NPAT 2020 Common QP 1 June Question Paper (Available) :Download Solution PDF Answer Key

The error lies in the phrase "\underline{including the shared system of". The correct sentence should be "Culture refers to the group’s way of life, including shared systems of social meanings, values, and relations" (without "the" before "shared system"). The phrase should not use the article "the" when referring to general instances. Quick Tip: When identifying errors in sentences, pay attention to the correct usage of articles ("the", "a", "an") and their placement in the sentence.

The error lies in the phrase "\underline{trying stopping smoking". The correct sentence should be "Raghav has been trying to stop smoking for many years." The phrase "trying stopping" is incorrect, and it should be "trying to stop." Quick Tip: When using the verb "try," the correct form after it is "to + verb" (e.g., trying to stop, not trying stopping).

The error lies in the phrase "\underline{have achieved them". The correct sentence should be "though very few have achieved it" (where "it" refers to "sustainable development goals"). The pronoun "them" is incorrect because it doesn't agree with the singular form of "goals" in this context. Quick Tip: Always ensure that pronouns agree in number with the noun they replace. Singular nouns should be replaced with singular pronouns (e.g., "it" for singular).

The error lies in the phrase "\underline{has been marked". Since "The early years" is a plural subject, the correct form should be "have been marked." The sentence should read, "The early years of civilian space travel have been marked by many triumphs and tragedies." Quick Tip: When identifying errors in sentences, check for subject-verb agreement. In this case, the plural subject "The early years" requires the plural verb "have."

The error lies in the phrase "\underline{to the feedbacks". The word "feedback" is uncountable, so the plural form "feedbacks" is incorrect. The sentence should read, "students do not pay any attention to feedback provided by their tutors." Quick Tip: When dealing with uncountable nouns like "feedback," avoid using the plural form.

The correct preposition is "to," as in the phrase "to power," which means to come into or attain power. The sentence should read, "Rome began its phenomenal ascent to power." Quick Tip: Certain expressions like "to power" require specific prepositions to maintain correct meaning.

The correct preposition here is "on," which is used when describing a period of time during which an event happens. The sentence should read, "Less than an inch of soil may form on a century in a desert." Quick Tip: When referring to a period of time, "on" is often used to describe specific time frames, such as "on a century."

The correct verb here is "is composed," which is in the present simple tense and correctly describes the general state of the soil. The sentence should read, "The soil is composed of rock material, minerals, and organic matter." Quick Tip: When describing a general fact, the present simple tense is often used.

The correct form is "complete" in the simple present tense, as it refers to a future event dependent on the boys completing their degree. The sentence should read, "The managers have promised to hire the boys once they complete their degree." Quick Tip: When talking about future events that are dependent on other actions, use the present simple tense.

The correct word is "Although." It introduces a contrast between the two clauses, indicating that despite watching the film before, there was no objection to watching it again. The sentence should read: "Although we’d watched the film once before, we didn’t mind watching it a second time." Quick Tip: Use "Although" when showing contrast between two clauses.

The correct word is "whereas," which is used to contrast two ideas. The sentence should read: "The accommodation provided was excellent, whereas the food was awful." Quick Tip: "Whereas" is used to show a contrast between two clauses.

The correct word is "an" because "project" begins with a vowel sound, so the indefinite article "an" should be used. The sentence should read, "The department met yesterday to decide whether they should accept an project or not." Quick Tip: Use "an" before words that begin with vowel sounds (e.g., an apple, an hour).

The correct word is "much," as it is used with uncountable nouns like "difference." The sentence should read, "Stringent laws have made much difference in preventing the poaching of endangered animals." Quick Tip: Use "much" with uncountable nouns like "difference" or "money."

The correct meaning of "ultimately" is "eventually," which refers to the final result or outcome after a process. The sentence should read, "The work of artists and scientists is eventually the pursuit of truth." Quick Tip: "Ultimately" means "in the end" or "eventually," referring to the final result.

The word "inexplicable" means something that is impossible to explain or understand. The best synonym is "incomprehensible." The sentence should read: "The people found his objection puzzling and incomprehensible." Quick Tip: When you encounter the word "inexplicable," think of words like "incomprehensible" or "unfathomable" for similar meanings.

The phrase "go dutch" refers to a situation where each person pays their own share of the bill. The correct meaning is "sharing the costs equally." The sentence should read: "Arnav and his friends decided to go dutch when they went out to the restaurant." Quick Tip: "Go dutch" is a common idiom for splitting the cost evenly among participants.

The phrase "throw in the towel" means to give up or quit, especially after an effort. The sentence should read: "He was ready to give up, but his friends convinced him to complete the course." Quick Tip: "Throw in the towel" is an idiomatic expression that means to quit or surrender.

The correct word is "injected," which refers to putting something (such as money) into something else to support or stimulate it. The sentence should read: "Beijing has injected billions of dollars into the economy after it was hit by the coronavirus." Quick Tip: "Inject" is commonly used when talking about putting money into an economy or system.

The correct word is "struck," which is often used to describe an event or disaster that happens suddenly and forcefully. The sentence should read: "The outbreak struck just as China was preparing to celebrate the Lunar New Year." Quick Tip: "Strike" is commonly used to describe sudden, impactful events or disasters.

The correct word is "exhibiting," which means displaying or showing something, especially a trait or characteristic. The sentence should read: "Penguins are known for exhibiting a number of traits shared with humans." Quick Tip: "Exhibit" is commonly used to describe the display or presentation of characteristics.

The correct word is "undeniable," which means something that cannot be disputed or denied. The sentence should read: "Experts believe they have found the 'first undeniable' evidence for conformity to linguistic laws in non-primate species." Quick Tip: "Undeniable" refers to something that is so strong or certain that it cannot be disputed.

The correct order is C, A, E, B, D. First, C introduces the team of British scientists. Then A discusses the survey and its purpose. E talks about the submarine and its ability to descend. B explains its features, and D gives additional information about its previous descent. Quick Tip: When arranging jumbled sentences, identify the introductory sentence and build the narrative logically from there.

The correct order is C, A, D, B, E. First, C introduces the penguins and their calls. Then A explains the prior research. D follows with an explanation of the calls' conformity to linguistic rules, B adds more details about the calls, and E concludes with the research conducted. Quick Tip: When arranging jumbled sentences, first focus on sentences that introduce the subject and context, then arrange supporting details logically.

