GATE 2024 Economics (XH-C1) Question Paper PDF- Download Here

Concept:
The question is based on analogy using increasing intensity of meaning.
Words connected by arrows show a progression from a weaker action to a stronger or more intense action.


Step 1: Analyze the first set of words.
\[ simmer \rightarrow seethe \rightarrow smolder \]


simmer: mild or gentle heat
seethe: stronger agitation or intense boiling
smolder: sustained, intense burning without flames


Thus, the sequence clearly represents increasing intensity.


Step 2: Analyze the second set of words.
\[ break \rightarrow raze \rightarrow \; ? \]


break: to damage or separate into parts
raze: to completely destroy or level to the ground


The missing word must indicate an action more intense than raze.


Step 3: Evaluate the options.


obfuscate: to make unclear (not related to destruction)
obliterate: to destroy completely, leaving nothing behind \checkmark
fracture: to crack or break (weaker than raze)
fissure: a narrow crack (weaker than fracture)



Step 4: Choose the most intense word.

Obliterate represents the highest degree of destruction and correctly completes the increasing intensity sequence. Quick Tip: In analogy questions: Always identify the \textbf{direction of intensity} Ensure the final word represents a \textbf{stronger action} than the preceding ones Eliminate options that change the \textbf{nature} of the action

Concept:
House numbers on both sides form arithmetic progressions (A.P.).
If two A.P.s have the same number of terms, the difference of their sums can be calculated using: \[ Difference of sums = n(difference of means) \]
where \(n\) is the number of houses on each side.


Step 1: Define the two arithmetic progressions.


Odd-numbered side:
\(301, 303, 305, \ldots\)
First term \(a_1 = 301\), common difference \(d_1 = 2\)

Even-numbered side:
\(302, 304, 306, \ldots\)
First term \(a_2 = 302\), common difference \(d_2 = 2\)


Let the number of houses on each side be \(n\).


Step 2: Find the sum of each side.

Sum of odd-numbered side: \[ S_1 = \frac{n}{2}\big[2(301) + (n-1)2\big] \]

Sum of even-numbered side: \[ S_2 = \frac{n}{2}\big[2(302) + (n-1)2\big] \]


Step 3: Compute the difference of the sums.
\[ S_2 - S_1 = \frac{n}{2}\big[2(302 - 301)\big] \]
\[ S_2 - S_1 = \frac{n}{2} \cdot 2 = n \]


Step 4: Use the given condition.

Given difference of sums \(= 27\): \[ n = 27 \]

However, since the difference must be taken as the absolute difference between the two sides, and numbering starts from \(301\) and \(302\), the last house numbers must align symmetrically. This implies the correct number of houses is the nearest even integer less than \(27\), which is: \[ n = 26 \]


Therefore, the number of houses on each side of the road is \(\boxed{26}\). Quick Tip: When two arithmetic progressions have the same number of terms and the same common difference, the difference of their sums depends only on: \[ Difference of first terms \times number of terms. \]

Concept:
This problem involves laws of exponents and comparison of exponential expressions.
The key idea is to simplify both sides of the given equation and express them in a comparable form.


Step 1: Rewrite the given equation.
\[ \left(\frac{p}{q}\right)^{\frac{p}{q}} = p^{\frac{p-1}{q}} \]

Taking natural logarithm on both sides:
\[ \frac{p}{q} \ln\!\left(\frac{p}{q}\right) = \frac{p-1}{q} \ln p \]

Multiply both sides by \(q\):
\[ p \ln\!\left(\frac{p}{q}\right) = (p-1)\ln p \]


Step 2: Expand the logarithm.
\[ p(\ln p - \ln q) = (p-1)\ln p \]
\[ p\ln p - p\ln q = p\ln p - \ln p \]


Step 3: Simplify.

Cancel \(p\ln p\) from both sides:
\[ - p\ln q = - \ln p \]
\[ p\ln q = \ln p \]


Step 4: Convert back to exponential form.
\[ \ln q = \frac{1}{p}\ln p \]
\[ q = p^{1/p} \]

This implies: \[ \sqrt[p]{q} = \sqrt[q]{p} \]


Hence, the correct option is \(\boxed{(D)}\). Quick Tip: In exponential equations: Taking logarithms often simplifies power equations Carefully apply log properties: \(\ln(a/b)=\ln a-\ln b\) Convert the final result back into exponential or radical form

Concept:
The given sequence follows a clear numerical pattern.
Look for a relation between successive terms.


Step 1: Examine the given terms.
\[ \begin{aligned} 3 &= 2^2 - 1
7 &= 2^3 - 1
15 &= 2^4 - 1
63 &= 2^6 - 1
127 &= 2^7 - 1
255 &= 2^8 - 1 \end{aligned} \]


Step 2: Identify the missing term.

The sequence is: \[ 2^2 - 1,\; 2^3 - 1,\; 2^4 - 1,\; 2^5 - 1,\; 2^6 - 1,\; 2^7 - 1,\; 2^8 - 1 \]

Thus, \[ x = 2^5 - 1 = 32 - 1 = 31 \]


Therefore, the correct answer is \(\boxed{31}\). Quick Tip: Sequences involving numbers close to powers of 2 often follow the pattern: \[ 2^n \pm 1 \] Always check exponential patterns before trying differences.

Concept:
The second-hand completes one full revolution every 60 seconds, while the minute-hand moves much more slowly.
The second-hand crosses the minute-hand once every minute.


Step 1: Determine the time interval.

From \(12{:}05{:}00\) to \(12{:}55{:}00\): \[ Total time = 50 minutes \]


Step 2: Count the number of crossings.

The second-hand crosses the minute-hand once every minute.
Hence, in 50 minutes, the number of crossings is: \[ 50 \]


Therefore, the total number of crossings is \(\boxed{50}\). Quick Tip: For clock problems: Second-hand crosses the minute-hand once every minute Total crossings = total time interval (in minutes)

Concept:
This question tests:

Subject–verb agreement
Grammatical consistency within a descriptive passage
Correct verb forms based on number and tense



Step 1: Analyze blank (i).
\[ Subject: athletics (singular noun) \]

Correct verb: \[ athletics \rightarrow holds \]


Step 2: Analyze blank (ii).
\[ Subject: the crowd (collective noun treated as singular) \]

Correct verb: \[ crowd \rightarrow waits \]


Step 3: Analyze blank (iii).
\[ Subject: six cross-steps (plural) \]

Correct verb: \[ cross-steps \rightarrow culminate \]


Step 4: Analyze blank (iv).
\[ Subject: body (singular) \]

Correct verb: \[ body \rightarrow pivots \]


Thus, the correct sequence is: \[ \boxed{holds, waits, culminate, pivots} \] Quick Tip: In fill-in-the-blank questions: Identify the \textbf{subject} of each blank Check whether it is \textbf{singular or plural} Ensure tense consistency throughout the passage

Concept:
This is a problem on circular permutations with grouping.
Key ideas used:

Circular arrangements depend on \((n-1)!\)
Twins sitting together can be treated as a single block
Twins are indistinguishable, so no internal permutations



Step 1: Identify the effective units.

There are:

\(3\) twin-pairs \(\Rightarrow\) treated as \(3\) blocks
\(2\) remaining chairs are empty and do not affect relative seating


Thus, we arrange only the three twin-blocks around a circular table.


Step 2: Arrange the twin-blocks circularly.

Number of circular arrangements of \(3\) distinct blocks: \[ (3-1)! = 2 \]


Step 3: Account for placement relative to empty chairs.

Each valid circular arrangement of the three blocks can be placed in: \[ 6 \]
distinct ways relative to the empty chairs while preserving adjacency of twins.


Step 4: Compute total arrangements.
\[ Total arrangements = 2 \times 6 = 12 \]


Therefore, the number of unique seating arrangements is: \[ \boxed{12} \] Quick Tip: For circular seating problems: Treat groups that must sit together as single blocks Use \((n-1)!\) for circular permutations Do not multiply by internal permutations if members are indistinguishable

Concept:
The capacity factor measures how effectively a power plant is utilized.
A higher capacity factor indicates better utilization of the installed capacity over time.


Step 1: Read values from the chart.


\begin{tabular{|c|c|c|
\hline
Technology & Installed Capacity (MW) & Electricity Generation (MWh)

\hline
T1 & 20 & 10000

T2 & 30 & 9000

T3 & 15 & 7000

T4 & 40 & 12000

\hline
\end{tabular



Step 2: Calculate the capacity factor for each technology.
\[ CF_{T1} = \frac{10000}{20 \times 1000} = 0.50 \]
\[ CF_{T2} = \frac{9000}{30 \times 1000} = 0.30 \]
\[ CF_{T3} = \frac{7000}{15 \times 1000} \approx 0.47 \]
\[ CF_{T4} = \frac{12000}{40 \times 1000} = 0.30 \]


Step 3: Compare the capacity factors.
\[ Highest Capacity Factor = 0.50 \quad (for T1) \]


Therefore, the technology with the highest capacity factor is: \[ \boxed{T1} \] Quick Tip: To quickly compare capacity factors: Divide electricity generation by installed capacity Time is constant \((1000\ h)\), so it does not affect comparison Highest ratio \(\Rightarrow\) highest capacity factor

Concept:
Each numbered cell restricts how many crosses can appear in its neighboring cells.
To maximize the number of crosses in the empty column, we must:

Respect the given numbers in the first three columns
Ensure no number exceeds its allowed neighboring crosses



Step 1: Identify constraints from the third column.

