To find the ratio of the area of the uncut portion to the area of the cut portion, we first need to calculate the total area of the original circular sheet, the area of the three small circles cut off, and then the area of the remaining (uncut) portion. Finally, we will form the required ratio.
Correct Option: D) The ratio of the area of the uncut portion to the area of the cut portion is 73 : 27.
To find the least five-digit number exactly divisible by 9, 12, and 15, we first need to find the Least Common Multiple (LCM) of these numbers. Then, we will find the smallest multiple of this LCM that is a five-digit number.
Find the LCM of 9, 12, and 15:
First, find the prime factorization of each number:
The LCM is the product of the highest powers of all prime factors involved:
\[ \text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 \]Find the least five-digit number:
The least five-digit number is 10000.
Find the smallest multiple of the LCM that is a five-digit number:
Divide the least five-digit number (10000) by the LCM (180):
\[ 10000 \div 180 \]\[ 10000 = 180 \times q + r \]\[ 10000 = 180 \times 55 + 100 \]Here, \(q = 55\) and \(r = 100\).This means that \(10000 - 100 = 9900\) is exactly divisible by 180. However, 9900 is a four-digit number.
To find the least five-digit number exactly divisible by 180, we need to find the next multiple of 180. This can be done by adding the difference between the LCM and the remainder to the least five-digit number, or simply by adding the LCM to the previous multiple.
The required number is:
\[ 10000 + (180 - 100) = 10000 + 80 = 10080 \]Alternatively, the next multiple of 180 after 9900 is:\[ 180 \times (55 + 1) = 180 \times 56 = 10080 \]Verify the result:
10080 is a five-digit number.
Check divisibility:
Thus, 10080 is the least five-digit number exactly divisible by 9, 12, and 15.
Correct Option: D) 10080
The question asks to identify the organization responsible for publishing the Red List of Threatened Species. This requires knowledge of major international environmental and conservation organizations and their specific roles.
Correct Option: D) IUCN
The International Union for Conservation of Nature (IUCN) is the organization that publishes the Red List of Threatened Species. The IUCN Red List is the world's most comprehensive inventory of the global conservation status of biological species. It uses a set of precise criteria to evaluate the extinction risk of thousands of species and subspecies, providing a critical indicator of the health of the world's biodiversity.
The problem presents a ratio and proportion relationship. We need to find the missing term in the proportion \( A : B :: C : D \), which implies that \( A/B = C/D \). We will simplify the given algebraic expressions using standard algebraic identities and then solve for the unknown term.
The given proportion is \( (a^{2} - b^{2}) : (a^{2} - ab) :: (a^{3} + b^{3}) : ? \). Let the unknown term be \( X \).
This can be written as:
\[ \frac{a^{2} - b^{2}}{a^{2} - ab} = \frac{a^{3} + b^{3}}{X} \]Factorize the terms using algebraic identities:
Numerator of the left side: \( a^{2} - b^{2} = (a - b)(a + b) \)
Denominator of the left side: \( a^{2} - ab = a(a - b) \)
Numerator of the right side: \( a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) \)
Substitute these factored forms back into the equation:
\[ \frac{(a - b)(a + b)}{a(a - b)} = \frac{(a + b)(a^{2} - ab + b^{2})}{X} \]Simplify the left side by canceling out the common term \( (a - b) \) (assuming \( a \neq b \)):
\[ \frac{a + b}{a} = \frac{(a + b)(a^{2} - ab + b^{2})}{X} \]Now, solve for \( X \). We can cross-multiply:
\[ X(a + b) = a(a + b)(a^{2} - ab + b^{2}) \]Divide both sides by \( (a + b) \) (assuming \( a \neq -b \)):
\[ X = a(a^{2} - ab + b^{2}) \]Correct Option: C) \( a(a^{2} + b^{2} - ab) \)
The question asks for the specific height of the Allahabad pillar, which is a historical monument known for its inscriptions related to the Gupta emperor Samudragupta. This requires factual knowledge about ancient Indian history and architecture.
