General Mental Ability Practice Questions — State PSC Exam

Complete the given number series by identifying the logical pattern.

Fill in the blanks below

In the number series

5, 10, 20, 35, 55, \dots

, the value of the missing term is .

Show answer & explanation

Answer

80

The terms increase by successive multiples of 5:

+5, +10, +15, +20

. Continuing this pattern, the next increment must be

+25

, giving

55 + 25 = 80

.

Step-by-Step Solution

1

Calculate the differences between consecutive terms in the series.

10 - 5 = 5

,

20 - 10 = 10

,

35 - 20 = 15

,

55 - 35 = 20

.

Finding the differences between consecutive terms helps identify the underlying sequence rule.

2

Determine the pattern governing the differences.

The differences form a sequence that increases by

5

at each step (

5, 10, 15, 20, \dots

).

The next difference must be

20 + 5 = 25

.

3

Add the calculated difference to the preceding term to obtain the missing value.

55 + 25 = 80

.

Adding the next difference of

25

to the fifth term (

55

) produces the sixth term.

Key Concept

Number Series Completion via Increasing Differences

Question 2Question →

Clock A continuously loses $2$ minutes every hour, while Clock B continuously gains $1$ minute every hour. Both clocks are synchronized to the exact true time at 8:00 AM on Monday, February 26, 1900.

Which of the following represents the exact day of the week, calendar date, and the time displayed by Clock B at the earliest instance when the time displayed by Clock A and Clock B are exactly $12$ hours apart?

Thursday, March 7, 1900, at 12:00 Noon

Thursday, March 8, 1900, at 8:00 AM

Thursday, March 8, 1900, at 12:00 Noon

Wednesday, March 7, 1900, at 12:00 Noon

Show answer & explanation

Answer: Thursday, March 8, 1900, at 12:00 Noon

Answer

Thursday, March 8, 1900, at 12:00 Noon

The two clocks drift apart at a combined rate of 3 minutes per hour. To achieve a 12-hour (720-minute) difference, exactly 240 true hours (10 days) must pass. Since 1900 is not a leap year, February has 28 days. Advancing 10 days from Monday, February 26 brings the true date to Thursday, March 8. In those 240 hours, Clock B gains 240 minutes (4 hours). Since the true time is 8:00 AM, Clock B displays 12:00 Noon.

Step-by-Step Solution

1

Calculate the relative speed of separation between the two clocks.

Clock A loses 2 mins/hr and Clock B gains 1 min/hr. Relative separation = 2 + 1 = 3 minutes every hour.

Since they move in opposite directions relative to the true time, their rates are added to find how fast they drift apart.

2

Determine the true time required for the clocks to be 12 hours apart.

12 hours = 720 minutes. Time required = 720 / 3 = 240 hours. 240 hours = 10 full days.

Dividing the total required separation by the hourly separation rate yields the total elapsed true time.

3

Calculate the true calendar date and day of the week after 10 days.

The year 1900 is not a leap year, so February has 28 days. Starting from Feb 26, adding 2 days reaches Feb 28. The remaining 8 days land on March 8. Monday + 10 days (or Monday + 3 odd days) = Thursday.

Century years must be divisible by 400 to be leap years. 1900 does not meet this condition.

4

Determine the time displayed by Clock B after 240 hours.

Clock B gains 1 minute per hour. In 240 hours, it gains 240 minutes, which is exactly 4 hours. True end time is 8:00 AM. 8:00 AM + 4 hours = 12:00 Noon.

The question specifically asks for the time displayed by Clock B, not the true time.

Key Concept

Relative speed of faulty clocks combined with century leap year constraints.

Question 3Question →

Six family members—Arthur, Beatrice, Clara, David, Elena, and Felix—are sitting around a circular table facing the center for dinner.

The seating arrangement is based on the following conditions:
1. Arthur is sitting directly opposite David.
2. Beatrice is sitting immediately to the right of Arthur.
3. Clara is sitting exactly between David and Beatrice.
4. Elena is sitting immediately to the left of Arthur.

Based on this information, who is sitting immediately to the right of David?

Clara

Felix

Beatrice

Elena

Show answer & explanation

Answer: Felix

Answer

Felix

By successfully mapping the circular arrangement, all family members are placed. Since Felix is the only one left without an explicitly stated position, he takes the final empty seat. This seat is located immediately to the right of David, making Felix the correct answer.

Step-by-Step Solution

1

Visualize or draw a circle with 6 equally spaced seats and place Arthur at one of the seats.

Arthur is assigned a position. For reference, let's place Arthur at the bottom (6 o'clock).

Setting a fixed starting point makes relative placements much easier.

2

Place David based on the first condition.

David is placed at the top (12 o'clock), directly opposite Arthur.

The condition states Arthur and David sit directly opposite each other.

3

Place Beatrice and Elena relative to Arthur.

Beatrice goes to the immediate right of Arthur (4 o'clock, counter-clockwise). Elena goes to the immediate left of Arthur (8 o'clock, clockwise).