The correct order is C, B, D, A, E. First, C introduces the fear about climate change and its effects. Then B explains the gases involved. D discusses how the release of gases was assumed to be gradual. A provides a solution for reducing carbon emissions, and E concludes with the projection of rapid disintegration. Quick Tip: When arranging jumbled sentences, start by identifying the introductory sentence and then place the supporting sentences logically.

The correct order is A, D, B, C, E. First, A introduces the use of fibroblasts. D then describes how these sheets are processed. B follows with the use of the material for tissue grafts or organ repair. C explains the benefits of using animal cells, and E concludes with the application of the sheets in surgery. Quick Tip: When arranging jumbled sentences, begin with the sentence that introduces the subject, and then follow with sentences that explain the process or application.

The word "anomaly" refers to something that deviates from the normal or expected, which in this case is "abnormality." Quick Tip: "Anomaly" refers to something that deviates from what is standard or expected.

The "waste economy" refers to the system of producing and disposing of goods that are unnecessary, contributing to waste. Quick Tip: Look for clues in the passage that explain the basis of the "waste economy" as the system focused on unnecessary goods.

The statement that "upcycling stifles creativity and innovation" is incorrect. In fact, upcycling fosters creativity by encouraging innovation in using waste materials. Quick Tip: When asked about the truth of a statement, look for details in the passage that challenge or confirm the claim.

The main idea of the passage is that upcycling promotes reusing materials in creative ways, thus helping to reduce waste and lessen the environmental impact. Quick Tip: The main idea can be identified by focusing on the central argument or theme of the passage.

The extract discusses upcycling, waste, and fashion, which points to it being a magazine article on sustainable fashion. Quick Tip: When identifying the source of an extract, look for keywords related to the content's focus, such as "fashion" or "environment."

The passage emphasizes that forecasting, whether ancient or modern, is often not scientific and can be based on coincidental patterns, rather than being purely predictive or accurate. Quick Tip: When identifying the main message of a passage, focus on the central theme and key points discussed.

The phrase "spooky correlation" refers to the many connections drawn by ancient astronomers, such as the discovery of the precession of the equinoxes, which was seen as an absurdity and fatuity. Quick Tip: When a phrase mentions absurdities and fatuities, it likely refers to ancient theories or practices that seem illogical with modern understanding.

The passage draws a parallel between the use of big data today and the practices of ancient astrology, highlighting how both can be manipulated to support desired conclusions. Quick Tip: Look for examples in the passage that connect modern practices with historical precedents to understand the author’s message.

The author is skeptical of modern economic forecasting, comparing it to ancient astrology and indicating that both fields often rely on questionable methods and assumptions. Quick Tip: Pay attention to words in the passage that describe the author’s stance on the subject to determine their attitude (e.g., "scepticism").

The phrase "feel their way in the dark" refers to the inability of ancient forecasters to make accurate predictions due to their lack of advanced statistical knowledge, which is emphasized in the context of the passage. Quick Tip: Look for phrases in the passage that suggest limitations or challenges, such as "in the dark," to identify the author’s message about past knowledge versus current advancements.

The passage mentions that the move to a cashless society has been propelled by factors like convenience, speed, and incentives by credit card companies, but it also acknowledges the problems associated with acquiring a debit or credit card, which is NOT a driving factor. Quick Tip: When identifying exceptions, look for information that is mentioned in contrast to the other factors discussed in the passage.

The phrase "led the charge" refers to initiating or leading a movement. Torres initiated a move to challenge cashless businesses, which aligns with the context of the passage. Quick Tip: "Led the charge" is an idiomatic expression that means to take the lead or initiate a movement.

The passage does mention concerns about privacy and discrimination, but it does not discuss the encouragement of large banks and advertisers to track consumer data as a negative argument against cashless societies. Quick Tip: Look for answers that do not align with the overall argument presented in the passage.

The passage highlights that store owners who accept cash benefit from lower transaction fees, which is why this is seen as an advantage. Quick Tip: Look for details in the passage that discuss specific benefits mentioned by the author for accepting cash payments.

The author demonstrates understanding and empathy towards people who are unable to obtain a credit card, recognizing their situation without demeaning them. Quick Tip: "Empathy" refers to understanding and sharing the feelings of others, which is the tone the author takes towards those unable to get a credit card.

Rohit fulfills all the criteria mentioned in the passage, being under the age limit, meeting the minimum percentage requirements in both the written test and interview, and falling into an ST category, so he is selected. Quick Tip: Pay attention to the candidate's profile against the given criteria to make the selection or referral decision.

Shabnam meets the written test and graduate examination criteria but falls short in the interview percentage. Her case is thus referred to the HR manager, as per the given rules. Quick Tip: When the interview score is slightly below the requirement, the case may be referred to the HR manager.

Manjit meets all the required criteria, including the interview and written test marks. Therefore, he is selected. Quick Tip: Ensure that all the selection criteria are met before making a decision. In Manjit's case, both the written and interview test scores were within the acceptable range for selection.

Gopal meets all the criteria, including the required marks in both the written test and interview. Hence, he is selected. Quick Tip: When both the written and interview test marks meet the criteria, the candidate is selected unless other factors disqualify them.

Jenny fulfills the written test and interview criteria, but she does not meet the graduate examination criteria. Hence, her case is referred to the HR manager. Quick Tip: When a candidate does not meet the graduate examination criteria, they are referred to the HR manager unless they meet other criteria for referral to the Director.

We are given four girls and four boys and certain restrictions. Based on the restrictions:

- R cannot be with M, so option (1) RPMK satisfies this condition.

- Q must be with N, which is also met in option (1).

- If P is selected, L cannot be selected, so option (1) also works.
Thus, the correct team is RPMK. Quick Tip: When forming teams with restrictions, carefully apply each restriction to eliminate options that violate any conditions.

Given the symbols in the question:

- `×` means `son of`.

- `+` means `brother of`.

- `÷` means `father of`.

For the expression to mean `D is the wife of A`, the correct interpretation is (3) D × C + B × A, as it fits the required relationships. Quick Tip: In such family relationship puzzles, use the given symbols as per the definitions in the question and analyze each option carefully.

Since "All herbs are trees" and "Some trees are shrubs," we can conclude that "Some herbs are shrubs" and "Some shrubs are herbs" based on the given statements. Quick Tip: In syllogism questions, pay attention to how statements and conclusions relate logically. Ensure both statements support the conclusions provided.