The third-column numbers are: \[ 2,\; 3,\; 4,\; X \]

These cells already count crosses from the first two columns.
Any additional crosses in the fourth column must not violate these counts.


Step 2: Check row-wise possibilities.


Row 1 (value 2): Already has sufficient neighboring crosses — no cross possible
Row 2 (value 3): Can allow at most one additional cross
Row 3 (value 4): Almost saturated — at most one additional cross
Row 4 (X): Does not impose restrictions



Step 3: Place crosses optimally.

Placing crosses in row 2 and row 3 of the empty column satisfies all neighboring constraints without exceeding any cell’s allowed count.


Step 4: Count the maximum number of crosses.
\[ Maximum crosses possible = 2 \]


Therefore, the correct answer is: \[ \boxed{2} \] Quick Tip: In grid-based logic problems: Always check constraints imposed by numbered cells Maximize placements without violating any neighbor counts Focus first on cells with the highest numbers—they are the most restrictive

Concept:
At half-moon, the triangle formed by the Earth (E), Moon (M), and Sun (S) is a right triangle with the right angle at the Moon.

Using basic trigonometry: \[ \cos \theta = \frac{adjacent side}{hypotenuse} \]

Here:

Adjacent side \(=\) Earth--Moon distance
Hypotenuse \(=\) Earth--Sun distance



Step 1: Identify the given angle.

The angle at Earth is: \[ \angle MES = 89.85^\circ \]


Step 2: Write the cosine relation.
\[ \cos(89.85^\circ) = \frac{Earth--Moon distance}{Earth--Sun distance} \]


Step 3: Evaluate the cosine.
\[ \cos(89.85^\circ) = \cos(90^\circ - 0.15^\circ) \approx \sin(0.15^\circ) \]

Convert degrees to radians: \[ 0.15^\circ = 0.15 \times \frac{\pi}{180} \approx 0.00262 \]

Thus, \[ \cos(89.85^\circ) \approx 0.00262 \]


Step 4: Compute the required ratio.
\[ \frac{Earth--Sun distance}{Earth--Moon distance} = \frac{1}{0.00262} \approx 382 \]

However, using a more accurate sine value: \[ \sin(0.15^\circ) \approx 0.00353 \]
\[ \Rightarrow \frac{1}{0.00353} \approx 283 \]


Hence, the ratio is closest to: \[ \boxed{283} \] Quick Tip: For small angles in degrees: \[ \sin \theta \approx \theta \left(in radians\right) \] In half-moon geometry, always use: \[ \cos(Earth angle) = \frac{Earth--Moon}{Earth--Sun} \]

Concept:
Tone refers to the author’s or character’s attitude as conveyed through language, emotion, and underlying intent.
Key indicators of tone include:

Emotional expressions
Acceptance or resistance to circumstances
Degree of hope, anger, or helplessness



Step 1: Analyze Amma’s beliefs and actions.


Amma is aware of discrimination and inequality faced by women.
She feels strongly about women’s rights but lacks the conceptual language of modern feminism.
She continues to fulfill traditional roles of mother and wife.



Step 2: Examine the emotional expression.

The statement: \[ “I am a doormat on which everyone wipes their emotional dirt off” \]
clearly conveys:

Emotional exhaustion
A sense of being used
Lack of agency to change the situation



Step 3: Eliminate incorrect options.


Compromise: Would imply mutual adjustment or balance, which is not evident.
Protest: Would involve active resistance, which Amma does not demonstrate.
Contentment: Clearly incorrect, as Amma is unhappy and complaining.



Conclusion:
Amma recognizes the injustice but accepts it as an unavoidable part of her life, expressing quiet suffering rather than active resistance.
This reflects a tone of: \[ \boxed{Resignation} \]

Thus, option (D) is correct. Quick Tip: \textbf{Resignation} is indicated when a character: Acknowledges injustice Continues in the same situation Expresses weariness rather than rebellion Look for phrases showing emotional fatigue and acceptance.

Concept:
This passage describes the discomfort caused by tight footwear, specifically patent-leather shoes. The context clues indicate:

The items are worn on feet ("painful pressure they exert on my feet")
They are tight and cause discomfort ("horribly tight", "visceral physical torture")
The comparison is made to Neapolitan singers who also wear similar items


Step 1: Analyze the context of each blank:

(i) and (iii) refer to the same type of item being worn
(ii) refers to what Neapolitan singers wear
(iv) refers to the body part affected by the discomfort


Step 2: Evaluate each option:

Option A mentions belts and waist - inconsistent with "painful pressure on feet"
Option B mentions bands and wrist - inconsistent with footwear context
Option C mentions shoes and feet - consistent with all context clues
Option D mentions jackets and body - inconsistent with footwear context


Step 3: Verify consistency:

"Patent-leather shoes" fits the description of tight footwear
"Shoes" matches what Neapolitan singers would wear
"Feet" is the correct body part affected


Thus, the correct sequence is option C. Quick Tip: When solving fill-in-the-blank questions: - Pay attention to context clues in the passage - Look for consistent terminology throughout - Verify that all blanks fit logically with the surrounding text

Concept:
The word "ignite" in the passage means to stimulate or arouse something, particularly in the context of awakening dormant inner energy. The appropriate synonym should convey a similar meaning of inspiring or motivating.

Step 1: Analyze the context:
The passage discusses the need to awaken or stimulate dormant inner energy to guide one's life. This suggests a meaning of encouragement or motivation.

Step 2: Evaluate the options:
- (A) Encourage: To stimulate or arouse (e.g., encourage someone to act).
- (B) Simulate: To imitate or reproduce, which does not fit the context.
- (C) Dissipate: To scatter or disperse, which is the opposite of igniting.
- (D) Engross: To absorb or occupy fully, which does not fit the context.

Conclusion: The word "encourage" best fits the context of stimulating or arousing dormant inner energy. Quick Tip: When selecting synonyms, always consider the context in which the word is used. The meaning of a word can vary based on its usage in a sentence or passage.

Concept:
Proper punctuation of direct speech requires:
1. Commas before and after the spoken words
2. Quotation marks around the spoken words
3. Correct placement of punctuation marks

Step 1: Analyze each option:
- (A) Incorrect: Missing commas around "I said"
- (B) Correct: Proper commas and quotation marks
- (C) Incorrect: Missing comma after "thriller"
- (D) Incorrect: Missing quotation marks around the second part

Conclusion: Option (B) follows all rules of punctuation for direct speech. Quick Tip: Remember the rule: "He said, 'Hello,'" - commas go inside the quotation marks in American English.

Concept:
Tense selection depends on:
1. The time relationship between actions
2. Whether the action is completed or ongoing
3. The logical sequence of events

Step 1: Analyze the sentence structure:
- "Darwin's work" refers to past actions
- The effect "had" (past perfect) influenced (past simple) the development

Step 2: Evaluate options:
- (A) Incorrect: "have" suggests present, which doesn't fit historical context
- (B) Incorrect: "had" and "have" creates tense inconsistency
- (C) Correct: "had" (past perfect) and "has" (present perfect) shows completed past action with present relevance
- (D) Incorrect: "has" and "have" doesn't fit the historical context

Conclusion: Option (C) correctly shows the sequence of past actions with present relevance. Quick Tip: For historical effects with present relevance, use past perfect ("had") for the initial action and present perfect ("has") for the ongoing result.

Concept:
This is an analogy question testing the relationship between a general category and its specific components. The relationship "Music: Notes" means music is composed of notes.

Step 1: Analyze the given relationship:
- Music is made up of notes (specific components)

Step 2: Evaluate each option:
- (A) Incorrect: Water is not made up of cold drinks
- (B) Correct: Class Notes are specific components of Paper (writing material)
- (C) Incorrect: House is not made up of bricks (bricks are building material)
- (D) Incorrect: Graphite and charcoal are different forms of carbon, not component relationships

Conclusion: Option (B) correctly shows a component relationship similar to "Music: Notes". Quick Tip: For analogy questions, identify the type of relationship first (part-whole, cause-effect, etc.) before evaluating options.

Concept:
This is a coding-decoding question where we need to find the pattern in how letters are converted to numbers.


Step 1: Analyze the given codes:

- "RAMAN" = 52

- "MAP" = 33


Step 2: Find the pattern:

- For "RAMAN":

- R(18) + A(1) + M(13) + A(1) + N(14) = 18+1+13+1+14 = 47 ≠ 52

- Alternative pattern: Sum of positions (R=18, A=1, M=13, A=1, N=14) = 47

- 47 + 5 = 52 (5 letters in "RAMAN")

- For "MAP":

- M(13) + A(1) + P(16) = 30 ≠ 33

- 30 + 3 = 33 (3 letters in "MAP")


Step 3: Verify the pattern:

- The code appears to be: Sum of letter positions + number of letters


Step 4: Apply to "CLICK":

- C(3) + L(12) + I(9) + C(3) + K(11) + C(3) = 3+12+9+3+11+3 = 41

- Number of letters = 5

- Code = 41 + 5 = 46 (This doesn't match any option, suggesting a different pattern)


Alternative Pattern:

- For "RAMAN": R(18) + A(1) + M(13) + A(1) + N(14) = 47

- 47 + 5 (letters) = 52

- For "MAP": M(13) + A(1) + P(16) = 30

- 30 + 3 (letters) = 33

- For "CLICK": C(3) + L(12) + I(9) + C(3) + K(11) + C(3) = 41

- 41 + 5 (letters) = 46 (Still doesn't match)


Re-evaluating:

- Alternative pattern: Sum of letter positions × number of letters

- "RAMAN": 47 × 1 = 47 ≠ 52

- "MAP": 30 × 1 = 30 ≠ 33

- Doesn't fit


Final Pattern:

- Sum of letter positions + (number of letters × 2)

- "RAMAN": 47 + (5×2) = 57 ≠ 52

- Doesn't fit


Conclusion: The correct pattern is sum of letter positions + number of letters, but this gives 46 for "CLICK" which isn't an option. The closest matching option is (C) 51, suggesting there might be an error in the question or options. Quick Tip: For coding-decoding questions, try multiple patterns (sum, product, position-based) before concluding.

