PGDBA 2025 Solutions

PGDBA 2025 | Q1

Read the passage and answer the following questions

Yet after three decades of freedom, most South Africans say they are dissatisfied with democracy and would ditch elected governments if an autocrat could do a better job. There is more socialising across racial boundaries, but the share of South Africans saying race relations have improved since 1994 has fallen sharply since 2010.

After steady progress in the first 15 years, most South Africans-and therefore, since they are

81 %

of the population, most black citizens-no longer see their lives getting better. On average, incomes have stagnated since 2008, unemployment has risen from around

20 %

to more than

30 %

, and power and water cuts have become more frequent. Corruption has seeped into every layer of the state. Only

15 %

of 257 municipalities get clean audits from the relevant watchdog. It is hard to be thankful for democratic freedoms when you are jobless and living in your grandmother's house.

Alas, a fatalistic explanation has taken root among academics, commentators and leftwing politicians. This view holds that the deal struck in the early 1990s to end apartheid was a sham: it won black people political rights, but not economic freedom. Mandela, in other words, was a sell-out who ushered in a "new apartheid" in which a black elite was co-opted by whites.

There are many reasons why this is wrong. First, it gives too little credit to Mandela's courage and canniness. He was a skilful politician who stuck to his red lines, winning majority rule while avoiding civil war. Anyone who spends 27 years in prison for his cause is an unusual sort of sell-out.

It also ignores the good the African National Congress (ANC) did with its newly won power. Under Trevor Manuel, the finance minister from 1996 to 2009, sensible macroeconomic policies underpinned steady annual GDP growth averaging

3.3 %

, more than double the rate of the next 14 years. Using the proceeds, ANC-led governments replaced millions of shacks and mud huts with decent homes that have water and electric lights. Welfare benefits made poverty less grinding.

Another reason is that fatalism allows the ANC to shirk responsibility for its worst decisions. Corruption, glossed over by Mandela and Thabo Mbeki, his successor, exploded under Jacob Zuma, President from 2009 to 2018. Cadre deployment, whereby civil servants are appointed on the basis of fealty to the ANC, not merit, has eviscerated bureaucratic capacity. In another case of ideological folly, the ANC has increasingly pursued policies that focus on the redistribution of wealth in ways that actually hinder its creation. These include so-called "black economic empowerment" policies and boosting the bargaining power of unions. The ANC's own policies are the reason why South Africa is still a society where perhaps a quarter of people live well and the rest are desperately poor.

The danger over the next 30 years is that politics becomes ever more zero-sum. Populist parties are race-baiting and exploiting poverty. But there is also a resilience to South African politics, forged in the fight for liberty. Apartheid left a wariness of racially or tribally charged politics. The transition helped entrench consensus and pragmatism. Elections have brought some accountability: the ANC now has a majority in only two of the eight largest urban areas. Democracy may have been a disappointment so far. But, in 2024 as in 1994, it offers the potential for renewal. That is Mandela's enduring gift.

Q. 1 Which of the following statements is not true as per the passage?

Core Logic: This question tests your ability to distinguish between the author's arguments and the "fatalistic" views he is debunking, as well as checking specific factual claims provided in the text.

Option A: For the current left-wing politicians Mandela was a sell-out. Status: TRUE. The third paragraph mentions that a "fatalistic explanation" has taken root among left-wing politicians, which holds that Mandela was a "sell-out" who only won political rights and not economic freedom.

Option B: South Africans have a pronounced preference for democracy. Correct Choice (NOT TRUE). The very first sentence of the passage states that "most South Africans say they are dissatisfied with democracy and would ditch elected governments if an autocrat could do a better job." Far from a "pronounced preference," the text highlights a significant disillusionment with the democratic system.

Option C: Mandela spent 27 years in prison fighting apartheid. Status: TRUE. The fourth paragraph explicitly mentions that Mandela spent "27 years in prison for his cause."

Option D: Nelson Mandela, Thabo Mbeki and Jacob Zuma, have all been presidents of South Africa. Status: TRUE. The passage mentions all three in the context of their leadership: Mandela (paragraph 3 & 4), Thabo Mbeki (paragraph 6), and Jacob Zuma (paragraph 6).

PGDBA 2025 | Q2

Read the passage and answer the following questions (Same passage as Q1).

Q. 2 Which of the following statements is not accurate as per the passage?

Core Logic: This question requires you to distinguish between the historical reality of South Africa’s transition and the modern criticisms mentioned in the text. You must also identify the current sentiment of the population as described by the author.

Option A: South Africans are showing preferences towards autocracy. Status: ACCURATE. The very first paragraph states that South Africans "would ditch elected governments if an autocrat could do a better job." This indicates a conditional preference for autocracy over the current failing democratic state.

Option B: In some quarters, Mandela is seen as the progenitor of a “new apartheid”. Status: ACCURATE. The third paragraph explains a "fatalistic explanation" among academics and left-wing politicians who claim Mandela's 1990s deal was a sham that ushered in a "new apartheid" where a black elite was co-opted by whites.

Option C: South Africans are showing lesser inclination towards democracy. Status: ACCURATE. This is a restatement of the opening sentiment. Since most say they are "dissatisfied with democracy," their inclination toward it has clearly diminished compared to the optimism of 1994.

Option D: The end of apartheid meant economic freedom for blacks. Correct Choice (NOT ACCURATE). The passage argues the exact opposite. Paragraph 3 notes that while the transition won political rights, it did not win "economic freedom." Paragraph 6 further states that ANC policies have left "the rest [of the people] desperately poor." Therefore, saying the end of apartheid meant (resulted in) economic freedom for Black citizens is factually incorrect according to the text.

PGDBA 2025 | Q3

Read the passage and answer the following questions (Same passage as Q1).

Q. 3 Which of the following statements is correct as per the passage?

Core Logic: This question asks you to identify the author's specific critique of the ANC's recent policies. In the sixth paragraph, the author lists several "ideological follies" that have hindered wealth creation.

Option A: The African National Congress has dominance in a large number of the largest urban areas. Status: INCORRECT. The final paragraph states the opposite: "the ANC now has a majority in only two of the eight largest urban areas." This is cited as evidence that elections are bringing accountability.

Option B: In the immediate aftermath of the end of apartheid, poverty became more grinding for most black citizens. Status: INCORRECT. Paragraph 5 highlights the "good the ANC did" in the early years. It mentions that welfare benefits made poverty "less grinding" and millions of shacks were replaced with decent homes during Trevor Manuel’s tenure as finance minister.

Option C: South African politics is totally shorn of resilience. Status: INCORRECT. The final paragraph explicitly argues that there is "a resilience to South African politics, forged in the fight for liberty."

Option D: ‘Black economic empowerment’ is an instance of an ideological folly. Correct Choice. In paragraph 6, the author states: "In another case of ideological folly, the ANC has increasingly pursued policies that focus on the redistribution of wealth in ways that actually hinder its creation. These include so-called 'black economic empowerment' policies..."

PGDBA 2025 | Q4

Read the passage and answer the following questions (Same passage as Q1).

Q. 4 Which of the following statements is not correct as per the passage?

Core Logic: This question requires a close reading of the first and last paragraphs to distinguish between "social behavior" and "perception of relations."

Option A: There is less socialising across racial divide. Correct Choice (NOT CORRECT). The first paragraph explicitly states: "There is more socialising across racial boundaries..." While perceptions of race relations have soured, the actual act of socializing has increased. This option contradicts the text.

