General Mental Ability Alıştırma Soruları — State PSC Exam — Sayfa 12

Soru 221Soru →

A forest conservation officer starting from a watchtower walks $5\text{ km}$ due North. She then turns $135^\circ$ clockwise and walks $4\sqrt{2}\text{ km}$ . Next, she turns $45^\circ$ counter-clockwise and walks $8\text{ km}$ straight. Finally, she turns $90^\circ$ counter-clockwise and walks $4\text{ km}$ to reach a research station. What is the shortest straight-line distance between the watchtower and the research station, and in which direction is the research station relative to the watchtower?

13\text{ km}

, North-East

13\text{ km}

, South-East

21\text{ km}

, North-East

13\text{ km}

, South-West

Cevabı ve açıklamayı göster

Cevap: $13\text{ km}$ , North-East

Cevap

The research station is

13\text{ km}

away from the watchtower in the North-East direction.

The net horizontal displacement along the East-West axis is

4 + 8 = 12\text{ km}

East, and the net vertical displacement along the North-South axis is

5 - 4 + 4 = 5\text{ km}

North. Applying the Pythagorean theorem yields a direct direct distance of

\sqrt{12^2 + 5^2} = 13\text{ km}

. Because both net displacement coordinates are positive, the research station lies to the North-East of the starting watchtower.

Adım Adım Çözüm

1

Set up a Cartesian coordinate system with the starting watchtower at

(0, 0)

.

Initial position:

(0, 0)

, facing North.

Establishing standard coordinate axes (

x

-axis for East/West,

y

-axis for North/South) simplifies vector calculation.

2

Calculate position after the first leg (

5\text{ km}

North).

Position:

(0, 5)

, facing North (

90^\circ

relative to positive

x

-axis).

Moving

5\text{ km}

North adds

5

units along the positive

y

-axis.

3

Calculate position after turning

135^\circ

clockwise and walking

4\sqrt{2}\text{ km}

.

New orientation is South-East (

-45^\circ

). Displacement components are

\Delta x = 4\sqrt{2}\cos(-45^\circ) = 4\text{ km}

East,

\Delta y = 4\sqrt{2}\sin(-45^\circ) = -4\text{ km}

North (South). Position becomes

(0 + 4, 5 - 4) = (4, 1)

.

Turning

135^\circ

clockwise from North points facing to South-East.

4

Calculate position after turning

45^\circ

counter-clockwise and walking

8\text{ km}

.

New orientation is due East (

0^\circ

). Displacement

\Delta x = 8\text{ km}

,

\Delta y = 0

. Position becomes

(4 + 8, 1 + 0) = (12, 1)

.

Turning

45^\circ

counter-clockwise from South-East realigns facing direction to East.

5

Calculate position after turning

90^\circ

counter-clockwise and walking

4\text{ km}

.

New orientation is due North (

90^\circ

). Displacement

\Delta x = 0

,

\Delta y = 4\text{ km}

. Final position becomes

(12, 1 + 4) = (12, 5)

.

Turning

90^\circ

counter-clockwise from East points facing to North.

6

Compute direct shortest distance and final cardinal orientation relative to origin

(0, 0)

.

Shortest distance

d = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ km}

. Since both

x = 12 > 0

and

y = 5 > 0

, the final direction relative to the watchtower is North-East.

The Pythagorean theorem gives straight-line displacement in a 2D Euclidean plane.

Anahtar Kavram

Direction and Distance Test: 2D Vector Displacement and Angular Rotation

Soru 222Soru →

In a survey of $80$ state administrative officers, $50$ officers read Newspaper X, $40$ officers read Newspaper Y, and $15$ officers read both newspapers. How many officers read only Newspaper X?

35

50

25

65

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

35 officers read only Newspaper X.

The total number of officers who read Newspaper X is 50. Since 15 of these officers also read Newspaper Y, subtracting the overlapping group (15) from the total Newspaper X group (50) gives 35 officers who read strictly Newspaper X.

Adım Adım Çözüm

1

Identify the total count of readers for Newspaper X and the intersection count of readers who read both newspapers.

Total readers of Newspaper X = 50, Readers of both Newspaper X and Y = 15.

The group reading Newspaper X consists of those reading Newspaper X only and those reading both newspapers.

2

Subtract the intersection count from the total count of Newspaper X readers.

50 - 15 = 35.

Using set theory, n(Only X) = n(X) - n(X ∩ Y).

Anahtar Kavram

Two-set Venn diagram region calculation: n(Only A) = n(A) - n(A ∩ B)

Soru 223Soru →

In a specific coding system, the word PARDON is written as MRWUZS. Following the exact same logical transformation rule, what is the code for the word POSTURE?

Cevabı ve açıklamayı göster

Cevap: HIXGVLS; hixgvls

Cevap

HIXGVLS

The code is determined by applying two transformation rules based on letter positions followed by a string reversal. Letters at odd positions (1st, 3rd, 5th, 7th) undergo a

+3

forward alphabetical shift, while letters at even positions (2nd, 4th, 6th) are replaced by their reverse alphabetical partner (

27 - \text{alphabetical position}

). For the word POSTURE, this yields the intermediate string SLVGXIH. Reversing this intermediate string produces the final code HIXGVLS.

Adım Adım Çözüm

1

Analyze the position-based transformation rules for each letter of the original word.