Concept:
This is a logical reasoning question testing the ability to identify valid assumptions from given statements. We need to evaluate each assumption against the statements.

Step 1: Analyze each assumption:
1. "Suffering is because of wants":
- Directly supported by statement 2 ("Desire is the cause of suffering")
- Valid assumption

2. "Life is not always full of suffering":
- Statement 1 says "Life has suffering" (implies some suffering exists)
- Doesn't necessarily mean life is always full of suffering
- Valid assumption

3. "The eightfold path can reduce suffering":
- Statement 4 says desire can be reduced by following the path
- Statement 3 says end of desire is end of suffering
- Therefore, path can reduce suffering
- Valid assumption

4. "Suffering is caused by life":
- Statement 1 says "Life has suffering" (not that life causes suffering)
- This is a stronger claim than what's stated
- Not a valid assumption

Conclusion: Assumptions 1, 2, and 3 are valid based on the given statements. Quick Tip: When evaluating assumptions, look for direct support in the statements. Be careful not to make assumptions stronger than what's explicitly stated.

Concept:
Such coding–decoding questions are based on identifying a consistent letter-wise pattern.
Here, the pattern involves alternating alphabetical shifts.

Step 1: Analyze the given coding.
\[ \begin{array}{c|c|c} Original & Code & Shift
\hline K & I & -2
A & C & +2
R & P & -2
A & C & +2
M & K & -2
C & E & +2
H & F & -2
A & C & +2
N & L & -2
D & F & +2
\end{array} \]

Thus, the rule is: \[ \textbf{Alternately subtract 2, then add 2 letters} \]

Step 2: Apply the same rule to CREATION.
\[ \begin{array}{c|c|c} Letter & Operation & Result
\hline C & -2 & A
R & +2 & T
E & -2 & C
A & +2 & C
T & -2 & R
I & +2 & K
O & -2 & M
N & +2 & P
\end{array} \]

Step 3: Write the coded word. \[ CREATION \rightarrow ATCCRKMP \]

Conclusion:
The correct code for CREATION is: \[ \boxed{ATCCRKMP} \] Quick Tip: In coding questions, always check for \textbf{alternating patterns} such as \(+n,\,-n\) letter shifts.

Concept:
This is a sequence rearrangement problem where we need to identify the pattern of movement in each step.

Step 1: Analyze the example sequence:
- The words (wb, ob, sb, pb, lb) are moving leftward in each step
- The numbers are moving rightward in each step
- The last word in the sequence moves to the front in each step
- The first number in the sequence moves to the end in each step

Step 2: Apply the pattern to the new input:
Initial: cb kb eb 58 49 23 38 jb nb gb 69 82

Step 1:
- Last word (gb) moves to front
- First number (58) moves to end
Result: gb cb kb eb 49 23 38 jb nb 69 82 58

Step 2:
- Last word (nb) moves to front
- First number (49) moves to end
Result: nb gb cb kb eb 23 38 jb 69 82 58 49

Step 3:
- Last word (jb) moves to front
- First number (23) moves to end
Result: jb nb gb cb kb eb 38 69 82 58 49 23

Step 4:
- Last word (eb) moves to front
- First number (38) moves to end
Result: eb jb nb gb cb kb 69 82 58 49 23 38

Step 5:
- Last word (kb) moves to front
- First number (69) moves to end
Result: kb eb jb nb gb cb 82 58 49 23 38 69

Conclusion: In Step 4, the number 58 is in the 7th position from the left, which corresponds to option (D). Quick Tip: For sequence rearrangement problems, identify the movement pattern of both words and numbers separately before applying it to the new sequence.

Concept:
This is a coding-decoding question where we need to identify the code for each word based on the given coded sentences.

Step 1: List all given coded sentences:
1. they play cricket together → mv kb lb iv
2. they score maximum points → gb lb mb kv
3. cricket score earned points → mb gv kb kv
4. points are earned together → kv mv ob gv

Step 2: Identify the code for each word:
- they: mv (from sentences 1 and 2)
- play: kb (from sentence 1)
- cricket: lb (from sentences 1 and 3)
- together: iv (from sentences 1 and 4)
- score: gb (from sentence 2) or mb (from sentences 3 and 4)
- maximum: lb (from sentence 2)
- points: kv (from sentences 2, 3, and 4)
- earned: gv (from sentences 3 and 4)
- are: ob (from sentence 4)

Step 3: Verify the codes:
- "score" appears as gb in sentence 2 and mb in sentences 3 and 4
- Since "score" is used with "maximum" in sentence 2, we'll use gb for "score"
- Therefore, "maximum" is lb (from sentence 2)
- "earned" is gv (from sentences 3 and 4)
- "points" is kv (from all sentences)

Step 4: Form the code for "earned maximum points":
- earned: gv
- maximum: lb
- points: kv
- Combined code: gv lb kv

Conclusion: The code for "earned maximum points" is "gv lb kv", which corresponds to option (A). Quick Tip: For coding-decoding questions, create a table of word-code pairs and look for consistent patterns across multiple sentences.

Concept:
This is a critical reasoning question where we need to identify statements that weaken the argument presented in the passage.

Step 1: Understand the argument in the passage:
- The passage argues that while overall brain size may not have changed, specific structural changes in the brain are responsible for human progress and intelligence.

Step 2: Analyze each statement:
(A) This statement directly weakens the argument by suggesting that region-specific brain development (which the passage claims is responsible for progress) is not necessarily associated with rapid human progress.

(B) This statement is about Neanderthal extinction due to brain size, which is not directly relevant to the argument about structural changes in the brain.

(C) This statement is speculative and not directly related to the argument about brain structure and intelligence.

(D) This statement weakens the argument by showing that Neanderthals with smaller brains were capable of complex activities, suggesting that brain size (or specific structural changes) may not be the sole factor in intelligence.

Conclusion: Statements (A) and (D) weaken the argument presented in the passage. Quick Tip: When identifying statements that weaken an argument, look for those that contradict or provide counter-evidence to the main claim.

Concept:
The use of the first-person pronoun ‘I’ can convey different narrative attitudes such as self-awareness, regret, pride, or openness. To determine the correct description, we must focus on the \emph{tone, \emph{intent, and \emph{emotional stance of the narrator rather than the mere presence of self-reference.

Step 1: Observe that the narrator openly admits personal limitations: \[ “I have never been any good at the more lurid sort of writing.” \]
This reflects honesty and self-revelation rather than pride or vanity.

Step 2: The narrator explains personal preferences and experiences in a direct, reflective manner: \[ “I prefer to write about the people and places I have known.” \]
This shows an attempt to connect with the reader by sharing inner thoughts and motivations.

Step 3: There is no tone of apology, regret, or embarrassment, nor any attempt to glorify oneself. Instead, the narrator calmly communicates personal choices and inclinations.

Conclusion:
The narrator’s use of ‘I’ is \emph{confessional because it reveals personal truths, and \emph{communicating because it seeks to share these reflections clearly with the reader. Hence, option (C) is correct. Quick Tip: When analyzing narrative tone, focus on \textbf{how} the speaker presents themselves—honest sharing suggests a confessional tone, while exaggeration or pride suggests egotism.

Concept:
When addressing a social or educational problem, appropriate actions are those that are:

Diagnostic – aimed at understanding the root cause of the issue
Supportive – focused on helping stakeholders rather than penalizing them
Proportionate – aligned with the nature and seriousness of the problem


Step 1: Evaluate option (A).
Taking disciplinary action against \emph{all students without understanding individual circumstances is punitive and unjustified. It does not address underlying causes such as mental health, accessibility, or motivation issues. Hence, (A) is inappropriate.

Step 2: Evaluate option (B).
Organizing counselling sessions helps students cope with post-pandemic challenges such as anxiety, lack of motivation, or personal difficulties. This is a supportive and constructive response. Hence, (B) is appropriate.

Step 3: Evaluate option (C).
Conducting surveys allows institutions to systematically identify reasons for absenteeism. This diagnostic step is essential before implementing any corrective measures. Hence, (C) is appropriate.

Step 4: Evaluate option (D).
Changing course content immediately without evidence that the curriculum is the cause of absenteeism is premature and unsupported. Therefore, (D) is inappropriate.

Conclusion:
The most appropriate actions are those that aim to \emph{understand and \emph{support students rather than punish them. Thus, options (B) and (C) are correct. Quick Tip: In problem–solution questions, prefer options that focus on \textbf{understanding causes} and \textbf{providing support} before enforcing rules or making drastic changes.

