Tüm alıştırma soruları

Soru 461Soru →

A cybersecurity protocol requires transmitting a data file of exactly $5400$ kilobytes. The system must divide this file into equally sized data packets without leaving any remaining data. For the encryption algorithm to function correctly, the size of each packet (in kilobytes) must be an integer that is a multiple of $12$ . Based on these conditions, how many different packet sizes are possible?

18

48

30

6

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the number of factors of

5400

that are multiples of

12

, we first divide

5400

by

12

, which gives

450

. The number of possible packet sizes is exactly equal to the number of divisors of

450

. The prime factorization of

450

is

2^1 \times 3^2 \times 5^2

. Using the divisor-counting formula (adding

1

to each exponent and multiplying), we find there are

(1+1) \times (2+1) \times (2+1) = 2 \times 3 \times 3 = 18

valid packet sizes.

Adım Adım Çözüm

1

Find the prime factorization of the total file size (

5400

).

5400 = 54 \times 100 = 2^3 \times 3^3 \times 5^2

.

Prime factorization is required to find the total number of divisors.

2

Determine the mathematical condition for the packet size

S

.

S

must be a divisor of

5400

and a multiple of

12

(

2^2 \times 3^1

). This means

S = 12 \times k

.

Translates the real-world constraints into a mathematical relationship.

3

Find the maximum possible value and prime factorization of the multiplier

k

.

k

must be a divisor of

\frac{5400}{12} = 450

. The prime factorization of

450

is

2^1 \times 3^2 \times 5^2

.

The number of valid packet sizes

S

is directly equal to the number of divisors of

k

.

4

Calculate the total number of divisors of

450

.

(1+1) \times (2+1) \times (2+1) = 2 \times 3 \times 3 = 18

.

Applies the standard formula for counting factors (adding 1 to each prime exponent and multiplying them).

Anahtar Kavram

Finding the number of factors of a number that are multiples of another given number.

Soru 462Soru →

Seven investors—M, N, O, P, Q, R, and S—are sitting around a circular table for a portfolio review. All of them are facing away from the center of the table.

1. S sits second to the right of R.
2. M sits third to the left of R.
3. O sits to the immediate left of M.
4. P sits exactly between R and S.
5. Q is an immediate neighbor of N, but Q is not an immediate neighbor of M.

What is the position of Q with respect to R?

To the immediate left

To the immediate right

Second to the left

Second to the right

Cevabı ve açıklamayı göster

Cevap: To the immediate left

Cevap

To the immediate left

Based on the clues, the final clockwise arrangement starting from R is: R, P, S, O, M, N, Q. When checking Q's position relative to R, Q is situated exactly one position counter-clockwise from R. Because all investors are facing away from the center, counter-clockwise corresponds to the left direction. Therefore, Q sits to the immediate left of R.

Adım Adım Çözüm

1

Establish the directional rules based on the facing direction.

Since all investors face away from the center, 'Right' means moving clockwise and 'Left' means moving counter-clockwise around the table.

This is the fundamental rule of circular seating arrangements when individuals face outward.

2

Place R as the starting reference point and position S, M, and O.

Assign R to position 1. S is second to the right (clockwise), placing S at position 3. M is third to the left (counter-clockwise), placing M at position 5. O is to the immediate left (counter-clockwise) of M, placing O at position 4.

Applying direct positional clues 1, 2, and 3 establishes the primary framework.

3

Determine the position of P.

P sits exactly between R (position 1) and S (position 3). Therefore, P must sit at position 2.

Fulfilling clue 4 fills the gap between R and S.

4

Position Q and N using the remaining seats and constraints.

The remaining empty seats are 6 and 7. Q and N must occupy them since they are immediate neighbors. Clue 5 states Q is not a neighbor of M (who is at position 5). Therefore, Q cannot be at position 6 and must be at position 7, leaving N at position 6.

Applying the final clues and the process of elimination completes the seating arrangement.

5

Determine Q's position with respect to R.

R is at position 1 and Q is at position 7. Position 7 is one step counter-clockwise from position 1. Since counter-clockwise is left, Q is to the immediate left of R.

Answering the specific question asked in the stem based on the completed arrangement.

Anahtar Kavram

Circular Seating Arrangement (Outward Facing)

Soru 463Soru →

During a materials testing process, the density of four composite alloys (A, B, C, and D) is recorded in grams per cubic centimeter ( $\text{g/cm}^3$ ):

- Alloy A: $2.\overline{15}$
- Alloy B: $\frac{15}{7}$
- Alloy C: $2.1\overline{5}$
- Alloy D: $\frac{28}{13}$

Arrange the alloys in ascending order based on their density, from the lowest to the highest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is Alloy B, Alloy A, Alloy D, Alloy C.

By converting all values to their decimal or pure fractional equivalents, we can observe that Alloy B (

2.142...

) is the smallest. The remaining values all begin with

2.15

, but expanding the subsequent decimal places reveals that Alloy A is

2.1515...

, Alloy D is

2.1538...

, and Alloy C is

2.1555...

. This yields the exact ascending sequence: Alloy B < Alloy A < Alloy D < Alloy C.

