The LCM of two numbers is 66. The numbers are in the ratio 2:3. The sum of the numbers is
We are given the LCM of two numbers and their ratio. We need to find the sum of these numbers. We can represent the numbers using the given ratio and a common multiple, then use the property that for any two numbers \(a\) and \(b\), their product is equal to the product of their LCM and HCF (i.e., \(a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b)\)). Alternatively, we can directly use the relationship between the numbers, their ratio, and their LCM.
Let the two numbers be \(A\) and \(B\). We are given that their ratio is 2:3. So, we can write the numbers as \(A = 2x\) and \(B = 3x\), where \(x\) is a common factor (which is also their HCF).
The LCM of \(2x\) and \(3x\) can be calculated as follows:
\[ \text{LCM}(2x, 3x) = x \times \text{LCM}(2, 3) \]Since 2 and 3 are prime numbers, their LCM is their product:
\[ \text{LCM}(2, 3) = 2 \times 3 = 6 \]Therefore, the LCM of the two numbers is:
\[ \text{LCM}(2x, 3x) = x \times 6 = 6x \]We are given that the LCM of the two numbers is 66. So, we can set up the equation:
\[ 6x = 66 \]Solve for \(x\):
\[ x = \frac{66}{6} \] \[ x = 11 \]Now that we have the value of \(x\), we can find the two numbers:
\[ A = 2x = 2 \times 11 = 22 \] \[ B = 3x = 3 \times 11 = 33 \]Finally, find the sum of the numbers:
\[ \text{Sum} = A + B = 22 + 33 = 55 \]B) 55 is the sum of the two numbers (22 and 33).
If P means '÷', R means '×', Q means '+' and S means '−', then 36 P 6 Q 7 R 8 S 11 =
This problem involves substituting given symbols with mathematical operators and then evaluating the resulting expression using the order of operations (BODMAS/PEMDAS).
The given expression is \(36 \text{ P } 6 \text{ Q } 7 \text{ R } 8 \text{ S } 11\).
Substituting the operators, we get:
\[ 36 \div 6 + 7 \times 8 - 11 \]The order of operations is Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
B) The final result of the expression is 51.
The question asks for the specific definition of a multitasking system. We need to understand what "multitasking" implies in the context of operating systems.
Correct Option: B) More than one process.
The question asks about the purpose of the Union Government's School Nursery Yojana. To answer this, one needs to recall the primary objective of this specific government scheme.
Correct Option: A) Creating environmental awareness in young minds.
This option accurately reflects the core objective of the School Nursery Yojana, which is to involve school children in planting and nurturing saplings to foster environmental consciousness and a connection with nature.
Ms. Chandra said that Lakshmi's husband is the only son of my grandfather. How is Ms. Chandra related to Lakshmi?
This is a blood relation question. We need to trace the relationships described in the statement to determine how Ms. Chandra is related to Lakshmi. We will break down the statement step-by-step to identify the individuals and their connections.
Correct Option: B) Daughter — As derived from the step-by-step analysis, Ms. Chandra is Lakshmi's daughter.
The question asks about the removal process of the Chief Election Commissioner (CEC) of India. This is a constitutional provision designed to ensure the independence of the Election Commission. The removal process is similar to that of a Supreme Court judge.
A) Both houses of Parliament
The Chief Election Commissioner can be removed from office by a resolution passed by both Houses of Parliament (Lok Sabha and Rajya Sabha) with a special majority, on the grounds of proven misbehaviour or incapacity. This process is identical to the removal of a Supreme Court judge, ensuring the independence and impartiality of the Election Commission.
\( \sin ^{2} 90^{\circ} + \operatorname{cosec} ^{2} 90^{\circ} - \cot ^{2} 45^{\circ} = \) ?
To solve this problem, we need to recall the standard trigonometric values for specific angles, namely \(90^{\circ}\) and \(45^{\circ}\). We will then substitute these values into the given expression and simplify it.
Recall the trigonometric values for \(90^{\circ}\):
Recall the trigonometric value for \(45^{\circ}\):
Substitute these values into the given expression:
\[ \sin ^{2} 90^{\circ} + \operatorname{cosec} ^{2} 90^{\circ} - \cot ^{2} 45^{\circ} \] \[ = (1)^{2} + (1)^{2} - (1)^{2} \]Perform the squaring operations:
\[ = 1 + 1 - 1 \]Simplify the expression:
\[ = 2 - 1 \] \[ = 1 \]B) 1 is the correct answer, as derived from the step-by-step calculation.
A) 2 — This would be the result if the term \( \cot^2 45^{\circ} \) was not subtracted or if its value was incorrectly taken as 0. However, \( \cot 45^{\circ} = 1 \), so \( \cot^2 45^{\circ} = 1 \).
C) \( \frac{1}{2} \) — This value is incorrect. It suggests errors in recalling the fundamental trigonometric values or in the arithmetic simplification.
D) \( \frac{\sqrt{3}}{2} \) — This value is incorrect. \( \frac{\sqrt{3}}{2} \) is typically associated with \( \sin 60^{\circ} \) or \( \cos 30^{\circ} \), which are not involved in this calculation. This indicates a misapplication of trigonometric values.
As per Census of India 2011, the population density was about
382 people per km2
353 people per km2
402 people per km2
428 people per km2
The question asks for the population density of India as per the Census of India 2011. This is a factual question requiring knowledge of demographic data from the census.
For India in 2011, this figure was officially recorded as 382 people per square kilometer.
A) 382 people per km2 is the correct population density for India as per the Census of India 2011.
We need to find the age of the elder sister. We are given the father's age, the age difference between the father and mother, and the relationship between the younger sister's age and the mother's age. We are also given the age difference between the two sisters. We will work backward from the known ages to find the unknown ages.
Calculate the mother's age:
Father's age is 54 years.
Father is elder by 2 years to mother, so:
\[ \text{Mother's age} = \text{Father's age} - 2 \]\[ \text{Mother's age} = 54 - 2 = 52 \text{ years} \]Calculate the younger sister's age:
Younger sister's age is half of mother's age, so:
\[ \text{Younger sister's age} = \frac{\text{Mother's age}}{2} \]\[ \text{Younger sister's age} = \frac{52}{2} = 26 \text{ years} \]Calculate the elder sister's age:
The difference between the ages of the two sisters is 4 years.
Since the younger sister's age is 26 years, the elder sister's age will be:
\[ \text{Elder sister's age} = \text{Younger sister's age} + 4 \]\[ \text{Elder sister's age} = 26 + 4 = 30 \text{ years} \]D) 30
The elder sister's age is 30 years, as calculated step-by-step.
\( 2^{2} - 3^{2} + 4^{2} - 6 = \) ?
To solve this problem, we need to follow the order of operations (PEMDAS/BODMAS). First, calculate the squares, then perform the additions and subtractions from left to right.
First, evaluate \(2^2\) and \(3^2\) and \(4^2\).
\[ 2^2 = 2 \times 2 = 4 \]\[ 3^2 = 3 \times 3 = 9 \]\[ 4^2 = 4 \times 4 = 16 \]Replace the squared terms with their calculated values:
\[ 4 - 9 + 16 - 6 \]Start from the left and move to the right.
First, \(4 - 9\):
\[ 4 - 9 = -5 \]Next, \(-5 + 16\):
\[ -5 + 16 = 11 \]Finally, \(11 - 6\):
\[ 11 - 6 = 5 \]Thus, the value of the expression \( 2^{2} - 3^{2} + 4^{2} - 6 \) is \(5\).
Correct Option: D) 5