General Mental Ability Alıştırma Soruları — State PSC Exam — Sayfa 13

Soru 241Soru →

In the Gregorian calendar, a standard year has $365$ days, whereas a leap year has $366$ days. Based on the rules for determining leap years, which of the following years had exactly $366$ days?

1700

1800

1900

2000

Cevabı ve açıklamayı göster

Cevap: $2000$

Cevap

The year

2000

had exactly

366

days because it is a century year perfectly divisible by

400

.

The correct answer is

2000

. All the given options are century years (ending in

00

). According to the Gregorian calendar, a century year is a leap year with

366

days only if it is exactly divisible by

400

. Since

2000 \div 400 = 5

, it is a leap year.

Adım Adım Çözüm

1

Recall the criteria for determining a leap year in the Gregorian calendar.

A general year is a leap year if it is divisible by

4

. However, century years (ending in

00

) must be divisible by

400

to be considered leap years.

This special rule for centuries corrects the slight overestimation of the solar year caused by adding a leap day every

4

years.

2

Evaluate each given option using the century year rule.

Since

1700

,

1800

,

1900

, and

2000

all end in

00

, divide each by

400

. Only

2000

is perfectly divisible by

400

(

2000 \div 400 = 5

).

Applying the correct divisibility rule isolated the only year that contains an extra day in February.

Anahtar Kavram

Identifying leap years across century boundaries.

Soru 242Soru →

An automated transit gate authenticates employee badges by generating a daily encrypted code based on the employee's assigned sector. When an employee's sector is WINTER, the authentication code generated is ZKOWGS.

Applying the identical logical sequence, what authentication code will be generated if an employee's sector is SPRING?

Cevabı ve açıklamayı göster

Cevap: VRSLPH; vrslph

Cevap

VRSLPH

The system encrypts words by applying a repeating forward alphabetical shift of +3, +2, and +1 to consecutive letters. Applying this sequence to SPRING (S+3, P+2, R+1, I+3, N+2, G+1) results in VRSLPH.

Adım Adım Çözüm

1

Determine the numerical alphabetical positions of the letters in the reference word 'WINTER' and its code 'ZKOWGS'.

W=23, I=9, N=14, T=20, E=5, R=18. Z=26, K=11, O=15, W=23, G=7, S=19.

Converting letters to their numerical positions allows us to mathematically determine the shift pattern.

2

Calculate the difference between the numerical positions of corresponding letters.

26-23 = +3; 11-9 = +2; 15-14 = +1; 23-20 = +3; 7-5 = +2; 19-18 = +1.

This reveals the underlying logical pattern: a repeating sequence of +3, +2, +1 shifts.

3

Identify the numerical positions of the letters in the target word 'SPRING'.

S=19, P=16, R=18, I=9, N=14, G=7.

These are the base values to which the shift pattern must be applied.

4

Apply the +3, +2, +1 repeating shift sequence to the numerical positions of 'SPRING' and convert back to letters.

19+3=22(V); 16+2=18(R); 18+1=19(S); 9+3=12(L); 14+2=16(P); 7+1=8(H). The result is VRSLPH.

Applying the exact same logical transformation rule yields the final encrypted code.

Anahtar Kavram

Identifying and applying sequential mathematical shifts to alphabetical characters.

Soru 243Soru →

A demographic research algorithm maps family lineages using the following logical operators:

• $X \nabla Y$ means $X$ is the legally recognized spouse of $Y$ .
• $X \Delta Y$ means $X$ is the biological mother of $Y$ .
• $X \Omega Y$ means $X$ is the sibling of $Y$ .
• $X \Sigma Y$ means $X$ is the child of $Y$ .

Based strictly on the relational sequence $H \Sigma G \nabla F \Delta E \Omega D$ , which of the following conclusions is logically definitive?

H is the brother of E.

E is the sister of H.

G is the father of D.

D is the son of F.

Cevabı ve açıklamayı göster

Cevap: G is the father of D.

Cevap

G is the father of D.

The sequence establishes that F is the mother of E, and E is the sibling of D, making F the mother of D as well. Because G is the spouse of F, G is definitively male and thus the father of F's children. Consequently, G is definitively the father of D.

Adım Adım Çözüm

1

Analyze the relationship

F \Delta E

and

E \Omega D

.

F

is the mother of

E

, and

E

is the sibling of

D

. Therefore,

F

is also the mother of

D

.

To establish the maternal lineage for the right side of the sequence.

2

Analyze the relationship

G \nabla F

.

G

is the spouse of

F

. Since

F

is a mother (female),

G

must be male.

To determine the gender and parental role of

G

.

3

Combine the findings for

G

,

F

, and

D

.

G

is the husband of

F

, and

F

is the mother of

D

. Therefore,

G

is definitively the father of

D

.

To identify the exact relationship between the male spouse and the children.

4

Evaluate the genders of

H

,

E

, and

D

.

The operators

\Sigma

(child) and

\Omega

(sibling) do not assign a gender. Thus, the exact genders of

H

,

E

, and

D

remain unknown.

To eliminate any conclusions that make assumptions about unstated genders.

Anahtar Kavram

Deducing exact kinships and avoiding gender assumptions in coded relationships.