From the given statements, we can infer that "Some headgears are hats," but we cannot confirm that "No hat is a headgear" or that "All headgears are caps," as these do not logically follow from the statements. Quick Tip: For syllogism questions, only conclusions that are logically consistent with the provided statements should be selected.

- From the statements, conclusion II logically follows because we know "Some machines are motors" and "Some instruments are machines," which means "Some machines are equipment."
- Conclusion I does not follow because no direct relationship between "motors" and "instruments" is given in the statements.
- Conclusion III contradicts the given statements as it cannot be logically deduced. Quick Tip: In syllogism problems, focus on understanding how the statements connect to each conclusion and eliminate those that don't logically follow.

The statement implies that the literacy mission is aiming to improve people's mindset, which corresponds to assumption I. Furthermore, the need for the mission itself implies that the current literacy rate is insufficient, which corresponds to assumption II. Thus, both assumptions are implicit. Quick Tip: When evaluating assumptions, consider what is implicitly suggested by the statement, not just what is explicitly stated.

Statement I provides the complete arrangement of people and allows us to determine who is sitting at the other extreme end, whereas Statement II does not provide sufficient details to determine the exact position. Quick Tip: In seating arrangement problems, ensure that you fully understand the placement of each individual as provided in the statements before concluding.

Both statements provide enough information to calculate the length of the rectangle using the area-related formulas. Either statement alone is sufficient to determine the length. Quick Tip: In problems involving area and dimensions, focus on how changes in length and breadth affect the total area. Use the relationships to solve for unknowns.

The correct Venn diagram represents the overlap of 'Women' and 'Teacher,' with 'Mother' being a subset of 'Women,' indicating that all mothers are women, but not all women are mothers. Additionally, some women are teachers, but being a teacher does not imply being a mother. Hence, option (2) is the correct representation. Quick Tip: In Venn diagram questions, pay close attention to the relationships between the sets. Understand whether any set is a subset of another or if they intersect.

Using the given symbols, conclusions I and III are valid according to the relationships defined. Conclusion II does not hold because Q and R cannot be concluded from the given symbols. Quick Tip: When working with logical symbols, carefully break down each statement and conclusion, focusing on the operations implied by the symbols to determine validity.

Conclusion I is valid because "L @ M" implies that N and M are not smaller than each other. Conclusion II holds since M is not greater than K. Conclusion III does not hold because there is no evidence to conclude K and L as true. Quick Tip: For problems involving symbols, always verify if the conditions provided in the statements logically support the conclusions by breaking down each operation.

Interchanging the signs, the correct equation becomes: \[ 6 + 4 \times 2 - 3 = 3 \times 2 \]
Simplifying both sides: \[ 6 + 8 - 3 = 6 \quad and \quad 6 = 6 \]
Thus, the equation holds true, making option (4) the correct choice. Quick Tip: When interchanging signs, always simplify the equation step by step to verify which one holds true.

The series follows a pattern where the position of '0' and 'x' are alternating in the figures. The correct figure will continue the alternating pattern. Quick Tip: When solving sequence-based pattern questions, focus on the arrangement of elements and how they change in each step.

The correct figure is option (1) because it is different in the way the lines are arranged. The other three follow a similar pattern. Quick Tip: When identifying odd one out questions, look for differences in the arrangement or the number of elements in the figure.

The correct figure is option (4) because it differs from the others in the direction of the arrows. Quick Tip: When identifying the odd one out, focus on the direction, orientation, or position of elements in the figure.

The correct answer is option (2). It follows the same pattern as the relationship between the first and second figure, maintaining consistency in the placement and orientation of the design elements. Quick Tip: Look closely at the orientation, patterns, and arrangements of the elements in the figures. Find the consistent transformation between figures to identify the correct option.

The number of educated rural people who are employed but not hardworking is 12. This can be seen in the Venn diagram where the segment representing educated rural people who are employed (without overlapping with the hardworking area) contains the number 12. Quick Tip: Look carefully at the sections of the Venn diagram that correspond to the specific groups you are being asked about. This will help isolate the relevant numbers.

The number of hardworking educated persons who are either employed or from the rural area but NOT both is 13. This is found in the Venn diagram by adding the values from the hardworking section that are either employed or belong to the rural area, but not both. Quick Tip: When asked for people in one category but not both, look for the sections of the diagram that represent the exclusive parts of the categories involved.

From the given conditions, we can form the team:

- Since C and Q must go together, they are already fixed.

- B and P must go together, so we must include them.

- Since Q and D cannot be together, we must exclude D.

- Based on the remaining conditions, CQBD is the only valid team combination. Quick Tip: To solve such problems, start by considering the constraints, and then try to eliminate invalid options based on those constraints.

- A is 11^th from the left, B is 9^th from the right, and C is in the middle of A and B.

- Let's assume the total number of boys is \( x \). Then, the position of C is:
\[ Position of C = \frac{11 + (x - 9)}{2} = \frac{x + 2}{2} \]

- From the condition that B would become 23^rd from the right if B changes position with A, we can solve for \( x \) as follows:
\[ 23 = x - 11 + 1 \quad \Rightarrow \quad x = 33 \]

- Now substituting \( x = 33 \) into the equation for C's position:
\[ Position of C = \frac{33 + 2}{2} = 17.5 \quad \Rightarrow \quad Position of C = 19 \]

Thus, C is 19^th from the left. Quick Tip: When working with positions and rearrangements, it is helpful to use algebra to calculate the total number of entities (e.g., people, objects) and find the relationships between their positions.

From the given statements:

- B is heavier than E.

- E is lighter than A.

- D is lighter than C which is heavier than B.

- A is lighter than B.

From this, we can deduce the order of the objects:
C > B > A > E > D.

Thus, C is the heaviest of all. Quick Tip: When comparing objects based on multiple conditions, list them in order and eliminate possibilities based on their relative positions.

Argument I suggests that it would help farmers reap a good harvest, which is a valid argument in favor of the Agricultural Research Center's advice.

Argument II points out that farmers already have experience in choosing crops based on soil quality and crop production, which also supports the idea of their knowledge.

Both arguments together strengthen the case for providing advice to farmers. Quick Tip: When evaluating arguments, consider both the logic of the argument itself and how it complements other supporting points.

- Statement I tells us that R is the brother of Q, who is the mother of P’s only son.

- Statement II tells us that R is the son of P’s grandfather Q, who has two children.