Adım Adım Çözüm

1

Separate the whole number from the fractional part for all given values to simplify the comparison.

All values have a whole number part of 2. We only need to compare their fractional parts: A (

0.\overline{15}

), B (

\frac{1}{7}

), C (

0.1\overline{5}

), and D (

\frac{2}{13}

).

Since all numbers start with 2, isolating the fractional or decimal part reduces computational load.

2

Convert all fractional and decimal parts into a common format (decimal approximation) to establish initial bounds.

A =

0.1515...

, B

\approx 0.1428...

, C =

0.1555...

, D

\approx 0.1538...

Decimal expansion allows for rapid estimation and sequencing without finding a large common denominator.

3

Verify the exact order by cross-multiplying the fractional equivalents of the closest values to eliminate any rounding uncertainties.

Fractions to compare: A (

\frac{15}{99} = \frac{5}{33}

), B (

\frac{1}{7}

), C (

\frac{14}{90} = \frac{7}{45}

), D (

\frac{2}{13}

). Comparing B and A:

1 \times 33 = 33

,

7 \times 5 = 35

, so B < A. Comparing A and D:

5 \times 13 = 65

,

33 \times 2 = 66

, so A < D. Comparing D and C:

2 \times 45 = 90

,

13 \times 7 = 91

, so D < C.

Cross-multiplication of fractions provides absolute mathematical proof of the inequalities:

\frac{1}{7} < \frac{5}{33} < \frac{2}{13} < \frac{7}{45}

.

Anahtar Kavram

Comparing mixed fractions and recurring decimals via decimal expansion and cross-multiplication

Soru 464Soru →

Seven colleagues—T, U, V, W, X, Y, and Z—are seated in a straight line. Some of them are facing North while others are facing South.

- T sits exactly in the middle of the line and faces North.
- W sits second to the left of T.
- U sits third to the right of W.
- Only one person sits between U and X.
- Y sits to the immediate right of X.
- V sits third to the right of Y.
- The immediate neighbors of T face opposite directions (i.e., if one faces North, the other faces South, and vice versa).
- Persons sitting at the extreme ends of the line face opposite directions.
- Z and U face the same direction, which is opposite to the direction faced by V.

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

Geçerli olan tümünü seçin

Z sits to the immediate left of W.

Y sits to the immediate left of U.

Three people sit between V and X.

V sits to the immediate right of T.

Cevabı ve açıklamayı göster

Cevap: Z sits to the immediate left of W.; Three people sit between V and X.

Cevap

The correct statements are that Z sits to the immediate left of W, and three people sit between V and X.

Based on the step-by-step seating arrangement, the final left-to-right order is Z(North), W(North), V(South), T(North), U(North), Y(South), and X(South). Therefore, Z is to the immediate left of W, and there are three people (T, U, Y) sitting between V and X.

Adım Adım Çözüm

1

Place T in the middle of the 7 positions and determine W's position.

T is at position 4 facing North. W is at position 2.

The middle of 7 spots is the 4th spot. W is second to the left of T, who is facing North.

2

Determine the position and facing direction of W, then place U.

W faces North, and U is placed at position 5.

U sits third to the right of W. If W faced South, the 3rd right would be off the line to the left (position -1). Thus, W faces North, placing U at position 5.

3

Locate X and Y based on U's position.

X is at position 7 facing South, and Y is at position 6.

Only one person sits between U (position 5) and X, so X must be at position 3 or 7. However, if X is at 3, Y cannot be to the immediate right (positions 2 and 4 are occupied). Thus, X is at 7. For Y to be at position 6, X must face South.

4

Place V and deduce Y's facing direction.

Y faces South, and V is placed at position 3.

V sits third to the right of Y. If Y faced North, the 3rd right would be position 9 (invalid). Therefore, Y faces South, placing V at position 3.

5

Place Z and determine all remaining facing directions.

Z is at position 1 facing North. U faces North, and V faces South.

Z takes the only empty spot. Extreme ends face opposite directions; X faces South, so Z faces North. Z and U face the same direction (North), opposite to V (South). This perfectly matches the condition that T's neighbors (V and U) face opposite directions.

Anahtar Kavram

Analytical seating arrangements with mixed facing directions (North and South).

Soru 465Soru →

Six urban planners—P, Q, R, S, T, and U—are to be assigned to two project groups, Group Alpha and Group Beta, such that each group contains exactly three members. The assignment must strictly comply with the following conditions:
1. Planner R must be assigned to Group Alpha.
2. Planners P and Q cannot be in the same group.
3. Planners S and T must be assigned to the same group.
4. Planner U cannot be assigned to Group Beta unless planner S is also assigned to Group Beta.

Which of the following statements regarding the group assignments are definitely correct?

Geçerli olan tümünü seçin

Planner U must be assigned to Group Alpha.

Both planners S and T must be assigned to Group Beta.

Planner P and planner R must always be assigned to the same group.

Planners U and S can be assigned to the same group.

Cevabı ve açıklamayı göster

Cevap: Planner U must be assigned to Group Alpha.; Both planners S and T must be assigned to Group Beta.

Cevap

Planner U must be assigned to Group Alpha, and both planners S and T must be assigned to Group Beta.