Soru 244Soru →

Shortly after sunrise, a wildlife photographer begins tracking an animal. She starts walking in a straight line, noticing that her shadow falls exactly to her left. After walking $40\text{ m}$ , she turns $135^\circ$ to her right and walks $30\sqrt{2}\text{ m}$ . Finally, she turns $90^\circ$ to her right and walks another $10\sqrt{2}\text{ m}$ to reach the animal. What is her shortest distance and direction from her starting point?

20\text{ m}

towards East

20\text{ m}

towards West

20\sqrt{5}\text{ m}

towards North-East

20\text{ m}

towards South

Cevabı ve açıklamayı göster

Cevap: $20\text{ m}$ towards East

Cevap

The photographer is

20\text{ m}

towards East from her starting point.

Based on sunrise shadow rules, the initial direction is North. Walking

40\text{ m}

North to

(0, 40)

, turning

135^\circ

right to travel

30\sqrt{2}\text{ m}

South-East to

(30, 10)

, and finally turning

90^\circ

right to travel

10\sqrt{2}\text{ m}

South-West results in a final coordinate of

(20, 0)

. This point is exactly

20\text{ m}

East of the starting point.

Adım Adım Çözüm

1

Determine the initial facing direction using the position of the sun and the shadow.

The photographer is initially facing North.

At sunrise, the sun is in the East, so light travels West and shadows are cast to the West. Since her shadow falls to her left, her left side points West. Therefore, she must be facing North.

2

Plot the first movement on a coordinate plane.

Her position becomes

(0, 40)

.

Starting at the origin

(0,0)

, walking

40\text{ m}

North translates to an upward movement of

+40

on the y-axis.

3

Calculate the second movement after a

135^\circ

right turn.

Her new position is

(30, 10)

.

Turning

135^\circ

to the right (clockwise) from North puts her facing South-East. Moving

30\sqrt{2}\text{ m}

South-East yields a displacement of

\Delta x = 30\sqrt{2} \times \cos(-45^\circ) = 30

and

\Delta y = 30\sqrt{2} \times \sin(-45^\circ) = -30

. Adding this to

(0, 40)

gives

(30, 10)

.

4

Calculate the final movement after a

90^\circ

right turn.

Her final position is

(20, 0)

.

Turning

90^\circ

to the right from South-East puts her facing South-West. Moving

10\sqrt{2}\text{ m}

South-West yields a displacement of

\Delta x = -10

and

\Delta y = -10

. Adding this to

(30, 10)

gives

(20, 0)

.

5

Determine the final distance and direction from the origin.

The shortest distance is

20\text{ m}

, and the direction is East.

The final coordinate is

(20, 0)

. The distance from

(0,0)

is

\sqrt{20^2 + 0^2} = 20\text{ m}

. Since it lies on the positive x-axis, the direction is East.

Anahtar Kavram

Vector displacement and cardinal alignment using sun-shadow rules.

Soru 245Soru →

Six colleagues—Priya, Rahul, Sara, Tarun, Uma, and Varun—are seated in a single row facing North for a group photograph. They are seated according to the following conditions:

1. Priya sits at one of the extreme ends of the row.
2. Rahul sits immediately to the right of Priya.
3. There are exactly two persons sitting between Rahul and Uma.
4. Sara sits immediately to the left of Uma.
5. Tarun sits immediately to the right of Rahul.

Based on the seating arrangement, which of the following statements are true?

Geçerli olan tümünü seçin

Tarun is sitting exactly between Rahul and Sara.

Varun is sitting at the extreme right end of the row.

Sara is sitting second to the left of Rahul.

Varun is sitting immediately to the left of Uma.

Cevabı ve açıklamayı göster

Cevap: Tarun is sitting exactly between Rahul and Sara.; Varun is sitting at the extreme right end of the row.

Cevap

The correct statements are that Tarun is sitting exactly between Rahul and Sara, and Varun is sitting at the extreme right end of the row.

Based on the step-by-step logical deduction, the complete sequence from left to right is: Priya, Rahul, Tarun, Sara, Uma, Varun. Looking at this sequence, Tarun is positioned exactly in the middle of Rahul and Sara. Furthermore, Varun is indeed occupying the last seat on the extreme right.

Adım Adım Çözüm

1

Determine the positions of Priya and Rahul.

Priya is at position 1 (extreme left) and Rahul is at position 2.

Priya is at an extreme end. Since they face North, if she were at the right end, nobody could sit to her right. Because Rahul is to her immediate right, Priya must be at the extreme left.

2

Determine the position of Uma.

Uma is at position 5.

There are exactly two people sitting between Rahul (position 2) and Uma. Moving to the right, position 2 + 3 = position 5.

3

Determine the positions of Sara and Tarun.

Sara is at position 4 and Tarun is at position 3.

Sara is immediately left of Uma (position 5 - 1 = 4). Tarun is immediately right of Rahul (position 2 + 1 = 3).

4

Determine the position of Varun.

Varun is at position 6 (extreme right).

Varun is the only colleague left, and position 6 is the only remaining empty seat.

Anahtar Kavram

Linear seating arrangement with fixed positions and relative direction analysis.

Tahmini Süre:1m 0s

Soru 246Soru →

A municipal transport authority conducted a survey of $500$ daily commuters to analyze their usage of three public transport modes: Bus, Metro, and Train. The survey revealed the following data:
- $210$ commuters use the Bus.
- $190$ commuters use the Metro.
- $160$ commuters use the Train.
- $70$ commuters use exactly two of these transport modes.
- $20$ commuters use all three transport modes.

