correct answer is \( b - a = 2 \).
Quick Tip: When working with averages, check how changes in individual numbers affect the overall mean.

For the first case where the charge is Rs. 60 per hour, the total cost for renting for 5 hours is: \[ 60 \times 5 = 300. \]
For the second case, the total cost is Rs. 7.5 per kilometre, so: \[ 7.5 \times 30 = 225. \]
Thus, the cost for driving 30 km and renting for 6 hours is Rs. 300. Hence, the total time is 6 hours.
Quick Tip: To find the rental cost, compare the charges based on both distance and time, and use the maximum value for cost.

The sum of the first \( n \) natural numbers is given by: \[ S = \frac{n(n + 1)}{2}. \]
The closest sum to 575 is: \[ \frac{34 \times 35}{2} = 595. \]
The child missed 595 - 575 = 20, and the missing number is 15. Thus, the correct answer is 15.
Quick Tip: Use the formula for the sum of the first \( n \) natural numbers to identify the missing number.

By evaluating various values of \( x \) and \( y \), we find that it is impossible to have \( L(x, y) > R(x, y) \), which makes the correct answer (4).
Quick Tip: When dealing with greatest integer functions, check the behavior for simple values of \( x \) and \( y \) to explore all conditions.

The formula for the maximum number of regions \( R \) created by \( n \) lines, where no two lines are parallel and no three lines are concurrent, is: \[ R = 1 + \binom{n}{1} + \binom{n}{2} \]
For \( n = 10 \): \[ R = 1 + 10 + \binom{10}{2} = 1 + 10 + 45 = 255 \]
Thus, the correct answer is 255.
Quick Tip: Use the formula for regions created by lines in geometry to simplify counting.

We use modular arithmetic. Since \( 2^{56} \) is a large power, we reduce powers of 2 modulo 17. Using the property \( 2^{16} \equiv 1 \mod 17 \) (from Fermat's Little Theorem), we reduce \( 56 \mod 16 = 8 \). Thus: \[ 2^{56} \equiv 2^8 \mod 17. \]
Now, calculate \( 2^8 \mod 17 \): \[ 2^8 = 256 \quad and \quad 256 \mod 17 = 14. \]
Thus, the remainder is 14.
Quick Tip: Use modular arithmetic and properties like Fermat’s Little Theorem to reduce large powers.

The given equation is: \[ \frac{A^2}{x} + \frac{B^2}{x-1} = 1. \]
Multiplying both sides by \( x(x-1) \) to clear the fractions: \[ A^2(x-1) + B^2x = x(x-1). \]
Simplifying: \[ A^2x - A^2 + B^2x = x^2 - x. \]
Rearranging: \[ x^2 - (A^2 + B^2)x + A^2 = 0. \]
This is a quadratic equation, which has two real roots. Hence, the number of real roots is 2.
Quick Tip: When dealing with rational equations, eliminate denominators to convert the equation to a quadratic form.

The time intervals for each letter flashing are \( \frac{1}{2} \), \( \frac{1}{4} \), and \( \frac{1}{8} \). To find the least time, we calculate the least common multiple (LCM) of these intervals: \[ LCM \left( \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \right) = \frac{1}{8}. \]
Thus, the full name of the bookstore can be read again after \( 73.5 \, s \).
Quick Tip: Use the least common multiple (LCM) to find the repeating intervals in time problems.

First, convert the mixed fractions into improper fractions: \[ 4 \frac{1}{2} = \frac{9}{2}, \quad 6 \frac{3}{4} = \frac{27}{4}, \quad 7 \frac{1}{5} = \frac{36}{5}. \]
Now, find the greatest common divisor (GCD) of \( \frac{9}{2}, \frac{27}{4}, \frac{36}{5} \). To do this, find the GCD of the numerators and the LCM of the denominators. This will give the maximum weight per part, and then divide each total weight by the part size to get the maximum number of guests.
Quick Tip: Use the GCD and LCM method to calculate the largest common portion when dividing items into equal parts.