Correct Option: B) 35 feet
The question asks about the average percentage of energy transferred between trophic levels in a food chain. This concept is fundamental to understanding energy flow in ecosystems and is often referred to as the "10% Law" of energy transfer.
Correct Option: C) 10
The 10% Law is a well-established ecological principle stating that, on average, only about 10% of the energy from one trophic level is incorporated into the biomass of the next trophic level. The rest is lost, mainly as metabolic heat.
The question asks about the field for which the Indo-American scientist Har Gobind Khorana received the Nobel Prize. This requires knowledge of his significant scientific contributions and the specific field recognized by the Nobel Committee.
Har Gobind Khorana was a biochemist who made significant contributions to our understanding of the genetic code. He was instrumental in demonstrating how the order of nucleotides in nucleic acids determines the order of amino acids in proteins. His work helped to decipher the genetic code, showing that the code is triplet and how these triplets (codons) specify particular amino acids. For this groundbreaking work, he shared the Nobel Prize in Physiology or Medicine in 1968 with Marshall W. Nirenberg and Robert W. Holley.
The study of heredity, genes, and genetic information is known as genetics.
C) genetics — Har Gobind Khorana received the Nobel Prize in Physiology or Medicine in 1968 for his work on the interpretation of the genetic code and its function in protein synthesis. This falls squarely within the field of genetics and biochemistry.
This problem can be solved using the concept of work and time. We will calculate the individual work rates (amount of work done per day) for A, B, and C, and then use these rates to find the time C alone takes to complete the task.
Calculate the combined work rate of A, B, and C:
A, B, and C together complete the task in 8 days. So, their combined work rate is:
Calculate the individual work rate of A:
A alone completes the task in 20 days. So, A's work rate is:
Calculate the individual work rate of B:
B alone completes the task in 20 days. So, B's work rate is:
Calculate the work rate of C:
The combined work rate of (A + B + C) is the sum of their individual work rates:
Calculate the time C alone takes to complete the task:
If C's work rate is \(\frac{1}{40}\) task per day, then C alone can complete the task in:
C) 40
To find the ratio A : C, we need to establish relationships between A, B, and C based on the given percentage conditions. We can express A in terms of B, and B in terms of C, and then substitute to find A in terms of C.
Relate A and B: "A exceeds B by 30%" means A is 30% more than B. Mathematically, this can be written as:
\[ A = B + 0.30B \]
\[ A = 1.30B \]
\[ A = \frac{130}{100}B \]
\[ A = \frac{13}{10}B \]
Relate B and C: "B is less than C by 20%" means B is 20% less than C. Mathematically, this can be written as:
\[ B = C - 0.20C \]
\[ B = 0.80C \]
\[ B = \frac{80}{100}C \]
\[ B = \frac{4}{5}C \]
Substitute B in the equation for A: Now we substitute the expression for B from step 2 into the equation for A from step 1:
\[ A = \frac{13}{10} \times \left(\frac{4}{5}C\right) \]
\[ A = \frac{13 \times 4}{10 \times 5}C \]
\[ A = \frac{52}{50}C \]
Find the ratio A : C: From the above equation, we can write the ratio A : C:
\[ \frac{A}{C} = \frac{52}{50} \]
Simplify the ratio by dividing both numerator and denominator by their greatest common divisor, which is 2:
\[ \frac{A}{C} = \frac{52 \div 2}{50 \div 2} = \frac{26}{25} \]
So, A : C = 26 : 25.
Correct Option: B) 26 : 25
IBM 370 was a _____ generation computer.
The question asks to identify the generation of the IBM 370 computer. This requires knowledge of the history and classification of computer generations based on their core technology.
A) 3rd — The IBM System/370 series of computers, introduced in 1970, was a prominent example of a third-generation computer, characterized by the use of Integrated Circuits (ICs).