The question asks to identify the young economist involved in drafting India's First Five Year Plan (1951-1956). This requires knowledge of key figures associated with early economic planning in independent India.
A) KN Raj was a key young economist involved in drafting the First Five Year Plan. His contributions were significant, especially in the agricultural sector and the underlying economic model.
The question asks to identify the correct representation of Indian Standard Time (IST) relative to Greenwich Mean Time (GMT). This requires knowledge of India's standard meridian and the calculation of time difference based on longitude.
C) GMT + 5.30 is correct because Indian Standard Time (IST) is 5 hours and 30 minutes ahead of Greenwich Mean Time (GMT), corresponding to India's standard meridian at \(82^\circ 30'\) East longitude.
The question asks to identify the first Surveyor General of India appointed in 1815. This is a factual question related to the history of surveying in British India. Recalling key figures and their roles during the British colonial period is essential.
D) Colin Mackenzie was indeed appointed as the first Surveyor General of India in 1815. He played a pivotal role in the Great Trigonometrical Survey of India and was a noted orientalist and collector of manuscripts and artifacts.
The problem asks for which conditions the equation \( (a + b)^{2} = a^{2} + b^{2} \) is valid. We will expand the left side of the equation and simplify to find the relationship between \(a\) and \(b\) that makes the equality hold true.
Start with the given equation:
\[ (a + b)^{2} = a^{2} + b^{2} \]Expand the left side of the equation using the algebraic identity \( (x + y)^{2} = x^{2} + 2xy + y^{2} \):
\[ a^{2} + 2ab + b^{2} = a^{2} + b^{2} \]Subtract \( a^{2} \) from both sides of the equation:
\[ 2ab + b^{2} = b^{2} \]Subtract \( b^{2} \) from both sides of the equation:
\[ 2ab = 0 \]For the product \( 2ab \) to be \( 0 \), at least one of the factors \( a \) or \( b \) must be \( 0 \). That is, \( a = 0 \) or \( b = 0 \) (or both).
Consider the implications of \( a = 0 \) or \( b = 0 \):
Since there are infinitely many real numbers, there are infinitely many pairs \( (0, b) \) and \( (a, 0) \) that satisfy the given equation. For example, \( (0, 1), (0, 2), (0, -5), (1, 0), (2, 0), (-3, 0) \), etc.
Correct Option: B) May be true for infinite number of (a, b)
This is correct because the equation \( (a + b)^{2} = a^{2} + b^{2} \) simplifies to \( 2ab = 0 \). This condition is satisfied if \( a = 0 \) or \( b = 0 \). There are infinitely many pairs \( (a, b) \) where either \( a = 0 \) (e.g., \( (0, 1), (0, 2), (0, \pi) \)) or \( b = 0 \) (e.g., \( (1, 0), (2, 0), (\sqrt{2}, 0) \)).
The question asks for the specific month and year when India's first biofuel-powered flight landed in New Delhi. This is a factual question related to a significant event in India's aviation and environmental history. Recalling the exact date of this milestone is key to answering correctly.
D) August, 2018 - India's first biofuel-powered flight, operated by SpiceJet, landed at the Indira Gandhi International Airport in New Delhi from Dehradun on August 27, 2018.
The question asks to identify the field for which Theophrastus is known as the "father." This requires knowledge of historical figures and their contributions to various scientific disciplines.
Correct Option: B) Botany — Theophrastus is widely recognized as the "Father of Botany" for his comprehensive works on plants, which were foundational to the field.
The question asks about the ministry responsible for launching the 'Ganga Gram Project'. To answer this, one needs to recall the objectives of the project and which government ministry aligns with those objectives.
Correct Option: C) Ministry of Drinking Water and Sanitation
To find the greatest four-digit number divisible by 15, 20, 25, and 45, we first need to find the Least Common Multiple (LCM) of these numbers. Then, we will find the greatest four-digit number and divide it by the LCM. The remainder will help us determine the required number.
First, express each number as a product of its prime factors:
To find the LCM, take the highest power of each prime factor present in the numbers:
\[ \text{LCM}(15, 20, 25, 45) = 2^2 \times 3^2 \times 5^2 \]\[ = 4 \times 9 \times 25 \]\[ = 36 \times 25 \]\[ = 900 \]The greatest four-digit number is 9999.
Divide 9999 by 900:
\[ 9999 \div 900 \]\[ 9999 = 900 \times 11 + 99 \]To find the greatest four-digit number exactly divisible by 900, subtract the remainder from the greatest four-digit number:
\[ \text{Required number} = 9999 - 99 \]\[ = 9900 \]C) 9900 is the greatest four-digit number divisible by 15, 20, 25, and 45.
The question asks to identify the ruler who constructed the "Shahi (Royal) road" from the Indus Valley to the Sonar Valley in Bengal, which was later renamed the Grand Trunk (GT) Road during the British period. This requires knowledge of Indian history, specifically the medieval period and the contributions of various rulers to infrastructure development.
Sher Shah Suri, an Afghan ruler who established the Sur Empire, is famously credited with building and improving the Grand Trunk Road (then known as Sarak-e-Azam or "Great Road"). He extended and repaired the ancient route, which stretched from Kabul (in modern-day Afghanistan) to Sonargaon (in modern-day Bangladesh), passing through the Indus Valley and Bengal. His objective was to facilitate trade, communication, and military movement, thereby strengthening his empire. The British later renamed it the Grand Trunk Road.
B) Sher Shah Suri is the correct answer. He was the ruler responsible for constructing and improving the Shahi (Royal) road, which later became known as the Grand Trunk Road during the British period. His efforts in building this extensive road network were crucial for trade, administration, and military movements across his empire.
The question asks for the year India conducted its second nuclear test in Pokhran. This requires knowledge of India's nuclear history, specifically the dates of its nuclear tests.
Correct Option: A) 1998