Because everyone is facing the center, 'right' is counter-clockwise and 'left' is clockwise from the perspective of the person sitting.

4

Place Clara based on her position relative to David and Beatrice.

Clara is placed at 2 o'clock, which is the only seat exactly between David (12 o'clock) and Beatrice (4 o'clock).

The condition explicitly places Clara between these two specific family members.

5

Determine Felix's position and answer the final question.

Felix must occupy the last remaining seat at 10 o'clock. Since David is at 12 o'clock facing the center, his immediate right (counter-clockwise) is 10 o'clock, where Felix is sitting.

All other family members are placed. The question asks for the person to the immediate right of David.

Key Concept

Circular Seating Arrangements

Question 4Question →

A botanical research institute archives its seed collection by encrypting the plant category names using a specific two-step alphabetical transformation rule. According to this rule, the category name ORCHID is transformed into MJYTSX. Applying the identical transformation rule, which of the following will be the encrypted code for the category name BAMBOO?

ZAOZMM

YZNYLL

CBNCPP

XYMXKK

Show answer & explanation

Answer: ZAOZMM

Answer

The encrypted code for the category name BAMBOO is ZAOZMM.

The established coding rule follows a two-step process: first, find the reverse alphabetical pair for each letter (e.g., A ↔ Z, B ↔ Y). Second, apply a +1 forward shift to that reverse letter. For BAMBOO, the reverse letters are YZNYLL. Adding a +1 shift to each yields ZAOZMM.

Step-by-Step Solution

1

Analyze the transformation of the word ORCHID to MJYTSX.

O (15) → Opposite is L (12) → +1 shift → M (13).
R (18) → Opposite is I (9) → +1 shift → J (10).
C (3) → Opposite is X (24) → +1 shift → Y (25).

This establishes the two-step pattern: replace each letter with its reverse alphabetical counterpart, then shift forward by one position (+1).

2

Apply the first step of the pattern (reverse alphabetical counterpart) to the target word BAMBOO.

B → Y, A → Z, M → N, B → Y, O → L, O → L. The intermediate string is YZNYLL.

The sum of forward and reverse positional values of complementary letters is always 27.

3

Apply the second step of the pattern (+1 forward shift) to the intermediate string YZNYLL.

Y + 1 → Z, Z + 1 → A, N + 1 → O, Y + 1 → Z, L + 1 → M, L + 1 → M. The final string is ZAOZMM.

This completes the identical transformation rule applied to the reference word.

Key Concept

Identifying complex coding patterns involving reverse alphabetical indexing (opposite letters) combined with sequential positional shifts.

Estimated Time:1m 30s

Question 5Question →

An environmental agency reviewed the compliance of $380$ industrial plants across three pollution control standards: Air Emissions ( $A$ ), Water Discharge ( $W$ ), and Solid Waste ( $S$ ). The review revealed the following data:
- $170$ plants met standard $A$ .
- $150$ plants met standard $W$ .
- $160$ plants met standard $S$ .
- $60$ plants met both $A$ and $W$ .
- $55$ plants met both $W$ and $S$ .
- $50$ plants met both $A$ and $S$ .
- $20$ plants failed to meet any of the three standards.

Based on this information, how many industrial plants met EXACTLY TWO of the pollution control standards?

30

45

120

165

Show answer & explanation

Answer: 30

Answer

The number of industrial plants that met exactly two pollution control standards is 30.

The number of plants that met at least one standard is

380 - 20 = 360

. Using the inclusion-exclusion principle for three sets,

360 = 170 + 150 + 160 - 60 - 55 - 50 + N(\text{all three})

, which yields

N(\text{all three}) = 45

. The intersection values given (

60, 55, 50

) include these

45

plants. To isolate the plants meeting *exactly two* standards, we subtract

45

from each intersection and sum the remaining values:

(60 - 45) + (55 - 45) + (50 - 45) = 15 + 10 + 5 = 30

.

Step-by-Step Solution

1

Calculate the total number of plants that met at least one standard.

380 - 20 = 360

To find the union of the three sets, we must subtract the plants that met none of the standards from the total number of surveyed plants.

2

Calculate the number of plants that met all three standards using the inclusion-exclusion principle.

N(A \cap W \cap S) = 45

Substituting the known values into the formula

N(A \cup W \cup S) = N(A) + N(W) + N(S) - N(A \cap W) - N(W \cap S) - N(A \cap S) + N(A \cap W \cap S)

gives

360 = 170 + 150 + 160 - 60 - 55 - 50 + N(A \cap W \cap S)

, which simplifies to

360 = 315 + N(A \cap W \cap S)

.

3

Calculate the number of plants that met exactly two standards.

(60 - 45) + (55 - 45) + (50 - 45) = 15 + 10 + 5 = 30

The 'both' values provided in the problem include the plants that met all three standards. We must subtract the 'all three' value (

45

) from each of the three intersections and sum the results.