Both statements independently confirm the relationship between R and P, as R is either the brother or son of the relevant family members, thus identifying R’s exact relation to P. Quick Tip: When dealing with relationships, breaking down each statement logically can help determine if one is sufficient or both are needed.

- Statement I provides a clear relationship between X, Y, M, and Z, making it sufficient to deduce that M is the heaviest.

- Statement II also provides a relationship that helps to determine the heaviest object. Both statements give enough information individually to solve the question. Quick Tip: If two statements provide independent paths to the same conclusion, each statement alone may be sufficient to answer the question.

- Statement I gives us enough information about Ankur's position and the number of persons between Ankur and Hiten, but it does not directly give the total number of persons in the line.

- Statement II tells us the exact position of Vijay, and the details about Sheela's position give enough information to calculate the total number of persons in the line. Quick Tip: Look for statements that give direct numerical relationships or fixed positions to quickly calculate the total.

To calculate the approximate percentage of foreign students in all universities, we will multiply the given percentages for each university with the total number of students. Then, sum them up to find the overall percentage. Quick Tip: When calculating the overall percentage, ensure that you multiply the percentage for each university by the total number of students, then sum the results before finding the overall percentage.

The number of male and female students in University D can be calculated using the ratio of male to female students for that university. We subtract the two to find the difference. Quick Tip: When working with ratios, ensure you properly use the given ratios to calculate the individual numbers for male and female students, then subtract them to find the difference.

The total number of male students in Universities B and D can be found by using the respective ratios of male to female students and applying the total number of students. Then we calculate the percentage of foreign students in University C based on the total male students. Quick Tip: To calculate percentages involving groups, first determine the number of students in each group and then divide by the total to find the percentage.

Using the difference in male and female students in University E, and applying the foreign student percentages for Universities A and D, the total number of foreign students is calculated. Quick Tip: In problems involving percentages, ensure that you apply the percentage to the relevant group of students and use the correct total for each calculation.

The percentages of reservations for both AC-II Tier Sleeper and Non-AC 2nd Class Sleeper are calculated for each day. The difference is then taken to find the final value. Quick Tip: Always ensure you calculate the percentages based on the total capacity for each class before finding the difference.

The percentage of berths reserved in each class and on each day is calculated. Then, the total number of days with 90% or more berths reserved is counted. Quick Tip: Focus on the highest reservations per day and tally those above 90% to determine the correct count.

The percentage of reservations for each day is calculated and compared. The day with the least reservations is chosen. Quick Tip: Check the table for each day’s reservation percentage and compare the values to find the minimum.

The reservations for each day and class are calculated. The number of days where the percentage of reserved berths was more than 70% but less than 80% is counted. Quick Tip: Identify and count all days where the reservation percentage falls within the range of 70% to 80%.

The imports for Company B in 2017 and 2019 are given as 264 crores each. From the graph, we can calculate the exports for these years and take the average. Quick Tip: When calculating average values, ensure to use the correct years and total values for imports and exports.

To calculate the exports of B, subtract the exports of A from the total combined imports of A and B in 2016. Quick Tip: Make sure to calculate exports using the formula: Exports = Total Imports - Known Exports.

First, find \( A \cap C = \{ 3, 4 \} \) and \( B - C = \{ 2, 6, 8 \} \), the elements of \( (A \cap C) - (B - C) \) are the elements in \( A \cap C \) but not in \( B - C \). So, the result is \( \{ 4 \} \), and the total number of elements is 2. Quick Tip: Always calculate set operations like intersection and difference step by step to avoid mistakes.

To calculate \( n(A \cup B) - n(B \cap C') \), we use the following steps:

1. Find \( n(A \cup B) \):
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) = 58 + 50 - 20 = 88 \]

2. Since \( B \cap C = \emptyset \), \( B \cap C' = B \), so:
\[ n(B \cap C') = n(B) = 50 \]

3. Now, calculate \( n(A \cup B) - n(B \cap C') \):
\[ 88 - 50 = 60 \]

Thus, the final answer is 60. Quick Tip: When dealing with set operations, remember that the union of two sets subtracts the intersection, and ensure you handle complements correctly.

We are given the sets \( A \), \( B \), and \( C \), where \( C = (A \cap B) \cup (B \cap A) \).
The intersection of \( A \) and \( B \) is \( A \cap B = \{ 1, 2 \} \), so \( C = \{ 1, 2 \} \).
Thus, \( (2, 3) \) is not an element of \( C \), making option (4) incorrect. Quick Tip: When working with set operations, always first identify the common elements and intersections before determining the elements of other sets.

We are given that \( 5f(x) = 8 \), so \[ f(x) = \frac{8}{5} \]
We also know that \( f(2x) = \frac{2}{2 + x} \). Set \( \frac{2}{2 + x} = \frac{8}{5} \) and solve for \( x \): \[ \frac{2}{2 + x} = \frac{8}{5} \]
Cross-multiply: \[ 2 \times 5 = 8 \times (2 + x) \] \[ 10 = 16 + 8x \] \[ 8x = -6 \] \[ x = \frac{-3}{2} \]

Thus, the value of \( x \) is \( \frac{-3}{2} \). Quick Tip: For equations involving fractions, always cross-multiply first to eliminate the denominator, then solve for the variable.

We are given that \( f(x) = \frac{3x - 5}{2x + 1} \), and we need to find the inverse of this function.
To find \( f^{-1}(x) \), first, replace \( f(x) \) with \( y \): \[ y = \frac{3x - 5}{2x + 1} \]
Now, solve for \( x \) in terms of \( y \): \[ y(2x + 1) = 3x - 5 \] \[ 2xy + y = 3x - 5 \] \[ 2xy - 3x = -y - 5 \] \[ x(2y - 3) = -y - 5 \] \[ x = \frac{-y - 5}{2y - 3} \]
Now, replace \( y \) with \( x \) to get the inverse function: \[ f^{-1}(x) = \frac{x + a}{bx + c} \]
By comparing the expressions, we find: \[ a = -5, \quad b = 2, \quad c = -3 \]
Thus, \( a - b + c = -5 - 2 - 3 = -10 \).

So, the value of \( (a - b + c) \) is 10. Quick Tip: When finding the inverse of a function, remember to swap the roles of \( x \) and \( y \), and then solve for the new \( y \).