How many commuters in the surveyed group use none of these three transport modes?

30

50

70

140

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The number of commuters using none of the three transport modes is 50.

The correct answer is found by determining the total number of commuters using at least one transport mode (the union). The sum of the individual sets (

210 + 190 + 160 = 560

) counts people using exactly one mode once, exactly two modes twice, and exactly three modes three times. Therefore,

560 = (\text{Exactly One}) + 2 \times (\text{Exactly Two}) + 3 \times (\text{Exactly Three})

. Substituting the given values gives

560 = (\text{Exactly One}) + 2(70) + 3(20)

, which simplifies to

560 = (\text{Exactly One}) + 140 + 60

. Thus, Exactly One equals

360

. The total union is Exactly One

+

Exactly Two

+

Exactly Three, which is

360 + 70 + 20 = 450

. Finally, the number of commuters using none of the modes is the total surveyed minus the union:

500 - 450 = 50

.

Adım Adım Çözüm

1

Calculate the sum of commuters in the individual transport mode categories.

210 + 190 + 160 = 560

.

This establishes the gross total before adjusting for overlaps, where multi-mode commuters are counted multiple times.

2

Set up the inclusion-exclusion relationship for disjoint regions.

560 = n(\text{Exactly 1}) + 2 \times n(\text{Exactly 2}) + 3 \times n(\text{Exactly 3})

.

When summing the individual sets, commuters using exactly two modes are counted twice, and those using all three are counted three times.

3

Solve for the number of commuters using exactly one mode.

560 = n(\text{Exactly 1}) + 2(70) + 3(20) \Rightarrow 560 = n(\text{Exactly 1}) + 200 \Rightarrow n(\text{Exactly 1}) = 360

.

We need the 'exactly one' count to piece together the total union of commuters using at least one mode.

4

Calculate the total number of commuters using at least one mode (the union).

\text{Union} = 360 + 70 + 20 = 450

commuters.

The union is simply the sum of the disjoint regions: exactly one, exactly two, and exactly three.

5

Subtract the union from the total surveyed commuters to find those using none.

500 - 450 = 50

commuters.

Commuters outside the union represent those who use zero of the surveyed transport modes.

Anahtar Kavram

Solving 3-set Venn diagrams using disjoint region equations rather than standard intersection formulas.

Soru 247Soru →

In a covert military operation, communication officers use a specific cryptographic algorithm to secure tactical locations. If the location BRIDGE is encoded as VTWRIY, which of the following represents the code for the location HARBOR?

ILYIZS

SZIYLI

HKXHYR

HLYHZS

Cevabı ve açıklamayı göster

Cevap: ILYIZS

Cevap

The cryptographic code for HARBOR is ILYIZS.

The correct answer accurately applies both steps of the established encryption algorithm: first replacing each letter in HARBOR with its alphabetical opposite (yielding SZIYLI), and then completely reversing the resulting string to arrive at ILYIZS.

Adım Adım Çözüm

1

Determine the transformation rule applied to the letters of the given word BRIDGE.

Each letter is replaced by its opposite in the alphabet (Reverse Alphabetical Index, where A=Z, B=Y, etc.). B(2) becomes Y(25), R(18) becomes I(9), I(9) becomes R(18), D(4) becomes W(23), G(7) becomes T(20), and E(5) becomes V(22). The resulting sequence is YIRWTV.

Establishing the initial letter substitution mapping is the first step in decoding the algorithm.

2

Compare the sequence of opposite letters to the final encrypted code.

The intermediate sequence YIRWTV is reversed to become VTWRIY, matching the given code.

Identifying any secondary transformations (like string reversal) ensures the complete algorithm is understood.

3

Apply the reverse alphabetical index to the target word HARBOR.

H(8) becomes S(19), A(1) becomes Z(26), R(18) becomes I(9), B(2) becomes Y(25), O(15) becomes L(12), and R(18) becomes I(9). The intermediate sequence is SZIYLI.

This executes the first half of the decoded algorithmic rule on the required word.

4

Apply the string reversal to the intermediate sequence.

Reversing SZIYLI results in ILYIZS.

This executes the final step of the algorithm to obtain the exact encrypted code.

Anahtar Kavram

Reverse Alphabetical Indexing and String Reversal

Soru 248Soru →

If today is Wednesday, what day of the week will it be after exactly $45$ days?

Cevabı ve açıklamayı göster

Cevap: Saturday; saturday

Cevap

Saturday

Because a week consists of

7

days, the day of the week repeats every

7

days. To find the day after

45

days, divide

45

by

7

, which yields a quotient of

6

and a remainder of

3

. This means exactly

6

full weeks and

3

additional days will pass. The

6

full weeks return the day to Wednesday. Counting forward

3

extra days from Wednesday results in Saturday.

Adım Adım Çözüm

1

Calculate the number of odd days by finding the remainder when 45 is divided by 7.

45 \div 7 = 6

with a remainder of

3

.

The days of the week repeat every 7 days, meaning full weeks do not change the day. We only need the remainder (odd days).

2

Count forward by the number of odd days from the given day.

Wednesday +

3

days = Saturday.

Moving

3

days forward from Wednesday sequentially gives Thursday, Friday, and Saturday.