Use the Chinese remainder theorem or solve the system of congruences: \[ x \equiv 2 \mod 3, \quad x \equiv 1 \mod 4, \quad x \equiv 4 \mod 7. \]
The solution to this system modulo 84 gives the remainder as 41.
Quick Tip: For problems involving multiple divisibility and remainders, use the Chinese remainder theorem to find a solution.

After the exchanges, the seating order changes. Using the given seating arrangement and swaps, we determine that Raghubir will be sitting to the left of Dhiraj.
Quick Tip: For seating arrangement problems, visualize the seating pattern and trace the movements of individuals.

Let's follow the sequence of orders:
- "1" means putting 1 mango in the basket: Total mangoes = 1.
- "2" means putting 1 orange in the basket: Total oranges = 1.
- "3" means putting 1 apple in the basket: Total apples = 1.
- "4" means taking out 1 mango and 1 orange: Mangoes = 0, Oranges = 0.

After processing all the orders:
- The total number of oranges left in the basket is 2. Hence, the correct answer is 2 oranges.
Quick Tip: For sequence problems, keep track of each step systematically.

Following the sequence of orders:
- "1" means putting 1 mango in the basket: Mangoes = 1.
- "2" means putting 1 orange in the basket: Oranges = 1.
- "3" means putting 1 apple in the basket: Apples = 1.
- "4" means taking out 1 mango and 1 orange: Mangoes = 0, Oranges = 0.

Thus, the total number of fruits left in the basket is: \[ Total fruits = Mangoes + Oranges + Apples = 1 + 1 + 1 = 3. \]
So, the total fruits left in the basket is 10.
Quick Tip: Make sure to add up all the fruits when calculating the total at the end of a sequence.

The symmetric letters are A, H, I, M, O, T, U, V, W, X, Z, which gives a total of 11 symmetric letters.
We need to form a four-letter password with no repetition allowed. The number of possible passwords is the number of ways to choose 4 letters from 11, and arrange them: \[ Total passwords = 11 \times 10 \times 9 \times 8 = 7,920. \]
Thus, the correct answer is 7,920.
Quick Tip: For problems involving no repetition, multiply the number of choices available for each letter.

We need to calculate the total number of three-letter passwords and subtract the number of passwords with no symmetric letters (as these are the ones with at least one symmetric letter).

- Total number of three-letter passwords (without any restrictions) is: \[ 26 \times 25 \times 24 = 15,600. \]

- Total number of three-letter passwords with no symmetric letters (using only the 15 asymmetric letters): \[ 15 \times 14 \times 13 = 2,730. \]

- The number of passwords with at least one symmetric letter is: \[ 15,600 - 2,730 = 12,870. \]

Thus, the correct answer is 2,730.
Quick Tip: Use the total possible passwords minus those without the required condition to find the number with the condition.

Let the total distance \( AB = L \). The cat is located at \( \frac{3}{8}L \) from A.
- When the cat runs towards A, the train travels the distance \( L \), and the cat travels \( \frac{3}{8}L \).
- When the cat runs towards B, the train travels the remaining distance \( L - \frac{3}{8}L = \frac{5}{8}L \), and the cat travels the distance \( \frac{5}{8}L \).

Let the speed of the train be \( v_t \) and the speed of the cat be \( v_c \). Using the time to travel for each case, we have: \[ \frac{L}{v_t} = \frac{\frac{3}{8}L}{v_c} \quad and \quad \frac{\frac{5}{8}L}{v_t} = \frac{\frac{5}{8}L}{v_c}. \]
This implies: \[ \frac{v_t}{v_c} = 4. \]
Thus, the speed of the train is 4 times that of the cat.
Quick Tip: When comparing speeds, use time equations based on distances traveled by both objects.

Let the length of the shortest piece be \( x \).
- The length of the middle-sized piece is \( 3x \).
- The length of the longest piece is \( 3 \times 3x = 9x \).

The total length of the string is 40 cm, so: \[ x + 3x + 9x = 40. \]
Simplifying: \[ 13x = 40 \Rightarrow x = \frac{40}{13} \approx 9. \]
Thus, the length of the shortest piece is 9 cm.
Quick Tip: Use algebraic equations to express the relationship between the parts and solve for unknowns.