Letters at odd positions (1st, 3rd, 5th...) are shifted forward by

+3

positions in the English alphabetical series. Letters at even positions (2nd, 4th, 6th...) are replaced by their reverse alphabetical partner (where position index sum equals

27

).

For PARDON: P (1st, odd)

\rightarrow

S; A (2nd, even)

\rightarrow

Z; R (3rd, odd)

\rightarrow

U; D (4th, even)

\rightarrow

W; O (5th, odd)

\rightarrow

R; N (6th, even)

\rightarrow

M. The intermediate string is SZUWRM.

2

Identify the final structural modification step.

The intermediate letter sequence is reversed from right to left.

Reversing SZUWRM character-by-character gives MRWUZS.

3

Apply the odd/even position rules to the target word POSTURE.

P (1st, odd)

+3 \rightarrow

S; O (2nd, even)

27-15=12 \rightarrow

L; S (3rd, odd)

+3 \rightarrow

V; T (4th, even)

27-20=7 \rightarrow

G; U (5th, odd)

+3 \rightarrow

X; R (6th, even)

27-18=9 \rightarrow

I; E (7th, odd)

+3 \rightarrow

H. Intermediate string is SLVGXIH.

Odd indices shift by

+3

, and even indices transform to reverse alphabetical complements.

4

Reverse the intermediate sequence to obtain the final coded answer.

HIXGVLS

Reversing SLVGXIH gives HIXGVLS.

Anahtar Kavram

Position-dependent shift combined with reverse alphabetical indexing and string reversal

Tahmini Süre:2m 0s

Soru 224Soru →

Four letter-group pair analogies are given below. For each analogy pair, calculate the total sum of forward alphabetical shift steps required to transform the first word into the second word (using standard position values $A=1$ to $Z=26$ , and wrapping around from $Z$ to $A$ where necessary). Arrange the four items in descending order based on their total shift sum value.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence of items arranged in descending order of their total forward shift sum is:

\text{MINT} : \text{SRAZ}

(34), followed by

\text{BARK} : \text{GGUQ}

(20), then

\text{DUST} : \text{HXZX}

(18), and finally

\text{GLOW} : \text{JOSB}

(15).

Calculating the total forward alphabetical shift steps for each pair yields:

\text{MINT} : \text{SRAZ}

has a sum of 34,

\text{BARK} : \text{GGUQ}

has a sum of 20,

\text{DUST} : \text{HXZX}

has a sum of 18, and

\text{GLOW} : \text{JOSB}

has a sum of 15. Sorting these values in descending order (

34 > 20 > 18 > 15

) produces the correct sequence.

Adım Adım Çözüm

1

Calculate the total forward alphabetical shift sum for

\text{DUST} : \text{HXZX}

D(4) \rightarrow H(8) = +4

,

U(21) \rightarrow X(24) = +3

,

S(19) \rightarrow Z(26) = +7

,

T(20) \rightarrow X(24) = +4

. Sum =

4 + 3 + 7 + 4 = 18

.

Determine the step-by-step positional shift for each letter position.

2

Calculate the total forward alphabetical shift sum for

\text{GLOW} : \text{JOSB}

G(7) \rightarrow J(10) = +3

,

L(12) \rightarrow O(15) = +3

,

O(15) \rightarrow S(19) = +4

,

W(23) \rightarrow B(28) = +5

(since

23 + 5 = 28 \equiv 2 \text{ mod } 26

). Sum =

3 + 3 + 4 + 5 = 15

.

Account for wrap-around shift past letter

Z

.

3

Calculate the total forward alphabetical shift sum for

\text{MINT} : \text{SRAZ}

M(13) \rightarrow S(19) = +6

,

I(9) \rightarrow R(18) = +9

,

N(14) \rightarrow A(27) = +13

(since

14 + 13 = 27 \equiv 1 \text{ mod } 26

),

T(20) \rightarrow Z(26) = +6

. Sum =

6 + 9 + 13 + 6 = 34

.

Determine wrap-around shift from

N(14)

to

A(1)

.

4

Calculate the total forward alphabetical shift sum for

\text{BARK} : \text{GGUQ}

B(2) \rightarrow G(7) = +5

,

A(1) \rightarrow G(7) = +6

,

R(18) \rightarrow U(21) = +3

,

K(11) \rightarrow Q(17) = +6

. Sum =

5 + 6 + 3 + 6 = 20

.

Evaluate individual letter shifts.

5

Order the computed values in descending sequence

Comparing

34 > 20 > 18 > 15

gives the sequence:

\text{MINT} : \text{SRAZ}

,

\text{BARK} : \text{GGUQ}

,

\text{DUST} : \text{HXZX}

,

\text{GLOW} : \text{JOSB}

.

Arrange from the highest calculated shift total to the lowest.

Anahtar Kavram

Positional Alphabetical Shift and Cyclic Wrap-Around Analysis in Letter Analogies

Soru 225Soru →

Four individuals ( $P$ , $Q$ , $R$ , and $S$ ) start from the same central point and walk according to different movement paths. Match each person in List-I with their final direction relative to the starting point given in List-II.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Person P matches North-East; Person Q matches South-East; Person R matches North-West; Person S matches South-West.