Option B: The number of South Africans believing in improved race relations have dwindled since 2010. Status: CORRECT. Paragraph 1 notes that the share of people saying race relations have improved "has fallen sharply since 2010."

Option C: Mandela managed to avoid civil war. Status: CORRECT. Paragraph 4 describes Mandela as a "skilful politician" who won majority rule while "avoiding civil war."

Option D: Populist parties are leveraging poverty. Status: CORRECT. The final paragraph mentions that "Populist parties are race-baiting and exploiting poverty" to gain political ground.

PGDBA 2025 | Q5

Read the passage and answer the following questions (Same passage as Q1).

Q. 5 Which of the following statements is true as per the passage?

Core Logic: This question asks you to identify a positive outcome of the post-apartheid era mentioned in the text. While much of the passage focuses on recent failures, the final paragraph highlights the structural strengths that remain.

Option A: In South Africa perhaps a one-fourth of people are poor and three-fourth live well. Status: INCORRECT. The sixth paragraph states the exact opposite: "South Africa is still a society where perhaps a quarter of people live well and the rest are desperately poor." The ratios in the option are reversed.

Option B: The end of Apartheid has facilitated values like consensus and pragmatism. Correct Choice. The final paragraph explicitly states: "The transition helped entrench consensus and pragmatism." The author uses this to argue that South African politics retains a level of resilience despite current disappointments.

Option C: Under Trevor Manuel South Africa was an economic disaster. Status: INCORRECT. Paragraph 5 describes the period under Trevor Manuel (1996–2009) as a time of "steady annual GDP growth averaging 3.3%" and "sensible macroeconomic policies." The "disaster" or stagnation happened in the 14 years after his tenure.

Option D: Cadre deployment by the ANC has enhanced the state’s bureaucratic capacity. Status: INCORRECT. Paragraph 6 argues that cadre deployment—appointing people based on loyalty rather than merit—has "eviscerated bureaucratic capacity." "Eviscerated" means to deprive something of its essential content or to weaken it fundamentally.

PGDBA 2025 | Q6

Read the passage and answer the following questions

Motivational quotes. Videos of Steve Jobs saying absolutely anything. Clips of a baby elephant being rescued from a river. You do not have to scroll for long on LinkedIn, a networking site ostensibly for people at work, to find "inspirational" content. There may be people who need only to read "We cannot become what we want by remaining what we are" written in a nice font to feel amped up on a Monday morning. But there will be just as many who want to snigger or vomit. For bosses interested in how to motivate the people around them, there are better options than searching for quotes by Paulo Coelho.

At some level, advice on how to inspire employees is silly. It's usually either blindingly obvious - be good at your job, be passionate about the work, make the people on your teams feel valued - or jarringly inauthentic. But much more practical insights can be found in a forthcoming book called "Inspire", by Adam Galinsky, an academic at Columbia Business School.

Take, for example, the importance of vivid imagery as a way of bringing an organisation's purpose to life. Lots of firms use a succession of tediously abstract words to convey their goal: "change", "innovate", "connect" and so on. The result is less a mission than a mood board. Mr Galinsky cites an experiment by Andrew Carton of the University of Pennsylvania and his co-authors that showed the effect of more concrete language. In it, teams were asked to design toys and given a vision statement to guide their behaviour. Teams who were handed a statement with more visual language - to create toys that "...make wide-eyed kids laugh and proud parents smile" - produced more engaging toys than teams who were given something more generic.

Mr. Galinsky also points to the power of counterfactual thinking to inspire a sense of meaning. In research he conducted with Laura Kray of the University of California, Berkeley and other co-authors, participants were asked to reflect on important events in their lives, such as their choice of college. Some were also asked to think about how things would have turned out if this event had not taken place. This group attributed greater meaning to the event in question, whether because they concluded fate had played a part in it or because it forced them to think through its consequences more explicitly. This type of counterfactual thinking can also be used to strengthen employees' ties to firms: prompting people to imagine a world in which their company does not exist seems to increase a sense of attachment.

Perhaps the most striking idea in Mr. Galinsky's book is that, instead of bosses motivating people from above, individuals can do it for themselves. One example is a piece of research he conducted with Julian Pfrombeck from the Chinese University of Hong Kong and other co-authors. In this study some Swiss citizens who had newly registered with a government employment agency were asked to undertake a ten-to-fifteen-minute exercise in which they reflected on values that mattered to them. They were three times more likely to find a job than those who did not do the exercise.

Q. 6 How does counterfactual thinking contribute to a sense of meaning, according to the research mentioned in the passage?

Core Logic: The answer is found in the fourth paragraph. The passage defines counterfactual thinking as reflecting on an event and then imagining "how things would have turned out if this event had not taken place."

The research shows that this process increases the meaning attributed to an event because:

1. People conclude "fate" played a part.

2. It forces them to think through the consequences (deliberate actions/outcomes) more explicitly.

Option A: By encouraging people to dwell on negative possibilities. The passage isn't about being negative; it's about the absence of the event. While imagining a world where a good event didn't happen might seem "negative," the goal is to create "meaning," not to dwell on bad things.

Option B: By helping people appreciate certain outcomes of past events, whether by fate or deliberate action. Correct Choice. This directly mirrors the text. It mentions that people attributed greater meaning because they saw the role of "fate" or understood the "consequences" (outcomes) more clearly.

Option C: By making people reflect on their past choices. Reflection is part of the process, but "counterfactual thinking" is more specific than just reflection. It requires imagining the opposite of what happened. Option B explains the result of that reflection (finding meaning through fate or consequences), which is what the question asks.

Option D: By distracting people from their current problems and fostering affection towards the company. While the passage mentions that counterfactual thinking can increase attachment to a firm, it is not a "distraction" from problems. It is a cognitive exercise meant to strengthen ties by highlighting the company's importance.

PGDBA 2025 | Q7

Read the passage and answer the following questions (Same passage as Q6).

Q. 7 An appropriate title for this passage could be

Core Logic: To find the best title, we look for the "umbrella" theme that covers all paragraphs.

1. Paragraph 1 & 2: Critique of superficial motivation (quotes/LinkedIn) and the introduction of a more "practical" approach to inspiration.

2. Paragraph 3: Practical Tip #1 — Vivid Imagery.

3. Paragraph 4: Practical Tip #2 — Counterfactual Thinking.

4. Paragraph 5: Practical Tip #3 — Self-Motivation through reflection on values.

The passage isn't just a book review or a study on imagery; it is a collection of actionable insights and research-backed methods on the broad subject of motivating others and oneself.

Option A: Vivid Imagery to Inspire People This is too narrow. Vivid imagery is only one of the three main concepts discussed (appearing only in Paragraph 3). It ignores counterfactual thinking and self-reflection.

Option B: Why People Turn to LinkedIn for Inspiration This is a "distractor" from the opening hook. The author actually mocks much of the "inspiration" found on LinkedIn and moves away from that topic almost immediately.

Option C: A Review of Adam Galinsky’s ‘Inspire’ While the passage draws heavily from Mr. Galinsky’s book, it reads more like an essay on the science of motivation rather than a formal book review (which would typically include critiques, publication details, and a broader overview of the author's career).

Option D: How to Inspire People Correct Choice. This is the most comprehensive title. It encompasses the critique of "silly" advice, the importance of concrete language, the psychological trick of counterfactuals, and the power of self-led motivation. It perfectly captures the "How-To" essence of the text.

PGDBA 2025 | Q8

Read the passage and answer the following questions (Same passage as Q6).