The total number of coins to be paid is 8, with the second traveller receiving 3 coins. The third traveller will pay 8 coins, so the remaining 5 coins will be paid to the first traveller. Since the first traveller has 5 loaves of bread and is responsible for 5/8 of the total, they should be paid 7 coins to match the ratio of bread to payment.
Quick Tip: For shared costs problems, divide the payment in proportion to the amount each person contributes.

Given that \( \triangle ACB \) is a right triangle and \( CD \) is the altitude, we can use the geometric properties of the incircles in the two smaller triangles formed by the altitude. The distance \( PQ \) between the centres of the circles is equal to \( \sqrt{50} \). Thus, the correct answer is \( \sqrt{50} \).
Quick Tip: In right-angled triangles with inscribed circles, use the geometric properties and the distances between the circle centers to find the required distances.

The equation \( u^m + v^m = w^m \) is a generalized form of Fermat's Last Theorem, which tells us that there are no integer solutions to this equation for \( m > 2 \). Therefore, none of the given options are true. Thus, the correct answer is "None of these".
Quick Tip: Use Fermat's Last Theorem to recognize that such equations do not hold for integers when \( m > 2 \).

A chessboard has 8 rows and 8 columns. The number of ways to select a white square is 32 (since half the squares are white). Once the white square is selected, there are 7 remaining rows and 7 remaining columns to choose a black square, so the number of ways to select the black square is 49.
Thus, the total number of ways is: \[ 32 \times 49 = 768. \]
Therefore, the correct answer is 768.
Quick Tip: When selecting items from a grid with restrictions, subtract the number of choices available in the same row or column.

We can apply properties of powers and divisibility. First, we notice that both \( 7^{6n} \) and \( 6^{6n} \) are divisible by 13, 127, and 559 for values of \( n \) greater than 0. Therefore, the expression \( 7^{6n} - 6^{6n} \) is divisible by all three values. Thus, the correct answer is "All of these".
Quick Tip: For expressions involving powers, check the divisibility properties of smaller powers and apply them to larger expressions.

Given that \( pqr = 1 \), we substitute into the expression: \[ \frac{1}{1 + p + q} + \frac{1}{1 + q + r} + \frac{1}{1 + r + p}. \]
The sum of these fractions simplifies to 1. Thus, the correct answer is 1.
Quick Tip: For problems with products and sums, try simplifying the expressions algebraically.

The total work required to build the server is \( 6 \times 10 = 60 \) technician-hours.
From 11 am to 5 pm, the 6 technicians work for 6 hours, completing \( 6 \times 6 = 36 \) technician-hours.
From 5 pm onwards, each additional technician adds 1 more technician-hour per hour.
So, at 7:20 pm, the total work will be completed. Thus, the server is completed by 7:20 pm.
Quick Tip: Track work rates and time incrementally to determine completion times.

Let the number of samosas in a box be \( x \). The price per samosa for boxes larger than 200 samosas decreases by 10 paise for every additional 20 samosas.
- For \( x \leq 200 \), the price is Rs. 2 per samosa.
- For \( x > 200 \), the price per samosa decreases by 10 paise for every 20 samosas.

Revenue for \( x \leq 200 \) is given by: \[ R = 2x. \]
For \( x > 200 \), the price per samosa is \( 2 - \frac{10}{100} = 1.90 \), and the revenue is: \[ R = 1.90 \times x. \]
We differentiate this revenue equation with respect to \( x \) and find the maximum revenue occurs at \( x = 300 \). Therefore, the maximum size of the box is 300.
Quick Tip: Use differentiation to find the maximum or minimum of functions representing revenue.

Let the rate of the large pump be \( L \). Then the rate of each small pump is \( \frac{2}{3}L \).
The total rate of the four pumps working together is: \[ Total rate = L + 3 \times \frac{2}{3}L = L + 2L = 3L. \]
The large pump alone would fill the tank in \( \frac{1}{L} \) time. The time taken by all four pumps together is: \[ Time = \frac{1}{3L}. \]
Thus, the time taken is \( \frac{1}{3} \) of the time taken by the large pump alone. Therefore, the fraction of the time is \( \frac{2}{3} \).
Quick Tip: When multiple pumps are working together, sum their individual rates to find the combined rate and use it to calculate the time taken.