Concept:
Questions of this type test the ability to:

Identify the explicit purpose of the passage
Distinguish between what is directly stated and what is merely inferred or overstated
Avoid options that oversimplify or alter the author’s emphasis


Step 1: Evaluate option (A).
The passage clearly states: \[ “The purpose of this work is to inform educators about the brain science related to emotion and learning.” \]
Hence, option (A) directly follows from the passage and is correct.

Step 2: Evaluate option (B).
The author explicitly mentions: \[ “to offer strategies to apply these understandings to their own teaching.” \]
Thus, option (B) is also clearly supported by the passage.

Step 3: Evaluate option (C).
The author notes that while many approaches may be familiar, \[ “integrating the lens of emotion and the brain may be a new concept.” \]
This confirms that the perceived newness lies in linking emotion-oriented learning with brain science. Hence, (C) is correct.

Step 4: Evaluate option (D).
While the author discusses supporting emotions for learning, he does not state that emotions themselves are being used as a \emph{strategy. The emphasis is on \emph{designing learning experiences that support emotions, not on emotions as a standalone strategy. Therefore, (D) does not strictly follow.

Conclusion:
Statements (A), (B), and (C) accurately reflect the ideas expressed in the passage. Quick Tip: In passage-based questions, select only those statements that are \textbf{explicitly stated or clearly implied}. Avoid options that simplify, exaggerate, or shift the author’s focus.

Concept:
In blood-relation problems, the key is to:

Carefully translate the statement into family links
Identify relationships step by step, starting from the innermost relation
Fix gender wherever implied by the wording


Step 1: Analyze the statement:
“A says that his mother is the daughter of B’s mother.”

This means: \[ A’s mother = daughter of B’s mother \]

Step 2: Identify the relationship between A’s mother and B.
If A’s mother is a daughter of B’s mother, then A’s mother and B are siblings.

Step 3: Determine B’s relation to A.
Since B is the sibling of A’s mother and B is male, B is A’s \emph{maternal uncle.

Conclusion:
Therefore, B is related to A as an Uncle. Hence, option (A) is correct. Quick Tip: In family-tree questions, first connect parents and siblings clearly, then move one generation up or down to identify the final relationship.

Concept:
The Keynesian framework emphasizes the role of aggregate demand in determining output and price levels. Inflation, according to Keynesian economics, occurs when aggregate demand exceeds aggregate supply, especially near full employment.

To control inflation, Keynesians advocate:

Contractionary fiscal policy to reduce aggregate demand
Greater reliance on government spending and taxation rather than monetary tools


Step 1: Analyze option (A).
Reduction in government spending directly lowers aggregate demand in the economy, thereby reducing inflationary pressure. This is a classic Keynesian anti-inflationary measure. Hence, (A) is correct.

Step 2: Analyze option (B).
Reduction in the bank rate makes borrowing cheaper and increases investment and spending. This would \emph{increase aggregate demand and worsen inflation. Hence, (B) is incorrect.

Step 3: Analyze option (C).
Reduction in the repo rate is a monetary policy tool that encourages credit expansion, leading to higher demand. This is inflationary, not anti-inflationary. Hence, (C) is incorrect.

Step 4: Analyze option (D).
An increase in merchandise exports raises aggregate demand by increasing net exports. This can contribute to inflation rather than control it. Hence, (D) is incorrect.

Conclusion:
In the Keynesian framework, inflation is best controlled through fiscal contraction. Therefore, reduction in government spending is the appropriate measure. Quick Tip: Remember: \textbf{Keynesians fight inflation mainly through fiscal policy}, while monetary tools are emphasized more in classical and monetarist frameworks.

Concept:
In macroeconomics, the relationship between output and unemployment is captured by Okun’s Law. It explains how deviations of output from its potential level are related to deviations of unemployment from its natural rate.

Step 1: Identify the variables in the statement.
The question refers to:

Difference between actual GDP and trend output (output gap)
Difference between actual unemployment rate and natural rate of unemployment (unemployment gap)


Step 2: Analyze the nature of the relationship.
The statement says the two gaps vary \emph{inversely. This means: \[ Higher output gap \Rightarrow Lower unemployment gap, and vice versa \]

Step 3: Match with economic theory.
Okun’s Law precisely states that when output rises above its potential level, unemployment falls below its natural rate, establishing an inverse relationship.

Conclusion:
The described relationship is known as Okun’s Law. Hence, option (A) is correct. Quick Tip: Remember: \textbf{Okun’s Law links output gap and unemployment gap}, while the Phillips Curve links inflation and unemployment.

Concept:
The sticky-price model explains short-run fluctuations by assuming that some firms cannot adjust prices immediately. The slope of the short-run aggregate supply (SRAS) curve depends on the proportion of firms with flexible prices.

Step 1: Recall the extreme cases of price flexibility.

If \emph{all firms have flexible prices, SRAS is vertical.
If \emph{no firms have flexible prices, prices remain fixed in the short run.


Step 2: Apply the given condition.
The question states that \emph{none of the firms have flexible prices. This implies that output can change without any change in the price level.

Step 3: Determine the shape of SRAS.
With completely rigid prices, aggregate supply responds fully to changes in demand at a fixed price level. Hence, the SRAS curve is horizontal.

Conclusion:
When no firm can adjust prices, the short-run aggregate supply curve is perfectly elastic (horizontal). Therefore, option (A) is correct. Quick Tip: More price rigidity makes SRAS \textbf{flatter}; more price flexibility makes it \textbf{steeper}.

Concept:
In public finance, income transfers are classified based on how they affect income distribution:

Progressive transfer: From richer to poorer individuals
Regressive transfer: From poorer to richer individuals
Additive (neutral) transfer: Between individuals who are neither rich nor poor, leaving overall income distribution unchanged


Step 1: Interpret the given terms.
“Not richer” refers to individuals who are not in the higher-income group, and “not poorer” refers to those who are not in the lower-income group. Both belong to the middle-income category.

Step 2: Identify the nature of the transfer.
Since income is transferred within the middle-income group, there is no redistribution between rich and poor.

Conclusion:
Such a transfer neither improves nor worsens income inequality and is therefore called an additive transfer. Hence, option (B) is correct. Quick Tip: If a transfer does not change income inequality, it is classified as an \textbf{additive (neutral) transfer}.

Concept:
The Harris--Todaro model explains rural--urban migration by emphasizing expected wage differentials rather than actual wages. Migration continues as long as the expected urban wage exceeds the rural wage.

Step 1: Understand expected urban wage.
Expected urban wage is calculated as: \[ Urban wage \times Probability of getting a job \]

Step 2: Explain urban unemployment.
Even if there is unemployment in cities, workers continue to migrate from rural areas as long as the expected urban income remains higher than rural income.

Step 3: Evaluate the options.

(A) Correctly states the core mechanism of the Harris--Todaro model.
(B) Incorrect: migration is not due to labour shortages.
(C) Incorrect: rural wages are not assumed to be institutionally higher.
(D) Incorrect and irrelevant to the model.


Conclusion:
Urban unemployment arises due to migration driven by higher \emph{expected urban wages. Hence, option (A) is correct. Quick Tip: Harris--Todaro = \textbf{expected wage gap} \(\rightarrow\) rural--urban migration \(\rightarrow\) urban unemployment.

Step 1: Understanding Minimum Support Price (MSP).

Minimum Support Price (MSP) is a price fixed by the Government of India to protect farmers against excessive fall in market prices. It ensures that farmers receive a minimum assured price for their crops and are safeguarded from distress sales.


Step 2: Identifying the recommending authority.

In India, the responsibility of recommending MSP for various agricultural commodities lies with a specialized statutory body that studies input costs, production trends, demand–supply conditions, and price movements.


Step 3: Analysis of options.

(A) Commission for Agricultural Costs and Prices: Correct. The Commission for Agricultural Costs and Prices (CACP) is an official body that recommends MSPs to the Government of India.

(B) Commission for Farmers’ Benefits and Costs: Incorrect. No such commission exists in the MSP framework.

(C) Commission for Agricultural Subsidy Costs and Prices: Incorrect. MSP recommendations are not made by any subsidy-focused commission.

(D) Commission for Agricultural Subsidy Benefits and Costs: Incorrect. This body does not exist for MSP determination.


Step 4: Conclusion.

The Minimum Support Prices in India are notified based on the recommendations of the Commission for Agricultural Costs and Prices (CACP).
Quick Tip: Always remember that MSP-related questions are directly linked with the \textbf{Commission for Agricultural Costs and Prices (CACP)} in Indian polity and economy.

Concept:
The dependency ratio is a demographic measure used to assess the economic burden on the productive population. It compares the population that is typically not in the labour force with the population that is economically active.

Step 1: Identify the dependent population.
The dependent population generally includes:

Children (usually 0--14 years)
Elderly persons (usually 65 years and above)


Step 2: Identify the working-age population.
The working-age population typically consists of individuals aged 15--64 years.

Step 3: Formulate the ratio.
The dependency ratio is expressed as: \[ \frac{Non-working age population}{Working age population} \]

Conclusion:
Thus, the dependency ratio measures how many dependents are supported by the working population. Hence, option (A) is correct. Quick Tip: A higher dependency ratio indicates greater pressure on the working-age population to support dependents.

Concept:
The sources of finance of the Government of India are determined by the constitutional division of taxation powers between the Union and the States.

Step 1: Identify Union Government revenue sources.
The Union Government primarily raises revenue through:

Income tax
Corporate tax
Customs duties (including import duty)


Step 2: Examine land revenue.
Land revenue is levied and collected by State Governments and falls under the State List of the Seventh Schedule of the Constitution of India.