Key Concept

Logical Venn Diagrams and the Principle of Inclusion-Exclusion for three sets.

Question 6Question →

In a specific cryptographic system, words are encoded using the following two distinct rules:
1. Consonants are replaced by the letter exactly two positions ahead in the English alphabet (e.g., B becomes D, K becomes M).
2. Vowels (A, E, I, O, U) are replaced by the immediate next vowel in the sequence (e.g., A becomes E, U becomes A).

Based on this coding logic, match the words in List I with their correct encoded forms in List II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are: MAGIC maps to OEOIE, BRAIN maps to DTEOP, FLUTE maps to HNAVI, and POWER maps to RUYIT.

Each word follows the stated logic precisely: consonants skip one letter forward (+2), and vowels transition to the next vowel in the standard sequence (A->E->I->O->U->A). Mapping each letter of the given words using these rules directly produces the coded forms identified in the correct pairings.

Step-by-Step Solution

1

Analyze the coding rules provided in the stem.

Rule 1 applies to Consonants (+2 shift). Rule 2 applies to Vowels (shifted to the next vowel in the A, E, I, O, U sequence).

Identifying the specific transformation for different letter types is required before attempting to encode the words.

2

Apply the coding logic to the word 'MAGIC'.

M (+2) -> O; A (vowel) -> E; G (+2) -> I; I (vowel) -> O; C (+2) -> E. Result: OEOIE.

To find the match for the first item.

3

Apply the coding logic to the word 'BRAIN'.

B (+2) -> D; R (+2) -> T; A (vowel) -> E; I (vowel) -> O; N (+2) -> P. Result: DTEOP.

To find the match for the second item.

4

Apply the coding logic to the word 'FLUTE'.

F (+2) -> H; L (+2) -> N; U (vowel) -> A; T (+2) -> V; E (vowel) -> I. Result: HNAVI.

To find the match for the third item.

5

Apply the coding logic to the word 'POWER'.

P (+2) -> R; O (vowel) -> U; W (+2) -> Y; E (vowel) -> I; R (+2) -> T. Result: RUYIT.

To find the match for the final item.

Key Concept

Mixed Alphanumeric Codes and Letter-Shift Rules

Estimated Time:1m 30s

Question 7Question →

A state health department surveyed $800$ rural clinics regarding the availability of three essential facilities: Telemedicine ( $T$ ), Maternal Care Units ( $M$ ), and $24/7$ Ambulance Services ( $A$ ).

The survey results revealed the following:
- $410$ clinics have Telemedicine.
- $460$ clinics have Maternal Care Units.
- $400$ clinics have $24/7$ Ambulance Services.
- $220$ clinics have both Telemedicine and Maternal Care Units.
- $230$ clinics have both Maternal Care Units and Ambulance Services.
- $210$ clinics have both Telemedicine and Ambulance Services.
- $50$ clinics do not have any of these three facilities.

Based on the data provided, which of the following statements are correct? (Select all that apply)

Select all that apply

Exactly

140

clinics have all three facilities.

Exactly

660

clinics have exactly two of the facilities.

Exactly

370

clinics have exactly one facility.

Exactly

220

clinics have Telemedicine and Maternal Care Units, but do not have Ambulance Services.

Show answer & explanation

Answer: Exactly $140$ clinics have all three facilities.; Exactly $370$ clinics have exactly one facility.

Answer

The correct statements are that exactly 140 clinics have all three facilities, and exactly 370 clinics have exactly one facility.

Based on the Venn diagram calculation, solving for the triple intersection yields exactly 140 clinics. Subsequently calculating the strictly single regions (Telemedicine only, Maternal Care only, and Ambulance only) yields 120, 150, and 100 respectively, which safely sum to exactly 370 clinics having only one facility.

Step-by-Step Solution

1

Determine the total number of clinics that have at least one facility.

Total clinics having at least one facility =

800 - 50 = 750

.

To apply the standard three-set inclusion-exclusion principle, the population must only include those inside the union of the three sets.

2

Calculate the number of clinics having all three facilities (

T \cap M \cap A

).

Apply the formula:

|T \cup M \cup A| = |T| + |M| + |A| - |T \cap M| - |M \cap A| - |T \cap A| + |T \cap M \cap A|

. Substituting the values gives:

750 = 410 + 460 + 400 - 220 - 230 - 210 + |T \cap M \cap A|

. Simplifying this yields

750 = 610 + |T \cap M \cap A|

, meaning

|T \cap M \cap A| = 140

.

This establishes the exact center of the Venn diagram, which is fundamentally required to find the exact values for all other independent regions.

3

Determine the number of clinics having exactly two facilities.

Exactly T and M:

220 - 140 = 80

. Exactly M and A:

230 - 140 = 90

. Exactly T and A:

210 - 140 = 70

. Total exactly two =

80 + 90 + 70 = 240

.