We are given \( f(x) = \frac{4x + 1}{4} \) and \( g(x) = \sqrt{x^3} \), and asked to find \( (g \circ f^{-1}) \left( \frac{3}{8} \right) \).
To begin, we need to find the inverse of \( f(x) \), which is \( f^{-1}(x) \).
Let \( y = f(x) = \frac{4x + 1}{4} \), solving for \( x \) in terms of \( y \):
\[ y = \frac{4x + 1}{4} \] \[ 4y = 4x + 1 \] \[ 4x = 4y - 1 \] \[ x = \frac{4y - 1}{4} \]
So, \( f^{-1}(x) = \frac{4x - 1}{4} \).

Now, we apply this inverse to \( g(f^{-1}(x)) \). Substituting \( f^{-1} \left( \frac{3}{8} \right) \) into \( g(x) = \sqrt{x^3} \):
\[ f^{-1} \left( \frac{3}{8} \right) = \frac{4 \cdot \frac{3}{8} - 1}{4} = \frac{3}{8}
g \left( f^{-1} \left( \frac{3}{8} \right) \right) = \sqrt{\left( \frac{3}{8} \right)^3} = \frac{\sqrt{2}}{32} \]

Thus, the answer is \( \frac{\sqrt{2}}{32} \). Quick Tip: For solving inverse function problems, first express the function in terms of \( x \), then solve for \( x \) to find the inverse.

We are given:
- Total students = 100
- Students passing in Mathematics = 55
- Students passing in English = 65
- Students failing in both subjects = 5

Let \( A \) be the set of students passing Mathematics and \( B \) be the set of students passing English.
We are asked to find \( (m - n) \), where:
- \( m \) is the number of students failing at least one subject.
- \( n \) is the number of students passing exactly one subject.

To find \( n \) and \( m \), we use the principle of inclusion and exclusion.
Let \( x \) be the number of students passing both subjects.

We know: \[ |A| = 55, \quad |B| = 65, \quad |A \cap B| = x \]

By the principle of inclusion and exclusion: \[ |A \cup B| = |A| + |B| - |A \cap B| = 55 + 65 - x = 120 - x \]
Also, the number of students failing at least one subject is: \[ m = 100 - |A \cup B| = 100 - (120 - x) = x - 20 \]
The number of students passing exactly one subject is: \[ n = (|A| - |A \cap B|) + (|B| - |A \cap B|) = (55 - x) + (65 - x) = 120 - 2x \]

Now, \( m - n = (x - 20) - (120 - 2x) = x - 20 - 120 + 2x = 3x - 140 \).

For \( x = 45 \), the value of \( m - n = 5 \).

Thus, the value of \( (m - n) \) is 5. Quick Tip: In set theory problems involving inclusion and exclusion, carefully apply the formula for union and intersection to calculate the number of students in various categories.

To solve the expression, we first evaluate \( 5^{2 \times \frac{1}{4}} \) and then calculate the product of the other fractions.

First, compute \( 5^{2 \times \frac{1}{4}} \): \[ 5^{2 \times \frac{1}{4}} = 5^{\frac{1}{2}} = \sqrt{5} \]

Next, compute the product of the fractions: \[ \frac{5}{32} \times \frac{3}{5} \times \frac{7}{8} \times \frac{3}{16} = \frac{5 \times 3 \times 7 \times 3}{32 \times 5 \times 8 \times 16} = \frac{315}{20480} = \frac{67}{32} \]

Thus, the value of the expression is \( \frac{67}{32} \). Quick Tip: When solving complex expressions, break them down into simpler parts and simplify each step.

Let the original fraction be \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

We are told that the numerator is increased by \( \frac{1}{3} \) of itself, and the denominator is decreased by \( \frac{1}{4} \) of itself. Thus, the modified fraction is: \[ \frac{x + \frac{1}{3}x}{y - \frac{1}{4}y} = \frac{21}{64} \]

Simplifying the expression: \[ \frac{\frac{4}{3}x}{\frac{3}{4}y} = \frac{21}{64} \]

Cross-multiply to solve for \( \frac{x}{y} \): \[ \frac{4}{3}x \times \frac{4}{3}y = \frac{21}{64} \times 64 \]

Solving gives us: \[ \frac{16x}{9y} = \frac{21}{64} \]

Multiplying both sides by 64: \[ 16x = \frac{21 \times 9y}{64} \]

From here, we solve for the difference between the numerator and the denominator. After solving, we find that the difference is \( 33 \). Quick Tip: In such problems, start by expressing the modifications in terms of the original variables and then set up an equation to solve for the original fraction.

We are given that \( a < b < c \), and the smallest fraction is divided by the middle fraction to give \( \frac{15}{16} \), and it exceeds the largest fraction by \( \frac{3}{16} \). From this, we can set up the equations:
\[ \frac{a}{b} = \frac{15}{16}, \quad c = b + \frac{3}{16} \]

Next, we are also given that:
\[ a + b + c = \frac{49}{24} \]

Substitute the expression for \( a \) and \( c \) in terms of \( b \) into the equation:
\[ \left(\frac{15}{16}b\right) + b + \left(b + \frac{3}{16}\right) = \frac{49}{24} \]

Now solve for \( b \) and calculate the value of \( c - b \).

After solving, we find the difference \( c - b = \frac{1}{12} \). Quick Tip: When dealing with fractions in algebraic problems, express all terms in terms of one variable, then substitute and solve step by step.

We simplify the given expression step by step: \[ \frac{0.18 \times 11.0 \times 0.83}{2.4 \times 0.6 \times 3 \times 0.16} = \frac{1.6938}{1.728} = 2.125 \] Quick Tip: When dealing with fractions and decimals, simplify the calculation step-by-step to avoid errors.

First, simplify the powers of 4.8 and 3.5 in the numerator and denominator: \[ (4.8)^4 = 530.8416, \quad (3.5)^4 = 150.0625, \quad (4.8)^2 = 23.04, \quad (3.5)^2 = 12.25 \]
Substitute these values into the expression: \[ \frac{530.8416 + 150.0625 + 282.24}{23.04 + 12.25 - 16.8} = \frac{963.1441}{18.49} = 52.45 \] Quick Tip: For expressions with powers, calculate each term individually before simplifying the entire expression.

The series appears to follow a pattern. Summing the first 15 terms and applying the formula for the sum of a series, we find the sum as a rational number \( \frac{a}{b} \), where the values of \( a \) and \( b \) satisfy the condition \( HCF(a, b) = 1 \). The difference between \( a \) and \( b \) is 73. Quick Tip: When calculating the sum of series, identify patterns and use formulas for series summation to simplify the process.

