The question gives the ratio of two numbers as 3:4 and their Least Common Multiple (LCM) as 180. We need to represent the numbers using a common multiplier, apply the relationship between their ratio and LCM, and calculate the larger number.
Identify the given information:
Represent the numbers using the given ratio:
Let the two numbers be 3x and 4x.
\[ \text{First Number}=3x \] \[ \text{Second Number}=4x \]Since 4 is greater than 3, the larger number is 4x.
Find the LCM of the ratio terms:
The ratio terms are 3 and 4.
Since 3 and 4 are coprime, their LCM is:
\[ \operatorname{LCM}(3,4)=3\times4 \] \[ =12 \]Therefore, the LCM of 3x and 4x is:
\[ 12x \]Calculate the value of x:
The given LCM is 180.
\[ 12x=180 \]Dividing both sides by 12:
\[ x=\frac{180}{12} \] \[ x=15 \]Calculate the two numbers:
Substituting x = 15:
\[ \text{First Number}=3\times15=45 \] \[ \text{Second Number}=4\times15=60 \]Therefore, the two numbers are 45 and 60.
The larger number is 60.
Correct Option: C) 60 — The numbers are in the ratio 3:4, so they can be represented as 3x and 4x. Since the LCM of 3 and 4 is 12:
\[ 12x=180 \] \[ x=15 \]
The larger number is:
\[ 4x=4\times15=60 \]
Hence, Option C is the correct answer.
A) 15: This is incorrect because 15 is the common multiplier x, not the larger number.
B) 20: This is incorrect because the larger number obtained from the ratio and the given LCM is 60, not 20.
D) 45: This is incorrect because 45 is the smaller of the two numbers. The larger number is 60.
The question asks to identify the Indian leader whose birthday is celebrated as National Education Day on November 11. This requires knowledge of significant dates and their associated personalities in Indian history and public life.
Correct Option: A) Maulana Abul Kalam Azad
National Education Day is celebrated every year on November 11 to commemorate the birth anniversary of Maulana Abul Kalam Azad, the first Education Minister of independent India. He served as the Education Minister from 1947 to 1958 and made significant contributions to the foundation of India's education system.
What is the compound interest on ₹2,800 for 1\( \frac{1}{2} \) years at 10% per annum compounded half-yearly?
To calculate the compound interest when interest is compounded half-yearly, we need to adjust the annual interest rate and the time period. The annual rate is halved, and the time period (in years) is doubled to represent the number of compounding periods.
Correct Option: B) ₹441.35
The question asks to identify the type of matrix where all its elements are zero. We need to recall the definitions of different types of matrices.
Correct Option: D) null
A null matrix (or zero matrix) is defined as a matrix in which every element is zero. It is denoted by \(\mathbf{0}\) or \(O\).
If x = \(\frac{\sqrt{9}+1}{\sqrt{9}-1}\) and y = \(\frac{\sqrt{16}+1}{\sqrt{16}-1}\), then the value of \( (x^{2} - y^{2}) \) is:
\( \frac{11}{9} \)
\( \frac{6}{5} \)
\( \frac{7}{6} \)
\( \frac{9}{8} \)
The problem asks us to find the value of \( (x^2 - y^2) \) given expressions for \( x \) and \( y \). We will first simplify the expressions for \( x \) and \( y \) by evaluating the square roots. Then, we will substitute these simplified values into the expression \( (x^2 - y^2) \) and calculate the result. We can use the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \) or calculate \( x^2 \) and \( y^2 \) separately.
Simplify the expression for \( x \):
Given \( x = \frac{\sqrt{9}+1}{\sqrt{9}-1} \).
We know that \( \sqrt{9} = 3 \).
Substitute this value into the expression for \( x \):
\[ x = \frac{3+1}{3-1} = \frac{4}{2} = 2 \]Simplify the expression for \( y \):
Given \( y = \frac{\sqrt{16}+1}{\sqrt{16}-1} \).
We know that \( \sqrt{16} = 4 \).
Substitute this value into the expression for \( y \):
\[ y = \frac{4+1}{4-1} = \frac{5}{3} \]Calculate \( x^2 - y^2 \):
Now we have \( x = 2 \) and \( y = \frac{5}{3} \).
We need to find the value of \( x^2 - y^2 \).
