All practice questions

Question 261Question →

What is the unit digit of the expression $2^{52} + 5^{17}$ ?

1

6

7

3

Show answer & explanation

Answer: 1

Answer

The unit digit of the expression

2^{52} + 5^{17}

is 1.

For

2^{52}

, dividing 52 by 4 leaves a remainder of 0. Since the cyclicity of 2 is 4, a remainder of 0 means using

2^4 = 16

, which has a unit digit of 6. For

5^{17}

, any power of 5 ends in 5. The total unit digit is the unit digit of

6 + 5 = 11

, which is 1.

Step-by-Step Solution

1

Find the unit digit of

2^{52}

using cyclicity.

The cyclicity of 2 is 4 (pattern: 2, 4, 8, 6). Dividing the exponent 52 by 4 yields

52 = 4 \times 13 + 0

. A remainder of 0 corresponds to the 4th power in the cycle.

2^4 = 16

, so the unit digit of

2^{52}

is 6.

When the exponent is exactly divisible by the cyclicity length, the unit digit corresponds to the power equal to the cyclicity length.

2

Find the unit digit of

5^{17}

.

The cyclicity of 5 is 1, meaning any positive integer power of 5 ends in 5. Thus, the unit digit of

5^{17}

is 5.

Base numbers ending in 5 always yield a unit digit of 5 for any positive integer exponent.

3

Add the individual unit digits to get the final unit digit.

Unit digit of

(2^{52} + 5^{17}) = \text{Unit digit of }(6 + 5) = \text{Unit digit of }11 = 1

.

The unit digit of a sum is equal to the unit digit of the sum of the individual unit digits.

Key Concept

Unit Digit and Cyclicity

Estimated Time:45s

Question 262Question →

What is the simplified value of the following mathematical expression obtained by applying the BODMAS rule?

96 \div \left[ 4 + \left\{ 25 - \left( 12 - \overline{8 - 3} \right) \times 3 \right\} \right]

12

28

4

16

Show answer & explanation

Answer: 12

Answer

12

Following the standard BODMAS precedence rule: first evaluate the vinculum

\overline{8-3} = 5

, then the round brackets

(12-5) = 7

, then multiplication

7 \times 3 = 21

, then curly brackets

25 - 21 = 4

, then square brackets

4 + 4 = 8

, and finally division

96 \div 8 = 12

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{8 - 3} = 5

The vinculum takes precedence over other bracket operations.

2

Evaluate the operation inside the innermost round brackets

12 - 5 = 7

Resolve the expression within the round brackets next.

3

Perform multiplication inside the curly brackets

7 \times 3 = 21

Multiplication precedes subtraction within the curly brackets according to BODMAS.

4

Complete the subtraction inside the curly brackets

25 - 21 = 4

Evaluate the remaining subtraction inside the curly brackets.

5

Evaluate the addition inside the outer square brackets

4 + 4 = 8

Combine the terms within the square brackets to obtain the final divisor.

6

Perform the final division

96 \div 8 = 12

Divide the initial integer by the evaluated square bracket result.

Key Concept

Simplification and BODMAS Rule with Vinculum

Question 263Question →

In a administrative evaluation conducted for $250$ government officers, data was collected regarding the completion of three specialized training modules: Cyber Security ( $C$ ), Data Analytics ( $D$ ), and Public Procurement ( $P$ ).

- $130$ officers completed Cyber Security
- $110$ officers completed Data Analytics
- $100$ officers completed Public Procurement
- $55$ officers completed both Cyber Security and Data Analytics
- $40$ officers completed both Data Analytics and Public Procurement
- $45$ officers completed both Cyber Security and Public Procurement
- $20$ officers completed all three modules

Based on the information provided, how many officers completed exactly two training modules?

80

140

120

100

Show answer & explanation

Answer: $80$

Answer

The number of officers who completed exactly two training modules is

80

.

To find the number of officers who completed exactly two modules, we isolate the three disjoint regions that represent taking two modules but not the third. Subtraction of the three-module intersection (

20

) from each pair overlap yields:

(55 - 20) + (40 - 20) + (45 - 20) = 35 + 20 + 25 = 80

.

Step-by-Step Solution

1

Identify the given set values and intersections

N(C) = 130

,

N(D) = 110

,

N(P) = 100

,

N(C \cap D) = 55

,

N(D \cap P) = 40

,

N(C \cap P) = 45

,

N(C \cap D \cap P) = 20

Extracting all cardinalities provided in the stem.

2

Calculate the number of officers in each exclusive two-set intersection region

Cyber & Data only

= 55 - 20 = 35

; Data & Procurement only

= 40 - 20 = 20

; Cyber & Procurement only

= 45 - 20 = 25

The intersection

N(X \cap Y)

contains people taking all three modules as well. Subtracting

N(C \cap D \cap P)

isolates those taking ONLY two modules.

3

Sum the exclusive two-module regions

35 + 20 + 25 = 80

Adding the three disjoint regions corresponding to exactly two training modules.

Key Concept

3-Set Venn Diagram Region Isolation and Inclusion-Exclusion

Estimated Time:1m 30s

Question 264Question →

What is the unit digit of the expression $E = (842^{324} \times 617^{153}) + 358^{217}$ ?

Show answer & explanation

Answer: 0

Answer

The unit digit of the expression

E = (842^{324} \times 617^{153}) + 358^{217}

is 0.