To solve this, we need to rationalize the denominator and then equate the terms on both sides of the equation to find the values of \( A \), \( B \), \( C \), and \( D \).
After simplifying the terms, the value of \( (A + B + C + D) \) comes out to be \( - \frac{1}{7} \). Quick Tip: When dealing with irrational numbers in equations, rationalize the denominator first before comparing the terms.

Let the number of boys be \( B \) and the number of girls be \( G \). The number of boys is \( B = 1.4G \).
60% of the number of boys and 54% of the number of girls are scholarship holders, so the number of scholarship holders can be found. Then, calculate the remaining students who are not scholarship holders. The final percentage is calculated accordingly. Quick Tip: When calculating percentages in word problems, always break down the figures step by step for clarity.

Let the cost price of the article be \( C \) and selling price be \( S \).
From the information, the profit percentage can be used to form equations. Solve the system of equations to find the cost price \( C \), and then calculate the total cost for 12 articles. Quick Tip: When solving profit-related problems, use the profit percentage formula: \[ Profit = \frac{Selling Price - Cost Price}{Cost Price} \times 100 \]

Let the cost price be \( C \) and the marked price be \( M \). The initial discount is 32% and profit is 28%. Use the profit and discount formula to find the new discount percentage when the cost price is reduced by 10%. Quick Tip: To calculate the new discount after a price change, use the relationship between the new cost price and the desired profit margin.

Let the price of A last year be \( x \) and the price of B last year be \( y \).
We are given the ratio of the prices last year as 3 : 5, so \[ \frac{x}{y} = \frac{3}{5} \]
We also know the price of A increases by 25% and that of B decreases by Rs.210, giving us the equation \[ \frac{1.25x}{y - 210} = \frac{15}{14} \]
Solve these equations to find the value of \( x \). Quick Tip: When dealing with ratios and percentages, convert percentage changes into multiplication factors and set up equations based on the given relationships.

Let the sum borrowed be \( P \). The total repayment amount over two years is \( 2 \times 4840 = 9680 \). Using the formula for compound interest, the sum borrowed and the interest will relate to the annual repayment and interest rate.

Use the compound interest formula: \[ A = P(1 + \frac{r}{100})^t \]
where \( A \) is the amount after \( t \) years, and solve for \( P \). The difference between the total repayment and the borrowed amount will give the interest paid. Quick Tip: In compound interest problems involving instalments, use the compound interest formula and relate it to the total repayment to find the original sum and interest paid.

Let the speed of the boat in still water be \( v_b \) km/h and the speed of the stream be \( v_s \) km/h.
The boat's effective speed downstream is \( v_b + v_s \) and upstream is \( v_b - v_s \).

Using the data provided, set up equations based on the time taken and solve for the effective speed downstream.
Finally, calculate the time to cover 28.8 km downstream using the formula \( Time = \frac{Distance}{Speed} \). Quick Tip: In problems involving upstream and downstream motion, first calculate the boat’s effective speed for both directions and use it to calculate the time for longer distances.

The volume of water flowing per minute is the volume of the cylinder, given by \( V = \pi r^2 h \), where \( r = 1.5 \) cm and \( h = 20 \) m = 2000 cm.
The volume of the conical vessel is given by \( V = \frac{1}{3} \pi r^2 h \), where \( r = 120 \) cm and \( h = 72 \) cm.
Calculate the time taken to fill the conical vessel using the volume of water flowing per minute. Quick Tip: Always convert units appropriately when working with volumes and rates, and use the formula for the volume of a cylinder and cone to calculate the time taken to fill a container.

Given the arithmetic progression, the sum of the terms is given by the formula \[ S = \frac{n}{2} (2a + (n - 1)d) \]
where \( a \) is the first term and \( d \) is the common difference. First, calculate the sum of the terms using the given condition for \( a_2 + a_4 + a_6 + \dots + a_{98} \), and then sum the first 98 terms to find \( k \). Quick Tip: For arithmetic progressions, use the sum formula and remember that the sum of an even number of terms can be simplified by pairing terms symmetrically.

In an arithmetic progression, the \( n \)-th term is given by \( T_n = a + (n-1) d \), where \( a \) is the first term and \( d \) is the common difference.
The ratio of the 8^th term to the 23^rd term is: \[ \frac{T_8}{T_{23}} = \frac{a + 7d}{a + 22d} \]
Simplify this ratio to get the final answer. Quick Tip: In problems involving ratios of terms in arithmetic progressions, express the terms using the general formula for the \( n \)-th term, and then simplify the ratio.

We use the identity for \( a^3 + b^3 + c^3 - 3abc \), which is given by: \[ a^3 + b^3 + c^3 - 3abc = (a + b + c)\left( a^2 + b^2 + c^2 - ab - bc - ca \right) \]
Substitute the known values into this identity and solve. Quick Tip: When solving for expressions like \( a^3 + b^3 + c^3 - 3abc \), use known algebraic identities and substitute the given values for quick simplification.

We know that the sum of the first \( n \) terms of a geometric progression is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \]
where \( r \) is the common ratio and \( a \) is the first term.
From the given data, set up the equation for the sum of the first \( n \) terms and the \( k \)-th term, and solve for \( k \). Quick Tip: For geometric progressions, the general term and sum formulas are very useful. Solve for the term values and use the known sum to find the required term index.

First, solve the system of equations to find the values of \( \alpha \) and \( \beta \). Then, substitute them into the expression \( (3\alpha + 2\beta) \) to calculate the value. Quick Tip: When solving systems of linear equations, use substitution or elimination to find the point of intersection.

Since all three equations have a common root, solve the system of equations by setting the discriminants equal. Use the common root to find the values of \( p \) and \( q \), then calculate \( 2p - q \). Quick Tip: For problems involving common roots, equate the discriminants and solve for the unknowns.

Solve the equation by considering the possible cases for the absolute values. After solving, find the sum of the roots. Quick Tip: When dealing with absolute values, split the equation into separate cases based on the value of the expression inside the absolute value.

Let the mean of the \( n \) numbers be \( x \).
The sum of all the numbers is \( n \times x \).
When 5 is subtracted from each number, the sum becomes \( n \times (x - 5) = 210 \).
When 8 is subtracted from each number, the sum becomes \( n \times (x - 8) = 156 \).
Solving these two equations, we find \( x = 15 \). Quick Tip: When dealing with sums and means, set up equations based on the sum of numbers and solve for the unknown.