Each person's movement path consists of an initial cardinal direction displacement followed by a single

90^\circ

turn. Analyzing each turn relative to their facing direction yields coordinates in four distinct ordinal quadrants: Person P at

(+10, +10)

[North-East], Person Q at

(+10, -10)

[South-East], Person R at

(-10, +10)

[North-West], and Person S at

(-10, -10)

[South-West].

Adım Adım Çözüm

1

Determine the final position coordinates for Person P.

Walking North gives

+10\text{ m}

along the Y-axis. Turning right while facing North directs facing East; walking

10\text{ m}

adds

+10\text{ m}

along the X-axis. Final position is

(+10, +10)

, which is North-East.

Determine spatial position relative to origin

(0,0)

.

2

Determine the final position coordinates for Person Q.

Walking South gives

-10\text{ m}

along the Y-axis. Turning left while facing South directs facing East; walking

10\text{ m}

adds

+10\text{ m}

along the X-axis. Final position is

(+10, -10)

, which is South-East.

Turning left when facing South points toward East.

3

Determine the final position coordinates for Person R.

Walking West gives

-10\text{ m}

along the X-axis. Turning right while facing West directs facing North; walking

10\text{ m}

adds

+10\text{ m}

along the Y-axis. Final position is

(-10, +10)

, which is North-West.

Turning right when facing West points toward North.

4

Determine the final position coordinates for Person S.

Walking West gives

-10\text{ m}

along the X-axis. Turning left while facing West directs facing South; walking

10\text{ m}

adds

-10\text{ m}

along the Y-axis. Final position is

(-10, -10)

, which is South-West.

Turning left when facing West points toward South.

Anahtar Kavram

Direction and Distance Test - Cardinal and Ordinal Directions

Soru 226Soru →

A municipal engineer begins inspecting a road pipeline from point $A$ by walking $12\text{ m}$ towards the East. She then turns to her right and walks $5\text{ m}$ to reach point $B$ . What is the shortest direct distance between point $A$ and point $B$ , and in which direction is point $B$ relative to point $A$ ?

13\text{ m}

, South-East

17\text{ m}

, South-East

13\text{ m}

, North-East

17\text{ m}

, North-West

Cevabı ve açıklamayı göster

Cevap: $13\text{ m}$ , South-East

Cevap

The shortest direct distance between point

A

and point

B

is

13\text{ m}

, and point

B

lies in the South-East direction relative to point

A

.

The engineer moves

12\text{ m}

East and then turns right (South) to walk

5\text{ m}

. The straight-line distance forms the hypotenuse of a right triangle with perpendicular sides of

12\text{ m}

and

5\text{ m}

, calculated as

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

. Since point

B

lies to the East and South of point

A

, its position relative to point

A

is South-East.

Adım Adım Çözüm

1

Determine the direction of movement.

Initial movement is

12\text{ m}

East. Turning right while facing East leads to facing South, followed by walking

5\text{ m}

South.

A right turn relative to East points directly South.

2

Identify the relative cardinal direction of point

B

from point

A

.

Point

B

is displaced Eastward and Southward from point

A

, placing it in the South-East quadrant relative to point

A

.

Combining an Eastward vector with a Southward vector results in a South-East displacement.

3

Calculate the direct straight-line distance using the Pythagorean theorem.

\text{Distance} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ m}

.

The perpendicular legs of

12\text{ m}

and

5\text{ m}

form a right-angled triangle where the hypotenuse represents the direct distance.

Anahtar Kavram

Direction reorientation and displacement calculation using the Pythagorean theorem

Soru 227Soru →

In a certain letter analogy pattern, the letter-group $\text{CRAFT}$ is related to $\text{ZKBWI}$ following a specific two-step positional rule combining reverse alphabet complementation and position shifting. Following the same logic, which of the following letter-groups replaces the question mark in the given analogy?

\text{CRAFT} : \text{ZKBWI} :: \text{PRISM} : ?

\text{MKTJP}

\text{RTKUO}

\text{KIRHN}

\text{IGPFL}

Cevabı ve açıklamayı göster

Cevap: $\text{MKTJP}$

Cevap

The letter-group that completes the analogy is

\text{MKTJP}

.

The pattern follows a two-step rule: first, replace each letter with its opposite letter in the reverse alphabetical order (such that the sum of positional values equals 27), and second, shift each obtained letter forward by

+2

positions in the standard alphabet. Applying this to

\text{PRISM}

: P(16)

\to

K(11)

+2 = \text{M}

, R(18)

\to

I(9)

+2 = \text{K}

, I(9)

\to

R(18)

+2 = \text{T}

, S(19)

\to

H(8)

+2 = \text{J}

, M(13)

\to

N(14)

+2 = \text{P}

, producing

\text{MKTJP}

.

Adım Adım Çözüm

1

Analyze the rule transforming

\text{CRAFT}

to

\text{ZKBWI}

.

Step 1a: Find reverse alphabet pairs (

27 - \text{position}

). C(3)

\to

X(24), R(18)

\to

I(9), A(1)

\to

Z(26), F(6)

\to

U(21), T(20)

\to

G(7).
Step 1b: Shift each resulting letter forward by

+2

positions: X(24)

+2 = Z(26)

, I(9)

+2 = K(11)

, Z(26)

+2 = B(2)

, U(21)

+2 = W(23)

, G(7)

+2 = I(9)

. Result:

\text{ZKBWI}

.

Establishes the two-step rule: reverse complement followed by

+2

positional shift.

2

Apply Step 1a (reverse alphabet pair) to each letter of

\text{PRISM}

.