Group Alpha starts with planner R and gets exactly one of P or Q, leaving only one open slot. Planners S and T must be in the same group, requiring two open slots, so they must be placed in Group Beta. This completely fills Group Beta with three members (S, T, and one of P or Q), forcing planner U into Group Alpha. Therefore, the statements that planner U must be in Group Alpha and that both S and T must be in Group Beta are unequivocally correct.

Adım Adım Çözüm

1

Analyze individual group capacities and fixed initial conditions.

Group Alpha has 3 spots, with R fixed in Group Alpha. Group Beta has 3 spots.

Establishing initial constraints establishes available space in each group.

2

Evaluate the placement of P and Q.

One of P or Q is in Group Alpha, and the other is in Group Beta.

Since P and Q cannot be in the same group, they must split across the two groups.

3

Determine the placement of S and T.

Group Alpha now contains {R, P/Q}, leaving only 1 available spot. Planners S and T must be together (requiring 2 spots), so S and T MUST both be placed in Group Beta.

Group Alpha does not have enough remaining capacity for the paired planners S and T.

4

Deduce the final group membership for planner U.

Group Beta is now full with {P/Q, S, T}. The remaining planner U must be placed in Group Alpha, giving Group Alpha = {R, U, P/Q}.

All 6 planners must be accounted for within their respective 3-member group capacities.

Anahtar Kavram

Analytical Grouping and Capacity Constraints

Soru 466Soru →

Match the public administration stakeholder conflict scenarios in List I with the most effective administrative negotiation strategy in List II to achieve sustainable resolution:

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct alignment pairs indigenous landholder protests with integrative spatial realignment, commercial shopkeeper strikes with compensatory phasing and operational concessions, inter-district water disputes with institutional mediation and formula-based volumetric release, and resident-waste picker clashes with participatory co-management and livelihood formalization.

The solution accurately pairs each administrative dilemma with its optimal negotiation mechanism. Indigenous cultural site conflicts demand integrative realignment protecting core heritage values. Commercial strikes require financial and logistical phasing to bridge transition periods. Water rights disputes necessitate objective, data-driven institutional monitoring. Urban civic clashes are resolved by empowering marginalized workers through formalization and consensus-building.

Adım Adım Çözüm

1

Analyze the core underlying driver for each stakeholder conflict scenario in List I.

Identified four distinct conflict drivers: non-negotiable cultural/identity values, short-term economic/logistical disruption, zero-sum resource scarcity, and social exclusion combined with localized environmental opposition.

Effective conflict resolution requires matching administrative strategies to the primary nature of the underlying stakeholder interest.

2

Evaluate the administrative options in List II against principled negotiation frameworks.

Matched non-negotiable identity concerns to boundary realignment; commercial disruption to economic phasing; water scarcity to objective formulaic governance; and social friction to participatory formalization.

Administrative decisions must balance public welfare, stakeholder rights, proportionality, and long-term policy feasibility.

3

Verify pair compatibility against administrative law and civil service conduct standards.

All four matched strategies resolve deadlocks through collaborative, constitutionally sound, and non-coercive mediation processes.

Avoids administrative pitfalls like unilateral enforcement, passive procrastination, or excessive deference to vocal interest groups.

Anahtar Kavram

Stakeholder Conflict Resolution and Negotiation Strategy

Soru 467Soru →

Analyze the numerical progression below to determine the rule governing the sequence. What is the value of the missing term that completes the series?

Aşağıdaki boşlukları doldurun

Consider the series: $$7, 16, 41, 90, 171, $$

Cevabı ve açıklamayı göster

Cevap

The missing term that completes the sequence is 292.

The difference between successive terms follows the sequence of squares of consecutive odd numbers:

3^2 = 9

,

5^2 = 25

,

7^2 = 49

,

9^2 = 81

, and

11^2 = 121

. Adding

121

to

171

gives the missing term

292

.

Adım Adım Çözüm

1

Calculate the differences between consecutive terms in the series.

16 - 7 = 9

,

41 - 16 = 25

,

90 - 41 = 49

,

171 - 90 = 81

Examining first-order differences helps uncover the underlying pattern.

2

Identify the mathematical relationship between the difference values.

The differences are

9, 25, 49, 81

, which correspond to

3^2, 5^2, 7^2, 9^2

.

The increments between consecutive terms represent the squares of consecutive odd integers starting from 3.

3

Calculate the next term in the sequence using the identified rule.

The next odd number after

9

is

11

, so the required increment is

11^2 = 121

. Adding this increment to the last given term gives

171 + 121 = 292

.

Applying the pattern's next step yields the correct missing term.

Anahtar Kavram

Series Completion using Differences of Squares of Consecutive Odd Numbers

Soru 468Soru →

In a given letter-analogy pattern, $\text{CAT}$ is related to $\text{FDW}$ following a constant positional shift rule. Following the exact same logical rule, which of the following letter-groups replaces the question mark in $\text{DOG} : \text{?}$ ?