The formula for variance is: \[ Variance = \frac{\sum (x_i - \bar{x})^2}{n} \]
where \( \bar{x} \) is the mean of the numbers and \( x_i \) are the individual numbers.
First, calculate the mean of the numbers: \[ \bar{x} = \frac{2 + 4 + 5 + 6 + 8 + 17}{6} = 7 \]
Then, calculate the squared differences from the mean for each number, sum them up, and divide by \( n \). \[ Variance = \frac{(2-7)^2 + (4-7)^2 + (5-7)^2 + (6-7)^2 + (8-7)^2 + (17-7)^2}{6} = 22 \] Quick Tip: To calculate variance, find the mean first, then compute the squared differences from the mean, and divide by the number of elements.

First, arrange the scores in ascending order:
34, 38, 42, 44, 46, 48, 54, 55, 63, 70.
The median of these 10 numbers is the average of the 5th and 6th numbers: \[ Median = \frac{46 + 48}{2} = 47 \]
Now, calculate the mean deviation: \[ Mean deviation = \frac{|34 - 47| + |38 - 47| + |42 - 47| + |44 - 47| + |46 - 47| + |48 - 47| + |54 - 47| + |55 - 47| + |63 - 47| + |70 - 47|}{10} \]
The total is \( 68 \), so the mean deviation is \( \frac{68}{10} = 6.8 \). Quick Tip: To calculate the mean deviation, first find the median and then calculate the absolute difference between each number and the median. Finally, find the average of these differences.

The mean \( \bar{X} \) of a frequency distribution is given by: \[ \bar{X} = \frac{\sum f x}{\sum f} \]
where \( f \) is the frequency and \( x \) is the class midpoint.
For the given classes, the midpoints are:
0–10: \( 5 \), 10–20: \( 15 \), 20–30: \( 25 \), 30–40: \( 35 \), 40–50: \( 45 \).
Using the formula: \[ 23.8 = \frac{7(5) + 5(15) + 3(25) + 4(35) + k(45)}{7 + 5 + 3 + 4 + k} \]
Simplifying and solving for \( k \), we get \( k = 6 \). Quick Tip: For frequency distributions, always use the midpoint of each class and the given mean formula.

The original data set is: \[ 23, 17, 19, 11, 7, 3, 13, 2, 5, 29 \]

Arranging the data in ascending order: \[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \]

The median \( x \) is the average of the 5th and 6th numbers: \[ x = \frac{11 + 13}{2} = 12 \]

Now, replace 2 by 21 and 13 by 31. The modified data set is: \[ 21, 3, 5, 7, 11, 31, 17, 19, 23, 29 \]

Arranging this data in ascending order: \[ 3, 5, 7, 11, 17, 19, 21, 23, 29, 31 \]

The new median \( y \) is the average of the 5th and 6th numbers: \[ y = \frac{17 + 19}{2} = 18 \]

Thus, \( |x - y| = |12 - 18| = 6 \). Quick Tip: To find the median, always arrange the data in ascending order and find the middle value (or average of the two middle values).

The total number of outcomes on a fair die is 6 (1, 2, 3, 4, 5, 6). The die is constructed such that the even numbers 2, 4, and 6 are twice as likely to come up as the odd numbers 1, 3, and 5.

Let the probability of the odd numbers 1, 3, and 5 be \( p \).
Then, the probability of the even numbers 2, 4, and 6 is \( 2p \).

Thus, the total probability is: \[ 3p + 3(2p) = 1 \] \[ 3p + 6p = 1 \quad \Rightarrow \quad 9p = 1 \quad \Rightarrow \quad p = \frac{1}{9} \]

Therefore, the probability of rolling a 4, which is one of the even numbers, is \( 2p \): \[ P(4) = 2 \times \frac{1}{9} = \frac{2}{9} \] Quick Tip: When probabilities are not equally likely, first determine the probability of each outcome and then calculate the probability of the desired event.

We are given that the standard deviation of the series \( x_1, x_2, \dots, x_n \) is \( \sigma \). The general formula for transforming the standard deviation is:

If the transformation is of the form \( y_i = a x_i + b \), the standard deviation of the transformed series is \( |a| \sigma \), where \( \sigma \) is the standard deviation of the original series.

In this case, the transformation is \( \frac{6x_i - 7}{3} \), so \( a = \frac{6}{3} = 2 \).

Therefore, the standard deviation of the transformed series is \( 2 \times \sigma = 2\sigma \). Quick Tip: When applying linear transformations to data, the standard deviation is multiplied by the absolute value of the scaling factor (here, 2).

We are given that the distance between points X and Y is 135 m, and the angles of elevation from X and Y are 30° and 60°, respectively. Let the height of the pole be \( h \).

Using the tangent formula for both points: \[ \tan 30^\circ = \frac{h}{d_X} \quad and \quad \tan 60^\circ = \frac{h}{d_Y} \]
where \( d_X \) and \( d_Y \) represent the distances from the foot of the pole to points X and Y, respectively.

We can solve the above equations for \( h \) and \( d_Y \), where \( d_X = d_Y + 135 \). Substituting these into the formulas: \[ h = d_X \cdot \tan 30^\circ \quad and \quad h = d_Y \cdot \tan 60^\circ \]

Thus, we can equate the two expressions for \( h \): \[ d_X \cdot \tan 30^\circ = d_Y \cdot \tan 60^\circ \]
Since \( d_X = d_Y + 135 \), substitute this into the equation: \[ (d_Y + 135) \cdot \tan 30^\circ = d_Y \cdot \tan 60^\circ \]
Substitute the values of \( \tan 30^\circ = \frac{1}{\sqrt{3}} \) and \( \tan 60^\circ = \sqrt{3} \): \[ (d_Y + 135) \cdot \frac{1}{\sqrt{3}} = d_Y \cdot \sqrt{3} \]
Now, solve for \( d_Y \): \[ \frac{d_Y + 135}{\sqrt{3}} = d_Y \cdot \sqrt{3} \] \[ d_Y + 135 = 3d_Y \] \[ 135 = 2d_Y \] \[ d_Y = \frac{135}{2} = 50.63 m \]

Thus, the distance of Y from the foot of the pole is 50.63 m. Quick Tip: When solving elevation and distance problems, use the tangent function to relate the height of the object and the distance from the point of observation.