The pairwise intersections given in the problem include the clinics that have all three facilities. We must subtract the triple intersection from each pairwise intersection to avoid double counting.

4

Calculate the total number of clinics having exactly one facility.

Only T =

410 - (80 + 70 + 140) = 120

. Only M =

460 - (80 + 90 + 140) = 150

. Only A =

400 - (70 + 90 + 140) = 100

. Total exactly one =

120 + 150 + 100 = 370

.

To find the strictly single categories, we must subtract the 'exactly two' regions and the 'all three' region from the original total of each designated set.

Key Concept

Inclusion-Exclusion Principle and 3-Set Logical Venn Diagrams

Question 8Question →

A researcher is cataloging environmental samples using a specific alphabetical labeling sequence. The first five sample labels are assigned as follows:

BKR, EJU, IHY, NED, TAJ

If the labeling system continues with the exact same underlying logic, what will be the exact label for the next sample in the sequence?

AVQ

AFQ

ZWP

WZM

Show answer & explanation

Answer: AVQ

Answer

AVQ

The correct sequence combines three distinct rules running in parallel across the three letter positions. The first position moves forward in increasing steps (+3, +4, +5, +6, +7) ending at A. The second position moves backward in increasing steps (-1, -2, -3, -4, -5) ending at V. The third position moves forward in increasing steps (+3, +4, +5, +6, +7) ending at Q. Combining these yields the label AVQ.

Step-by-Step Solution

1

Analyze the pattern of the first letter in each term: B, E, I, N, T.

The positional shifts are: B(+3) → E(+4) → I(+5) → N(+6) → T.

This establishes an increasing forward progression for the first position.

2

Calculate the next letter for the first position.

The next shift must be +7. T (20) + 7 = 27. Since the alphabet has 26 letters, 27 corresponds to A (1). The first letter is A.

Applying the established mathematical pattern to find the missing term.

3

Analyze the pattern of the second letter in each term: K, J, H, E, A.

The positional shifts are: K(-1) → J(-2) → H(-3) → E(-4) → A.

This establishes an increasing backward progression for the second position.

4

Calculate the next letter for the second position.

The next shift must be -5. A (1) - 5 = -4. Wrapping around the alphabet backwards (26 - 4) gives 22, which is V. The second letter is V.

Continuing the reverse alphabet indexing accurately.

5

Analyze the pattern of the third letter in each term: R, U, Y, D, J.

The positional shifts are: R(+3) → U(+4) → Y(+5) → D(+6) → J. (Note: Y(25)+5 = 30; 30-26 = 4 which is D).

This establishes an increasing forward progression for the third position.

6

Calculate the next letter for the third position.

The next shift must be +7. J (10) + 7 = 17. The 17th letter is Q. The third letter is Q.

Completing the final position using the deduced logical rule.

Key Concept

Identifying independent progressive positional shift patterns in alphanumeric coding sequences.

Estimated Time:1m 30s

Question 9Question →

Four delivery vans depart from a central depot. Each driver executes a specific sequence of movements and turns. Match each van's movement sequence with the final direction it is facing.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Van 1 matches with North-West, Van 2 matches with South-West, Van 3 matches with South-East, and Van 4 matches with North-East.

By mapping out each individual vehicle's rotational shifts on a standard compass rose, the exact final orientations correctly pair Van 1 to North-West, Van 2 to South-West, Van 3 to South-East, and Van 4 to North-East.

Step-by-Step Solution

1

Track Van 1's orientation sequentially.

North -> East -> North-West.

North is

0^\circ

. A

90^\circ

clockwise turn results in East (

90^\circ

). A

135^\circ

anti-clockwise turn from East subtracts

135^\circ

, ending at

315^\circ

, which is North-West.

2

Track Van 2's orientation sequentially.

South -> East -> South -> South-West.

South is

180^\circ

. A left turn from South faces East (

90^\circ

). A right turn from East returns to South (

180^\circ

). A

45^\circ

clockwise turn from South adds

45^\circ

, ending at

225^\circ

, which is South-West.

3

Track Van 3's orientation sequentially.

West -> North-East -> South-East.

West is

270^\circ

. A

135^\circ

clockwise turn adds

135^\circ

to reach

405^\circ

(equivalent to

45^\circ

, North-East). A

90^\circ

right turn from North-East adds

90^\circ

, ending at

135^\circ

, which is South-East.

4

Track Van 4's orientation sequentially.

East -> West -> North -> North-East.

East is

90^\circ

. A U-turn flips the direction to West (

270^\circ

). A right turn from West faces North (

0^\circ

). A

45^\circ

clockwise turn from North ends at

45^\circ

, which is North-East.

Key Concept

Angular Rotations and Cardinal Direction Mapping

Question 10Question →

A weather monitoring station encodes its daily operational logs to secure data integrity. In this specific coding system, the word MODULE is written as NQWWOG, and the word SYSTEM is written as HBHGGN.