Anahtar Kavram

Calculating future days of the week using odd days (modulo 7 arithmetic).

Soru 249Soru →

Four tourists (Alex, Blake, Casey, and Drew) start exploring a flat city from the exact same central plaza. Match each tourist's movement sequence with their final direction relative to the starting point.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Alex is Exactly East, Blake is South-East, Casey is Exactly West, and Drew is North-East of the starting point.

By placing the starting point at the origin (0,0) of a Cartesian plane, we can translate every movement into (x,y) coordinate changes. Alex ends at (40, 0), indicating a positive x-axis position (East). Blake ends at (30, -30), indicating a positive x and negative y position (South-East). Casey ends at (-60, 0), indicating a negative x-axis position (West). Drew ends at (40, 40), indicating a positive x and positive y position (North-East).

Adım Adım Çözüm

1

Map Alex's path on a coordinate plane starting at (0,0).

Moves

40\text{ m}

North to

(0, 40)

. A

135^\circ

clockwise turn from North faces South-East. Moving

40\sqrt{2}\text{ m}

SE results in

\Delta x = 40

and

\Delta y = -40

. Final position:

(40, 0)

.

Vector decomposition (using Pythagoras/trigonometry on

45^\circ

right triangles) accurately tracks diagonal displacement.

2

Map Blake's path starting at (0,0).

Moves West

30\text{ m}

to

(-30, 0)

. Turning

90^\circ

left (faces South) and moving

30\text{ m}

gives

(-30, -30)

. Turning left again (faces East) and moving

60\text{ m}

yields

(-30 + 60, -30) = (30, -30)

.

Step-by-step orthogonal tracking determines the final Cartesian coordinates.

3

Map Casey's path starting at (0,0).

Moves South

20\text{ m}

to

(0, -20)

. Turning

90^\circ

right (faces West) and moving

40\text{ m}

gives

(-40, -20)

. A

45^\circ

clockwise turn from West faces North-West. Moving

20\sqrt{2}\text{ m}

NW gives

\Delta x = -20

,

\Delta y = +20

. Final position:

(-40 - 20, -20 + 20) = (-60, 0)

.

Translating relative turns (right, clockwise) into cardinal headings ensures correct vector application.

4

Map Drew's path starting at (0,0).

Moves North

10\text{ m}

to

(0, 10)

. Turning

90^\circ

right (faces East) and moving

40\text{ m}

gives

(40, 10)

. Turning

90^\circ

left (faces North) and moving

30\text{ m}

yields

(40, 10 + 30) = (40, 40)

.

Summing the

x

and

y

vectors gives the exact final location in the first quadrant.

Anahtar Kavram

Direction and Distance Vector Mapping

Soru 250Soru →

A cryptographer is analyzing intercepted transmissions and has successfully aligned three coded messages with their English translations. Note that the order of words in the coded messages does not necessarily correspond to the order of words in the English translations.

1. 'alpha delta echo' translates to 'secure the perimeter'
2. 'echo bravo charlie' translates to 'the main target'
3. 'alpha charlie foxtrot' translates to 'secure main exit'

Based on these translations, match each English word with its corresponding coded term.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches are: 'secure' corresponds to 'alpha', 'the' corresponds to 'echo', 'perimeter' corresponds to 'delta', and 'main' corresponds to 'charlie'.

By cross-referencing phrases that share common words, we isolate specific codes. 'secure' is shared by phrases 1 and 3 (yielding 'alpha'). 'the' is shared by phrases 1 and 2 (yielding 'echo'). 'main' is shared by phrases 2 and 3 (yielding 'charlie'). Consequently, 'perimeter' in phrase 1 must be the remaining code 'delta'.

Adım Adım Çözüm

1

Analyze the first and second phrases for common words and codes.

Common English word: 'the'. Common coded term: 'echo'.

Finding elements common to two sets is the primary deductive method for deciphering substitution matrices.

2

Analyze the first and third phrases for common words and codes.

Common English word: 'secure'. Common coded term: 'alpha'.

This establishes another direct translation pair by isolating shared terms.

3

Use the known translations to deduce the remaining words in the first phrase.

Since 'secure' is 'alpha' and 'the' is 'echo', the final word 'perimeter' must map to the remaining code 'delta'.

Process of elimination within a fully mapped phrase confirms the final mapping.

4

Analyze the second and third phrases for common words and codes.

Common English word: 'main'. Common coded term: 'charlie'.

This confirms the translation for 'main' and completes the required matching.

Anahtar Kavram

Sentence Substitution Matrices

Tahmini Süre:1m 0s

Soru 251Soru →

In a straight row of cadets facing North, Vikram is positioned $13^{\text{th}}$ from the left end and Rahul is positioned $31^{\text{st}}$ from the right end. If they interchange their positions, Vikram becomes $39^{\text{th}}$ from the left end. Another cadet, Sanjay, is standing exactly in the middle of Vikram and Rahul's new positions. What is Sanjay's position from the right end?

44^{\text{th}}

26^{\text{th}}

43^{\text{rd}}

45^{\text{th}}

Cevabı ve açıklamayı göster

Cevap: $44^{\text{th}}$

Cevap

Sanjay's position is

44^{\text{th}}

from the right end.

The correct answer accurately determines the total number of cadets as

69

by combining the left and right ranks of the intersection seat. It identifies Sanjay's position as

26^{\text{th}}

from the left by averaging

13

and

39

, and finally correctly converts this to the

44^{\text{th}}

position from the right using the formula

N - L + 1

.