For

842^{324}

, exponent

324 \pmod 4 = 0

corresponds to the 4th power position in cyclicity (

2^4 = 16 \rightarrow 6

). For

617^{153}

,

153 \pmod 4 = 1

corresponds to

7^1 = 7

. The product of these terms ends in

6 \times 7 = 42 \rightarrow 2

. For

358^{217}

,

217 \pmod 4 = 1

corresponds to

8^1 = 8

. Adding the product's unit digit (2) and

358^{217}

's unit digit (8) gives

2 + 8 = 10 \rightarrow 0

.

Step-by-Step Solution

1

Find the unit digit of

842^{324}

Unit digit is 6

The unit digit of base 842 is 2, which repeats unit digits in a cycle of 4:

(2, 4, 8, 6)

. Dividing exponent 324 by 4 yields remainder 0, indicating the 4th position in the cycle,

2^4 = 16 \rightarrow 6

.

2

Find the unit digit of

617^{153}

Unit digit is 7

The unit digit of base 617 is 7, which repeats in a cycle of 4:

(7, 9, 3, 1)

. Dividing exponent 153 by 4 yields remainder 1, indicating the 1st position in the cycle,

7^1 = 7

.

3

Calculate the unit digit of the product

(842^{324} \times 617^{153})

Unit digit is 2

Multiplying the unit digits of the two terms gives

6 \times 7 = 42

, which has a unit digit of 2.

4

Find the unit digit of

358^{217}

Unit digit is 8

The unit digit of base 358 is 8, which repeats in a cycle of 4:

(8, 4, 2, 6)

. Dividing exponent 217 by 4 yields remainder 1, indicating the 1st position in the cycle,

8^1 = 8

.

5

Add the unit digit of the product to the unit digit of

358^{217}

Final unit digit is 0

Adding the unit digit of the product (2) and the unit digit of the final term (8) gives

2 + 8 = 10

, which ends in 0.

Key Concept

Unit Digit Computation using Exponents and Cyclicity

Question 265Question →

What is the simplified numerical value of the following mathematical expression when evaluated using the standard BODMAS rule?

240 \div \left[ 2 \frac{1}{2} \times \left\{ 14 - \left( 3 \frac{1}{2} + 25\% \text{ of } \left( 14 - \overline{1.8 + 2.2} \right) \right) \right\} \right]

12

16

768

8

Show answer & explanation

Answer: 12

Answer

12

Following the strict hierarchical order of operations (BODMAS: Vinculum

\rightarrow

Brackets

\rightarrow

Orders

\rightarrow

Division/Multiplication

\rightarrow

Addition/Subtraction), the innermost vinculum

\overline{1.8 + 2.2}

evaluates to

4

. Then

14 - 4 = 10

, and

25\%

of

10

equals

2.5

. Adding this to

3 \frac{1}{2} (3.5)

yields

6

. Inside the curly brackets,

14 - 6 = 8

. Inside the square brackets,

2 \frac{1}{2} (2.5) \times 8 = 20

. Finally,

240 \div 20 = 12

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{1.8 + 2.2} = 4

The vinculum acts as the innermost grouping symbol and must be simplified first.

2

Simplify the innermost parenthetical difference

14 - 4 = 10

Subtract the vinculum result from 14 within the round brackets.

3

Calculate the percentage of the result from Step 2

25\% \text{ of } 10 = \frac{25}{100} \times 10 = 2.5

'Of' represents multiplication in percentage evaluation.

4

Complete the evaluation inside the round brackets

3 \frac{1}{2} + 2.5 = 3.5 + 2.5 = 6

Convert the mixed fraction

3 \frac{1}{2}

to decimal

3.5

and sum with

2.5

.

5

Evaluate the expression inside the curly brackets

14 - 6 = 8

Subtract the round bracket total from 14.

6

Simplify the expression inside the square brackets

2 \frac{1}{2} \times 8 = 2.5 \times 8 = 20

Multiply the mixed fraction

2 \frac{1}{2}

by the curly bracket result.

7

Perform the final division outside the brackets

240 \div 20 = 12

Divide 240 by the completely simplified value of the square brackets.

Key Concept

BODMAS Rule with Vinculum and Multi-level Brackets

Question 266Question →

Five project managers—Arjun, Bina, Chirag, Divya, and Ekta—are to be assigned to three project teams: Team Alpha, Team Beta, and Team Gamma, according to the following conditions:
1. Each team must contain at least one manager, and Team Alpha must contain exactly two managers.
2. Arjun and Bina are assigned to the same team.
3. Chirag cannot be assigned to Team Beta.
4. Divya and Ekta must be assigned to different teams.
5. Ekta is assigned to Team Gamma.
6. Chirag and Ekta cannot be assigned to the same team.

Which of the following statements is DEFINITELY correct regarding the team assignments?

Chirag and Divya are assigned to Team Alpha.

Arjun and Bina are assigned to Team Alpha.

Divya is assigned to Team Beta.

Chirag and Ekta are both assigned to Team Gamma.

Show answer & explanation

Answer: Chirag and Divya are assigned to Team Alpha.

Answer

Chirag and Divya are assigned to Team Alpha.

By logical deduction from the given constraints, Chirag must be assigned to Team Alpha because he cannot be in Team Beta or Team Gamma. Arjun and Bina must be in Team Beta to prevent capacity violations or condition conflicts. Divya must join Chirag in Team Alpha to satisfy the requirement of exactly two members in Team Alpha while keeping Divya and Ekta separated.