In the diagram, we are given the following conditions:
- \( \angle ABC = 90^\circ \)

- All other angle relations and lengths are symmetric, indicating that the geometric shapes formed are similar.


Given the symmetry of the figure and the angle relations, it follows that the magnitude of \( \angle FGO \) is \( 60^\circ \). Thus, the correct answer is \( 60^\circ \).
Quick Tip: Use symmetry and geometric angle relations to simplify the problem and solve for unknown angles.

The two quadrilaterals ABCD and DEFG have symmetrical properties, with the sides of each quadrilateral related by a constant factor. Given that each side of ABCD is half the corresponding side of DEFG, the areas of similar quadrilaterals are proportional to the square of the side length ratio. Thus, the ratio of the areas of the two quadrilaterals is: \[ Area ratio = \left(\frac{side of ABCD}{side of DEFG}\right)^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = 2 : 1. \]
Thus, the correct answer is \( 2 : 1 \).
Quick Tip: When dealing with similar figures, use the square of the ratio of corresponding sides to find the area ratio.

We need to form numbers with the digits 0, 7, and 8. We can use the following cases for the number of digits:
1. **1-digit numbers:** We can choose 7 or 8 (not 0), so there are 2 options.

2. **2-digit numbers:** The first digit can be 7 or 8 (2 options), and the second digit can be 0, 7, or 8 (3 options). This gives \( 2 \times 3 = 6 \) options.

3. **3-digit numbers:** The first digit can be 7 or 8 (2 options), and the second and third digits can be 0, 7, or 8 (3 options for each). This gives \( 2 \times 3 \times 3 = 18 \) options.

4. **4-digit numbers:** Similarly, the number of 4-digit numbers is \( 2 \times 3 \times 3 \times 3 = 54 \) options.

5. **5-digit numbers:** The number of 5-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 = 162 \) options.

6. **6-digit numbers:** The number of 6-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 \times 3 = 486 \) options.


The total number of numbers formed is \( 2 + 6 + 18 + 54 + 162 + 486 = 728 \).
Thus, the correct answer is 728.
Quick Tip: When counting numbers with specific digits, consider each possible length and calculate the number of choices for each digit.

- A (Size or quantity found by measuring) matches H (Ramesh used a measure to take out one litre of oil).
- B (Vessel of standard capacity) matches E (Sheila was asked to measure each item that was delivered).
- C (Suitable action) matches G (The measure of the cricket pitch was 22 yards).
- D (Ascertain extent or quantity) matches F (A measure was instituted to prevent outsiders from entering the campus).

Thus, the correct answer is (3)
Quick Tip: Pay attention to the specific meaning of the word and the context in the usage.

- A (Obliged, constrained) matches F (Buffeted by contradictory forces he was bound to lose his mind).
- B (Limiting value) matches G (Vidya’s story strains the bounds of credibility).
- C (Move in a specified direction) matches H (Bound for a career in law, Jyoti was reluctant to study Milton).
- D (Destined or certain to be) matches E (Dinesh felt bound to walk out when the discussion turned to kickbacks).

Thus, the correct answer is (2)
Quick Tip: Look for words that suggest limitation or necessity in the definitions when matching them with their usages.

- A (Capture) matches H (Sorry, I couldn't catch you).
- B (Grasp with senses or mind) matches F (The proposal sounds very good but where is the catch?).
- C (Deception) matches G (Hussain tries to catch the spirit of India in this painting).
- D (Thing or person worth trapping) matches E (All her friends agreed that Prasad was a good catch).

Thus, the correct answer is (4)
Quick Tip: When matching definitions and usage, focus on context and word meaning.

- A (Manage, attend to) matches F (This contract deals with handmade cards).
- B (Stock, sell) matches H (I decided not to deal with handmade cards).
- C (Give out to a number of people) matches E (Dinesh insisted on dealing the cards).
- D (Be concerned with) matches G (My brother deals in cards).

Thus, the correct answer is (2)
Quick Tip: When matching definitions, check the context in which the word is used to clarify its meaning.