P(16)

\to

K(11), R(18)

\to

I(9), I(9)

\to

R(18), S(19)

\to

H(8), M(13)

\to

N(14). Intermediate group:

\text{KIRHN}

.

Calculates the reverse alphabetical complement for each letter of the target word.

3

Apply Step 1b (

+2

positional shift) to

\text{KIRHN}

.

K(11)

+2 = M(13)

, I(9)

+2 = K(11)

, R(18)

+2 = T(20)

, H(8)

+2 = J(10)

, N(14)

+2 = P(16)

. Final result:

\text{MKTJP}

.

Executes the final positional advancement to complete the analogy pattern.

Anahtar Kavram

Letter Analogy based on Compound Reverse Alphabet Pair and Constant Positional Shift

Soru 228Soru →

In a state revenue department consisting of $200$ inspectors, assignments are carried out across three specialized sectors: Taxation ( $T$ ), Customs ( $C$ ), and Financial Intelligence ( $F$ ). A departmental survey revealed that $105$ inspectors handle Taxation, $90$ handle Customs, and $85$ handle Financial Intelligence. Furthermore, $45$ inspectors handle both Taxation and Customs, $40$ handle both Customs and Financial Intelligence, and $35$ handle both Taxation and Financial Intelligence. If $15$ inspectors do not work in any of these three sectors, how many inspectors are engaged in EXACTLY TWO of the three sectors?

45

70

95

120

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

45 inspectors are engaged in exactly two of the three sectors.

The option stating 45 is correct. By using the principle of inclusion-exclusion, the total number of inspectors operating in at least one sector is

200 - 15 = 185

. Solving

185 = (105 + 90 + 85) - (45 + 40 + 35) + |T \cap C \cap F|

yields

|T \cap C \cap F| = 25

. To find those in exactly two sectors, we subtract 25 from each pairwise intersection (

45-25=20

,

40-25=15

,

35-25=10

), yielding a total of

20 + 15 + 10 = 45

.

Adım Adım Çözüm

1

Determine the total number of inspectors working in at least one sector

|T \cup C \cup F| = 200 - 15 = 185

Inspectors who work in at least one sector equal the total inspectors minus those in none of the sectors.

2

Apply the Principle of Inclusion-Exclusion to find the number of inspectors in all three sectors

|T \cup C \cup F| = |T| + |C| + |F| - (|T \cap C| + |C \cap F| + |T \cap F|) + |T \cap C \cap F| \Rightarrow 185 = 105 + 90 + 85 - (45 + 40 + 35) + |T \cap C \cap F| \Rightarrow 185 = 280 - 120 + |T \cap C \cap F| \Rightarrow |T \cap C \cap F| = 25

The standard set identity relates the union of three sets with their individual sizes and overlapping regions.

3

Calculate the inspectors working in EXACTLY two sectors for each pairwise combination

Taxation and Customs only:

45 - 25 = 20

; Customs and Financial Intelligence only:

40 - 25 = 15

; Taxation and Financial Intelligence only:

35 - 25 = 10

Inspectors in all three sectors are included in every pairwise intersection and must be subtracted to isolate those in exactly two sectors.

4

Sum the counts of inspectors in exactly two sectors

20 + 15 + 10 = 45

Adding these disjoint sets gives the total count of inspectors working in exactly two sectors.

Anahtar Kavram

Principle of Inclusion-Exclusion for 3 Sets

Tahmini Süre:2m 0s

Soru 229Soru →

In a family, $X$ is the sister of $Y$ . $Y$ is married to $Z$ , who is the only son of $W$ . $V$ is the mother-in-law of $Z$ . How is $V$ related to $X$ ?

Cevabı ve açıklamayı göster

Cevap: Mother; mother

Cevap

Mother

V

is the mother-in-law of

Z

, meaning

V

is the mother of

Z

's wife,

Y

. Since

X

is the sister of

Y

,

V

is also the mother of

X

.

Adım Adım Çözüm

1

Identify the relationship between

Z

,

Y

, and

V

V

is the mother-in-law of

Z

. Since

Z

is married to

Y

,

V

is the mother of

Y

.

A person's mother-in-law is the mother of their spouse.

2

Connect

Y

's mother to

X

X

is the sister of

Y

, which means

X

and

Y

share the same mother.

Siblings share the same parents.

3

Deduce the relation of

V

to

X

Since

V

is the mother of

Y

and

X

is

Y

's sister,

V

is the mother of

X

.

The mother of a person's sibling is that person's mother.

Anahtar Kavram

Kinship and In-Law Relationship Deduction

Soru 230Soru →

If January 1, 1900, was a Monday, what day of the week was January 1, 1901?

Wednesday

Sunday

Tuesday

Monday

Cevabı ve açıklamayı göster

Cevap: Tuesday

Cevap

Tuesday

The correct answer is Tuesday because the year 1900 is an ordinary year with 365 days, which produces exactly 1 odd day. Adding 1 day to the starting day of Monday brings us to Tuesday.

Adım Adım Çözüm

1

Determine if the starting year (1900) is a leap year or an ordinary year.

1900 is an ordinary year.

Although 1900 is divisible by 4, century years must be divisible by 400 to be considered leap years. Since 1900 is not divisible by 400, it is an ordinary year.

2

Calculate the number of odd days in an ordinary year.