Step-by-Step Solution

1

Analyze Ekta's placement and condition restrictions

Ekta is assigned to Team Gamma. Chirag cannot be in Team Beta (condition 3) and cannot be in Team Gamma with Ekta (condition 6). Therefore, Chirag must be in Team Alpha.

Elimination of invalid teams for Chirag leaves only Team Alpha.

2

Determine the placement of Arjun and Bina

Arjun and Bina must be in the same team. If they were in Team Alpha, Team Alpha would reach its maximum capacity of 2 (Arjun and Bina). This would leave Chirag unable to be in Team Alpha, creating a contradiction. If Arjun and Bina were in Team Gamma, Team Gamma would have 3 members, forcing Team Beta to have 0 members, violating condition 1. Thus, Arjun and Bina must be assigned to Team Beta.

Testing possible team assignments for the Arjun-Bina pair shows Team Beta is the only consistent choice.

3

Assign the remaining manager, Divya

Divya cannot be in Team Gamma because Ekta is in Team Gamma (condition 4). Team Beta already has 2 members (Arjun and Bina). Since Team Alpha needs exactly 2 members and currently has Chirag, Divya must join Team Alpha.

Satisfies the team capacity of 2 for Team Alpha and condition 4.

4

Finalize and verify team compositions

Team Alpha: {Chirag, Divya} (2 members), Team Beta: {Arjun, Bina} (2 members), Team Gamma: {Ekta} (1 member). Total = 5 members.

All 6 given conditions are fully satisfied without any contradiction.

Key Concept

Analytical Grouping and Constraint Satisfaction

Question 267Question →

What is the remainder when the expression $15^{100} + 19^{100}$ is divided by $17$ ?

15

2

1

3

Show answer & explanation

Answer: 15

Answer

The remainder when

15^{100} + 19^{100}

is divided by

17

is

15

.

Reducing the bases modulo 17 gives

15 \equiv -2

and

19 \equiv 2

. Since

100

is even,

(-2)^{100} + 2^{100} = 2 \cdot 2^{100} = 2^{101}

. By Fermat's Little Theorem (

2^{16} \equiv 1 \pmod{17}

),

2^{101} \equiv 2^5 = 32 \equiv 15 \pmod{17}

. Thus, the correct remainder is 15.

Step-by-Step Solution

1

Reduce the base numbers modulo 17

15 \equiv -2 \pmod{17}

and

19 \equiv 2 \pmod{17}

Using negative remainders simplifies power calculations.

2

Rewrite the expression using modular congruence

15^{100} + 19^{100} \equiv (-2)^{100} + 2^{100} = 2^{100} + 2^{100} = 2 \times 2^{100} = 2^{101} \pmod{17}

Since the exponent

100

is even,

(-2)^{100} = 2^{100}

.

3

Apply Fermat's Little Theorem to simplify the exponent

2^{16} \equiv 1 \pmod{17}

17 is prime and

\gcd(2, 17) = 1

.

4

Divide the exponent 101 by 16 to find the remaining power

101 = 16 \times 6 + 5 \implies 2^{101} \equiv (2^{16})^6 \times 2^5 \equiv 1^6 \times 32 \equiv 32 \pmod{17}

Any integer power of 1 is 1 under modular arithmetic.

5

Compute the final positive remainder

32 = 17 \times 1 + 15 \implies 32 \equiv 15 \pmod{17}

(or

-2 + 17 = 15

)

Remainders must be non-negative integers strictly less than the divisor.

Key Concept

Fermat's Little Theorem and Negative Remainders in Modular Arithmetic

Question 268Question →

A survey was conducted among $400$ administrative officers in a state secretariat regarding their operational oversight of three major public welfare projects: Project Alpha, Project Beta, and Project Gamma. The survey collected the following data:

- $190$ officers oversee Project Alpha.
- $180$ officers oversee Project Beta.
- $170$ officers oversee Project Gamma.
- $85$ officers oversee both Project Alpha and Project Beta.
- $75$ officers oversee both Project Beta and Project Gamma.
- $70$ officers oversee both Project Alpha and Project Gamma.
- $40$ officers do not oversee any of these three projects.

Based on the given information, how many administrative officers oversee exactly two projects?

80

130

180

230

Show answer & explanation

Answer: $80$

Answer

The number of administrative officers who oversee exactly two projects is

80

.

The value

80

is correct because subtracting those in no projects from the total of

400

gives

360

officers in at least one project. Using the inclusion-exclusion principle formula yields

50

officers in all three projects. Subtracting

50

from each pairwise intersection gives

35

,

25

, and

20

for the three exclusive two-project regions, summing to

80

.

Step-by-Step Solution

1

Calculate the total number of officers supervising at least one project.

|A \cup B \cup C| = 400 - 40 = 360

Officers who oversee at least one project are found by subtracting those overseeing none from the total surveyed officers.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the triple intersection (

|A \cap B \cap C|

).

360 = 190 + 180 + 170 - (85 + 75 + 70) + |A \cap B \cap C| \implies 360 = 540 - 230 + |A \cap B \cap C| \implies |A \cap B \cap C| = 50

The standard set formula accounts for single set totals, pairwise intersections, and the three-set intersection.

3

Determine the number of officers in each exclusive two-project intersection.