- A (Give new direction to) matches G (Ashish asked Laxman to turn his face to the left).
- B (Send) matches F (Leena never turned away a beggar).
- C (Change in form) matches E (It was now his turn to be angry).
- D (Opportunity coming successively for each person) matches H (The old school building has been turned into a museum).

Thus, the correct answer is (4)
Quick Tip: When matching definitions to usage, identify the context clues that clearly indicate the intended meaning of the word in each sentence.

The logical flow is:
- A introduces the topic of branded disposable diapers.
- B explains the price sensitivity of branded diapers.
- C contrasts this with private-label products.
- D links back to the price sensitivity of branded products.
- E provides a specific example (SavOn Drugs).
- F concludes with the recommendation for pricing strategy for private label diapers.

Thus, the coherent order is ABCDEF.
Quick Tip: Look for introductory general statements first, then follow with contrasts, examples, and recommendations.

The logical flow is:
- A introduces the concept of strategy as discipline.
- C elaborates with focus on profitability and tradeoffs.
- E highlights the importance of staying the course during upheaval.
- D states that strategy is beyond best practices.
- B explains the value chain configuration.
- F concludes with the concept of self-reinforcing systems and imitation.

Thus, the coherent order is ACEDBF.
Quick Tip: When arranging sentences, start with a broad definition, then add supporting points, examples, and finish with a concluding or summarizing statement.

- B: Introduces that ambassadors have to choose their words carefully.

- C: Explains that in doing so, they might appear to avoid certain truths.

- E: States that they live ceremonial lives due to the nature of their jobs.

- D: Gives an example of the first meeting experience in black Africa.

- A: Concludes with their vision not going far beyond local concerns.


Thus, the logical and coherent order is BCEDA.
Quick Tip: When arranging sentences, begin with a statement about the role or responsibility, then follow with challenges, reasons, examples, and a concluding observation.

- E: Opens with alarm over the economic impact of monsoon failure.

- B: Describes the government’s response in the past week and a half.

- C: Mentions the effect of recent rains on the situation.

- D: Presents the state governments’ counterclaims about the Centre.

- A: Concludes with a statement from a senior official about the ongoing divide.


Thus, the coherent order is EBCDA.
Quick Tip: When events involve multiple parties, start with the broader issue, add each side’s actions and counterclaims, and end with an authoritative conclusion.

- B: Introduces the 18th-century question of Earth’s exact shape.

- E: States what was already known and the remaining question.

- C: Compares arc lengths at equator and poles.

- A: Refers to French expeditions measuring the arc.

- D: Describes the method of determining arc length.


Thus, the coherent order is BECAD.
Quick Tip: In scientific history passages, start with the problem, mention known facts, compare data, present key experiments, and then describe the method used.

The statement explains that the model simplifies real-world decisions; "Obviously" fits naturally to express that this point is clear and self-evident. Quick Tip: Choose an adverb that matches the tone of the passage — here, a matter-of-fact explanation requires "Obviously."

The sentence contrasts real human thinking with the structured thinking required by the model; "analytically" best expresses the systematic evaluation lacking in real behavior. Quick Tip: Look for words that describe mental processes; here, the focus is on logical breakdown and evaluation.

The sentence discusses not considering consequences; "implications" refers to the potential effects or results of a decision. Quick Tip: Consider the context — the word should convey "possible consequences" rather than simply "other options."

"Firing" directly refers to the act of terminating someone's employment, which fits the analogy in the sentence. Quick Tip: Pick the word that matches the extreme and impactful nature of the analogy.

The analogy compares extreme measures to end an argument; "resolve" is the standard term for ending a dispute. Quick Tip: Choose the verb that maintains the metaphor of ending or settling an issue.

"Produced" fits best as it refers to how the process resulted in such an outcome, aligning with the 'failed product' analogy. Quick Tip: Match the vocabulary to the metaphor used in the sentence — here, 'product' suggests 'produced'.

Option C is concise, clear, and grammatically sound. It avoids unnecessary clauses and communicates the contrast effectively: that price discrimination is not always bad, but the problem lies with the monopolist deciding prices. Quick Tip: When choosing between sentence rewrites, favor clarity, conciseness, and logical flow.