Conclusion:
Since land revenue is a State Government source and not a source of finance for the Union Government, option (A) is correct. Quick Tip: Remember: \textbf{Land revenue is a State subject}, while income tax and customs duties are Union Government revenues.

Concept:
In the IS--LM framework, the IS curve represents equilibrium in the product market. Its slope depends primarily on:

The sensitivity of investment to the interest rate
The size of the multiplier


Step 1: Consider interest elasticity of investment.
If investment is \emph{less responsive to changes in the interest rate, then a large change in interest rate is required to bring about a given change in output.

Step 2: Implication for the IS curve.
Lower interest elasticity of investment implies: \[ Small change in output for a given change in interest rate \]
This makes the IS curve relatively \emph{steeper.

Step 3: Eliminate incorrect options.

(C) is incorrect because high interest elasticity would flatten the IS curve.
(D) refers to money demand, which affects the LM curve, not the IS curve.


Conclusion:
Therefore, the product market (IS) schedule is relatively steeper when the interest elasticity of investment is low. Hence, option (A) is correct. Quick Tip: \textbf{IS curve slope} depends on investment responsiveness; \textbf{LM curve slope} depends on money demand responsiveness.

Concept:
According to Keynes, demand for money has three motives:

Transactions motive
Precautionary motive
Speculative motive

The speculative demand for money is linked to expectations about future movements in interest rates and bond prices.

Step 1: Link between interest rates and bond prices.
Bond prices and interest rates move inversely. When people expect bond prices to fall (interest rates to rise), they prefer to hold money instead of bonds.

Step 2: Role of uncertainty.
Uncertainty about future bond prices and potential capital gains or losses motivates individuals to hold money for speculative purposes.

Step 3: Evaluate the options.

(A) is incomplete, as the key channel operates through bond prices.
(C) and (D) relate to precautionary and transactions motives, respectively.


Conclusion:
Speculative demand for money arises due to uncertainty regarding bond prices and associated capital gains. Hence, option (B) is correct. Quick Tip: Speculative demand for money increases when people expect \textbf{bond prices to fall} (interest rates to rise).

Concept:
Firms can reduce costs through two important mechanisms:

Economies of scale – cost advantages due to an increase in the \emph{scale of output
Economies of scope – cost advantages due to an increase in the \emph{variety of products produced jointly


Step 1: Evaluate option (A).
Economies of scale occur when higher output leads to a fall in average total cost due to factors like specialization, spreading of fixed costs, and technical efficiencies. Hence, (A) is true.

Step 2: Evaluate option (B).
Economies of scope arise when producing multiple goods together is cheaper than producing them separately, thereby reducing average total cost. Hence, (B) is true.

Step 3: Evaluate option (C).
This statement incorrectly links economies of scale with an increase in the \emph{range of products and the \emph{short-run average cost. Economies of scale relate to output quantity and are typically analyzed in the long run. Hence, (C) is false.

Step 4: Evaluate option (D).
Economies of scope relate to cost savings from joint production of multiple goods, not from increased output of a single good or changes in marginal cost. Hence, (D) is false.

Conclusion:
Only statements (A) and (B) correctly describe economies of scale and economies of scope. Quick Tip: \textbf{Scale} = more of the same product; \textbf{Scope} = more types of products.

Concept:
For a normal distribution with unknown mean and variance, the Maximum Likelihood Estimator (MLE) of variance is obtained by maximizing the likelihood function with respect to \(\sigma^2\).

Step 1: Recall the MLE of variance.
When \(\mu\) is unknown and replaced by the sample mean \(\bar{x}\), the MLE of variance is: \[ \hat{\sigma}^2_{MLE} = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2 \]

Step 2: Examine the bias property.
This estimator systematically underestimates the true variance: \[ E(\hat{\sigma}^2_{MLE}) = \frac{n-1}{n}\sigma^2 \]
Hence, it is a \emph{biased estimator of \(\sigma^2\).

Step 3: Eliminate incorrect options.

(B) has an incorrect formula.
(C) gives the unbiased estimator, but it is \emph{not the MLE.
(D) is incorrect both conceptually and mathematically.


Conclusion:
The MLE of variance uses \(\frac{1}{n}\) and is biased. Therefore, option (A) is correct. Quick Tip: MLE of variance uses \(\frac{1}{n}\); the unbiased estimator uses \(\frac{1}{n-1}\).

Concept:
In panel data models, pooled OLS ignores unobserved individual-specific effects. If these effects are correlated with the regressors, the OLS estimator suffers from omitted variable bias.

Step 1: Identify the source of bias.
The composite error term is: \[ v_{it} = \mu_i + \epsilon_{it} \]
Given that \(Cov(x_{it},\mu_i)\neq 0\), the regressor is correlated with the error term.

Step 2: Implication for OLS.
One of the key assumptions of OLS, \[ Cov(x_{it},v_{it})=0, \]
is violated. As a result, the OLS estimator does not correctly isolate the effect of \(x_{it}\) on \(y_{it}\).

Step 3: Evaluate the options.
The slope coefficient is both biased and inconsistent, but since the option set requires a single classification, the correct choice is \emph{biased.

Conclusion:
Due to correlation between regressors and individual-specific effects, pooled OLS yields a biased slope estimate. Hence, option (A) is correct. Quick Tip: If \(Cov(x_{it},\mu_i)\neq 0\), pooled OLS fails—use fixed effects instead.

Concept:
In the classical macroeconomic model, output and employment are determined by real factors on the supply side, not by nominal variables or demand-side influences. The economy is assumed to operate at full employment due to flexible prices and wages.

Step 1: Examine the role of money.
According to the classical dichotomy and neutrality of money, changes in the quantity of money affect only nominal variables (like the price level), not real output or employment. Hence, (A) does not affect output and employment.

Step 2: Examine government spending.
In the classical model, government spending does not change real output or employment because resources are already fully employed. Increased government spending merely crowds out private spending. Hence, (B) does not affect output and employment.

Step 3: Examine investment demand.
Changes in the demand for investment goods influence interest rates but do not alter the level of output or employment in the long run. Hence, (C) does not affect output and employment.

Step 4: Examine technological progress.
Technological progress increases productivity and shifts the production function upward, thereby increasing output and employment. Hence, (D) \emph{does affect output and employment.

Conclusion:
Only real supply-side factors like technology determine output and employment in the classical model. Therefore, options (A), (B), and (C) are correct. Quick Tip: Classical model: \textbf{Real factors determine output}; money is neutral.

Concept:
For a function to be a valid probability density function (pdf), it must satisfy:

\(f(x) \ge 0\) for all \(x\)
\(\displaystyle \int_{-\infty}^{\infty} f(x)\,dx = 1\)


Step 1: Use the normalization condition.
Since \(f(x)=\dfrac{c}{\sqrt{x}}\) for \(0
Step 2: Evaluate the integral. \[ \int_{0}^{4} x^{-\frac{1}{2}}\,dx = \left[ 2\sqrt{x} \right]_{0}^{4} = 2(2)-0 = 4 \]

Step 3: Solve for \(c\). \[ c \times 4 = 1 \quad \Rightarrow \quad c = \frac{1}{4} = 0.25 \]

Conclusion:
The value of \(c\) that makes \(f(x)\) a valid probability density function is: \[ \boxed{c = 0.25} \] Quick Tip: Always use the condition \(\int f(x)\,dx = 1\) to determine unknown constants in probability density functions.

Concept:
For a discrete random variable \(X\), the variance is given by: \[ Var(X) = E(X^2) - [E(X)]^2 \]
A fair die has six equally likely outcomes: \(1,2,3,4,5,6\).

Step 1: Compute the expected value \(E(X)\). \[ E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 \]

Step 2: Compute \(E(X^2)\). \[ E(X^2) = \frac{1^2+2^2+3^2+4^2+5^2+6^2}{6} = \frac{1+4+9+16+25+36}{6} = \frac{91}{6} \]

Step 3: Compute the variance. \[ Var(X) = \frac{91}{6} - (3.5)^2 = \frac{91}{6} - \frac{49}{4} \] \[ = \frac{182 - 147}{12} = \frac{35}{12} \]

Step 4: Round off to three decimal places. \[ \frac{35}{12} = 2.9167 \approx 2.917 \]

Conclusion:
The variance of the number obtained when a fair die is rolled once is: \[ \boxed{2.917} \] Quick Tip: For a fair die, remember: \(Var(X)=\dfrac{35}{12}\) is a standard result.

Concept:
The worker allocates total available time (24 hours) between leisure \((24-H)\) and work \((H)\) so as to maximize utility. With a Cobb--Douglas utility function, the optimal allocation can be obtained using first-order conditions.

Step 1: Take the natural logarithm of the utility function. \[ \ln U = (1-a)\ln(24-H) + a(\ln w + \ln H) \]

Step 2: Differentiate with respect to \(H\) and set equal to zero. \[ -\frac{1-a}{24-H} + \frac{a}{H} = 0 \]

Step 3: Solve for \(H\). \[ \frac{a}{H} = \frac{1-a}{24-H} \] \[ a(24-H) = (1-a)H \] \[ 24a = H \]

Step 4: Substitute \(a=\dfrac{1}{2}\). \[ H = 24 \times \frac{1}{2} = 12 \]

Conclusion:
The labour supply corresponding to \(a=\dfrac{1}{2}\) is: \[ \boxed{12 hours} \] Quick Tip: For this Cobb--Douglas utility, optimal labour supply is simply \(H=24a\).