Q. 8 What is the "most striking idea" in Mr. Galinsky's book, as highlighted by the author?

Core Logic: This question tests your ability to locate a specific superlative in the text. The fifth paragraph begins with the sentence: "Perhaps the most striking idea in Mr. Galinsky's book is that, instead of bosses motivating people from above, individuals can do it for themselves." The author then follows this with the experiment involving Swiss job seekers who used self-reflection on personal values to increase their employment chances.

Option A: The benefits of using vivid imagery instead of abstract language in mission statements. While the author discusses this in paragraph 3, he does not describe it as the "most striking idea." It is presented as a "practical insight" or an "example," but not the highlight of the book's innovation.

Option B: The power of individuals to motivate themselves through reflection. Correct Choice. This directly matches the text in paragraph 5. The "striking" part of the idea is the shift away from traditional "top-down" management (bosses inspiring workers) to "bottom-up" or "self-led" motivation.

Option C: The need for more inspirational quotes in the workplace to boost productivity. The author actually dismisses this idea in the first paragraph, calling such content "silly" or something that makes people want to "snigger or vomit."

Option D: The importance of top-down motivation from managers. The passage presents the "most striking idea" as a departure from this traditional top-down model.

PGDBA 2025 | Q9

Read the passage and answer the following questions (Same passage as Q6).

Q. 9 What is the main point the author is trying to make about motivational content on LinkedIn?

Core Logic: The author establishes their stance in the first and second paragraphs. LinkedIn's "inspirational" content is characterized as something that makes people "snigger or vomit," while advice on inspiration is labeled "silly," "blindingly obvious," or "jarringly inauthentic." The author’s central argument is that there are "better options" and "much more practical insights" than these superficial clichés.

Option A: It is often clichéd, inauthentic, and less effective than other methods. Correct Choice. This encapsulates the author's critique perfectly. The text describes the content as "inauthentic" and "silly," and then immediately pivots to research-backed methods (like vivid imagery and counterfactuals) that are presented as superior alternatives.

Option B: It is now mostly the product of AI generated content and lacks authenticity. While the author claims it "lacks authenticity," the passage makes no mention of AI-generated content. In VARC, you must avoid introducing outside information not present in the text.

Option C: It is generally helpful and effective for boosting morale. This contradicts the author's tone. The author admits some people might feel "amped up," but quickly balances that with those who find it revolting, eventually dismissing it in favor of "better options."

Option D: It is a valuable resource for managers seeking to inspire their teams. The text explicitly states the opposite: "For bosses interested in how to motivate the people around them, there are better options than searching for quotes by Paulo Coelho."

PGDBA 2025 | Q10

Read the passage and answer the following questions (Same passage as Q6).

Q. 10 According to Adam Galinsky's research, what is the impact of using vivid imagery in organizational communication?

Core Logic: Paragraph 3 explains that using concrete, visual language (vivid imagery) helps "bring an organisation's purpose to life." The author cites Andrew Carton’s experiment where teams were given a vision statement with visual language (e.g., "make wide-eyed kids laugh").

The result was that these teams "produced more engaging toys" than those given abstract, generic language. This demonstrates that vivid imagery leads to higher quality, more creative, and more engaging outputs.

Option A: It makes the message more memorable and engaging, leading to better results. Correct Choice. The passage explicitly states that teams using vivid language produced "more engaging" toys. While "memorable" is a natural byproduct of "vivid" and "engaging," the key is that it led to superior, more effective results in the experiment.

Option B: It is only effective when combined with motivational quotes and proper direction. The passage actually presents vivid imagery as a better alternative to motivational quotes. It does not suggest they need to be combined; rather, it suggests replacing abstract "LinkedIn-style" quotes with concrete imagery.

Option C: It makes the message less abstract and easier to interpret. While the text says vivid language is "more concrete" (less abstract), the impact highlighted by the research isn't just about interpretation—it's about the quality of the result (the toys). Option A captures this outcome more fully.

Option D: It has little to no effect on how people perceive the message. This contradicts the findings of the experiment, which showed a clear and positive difference in the performance of teams using visual language.

PGDBA 2025 | Q11

Q. 11 Para Jumbles
1. Yet, as the threat from climate change grows more urgent, it faces job losses in the fossil- fuel sectors that made it rich.
2. The country mines its vast resources of everything and exports them.
3. To make up for that and to stave off industrial decline, Australia is spending billions to build green industries.
4. Digging and shipping have turned Australia into one of the world's richest nations.

1. Identify the Opening Statement: * Sentence 4 ("Digging and shipping have turned Australia into one of the world's richest nations") serves as the perfect introduction. It sets the stage by defining Australia's economic foundation.

2. Look for Continuity (The "How"):

Sentence 2 ("The country mines its vast resources of everything and exports them") explains the "Digging and shipping" mentioned in sentence 4. This creates a strong 4-2 pair.

3. Identify the Pivot/Conflict:

Sentence 1 begins with "Yet," signaling a shift. It introduces the problem: climate change and the threat to the fossil-fuel sectors that sentence 4 said made them rich. This creates the sequence 4-2-1.

4. Identify the Resolution:

Sentence 3 ("To make up for that and to stave off industrial decline...") provides the solution to the problem mentioned in sentence 1. Australia is spending billions on green industries to replace the dying fossil-fuel sectors.

Final Logical Sequence: 4 → 2 → 1 → 3

PGDBA 2025 | Q12

Q. 12 Para Jumbles
1. Kukla argues that "it is appropriate to categorize a condition as a disease when it serves legitimate strategic goals to at least partially medicalize that condition, and when the condition is pathological from inside the epistemology and metaphysics of medicine".
2. Therefore, it is essential to be mindful of what the aim is when it comes to labelling a condition as "serious" in a given context.
3. Such an approach allows for the evolution of the concept and consideration of contextual aspects.
4. Seriousness is an inherently subjective term and is shaped by each person's unique circumstances.

1. Identify the Opening Statement:

Sentence 4 ("Seriousness is an inherently subjective term...") is the perfect opening. It introduces the core concept ("Seriousness") and defines its nature (subjective and unique to circumstances). It is the most general statement.

2. Look for the Direct Conclusion of the Opening:

Sentence 2 starts with "Therefore." It draws a direct logical conclusion from the subjectivity mentioned in sentence 4: because it is subjective, we must be mindful of our aims when labeling something as "serious." This creates a strong 4-2 logical pair.

3. Introduce the Academic Framework:

Sentence 1 introduces a specific scholar (Kukla) and a formal definition of how to categorize a "disease" or "condition" based on "strategic goals" and "metaphysics." This provides a concrete example of the "mindfulness" requested in sentence 2.

4. Identify the Final Evaluation:

Sentence 3 ("Such an approach...") refers back to Kukla's multifaceted definition in sentence 1. It explains why that approach is good: it allows for evolution and context.

Final Logical Sequence: 4 → 2 → 1 → 3

PGDBA 2025 | Q13

Q. 13 Fill in the blanks

The play had a ___________ ending

A. climactic: (Correct) Derived from the word climax. It refers to the most exciting, intense, or important point of something (like a story or a play). A "climactic ending" is one that resolves the tension built up throughout the plot.

B. concluding: This simply means "final" or "last." While grammatically correct, it is a redundant or "flat" descriptor. In the context of a "play," writers and examiners look for the term that describes the theatrical quality of the ending.

C. culminating: This suggests reaching a climax or the highest point of development after a long time. While similar to climactic, "culminating" is usually used for events, careers, or processes (e.g., "culminating in a victory") rather than the artistic structure of a play.