\text{GRJ}

\text{GRI}

\text{FQI}

\text{ALI}

Cevabı ve açıklamayı göster

Cevap: $\text{GRJ}$

Cevap

\text{GRJ}

Each letter in the first term is shifted forward by 3 places in the English alphabet (

\text{C}+3=\text{F}

,

\text{A}+3=\text{D}

,

\text{T}+3=\text{W}

). Applying the exact same

+3

positional shift to

\text{DOG}

yields

\text{D}+3=\text{G}

,

\text{O}+3=\text{R}

, and

\text{G}+3=\text{J}

, forming

\text{GRJ}

.

Adım Adım Çözüm

1

Determine the pattern established by the reference pair

\text{CAT} : \text{FDW}

.

C (

3

)

\rightarrow

F (

6

) [Shift

+3

]; A (

1

)

\rightarrow

D (

4

) [Shift

+3

]; T (

20

)

\rightarrow

W (

23

) [Shift

+3

]. The rule is a forward shift of

+3

for every letter.

Establishing the exact letter position transformation rule from the given pair.

2

Apply the

+3

forward shift rule to each letter of

\text{DOG}

.

D (

4 + 3 = 7 \rightarrow \text{G}

); O (

15 + 3 = 18 \rightarrow \text{R}

); G (

7 + 3 = 10 \rightarrow \text{J}

).

Translating the rule to find the corresponding completing term.

Anahtar Kavram

Letter Analogy via Constant Forward Positional Shift

Tahmini Süre:45s

Soru 469Soru →

Read the following information carefully:
In a family of seven members— $A, B, C, D, E, F,$ and $G$ —spanning three generations:
• $A$ is the father-in-law of $D$ , who is married to $B$ .
• $C$ is the only brother-in-law of $B$ and has no children.
• $E$ is the paternal grandmother of $F$ .
• $G$ is the daughter of $C$ 's sister.
• $F$ is the brother of $G$ .

What is the relationship of $C$ to $F$ ?

Cevabı ve açıklamayı göster

Cevap: Maternal uncle; Maternal Uncle; maternal uncle; Uncle; uncle

Cevap

Maternal uncle

Based on the given conditions,

D

is married to

B

, and

C

is

D

's brother. Since

G

is the daughter of

C

's sister (

D

) and

F

is the brother of

G

,

D

is the mother of

F

. Therefore,

C

, being the brother of

F

's mother, is the maternal uncle of

F

.

Adım Adım Çözüm

1

Analyze the first-generation and second-generation parent-in-law link.

A

is the father-in-law of

D

, who is married to

B

. This implies

B

is the child of

A

, and

D

is

B

's spouse.

A father-in-law is the father of one's spouse.

2

Deduce the sibling relationship and gender of

D

and

B

.

C

is the brother-in-law of

B

and has a sister whose daughter is

G

. Since

D

is married to

B

,

D

must be

C

's sister, establishing

D

as female and

B

as male.

Since

C

is

B

's brother-in-law and has a sister with children,

D

is that sister.

3

Determine the third-generation parentage and verify the paternal grandmother clue.

G

is the daughter of

D

, and

F

is the brother of

G

, making both

F

and

G

the children of

B

(father) and

D

(mother).

E

is the paternal grandmother of

F

, so

E

is

B

's mother married to

A

.

Siblings

F

and

G

share parents

B

and

D

.

4

Establish the relationship of

C

to

F

.

Since

C

is the brother of

D

(who is

F

's mother),

C

is the maternal uncle of

F

.

The brother of one's mother is defined as a maternal uncle.

Anahtar Kavram

Blood Relations - Multi-generational Family Tree Deduction

Soru 470Soru →

In a technology seminar attended by $60$ software engineers, $35$ engineers know Python and $25$ engineers know Java. If $10$ engineers know both Python and Java, how many engineers know neither of these two programming languages?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The number of engineers who know neither Python nor Java is 10.

The correct answer is 10. By taking the sum of Python knowers (

35

) and Java knowers (

25

) and subtracting those who know both (

10

), we obtain

50

engineers who know at least one language. Subtracting this from the total

60

seminar participants leaves

10

engineers who know neither language.

Adım Adım Çözüm

1

Find the total number of engineers who know at least one programming language

50 engineers know at least one language

Apply the two-set union formula:

n(A \cup B) = n(A) + n(B) - n(A \cap B) = 35 + 25 - 10 = 50

.

2

Calculate the number of engineers who know neither language

10 engineers know neither language

Subtract the number of engineers knowing at least one language from the total group size:

60 - 50 = 10

.

Anahtar Kavram

Two-Set Inclusion-Exclusion Principle and Complement of Sets

Soru 471Soru →

An administrative officer is appointed to resolve a growing dispute among local villagers, a private infrastructure company, and environmental regulators regarding a proposed land development project. Arrange the following conflict-resolution steps in the logical administrative order in which they should be executed, from first to last.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The optimal administrative sequence is: First, conduct baseline fact-finding and stakeholder mapping; Second, hold separate bilateral consultations; Third, facilitate joint multi-party negotiations; Fourth, formalize the agreement and establish a joint monitoring mechanism.

In public administration, effective negotiation follows a structured, progressive process: preliminary research and stakeholder mapping must precede direct engagement; separate bilateral dialogues must be conducted to establish trust and clarify positions before bringing hostile groups into the same room; joint negotiations are then used to build consensus; and formalization with joint monitoring secures long-term compliance.