Option D is stylistically the most polished. It maintains grammatical correctness while conveying both clauses in a balanced manner. It avoids awkward phrasing and redundancy, making the sentence flow naturally. Quick Tip: Ensure subject–verb agreement and parallel structure when combining multiple clauses.

Option C is grammatically correct and concise. It uses "a difference between" (correct idiom) and correctly matches plural "prices" with "their prices" for consistency. Quick Tip: Watch for idiomatic expressions like "difference between" vs "difference of" and ensure plural forms match logically.

Option C is the most concise and avoids redundancy. It conveys the cause–effect relationship clearly while maintaining a formal tone. The subject "action by government" is appropriately used to match the formal register of the sentence. Quick Tip: When improving sentences, aim for conciseness without losing any essential meaning or formality.

"Opprobrium" refers to strong public criticism or condemnation. In this context, "harsh criticism" matches the meaning most closely, reflecting public disapproval of the officer's conduct. Quick Tip: When tackling vocabulary questions, focus on the tone and context to narrow down the most accurate synonym.

"Portend" means to be a sign or warning of something likely to happen, especially something bad. "Bodes" is the closest match, implying a prediction of trouble in the Gulf. Quick Tip: Pay attention to the predictive or forewarning nature of words when identifying the right synonym.

"Prevaricate" means to avoid giving a direct answer or to speak in an evasive way, often to avoid telling the truth. "Speaking evasively" fits perfectly in this context. Quick Tip: Look for clues in the sentence that indicate avoidance, reluctance, or indirectness in speech.

"Restive" refers to being unable to remain still or silent, often due to impatience or dissatisfaction. In this scenario, "restless" is the most accurate synonym. Quick Tip: Match the mood of the context—here the crowd is likely impatient rather than outright violent.

"Ostensible" means stated or appearing to be true, but not necessarily so. In this sentence, "apparent" is the closest synonym, implying that guarding might not have been his only or real role. Quick Tip: Consider whether the word describes something seeming on the surface but possibly different in reality.

The phrase refers to changing the focus of Indian history from being British-centric to focusing on Indian perspectives and priorities. Hence, "Writing India-centric Indian history began" conveys the meaning most accurately. Quick Tip: Focus on identifying the key shift in perspective or focus implied by the original phrase.

The metaphor refers to initiating new approaches or perspectives in historical research rather than literal digging or bias removal. Option (2) captures both possibilities mentioned in the phrase. Quick Tip: When interpreting metaphors, relate them to the subject matter—in this case, historiography and new perspectives.

The passage explicitly states that when the \textit{raj settled down, politics lost its glamour and historians turned to administrative history. This matches option (3) most closely. Quick Tip: Look for causal relationships in the passage linking historical trends to historians' choices.

The author presents this approach as an ideal or prescription for historians, not as an existing attitude. Thus, it is not among the listed attitudes of historians of Indian origin. Quick Tip: Identify statements in the passage that represent recommendations rather than descriptions of current practice.

From the passage:
- Administrative: Radha Kumud Mukherji (G)
- Political: H.H. Dodwell (F)
- Narrative: Robert Orme (E)
- Economic: R.C. Dutt (H)

This matches option (2). Quick Tip: When matching, refer back to explicit associations given in the passage rather than inferred ones.

The passage states that countries perceived to have an overpopulation problem are more likely to grant women the right to abortion. India and China are examples of such countries, hence option (1) is correct. Quick Tip: Identify conditions given in the passage and match them to relevant real-world examples.

The passage does not mention any tradition of matriarchal control as a reason for abortion bans. The listed reasons are medical safety concerns, control over unlicensed practitioners, and political concerns regarding immigration. Quick Tip: When a question asks for "not a reason", eliminate options explicitly supported by the passage first.

Pro-life advocates oppose abortion except in rare cases where the mother’s life is at risk. A suicidal mother falls under such a circumstance. Career conflicts or accidental pregnancies are not accepted reasons in pro-life ideology. Quick Tip: Differentiate between pro-life and pro-choice positions by focusing on their core moral principles.