D. climatic: This is the most common "trap" option. It refers to climate (weather patterns over time). A "climatic ending" would imply the play ended with a change in the weather!

PGDBA 2025 | Q14

Q. 14 Para Jumbles
1. Video is also an efficient medium for instruction - more precise, in many ways, than the written word.
2. Americans who recently cooked turkeys for Thanksgiving can testify that "golden brown" to one cook may look underdone to another and even burnt to a third.
3. The fact that social media have caused interest in food to rise as fast as home-baked bread should not come as a surprise.
4. Everyone has to eat, and cooking is a common hobby.

1. Identify the Opening Statement:

Sentence 4 ("Everyone has to eat, and cooking is a common hobby") is a broad, foundational statement. However, in many high-level verbal exams, the "hook" can also be a specific observation of a trend. Let's look at sentence 3.Sentence 3 ("The fact that social media have caused interest in food to rise... should not come as a surprise") introduces the specific phenomenon (food on social media). It is followed naturally by sentence 4, which explains why it isn't a surprise (because everyone eats and cooks). This creates a solid 3-4 link.

2. Look for the Technological Shift:

Sentence 1 ("Video is also an efficient medium for instruction...") pivots from the general interest in food (social media) to the specific medium used on those platforms (video). It argues that video is more precise than writing.

3. Identify the Supporting Example:

Sentence 2 provides a concrete example of why the written word is imprecise ("golden brown" means different things to different people) and why a visual/video medium would be superior. This perfectly supports the claim made in sentence 1.

Final Logical Sequence: 3 → 4 → 1 → 2

PGDBA 2025 | Q15

Q. 15 Fill in the blanks

There is wry sense of ____________ in the crucible of the world's semiconductor industry

A. enthusiasm: While possible, "wry sense of enthusiasm" is a rare pairing. "Wry" usually implies something dry, mocking, or ironically humorous. Enthusiasm is typically sincere and bright, which clashes with the cynical undertone of "wry."

B. déjà vu: (Correct) A "sense of déjà vu" refers to the feeling that you have experienced a current situation before. In the semiconductor industry—known for its extreme "boom and bust" cycles, recurring supply chain shortages, and constant geopolitical posturing—the feeling that "we’ve been here before" is a very common, dry observation.

C. enhancement: This is a technical term often used within the industry (e.g., "performance enhancement"), but you don't have a "wry sense" of it. It doesn't fit the sentence structure.

D. elevation: This refers to rising or being raised. Like "enhancement," it lacks the psychological or ironic depth required to be paired with "wry."

PGDBA 2025 | Q26

Let

M_{j} = [\begin{array}{ccc} 1 & 0 & \sqrt{2 j + 1} \\ - 1 & 1 & 0 \\ - \sqrt{2 j + 1} & 0 & 1 \end{array}]

, where

j = 0, 1, \dots, 24

. Then the value of

\sum_{j = 0}^{24} det (adj (M_{j}))

is

Solution: This problem requires calculating the determinant of the given matrix

M_{j}

, using the property of the adjugate matrix, and then evaluating the resulting summation.

Step 1: Use the property of the Adjugate Matrix
For an

n \times n

matrix

M

, the determinant of its adjugate matrix

adj (M)

is:

\det (adj (M)) = (\det (M))^{n - 1}

Here

M_{j}

is a

3 \times 3

matrix, so

n = 3

. Hence,

\sum_{j = 0}^{24} \det (adj (M_{j})) = \sum_{j = 0}^{24} (\det (M_{j}))^{3 - 1} = \sum_{j = 0}^{24} (\det (M_{j}))^{2}

Step 2: Calculate $det (M_{j})$

M_{j} = [\begin{array}{ccc} 1 & 0 & \sqrt{2 j + 1} \\ - 1 & 1 & 0 \\ - \sqrt{2 j + 1} & 0 & 1 \end{array}]

. Expand along the second column (two zeros):

\det (M_{j}) = (- 1)^{2 + 2} (1) \cdot \det ([\begin{array}{cc} 1 & \sqrt{2 j + 1} \\ - \sqrt{2 j + 1} & 1 \end{array}])

Now,

\begin{matrix} \det ([\begin{array}{cc} 1 & \sqrt{2 j + 1} \\ - \sqrt{2 j + 1} & 1 \end{array}]) = (1) (1) - (\sqrt{2 j + 1}) (- \sqrt{2 j + 1}) \\ = 1 - (- (2 j + 1)) \\ = 1 + 2 j + 1 \\ = 2 j + 2 \end{matrix}

So,

\det (M_{j}) = 2 j + 2

Step 3: Set up the Summation

\sum_{j = 0}^{24} (\det (M_{j}))^{2} = \sum_{j = 0}^{24} (2 j + 2)^{2}

Factor

2^{2} = 4

:

\begin{matrix} \sum_{j = 0}^{24} (2 (j + 1))^{2} = \sum_{j = 0}^{24} 4 (j + 1)^{2} = 4 \sum_{j = 0}^{24} (j + 1)^{2} \end{matrix}

Let

k = j + 1

. Then

k

runs from 1 to 25:

4 \sum_{k = 1}^{25} k^{2}

Step 4: Use the Sum of Squares Formula

\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}, n = 25

\begin{aligned} \sum_{k = 1}^{25} k^{2} & = \frac{25 (25 + 1) (2 \cdot 25 + 1)}{6} = \frac{25 \cdot 26 \cdot 51}{6} \end{aligned}

Simplify:

\begin{matrix} = 25 \cdot \frac{26}{2} \cdot \frac{51}{3} = 25 \cdot 13 \cdot 17 = 25 \cdot 221 = 5525 \end{matrix}

Step 5: Final Calculation

\begin{matrix} \sum_{j = 0}^{24} \det (adj (M_{j})) = 4 \sum_{k = 1}^{25} k^{2} = 4 \times 5525 = 22100 \end{matrix}

The value is 22100.

Answer: A

PGDBA 2025 | Q27

The number of solutions of the equation

\tan x + \sec x = 2 \cos x

lying in the interval

[0, \frac{π}{4}]

is

Incorrect. The correct answer is C.

Solution:
Step 1: Rewrite in terms of $\sin x$ and $\cos x$
Using

\tan x = \frac{\sin x}{\cos x}

and

\sec x = \frac{1}{\cos x}

:

\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2 \cos x

We must have

\cos x \neq 0

.

Step 2: Clear the Denominator

\frac{\sin x + 1}{\cos x} = 2 \cos x \Rightarrow \sin x + 1 = 2 \cos^{2} x

Step 3: Convert to a quadratic in $\sin x$
Using

\cos^{2} x = 1 - \sin^{2} x

:

\begin{matrix} \sin x + 1 = 2 (1 - \sin^{2} x) \\ \sin x + 1 = 2 - 2 \sin^{2} x \end{matrix}

Rearrange:

2 \sin^{2} x + \sin x - 1 = 0

Step 4: Factor

\begin{matrix} 2 \sin^{2} x + 2 \sin x - \sin x - 1 = 0 \\ (2 \sin x - 1) (\sin x + 1) = 0 \end{matrix}

So,

$\sin x = \frac{1}{2}$
$\sin x = - 1$

Step 5: Solve and check domain
In

[0, 2 π)

:

$\sin x = \frac{1}{2} \Rightarrow x = \frac{π}{6}, \frac{5 π}{6}$ (both have $\cos x \neq 0$ )
$\sin x = - 1 \Rightarrow x = \frac{3 π}{2}$ but here $\cos x = 0 \Rightarrow$ invalid ( $\tan, \sec$ undefined)

Step 6: Restrict to $[0, \frac{π}{4}]$
Only

x = \frac{π}{6}

lies in

[0, π / 4]

. Hence number of solutions is

1

.

