All practice questions

Three different wooden boards have lengths of $\frac{3}{4}$ meters, $\frac{9}{10}$ meters, and $\frac{15}{8}$ meters, respectively. A carpenter wants to cut them into smaller pieces of equal length such that no material is wasted. What is the maximum possible length of each smaller piece?

Arrange the following words in a logical sequence as they would appear in an English dictionary:

Consider the following $8$ mathematical expressions:

I. $\pi - \frac{22}{7}$
II. $\frac{\sqrt{27}}{\sqrt{3}}$
III. $(\sqrt{3} - \sqrt{2})^2$
IV. The infinite decimal $0.101101110\dots$ (where the number of consecutive $1$ s increases by one each time)
V. $1.414$
VI. $\sqrt{10} \times \sqrt{2.5}$
VII. $e^0$
VIII. $\sqrt{2} + \sqrt{8}$

How many of the above expressions evaluate to a rational number?

Six friends—J, K, L, M, N, and O—are seated around a circular table facing the center. Each has a different profession: Chef, Doctor, Engineer, Lawyer, Pilot, and Teacher. The seating arrangement follows these conditions:

- The Doctor sits second to the left of M.
- K is an Engineer and sits immediately to the right of M.
- The Pilot sits exactly between J and the Lawyer.
- N sits exactly opposite the Teacher.
- O sits adjacent to the Doctor.
- The Chef sits immediately to the left of the Teacher.

Based on the given information, which of the following statements is true?

Four marine exploration vessels (Alpha, Beta, Gamma, and Delta) are deployed for oceanographic surveying. Each vessel begins facing a specific cardinal direction and executes a sequence of angular maneuvers as described below:

- Vessel Alpha: Starts facing North, rotates $135^{\circ}$ clockwise, moves forward, and then rotates $45^{\circ}$ anti-clockwise.
- Vessel Beta: Starts facing East, rotates $90^{\circ}$ clockwise, moves forward, and then rotates $135^{\circ}$ clockwise.
- Vessel Gamma: Starts facing South, rotates $45^{\circ}$ anti-clockwise, moves forward, rotates $180^{\circ}$ clockwise, and finally rotates $90^{\circ}$ anti-clockwise.
- Vessel Delta: Starts facing West, rotates $135^{\circ}$ anti-clockwise, moves forward, rotates $90^{\circ}$ clockwise, and finally rotates $45^{\circ}$ anti-clockwise.

Match each vessel with its final facing direction.

As the District Magistrate overseeing land acquisition for a new National Highway bypass, you are confronted with a multifaceted conflict halting the project. Different stakeholder groups have raised distinct objections and demands. Match the following stakeholder positions with the most constitutionally aligned and proportionate administrative intervention you should initiate to de-escalate and resolve the conflict.

Consider the following statement and the two assumptions that follow it.

Statement:
"The National Meteorological Authority has issued an official advisory urging coastal farmers to delay the sowing of their Kharif crops by two weeks due to a predicted delay in the arrival of the southwest monsoon."

Assumptions:
I. Coastal farmers generally rely on the rainfall from the southwest monsoon for the successful cultivation of their Kharif crops.
II. Delaying the sowing of crops by two weeks will inevitably result in a lower agricultural yield for the upcoming harvest season.

Based on the statement provided, which of the following is correct?

Six colleagues—Farah, Gauri, Hasan, Imran, Jamal, and Kiran—are seated in a single straight row facing North for a team meeting.

Read the following information to determine their exact seating positions:
1. Jamal sits third to the right of Farah.
2. Neither Jamal nor Farah sits at the extreme ends of the row.
3. Hasan sits to the immediate left of Gauri.
4. Kiran does not sit at the extreme right end of the row.

Arrange the colleagues in their correct seating order from left (Position 1) to right (Position 6).