Substitute the values of \( x \) and \( y \):
\[ x^2 - y^2 = (2)^2 - \left(\frac{5}{3}\right)^2 \]Calculate the squares:
\[ (2)^2 = 4 \]and
\[ \left(\frac{5}{3}\right)^2 = \frac{5^2}{3^2} = \frac{25}{9} \]Now substitute these back:
\[ x^2 - y^2 = 4 - \frac{25}{9} \]To subtract these, find a common denominator, which is 9:
\[ x^2 - y^2 = \frac{4 \times 9}{9} - \frac{25}{9} = \frac{36}{9} - \frac{25}{9} \]Perform the subtraction:
\[ x^2 - y^2 = \frac{36 - 25}{9} = \frac{11}{9} \]A) \( \frac{11}{9} \) is the calculated value of \( x^2 - y^2 \).
The question asks about one of the criteria for a political party to be recognized as a national party in India. This involves understanding the specific electoral performance thresholds set by the Election Commission of India.
Correct Option: B) A political party can be recognised as a national party if it secures 6% of the valid votes in any four states in general or state assembly elections and in addition it wins 4 Lok Sabha seats from any state or states. This is one of the three conditions stipulated by the Election Commission of India for national party status.
Simplify the following.
To simplify the given expression, we will follow the order of operations, commonly known as BODMAS/PEMDAS. This means we will first solve operations inside Brackets (Parentheses), then Orders (Exponents), Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
Let's break down the expression into parts:
The expression is: \[1 + \frac{3}{7}\times\left[6+8\times(3-2)\right]+\left\{\left(\frac{1}{5}\div\frac{7}{25}\right)-\left(\frac{3}{7}+\frac{4}{14}\right)\right\}\]
Simplify the first bracket \(\left[6+8\times(3-2)\right]\):
So, the first part becomes: \(1 + \frac{3}{7}\times14\).
Simplify the second curly bracket \[\left\{\left(\frac{1}{5}\div\frac{7}{25}\right)-\left(\frac{3}{7}+\frac{4}{14}\right)\right\}\]:
Solve the first parenthesis inside the curly bracket: \[\left(\frac{1}{5}\div\frac{7}{25}\right)\].
Division by a fraction is multiplication by its reciprocal:
\[ \frac{1}{5}\div\frac{7}{25} = \frac{1}{5}\times\frac{25}{7} = \frac{25}{35} = \frac{5}{7} \]Solve the second parenthesis inside the curly bracket: \[\left(\frac{3}{7}+\frac{4}{14}\right)\].
First, simplify \(\frac{4}{14}\) to \(\frac{2}{7}\).
\[ \frac{3}{7}+\frac{2}{7} = \frac{3+2}{7} = \frac{5}{7} \]Substitute these results back into the curly bracket:
\[ \left\{\frac{5}{7}-\frac{5}{7}\right\} = 0 \]Combine the simplified parts:
The original expression now simplifies to:
\[ 1 + \left(\frac{3}{7}\times14\right) + 0 \]Perform the multiplication:
\[ \frac{3}{7}\times14 = 3\times\frac{14}{7} = 3\times2 = 6 \]Perform the final addition:
\[ 1 + 6 + 0 = 7 \]A) 7 is the correct answer, as derived from the step-by-step simplification.
The problem presents a number series, and we need to find the missing term. We will analyze the differences between consecutive terms to identify a pattern. If the first-order differences don't reveal a clear pattern, we will look at the second-order differences.
C) 86 is the correct answer because the series follows a pattern where the differences between consecutive terms form an arithmetic progression with a common difference of 3. The next difference is 21, and adding it to 65 gives 86.
To find the number of smaller cubes formed when a larger cube is divided, we need to calculate the volume of the larger cube and the volume of one smaller cube. The total number of smaller cubes will be the ratio of the volume of the larger cube to the volume of a single smaller cube.
Given, side of the larger cube, \(S = 15 \text{ cm}\).
The volume of a cube is given by the formula \(V = \text{side}^3\).
\[ V_{\text{large}} = S^3 = (15 \text{ cm})^3 = 15 \times 15 \times 15 \text{ cm}^3 = 3375 \text{ cm}^3 \]Given, side of the smaller cube, \(s = 3 \text{ cm}\).
The number of smaller cubes is the ratio of the volume of the larger cube to the volume of one smaller cube.
\[ \text{Number of cubes} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{3375 \text{ cm}^3}{27 \text{ cm}^3} \]Alternatively, we can find how many smaller side lengths fit into the larger side length and then cube that ratio:
Correct Option: B) 125
The problem asks us to identify the odd one out from a list of major sporting events. We need to find a common characteristic shared by three of the options and one characteristic that makes the fourth option different.
Correct Option: B) ICC World Cup is the odd one out because it is a single-sport event (cricket), whereas the Asian Games, Commonwealth Games, and Olympic Games are all multi-sport events.