Simplify:
\(1 \div \left[\left\{\frac{p^2}{p+6}\right\}+\left\{\frac{6p}{p+6}\right\}\right]\)
The problem requires simplifying a complex algebraic expression involving fractions. The key steps involve first simplifying the terms inside the square brackets by adding the fractions, and then performing the division.
The given expression is:
\[ 1 \div \left[\left\{\frac{p^2}{p+6}\right\}+\left\{\frac{6p}{p+6}\right\}\right] \]The terms inside the square brackets are two fractions with the same denominator \((p+6)\). We can add them directly:
\[ \left\{\frac{p^2}{p+6}\right\}+\left\{\frac{6p}{p+6}\right\} = \frac{p^2 + 6p}{p+6} \]Factor out \(p\) from the numerator \((p^2 + 6p)\):
\[ p^2 + 6p = p(p+6) \]Now the sum of fractions becomes:
\[ \frac{p(p+6)}{p+6} \]Assuming \(p+6 \neq 0\), we can cancel out \((p+6)\) from the numerator and the denominator:
\[ \frac{p(p+6)}{p+6} = p \]Now substitute this simplified result back into the original expression:
\[ 1 \div p = \frac{1}{p} \]A) \( \frac{1}{p} \) is the simplified form of the given expression.
The problem requires us to evaluate a mathematical expression involving multiplication, division, and subtraction. We need to follow the order of operations (BODMAS/PEMDAS) to solve it correctly. First, perform the operations inside the parentheses, then division, and finally subtraction.
The expression inside the parentheses is \((169 \times 121) \div (11 \times 13)\).
First, calculate the products:
\[ 169 \times 121 = 20469 \]\[ 11 \times 13 = 143 \]Now, divide the first product by the second product:
\[ 20469 \div 143 = 143 \]The original expression was \([(169 \times 121) \div (11 \times 13)] - 3\).
Substituting the result from step 2:
\[ 143 - 3 \]Thus, the value of the expression is 140.
B) 140 is the correct answer, as calculated by following the order of operations.
If tanθ = \(\frac{1}{\sqrt{5}}\), \( \operatorname{cosec} ^{2}\theta = \) ?
The problem asks us to find the value of \( \operatorname{cosec}^2\theta \) given \( \tan\theta = \frac{1}{\sqrt{5}} \). We can solve this by first finding \( \sin\theta \) or by using trigonometric identities to relate \( \tan\theta \) to \( \operatorname{cosec}^2\theta \).
D) 6
The question asks to identify the sport associated with the Indian personality Gourav Ghei. This requires general knowledge about prominent Indian sports figures.
Correct Option: C) Gourav Ghei is a renowned Indian professional golfer. He has won multiple titles on the Asian Tour and represented India in various international golf tournaments.
The question asks for the year in which the Indian Railway Board Act was passed. This is a factual question related to the history of Indian Railways and its administrative structure. We need to recall the specific historical date for the establishment of the Railway Board.
Correct Option: A) 1905. The Indian Railway Board Act was passed in 1905, leading to the formation of the Railway Board to oversee the administration of railways in India.
Choose the most appropriate description:
National Bank for Agriculture and Rural Development is a _______.
Self Help Group
Commercial Bank
Small and Medium Enterprise
Supervisory Institution
The question asks for the most appropriate description of the National Bank for Agriculture and Rural Development (NABARD). We need to understand NABARD's primary functions and role in the Indian financial system to correctly identify its nature.
Correct Option: D) Supervisory Institution
The question asks about the native region of Lala Lajpat Rai, a prominent figure in the Indian independence movement. To answer this, one needs to recall basic biographical facts about him.
Correct Option: C) Lala Lajpat Rai hailed from Punjab.
The question asks to identify the field with which Ben Kingsley is associated. This requires general knowledge about prominent personalities and their professions.
Correct Option: D) Motion pictures
This problem can be solved using the concept of work and time. We will calculate the individual work rates of G and H and then use these rates to find the time G takes to complete the work alone.
Calculate the combined work rate of G and H:
If G and H together can do a work in 30 days, their combined work rate (amount of work done per day) is:
Calculate the work rate of H alone:
If H alone can do the work in 50 days, H's individual work rate is:
Calculate the work rate of G alone:
The work rate of G alone can be found by subtracting H's work rate from the combined work rate of G and H:
To subtract these fractions, find a common denominator, which is 150:
\[ \text{Work rate of G} = \frac{5}{150} - \frac{3}{150} = \frac{5 - 3}{150} = \frac{2}{150} = \frac{1}{75} \text{ work/day} \]Calculate the time G alone takes to complete the work:
If G's work rate is \( \frac{1}{75} \) work per day, then G alone will take the reciprocal of this rate to complete the entire work:
Correct Option: A) 75
To find the profit percentage, we first need to calculate the cost price (CP) of the mixed variety of rice and then compare it with the selling price (SP). The cost price of the mixture will be a weighted average of the costs of the individual varieties based on their mixing ratio.
Let the quantity of rice A be \(3x\) kg and rice B be \(4x\) kg, as they are mixed in the ratio 3:4.
Total quantity of mixture = \(3x + 4x = 7x\) kg
Total cost of the mixture = \(630x + 1120x = \text{Rs. } 1750x\)
Cost Price (CP) per kg of the mixture is:
\[ \text{CP per kg} = \frac{\text{Total cost}}{\text{Total quantity}} = \frac{1750x}{7x} = \text{Rs. } 250 \text{ per kg} \]The mixed variety is sold at Rs. 260 per kg. So, \(\text{SP} = \text{Rs. } 260 \text{ per kg}\).
Profit per kg = \(\text{SP} - \text{CP} = 260 - 250 = \text{Rs. } 10 \text{ per kg}\).
The profit percentage is calculated as:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100 \] \[ \text{Profit Percentage} = \frac{10}{250} \times 100 = \frac{1}{25} \times 100 = 4\% \]Correct Option: D) 4%