You are the Municipal Commissioner of a rapidly expanding city. A severe outbreak of a vector-borne disease has occurred in a densely populated, unauthorized slum settlement. The slum is situated on prime municipal land previously earmarked for a commercial complex. While local health workers request the immediate deployment of emergency medical camps, some municipal officials advise you to use this health crisis as an opportunity to evict the squatters, arguing that the area is inherently unhygienic and illegal.

Arrange the following courses of administrative action from the most ethically sound to the least ethically sound.

At exactly 10:10 AM, the hour hand of a standard analog clock is mechanically locked in its precise current position. The minute hand is then manually rotated forward (clockwise) by exactly $210^{\circ}$. Calculate the new angle (in degrees) formed between the hour and minute hands after this adjustment. Provide the smaller of the two possible angles.

Trees are planted in a single straight line along a newly constructed highway. An Oak tree is the $42^{\text{nd}}$ tree from the northern end of the line, and a Maple tree is the $58^{\text{th}}$ tree from the southern end. If there are exactly $18$ trees planted between the Oak tree and the Maple tree, what is the minimum possible total number of trees in this line?

Evaluate the mathematical logic governing the number pairs in Group X. Match each pair with its analog in Group Y that follows the exact same logical transformation.

Analyze the following three mathematical assertions concerning the classification and properties of numbers:

Assertion I: The expression $\sqrt{32} - 16 \div 4 \times \sqrt{2}$ evaluates to an irrational number.
Assertion II: For any prime number $p$ that is strictly greater than $2$, the value of $p^2 - 1$ is always a multiple of $8$.
Assertion III: The number $0$ is classified as a rational number and an even integer, but it is neither a prime nor a composite number.

Which of the assertions provided above are logically correct?

A genetic researcher is analyzing a three-generation pedigree chart containing six individuals: K, V, W, X, Y, and Z. The clinical notes reveal the following structure:

- Z is the paternal grandfather of Y.
- X is the only son of Z.
- W is the sister of X.
- V is the daughter-in-law of Z.
- K is the son of W.
- Y is the daughter of V.

Match the individuals on the left with their exact relationship to Y on the right.

A positive integer $X$ is of the form $X = 2^a \times 3^b$, where $a$ and $b$ are positive integers. It is known that $X$ has exactly $15$ positive factors. If $X$ is a multiple of $9$ but is NOT a multiple of $27$, which of the following is the exact value of $X$?

Four identical delivery vans have different amounts of fuel remaining in their tanks. Arrange the vehicles based on their remaining fuel fraction from the least amount to the greatest amount.

A policy statement is given below, followed by two candidate assumptions. An assumption is something supposed or taken for granted. Identify which of the assumptions is/are implicit in the statement.

Statement:
"To alleviate the increasing load on the municipal electricity grid, the State Energy Commission has directed that all future commercial building projects must incorporate rooftop solar power systems."

Assumptions:
I. Rooftop solar systems can produce enough energy to significantly lower a commercial facility's dependence on the municipal grid.
II. The State Energy Commission plans to enforce similar solar power requirements for existing residential homes in the near future.

Which of the following is correct?

In a fully automated logistics center, a single row of delivery robots is lined up facing the loading bay. Robot $M$ is positioned $27^{\text{th}}$ from the left end of the row, while Robot $P$ is positioned $31^{\text{st}}$ from the right end. If Robot $M$ and Robot $P$ interchange their positions, Robot $P$ becomes $48^{\text{th}}$ from the right end. Based on this rearrangement, what is the new position of Robot $M$ from the left end?

Examine the policy details provided below and evaluate the logical validity of the subsequent inferences.

Policy Directives:
- Every public welfare initiative launched by the incumbent government mandates that recipients synchronize their national identity documents with their bank profiles to prevent fund misappropriation.
- The 'Urban Housing Assistance' framework is a newly launched public welfare initiative under the incumbent government.

Proposed Inferences:
I. Recipients of the 'Urban Housing Assistance' framework must synchronize their national identity documents with their bank profiles.
II. Welfare initiatives launched before the incumbent government experienced severe fund misappropriation.

Based exclusively on the policy directives provided, which inference can be logically deduced?