1 odd day.

An ordinary year has 365 days. Dividing 365 by 7 yields 52 weeks and a remainder of 1 day. This remainder represents the odd day.

3

Add the number of odd days to the given day of the week.

Tuesday.

Since January 1, 1900, was a Monday, moving forward by 1 odd day results in Tuesday.

Anahtar Kavram

Century Leap Year Rules and Odd Days Calculation

Soru 231Soru →

A logical relationship is established between two letter-groups through a compound alphabetical transformation. The transformation linking the word $\text{FORGE}$ to the sequence $\text{GDWHQ}$ relies on a specific structural rearrangement of the letters, followed immediately by a sequence of position-dependent alphabetic shifts. Assuming the exact same multi-step transformation rules are applied, which of the following sequences logically pairs with the word $\text{VAULT}$ ?

\text{XXZEE}

\text{VOZHG}

\text{VIZTG}

\text{UIZTE}

Cevabı ve açıklamayı göster

Cevap: \text{VIZTG}

Cevap

The correct sequence is \text{VIZTG}.

The established logical rule consists of two distinct phases. First, the input word is spelled backwards. Second, the letters are shifted alphabetically by an alternating sequence of addition and subtraction using consecutive prime numbers (+2, -3, +5, -7, +11). Applying this to

\text{VAULT}

: reversing the word yields

\text{TLUAV}

. Applying the shifts:

\text{T}

shifted forward by 2 is

\text{V}

;

\text{L}

shifted backward by 3 is

\text{I}

;

\text{U}

shifted forward by 5 is

\text{Z}

;

\text{A}

shifted backward by 7 wraps around to

\text{T}

; and

\text{V}

shifted forward by 11 wraps around to

\text{G}

. This produces the correct pairing

\text{VIZTG}

.

Adım Adım Çözüm

1

Analyze the structural rearrangement in the given example.

The word

\text{FORGE}

is reversed to form

\text{EGROF}

.

Direct position-to-position mapping between

\text{FORGE}

and

\text{GDWHQ}

yields no logical mathematical sequence, indicating a structural shift occurs first.

2

Determine the alphabetic shifts between the reversed word

\text{EGROF}

and the target sequence

\text{GDWHQ}

.

The shifts are:

\text{E} \xrightarrow{+2} \text{G}

,

\text{G} \xrightarrow{-3} \text{D}

,

\text{R} \xrightarrow{+5} \text{W}

,

\text{O} \xrightarrow{-7} \text{H}

,

\text{F} \xrightarrow{+11} \text{Q}

.

By calculating the numerical difference between the alphabet positions, a clear pattern emerges: the operations alternate between addition and subtraction (+, -, +, -, +), and the magnitudes follow the sequence of prime numbers (2, 3, 5, 7, 11).

3

Apply the identical structural rearrangement to the target word

\text{VAULT}

.

Reversing

\text{VAULT}

gives the sequence

\text{TLUAV}

.

The analogy rule requires the exact same multi-step logic to be applied to the new input.

4

Apply the derived alternating prime number shift sequence to

\text{TLUAV}

.

\text{T} (+2) = \text{V}

;

\text{L} (-3) = \text{I}

;

\text{U} (+5) = \text{Z}

;

\text{A} (-7) = \text{T}

;

\text{V} (+11) = \text{G}

.

\text{A}

(position 1 or 27) minus 7 lands on

\text{T}

(position 20).

\text{V}

(position 22) plus 11 equals 33, wrapping around the 26-letter alphabet to land on

\text{G}

(position 7).

Anahtar Kavram

Compound Letter Analogy involving structural reversal and alternating prime number alphabetic shifts.

Soru 232Soru →

A smart grid feasibility study categorized $1000$ commercial buildings based on their adoption of three energy efficiency upgrades: Smart Lighting ( $L$ ), HVAC Optimization ( $H$ ), and Automated Shading ( $A$ ).

The survey revealed the following:
- $150$ buildings have not adopted any of these three upgrades.
- The number of buildings that have adopted exactly one type of upgrade is identical across all three categories.
- The number of buildings with both $L$ and $H$ , but not $A$ , is twice the number of buildings with all three upgrades.
- The number of buildings with both $H$ and $A$ , but not $L$ , is three times the number of buildings with all three upgrades.
- The number of buildings with both $L$ and $A$ , but not $H$ , is equal to the number of buildings with all three upgrades.
- The total number of buildings that have adopted HVAC Optimization ( $H$ ) is $430$ .

Based on this data, what is the total number of buildings that have adopted exactly two types of upgrades?

240

280

330

40

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

240 buildings

By defining the 'exactly one' regions as

x

and the 'all three' region as

y

, we can model the 'exactly two' regions as

2y

,

3y

, and

y

. This gives us two linear equations based on the given totals:

3x + 7y = 850

(from the union of

1000 - 150

) and

x + 6y = 430

(from the components of set

H

). Solving this system yields

y = 40

. The question asks for the total number of buildings with exactly two upgrades, which is

2y + 3y + y = 6y

. Multiplying

6 \times 40

gives

240

.

Adım Adım Çözüm

1

Determine the total number of buildings with at least one upgrade.

Total Union

= 1000 - 150 = 850

buildings.

Subtracting the buildings with none of the upgrades from the total surveyed gives the union of the three sets.

2

Define algebraic variables for the distinct regions of the Venn diagram.