Based on the logic used in this system, how will the word FORMAT be encoded?

TQHMCF

UQINCG

HLTOZV

HQTOCV

Show answer & explanation

Answer: UQINCG

Answer

The correct answer is UQINCG.

The encoding logic applies two distinct rules based on letter type. Consonants are replaced by their reverse alphabetical equivalent (e.g., A=26, Z=1, so F=6 becomes U=21). Vowels are shifted forward by exactly 2 positions in the alphabet (e.g., O=15 becomes Q=17). Applying this to the target word FORMAT: The consonants F, R, M, and T reverse to U, I, N, and G respectively. The vowels O and A shift forward +2 to become Q and C. Reassembling these gives the encoded word UQINCG.

Step-by-Step Solution

1

Analyze the first example (MODULE -> NQWWOG) to identify distinct patterns for vowels and consonants.

Consonants (M, D, L) become (N, W, O). Vowels (O, U, E) become (Q, W, G).

Recognizing that letters may follow different rules based on their type is a key strategy in coding-decoding.

2

Determine the mathematical shifts for each letter type.

Consonants are replaced by their reverse alphabetical equivalent (M=13, N=14; 13+14=27). Vowels are shifted forward by exactly 2 positions (O=15 to Q=17).

Establishing the exact transformation formulas is required before applying them to a new word.

3

Verify the established logic against the second example (SYSTEM -> HBHGGN).

Consonants S(19)->H(8), Y(25)->B(2), T(20)->G(7), M(13)->N(14) match the reverse rule. Vowel E(5)->G(7) matches the +2 shift rule.

Confirming the pattern ensures the derived logic is consistently applied across the entire system.

4

Apply the verified rules to the target word (FORMAT).

Identify consonants: F, R, M, T. Identify vowels: O, A.

Categorizing the letters of the target word correctly dictates which rule to apply to each letter.

5

Calculate the encoded letters for FORMAT using the formulas.

Consonants: F(6)->U(21), R(18)->I(9), M(13)->N(14), T(20)->G(7). Vowels: O(15)->Q(17), A(1)->C(3). Combining these yields UQINCG.

Executing the mathematical transformations provides the final decoded string.

Key Concept

Differentiating transformation rules between vowels and consonants, and accurately applying reverse alphabetical indexing.

Question 11Question →

Arrange the following stages of the scientific method into a logical sequence, proceeding from the initial foundation to the most universally established outcome.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence is: Observation -> Hypothesis -> Experimentation -> Theory -> Scientific Law.

The correct logical progression builds from raw data to universal truth. First, one makes an Observation, which leads to a testable Hypothesis. This hypothesis is then subjected to Experimentation. Repeated successful experiments elevate the hypothesis to a scientific Theory, which, upon achieving universal fundamental consistency, is recognized as a Scientific Law.

Step-by-Step Solution

1

Identify the foundational starting point of the sequence.

Observation is selected as the first step.

All scientific inquiry begins with noticing a natural phenomenon before any formal explanation can be attempted.

2

Determine the immediate logical consequence of the initial step.

Hypothesis follows Observation.

After observing a phenomenon, an initial, untested explanatory prediction is formed to explain it.

3

Identify how the tentative prediction is validated.

Experimentation follows Hypothesis.

A hypothesis must be tested through physical trials or empirical data analysis to check its validity.

4

Determine the outcome of successful validation.

Theory follows Experimentation.

Once experiments consistently support the hypothesis over time, it is elevated to a comprehensive explanatory framework.

5

Identify the final, universal stage of establishment.

Scientific Law is placed last.

A theory that describes universally accepted and consistent relationships (often mathematically) becomes a scientific law.

Key Concept

Logical Sequence and Classification

Question 12Question →

A city library conducted a reading preferences survey among $600$ of its registered members. The survey focused on three genres: Mystery ( $M$ ), Sci-Fi ( $S$ ), and Biography ( $B$ ). The results showed that $120$ members do not read any of these three genres.

Furthermore, the survey revealed the following:
- $250$ members read Mystery.
- $210$ members read Sci-Fi.
- $180$ members read Biography.
- $70$ members read both Mystery and Biography.
- $60$ members read both Sci-Fi and Biography.
- $40$ members read all three genres.

Based on the survey results, how many members read both Mystery and Sci-Fi, but NOT Biography?

70

110

30

150

Show answer & explanation

Answer: 30

Answer

The correct answer is 30, which is the number of members who read both Mystery and Sci-Fi, but not Biography.

By applying the inclusion-exclusion principle, we find that the total number of readers for both Mystery and Sci-Fi is 70. However, this 70 includes the 40 readers who also read Biography. Since the question specifies readers who do NOT read Biography, we must subtract the 40 who read all three, leaving 30 readers who read exactly Mystery and Sci-Fi.

Step-by-Step Solution

1

Determine the total number of members who read at least one of the three genres.

600 - 120 = 480

members read at least one genre.