Adım Adım Çözüm

1

Analyze initial project information and identify all affected parties.

Establishes a clear, impartial understanding of legal rights and stakeholder interests.

Administrative decisions require accurate baseline data prior to active intervention.

2

Engage stakeholders individually through bilateral discussions.

Uncovers unstated interests and potential compromise areas without public confrontation.

Direct joint meetings without prior separate consultations often escalate friction.

3

Bring all groups together in a structured negotiation forum.

Allows stakeholders to co-create solution terms and reach consensus.

Multi-party consensus ensures ownership of the final resolution by all sides.

4

Draft the final MoU/policy and institute an oversight body.

Provides legal enforceability and ongoing grievance redressal.

Without formalization and monitoring mechanisms, negotiated agreements frequently fall apart.

Anahtar Kavram

Phased Administrative Stakeholder Negotiation Strategy

Soru 472Soru →

Match each autonomous robotic surveyor (P, Q, R, S) executing a multi-step navigation sequence with its exact shortest straight-line displacement from its initial starting point.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matching pairs are: Robotic Surveyor P matches with

13\text{ m}

, Robotic Surveyor Q matches with

15\text{ m}

, Robotic Surveyor R matches with

20\text{ m}

, and Robotic Surveyor S matches with

25\text{ m}

.

Each robotic surveyor's final position is determined by establishing a Cartesian coordinate system with the starting point at

(0,0)

. By breaking each movement into orthogonal components

(x, y)

using trigonometry for diagonal movements (

45^\circ, 135^\circ

) and applying the Pythagorean theorem

d = \sqrt{x^2 + y^2}

, we obtain exact straight-line displacements of

13\text{ m}

for P,

15\text{ m}

for Q,

20\text{ m}

for R, and

25\text{ m}

for S.

Adım Adım Çözüm

1

Calculate displacement coordinates for Robotic Surveyor P

Path vector components:
1.

15\text{ m}

North:

(0, 15)

2. Turn

135^\circ

CW (facing SE), walk

10\sqrt{2}\text{ m}

:

\Delta x = 10\sqrt{2}\sin(135^\circ) = 10

,

\Delta y = 10\sqrt{2}\cos(135^\circ) = -10 \implies (10, 5)

3. Turn

90^\circ

right (facing SW), walk

5\sqrt{2}\text{ m}

:

\Delta x = -5

,

\Delta y = -5 \implies (5, 0)

4. Walk

12\text{ m}

North:

\Delta x = 0

,

\Delta y = 12 \implies (5, 12)

Final distance:

d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13\text{ m}

.

Decomposing vectors into

x

(East) and

y

(North) components allows precise tracking of position.

2

Calculate displacement coordinates for Robotic Surveyor Q

Path vector components:
1.

20\text{ m}

East:

(20, 0)

2. Turn

135^\circ

CCW (facing NW), walk

8\sqrt{2}\text{ m}

:

\Delta x = -8

,

\Delta y = 8 \implies (12, 8)

3. Turn

45^\circ

CW (facing North), walk

7\text{ m}

:

\Delta x = 0

,

\Delta y = 7 \implies (12, 15)

4. Walk

6\text{ m}

South:

\Delta x = 0

,

\Delta y = -6 \implies (12, 9)

Final distance:

d = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = 15\text{ m}

.

Using trigonometry and coordinate tracking ensures correct evaluation of net displacement.

3

Calculate displacement coordinates for Robotic Surveyor R

Path vector components:
1.

10\text{ m}

West:

(-10, 0)

2. Turn

45^\circ

CCW (facing SW), walk

6\sqrt{2}\text{ m}

:

\Delta x = -6

,

\Delta y = -6 \implies (-16, -6)

3. Turn

135^\circ

CW from SW (facing North), walk

18\text{ m}

:

\Delta x = 0

,

\Delta y = 18 \implies (-16, 12)

Final distance:

d = \sqrt{(-16)^2 + 12^2} = \sqrt{256 + 144} = 20\text{ m}

.

Angular rotations must be measured relative to the current heading, not absolute cardinal directions.

4

Determine initial orientation and calculate displacement for Robotic Surveyor S

At sunrise, the Sun is in the East, so shadows fall toward the West. Facing its own shadow means S initially faces West.
1. Turn

90^\circ

right (facing North), walk

10\text{ m}

:

(0, 10)

2. Turn

45^\circ

right (facing NE), walk

7\sqrt{2}\text{ m}

:

\Delta x = 7

,

\Delta y = 7 \implies (7, 17)

3. Turn North, walk

7\text{ m}

:

\Delta x = 0

,

\Delta y = 7 \implies (7, 24)

Final distance:

d = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25\text{ m}

.

Shadow orientation establishes the initial cardinal direction from which relative turns are executed.

Anahtar Kavram

Vector Displacement, Angular Rotation, and Shadow Orientation

Soru 473Soru →

Read the following family scenario carefully and match each person in Column I with their exact relationship to $B$ listed in Column II.