Let the number of buildings with exactly one upgrade in any category be

x

. Let the number of buildings with all three upgrades be

y

.

Parameterizing the unknown regions allows us to express the given relationships algebraically.

3

Express the regions representing exactly two upgrades in terms of

y

.

n(L \cap H \text{ only}) = 2y

,

n(H \cap A \text{ only}) = 3y

, and

n(L \cap A \text{ only}) = y

. The total for exactly two upgrades is

2y + 3y + y = 6y

.

Translating the comparative statements in the prompt into exact algebraic expressions.

4

Formulate a system of linear equations using the Total Union and Total H.

Total Union equation:

3x + (2y + 3y + y) + y = 850 \Rightarrow 3x + 7y = 850

. Total H equation:

x + 2y + 3y + y = 430 \Rightarrow x + 6y = 430

.

The union includes all distinct regions. The Total H set includes H only, all intersections involving H, and the central intersection.

5

Solve the system of equations to find

y

and calculate the final target (

6y

).

From the second equation,

x = 430 - 6y

. Substituting into the first:

3(430 - 6y) + 7y = 850 \Rightarrow 1290 - 18y + 7y = 850 \Rightarrow 11y = 440 \Rightarrow y = 40

. The total for exactly two is

6(40) = 240

.

This isolates

y

, giving the base unit needed to find the number of buildings with exactly two upgrades.

Anahtar Kavram

Algebraic Formulation of Multi-Set Logical Venn Diagrams

Soru 233Soru →

At an annual fitness convention, $220$ individuals registered for various sessions. The convention offered three main workshop categories: Yoga ( $Y$ ), Pilates ( $P$ ), and Aerobics ( $A$ ).

According to the attendance records:
- $100$ individuals participated in Yoga.
- $110$ individuals participated in Pilates.
- $90$ individuals participated in Aerobics.
- $50$ individuals participated in exactly two of these workshop categories.
- $20$ individuals did not participate in any of the three workshops.

Based on this information, how many individuals participated in all three workshop categories?

15

25

50

75

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

To find the number of people in all three categories, we analyze the structure of the sets. The sum of people in the individual categories is

100 + 110 + 90 = 300

. The actual number of active participants (the union) is the total population minus those who did not participate:

220 - 20 = 200

. When we subtract the union from the sum of the individual sets (

300 - 200 = 100

), this difference represents the overcounted overlaps. Specifically, people in exactly two categories (

E_2

) were counted twice, leaving one extra count. People in all three categories (

E_3

) were counted three times, leaving two extra counts. Therefore,

100 = E_2 + 2E_3

. Since we are given

E_2 = 50

, we substitute to get

100 = 50 + 2E_3

, which simplifies to

2E_3 = 50

, meaning

E_3 = 25

.

Adım Adım Çözüm

1

Determine the number of individuals who participated in at least one workshop (the union of all three sets).

220 - 20 = 200

individuals.

The universal set contains

220

people, but

20

participated in none. The union (

N(Y \cup P \cup A)

) is the total minus the 'none' group.

2

Calculate the total sum of individuals across the three individual sets.

100 + 110 + 90 = 300

.

This sum counts individuals in exactly one category once, individuals in exactly two categories twice, and individuals in all three categories three times.

3

Set up an equation relating the sum of individual sets, the union, the 'exactly two' group (

E_2

), and the 'all three' group (

E_3

).

\text{Sum of Sets} - \text{Union} = E_2 + 2E_3

. Substituting values:

300 - 200 = 50 + 2E_3

.

Subtracting the union (which counts every participant exactly once) from the sum of sets leaves behind the extra counts: one extra count for everyone in exactly two, and two extra counts for everyone in all three.

4

Solve the algebraic equation for

E_3

.

100 = 50 + 2E_3 \implies 50 = 2E_3 \implies E_3 = 25

.

Isolating

E_3

provides the exact number of individuals who participated in all three workshops.

Anahtar Kavram

Logical Venn Diagrams and the application of categorical region sums (Inclusion-Exclusion Principle).

Soru 234Soru →

A cryptography algorithm generates a continuous stream of security tokens based on a polynomial progression. The first five tokens generated by the system are $5$ , $12$ , $31$ , $68$ , and $129$ .

What is the numerical value of the sixth token generated by this algorithm?

Cevabı ve açıklamayı göster

Cevap: 220

Cevap

220

The sequence follows the mathematical rule where the

n

-th term is equal to

n^3 + 4

. The first term is

1^3 + 4 = 5

, the second is

2^3 + 4 = 12

, up to the fifth which is

5^3 + 4 = 129

. Applying this rule to the sixth position yields

6^3 + 4 = 216 + 4 = 220

. Alternatively, solving via successive differences confirms that the constant third difference is 6, which accurately points to 220.

Adım Adım Çözüm

1

Calculate the first-level differences between the given consecutive tokens.

The sequence of differences is 7, 19, 37, and 61.

Establishing the initial rate of change helps identify if a linear or higher-order polynomial pattern exists.

2

Calculate the second-level differences from the results of Step 1.

The differences between the differences are 12, 18, and 24.

Finding the differences of the differences reveals simpler underlying arithmetic progressions.

3

Identify the pattern in the second-level differences and extrapolate the next value.

The values (12, 18, 24) increase by exactly 6 each time. The next second-level difference is 24 + 6 = 30.