The union of all three sets (

|M \cup S \cup B|

) is equal to the total surveyed population minus those who belong to none of the sets.

2

Use the inclusion-exclusion principle for three sets to find the total number of members who read both Mystery and Sci-Fi (

|M \cap S|

).

480 = 250 + 210 + 180 - (|M \cap S| + 70 + 60) + 40

. Simplifying this yields

480 = 550 - |M \cap S|

, which means

|M \cap S| = 70

.

The formula

|M \cup S \cup B| = |M| + |S| + |B| - |M \cap S| - |M \cap B| - |S \cap B| + |M \cap S \cap B|

links all individual totals, pairwise intersections, and the central three-way intersection.

3

Isolate the members who read exactly Mystery and Sci-Fi by excluding those who also read Biography.

70 - 40 = 30

members.

The total intersection

|M \cap S|

includes the members who read all three genres. Subtracting

|M \cap S \cap B|

leaves only those who read exactly Mystery and Sci-Fi.

Key Concept

Applying the inclusion-exclusion principle for three sets to solve for an unknown pairwise intersection, then isolating a specific exclusive region in a logical Venn diagram.

Question 13Question →

An analog $12$ -hour clock is set to the exact correct standard time at 12:00 NOON on Sunday, February 26, 1896. This particular clock consistently gains exactly $4$ minutes every $24$ hours. Assuming standard time follows the Gregorian calendar without any daylight saving adjustments, what time will this faulty clock display, and what will be the true day of the week, when exactly $8$ standard years have passed (i.e., on February 26, 1904 at 12:00 NOON standard time)?

2:44 PM, Tuesday

2:48 PM, Wednesday

2:52 PM, Thursday

2:40 PM, Monday

Show answer & explanation

Answer: 2:44 PM, Tuesday

Answer

The faulty clock will display 2:44 PM, and the true day of the week will be Tuesday.

Between February 26, 1896, and February 26, 1904, there is exactly one leap day crossed (February 29, 1896). The year 1900 is not a leap year due to the century rule, and the period ends before February 29, 1904. This results in precisely

2921

elapsed days. Adding

2

odd days (

2921 \pmod 7

) to Sunday determines the true day is Tuesday. The clock gains

11684

minutes (

2921 \times 4

), which translates to

16

full

12

-hour cycles plus an extra

164

minutes (

2

hours and

44

minutes). Advancing 12:00 NOON by this remainder yields exactly 2:44 PM.

Step-by-Step Solution

1

Calculate the total number of true days elapsed between February 26, 1896, and February 26, 1904.

Identify that the period covers exactly 8 years, but requires careful evaluation of leap days.

Calendar and clock drift problems require the exact number of 24-hour periods that have passed.

2

Determine how many leap days (February 29ths) fall inside this exact date range.

Exactly 1 leap day is included (February 29, 1896).

1896 is a leap year and its leap day occurs after Feb 26. 1900 is a century year not divisible by 400, so it is NOT a leap year. 1904 is a leap year, but the period ends on Feb 26, before its leap day occurs.

3

Compute the total elapsed days.

8 \text{ years} \times 365 \text{ days} + 1 \text{ leap day} = 2921 \text{ days}

.

This establishes the precise duration for both the day-of-week shift and the total minutes gained by the clock.

4

Calculate the true day of the week.

Tuesday.

2921 \pmod 7 = 2

odd days. Adding 2 days to the starting day (Sunday) yields Tuesday.

5

Calculate the total time gained by the faulty clock.

2921 \text{ days} \times 4 \text{ minutes/day} = 11684 \text{ minutes}

.

The clock continuously gains 4 minutes every 24 hours over the exact elapsed duration.

6

Determine the time displayed on the 12-hour analog dial.

2:44 PM.

A 12-hour clock resets every 720 minutes.

11684 \pmod{720} = 164

minutes.

164

minutes is exactly 2 hours and 44 minutes. Adding this shift to 12:00 NOON gives 2:44 PM.

Key Concept

Integration of Calendar Leap Exceptions with Continuous Clock Drift

Estimated Time:4m 0s

Question 14Question →

Find the next term in the following number sequence:

3, 5, 8, 13, 22, 39, 72, \dots

Show answer & explanation

Answer: 137

Answer

The next number in the given series is 137.

Computing second-level differences between terms reveals a doubling sequence (

1, 2, 4, 8, 16

). The next second-level difference is

32

. Adding

32

to the last first-level difference (

33

) yields

65

. Finally, adding

65

to the last term (

72

) gives the correct value of

137

.

Step-by-Step Solution

1

Compute the first-order differences between consecutive terms in the series.

The first-order differences are

2, 3, 5, 9, 17, 33

.

Analyzing differences helps determine whether the growth rate follows an arithmetic, geometric, or polynomial rule.

2

Compute the second-order differences of the first-order difference sequence.

The second-order differences are

1, 2, 4, 8, 16

.