Adım Adım Çözüm

1

Calculate the total number of cadets in the row.

Total =

39 + 31 - 1 = 69

cadets.

After the swap, Vikram is

39^{\text{th}}

from the left. This physical seat is the same as Rahul's original seat, which was

31^{\text{st}}

from the right. The formula for total is

L + R - 1

.

2

Determine the new positions of Vikram and Rahul from the left end.

Vikram's new position =

39^{\text{th}}

from the left. Rahul's new position =

13^{\text{th}}

from the left.

Since they swapped, Rahul occupies Vikram's original seat, which is

13^{\text{th}}

from the left.

3

Calculate Sanjay's position from the left end.

Sanjay's position =

(13 + 39) \div 2 = 26^{\text{th}}

from the left.

Sanjay is exactly in the middle of

13

and

39

, so we average the two positions.

4

Convert Sanjay's position from the left end to the right end.

Sanjay's position from the right =

69 - 26 + 1 = 44^{\text{th}}

.

The formula to find the rank from the opposite end is

\text{Total} - \text{Current Rank} + 1

.

Anahtar Kavram

Position Interchange and Midpoint Calculations

Soru 252Soru →

During a cybersecurity drill, passwords are obfuscated using a specific two-step alphabetic logic. If the system transforms the access word TABLE into HAZPW, what will the access word CHAIR be transformed into under the same logic?

YTASJ

XSZRI

DIBJS

WRYQH

Cevabı ve açıklamayı göster

Cevap: YTASJ

Cevap

The correct transformation for the word CHAIR is YTASJ.

The encryption logic applies a reverse alphabetical index to each letter (where A=Z, B=Y, etc.) and then shifts that result forward by one letter (+1). Applying this to CHAIR gives the intermediate letters XSZRI, which then shift +1 to become YTASJ.

Adım Adım Çözüm

1

Analyze the transformation from TABLE to HAZPW to identify the first logical step.

The reverse alphabetical equivalents for T-A-B-L-E are G-Z-Y-O-V.

Finding the foundational pattern requires testing common cryptographic shifts, starting with opposite letters.

2

Determine the second logical step by comparing the intermediate reversed letters to the final code.

Shifting G-Z-Y-O-V forward by one letter (+1) in the alphabet results in H-A-Z-P-W.

This confirms the two-step logic: reverse the letter, then shift +1.

3

Apply the first step (reverse alphabet) to the target word CHAIR.

The reverse alphabetical equivalents for C-H-A-I-R are X-S-Z-R-I.

To begin encrypting the new access word according to the established rule.

4

Apply the second step (+1 shift) to the intermediate result.

Shifting X-S-Z-R-I forward by one letter (+1) results in Y-T-A-S-J.

To complete the obfuscation logic and determine the final coded sequence.

Anahtar Kavram

Reverse Alphabetical Indexing and Positional Shifting

Soru 253Soru →

An archivist intends to perfectly reuse a daily calendar planner originally printed for the year $2096$ . In order for every single date in the planner to fall on the exact same day of the week for the entire year, what is the earliest subsequent year for which this planner can be used again?

2124

2108

2102

2136

Cevabı ve açıklamayı göster

Cevap: $2108$

Cevap

The earliest subsequent year with an identical calendar is

2108

.

A perfectly repeating calendar must satisfy two strict conditions: the starting weekday must be identical (meaning the accumulated odd days between them must be a multiple of

7

), and both years must share the same leap status. The year

2096

is a leap year. Because

2100

is not a leap year (it is not divisible by

400

), the standard

28

-year repetition cycle is interrupted. We must count the odd days manually. Between

2096

and

2107

, there are

12

years in total:

2

leap years (

2096, 2104

) and

10

normal years. The sum of odd days is

(2 \times 2) + (10 \times 1) = 14

. Since

14

is exactly divisible by

7

, the year

2108

starts on the same day as

2096

. Because

2108

is also a leap year, every date including February 29 perfectly aligns.

Adım Adım Çözüm

1

Determine the leap year status of the original year.

The year

2096

is divisible by

4

, so it is a leap year with

366

days.

A leap year calendar can only perfectly repeat in another leap year. Otherwise, dates from March 1 onwards will shift out of alignment.

2

Identify the century boundary behavior for the upcoming century.

The upcoming century year is

2100

. Since

2100

is not perfectly divisible by

400

, it is a normal year (not a leap year).

Crossing a non-leap century breaks the standard

28

-year repetition cycle for leap years, necessitating a manual calculation of odd days.

3

Calculate cumulative odd days year-by-year starting from

2096

.

Normal years contribute

1

odd day (

365 \pmod 7 = 1

); leap years contribute

2

odd days (

366 \pmod 7 = 2

). The sum must reach a multiple of

7

exactly at the start of a leap year.

A cumulative odd day sum that is a multiple of

7

ensures the starting day of the week exactly matches the original year.

4

Trace the odd days until both the weekday and leap year conditions are met.

From

2096

to

2107

(a span of

12

years), there are

10

normal years and

2

leap years (

2096

and

2104

). Total odd days =

(10 \times 1) + (2 \times 2) = 14

. The sum

14

is a multiple of

7

.