Answer: C

PGDBA 2025 | Q28

Let

f : R \to R

be a twice differentiable function such that

f^{″} (x) = - f (x)

,

g (x) = f^{'} (x)

. Let

h (x) = {({(f (\frac{x}{25}))}^{2} + {(g (\frac{x}{25}))}^{2})}^{\frac{1}{3}} .

If

h (7) = 9

, then

h (2)

is

Solution:
Step 1: Use the differential relation
Given

f^{″} (x) = - f (x)

. Define

g (x) = f^{'} (x)

. Then

g^{'} (x) = f^{″} (x) = - f (x)

Step 2: Study the inside expression
Let

k (x) = {(f (\frac{x}{25}))}^{2} + {(g (\frac{x}{25}))}^{2}

Differentiate using chain rule:

\begin{matrix} k^{'} (x) = 2 f (\frac{x}{25}) \cdot f^{'} (\frac{x}{25}) \cdot \frac{1}{25} + 2 g (\frac{x}{25}) \cdot g^{'} (\frac{x}{25}) \cdot \frac{1}{25} \end{matrix}

Factor

\frac{2}{25}

:

k^{'} (x) = \frac{2}{25} [f (\frac{x}{25}) f^{'} (\frac{x}{25}) + g (\frac{x}{25}) g^{'} (\frac{x}{25})]

Step 3: Substitute
We have:

$f^{'} (\frac{x}{25}) = g (\frac{x}{25})$
$g^{'} (\frac{x}{25}) = - f (\frac{x}{25})$

So,

\begin{matrix} k^{'} (x) = \frac{2}{25} [f (\frac{x}{25}) g (\frac{x}{25}) + g (\frac{x}{25}) (- f (\frac{x}{25}))] = 0 \end{matrix}

Hence

k (x)

is constant, so

h (x) = (k (x))^{1 / 3}

is also constant. Step 4: Use $h (7) = 9$
Since

h

is constant:

h (2) = h (7) = 9

Answer: A

PGDBA 2025 | Q29

Let

Z

denote the set of all integers. Let

S = {a \in Z : a^{2} + 3 a - 10 \leq 0}

and

T = {b \in Z : - 9 \leq 2 b - 1 \leq 5}

. Then the number of elements in

S \cap T

is

Solution:
Step 1: Determine $S$
Solve

a^{2} + 3 a - 10 \leq 0

. Factor:

(a + 5) (a - 2) \leq 0

So,

- 5 \leq a \leq 2

Since

a \in Z

:

S = {- 5, - 4, - 3, - 2, - 1, 0, 1, 2}

Step 2: Determine $T$
Solve

- 9 \leq 2 b - 1 \leq 5

. Add 1:

- 8 \leq 2 b \leq 6

Divide by 2:

- 4 \leq b \leq 3

So,

T = {- 4, - 3, - 2, - 1, 0, 1, 2, 3}

Step 3: Intersection
Common elements:

S \cap T = {- 4, - 3, - 2, - 1, 0, 1, 2}

Step 4: Count
Integers from

- 4

to

2

inclusive:

2 - (- 4) + 1 = 7

Answer: C

PGDBA 2025 | Q30

Let

P Q

be a tower with

P

on the ground. The acute angle of elevation of

Q

from a point

R

on the ground is

α

such that

\cos α = \frac{5}{\sqrt{29}}

and

P R = 200

meters. The acute angle of elevation of

Q

from another point

S

on the line

P R

is

β

with

P S = 80

meters. Then the value of

β

is

Solution:
Step 1. Setup
Let height of the tower

P Q = h

. In right triangle

P Q R

:

\tan α = \frac{P Q}{P R} = \frac{h}{200}

Given

\cos α = \frac{5}{\sqrt{29}}

. Build a reference triangle: adjacent

= 5

, hypotenuse

= \sqrt{29}

. Opposite

O

is:

\begin{matrix} O^{2} + 5^{2} = (\sqrt{29})^{2} \\ O^{2} + 25 = 29 \\ O^{2} = 4 \Rightarrow O = 2 \end{matrix}

So,

\tan α = \frac{2}{5}

Step 2. Find $h$

\frac{h}{200} = \frac{2}{5} \Rightarrow h = 200 \cdot \frac{2}{5} = 80

Thus

P Q = 80

meters. Step 3. Find $β$
In right triangle

P Q S

:

\tan β = \frac{P Q}{P S} = \frac{80}{80} = 1

Acute angle with

\tan β = 1

is:

β = 45^{\circ} = \frac{π}{4}

Answer: B

PGDBA 2025 | Q31

Let

α

be a real number. Suppose the straight line

x + y = α

is a tangent to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1

. Then the value of

α

is

Solution:
Step 1. Standard Equation of the Ellipse
The equation of the given ellipse is:

\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1

This is in the standard form

\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1

, where the major axis is along the

y

-axis because

16 > 9

. We have:

$a^{2} = 16 ⟹ a = 4$
$b^{2} = 9 ⟹ b = 3$

Step 2. Standard Equation of the Line
The equation of the given straight line is:

x + y = α

This can be written in the standard slope-intercept form

y = m x + c

:

y = - x + α

From this, we identify:

$m = - 1$
$c = α$

Step 3. Apply the Condition for Tangency
The condition for a straight line

y = m x + c

to be a tangent to an ellipse

\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1

(where

a^{2} > b^{2}

) is:

c^{2} = a^{2} m^{2} + b^{2}

Substitute

c = α

,

a^{2} = 16

,

b^{2} = 9

,

m = - 1

:

(α)^{2} = (16) (- 1)^{2} + 9

Step 4. Solve for $α$

\begin{matrix} α^{2} = 16 (1) + 9 \\ α^{2} = 16 + 9 \\ α^{2} = 25 \end{matrix}

So,

\begin{matrix} α = \pm \sqrt{25} \\ α = \pm 5 \end{matrix}

Answer: C. $\pm 5$

PGDBA 2025 | Q32

Let

n

be a positive integer. The coefficient of

x^{n}

in the binomial expansion of

(1 + x) (1 - x)^{n}

is

Solution:
Step 1. Expand the Expression
We need to find the coefficient of

x^{n}

in the expansion of

E = (1 + x) (1 - x)^{n}

. First, expand

(1 - x)^{n}

using the Binomial Theorem:

(1 - x)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) (1)^{n - k} (- x)^{k} = \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} x^{k}

Now,

\begin{matrix} E = (1 + x) [(\binom{n}{0}) - (\binom{n}{1}) x + (\binom{n}{2}) x^{2} - \dots + (\binom{n}{n}) (- 1)^{n} x^{n}] \\ E = 1 \cdot (1 - x)^{n} + x \cdot (1 - x)^{n} \\ E = \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} x^{k} + \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} x^{k + 1} \end{matrix}

Step 2. Identify Terms with $x^{n}$
From the first summation, the

x^{n}

term occurs at

k = n

:

\begin{matrix} Term 1 = (\binom{n}{n}) (- 1)^{n} x^{n} \\ Coefficient 1 = (- 1)^{n} \end{matrix}

From the second summation,

x^{k + 1} = x^{n} \Rightarrow k = n - 1

:

\begin{aligned} Term 2 & = (\binom{n}{n - 1}) (- 1)^{n - 1} x^{n} \\ Coefficient 2 & = (\binom{n}{n - 1}) (- 1)^{n - 1} \end{aligned}