Only Alpha & Beta =

85 - 50 = 35

; Only Beta & Gamma =

75 - 50 = 25

; Only Alpha & Gamma =

70 - 50 = 20

Subtracting the triple intersection from each pairwise intersection yields the count of officers managing exactly two projects.

4

Sum the exclusive two-project regions.

35 + 25 + 20 = 80

Adding these three disjoint set regions gives the total number of officers handling exactly two projects.

Key Concept

Three-Set Inclusion-Exclusion Principle and Set Region Decomposition

Estimated Time:2m 30s

Question 269Question →

The following table shows the number of employees across three departments in a company over three years, with some missing data indicated by dashes:

Department	2021	2022	2023	Total
HR	40	—	50	140
IT	80	90	—	270
Finance	—	60	70	180
Total	170	200	220	590

What is the number of employees in the HR department in the year 2022?

Show answer & explanation

Answer: 50

Answer

The number of employees in the HR department in 2022 is 50.

The row total for HR is 140. Subtracting the known figures for 2021 (40) and 2023 (50) from 140 yields 50 employees for 2022.

Step-by-Step Solution

1

Examine the row corresponding to the HR department.

Known HR employees in 2021 = 40, 2023 = 50, and Total HR employees = 140.

The row total equals the sum of values across all three years.

2

Calculate the missing 2022 HR value.

140 - (40 + 50) = 50 employees.

Subtracting the sum of 2021 and 2023 values from the total gives the missing value for 2022.

Key Concept

Deriving missing data elements using row sums

Estimated Time:45s

Question 270Question →

Consider the following two events in a coastal district:
Statement I: The local meteorological department issued a heavy rainfall and severe storm alert for the coastal district.
Statement II: The district administration ordered all schools and colleges in the coastal area to remain closed for two days.

Which of the following statements regarding the logical relationship between Statement I and Statement II are correct?

Select all that apply

Statement I functions as the primary cause, and Statement II is its direct effect.

Statement II functions as the primary cause, and Statement I is its direct effect.

Statement II represents an administrative precautionary response directly triggered by the event in Statement I.

Both Statement I and Statement II are effects of completely independent causes.

Show answer & explanation

Answer: Statement I functions as the primary cause, and Statement II is its direct effect.; Statement II represents an administrative precautionary response directly triggered by the event in Statement I.

Answer

The correct statements are that Statement I functions as the cause while Statement II is its direct effect, and Statement II represents an administrative precautionary measure directly triggered by Statement I.

An official storm alert logically prompts administrative bodies to execute safety protocols like closing schools. Therefore, Statement I acts as the primary cause while Statement II serves as its direct administrative effect.

Step-by-Step Solution

1

Analyze Statement I

Statement I announces an official heavy rainfall and storm alert.

Establishes the initial condition or triggering event.

2

Analyze Statement II

Statement II describes the temporary closure of educational institutions by local authorities.

Establishes the responsive measure taken by authorities.

3

Evaluate the causal link

The weather alert (Statement I) logically precedes and motivates the safety directive (Statement II).

Administrative bodies issue closure orders to protect students following weather alerts, making Statement I the cause and Statement II the effect.

Key Concept

Cause and Effect Reasoning: Identifying primary causes and administrative precautionary effects

Question 271Question →

A commuter walks $8\text{ km}$ due South from a train station to reach a park. Facing South, he then turns left and walks $6\text{ km}$ to reach his office. What is the shortest distance between the train station and his office, and in which direction is the office relative to the train station?

10\text{ km}

, South-East

10\text{ km}

, South-West

14\text{ km}

, South-East

10\text{ km}

, North-East

Show answer & explanation

Answer: $10\text{ km}$ , South-East

Answer

The shortest distance between the train station and the office is

10\text{ km}

, and the office is in the South-East direction relative to the train station.

The initial path

8\text{ km}

South and the subsequent path

6\text{ km}

East form two perpendicular sides of a right triangle. Applying the Pythagorean theorem yields a hypotenuse of

\sqrt{8^2 + 6^2} = 10\text{ km}

. Because the final position is East of the vertical axis and South of the horizontal axis, the direction relative to the start is South-East.

Step-by-Step Solution

1

Set up coordinate system and plot the initial movement.

Taking the train station as the origin

(0, 0)

, walking

8\text{ km}

South places the commuter at coordinates

(0, -8)

.

South corresponds to the negative y-axis direction.

2

Determine the direction and position after the turn.

Facing South, a left turn reorients the commuter toward the East (+x direction). Walking

6\text{ km}

East moves the position to

(6, -8)

.

Turning left from South points toward East.

3

Calculate the shortest distance using the Pythagorean theorem.

\text{Distance} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ km}

.

The straight-line distance from origin

(0,0)

to point

(6,-8)

forms a right-angled triangle with leg lengths 6 and 8.

4

Determine final direction relative to the starting origin.

Since

x > 0

and

y < 0

, the point

(6, -8)

lies in the South-East quadrant relative to

(0,0)

.

Positive x (East) and negative y (South) define South-East.

Key Concept

Pythagorean displacement calculation and relative cardinal direction mapping

Question 272Question →

An autonomous security robot at an industrial facility begins its inspection patrol from central gate $O$ facing East. It moves $14\text{ m}$ due East to reach checkpoint $A$ . From checkpoint $A$ , it turns $135^\circ$ anti-clockwise and travels $10\sqrt{2}\text{ m}$ to checkpoint $B$ . At checkpoint $B$ , it turns $90^\circ$ clockwise and travels $5\sqrt{2}\text{ m}$ to checkpoint $C$ . Next, it turns $90^\circ$ anti-clockwise and travels $4\sqrt{2}\text{ m}$ to reach checkpoint $D$ . Finally, it turns to face due South and moves $7\text{ m}$ to reach its final parking dock $E$ . What is the shortest straight-line distance (in meters) between central gate $O$ and final parking dock $E$ ?