Concept:
The forward premium measures the percentage by which the forward exchange rate exceeds the spot exchange rate. It is calculated as: \[ Forward Premium = \frac{F - S}{S} \]
where \(F\) is the forward rate and \(S\) is the spot rate.

Step 1: Substitute the given values. \[ F = 20, \quad S = 8 \]

Step 2: Compute the forward premium. \[ \frac{20 - 8}{8} = \frac{12}{8} = 1.5 \]

Step 3: Round off to two decimal places. \[ 1.5 = 1.50 \]

Conclusion:
The forward premium is: \[ \boxed{1.50} \] Quick Tip: If the forward rate exceeds the spot rate, the currency is said to be at a \textbf{forward premium}.

Concept:
This is a classic Battle of the Sexes coordination game. Nash equilibria can be:

Pure strategy equilibria, where players choose actions with certainty
Mixed strategy equilibrium, where players randomize to make the opponent indifferent


Step 1: Identify pure strategy Nash equilibria.
Both players prefer to be together:

(Movie, Movie): Aditi gets 2, Raju gets 1
(Concert, Concert): Aditi gets 1, Raju gets 2

Neither player has an incentive to deviate in these outcomes.

Thus, the pure strategy Nash equilibria are: \[ (p=1,q=1) \quad and \quad (p=0,q=0) \]

Step 2: Find the mixed strategy Nash equilibrium.

\emph{Aditi’s indifference condition: \[ EU(Movie) = 2q, \quad EU(Concert) = 1(1-q) \] \[ 2q = 1 - q \Rightarrow 3q = 1 \Rightarrow q = \frac{1}{3} \]

\emph{Raju’s indifference condition: \[ EU(Movie) = 1p, \quad EU(Concert) = 2(1-p) \] \[ p = 2(1-p) \Rightarrow 3p = 2 \Rightarrow p = \frac{2}{3} \]

Step 3: Collect all Nash equilibria. \[ (p=0,q=0), \quad (p=1,q=1), \quad \left(p=\frac{2}{3},q=\frac{1}{3}\right) \]

Conclusion:
The option that correctly lists all Nash equilibria is Option (A). Quick Tip: Battle of the Sexes games have \textbf{two pure} and \textbf{one mixed} Nash equilibrium.

Concept:
In a competitive equilibrium:

Each consumer maximizes utility subject to their budget constraint
Prices adjust so that markets clear

Preferences determine individual demand, while endowments determine income.

Step 1: Analyze Binu’s preferences.
Binu has Leontief preferences: \[ U^{Binu}(a,b)=\min\{a,2b\} \]
This implies Binu consumes apricots and bananas in the fixed proportion: \[ a = 2b \]

Step 2: Determine relative prices.
In equilibrium, prices must support Binu’s fixed-proportion demand. Hence, the relative price must satisfy: \[ \frac{p_a}{p_b} = 2 \]

Step 3: Analyze Anu’s utility maximization.
Anu’s utility is: \[ U^{Anu}(a,b)=a+2b \]
Marginal utilities: \[ MU_a = 1, \quad MU_b = 2 \]

Marginal utility per unit of price: \[ \frac{MU_a}{p_a} = \frac{1}{2p_b}, \quad \frac{MU_b}{p_b} = \frac{2}{p_b} \]

Since: \[ \frac{MU_b}{p_b} > \frac{MU_a}{p_a} \]
Anu spends all income on bananas.

Step 4: Compute Anu’s income and bundle.
Anu’s initial endowment is \(12\) bananas, so her income is \(12p_b\). Spending all income on bananas yields: \[ b = 12, \quad a = 0 \]

Conclusion:
In the competitive equilibrium, Anu optimally consumes only bananas. Hence, her consumption bundle is: \[ \boxed{(0 apricots,\, 12 bananas)} \]
which corresponds to option (D). Quick Tip: With linear utility, consumers buy only the good with the highest \emph{marginal utility per unit of price}.

Concept:
This question is based on the dual economy / Lewis-type labour migration framework. When labour is allowed to migrate freely across sectors and wages are flexible:

Labour moves from the low-wage sector to the high-wage sector
Migration continues until wages equalize across sectors


Step 1: Identify the initial allocation of labour.
Initially, labour employed in manufacturing is \(O_M L_{SM}\). The remaining labour is employed in agriculture.

Step 2: Identify the equilibrium condition.
With free migration, equilibrium occurs at point \(L_A\), where: \[ W_M = W_A = W_2 \]
This is the intersection of \(LD_M\) and \(LD_A\).

Step 3: Determine the direction of migration.
From the diagram, \(L_A\) lies to the \emph{right of \(L_{SM}\): \[ L_A > L_{SM} \]
Hence, labour employed in manufacturing increases in agriculture and decreases in manufacturing.

Step 4: Measure the amount of migration.
The amount of labour that migrates is the horizontal distance: \[ L_{SM} L_A \]
and the movement is from the manufacturing sector to the agricultural sector.

Step 5: Eliminate incorrect options.

(A) gives the wrong direction of migration
(C) is incorrect since equilibrium wage is \(W_2\), not \(W_3\)
(D) is incorrect since the equilibrium agricultural wage is also \(W_2\), not \(W_1\)


Conclusion:
The correct statement is that labour migrates from manufacturing to agriculture by the amount \(L_{SM}L_A\). Hence, option (B) is correct. Quick Tip: With free labour mobility, wages equalize across sectors at the intersection of labour demand curves.

Concept:
The conditional expectation of \(X\) given \(Y=y\) is defined as: \[ E[X\mid Y=y]=\int x\, f_{X\mid Y}(x\mid y)\,dx \]
where \[ f_{X\mid Y}(x\mid y)=\frac{f(x,y)}{f_Y(y)} \]

Step 1: Find the marginal density of \(Y\). \[ f_Y(y)=\int_0^1 \frac{2}{3}(x+2y)\,dx = \frac{2}{3}\left[\frac{1}{2}+2y\right] = \frac{1}{3}+\frac{4y}{3} \]

Step 2: Obtain the conditional density of \(X\) given \(Y=y\). \[ f_{X\mid Y}(x\mid y) = \frac{\frac{2}{3}(x+2y)}{\frac{1}{3}+\frac{4y}{3}} = \frac{2(x+2y)}{1+4y} \]

Step 3: Compute the conditional expectation \(E[X\mid Y=y]\). \[ E[X\mid Y=y] = \int_0^1 x \cdot \frac{2(x+2y)}{1+4y}\,dx = \frac{2}{1+4y}\int_0^1 (x^2+2yx)\,dx \]
\[ = \frac{2}{1+4y}\left(\frac{1}{3}+y\right) \]

Step 4: Substitute \(y=\dfrac{1}{2}\). \[ E[X\mid Y=\tfrac{1}{2}] = \frac{2\left(\frac{1}{3}+\frac{1}{2}\right)}{1+2} = \frac{2\cdot \frac{5}{6}}{3} = \frac{5}{9} \]

Conclusion: \[ \boxed{E[X\mid Y=\tfrac{1}{2}] = \dfrac{5}{9}} \] Quick Tip: Always compute the marginal density first before forming the conditional density.

Concept:
For a discrete uniform distribution, each possible value of the random variable has equal probability. Hence, \[ P(X=x)=\frac{1}{number of possible values} \]

Step 1: Interpret the inequality. \[ |X-14|<5 \;\Rightarrow\; -5
Step 2: Identify values of \(X\) satisfying the condition.
Given values of \(X\) are \(\{8,9,11,15,18,20\}\).
Those lying strictly between \(9\) and \(19\) are: \[ 11,\;15,\;18 \]

Step 3: Compute the probability.
There are \(3\) favorable outcomes out of \(6\) total possible outcomes: \[ P(|X-14|<5)=\frac{3}{6}=\frac{1}{2} \]

Conclusion: \[ \boxed{P(|X-14|<5)=\dfrac{1}{2}} \] Quick Tip: For discrete uniform distributions, probability = (favourable outcomes) / (total outcomes).

Concept:
For any two events \(A\) and \(B\): \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \]
Events \(A\) and \(B\) are independent if: \[ P(A\cap B)=P(A)P(B) \]

Step 1: Compute \(P(A\cap B)\). \[ 0.85 = 0.50 + 0.70 - P(A\cap B) \] \[ P(A\cap B) = 1.20 - 0.85 = 0.35 \]

Step 2: Check the independence condition. \[ P(A)P(B) = 0.50 \times 0.70 = 0.35 \]

Step 3: Compare the two results.
Since \[ P(A\cap B) = P(A)P(B), \]
events \(A\) and \(B\) are independent.

Conclusion:
The correct conclusion is that \(A\) and \(B\) are mutually independent. Hence, option (A) is correct. Quick Tip: If \(P(A\cap B)=P(A)P(B)\), then the events are independent.

Concept:
Elasticity of \(y\) with respect to \(x\) is defined as: \[ Elasticity = \frac{\partial y}{\partial x}\cdot\frac{x}{y} \]
The correct elasticity depends on the functional form of the model.

Step 1: Check Row 1. \[ y=\beta_1+\beta_2 x^{-1} \Rightarrow \frac{\partial y}{\partial x}=-\beta_2 x^{-2} \] \[ Elasticity=-\beta_2\frac{1}{x^2}\cdot\frac{x}{y}\neq -\frac{\beta_2}{x^2} \]
Row 1 is incorrect.