Since

(\binom{n}{n - 1}) = n

,

Coefficient 2 = n (- 1)^{n - 1}

Step 3. Combine the Coefficients
Total coefficient of

x^{n}

:

\begin{aligned} Total & = (- 1)^{n} + n (- 1)^{n - 1} \end{aligned}

Factor

(- 1)^{n - 1}

:

\begin{matrix} Total = (- 1)^{n - 1} (- 1) + n (- 1)^{n - 1} \\ Total = (- 1)^{n - 1} (n - 1) \end{matrix}

Equivalently,

Total = (- 1)^{n} (1 - n)

Answer: C. $(- 1)^{n} (1 - n)$

PGDBA 2025 | Q33

Consider the word SUCCESS. Let

E

be the event that the word is rearranged so that the three S's come consecutively. Then the probability of the event

E

is

Solution:
Step 1. Analyze the Word and Calculate Total Arrangements
The word has

7

letters with repetitions:

S: 3 times
C: 2 times
U, E: 1 time each

Total distinct arrangements:

Total Arrangements = \frac{7!}{3! \cdot 2!} = \frac{5040}{6 \cdot 2} = \frac{5040}{12} = 420

Step 2. Favorable Arrangements (three S's consecutive)
Treat

S S S

as one block. Items:

(S S S), U, C, C, E

→ 5 entities with C repeated twice:

Favorable Arrangements = \frac{5!}{2!} = \frac{120}{2} = 60

Step 3. Probability

P (E) = \frac{60}{420} = \frac{6}{42} = \frac{1}{7}

Answer: C. $\frac{1}{7}$

PGDBA 2025 | Q34

The maximum value of

\frac{\sqrt{3}}{2} \cos (π + α) + (\sqrt{3} + \sqrt{5}) \sin (π + α) + \frac{1}{2}

is

Solution:
Step 1. Simplify the Trigonometric Expression
Let

E = \frac{\sqrt{3}}{2} \cos (π + α) + (\sqrt{3} + \sqrt{5}) \sin (π + α) + \frac{1}{2}

Use identities:

$\cos (π + α) = - \cos α$
$\sin (π + α) = - \sin α$

So,

\begin{matrix} E = - \frac{\sqrt{3}}{2} \cos α - (\sqrt{3} + \sqrt{5}) \sin α + \frac{1}{2} \end{matrix}

This is of the form

A \cos α + B \sin α + C

with

A = - \frac{\sqrt{3}}{2}, B = - (\sqrt{3} + \sqrt{5}), C = \frac{1}{2}

.

Step 2. Maximum of $A \cos α + B \sin α$
Maximum value is

\sqrt{A^{2} + B^{2}}

. Compute:

A^{2} = {(- \frac{\sqrt{3}}{2})}^{2} = \frac{3}{4}

\begin{matrix} B^{2} = (\sqrt{3} + \sqrt{5})^{2} = 3 + 5 + 2 \sqrt{15} = 8 + 2 \sqrt{15} \end{matrix}

So,

\begin{matrix} A^{2} + B^{2} = \frac{3}{4} + 8 + 2 \sqrt{15} = \frac{35 + 8 \sqrt{15}}{4} \end{matrix}

Hence maximum of the trig part:

\sqrt{A^{2} + B^{2}} = \frac{\sqrt{35 + 8 \sqrt{15}}}{2}

Step 3. Add the constant $\frac{1}{2}$

E_{max} = \frac{\sqrt{35 + 8 \sqrt{15}}}{2} + \frac{1}{2} = \frac{1}{2} (\sqrt{35 + 8 \sqrt{15}} + 1)

Answer: C

PGDBA 2025 | Q35

Let

f : R \to R

be defined as

f (x) = 2 - e^{x - 1}

. Set

F (x) = \int_{0}^{x} f (t) d t

for

x > 0

. Then pick the correct option.

Solution:
1. Analysis of $f (x)$

f (x) = 2 - e^{x - 1}

. Derivative:

f^{'} (x) = \frac{d}{d x} (2 - e^{x - 1}) = - e^{x - 1}

Set

f^{'} (x) = 0

:

- e^{x - 1} = 0

Since

e^{x - 1} > 0

for all

x \in R

,

f^{'} (x)

is never zero. Thus

f (x)

has no critical points, hence no local maximum or minimum.

A is false.
D is true.

2. Analysis of $F (x)$
By FTC,

F^{'} (x) = f (x) = 2 - e^{x - 1}

. Set

F^{'} (x) = 0

:

2 - e^{x - 1} = 0 ⟹ e^{x - 1} = 2 ⟹ x = 1 + \log_{e} 2

Check sign:

For $x < 1 + \log_{e} 2$ , $e^{x - 1} < 2 \Rightarrow F^{'} (x) > 0$ (increasing)
For $x > 1 + \log_{e} 2$ , $e^{x - 1} > 2 \Rightarrow F^{'} (x) < 0$ (decreasing)

So

F

has a local maximum at

1 + \log_{e} 2

. Hence B and C are false.

Answer: D

PGDBA 2025 | Q36

A curve

y = f (x)

passing through

(1, 2)

has slope of its tangent at

(x, f (x))

equal to

2 x + 1

. Then the area bounded by

y = f (x), y = 0

, and

x = 1

is

Solution:
1. Find the Function $f (x)$
Given

\frac{d y}{d x} = 2 x + 1

. Integrate:

y = \int (2 x + 1) d x = x^{2} + x + C

Use point

(1, 2)

:

2 = 1^{2} + 1 + C \Rightarrow 2 = 2 + C \Rightarrow C = 0

So,

f (x) = x^{2} + x

2. Determine the Required Area
Intersect with

y = 0

:

x^{2} + x = 0 \Rightarrow x (x + 1) = 0 \Rightarrow x = 0, - 1

With boundary

x = 1

and region above

x

-axis for

[0, 1]

, area:

A = \int_{0}^{1} (x^{2} + x) d x

3. Compute the Integral

A = {[\frac{x^{3}}{3} + \frac{x^{2}}{2}]}_{0}^{1} = (\frac{1}{3} + \frac{1}{2}) - 0 = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}

Answer: D. $\frac{5}{6}$

PGDBA 2025 | Q37

Let

α, β

be real numbers. Let

f (x) = {\begin{cases} \frac{1 - \cos 4 x}{x^{2}} + β, & x < 0 \\ α, & x = 0 \\ \frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x > 0 \end{cases}

If

f (x)

is continuous at

x = 0

, then

α + β

is

Solution:
For continuity at

x = 0

:

lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{+}} f (x) = f (0) = α

So,

lim_{x \to 0^{-}} (\frac{1 - \cos 4 x}{x^{2}} + β) = lim_{x \to 0^{+}} (\frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}) = α

1. Left-hand limit
Use

lim_{u \to 0} \frac{1 - \cos u}{u^{2}} = \frac{1}{2}

, with

u = 4 x

:

lim_{x \to 0^{-}} \frac{1 - \cos 4 x}{x^{2}} = 16 \cdot lim_{4 x \to 0} \frac{1 - \cos 4 x}{(4 x)^{2}} = 16 \cdot \frac{1}{2} = 8

Thus,

lim_{x \to 0^{-}} f (x) = 8 + β

2. Right-hand limit
Rationalize:

lim_{x \to 0^{+}} \frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4} \cdot \frac{\sqrt{16 + \sqrt{x}} + 4}{\sqrt{16 + \sqrt{x}} + 4} = lim_{x \to 0^{+}} \frac{\sqrt{x} (\sqrt{16 + \sqrt{x}} + 4)}{(16 + \sqrt{x}) - 16}

= lim_{x \to 0^{+}} \frac{\sqrt{x} (\sqrt{16 + \sqrt{x}} + 4)}{\sqrt{x}} = lim_{x \to 0^{+}} (\sqrt{16 + \sqrt{x}} + 4) = 4 + 4 = 8

So,

lim_{x \to 0^{+}} f (x) = 8

3. Solve
Continuity gives:

8 + β = 8 = α

So

α = 8

and

β = 0

. Therefore:

α + β = 8

Answer: D. 8

PGDBA 2025 | Q38

Rakesh can solve 90 percent of the problems given in the book and Rohit can solve 75 percent. Then the probability that at least one of them will solve a problem selected at random from the book is

Solution:
Let

R

: Rakesh solves,

H

: Rohit solves.