Scenario:
In a family of six members— $A, B, C, D, E,$ and $F$ —spanning three generations:
- $D$ is married to $F$ , and $A$ is the father of $D$ .
- $C$ is the mother-in-law of $F$ and has only one child.
- $E$ is the daughter of $F$ , and $B$ is the sibling of $E$ .

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches are:

A

matches with Paternal Grandfather,

C

matches with Paternal Grandmother,

D

matches with Father, and

E

matches with Sister.

By analyzing the three-generation structure:

A

(male) and

C

(female) are married parents of

D

(male).

D

is married to

F

(female). Their children are

E

(female) and

B

. Therefore, with respect to

B

:

A

is the paternal grandfather,

C

is the paternal grandmother,

D

is the father, and

E

is the sister.

Adım Adım Çözüm

1

Determine the parental generation relationships

A

is the father of

D

.

C

is the mother-in-law of

F

, who is married to

D

, which means

C

is the mother of

D

.

The mother-in-law of a spouse is the mother of the other spouse.

2

Identify the genders and marital tie of the middle generation

D

(male) and

F

(female) are married to each other.

D

is the only child of

A

and

C

.

A

is father and

C

is mother of

D

, making

D

their male child.

3

Determine the third generation relationships

E

(female) and

B

are children of

D

and

F

.

E

is the sister of

B

.

E

is the daughter of

F

and sibling of

B

.

4

Deduce each person's relationship to

B

A

is paternal grandfather,

C

is paternal grandmother,

D

is father, and

E

is sister.

Tracing lineage through

B

's father

D

yields paternal grandparents

A

and

C

, father

D

, and sister

E

.

Anahtar Kavram

Multi-generational blood relation tree deduction and kinship mapping

Tahmini Süre:1m 30s

Soru 474Soru →

An archaeological surveyor starting from a central benchmark walks $20\text{ m}$ due East. He then turns $135^\circ$ to his left and walks $14\sqrt{2}\text{ m}$ . Next, he turns $45^\circ$ to his left and walks $18\text{ m}$ . Finally, he turns $90^\circ$ to his right and walks $2\text{ m}$ to reach an excavation site. What is the shortest direct distance (in meters) between the starting benchmark and the excavation site?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The shortest direct distance between the starting benchmark and the excavation site is 20 meters.

By resolving each sequential movement into Cartesian coordinate components

(x, y)

, the final position relative to origin

(0,0)

is found to be

(-12\text{ m}, 16\text{ m})

. The straight-line distance is computed using the distance formula

\sqrt{(-12)^2 + (16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Adım Adım Çözüm

1

Plot the initial displacement on a Cartesian coordinate plane with origin

(0,0)

at the starting benchmark.

Position after walking

20\text{ m}

East is

(20, 0)

facing East (

0^\circ

).

East represents the positive x-axis direction.

2

Calculate displacement after turning

135^\circ

left (facing North-West) and walking

14\sqrt{2}\text{ m}

.

Position becomes

(20 - 14, 0 + 14) = (6, 14)

.

In North-West direction, horizontal component is

-14\sqrt{2} \times \frac{1}{\sqrt{2}} = -14\text{ m}

and vertical component is

+14\sqrt{2} \times \frac{1}{\sqrt{2}} = +14\text{ m}

.

3

Calculate displacement after turning

45^\circ

left from North-West (facing West) and walking

18\text{ m}

.

Position becomes

(6 - 18, 14) = (-12, 14)

.

Turning

45^\circ

left from North-West aligns facing direction with West (negative x-axis).

4

Calculate displacement after turning

90^\circ

right from West (facing North) and walking

2\text{ m}

.

Final position is

(-12, 14 + 2) = (-12, 16)

.

Turning

90^\circ

right from West aligns facing direction with North (positive y-axis).

5

Calculate the direct straight-line distance from starting point

(0,0)

to final position

(-12, 16)

.

Distance

D = \sqrt{(-12)^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Applying the Pythagorean theorem

D = \sqrt{x^2 + y^2}

yields the shortest distance.

Anahtar Kavram

Coordinate Geometry and Vector Resolution in Direction and Distance Problems

Tahmini Süre:2m 0s

Soru 475Soru →

In a certain code language, the word GLIMMER is written as INKMOUT. Following the exact same logical transformation rule, what is the code for the word PROJECT?

RHQGPWV

RIQQGXV

RJQRGYV

JTKLUEF

Cevabı ve açıklamayı göster

Cevap: RHQGPWV

Cevap

The coded word for PROJECT is RHQGPWV.

The code follows a dual positional rule: letters at odd positions (1st, 3rd, 5th, 7th) are shifted forward by 2 positions in alphabetical order, while letters at even positions (2nd, 4th, 6th) are replaced by their reverse alphabetical counterpart (A↔Z, B↔Y, etc.) followed by a 1-letter backward shift. Applying these exact steps to PROJECT gives RHQGPWV.

Adım Adım Çözüm

1

Analyze the rule applied to letters at odd positions (1st, 3rd, 5th, 7th) in GLIMMER.

G (1st) → I (+2 steps forward), I (3rd) → K (+2 steps forward), M (5th) → O (+2 steps forward), R (7th) → T (+2 steps forward).