Because the first-order differences do not show an obvious linear pattern, evaluating higher-order differences is necessary.

3

Identify the underlying rule governing the second-order differences.

The second-order sequence doubles with each step (

2^0, 2^1, 2^2, 2^3, 2^4, \dots

). The next second-order difference is

16 \times 2 = 32

.

Recognizing geometric doubling (

2^n

) allows for projecting the next step accurately.

4

Reconstruct the next first-order difference and calculate the required next sequence value.

Next first-order difference

= 33 + 32 = 65

. Next series term

= 72 + 65 = 137

.

Working backward through the difference layers yields the exact value of the next term.

Key Concept

Double-Difference Series with Geometric Progression

Estimated Time:2m 0s

Question 15Question →

Consider the following numerical series:

4, 11, 30, 85, 248, \dots

What is the numerical value of the next term in this series?

Show answer & explanation

Answer: 735

Answer

735

Each term is generated by multiplying the preceding term by 3 and subtracting consecutive odd numbers starting from 1 (

1, 3, 5, 7, 9

). Therefore, the term following 248 is computed as

(248 \times 3) - 9 = 735

.

Step-by-Step Solution

1

Examine the multiplier and subtractor progression between consecutive terms.

Multiplier is constantly 3, while subtractors follow the odd number sequence

1, 3, 5, 7, \dots

Establishing a standard recurrence rule

T_n = 3 \cdot T_{n-1} - (2n - 3)

for

n \ge 2

.

2

Calculate the sixth term using the derived rule.

T_6 = (248 \times 3) - 9 = 744 - 9 = 735

The fifth term is 248, and the fifth odd integer to subtract is 9.

Key Concept

Mixed operation numerical series involving constant multiplication and arithmetic progression of subtractors

Estimated Time:2m 0s

Question 16Question →

In the following number sequence, a specific mathematical pattern is followed:

7, 14, 25, 42, 67, ?

What is the value of the missing term represented by the question mark?

Show answer & explanation

Answer: 102

Answer

The missing term in the series is 102.

The correct answer is 102. The sequence is governed by a double difference pattern. The consecutive differences between terms are 7, 11, 17, and 25. The differences of these differences are 4, 6, and 8, which increase by 2 at each step. Thus, the next second difference is 10, the next term difference is 35, and the next term in the sequence is 67 + 35 = 102.

Step-by-Step Solution

1

Find the first-order differences between consecutive terms.

The differences are 7, 11, 17, and 25.

Analyzing first differences helps determine if the growth rate follows a recognizable secondary pattern.

2

Find the second-order differences.

The differences between the consecutive first differences are 4, 6, and 8.

Since the first differences are non-linear, second-order differences reveal the underlying constant acceleration (+2).

3

Project the next first-order difference using the second-order pattern.

The next second-order difference is 10 (since 8 + 2 = 10), making the next first-order difference 25 + 10 = 35.

Extrapolating the arithmetic pattern of the second differences yields the correct next increment.

4

Add the projected difference to the last term of the sequence.

67 + 35 = 102.

Applying the calculated increment to the last term completes the series.

Key Concept

Second-order arithmetic sequence (Double Difference Series)

Question 17Question →

In the given alphanumeric sequence, which term logically comes next to complete the series: $A2, D4, G8, J16, \dots$ ?

M32

L32

M30

N32

Show answer & explanation

Answer: $M32$

Answer

M32

The correct answer combines the letter 'M' (obtained by adding 3 forward steps from 'J' in the English alphabet) and the number 32 (obtained by multiplying 16 by 2).

Step-by-Step Solution

1

Analyze the letter progression pattern

The letters follow a pattern of adding 3 positions:

A (1) + 3 = D (4)

,

D (4) + 3 = G (7)

,

G (7) + 3 = J (10)

, and

J (10) + 3 = M (13)

.

Identify the rule governing the alphabetical component of the sequence.

2

Analyze the number progression pattern

The numbers follow a pattern of doubling:

2 \times 2 = 4

,

4 \times 2 = 8

,

8 \times 2 = 16

, and

16 \times 2 = 32

.

Identify the rule governing the numerical component of the sequence.

3

Combine the letter and number terms to find the next element

Combining the 13th letter (

M

) and the calculated number (

32

) gives

M32

.

Formulate the final term of the series.

Key Concept

Alphanumeric Series Completion

Question 18Question →

Complete the given number series by determining the missing term that logically fits the sequence.

Fill in the blanks below

In the numerical series $6, 13, 27, 48, 76, $, the value of the missing term is .

Show answer & explanation

Answer

The missing term in the series is 111.

The differences between successive terms form an increasing sequence of multiples of 7:

+7, +14, +21, +28, \dots

The next difference must be

+35

. Adding

35

to

76

yields

111

.

Step-by-Step Solution

1

Calculate the first-order differences between consecutive terms in the series.

13 - 6 = 7

,

27 - 13 = 14

,

48 - 27 = 21

,

76 - 48 = 28

To identify the underlying pattern governing the progression of the sequence.