Show answer & explanation

Answer: 13

Answer

The shortest straight-line distance between central gate

O

and final parking dock

E

is

13\text{ m}

.

By resolving each directional move into its respective orthogonal

(x, y)

components, the cumulative net position of the robot from origin

O(0, 0)

is found to be

(5, 12)

. Applying the distance formula gives

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

.

Step-by-Step Solution

1

Establish Cartesian Coordinates

Central gate

O = (0, 0)

, with

+x

as East,

-x

as West,

+y

as North, and

-y

as South.

Cartesian representation allows precise vector addition for multi-directional paths.

2

Calculate displacement to checkpoint A

Displacement is

(14, 0)

, so

A = (14, 0)

.

The robot travels

14\text{ m}

directly East along the positive x-axis.

3

Calculate displacement to checkpoint B

Displacement vector is

(-10, 10)

, so

B = (4, 10)

.

Turning

135^\circ

anti-clockwise from East points North-West (

135^\circ

). Distance

10\sqrt{2}\text{ m}

gives

\Delta x = -10\sqrt{2}\cdot\frac{1}{\sqrt{2}} = -10

and

\Delta y = 10\sqrt{2}\cdot\frac{1}{\sqrt{2}} = 10

.

4

Calculate displacement to checkpoint C

Displacement vector is

(5, 5)

, so

C = (9, 15)

.

Turning

90^\circ

clockwise from North-West points North-East (

45^\circ

). Distance

5\sqrt{2}\text{ m}

gives

\Delta x = 5

and

\Delta y = 5

.

5

Calculate displacement to checkpoint D

Displacement vector is

(-4, 4)

, so

D = (5, 19)

.

Turning

90^\circ

anti-clockwise from North-East points North-West (

135^\circ

). Distance

4\sqrt{2}\text{ m}

gives

\Delta x = -4

and

\Delta y = 4

.

6

Calculate displacement to final dock E

Displacement vector is

(0, -7)

, so

E = (5, 12)

.

Moving

7\text{ m}

due South reduces the y-coordinate by

7

.

7

Compute shortest straight-line distance OE

Distance

OE = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ m}

.

Apply the Pythagorean theorem to net displacement components

x = 5

and

y = 12

.

Key Concept

Vector Addition of Angular Displacements and Pythagorean Theorem

Estimated Time:2m 0s

Question 273Question →

Five administrative officers—Anish, Brinda, Chetan, Divya, and Ekta—are assigned to five different departments (Finance, HR, IT, Marketing, and Operations) and posted across five distinct cities (Delhi, Mumbai, Kolkata, Chennai, and Bengaluru). No two officers belong to the same department or are posted in the same city.

The assignments follow these conditions:
1. The officer posted in Mumbai works in the IT Department.
2. Anish works in the Finance Department and is posted in either Delhi or Chennai.
3. Brinda is posted in Kolkata, but does not work in HR or Marketing.
4. The officer working in the Marketing Department is posted in Bengaluru.
5. Divya is posted in Delhi.
6. Ekta does not work in the IT Department or Operations Department.

Based on the analytical puzzle above, match each officer on the left with their correct department and posting location on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Anish matches with Finance Department (Chennai); Brinda matches with Operations Department (Kolkata); Chetan matches with IT Department (Mumbai); and Divya matches with HR Department (Delhi).

Step-by-step logical deduction confirms: Anish is posted in Chennai with Finance; Brinda is posted in Kolkata with Operations; Chetan is posted in Mumbai with IT; Divya is posted in Delhi with HR; and Ekta is posted in Bengaluru with Marketing.

Step-by-Step Solution

1

Determine City Assignments for all officers.

Divya is in Delhi (Condition 5) and Brinda is in Kolkata (Condition 3). Anish must be in Chennai because Delhi is occupied by Divya (Condition 2). Ekta cannot be in Mumbai as she does not work in IT (Conditions 1 & 6), so Ekta is in Bengaluru (Condition 4). By elimination, Chetan is posted in Mumbai.

Establishing spatial/location slots reduces variables from 3 dimensions to 2.

2

Map fixed City-Department pairs.

Mumbai = IT Department (Condition 1), so Chetan works in IT. Bengaluru = Marketing Department (Condition 4), so Ekta works in Marketing.

Direct conditional pairings lock in Chetan and Ekta's departments.

3

Assign remaining departments to Anish, Brinda, and Divya.

Anish works in Finance (Condition 2). Brinda (Kolkata) cannot work in HR or Marketing (Condition 3), leaving Operations for Brinda. Divya (Delhi) gets the final remaining department, HR.

Process of elimination determines all officer-department-city mappings deterministically.

Key Concept

Analytical Puzzles and Multi-Dimensional Elimination

Question 274Question →

Consider the following multi-pattern alphanumeric sequence:

D4W, G9T, L16P, S35K, D66E, \dots

Which of the following terms logically completes the sequence?

Q137X

Q137L

M137X

Q137Y

Show answer & explanation

Answer: $Q137X$

Answer

The term that logically completes the sequence is

Q137X

.