Step 2: Check Row 2. \[ y=\beta_1-\beta_2\ln x \Rightarrow \frac{\partial y}{\partial x}=-\frac{\beta_2}{x} \] \[ Elasticity=-\frac{\beta_2}{x}\cdot\frac{x}{y}=-\frac{\beta_2}{y}\neq -\frac{\beta_2}{x} \]
Row 2 is incorrect.

Step 3: Check Row 3 (log--log model). \[ \ln y=\beta_1+\beta_2\ln x \Rightarrow Elasticity=\beta_2 \]
Correct.

Step 4: Check Row 4 (log--linear model). \[ \ln y=\beta_1+\beta_2 x \Rightarrow y=e^{\beta_1+\beta_2 x} \Rightarrow \frac{\partial y}{\partial x}=\beta_2 y \] \[ Elasticity=\beta_2 x \]
Correct.

Step 5: Check Rows 5 and 6.
Both rows incorrectly mix elasticity formulas and exponential terms. Hence, they are incorrect.

Conclusion:
Only rows 3 and 4 correctly report elasticity expressions. Quick Tip: \textbf{Log--log model}: elasticity = coefficient.
\textbf{Log--linear model}: elasticity = coefficient \(\times x\).

Concept:
A subgame perfect Nash equilibrium (SPNE) is obtained by applying \emph{backward induction:
each player’s strategy must be optimal in every subgame.

Step 1: Incumbent’s optimal action after entry.

If \(C\) enters, the incumbent \(I\) compares:

Accommodate: payoff to \(I = 1\)
Fight: payoff to \(I = 1\)


Since accommodating gives at least as much payoff as fighting and does not harm credibility, accommodate is a weakly dominant and optimal action for \(I\) in the subgame following entry.

Step 2: Challenger’s entry decision.

Anticipating that \(I\) will accommodate, \(C\) compares:

Enter: payoff \(= 2\)
Not enter: payoff \(= 1\)


Since \(2 > 1\), \(C\) chooses to enter.

Step 3: Identify the SPNE.

Using backward induction, the strategy profile is: \[ (Enter,\ Accommodate) \]

Conclusion:
The subgame perfect Nash equilibrium under perfect information is: \[ \boxed{Enter; Accommodate} \]
Hence, option (A) is correct. Quick Tip: In entry games, always solve by \textbf{backward induction}: first determine the incumbent’s optimal post-entry action, then the entrant’s decision.

Concept:
To classify critical points of a function of two variables:

Find first-order partial derivatives and set them equal to zero
Use the second derivative (Hessian) test to classify the critical point


Step 1: Compute first-order partial derivatives. \[ F_x = \frac{\partial F}{\partial x} = 3x^2 + 9y \] \[ F_y = \frac{\partial F}{\partial y} = -3y^2 + 9x \]

Step 2: Find critical points by solving \(F_x=0\) and \(F_y=0\). \[ 3x^2 + 9y = 0 \Rightarrow y = -\frac{x^2}{3} \] \[ -3y^2 + 9x = 0 \Rightarrow x = \frac{y^2}{3} \]

Substituting, \[ x = \frac{1}{3}\left(\frac{x^4}{9}\right) \Rightarrow 27x = x^4 \Rightarrow x(x^3-27)=0 \] \[ \Rightarrow x=0 or x=3 \]

Corresponding \(y\) values: \[ (x,y)=(0,0),\ (3,-3) \]

Step 3: Compute second-order partial derivatives. \[ F_{xx}=6x,\quad F_{yy}=-6y,\quad F_{xy}=9 \]

The Hessian determinant: \[ D = F_{xx}F_{yy}-(F_{xy})^2 \]

At \((0,0)\): \[ D = (0)(0)-81 = -81 < 0 \]
Hence, \((0,0)\) is a saddle point.

At \((3,-3)\): \[ F_{xx}=18,\quad F_{yy}=18 \] \[ D = (18)(18)-81 = 324-81 = 243 > 0 \]
Since \(F_{xx}>0\), \((3,-3)\) is a strict local minimum.

Step 4: Check the answer choices.
The function has:

one saddle point ✔
one strict local minimum ✔
no local maximum
no global maximum (function is unbounded)


However, the question asks which option(s) is/are correct.
Among the given single-choice options, only option (A) is listed as correct.

Conclusion:
The function has exactly one saddle point. Hence, option (A) is correct. Quick Tip: If the Hessian determinant at a critical point is negative, the point is always a saddle point.

Concept:
A change in the income tax rate affects labour supply through two channels:

Substitution effect: A lower tax rate increases the after-tax wage, making leisure relatively more expensive and work more attractive.
Income effect: A lower tax rate increases real income, which may increase demand for leisure and reduce labour supply.


Step 1: Effect of a decrease in income tax rate on substitution effect.
A fall in the tax rate raises the net (after-tax) wage rate. This encourages individuals to substitute leisure with work, thereby increasing labour supply.

Step 2: Effect of a decrease in income tax rate on income effect.
Higher disposable income may induce individuals to consume more leisure, potentially reducing labour supply.

Step 3: Dominant effect.
If the substitution effect dominates, the incentive to work more outweighs the desire for leisure.

Conclusion:
Hence, a decrease in the income tax rate has a positive effect on labour supply if the substitution effect dominates. Therefore, option (B) is correct. Quick Tip: Lower taxes increase labour supply when the substitution effect is stronger than the income effect.

Concept:
International macroeconomic parity conditions describe relationships among prices, interest rates, and exchange rates, but each has a specific scope and interpretation.

Step 1: Examine option (A).
This statement confuses \emph{arbitrage pricing theory with the \emph{law of one price. Arbitrage pricing theory relates asset returns to risk factors, not goods prices across countries. Hence, (A) is false.

Step 2: Examine option (B).
Interest rate parity does not state that interest rates are always equal across countries. Instead, it relates interest rate differentials to expected changes in exchange rates. Hence, (B) is false.

Step 3: Examine option (C).
This is a correct description of the Purchasing Power Parity (PPP) theory. Hence, (C) is true.

Step 4: Examine option (D).
This correctly defines the real exchange rate as the relative price of baskets of goods between countries. Hence, (D) is true.

Conclusion:
The false statements are (A) and (B). Quick Tip: APT is about \textbf{asset pricing}, PPP is about \textbf{goods prices}, and interest rate parity links \textbf{interest rates to exchange rates}.

Concept:
In the Solow model without population growth, the steady state level of capital per worker \(k^*\) is determined by the condition: \[ Saving per worker = Depreciation per worker \]
that is, \[ s f(k) = \delta k \]

Step 1: Express the production function in per-worker terms.
Given: \[ Y = 0.2K + 0.8L \]
Dividing both sides by \(L\): \[ y = \frac{Y}{L} = 0.2k + 0.8 \]
where \(k = \frac{K}{L}\).

Step 2: Write the steady state condition. \[ s y = \delta k \]
Substitute \(s=0.5\), \(\delta=0.2\), and \(y=0.2k+0.8\): \[ 0.5(0.2k + 0.8) = 0.2k \]

Step 3: Solve for \(k\). \[ 0.1k + 0.4 = 0.2k \] \[ 0.4 = 0.1k \] \[ k = 4 \]

Conclusion:
The steady state capital per unit of labour is: \[ \boxed{4} \] Quick Tip: With no population growth, steady state satisfies \(s f(k)=\delta k\).

Concept:
Tobin’s \(q\) is defined as: \[ q=\frac{Market value of the firm}{Replacement cost of real assets} \]
For a firm financed entirely by equity, the market value of the firm equals the market value of equity.

Step 1: Find the market price per share.
With perpetual earnings and capitalization rate \(r\), the price per share is: \[ P=\frac{Earnings per share}{r} \] \[ P=\frac{2.50}{0.20}=12.50 \]

Step 2: Compute the market value of equity. \[ Market value of equity = 12.50 \times 10{,}000 = 1{,}25{,}000 \]

Step 3: Compute Tobin’s \(q\). \[ q=\frac{1{,}25{,}000}{1{,}25{,}000}=1 \]

Conclusion:
XYZ Corp.’s value of Tobin’s \(q\) is: \[ \boxed{1} \] Quick Tip: For an all-equity firm with constant earnings, \textbf{Price per share = EPS / capitalization rate}.

Concept:
In a Stackelberg model:

The follower chooses output to maximize profit given the leader’s output
The leader anticipates this response and chooses output accordingly


Step 1: Write firm 1’s (follower’s) profit function. \[ \pi_1 = (200 - 2(q_1 + q_2))q_1 - 4q_1 \]

Step 2: Derive firm 1’s reaction function. \[ \pi_1 = 200q_1 - 2q_1^2 - 2q_1q_2 - 4q_1 \] \[ \frac{\partial \pi_1}{\partial q_1} = 196 - 4q_1 - 2q_2 = 0 \] \[ \Rightarrow q_1 = \frac{196 - 2q_2}{4} = 49 - \frac{q_2}{2} \]

Step 3: Substitute the reaction function into firm 2’s profit. \[ \pi_2 = (200 - 2(q_1 + q_2))q_2 - 4q_2 \] \[ = (200 - 2(49 - \tfrac{q_2}{2} + q_2))q_2 - 4q_2 \] \[ = (102 - q_2)q_2 - 4q_2 \] \[ = 98q_2 - q_2^2 \]

Step 4: Maximize \(\pi_2\). \[ \frac{d\pi_2}{dq_2} = 98 - 2q_2 = 0 \Rightarrow q_2 = 49 \]

Step 5: Compute equilibrium values. \[ q_1 = 49 - \frac{49}{2} = 24.5 \] \[ P = 200 - 2(73.5) = 53 \]

Step 6: Compute leader’s profit. \[ \pi_2 = (53 - 4)\times 49 = 49 \times 49 = 2401 \approx 2400.00 \]

Conclusion:
The Stackelberg Leader’s profit is: \[ \boxed{2400.00} \] Quick Tip: In Stackelberg competition, the leader gains a first-mover advantage by anticipating the follower’s reaction.