Because the odd days perfectly balance out to

0 \pmod 7

, the next year (

2108

) starts on the identical weekday. Since

2108

is also a leap year, its entire calendar matches

2096

.

Anahtar Kavram

Calendar Repetition Rules and Non-400 Century Boundaries

Soru 254Soru →

An autonomous delivery drone is programmed to navigate from a central warehouse to a remote drop-off location. The drone executes the following sequence of movements:

1. It takes off and flies $10\text{ km}$ due North.
2. It turns $90^\circ$ to its right and flies $20\text{ km}$ .
3. It then rotates $135^\circ$ anti-clockwise and flies $10\sqrt{2}\text{ km}$ .
4. Finally, it turns $135^\circ$ clockwise and flies $5\text{ km}$ to reach the destination.

What is the shortest direct distance between the warehouse and the drop-off destination?

15\text{ km}

20\text{ km}

35\text{ km}

25\text{ km}

Cevabı ve açıklamayı göster

Cevap: $25\text{ km}$

Cevap

The shortest direct distance between the warehouse and the drop-off destination is

25\text{ km}

.

The correct answer is found by tracking the drone's position on a 2D plane. By applying the vector components of each movement, the net displacement is

15\text{ km}

East and

20\text{ km}

North. The shortest direct distance is the hypotenuse of these components:

\sqrt{15^2 + 20^2} = 25\text{ km}

.

Adım Adım Çözüm

1

Determine the drone's position after the first two movements.

Starting at origin

(0,0)

, the drone flies

10\text{ km}

North to

(0, 10)

. Turning

90^\circ

right faces it East; flying

20\text{ km}

brings it to

(20, 10)

.

Tracking coordinates on a plane allows for precise calculation of vector displacements.

2

Calculate the coordinates after the

135^\circ

anti-clockwise rotation.

Facing East, a

135^\circ

anti-clockwise turn points the drone North-West. A distance of

10\sqrt{2}\text{ km}

NW means moving

10\text{ km}

West and

10\text{ km}

North. The new position is

(20-10, 10+10) = (10, 20)

.

The North-West vector components of

10\sqrt{2}

are split equally into

(-10, +10)

on a standard coordinate grid.

3

Calculate the final position after the

135^\circ

clockwise rotation.

Facing North-West, a

135^\circ

clockwise turn points the drone East. Flying

5\text{ km}

East adds

+5

to the x-coordinate. The final position is

(10+5, 20) = (15, 20)

.

Establishing the final drop-off location coordinates relative to the warehouse to allow for distance calculation.

4

Compute the shortest direct distance using the final coordinates.

Using the Pythagorean theorem: Distance

= \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25\text{ km}

.

The shortest direct distance corresponds to the hypotenuse of the right-angled triangle formed by the net horizontal and vertical displacements.

Anahtar Kavram

Vector displacement, angular rotation tracking, and Pythagorean theorem application.

Soru 255Soru →

A cybersecurity agency audited $250$ government web portals and found vulnerabilities across three categories: Data Breach ( $D$ ), Injection Attacks ( $I$ ), and Authentication Flaws ( $A$ ). The audit report stated:
- $110$ portals were vulnerable to Data Breach
- $130$ portals were vulnerable to Injection Attacks
- $100$ portals were vulnerable to Authentication Flaws
- $45$ portals had both $D$ and $I$ vulnerabilities
- $55$ portals had both $I$ and $A$ vulnerabilities
- $40$ portals had both $D$ and $A$ vulnerabilities
- $20$ portals had all three types of vulnerabilities

Based on this data, how many portals had exactly one type of vulnerability?

120

60

220

80

Cevabı ve açıklamayı göster

Cevap: $120$

Cevap

There are

120

portals with exactly one type of vulnerability.

To find the number of portals with exactly one vulnerability, we isolate the non-overlapping portion of each set. For set

D

, this is

110 - 45 - 40 + 20 = 45

. For set

I

, it is

130 - 45 - 55 + 20 = 50

. For set

A

, it is

100 - 40 - 55 + 20 = 25

. Adding these exclusive regions gives

45 + 50 + 25 = 120

.

Adım Adım Çözüm

1

Calculate the number of portals vulnerable ONLY to Data Breach (

D

).

|D_{only}| = |D| - |D \cap I| - |D \cap A| + |D \cap I \cap A| = 110 - 45 - 40 + 20 = 45

Subtracting the pairwise intersections removes the portals with multiple vulnerabilities, but it subtracts the triple intersection twice, so it must be added back once.

2

Calculate the number of portals vulnerable ONLY to Injection Attacks (

I

).

|I_{only}| = |I| - |D \cap I| - |I \cap A| + |D \cap I \cap A| = 130 - 45 - 55 + 20 = 50

Applying the same region isolation principle to set

I

.

3

Calculate the number of portals vulnerable ONLY to Authentication Flaws (

A

).

|A_{only}| = |A| - |D \cap A| - |I \cap A| + |D \cap I \cap A| = 100 - 40 - 55 + 20 = 25

Applying the same region isolation principle to set

A

.

4

Sum the three isolated regions to find the total number of portals with exactly one vulnerability.

45 + 50 + 25 = 120

These regions represent portals that have one and only one type of vulnerability.

Anahtar Kavram

Applying set theory and the inclusion-exclusion principle to isolate specific regions within a three-set Venn diagram.