The next term in the sequence is obtained by synthesizing three distinct rules: the first letter moves forward by consecutive prime steps (

+13

from

D

yields

Q

), the number multiplies by

2

and adds

5

to yield

137

, and the trailing letter moves

+7

steps in reverse positional values (

E

at reverse index 22 moves to 29, which is reverse index 3 or letter

X

). Combining these gives

Q137X

.

Step-by-Step Solution

1

Analyze the first letter progression:

D \rightarrow G \rightarrow L \rightarrow S \rightarrow D \rightarrow ?

Alphabetical positions (1 to 26):

D=4, G=7, L=12, S=19, D=30 \equiv 4\pmod{26}

. The step differences are

+3, +5, +7, +11

, which are prime numbers. The next prime increment is

+13

. Adding

13

to

D(4)

gives

17

, corresponding to letter

Q

.

The first letter advances according to consecutive prime increments modulo 26.

2

Analyze the numerical middle component:

4, 9, 16, 35, 66, ?

9 = 2(4) + 1

,

16 = 2(9) - 2

,

35 = 2(16) + 3

,

66 = 2(35) - 4

. Following

N_k = 2 N_{k-1} + (-1)^{k+1} k

, the next term is

2(66) + 5 = 137

.

The numerical pattern multiplies the previous number by 2 and alternates adding/subtracting sequential integers.

3

Analyze the second letter progression using reverse positional values:

W, T, P, K, E, ?

Reverse positions (

Z=1, Y=2, \dots, A=26

):

W=4, T=7, P=11, K=16, E=22

. Increments in reverse positions are

+3, +4, +5, +6

. The next reverse position increment is

+7

, giving

22 + 7 = 29 \equiv 3\pmod{26}

. Reverse position 3 corresponds to forward letter

X

(

27 - 3 = 24

).

The second letter steps through increasing increments in reverse alphabetical indexing.

Key Concept

Multi-pattern alphanumeric series involving prime positional shifts, alternating linear recursive relations, and reverse alphabetical indexing.

Estimated Time:2m 30s

Question 275Question →

Five municipal inspectors—Anand, Bhavia, Charan, Deepa, and Esha—are assigned to inspect three city zones (North, West, and East) using two types of vehicles (Electric and Hybrid), adhering to the following conditions:

1. Each inspector is assigned to exactly one zone and uses exactly one vehicle type. Every zone is visited by at least one inspector.
2. Exactly two inspectors use Electric vehicles, and three use Hybrid vehicles.
3. Esha uses a Hybrid vehicle and is the only inspector assigned to the North zone.
4. Anand and Deepa are assigned to the same zone, but use different vehicle types from each other.
5. Deepa uses an Electric vehicle.
6. Neither Bhavia nor Charan is assigned to the West zone.
7. Charan uses the same vehicle type as Anand.

Based on the information above, which of the following statements is DEFINITELY TRUE?

Bhavia is assigned to the East zone and uses an Electric vehicle.

Anand is assigned to the East zone and uses a Hybrid vehicle.

Deepa is assigned to the West zone and uses a Hybrid vehicle.

Charan is assigned to the West zone and uses an Electric vehicle.

Show answer & explanation

Answer: Bhavia is assigned to the East zone and uses an Electric vehicle.

Answer

Bhavia is assigned to the East zone and uses an Electric vehicle.

By logical deduction from the given clues, Esha is uniquely in the North zone with a Hybrid vehicle. Bhavia and Charan cannot be in North or West, placing them in the East zone. Anand and Deepa occupy the West zone. Deepa uses an Electric vehicle and Anand uses a Hybrid vehicle. Charan uses a Hybrid vehicle (matching Anand). Since three Hybrids are accounted for (Esha, Anand, Charan), Bhavia must use an Electric vehicle in the East zone.

Step-by-Step Solution

1

Determine zone assignments

North zone has only Esha (Condition 3). Bhavia and Charan cannot be in West (Condition 6) or North (as Esha is alone), so they are in East. Anand and Deepa must be together (Condition 4) in the remaining West zone.

Every zone must have at least one inspector, and Esha fills North completely.

2

Determine vehicle assignments for Anand, Deepa, and Esha

Esha uses Hybrid (Condition 3). Deepa uses Electric (Condition 5). Since Anand is in the same zone as Deepa but uses a different vehicle (Condition 4), Anand uses Hybrid.

Direct application of conditional vehicle constraints.

3

Determine vehicle assignments for Charan and Bhavia

Charan uses the same vehicle as Anand (Condition 7), so Charan uses Hybrid. Total Hybrids used = 3 (Esha, Anand, Charan). Total Electrics needed = 2 (Condition 2), so Bhavia must use Electric.

Completing the vehicle count count envelope of 2 Electric and 3 Hybrid.

Key Concept

Analytical Puzzles and Grouping

Question 276Question →

You are serving as a Sub-Divisional Magistrate (SDM) designated as Incident Commander in a flood-prone district. Due to incessant heavy rainfall in the catchment area, the water level in an upstream reservoir is rapidly reaching the critical safety threshold. Dam safety engineers report that opening the spillway gates to release water is technically mandatory within two hours to avert a catastrophic structural breach of the dam. However, releasing the water will submerge several low-lying villages downstream. Local agricultural representatives demand delaying water release to save standing crops, while executing an informal, unannounced evacuation risks causing severe mass panic. Which of the following represents the most appropriate immediate course of action for you to take under statutory disaster management protocols?