2

Analyze the sequence of differences.

The differences are

7, 14, 21, 28

, which form an arithmetic progression with a common difference of

7

(multiples of

7

).

To determine the next required addition to the sequence.

3

Determine the next difference and compute the missing term.

The next difference is

28 + 7 = 35

. Adding this to the last term gives

76 + 35 = 111

.

Applying the identified pattern yields the correct missing term.

Key Concept

Double Difference / Arithmetic Progression of Differences

Estimated Time:1m 0s

Question 19Question →

Determine the underlying double-pattern rule governing the alphanumeric series below and find the term that logically completes it.

Fill in the blanks below

In the sequence

Z12, X25, U52, Q107, L218, \dots

, the missing term that should replace the blank is .

Show answer & explanation

Answer

The missing term in the sequence is F441.

The sequence consists of two distinct intertwined logic rules: the letter component decreases positional value by progressive increments (-2, -3, -4, -5, -6), turning L (12th letter) into F (6th letter). The numerical component follows the recurrence formula

T_{n} = 2 \times T_{n-1} + (n-1)

, where the additive constant increases by 1 each step (+1, +2, +3, +4, +5), yielding

218 \times 2 + 5 = 441

. Combining both gives F441.

Step-by-Step Solution

1

Analyze the alphabetical component by converting letters to their corresponding numerical positions in the alphabet (A=1, B=2, ..., Z=26).

The letter sequence corresponds to positions: Z (26), X (24), U (21), Q (17), L (12).

Tracking positional values helps identify decremental positional shifts.

2

Determine the difference pattern between consecutive letter positions.

26 - 2 = 24 (X); 24 - 3 = 21 (U); 21 - 4 = 17 (Q); 17 - 5 = 12 (L). The positional shift decreases by consecutive integers (-2, -3, -4, -5).

The next shift must be -6, giving position 12 - 6 = 6, which corresponds to the letter F.

3

Analyze the numerical component of the series: 12, 25, 52, 107, 218.

12 × 2 + 1 = 25; 25 × 2 + 2 = 52; 52 × 2 + 3 = 107; 107 × 2 + 4 = 218.

The numerical pattern multiplies the preceding term by 2 and adds an incrementing integer starting from +1.

4

Calculate the next numerical term using the derived recurrence rule.

218 × 2 + 5 = 436 + 5 = 441.

Applying the next multiplier and additive step (×2 + 5) yields 441.

5

Combine the letter and number components.

The completed missing term is F441.

Combining the 6th letter 'F' and calculated number '441' forms the full missing term.

Key Concept

Interleaved positional decrements combined with linear multiplicative-additive recurrence relation.

Estimated Time:2m 0s

Question 20Question →

Determine the missing term that logically completes the given alphanumeric series.

Fill in the blanks below

Consider the following multi-pattern sequence: $$A0, D6, I24, P60, $$ What term should replace the blank to complete the series?

Show answer & explanation

Answer

The term that completes the series is Y120.

The sequence operates on two synchronized rules: the letter in the

n

-th position corresponds to the

n^2

-th letter of the alphabet (

1^2=1 \rightarrow A

,

2^2=4 \rightarrow D

,

3^2=9 \rightarrow I

,

4^2=16 \rightarrow P

,

5^2=25 \rightarrow Y

), while the numerical portion follows the cubic sequence

n^3 - n

(

1^3-1=0

,

2^3-2=6

,

3^3-3=24

,

4^3-4=60

,

5^3-5=120

). Combining both results in Y120.

Step-by-Step Solution

1

Analyze the pattern governing the letter component (

A, D, I, P, \dots

).

Alphabetical positions are

A=1, D=4, I=9, P=16

, which follow perfect squares

1^2, 2^2, 3^2, 4^2

.

Determining positional index patterns establishes the letter rule for the

n

-th term.

2

Determine the letter for the 5th term (

n=5

).

The position is

5^2 = 25

, corresponding to the letter

Y

.

Applying the identified square positional rule yields the 25th alphabet letter.

3

Analyze the pattern governing the numerical component (

0, 6, 24, 60, \dots

).

The numerical terms follow the polynomial formula

n^3 - n

for term indices

n = 1, 2, 3, 4

.

Evaluating differences (

6, 18, 36\dots

) reveals a cubic relation

n^3 - n

(

1^3-1=0, 2^3-2=6, 3^3-3=24, 4^3-4=60

).

4

Calculate the numerical value for

n=5

.

5^3 - 5 = 125 - 5 = 120

.

Substituting

n=5

into the cubic polynomial gives the 5th numerical term.

5

Combine the letter and numerical components.

The completed term is Y120.

Joining the 5th letter

Y

with the 5th numerical value

120

provides the final answer.

Key Concept

Alphanumeric Series Integration with Perfect Square Letter Indexing and Cubic Polynomial Sequences

General Mental Ability

Answer

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