Establishes that letters at odd positions undergo a direct +2 forward alphabetical shift.

2

Analyze the rule applied to letters at even positions (2nd, 4th, 6th) in GLIMMER.

L (2nd, reverse letter O) → N (-1 step backward), M (4th, reverse letter N) → M (-1 step backward), E (6th, reverse letter V) → U (-1 step backward).

Establishes that letters at even positions are replaced by their opposite alphabetical letter (where A=Z, B=Y, etc.) and then shifted 1 letter backward.

3

Apply the odd-position transformation (+2 forward shift) to the target word PROJECT.

P (1st) → R, O (3rd) → Q, E (5th) → G, T (7th) → V.

Determines the output letters for positions 1, 3, 5, and 7.

4

Apply the even-position transformation (reverse letter - 1 step) to the target word PROJECT.

R (2nd, reverse I) → H, J (4th, reverse Q) → P, C (6th, reverse X) → W.

Determines the output letters for positions 2, 4, and 6.

5

Combine all transformed positions sequentially.

Position 1 to 7 yields RHQGPWV.

Finalizes the code for PROJECT.

Anahtar Kavram

Positional parity-based multi-step coding combining forward shift and reverse alphabetical indexing.

Soru 476Soru →

Which of the following terms logically completes the given alphanumeric series?

C24X, \quad F20U, \quad I16R, \quad L12O, \quad ?

O8L

N8L

O8M

O10L

Cevabı ve açıklamayı göster

Cevap: $O8L$

Cevap

O8L

The term containing

O8L

is correct because it satisfies all three independent series rules: the first letter advances by 3 (

L + 3 = O

), the middle number decreases by 4 (

12 - 4 = 8

), and the third letter shifts backward by 3 (

O - 3 = L

).

Adım Adım Çözüm

1

Analyze the pattern of the first letter in each term

The first letters follow the sequence:

C (3) \xrightarrow{+3} F (6) \xrightarrow{+3} I (9) \xrightarrow{+3} L (12) \xrightarrow{+3} O (15)

.

The first letter advances by 3 positions in alphabetical order.

2

Analyze the pattern of the numerical term

The numbers follow the sequence:

24 \xrightarrow{-4} 20 \xrightarrow{-4} 16 \xrightarrow{-4} 12 \xrightarrow{-4} 8

.

Each number decreases by a constant difference of 4.

3

Analyze the pattern of the last letter in each term

The last letters follow the sequence:

X (24) \xrightarrow{-3} U (21) \xrightarrow{-3} R (18) \xrightarrow{-3} O (15) \xrightarrow{-3} L (12)

.

The last letter moves backward by 3 positions in alphabetical order.

4

Combine the resulting components

Combining

O

,

8

, and

L

yields the missing term

O8L

.

Synthesizing all three independent rules completes the sequence.

Anahtar Kavram

Alphanumeric Series Completion with Simultaneous Forward-Reverse Letter Shifts and Arithmetic Decrement

Tahmini Süre:1m 0s

Soru 477Soru →

In a single row of students facing North, Akash is ranked $18^{\text{th}}$ from the left end, while Banya is ranked $24^{\text{th}}$ from the right end. Chetan is sitting $6^{\text{th}}$ to the right of Akash and $4^{\text{th}}$ to the left of Banya. If Akash and Banya interchange their positions while Chetan remains in his place, and then Deepa sits exactly midway between Akash's new position and Chetan's position, what is Deepa's rank from the left end of the row?

26^{\text{th}}

21^{\text{st}}

25^{\text{th}}

27^{\text{th}}

Cevabı ve açıklamayı göster

Cevap: $26^{\text{th}}$

Cevap

The rank of Deepa from the left end of the row is

26^{\text{th}}

.

By following the relative linear placements, Chetan's left rank is

18 + 6 = 24

, and Banya's left rank is

24 + 4 = 28

. After Akash swaps positions with Banya, Akash's new position from the left becomes

28

. Deepa sits directly midway between Chetan (

24^{\text{th}}

) and Akash's new position (

28^{\text{th}}

), which corresponds to the rank

\frac{24 + 28}{2} = 26^{\text{th}}

.

Adım Adım Çözüm

1

Determine Chetan's position from the left end.

Since Akash is

18^{\text{th}}

from the left and Chetan is

6^{\text{th}}

to his right, Chetan's left-rank

= 18 + 6 = 24^{\text{th}}

.

Moving right along a row facing North increases the positional index from the left.

2

Determine Banya's position from the left end.

Chetan is

4^{\text{th}}

to the left of Banya, so Banya's left-rank

= 24 + 4 = 28^{\text{th}}

.

Since Chetan is to the left of Banya, adding

4

to Chetan's position gives Banya's position from the left.

3

Find Akash's position after interchanging with Banya.

Akash takes Banya's original place, making Akash's new left-rank

= 28^{\text{th}}

.

Interchanging positions swaps their exact ranks from the left end.

4

Calculate Deepa's position midway between Chetan's position (

24^{\text{th}}

) and Akash's new position (

28^{\text{th}}

).

Midpoint

= \frac{24 + 28}{2} = 26^{\text{th}}

from the left end.