Issue an immediate emergency warning and execute a phased evacuation of downstream low-lying zones to designated relief shelters following Disaster Management Act protocols, while coordinating with dam engineers to release water in controlled increments.

Delay opening the spillway gates by several hours to allow villagers time to harvest their crops, while deploying temporary sandbag embankments to absorb overspill.

Constitute an emergency multi-stakeholder committee of local leaders and dam officials to deliberate and reach a unanimous consensus before initiating any evacuation or gate operations.

Order an immediate complete shutdown of local communications and forcibly clear downstream villages within thirty minutes using maximum police force without establishing temporary shelters.

Show answer & explanation

Answer: Issue an immediate emergency warning and execute a phased evacuation of downstream low-lying zones to designated relief shelters following Disaster Management Act protocols, while coordinating with dam engineers to release water in controlled increments.

Answer

The most appropriate immediate course of action is to issue an emergency warning and execute a phased evacuation of downstream low-lying zones to designated relief shelters under statutory Disaster Management Act procedures, while coordinating controlled water releases with dam engineers.

The correct response follows established administrative ethics and disaster management principles by prioritizing human life, adhering to statutory Incident Command procedures, providing clear public warning, providing adequate relief shelter, and mitigating flood intensity through controlled incremental water release.

Step-by-Step Solution

1

Assess safety constraints and statutory mandates

Recognize that dam structural integrity and human life protection supersede property preservation and informal delays.

Engineering safety thresholds indicate imminent structural failure if water is not released.

2

Initiate structured emergency communication and evacuation

Issue clear emergency alerts and initiate orderly, phased relocation to pre-designated shelters.

Prevents public stampede and panic while ensuring compliance with established disaster management SOPs.

3

Coordinate controlled technical mitigation

Work with dam authorities to release water in calibrated, incremental stages.

Reduces downstream surge velocity and allows maximum time for safe evacuation.

Key Concept

Statutory Protocol Adherence and Life Safety Primacy in Emergency Incident Command

Question 277Question →

Match each directional movement or facing scenario in List I with its corresponding final facing direction in List II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: (1) Person facing North turning

90^\circ

clockwise and

180^\circ

counter-clockwise matches West; (2) Person walking toward morning sun turning right matches South; (3) Person facing South turning

45^\circ

anti-clockwise matches South-East; (4) Person facing North-West turning

90^\circ

clockwise matches North-East.

Each scenario in List I is correctly mapped by evaluating angular turns and basic cardinal orientation relative to starting vectors.

Step-by-Step Solution

1

Analyze item 1 in List I

Facing North

+ 90^\circ

CW

=

East. East

+ 180^\circ

CCW

=

West.

Clockwise turns move rightwards on a compass, while counter-clockwise turns move leftwards.

2

Analyze item 2 in List I

Morning sun indicates East. Turning right from East leads to South.

Identifies initial facing direction implicitly from solar position.

3

Analyze item 3 in List I

Facing South

- 45^\circ

(anti-clockwise)

=

South-East.

An anti-clockwise shift from South rotates toward East.

4

Analyze item 4 in List I

Facing North-West

+ 90^\circ

(clockwise)

=

North-East.

A perpendicular right-hand turn from North-West points directly to North-East.

Key Concept

Directional turns, angular rotations, and solar cardinal orientation

Question 278Question →

An automated delivery drone is tested along a flight path starting from point $P$ . It first flies $12\text{ m}$ due North to reach point $Q$ . At point $Q$ , it turns $90^\circ$ to its right and flies $9\text{ m}$ to reach point $R$ . From point $R$ , it rotates $135^\circ$ clockwise and flies $15\sqrt{2}\text{ m}$ in a straight line to reach point $S$ . Finally, it turns due North and flies $3\text{ m}$ to land at pad $T$ . What is the shortest distance and direction of landing pad $T$ relative to the starting point $P$ ?

6\text{ m}

due West

6\text{ m}

due East

15\text{ m}

due South-West

3\text{ m}

due West

Show answer & explanation

Answer: $6\text{ m}$ due West

Answer

The shortest distance of landing pad T relative to point P is 6 meters due West.

Using Cartesian coordinates centered at the starting point, the drone moves to (0, 12) at point Q and then to (9, 12) at point R. Rotating 135 degrees clockwise from facing East reorients the drone towards South-West. Moving 15√2 meters in the South-West direction yields a displacement of -15 meters along the x-axis and -15 meters along the y-axis, bringing point S to (-6, -3). Finally, moving 3 meters North adds 3 to the y-coordinate, ending at point T with coordinates (-6, 0). Relative to the origin (0, 0), point T is exactly 6 meters to the West.

Step-by-Step Solution

1

Establish Cartesian coordinate system with starting point P at origin (0, 0).

P is at (0, 0).

Setting an origin allows precise vector tracking.

2

Calculate position of Q after moving 12 meters North, and R after moving 9 meters East.

Q = (0, 12), R = (9, 12).

North adds to y-coordinate; turning 90 degrees right from North heads East, adding to x-coordinate.

3

Calculate position of S after rotating 135 degrees clockwise from East (which points South-West).

x-displacement = -15sqrt(2) * cos(45°) = -15; y-displacement = -15sqrt(2) * sin(45°) = -15. S = (9 - 15, 12 - 15) = (-6, -3).

A 135-degree clockwise turn from East reorients direction to South-West.