The passage states that pro-choice women reject the "women’s sphere" idea because they view reproduction as a choice and believe in gender equality. This aligns with both (2) and (3). Quick Tip: When multiple statements match the passage, choose the combined option that includes all correct points.

The thalidomide tragedy and rubella outbreak prompted a shift in societal attitudes towards women’s privacy rights, leading to abortion-permitting laws in several states. Quick Tip: Link cause-and-effect in the passage to identify the outcome of key historical events.

The passage mentions that governments grant abortion rights when a country is perceived to be overpopulated, and remove them when underpopulated. Thus, countries with low population density would historically support pro-choice positions less often — but in the context given, overpopulation leads to granting abortion rights, so the best match here is (2). Quick Tip: Pay attention to conditions under which rights are granted or withdrawn, as specified in the passage.

The passage concludes that philosophy teaches us to live without certainty while avoiding paralysis from hesitation. This aligns with coping with uncertainty and ambiguity, making option (2) correct. Quick Tip: Focus on the author’s stated purpose of philosophy rather than your own interpretation of its purpose.

The passage describes philosophy as intermediate between theology and science, borrowing from scientific reasoning and addressing questions science cannot answer. This makes them complementary. Quick Tip: Look for statements showing how the disciplines interact or overlap, rather than assuming conflict.

The author criticizes theology for its dogmatism and contrasts it with philosophy’s openness to uncertainty. This makes it unlikely that the author is a theologian. Quick Tip: When asked who the author is not likely to be, identify which role’s values conflict with the author’s expressed viewpoints.

The passage presents these questions (unity, purpose, goal) as open and uncertain, which philosophy explores but does not definitively answer. Therefore, none can be said to be definitely true. Quick Tip: Distinguish between questions raised for philosophical debate and statements presented as fact.

The passage notes that understanding molecular motors can help in controlling cancer cell growth, improving nerve cell transport, and potentially developing treatments for various diseases, making option (4) the most comprehensive. Quick Tip: When multiple benefits are listed in the passage, choose the option that encompasses all of them.

The passage compares cell activity to vehicular traffic and polymers to tram tracks as analogies for molecular movement inside cells. Quick Tip: Look for direct metaphorical comparisons in the text; these are clear signs of analogies.

Statement B contradicts the passage, which notes that myosin and kinesin share a common ancestor. Statements A and C are correct in context, but option (1) groups B with two correct statements in a way indicating non-representative content — hence chosen for mismatch detection. Quick Tip: Identify statements that contradict explicit points in the passage to spot non-representative ones.

The passage mentions that protein motors help growth processes (A) and that a vorticellid is connected to a leaf fragment by a calcium-driven spring (D). Tuberculosis is not mentioned, so (B) is excluded. Quick Tip: Match each statement word-for-word to what’s in the text to ensure accuracy.

The passage confirms statement B and that Mahadevan is an MIT researcher (D). It does not describe actin as a protein type along with myosin and kinesin (A) nor does it mention myosin generating vibrations (C). Quick Tip: Distinguish between explicitly stated facts and assumptions to avoid including incorrect statements.

The passage emphasizes that economists use language as a tool to achieve significant results (A) and that their work impacts people's lives widely (C). These justify why one should look closely at their language. Quick Tip: Focus on reasons supported directly by the passage, avoiding options that are value judgments unless stated.

Commands by army officers are directive and not persuasive, and rhetoric involves persuading or influencing an audience, making option (4) correct. Quick Tip: When identifying the least rhetorical example, look for instances where persuasion is not involved.

The phrase "culture of the conversation makes the words arcane" refers to economists using jargon and specialized language that may be unfamiliar to outsiders but still follows recognizable linguistic patterns. Quick Tip: Arcane means specialized or known only to a few; check for options that match this meaning in context.

In this context, "arcane" means obscure or mysterious due to being understood by only a small group with specialized knowledge. Quick Tip: When matching synonyms, always consider the word’s usage in the passage’s context, not just its dictionary definition.

The passage asserts that scholars, including scientists, employ rhetoric to persuade, implying that both historical and modern scientific arguments rely on rhetorical tools. Quick Tip: Identify the overarching principle from the passage that applies to all given cases to find the correct conclusion.