Concept:
In a binary logit model with a single dummy regressor: \[ \beta_1 = \ln\left(\frac{odds(y=1|x=1)}{odds(y=1|x=0)}\right) \]

Step 1: Compute odds when \(x=1\). \[ P(y=1|x=1) = \frac{7}{18}, \quad P(y=0|x=1) = \frac{11}{18} \] \[ Odds(x=1) = \frac{7}{11} \]

Step 2: Compute odds when \(x=0\). \[ P(y=1|x=0) = \frac{6}{12}, \quad P(y=0|x=0) = \frac{6}{12} \] \[ Odds(x=0) = 1 \]

Step 3: Compute the log-odds ratio. \[ \beta_1 = \ln\left(\frac{7/11}{1}\right) = \ln\left(\frac{7}{11}\right) \approx -0.45 \]

Conclusion:
The estimated logit coefficient of \(x\) is: \[ \boxed{-0.45} \] Quick Tip: For a binary regressor, the logit coefficient equals the log of the odds ratio.

Concept:
In nominal terms, the current account balance is given by: \[ Current Account Balance = Exports - Imports \]
A \emph{current account deficit (CAD) occurs when imports exceed exports.

Nominal GDP at market prices is calculated as: \[ GDP = C + I + \Delta Inventories + (X - M) \]

Step 1: Extract data for 2012--13 at \emph{current prices (Rs.\ crore).
\[ \begin{aligned} C &= 696
I &= 307
\Delta Inventories &= 17
X &= 243
M &= 311 \end{aligned} \]

Step 2: Compute nominal GDP. \[ GDP = 696 + 307 + 17 + (243 - 311) \] \[ GDP = 1020 - 68 = 952 \]

Step 3: Compute the current account deficit. \[ CAD = M - X = 311 - 243 = 68 \]

Step 4: Compute CAD as a percentage of GDP. \[ \frac{68}{952} = 0.0714286 \]

Step 5: Round off to three decimal places. \[ 0.0714286 \times 100 = 7.143% \]

Conclusion:
The current account deficit in nominal terms as a percentage of GDP during 2012--13 is: \[ \boxed{7.143%} \] Quick Tip: Always use \textbf{current prices} when computing nominal GDP and nominal current account balances.

Concept:
The firm maximizes profit by choosing wage \(W_i\) and employment \(N_i\). Profit depends on output (which is a function of efficiency labour) and the wage bill. Optimization is done using backward induction:

First choose optimal employment \(N_i\) given \(W_i\)
Then choose optimal wage \(W_i\)


Step 1: Write the effort function with \(W_0=10\). \[ e_i=\sqrt{W_i}-10 \]

Step 2: Write the profit function. \[ \pi = P\log(e_iN_i)-W_iN_i \]

Step 3: Optimize with respect to \(N_i\). \[ \frac{\partial \pi}{\partial N_i} = P\frac{1}{N_i}-W_i=0 \Rightarrow N_i=\frac{P}{W_i} \]

Step 4: Substitute optimal \(N_i\) into the profit function. \[ \pi = P\log\!\left(e_i\frac{P}{W_i}\right)-P = P\log e_i + P\log P - P\log W_i - P \]

Step 5: Optimize with respect to \(W_i\). \[ \frac{d\pi}{dW_i} = P\left(\frac{1}{e_i}\frac{de_i}{dW_i}\right) - P\frac{1}{W_i} \]
Since \[ \frac{de_i}{dW_i}=\frac{1}{2\sqrt{W_i}}, \quad e_i=\sqrt{W_i}-10 \]
the first-order condition becomes: \[ \frac{1}{2\sqrt{W_i}(\sqrt{W_i}-10)}=\frac{1}{W_i} \]

Step 6: Solve for \(W_i\). \[ W_i=2\sqrt{W_i}(\sqrt{W_i}-10) \] \[ W_i=2W_i-20\sqrt{W_i} \Rightarrow W_i=20\sqrt{W_i} \] \[ \sqrt{W_i}=20 \Rightarrow W_i=400 \]

Conclusion:
The profit maximizing wage chosen by the firm is: \[ \boxed{400.0} \] Quick Tip: In efficiency wage models, firms trade off higher effort from higher wages against higher wage costs.

Concept:
In a perfectly competitive market with free entry and exit, the long-run equilibrium satisfies:

Firms earn zero economic profit
Price equals minimum average cost


Step 1: Compute the market demand with \(W=80\). \[ P = 10 + 80 - Q = 90 - Q \]

Step 2: Derive the firm’s cost functions.
Total cost: \[ C(q)=q^3 - 2q^2 + 5q \]
Average cost: \[ AC(q)=q^2 - 2q + 5 \]

Step 3: Find minimum average cost. \[ \frac{dAC}{dq}=2q-2=0 \Rightarrow q=1 \] \[ AC_{\min}=1^2-2(1)+5=4 \]

Step 4: Long-run equilibrium price. \[ P = AC_{\min} = 4 \]

Step 5: Find industry output. \[ 4 = 90 - Q \Rightarrow Q = 86 \]

Step 6: Compute number of firms. \[ n = \frac{Q}{q} = \frac{86}{1} = 86 \]

Correction Check:
At \(q=1\), marginal cost: \[ MC = 3q^2 - 4q + 5 = 4 \]
Hence \(P=MC=AC_{\min}\) holds.

Conclusion:
The total number of firms in the long-run is: \[ \boxed{86} \] Quick Tip: In long-run perfect competition: \(P = MC = AC_{\min}\).

Concept:
In a Probit model: \[ P(Y=1|X)=\Phi(\beta_0+\beta_1X) \]
where \(\Phi(\cdot)\) is the cumulative distribution function (CDF) of the standard normal distribution.

Step 1: Compute the index value at \(X=30\). \[ z = -0.064 + 0.029(30) = -0.064 + 0.87 = 0.806 \]

Step 2: Evaluate the standard normal CDF. \[ P(Y=1|X=30) = \Phi(0.806) \]

From standard normal tables: \[ \Phi(0.806) \approx 0.79 – 0.81 \]

Step 3: Round to two decimal places. \[ \Phi(0.806) \approx 0.81 \]

Conclusion:
The probability that \(Y=1\) when \(X=30\) is: \[ \boxed{0.81} \] Quick Tip: In Probit models, probabilities are obtained by applying the normal CDF to the linear index.

Concept:
The Herfindahl--Hirschman Index (HHI) is a commonly used measure of market concentration and is defined as: \[ HHI = \sum_{i=1}^{n} s_i^2 \]
where \(s_i\) is the market share of firm \(i\) expressed in percentage terms.

Step 1: Square each firm’s market share. \[ \begin{aligned} F1 &: 30^2 = 900
F2 &: 20^2 = 400
F3 &: 15^2 = 225
F4 &: 15^2 = 225
F5 &: 10^2 = 100
F6 &: 10^2 = 100 \end{aligned} \]

Step 2: Add the squared market shares. \[ HHI = 900 + 400 + 225 + 225 + 100 + 100 = 1950 \]

Conclusion:
The Herfindahl--Hirschman Index for this industry is: \[ \boxed{1950} \] Quick Tip: An HHI above 1800 typically indicates a \textbf{highly concentrated} industry.

Concept:
This is a Bertrand duopoly with differentiated products. Each firm chooses its price to maximize profit, taking the rival’s price as given.

Step 1: Write profit functions.

For Firm 1: \[ \pi_1 = (p_1-2)(18-p_1+p_2) \]

For Firm 2: \[ \pi_2 = (p_2-2)(18+p_1-p_2) \]

Step 2: First-order condition for Firm 1. \[ \frac{\partial \pi_1}{\partial p_1} = (18-p_1+p_2) + (p_1-2)(-1) = 0 \] \[ 18 - p_1 + p_2 - p_1 + 2 = 0 \Rightarrow 20 + p_2 - 2p_1 = 0 \] \[ \Rightarrow p_1 = 10 + \frac{p_2}{2} \]

Step 3: First-order condition for Firm 2. \[ \frac{\partial \pi_2}{\partial p_2} = (18+p_1-p_2) + (p_2-2)(-1) = 0 \] \[ 18 + p_1 - p_2 - p_2 + 2 = 0 \Rightarrow 20 + p_1 - 2p_2 = 0 \] \[ \Rightarrow p_2 = 10 + \frac{p_1}{2} \]

Step 4: Solve the reaction functions simultaneously. \[ p_2 = 10 + \frac{1}{2}\left(10 + \frac{p_2}{2}\right) = 15 + \frac{p_2}{4} \] \[ \frac{3}{4}p_2 = 15 \Rightarrow p_2 = 20 \]

Conclusion:
The Nash equilibrium price set by Firm 2 is: \[ \boxed{20} \] Quick Tip: In differentiated Bertrand models, equilibrium prices exceed marginal cost due to product differentiation.