Soru 256Soru →

Six diplomats representing Brazil, Canada, Denmark, Egypt, France, and Ghana are seated in a single straight row facing North during an international summit. Based on the following conditions, arrange the diplomats in the correct order from the extreme left (Position 1) to the extreme right (Position 6) of the row:

- The diplomat from Brazil sits at the extreme left of the row.
- The diplomat from Canada sits exactly between the diplomats from Brazil and Denmark.
- The diplomat from Ghana sits at the extreme right of the row.
- The diplomat from France sits immediately to the left of the diplomat from Ghana.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from left to right is: Brazil, Canada, Denmark, Egypt, France, and Ghana.

By following the direct placement clues sequentially, we can fill the 6 seats without any ambiguity. Brazil takes the far left (1) and Ghana takes the far right (6). Canada and Denmark follow Brazil in that order to satisfy the 'between' condition, filling seats 2 and 3. France goes just before Ghana into seat 5. The only remaining empty seat is seat 4, which is filled by Egypt.

Adım Adım Çözüm

1

Place the diplomats at the extreme ends of the row.

Brazil is at Position 1 and Ghana is at Position 6.

The first and third clues explicitly state the positions for Brazil and Ghana.

2

Determine the positions of Canada and Denmark.

Canada is at Position 2 and Denmark is at Position 3.

The second clue states Canada is exactly between Brazil and Denmark. Since Brazil is at Position 1, Canada must be at Position 2 and Denmark at Position 3.

3

Determine the position of France.

France is at Position 5.

The fourth clue states France is immediately to the left of Ghana. Since Ghana is at Position 6, France must be at Position 5.

4

Assign the final remaining seat.

Egypt is at Position 4.

Positions 1, 2, 3, 5, and 6 are filled. The only diplomat left is Egypt, so they must sit in the only remaining seat, which is Position 4.

Anahtar Kavram

Linear seating arrangement with fixed positions and relative placement clues.

Tahmini Süre:1m 0s

Soru 257Soru →

A metropolitan transport network assigns an identification string to each station name using a consistent alphabetic transformation logic. Under this system, the station named NORTH is encoded as OQUXM, and the station named SOUTH is encoded as TQXXM.

What will be the encoded identification string for the station named CENTRAL?

Cevabı ve açıklamayı göster

Cevap: DGQXWGS; dgqxwgs

Cevap

DGQXWGS

The coding logic applies an increasing forward shift to the English alphabet based on the letter's position in the word (1st letter +1, 2nd letter +2, 3rd letter +3, etc.). Applying this sequence to CENTRAL (C+1, E+2, N+3, T+4, R+5, A+6, L+7) yields DGQXWGS.

Adım Adım Çözüm

1

Analyze the transformation pattern from the first example, NORTH to OQUXM.

N(+1) = O; O(+2) = Q; R(+3) = U; T(+4) = X; H(+5) = M.

To identify the underlying coding rule, which involves an increasing positional shift.

2

Verify the identified pattern using the second example, SOUTH to TQXXM.

S(+1) = T; O(+2) = Q; U(+3) = X; T(+4) = X; H(+5) = M.

To confirm that the logic applies consistently across different inputs.

3

Calculate the shifted letters for the target word CENTRAL by applying the verified pattern.

C(+1)=D, E(+2)=G, N(+3)=Q, T(+4)=X, R(+5)=W, A(+6)=G, L(+7)=S.

To determine the final encoded string according to the established algorithm.

Anahtar Kavram

Positional Alphabet Shifting

Soru 258Soru →

A language institute conducted a survey among $400$ civil service aspirants to assess their proficiency in three foreign languages: French, German, and Spanish. The survey revealed the following data:
- $180$ aspirants are proficient in French.
- $150$ aspirants are proficient in German.
- $160$ aspirants are proficient in Spanish.
- $40$ aspirants are proficient in exactly French and German, but not Spanish.
- $30$ aspirants are proficient in exactly German and Spanish, but not French.
- $50$ aspirants are proficient in exactly French and Spanish, but not German.
- $70$ aspirants are not proficient in any of these three languages.

What is the number of aspirants who are proficient in all three languages?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

By determining the union of the three sets (

400 - 70 = 330

) and applying the inclusion-exclusion principle while correctly distinguishing between 'exactly two' and the full intersection of two sets, we find that

20

aspirants are proficient in all three languages.

Adım Adım Çözüm

1

Determine the number of aspirants proficient in at least one of the three languages.

n(F \cup G \cup S) = 400 - 70 = 330

The total population consists of those who speak at least one language and those who speak none.

2

Set up an equation using the Principle of Inclusion-Exclusion for three sets. Let

x

be the number of aspirants proficient in all three languages.

n(F \cap G) = 40 + x

,

n(G \cap S) = 30 + x

, and

n(F \cap S) = 50 + x

The total intersection of any two sets includes those in exactly those two sets plus those in all three sets.

3

Substitute all values into the union formula.

330 = 180 + 150 + 160 - (40 + x) - (30 + x) - (50 + x) + x

The formula

n(F \cup G \cup S) = n(F) + n(G) + n(S) - n(F \cap G) - n(G \cap S) - n(F \cap S) + n(F \cap G \cap S)

accounts for all overlapping regions.

4

Simplify the equation and solve for

x

.