The exact midway index between two positions

p_1

and

p_2

is their arithmetic mean

(p_1 + p_2) / 2

.

Anahtar Kavram

Order and Ranking involving multi-person relative placement and position interchange

Tahmini Süre:2m 0s

Soru 478Soru →

Analyze the pattern in the numerical sequence given below:

4, \quad 9, \quad 20, \quad 43, \quad 90, \quad ?

What is the value of the missing term that completes the sequence?

Cevabı ve açıklamayı göster

Cevap: 185

Cevap

185

Each consecutive term is calculated by doubling the previous term and adding an increasing integer (

+1, +2, +3, +4, +5

). Following this rule,

90 \times 2 + 5 = 185

.

Adım Adım Çözüm

1

Examine differences or operations between adjacent terms

9 - 4 = 5

,

20 - 9 = 11

,

43 - 20 = 23

,

90 - 43 = 47

Identify the primary numerical increment pattern

2

Analyze the rule for term progression

The operations follow

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

for

n \ge 2

Confirm consistency across all given terms

3

Compute the sixth term (

T_6

)

T_6 = 90 \times 2 + 5 = 185

Apply the addition increment

+5

following

+4

Anahtar Kavram

Number and Alphabet Series

Soru 479Soru →

A wholesale merchant has three batches of specialty syrups measuring $\frac{63}{5}$ liters, $\frac{72}{25}$ liters, and $\frac{81}{10}$ liters in volume. To distribute them in uniform sample sizes, the merchant wants to repackage all the syrup into identical smaller bottles filled exactly to the brim, ensuring no syrup is left over from any of the original batches. What is the maximum possible capacity (in liters) of each sample bottle?

Cevabı ve açıklamayı göster

Cevap: 0.18

Cevap

The maximum possible capacity of each sample bottle is 0.18 liters.

To find the maximum capacity of uniform bottles that can hold the syrup from any batch without leftover, we must find the Highest Common Factor (HCF) of the respective volumes. For fractions, the rule is

\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

. The HCF of the numerators (

63, 72, 81

) is

9

, and the LCM of the denominators (

5, 25, 10

) is

50

. The resulting maximum capacity is

\frac{9}{50}

, which equals

0.18

liters.

Adım Adım Çözüm

1

Determine the mathematical operation required.

Calculate the HCF of

\frac{63}{5}

,

\frac{72}{25}

, and

\frac{81}{10}

.

Finding the largest equal container size that leaves no remainder requires computing the Highest Common Factor.

2

Apply the rule for finding the HCF of fractions.

\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

This standard formula correctly identifies the greatest common divisor for a set of rational numbers.

3

Find the HCF of the numerators.

\text{HCF}(63, 72, 81) = 9

The largest integer that perfectly divides 63, 72, and 81 is 9.

4

Find the LCM of the denominators.

\text{LCM}(5, 25, 10) = 50

The smallest integer that is a multiple of 5, 25, and 10 is 50.

5

Compute the final fraction and convert it to a decimal.

\frac{9}{50} = 0.18

Dividing the HCF of the numerators by the LCM of the denominators yields the exact capacity in liters.

Anahtar Kavram

HCF and LCM of fractions

Soru 480Soru →

Three environmental monitoring sensors transmit data packets at regular intervals of $\frac{8}{3}$ seconds, $\frac{12}{5}$ seconds, and $\frac{24}{7}$ seconds, respectively. If all three sensors transmit a packet simultaneously at exactly 10:00:00 AM, how many seconds will pass before they next transmit a packet together at the exact same instant?

24

\frac{105}{4}

\frac{4}{105}

\frac{8}{35}

Cevabı ve açıklamayı göster

Cevap: $24$

Cevap

24 seconds

The sensors will sync up again at a time that is a multiple of all three individual intervals. To find this, we calculate the LCM of the fractions

\frac{8}{3}

,

\frac{12}{5}

, and

\frac{24}{7}

. The mathematical rule for the LCM of fractions is the LCM of the numerators divided by the HCF of the denominators. We find

\text{LCM}(8, 12, 24) = 24

and

\text{HCF}(3, 5, 7) = 1

. Thus, the total LCM is

\frac{24}{1} = 24

seconds.

Adım Adım Çözüm

1

Identify the mathematical requirement

The time until the next simultaneous transmission is the Least Common Multiple (LCM) of the three interval times.

Simultaneous events that repeat at regular intervals align perfectly at the LCM of those individual intervals.

2

Apply the mathematical formula for the LCM of fractions

\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

Calculating the LCM for fractions requires evaluating the numerators and denominators cross-layer using this specific formula.

3

Calculate the LCM of the numerators

\text{LCM}(8, 12, 24) = 24

The smallest integer that is perfectly divisible by 8, 12, and 24 is 24.

4

Calculate the HCF of the denominators

\text{HCF}(3, 5, 7) = 1

The numbers 3, 5, and 7 are all prime numbers and share no common divisor other than 1.

5

Compute the final result

\frac{24}{1} = 24

seconds

Dividing the LCM of the numerators by the HCF of the denominators gives the synchronized time interval.

Anahtar Kavram

Calculating the Lowest Common Multiple (LCM) of fractional values for synchronization problems.

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