Extending this constant third-level difference (+6) is required to build the sequence forward.

4

Calculate the next first-level difference and the final sequence term.

Next first-level difference = 61 + 30 = 91. Next sequence term = 129 + 91 = 220.

Applying the extrapolated values back up the chain yields the target term in the main sequence.

Anahtar Kavram

Number series completion using the method of successive differences or cube offsets.

Tahmini Süre:1m 15s

Soru 235Soru →

A geological survey team is navigating an uncharted terrain using a miscalibrated digital compass. The compass incorrectly displays 'North' when it is actually pointing towards true North-West.

According to the readings on this faulty compass, the team leader executes the following sequence of movements:
- First, he travels $12\sqrt{2}$ km towards the 'East' indicated by the compass.
- Second, he rotates $135^\circ$ to his right and travels $12$ km straight.
- Finally, he rotates $90^\circ$ to his left and travels another $12$ km.

What is the team's shortest straight-line distance and true geographic direction from their starting point?

24

km, True South-East

24

km, True South

24

km, True East

24

km, True North

Cevabı ve açıklamayı göster

Cevap: $24$ km, True East

Cevap

The team's shortest straight-line distance is

24

km, and their true geographic direction from the starting point is True East.

The miscalibrated compass means all readings are shifted. Compass North is True North-West, so Compass East is True North-East. A

12\sqrt{2}

km movement True North-East equals

(12, 12)

on a coordinate plane. A

135^\circ

right turn from Compass East faces Compass South-West, which is True South. Moving

12

km True South changes the position to

(12, 0)

. A

90^\circ

left turn from Compass South-West faces Compass South-East, which is True East. Moving

12

km True East changes the position to

(24, 0)

. This represents a distance of

24

km in the True East direction.

Adım Adım Çözüm

1

Determine the true geographic bearing for the initial movement.

The compass reads 'North' when pointing True North-West. This means true directions are

45^\circ

counter-clockwise from compass readings. The team moves towards Compass 'East', which corresponds to True North-East. The displacement of

12\sqrt{2}

km True North-East translates to

12

km True East and

12

km True North.

Establishing the relationship between the faulty compass and true geographic directions is required to map the exact path correctly.

2

Calculate the true bearing and displacement for the second movement.

Turning

135^\circ

right (clockwise) from Compass East results in facing Compass South-West. Compass South-West corresponds to True South. Moving

12

km True South cancels out the previous

12

km True North, placing the team at exactly

12

km True East from the starting point.

Angular turns apply to the current facing direction. A

135^\circ

clockwise rotation from East lands on South-West.

3

Calculate the true bearing and displacement for the final movement.

Turning

90^\circ

left (counter-clockwise) from Compass South-West results in facing Compass South-East. Compass South-East corresponds to True East. Moving

12

km True East adds to the previous position, resulting in a total displacement of

24

km True East.

A

90^\circ

counter-clockwise rotation from South-West lands on South-East. Summing the true vectors provides the final destination.

Anahtar Kavram

Vector Addition and Relative Bearings in Miscalibrated Systems

Soru 236Soru →

In a vertical stack of passport applications, Amit's folder is placed $8^{\text{th}}$ from the top and $16^{\text{th}}$ from the bottom. What is the total number of passport applications in this stack?

22

23

24

25

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

The total number of passport applications in the stack is 23.

To find the total number of items in a linear arrangement when one item's position is given from both ends, you add the two position numbers and subtract 1. Adding the top rank (8) and the bottom rank (16) gives 24. We must then subtract 1 because Amit's folder is included in both the '8' and the '16'. Therefore,

8 + 16 - 1 = 23

.

Adım Adım Çözüm

1

Identify the top and bottom rank of the specific item.

Top position = 8, Bottom position = 16.

Both positions of the same object are required to determine the overall length of the sequence.

2

Apply the dual rank formula for a single item: Total = Top + Bottom - 1.

8 + 16 - 1 = 23.

Subtracting 1 is necessary because Amit's folder is counted twice—once when counting from the top, and once when counting from the bottom.

Anahtar Kavram

Single-person dual rank total calculation

Soru 237Soru →

Four runners participated in a 100-meter sprint. Based on the information provided, arrange the runners in order of their finishing positions, starting from 1st place (fastest) to 4th place (slowest).

- David finished ahead of Michael but behind John.
- Robert finished behind Michael.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence from fastest to slowest is John, David, Michael, and Robert.

Based on the provided statements, John is faster than David, who is faster than Michael, who is faster than Robert. Therefore, the continuous sequence from 1st place to 4th place is John, David, Michael, and Robert.

Adım Adım Çözüm

1

Analyze the first statement regarding David, Michael, and John.

John > David > Michael

David finished ahead of Michael (David is faster than Michael) but behind John (John is faster than David).

2

Analyze the second statement regarding Robert and Michael.

Michael > Robert

Robert finished behind Michael, meaning Michael is faster than Robert.

3

Combine the relative rankings.

John > David > Michael > Robert

Using transitive logic on the established sequences determines the final absolute order.