330 = 490 - 120 - 2x \implies 330 = 370 - 2x \implies 2x = 40 \implies x = 20

Basic algebraic simplification yields the final value for the intersection of all three sets.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Alternatif Yöntem

Instead of using the union formula, use a region-based approach in a Venn diagram. Let the central 'all three' region be

x

. Calculate the 'only one' regions in terms of

x

: Only French =

180 - (40 + 50 + x) = 90 - x

. Only German =

150 - (40 + 30 + x) = 80 - x

. Only Spanish =

160 - (50 + 30 + x) = 80 - x

. The sum of all disjoint regions inside the union is

(90 - x) + (80 - x) + (80 - x) + 40 + 30 + 50 + x = 370 - 2x

. Since the union is

400 - 70 = 330

, we have

370 - 2x = 330

, which gives

x = 20

.

Tahmini Süre:2m 0s

Soru 259Soru →

A maritime patrol boat departs from its home harbor to secure coastal waters. It first sails $24\text{ km}$ strictly towards the West to investigate a radar blip. Finding nothing, it turns North and travels $7\text{ km}$ to a reported location. It then changes course, sailing $45\text{ km}$ East to follow a distress signal. Finally, it turns South and travels $27\text{ km}$ to successfully intercept the target. Calculate the exact straight-line distance from the home harbor to the interception point.

Cevabı ve açıklamayı göster

Cevap: 29

Cevap

29

By resolving the boat's path into net vertical and horizontal components, we find its final position is

21\text{ km}

East and

20\text{ km}

South of its starting point. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the shortest straight-line distance is the hypotenuse:

\sqrt{21^2 + 20^2} = 29\text{ km}

.

Adım Adım Çözüm

1

Calculate the net horizontal (East-West) displacement.

21\text{ km}

East

The boat initially travels

24\text{ km}

West, then later travels

45\text{ km}

East. The net horizontal movement is

45 - 24 = 21\text{ km}

in the East direction.

2

Calculate the net vertical (North-South) displacement.

20\text{ km}

South

The boat travels

7\text{ km}

North, and later travels

27\text{ km}

South. The net vertical movement is

27 - 7 = 20\text{ km}

in the South direction.

3

Apply the Pythagorean theorem to find the shortest straight-line distance.

\sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29\text{ km}

The net East and South displacements form the two perpendicular legs of a right-angled triangle relative to the starting point. The straight-line distance is the hypotenuse.

Anahtar Kavram

Vector displacement across cardinal directions and Pythagorean theorem calculation.

Soru 260Soru →

On Wednesday, August 12, $1761$ , a rare astronomical alignment was recorded in a historical journal. Exactly $200$ years later, on August 12, $1961$ , the alignment was observed again. Based on the rules of the Gregorian calendar, what day of the week was August 12, $1961$ ?

Cevabı ve açıklamayı göster

Cevap: Saturday; saturday; Sat; sat

Cevap

Saturday

The total day shift over a 200-year span is the sum of standard year shifts (200 days) and the true number of leap days (48 days, which excludes the non-leap century years 1800 and 1900). A total shift of 248 days is equivalent to 3 days modulo 7, meaning the starting day of Wednesday shifts forward by 3 days to land on Saturday.

Adım Adım Çözüm

1

Determine the basic shift in days for the total number of years.

The timeframe is exactly

200

years, which provides a base shift of

200

days.

A standard year of

365

days shifts the day of the week by

1

day (

365 \equiv 1 \pmod 7

). Therefore, each year contributes at least

1

day of shift.

2

Count the total multiples of 4 in the given year range to estimate leap years.

There are

50

multiples of

4

between

1761

and

1961

.

The first multiple of

4

after

1761

is

1764

, and the last multiple of

4

before

1961

is

1960

. The number of multiples is calculated as

\frac{1960 - 1764}{4} + 1 = 50

.

3

Apply the Gregorian calendar rules for century years to determine the true number of leap days.

The true number of leap days is

48

.

Century years are leap years only if they are divisible by

400

. The years

1800

and

1900

fall within this range and are divisible by

100

but not

400

. Thus, they are standard

365

-day years. Subtracting these

2

exceptions from the initial

50

yields

48

true leap days.

4

Calculate the total shift in days and find its equivalent modulo 7.

The total shift is

248

days, which is equivalent to a shift of

3

days.

The total shift equals the base shift (

200

days) plus the extra leap days (

48

days). Dividing

248

by

7

gives

35

with a remainder of

3

. So,

248 \equiv 3 \pmod 7

.

5

Add the calculated modular shift to the original day of the week.

Wednesday +

3

days = Saturday.

Moving

3

days forward from the starting day of Wednesday (Thursday, Friday, Saturday) provides the correct day of the week for August 12, 1961.

Anahtar Kavram

Gregorian calendar leap year century rules and modular arithmetic for day shifting

Alternatif Yöntem

You can calculate the shift per century. In a 100-year span crossing a non-leap century year (like 1761-1861 or 1861-1961), there are exactly 24 leap years. Each 100-year span shifts the day by 124 days. Since 124 modulo 7 is 5, each century shifts the date by 5 days. For two centuries, the total shift is 5 + 5 = 10 days. Finally, 10 modulo 7 results in a 3-day forward shift.

Tahmini Süre:3m 0s

General Mental Ability

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Alternatif Yöntem

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Anahtar Kavram

Sık Yapılan Hatalar

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