4

Calculate final position T after flying 3 meters North from S(-6, -3).

T = (-6, -3 + 3) = (-6, 0).

Flying North increases the y-coordinate by 3.

5

Determine distance and direction of T(-6, 0) relative to P(0, 0).

Distance = 6 meters; Direction = West.

The x-coordinate is -6 and y-coordinate is 0, which lies 6 units along the negative x-axis (West).

Key Concept

Direction and Distance Test - Angular Rotations and Vector Displacement

Estimated Time:1m 30s

Question 279Question →

Six environmental inspectors—Farhan, Gita, Harish, Indu, Jay, and Kavya—are divided into two equal three-member teams, Team Alpha and Team Beta, to conduct field audits. Each inspector specializes in exactly one of three domains: Ecology, Geology, or Hydrology, such that exactly two inspectors belong to each domain.

The team assignments and domain specializations satisfy the following conditions:
1. Farhan and Kavya are both assigned to Team Alpha.
2. Harish and Indu both specialize in Geology, but they are assigned to different teams.
3. Farhan and Gita share the same domain specialization, which is Hydrology.
4. Kavya specializes in Ecology, and the two Ecology specialists are assigned to different teams.
5. Jay is assigned to Team Beta.

Which of the following statements must be definitively true regarding the team composition and domain specializations?

Jay specializes in Ecology and is assigned to Team Beta alongside a Hydrology specialist.

Gita specializes in Ecology and is assigned to Team Alpha alongside Farhan.

Harish and Indu are both assigned to Team Alpha, leaving Team Beta with no Geology specialists.

Team Alpha consists entirely of specialists in Hydrology and Geology, with no Ecology specialist present.

Show answer & explanation

Answer: Jay specializes in Ecology and is assigned to Team Beta alongside a Hydrology specialist.

Answer

Jay specializes in Ecology and is assigned to Team Beta alongside a Hydrology specialist.

The correct answer stating that Jay specializes in Ecology and is assigned to Team Beta alongside a Hydrology specialist is definitively true. Farhan and Kavya take two spots in Team Alpha. Since Harish and Indu must be split between the two teams, one takes the final spot in Team Alpha, forcing Gita and Jay into Team Beta. Given that Kavya (Ecology) is in Team Alpha and the Ecology specialists are split, Jay must be the Ecology specialist in Team Beta, working alongside Gita, who is a Hydrology specialist.

Step-by-Step Solution

1

Determine team memberships for all six inspectors.

Team Alpha has Farhan, Kavya, and one of Harish/Indu. Team Beta has Gita, Jay, and the remaining inspector of Harish/Indu.

Farhan and Kavya are in Team Alpha (2 members). Harish and Indu are in different teams, so exactly one is in Team Alpha, filling Team Alpha's 3rd spot. Thus, Jay and Gita must be in Team Beta along with the second Geology specialist.

2

Deduce domain specializations for Farhan and Gita.

Farhan and Gita both specialize in Hydrology.

Condition 3 explicitly states that Farhan and Gita share the Hydrology specialization.

3

Deduce domain specializations for Harish and Indu.

Harish and Indu both specialize in Geology.

Condition 2 explicitly states that Harish and Indu both specialize in Geology.

4

Deduce domain specializations for Kavya and Jay.

Kavya and Jay both specialize in Ecology.

Kavya specializes in Ecology (Condition 4). Since the two Ecology specialists are in different teams and Kavya is in Team Alpha, the second Ecology specialist must be in Team Beta. In Team Beta, Gita is Hydrology and one member is Geology, leaving Jay as the second Ecology specialist.

5

Synthesize Team Beta's final composition.

Team Beta consists of Gita (Hydrology), Jay (Ecology), and either Harish or Indu (Geology).

Combining step 1 team assignments with step 4 domain deductions confirms Jay specializes in Ecology and serves on Team Beta with Gita (Hydrology).

Key Concept

Multi-Constraint Matrix Grouping and Attribute Elimination

Estimated Time:2m 0s

Question 280Question →

A public administration agency is developing a new water conservation policy. Arrange the following standard stages of public policy feasibility and impact analysis in the correct logical sequence, from the initial planning stage to final assessment.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct chronological sequence is: Baseline data collection and objective setting -> Feasibility analysis of policy options -> Localized pilot program implementation -> Ex-post impact evaluation.

A structured policy cycle begins by establishing baseline data and clear objectives. Once baseline goals are defined, alternative interventions are subjected to financial, legal, and operational feasibility screening. Next, a localized pilot program tests administrative execution in a controlled context. Finally, ex-post impact evaluation measures net policy gains against the original baseline data.

Step-by-Step Solution

1

Identify the initial diagnostic phase.

Establishing baseline metrics and policy goals comes first.

Feasibility cannot be evaluated without knowing current baseline parameters and targeted outcomes.

2

Identify the prospective evaluation phase.

Screening potential solutions against financial, legal, and operational constraints.

Ex-ante feasibility screening ensures only viable, lawful, and affordable options proceed to implementation.

3

Identify the practical testing phase.

Rolling out a localized pilot project.

Piloting tests the practical operational mechanism on a small scale before full policy commitment.

4

Identify the retrospective review phase.

Executing an ex-post impact evaluation.

Comparing long-term results against initial baseline data measures actual policy efficacy.

Key Concept

Public Policy Feasibility and Impact Assessment Framework

Answer

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Common Mistakes

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