Anahtar Kavram

Relative position and transitive ranking

Soru 238Soru →

During an archaeological excavation, researchers found a set of tablets with inscriptions. By comparing contextual texts, they established the following phrase translations:

'ancient hidden temple' is coded as 'zola mizu kanto'
'hidden gold treasure' is coded as 'mizu pelta vosh'
'ancient stone treasure' is coded as 'kanto nubo vosh'
'stone temple pathway' is coded as 'nubo zola rima'

Based on this information, match each English word with its corresponding ancient code.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matching is: 'ancient' matches with 'kanto', 'hidden' matches with 'mizu', 'treasure' matches with 'vosh', and 'temple' matches with 'zola'.

By systematically comparing pairs of phrases that share exactly one common English word, we can isolate the corresponding common code word regardless of its position in the sentence. This logical intersection process reveals that 'ancient' = 'kanto', 'hidden' = 'mizu', 'treasure' = 'vosh', and 'temple' = 'zola'.

Adım Adım Çözüm

1

Compare the first phrase ('ancient hidden temple') and the third phrase ('ancient stone treasure').

The common English word is 'ancient' and the common code is 'kanto'.

Finding the intersection of words and codes between two phrases isolates the translation for the shared word.

2

Compare the first phrase ('ancient hidden temple') and the second phrase ('hidden gold treasure').

The common English word is 'hidden' and the common code is 'mizu'.

Applying the same intersection method identifies the code for the next target word.

3

Compare the second phrase ('hidden gold treasure') and the third phrase ('ancient stone treasure').

The common English word is 'treasure' and the common code is 'vosh'.

This confirms the translation for 'treasure' by isolating it across two different contexts.

4

Compare the first phrase ('ancient hidden temple') and the fourth phrase ('stone temple pathway').

The common English word is 'temple' and the common code is 'zola'.

This completes the required mapping for all four target words.

Anahtar Kavram

Phrase Substitution and Matrix Decoding

Soru 239Soru →

Six friends (M, N, O, P, Q, and R) are sitting in a straight row facing North. Based on the following conditions, arrange the friends in the correct order from left to right:

1. M sits at the extreme left end of the row.
2. R sits at the extreme right end of the row.
3. There are exactly two people sitting between M and P.
4. N sits immediately to the left of P.
5. Q sits immediately to the left of R.
6. O sits immediately to the right of M.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence from left to right is M, O, N, P, Q, and R.

By systematically applying the absolute boundary conditions first (M and R at the ends) and then mapping out the relative conditions (gaps and immediate neighbors), only one unique arrangement is possible: M, O, N, P, Q, R.

Adım Adım Çözüm

1

Determine the extreme ends of the row.

M is placed at position 1, and R is placed at position 6. (M, _, _, _, _, R)

Conditions 1 and 2 explicitly state the exact positions of M and R at the left and right ends respectively.

2

Find P's position relative to M.

P is placed at position 4. (M, _, _, P, _, R)

Condition 3 dictates there are exactly two people between M (position 1) and P. The only valid position for P is position 4.

3

Place N relative to P.

N is placed at position 3. (M, _, N, P, _, R)

Condition 4 states N sits immediately to the left of P. Since P is 4th, N is 3rd.

4

Place Q relative to R.

Q is placed at position 5. (M, _, N, P, Q, R)

Condition 5 states Q sits immediately to the left of R. Since R is 6th, Q is 5th.

5

Place O relative to M.

O is placed at position 2. (M, O, N, P, Q, R)

Condition 6 states O sits immediately to the right of M, which fills the last remaining seat.

Anahtar Kavram

Linear Seating Arrangement

Soru 240Soru →

Six executives—Hasan, Imran, Julia, Kevin, Liam, and Maria—are seated around a circular table facing the center for a board meeting. The seating arrangement follows these conditions:

- Hasan is seated immediately to the right of Kevin.
- Julia is seated exactly opposite to Kevin.
- Liam is seated immediately to the left of Julia.
- Imran is seated immediately to the left of Kevin.

Based on the information provided, who is seated immediately to the right of Maria?

Imran

Julia

Liam

Hasan

Cevabı ve açıklamayı göster

Cevap: Imran

Cevap

Imran is seated immediately to the right of Maria.

By mapping out the seats one by one, we find that Imran sits to the left of Kevin, and Julia sits opposite Kevin. This leaves the seat between Imran and Julia for Maria. Since Maria is facing the center, the person to her immediate right (in the counter-clockwise direction) is Imran.

Adım Adım Çözüm

1

Establish a baseline position and determine left/right directions for facing the center.

Place Kevin at Position 1 (bottom of the circle). When facing the center, 'Left' is clockwise and 'Right' is counter-clockwise.

Setting a fixed reference point prevents orientation mistakes in circular arrangements.

2

Place Hasan and Julia using their relationships to Kevin.

Hasan is at Position 6 (immediately to the right/counter-clockwise of Kevin). Julia is at Position 4 (exactly opposite Kevin).

These are direct placements based on the first two clues.

3

Place Liam and Imran based on the remaining specific clues.

Liam is at Position 5 (immediately left/clockwise of Julia). Imran is at Position 2 (immediately left/clockwise of Kevin).

Applying the clockwise rule for 'left' locks these individuals into their exact seats.

4

Determine Maria's seat and answer the final question.

Maria must be in the only empty seat, Position 3. The person immediately to the right (counter-clockwise) of Maria is Imran (at Position 2).

After 5 people are placed, the 6th person takes the last seat. We then apply the direction rule one final time.

Anahtar Kavram

Circular arrangement with inward-facing directional logic.

Tahmini Süre:1m 0s

General Mental Ability

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