To find the Least Common Multiple (LCM) of two numbers, we can use the prime factorization method or the formula relating LCM, HCF, and the product of the numbers. Since 48 and 65 are relatively prime (their Highest Common Factor is 1), their LCM will simply be their product.
The prime factors of 48 are 2 and 3. The prime factors of 65 are 5 and 13. There are no common prime factors between 48 and 65. This means that 48 and 65 are coprime (or relatively prime).
When two numbers are coprime, their LCM is simply the product of the numbers.
\[ \text{LCM}(48, 65) = 48 \times 65 \] \[ 48 \times 65 = 48 \times (60 + 5) = 48 \times 60 + 48 \times 5 \] \[ 48 \times 60 = 2880 \] \[ 48 \times 5 = 240 \] \[ 2880 + 240 = 3120 \]Alternatively, using the formula \(\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}\):
Since 48 and 65 are coprime, their HCF (Highest Common Factor) is 1.
\[ \text{LCM}(48, 65) = \frac{48 \times 65}{1} = 48 \times 65 = 3120 \]B) 3120 is the correct LCM of 48 and 65.
The problem provides a ratio between two variables, \(x\) and \(y\), and asks for the ratio of two expressions involving \(x\) and \(y\). We can express one variable in terms of the other using the given ratio and then substitute it into the expression to find the required ratio.
We are given that \(x : y = 2 : 3\).
This can be written as a fraction:
\[ \frac{x}{y} = \frac{2}{3} \]From the above equation, we can express \(x\) in terms of \(y\):
\[ x = \frac{2}{3}y \]We need to find the value of the ratio \((5x + 3y) : (5x - 3y)\). Let's substitute \(x = \frac{2}{3}y\) into both parts of the ratio:
For the first part, \(5x + 3y\):
\[ 5x + 3y = 5\left(\frac{2}{3}y\right) + 3y \]\[ = \frac{10}{3}y + 3y \]\[ = \frac{10y + 9y}{3} \]\[ = \frac{19y}{3} \]For the second part, \(5x - 3y\):
\[ 5x - 3y = 5\left(\frac{2}{3}y\right) - 3y \]\[ = \frac{10}{3}y - 3y \]\[ = \frac{10y - 9y}{3} \]\[ = \frac{y}{3} \]Now, we form the ratio of these two simplified expressions:
\[ (5x + 3y) : (5x - 3y) = \frac{19y}{3} : \frac{y}{3} \]To simplify, we can divide both sides of the ratio by \(\frac{y}{3}\) (assuming \(y \neq 0\)):
\[ \frac{\frac{19y}{3}}{\frac{y}{3}} = \frac{19y}{3} \times \frac{3}{y} = 19 \]So, the ratio is \(19 : 1\).
B) 19 : 1 — The step-by-step calculation shows that the ratio \((5x + 3y) : (5x - 3y)\) simplifies to \(19 : 1\).
As of 31st October 2020, the United Nations comprises ______ member states.
The question asks for the number of member states in the United Nations as of a specific date (October 31, 2020). This is a factual question requiring knowledge of international organizations.
Correct Option: B) 193 is the correct number of member states in the United Nations as of October 31, 2020.
The value of x, if 3x + 4 × 8 ÷ 9 = x ÷ 3 − 1, is:
To find the value of \(x\), we need to solve the given linear equation. This involves simplifying both sides of the equation, combining like terms, and isolating \(x\).
Write down the given equation:
\[ 3x + 4 \times 8 \div 9 = x \div 3 - 1 \]Perform the multiplication and division operations on the left side of the equation according to the order of operations (BODMAS/PEMDAS). First, \(4 \times 8 = 32\).
\[ 3x + 32 \div 9 = x \div 3 - 1 \]Next, perform the division \(32 \div 9\). This results in a fraction \(\frac{32}{9}\).
\[ 3x + \frac{32}{9} = \frac{x}{3} - 1 \]Now, gather all terms containing \(x\) on one side of the equation and constant terms on the other side. Subtract \(\frac{x}{3}\) from both sides and subtract \(\frac{32}{9}\) from both sides:
\[ 3x - \frac{x}{3} = -1 - \frac{32}{9} \]To combine the terms with \(x\) on the left side, find a common denominator, which is 3:
\[ \frac{3x \times 3}{3} - \frac{x}{3} = \frac{9x - x}{3} = \frac{8x}{3} \]To combine the constant terms on the right side, find a common denominator, which is 9:
\[ -1 - \frac{32}{9} = -\frac{1 \times 9}{9} - \frac{32}{9} = -\frac{9}{9} - \frac{32}{9} = -\frac{9 + 32}{9} = -\frac{41}{9} \]Now the equation becomes:
\[ \frac{8x}{3} = -\frac{41}{9} \]To solve for \(x\), multiply both sides by \(\frac{3}{8}\):
\[ x = -\frac{41}{9} \times \frac{3}{8} \]Simplify the multiplication. The 3 in the numerator and 9 in the denominator can be cancelled (\(9 \div 3 = 3\)):
\[ x = -\frac{41}{3 \times 8} \]\[ x = -\frac{41}{24} \]Correct Option: D) -\( \frac{41}{24} \)
Which infantry battalion was involved in the killing of all its white officers in the Revolt of 1857?
20th Native Infantry
1st Native Infantry
41st Native Infantry
21st Native Infantry
The question asks to identify the specific infantry battalion that was responsible for the killing of all its white officers during the Revolt of 1857. This requires knowledge of specific incidents and units involved in the early stages of the rebellion.
C) 41st Native Infantry — The 41st Native Infantry is historically documented to have mutinied at Sitapur on June 3, 1857, and killed all its European officers, making it the correct answer.
To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers, we can use the prime factorization method. The HCF is the product of the common prime factors raised to the lowest power, and the LCM is the product of all prime factors (common and uncommon) raised to the highest power.
Correct Option: D) 81 ; 27
The problem describes a scenario of preferences for working together among several individuals. We need to identify who is isolated, meaning someone who does not like to work with anyone, and no one likes to work with them. We will analyze each statement to determine the connections between individuals.
Correct Option: C) L is isolated because L does not like to work with anyone, and no other person mentioned (E, C, A, H) has expressed a liking to work with L. This makes L completely disconnected from the working preferences of others.
The question asks to identify the city known as 'The Queen of the Arabian Sea'. This is a general knowledge question related to geographical epithets of Indian cities. One needs to recall the specific city associated with this title.
Correct Option: C) Kochi is famously known as 'The Queen of the Arabian Sea' due to its historical importance as a major port and its picturesque location on the Arabian Sea coast.
The question asks for the specific year when the East India Company obtained 'Diwani' rights over Bengal and Bihar. This is a direct historical fact related to the British colonial expansion in India. Recalling key events and their dates in Indian history is crucial for answering such questions.
Correct Option: B) 1765 — This is the year the Treaty of Allahabad was signed, granting the East India Company the Diwani rights over Bengal and Bihar.
The problem asks us to identify the odd one out from a list of four words. This type of question tests our ability to categorize words based on their common characteristics and identify the one that does not fit the pattern.
Correct Option: B) Intelligence — Anger, Happiness, and Fear are all emotions. Intelligence is a cognitive ability, not an emotion. Therefore, it is the odd one out.