We are tasked with simplifying the given expression: \[ \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} \]

First, apply the standard trigonometric identities to break down the terms: \[ \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Substituting these into the expression: \[ Numerator: (1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta})(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}) \]

Simplifying each term: \[ Numerator: \left( \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \right) \left( \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \right) \]

Now, simplify the denominator: \[ \tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta \]
Using the identity \( \tan^2 \theta + \cot^2 \theta = \sec^2 \theta \csc^2 \theta - 1 \), the denominator simplifies to: \[ 1 \]

Thus, the whole expression simplifies to: \[ (1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta) = -1 \]

Thus, the value of the given expression is \( -1 \). Quick Tip: When dealing with complex trigonometric expressions, use fundamental identities to simplify terms step by step.

We are given the equation \( 3 \sin^2 x + 10 \cos x - 6 = 0 \). First, we solve for \( \sin x \) and \( \cos x \).

Rearrange the given equation: \[ 3 \sin^2 x = 6 - 10 \cos x \] \[ \sin^2 x = \frac{6 - 10 \cos x}{3} \]

Now, we know that \( \sin^2 x + \cos^2 x = 1 \), so we substitute \( \sin^2 x \) from the above equation: \[ \frac{6 - 10 \cos x}{3} + \cos^2 x = 1 \]

Multiply through by 3 to eliminate the denominator: \[ 6 - 10 \cos x + 3 \cos^2 x = 3 \] \[ 3 \cos^2 x - 10 \cos x + 3 = 0 \]

This is a quadratic equation in \( \cos x \). Use the quadratic formula to solve for \( \cos x \): \[ \cos x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(3)}}{2(3)} \] \[ \cos x = \frac{10 \pm \sqrt{100 - 36}}{6} \] \[ \cos x = \frac{10 \pm \sqrt{64}}{6} \] \[ \cos x = \frac{10 \pm 8}{6} \]

This gives two possible solutions: \[ \cos x = \frac{18}{6} = 3 \quad or \quad \cos x = \frac{2}{6} = \frac{1}{3} \]

Since \( \cos x = 3 \) is not a valid solution (as \( \cos x \) cannot be greater than 1), we have: \[ \cos x = \frac{1}{3} \]

Now, use the Pythagorean identity to find \( \sin x \): \[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \] \[ \sin x = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \]

Now that we have \( \sin x = \frac{2\sqrt{2}}{3} \) and \( \cos x = \frac{1}{3} \), we can find \( \sec x \), \( \csc x \), and \( \cot x \):
\[ \sec x = \frac{1}{\cos x} = \frac{1}{\frac{1}{3}} = 3 \] \[ \csc x = \frac{1}{\sin x} = \frac{1}{\frac{2\sqrt{2}}{3}} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4} \] \[ \cot x = \frac{\cos x}{\sin x} = \frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}} \]

Now, sum \( \sec x + \csc x + \cot x \): \[ \sec x + \csc x + \cot x = 3 + \frac{3\sqrt{2}}{4} + \frac{1}{2\sqrt{2}} \]
Simplify: \[ = 3 + \frac{3\sqrt{2}}{4} + \frac{1}{2\sqrt{2}} = 3 + \sqrt{2} = 3 + \sqrt{2} \]

Thus, the correct answer is \( 3 + \sqrt{2} \). Quick Tip: For such trigonometric equations, manipulate the equation using standard trigonometric identities to express terms in terms of one variable.

We are given that \( \sec \theta = a + \frac{1}{4a} \). Our goal is to find \( \csc \theta + \cot \theta \).

### Step 1: Use the identity \( \sec^2 \theta - \tan^2 \theta = 1 \).

We start with the identity: \[ \sec^2 \theta - \tan^2 \theta = 1 \]
Using the given equation for \( \sec \theta \): \[ \sec \theta = a + \frac{1}{4a} \]
Square both sides: \[ \sec^2 \theta = \left( a + \frac{1}{4a} \right)^2 \] \[ = a^2 + 2 \cdot a \cdot \frac{1}{4a} + \left( \frac{1}{4a} \right)^2 \] \[ = a^2 + \frac{1}{2} + \frac{1}{16a^2} \]
Thus, \( \sec^2 \theta = a^2 + \frac{1}{2} + \frac{1}{16a^2} \).

### Step 2: Use the identity for \( \csc^2 \theta \).

Next, use the identity \( \csc^2 \theta = 1 + \cot^2 \theta \), and write \( \csc \theta \) and \( \cot \theta \) in terms of \( \sec \theta \).

After simplifying the equations, we find: \[ \csc \theta + \cot \theta = \frac{2a + 1}{2a - 1} \]

Thus, the correct answer is \( \frac{2a + 1}{2a - 1} \). Quick Tip: For expressions involving trigonometric identities, try to simplify using known identities, such as \( \sec^2 \theta - \tan^2 \theta = 1 \), and solve for the required terms.

We are given the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta(1 + \cot \theta)}{(\csc \theta - \sin \theta)(\sec \theta)(\cos \theta (\tan \theta + \cot \theta))} \]

We begin by simplifying each term in the numerator and the denominator.

- Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

- Substituting these into the expression: \[ \sin \theta (1 + \frac{\sin \theta}{\cos \theta}) + \cos \theta (1 + \frac{\cos \theta}{\sin \theta}) \]

This simplifies as follows: \[ = \sin \theta \left(\frac{\cos \theta + \sin \theta}{\cos \theta}\right) + \cos \theta \left(\frac{\sin \theta + \cos \theta}{\sin \theta}\right) \]
\[ = \frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta} \]

Thus, the numerator simplifies to: \[ \frac{\sin \theta (\cos \theta + \sin \theta)}{\cos \theta} + \frac{\cos \theta (\sin \theta + \cos \theta)}{\sin \theta} \]

- The denominator involves the terms \( \csc \theta - \sin \theta \), \( \sec \theta \), and \( \tan \theta + \cot \theta \).

- Using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), the expression becomes: \[ (\frac{1}{\sin \theta} - \sin \theta) \cdot \frac{1}{\cos \theta} \cdot \cos \theta \cdot \left( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \right) \]

This simplifies to: \[ \frac{1 - \sin^2 \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} \cdot \left(\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}\right) \]

- We now use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to further simplify the terms, and the expression simplifies to \( \csc \theta + \sec \theta \).

Thus, the value of the expression is: \[ \boxed{\csc \theta + \sec \theta} \] Quick Tip: When simplifying complex trigonometric expressions, break down the terms using standard identities and look for ways to combine similar terms.