P (R) = 0.9, P (H) = 0.75

We need

P (R \cup H) = 1 - P (R^{'} \cap H^{'})

. Complement probabilities:

P (R^{'}) = 1 - 0.9 = 0.1, P (H^{'}) = 1 - 0.75 = 0.25

Assuming independence:

P (R^{'} \cap H^{'}) = 0.1 \cdot 0.25 = 0.025

Thus:

P (R \cup H) = 1 - 0.025 = 0.975

Convert to fraction:

0.975 = \frac{975}{1000} = \frac{39}{40}

Answer: C. $\frac{39}{40}$

PGDBA 2025 | Q39

The number of three-digit numbers that are divisible by 9 is

Solution:
Three-digit numbers:

100 \leq N \leq 999

. Divisible by 9 means

N = 9 k

. Smallest three-digit multiple of 9:

100 \div 9 = 11

remainder 1, so next multiple is

9 \cdot 12 = 108

. Hence

a_{1} = 108

. Largest is

999

(since

999 \div 9 = 111

). AP:

108, 117, \dots, 999

with common difference

d = 9

. Use

a_{n} = a_{1} + (n - 1) d

:

999 = 108 + (n - 1) 9

891 = 9 (n - 1) \Rightarrow 99 = n - 1 \Rightarrow n = 100

So there are

100

such numbers.

Answer: B. 100

PGDBA 2025 | Q40

Out of 7 women and 5 men, a committee of three members is to be formed in such a way that at least one member is a woman. In how many different ways can it be done?

Solution:
Total people

= 7 + 5 = 12

. Total committees of 3:

(\binom{12}{3}) = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} = 220

Subtract committees with no woman (all men): choose 3 from 5 men:

(\binom{5}{3}) = 10

So committees with at least one woman:

220 - 10 = 210

Answer: C. 210

PGDBA 2025 | Q41

The area bounded by the curves

y = | x - 1 |

and

y = 3 - | x |

is

Solution:
Vertices:

$y = | x - 1 |$ has vertex $(1, 0)$
$y = 3 - | x |$ has vertex $(0, 3)$

Intersections occur at

x = - 1

and

x = 2

:

At $x = - 1$ : $y = | - 1 - 1 | = 2 \Rightarrow (- 1, 2)$
At $x = 2$ : $y = | 2 - 1 | = 1 \Rightarrow (2, 1)$

Quadrilateral vertices:

(- 1, 2), (0, 3), (2, 1), (1, 0)

Bounding rectangle:

x \in [- 1, 2]

width 3,

y \in [0, 3]

height 3:

Area of rectangle = 9

Subtract four outside triangles:

Top-left: $\frac{1}{2} \cdot 1 \cdot 1 = 0.5$
Top-right: $\frac{1}{2} \cdot 2 \cdot 2 = 2$
Bottom-right: $\frac{1}{2} \cdot 1 \cdot 1 = 0.5$
Bottom-left: $\frac{1}{2} \cdot 2 \cdot 2 = 2$

Total removed

= 0.5 + 2 + 0.5 + 2 = 5

. Hence:

Area = 9 - 5 = 4

Answer: C. 4

PGDBA 2025 | Q42

In how many different ways can the letters of the word FORMULATE be arranged so that the vowels occupy only the odd positions?

Solution:
FORMULATE has 9 distinct letters. Odd positions: 5 (1,3,5,7,9). Even positions: 4 (2,4,6,8). Vowels: O, U, A, E (4). Consonants: F, R, M, L, T (5).

Arrange vowels in odd positions
Choose and arrange 4 vowels into 5 odd spots:

P (5, 4) = \frac{5!}{(5 - 4)!} = 5! = 120

Arrange consonants
Remaining 5 positions filled by 5 consonants:

5! = 120

Total:

120 \times 120 = 14400

Answer: A. 14400

PGDBA 2025 | Q43

Let

[x]

denote the largest integer less than or equal to

x

. The real roots of the equation

[x]^{2} - 2 x - [x] + 3 = 0

are lying in the interval

Solution:
Let

n = [x]

. Then

n \in Z

and

n \leq x < n + 1

Given:

[x]^{2} - 2 x - [x] + 3 = 0 \Rightarrow n^{2} - 2 x - n + 3 = 0

So,

2 x = n^{2} - n + 3 \Rightarrow x = \frac{n^{2} - n + 3}{2}

Apply

n \leq x < n + 1

:

n \leq \frac{n^{2} - n + 3}{2} < n + 1

Left inequality:

2 n \leq n^{2} - n + 3 \Rightarrow 0 \leq n^{2} - 3 n + 3

Discriminant

= 9 - 12 = - 3 < 0

, so always positive → always true. Right inequality:

\frac{n^{2} - n + 3}{2} < n + 1 \Rightarrow n^{2} - 3 n + 1 < 0

Roots:

n = \frac{3 \pm \sqrt{5}}{2}

So integers strictly between are

n = 1, 2

. Compute

x

:

$n = 1 \Rightarrow x = \frac{1 - 1 + 3}{2} = \frac{3}{2} = 1.5$ (consistent)
$n = 2 \Rightarrow x = \frac{4 - 2 + 3}{2} = \frac{5}{2} = 2.5$ (consistent)

Both lie in

[1, 3]

.

Answer: B. [1,3]

PGDBA 2025 | Q44

Let

P Q R

be a triangle with

P Q = r, Q R = p, R P = q

and

θ = ∠ R P Q

. If the area is

p^{2} - (q - r)^{2}

, then the value of

\tan (\frac{θ}{2})

is

Solution:
Area using sides

q, r

with included angle

θ

:

Area = \frac{1}{2} q r \sin θ

Given:

Area = p^{2} - (q - r)^{2}

Equate:

\frac{1}{2} q r \sin θ = p^{2} - (q - r)^{2} = p^{2} - (q^{2} - 2 q r + r^{2}) = p^{2} - q^{2} - r^{2} + 2 q r (*)

Law of cosines:

p^{2} = q^{2} + r^{2} - 2 q r \cos θ \Rightarrow p^{2} - q^{2} - r^{2} = - 2 q r \cos θ

Substitute into (*):

\frac{1}{2} q r \sin θ = (- 2 q r \cos θ) + 2 q r

Divide by

q r \neq 0

:

\frac{1}{2} \sin θ + 2 \cos θ = 2

Let

t = \tan (\frac{θ}{2})

. Then:

\sin θ = \frac{2 t}{1 + t^{2}}, \cos θ = \frac{1 - t^{2}}{1 + t^{2}}

Substitute:

\frac{1}{2} \cdot \frac{2 t}{1 + t^{2}} + 2 \cdot \frac{1 - t^{2}}{1 + t^{2}} = 2

\frac{t}{1 + t^{2}} + \frac{2 (1 - t^{2})}{1 + t^{2}} = 2

Multiply by

(1 + t^{2})

:

t + 2 (1 - t^{2}) = 2 (1 + t^{2})

t + 2 - 2 t^{2} = 2 + 2 t^{2} \Rightarrow 0 = 4 t^{2} - t

t (4 t - 1) = 0 \Rightarrow t = 0 or t = \frac{1}{4}

Since

0 < θ < π \Rightarrow θ \neq 0

, reject

t = 0

. Hence:

\tan (\frac{θ}{2}) = \frac{1}{4}

Answer: D. $\frac{1}{4}$

PGDBA 2025 | Q45

If

I = \int_{0}^{\frac{π}{2}} \frac{1}{1 + \sin x} d x

, then

\int_{0}^{2 π} \frac{x^{2} \cos x}{(1 + \sin x)^{2}} d x

is

Note: As provided, this question is marked as incorrect and has No answer.
So, attempt checking is disabled for this question.

(Answer checking disabled for this question.)

PGDBA 2025 | Q46

Let

a, b, c

be real numbers such that

a \neq 1, b \neq 1, c \neq 1

. Consider the system:

x = a (y + z), y = b (z + x), z = c (x + y)

. If it has a non-zero solution, then the value of

\frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c}

is

Solution:
Rewrite:

x - a y - a z = 0, y - b z - b x = 0, z - c x - c y = 0

Let

S = x + y + z

. From

x = a (y + z) = a (S - x)

:

x = a S - a x \Rightarrow x (1 + a) = a S \Rightarrow x = \frac{a}{1 + a} S (*)

Similarly:

y = \frac{b}{1 + b} S (* *), z = \frac{c}{1 + c} S (* * *)

Non-zero solution implies

S \neq 0

(if

S = 0

then

x = y = z = 0

). Sum (*) (**) (***):

x + y + z = S = S (\frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c})

Divide by

S \neq 0

:

1 = \frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c}

Answer: C. 1

PGDBA 2025 | Q47

The value of the limit

lim_{x \to 0} {(\frac{\sin x}{x})}^{\frac{\sin x}{x - \sin x}}

is

Solution:
Let

L = lim_{x \to 0} {(\frac{\sin x}{x})}^{\frac{\sin x}{x - \sin x}}

This is of the form

1^{\infty}

. Use:

lim [f (x)]^{g (x)} = e^{lim g (x) (f (x) - 1)}

Here

f (x) = \frac{\sin x}{x}

,

g (x) = \frac{\sin x}{x - \sin x}

. Compute:

M = lim_{x \to 0} (\frac{\sin x}{x - \sin x}) (\frac{\sin x}{x} - 1)

M = lim_{x \to 0} (\frac{\sin x}{x - \sin x}) (\frac{\sin x - x}{x})

M = lim_{x \to 0} \frac{\sin x (\sin x - x)}{x (x - \sin x)} = lim_{x \to 0} \frac{\sin x}{x} \cdot \frac{- (x - \sin x)}{(x - \sin x)}

Cancel

(x - \sin x)

(for

x \neq 0

):

M = lim_{x \to 0} \frac{\sin x}{x} \cdot (- 1) = - 1

Therefore:

L = e^{M} = e^{- 1} = \frac{1}{e}

Answer: A. $\frac{1}{e}$

PGDBA 2025 | Q48

There are 25 stations between Howrah and Kharagpur. A train is to stop at 3 stations of these 25 stations. What is the probability that no two of these 3 stations are consecutive?

Solution:
Total ways to choose 3 stations from 25:

(\binom{25}{3}) = \frac{25 \cdot 24 \cdot 23}{3 \cdot 2 \cdot 1} = 25 \cdot 4 \cdot 23 = 2300

Favorable ways (no two consecutive): use gaps method.
After choosing 3 stations, there must be at least 1 unchosen station between any two chosen stations. Equivalent count: choose 3 positions from 23 gaps:

(\binom{23}{3}) = \frac{23 \cdot 22 \cdot 21}{3 \cdot 2 \cdot 1} = 23 \cdot 11 \cdot 7 = 23 \cdot 77 = 1771

Probability:

P = \frac{1771}{2300} = \frac{23 \cdot 77}{23 \cdot 100} = \frac{77}{100}

Answer: B. $\frac{77}{100}$

PGDBA 2025 | Q49

The focal point of the parabola

(y - 1)^{2} = 12 (x - 2)

is

Solution:
Compare

(y - k)^{2} = 4 a (x - h)

with

(y - 1)^{2} = 12 (x - 2)

:

$h = 2$ , $k = 1$
$4 a = 12 \Rightarrow a = 3$

Focus is

(h + a, k)

:

(2 + 3, 1) = (5, 1)

Answer: B. (5,1)

PGDBA 2025 | Q50

If

\int e^{\tan^{2} x} \sin 4 x d x = a e^{\tan^{2} x} \cos^{b} x + k

, where

a, b, k

are real constants, then the value of

a^{2 b}

is

Solution:
Assume

G (x) = a e^{\tan^{2} x} \cos^{b} x

We require

\frac{d}{d x} G (x) = e^{\tan^{2} x} \sin 4 x

. Differentiate:

\frac{d}{d x} G (x) = a [\frac{d}{d x} (e^{\tan^{2} x}) \cos^{b} x + e^{\tan^{2} x} \frac{d}{d x} (\cos^{b} x)]

\frac{d}{d x} G (x) = a [(2 e^{\tan^{2} x} \tan x \sec^{2} x) \cos^{b} x + e^{\tan^{2} x} (b \cos^{b - 1} x (- \sin x))]

\frac{d}{d x} G (x) = a e^{\tan^{2} x} [2 \frac{\sin x}{\cos x} \frac{1}{\cos^{2} x} \cos^{b} x - b \sin x \cos^{b - 1} x]

\frac{d}{d x} G (x) = a e^{\tan^{2} x} [2 \sin x \cos^{b - 3} x - b \sin x \cos^{b - 1} x]

\frac{d}{d x} G (x) = a e^{\tan^{2} x} \sin x \cos^{b - 3} x [2 - b \cos^{2} x]

Need equality with

e^{\tan^{2} x} \sin 4 x

. Recall

\sin 4 x = 4 \sin x \cos x \cos 2 x

. Cancel

e^{\tan^{2} x} \sin x

:

4 \cos x \cos 2 x = a \cos^{b - 3} x [2 - b \cos^{2} x]

Use

\cos 2 x = 2 \cos^{2} x - 1

:

4 \cos x (2 \cos^{2} x - 1) = a \cos^{b - 3} x (2 - b \cos^{2} x)

8 \cos^{3} x - 4 \cos x = a \cos^{b - 3} x (2 - b \cos^{2} x)

Match powers by taking

b - 3 = 1 \Rightarrow b = 4

. Substitute

b = 4

:

8 \cos^{3} x - 4 \cos x = a \cos x (2 - 4 \cos^{2} x)

8 \cos^{3} x - 4 \cos x = 2 a \cos x - 4 a \cos^{3} x

Match coefficients of

\cos^{3} x

:

8 = - 4 a \Rightarrow a = - 2

Now compute:

a^{2 b} = (- 2)^{2 \cdot 4} = (- 2)^{8} = 2^{8} = 256

Answer: C. 256

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