If two quadratic equations have both roots in common, then their corresponding coefficients must be in proportion. We will first find the roots of the given equation \( x^{2} + 3x + 2 = 0 \). Then, we will use the property of common roots to establish a relationship between the coefficients \( a, b, c \) and use this relationship to check each option.
Correct Option: C) \( 4a = 2c + 2b \)
This problem involves substituting new mathematical operations for given symbols and then evaluating the resulting expression using the order of operations (BODMAS/PEMDAS).
The original equation is: \(171 \div 95 \times 5 - 4 + 33 = ?\)
Applying the transformations:
\[ 171 + 95 \div 5 \times 4 - 33 = ? \]BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. We perform operations in this order from left to right.
D) 214 is the correct answer, as derived from the step-by-step evaluation of the transformed expression.
The problem states that a man spends a certain percentage of his salary on house rent and fixed amounts on conveyance and groceries, saving the rest. We need to find his total monthly salary. We can set up an equation where the total salary is equal to the sum of all expenses and savings. By calculating the total amount spent on conveyance, groceries, and savings, we can determine the percentage of salary this amount represents, which is the remaining percentage after house rent. This will allow us to find the total salary.
Let this total be \( \text{Amount}_{\text{other}} \).
\[ \text{Amount}_{\text{other}} = \text{Conveyance} + \text{Groceries} + \text{Savings} \]\[ \text{Amount}_{\text{other}} = \text{₹}385 + \text{₹}2,725 + \text{₹}621 \]\[ \text{Amount}_{\text{other}} = \text{₹}3,731 \]House rent = 74% of monthly salary.
The remaining percentage of salary covers conveyance, groceries, and savings.
\[ \text{Remaining percentage} = 100\% - \text{Percentage spent on house rent} \]\[ \text{Remaining percentage} = 100\% - 74\% \]\[ \text{Remaining percentage} = 26\% \]We know that ₹3,731 represents 26% of his monthly salary.
Let the monthly salary be \( S \).
\[ 26\% \text{ of } S = \text{₹}3,731 \]\[ \frac{26}{100} \times S = 3,731 \]B) ₹14,350 is the correct monthly salary, calculated by determining that the sum of conveyance, groceries, and savings (₹3,731) represents 26% of the total salary.
To solve this problem, we need to calculate two values: the fourth proportional of 2, 8, and 12, and the mean proportional of 9 and 4. Once these two values are found, we will determine their ratio.
Calculate the fourth proportional of 2, 8, and 12:
If \(a, b, c\) are three numbers, and \(x\) is their fourth proportional, then the relationship is given by:
\[ \frac{a}{b} = \frac{c}{x} \]In this case, \(a = 2\), \(b = 8\), and \(c = 12\). So, we have:
\[ \frac{2}{8} = \frac{12}{x} \]To find \(x\), we cross-multiply:
\[ 2x = 8 \times 12 \]\[ 2x = 96 \]\[ x = \frac{96}{2} \]\[ x = 48 \]So, the fourth proportional is 48.
Calculate the mean proportional of 9 and 4:
If \(a\) and \(b\) are two numbers, and \(y\) is their mean proportional, then the relationship is given by:
\[ \frac{a}{y} = \frac{y}{b} \]This simplifies to \(y^2 = ab\), so \(y = \sqrt{ab}\).
In this case, \(a = 9\) and \(b = 4\). So, we have:
\[ y = \sqrt{9 \times 4} \]\[ y = \sqrt{36} \]\[ y = 6 \]So, the mean proportional is 6.
Find the ratio between the fourth proportional and the mean proportional:
The ratio required is (fourth proportional) : (mean proportional).
Ratio \( = 48 : 6 \)
To simplify the ratio, divide both numbers by their greatest common divisor, which is 6:
\[ \frac{48}{6} : \frac{6}{6} \]\[ 8 : 1 \]C) 8 : 1 is the correct ratio between the fourth proportional of 2, 8, and 12 (which is 48) and the mean proportional of 9 and 4 (which is 6).
To solve this problem, we need to process each digit of the given number according to the specified rules: add 2 to odd digits and subtract 2 from even digits. After modifying all digits, we will form the new number and then identify the digit that is third from the right.
B) 5 — The third digit from the right in the new number 4309567 is 5.
To find the average amount spent per week, we need to sum up all the amounts spent over the five weeks and then divide by the total number of weeks.
The average amount spent per week is ₹300.
C) ₹300 is the correct average amount spent per week.
The question asks about the phenomenon responsible for winter rainfall in the northern plains of India. We need to identify the atmospheric system that brings precipitation to this region during the winter months.
D) Western disturbances — Western disturbances are extratropical storms that originate in the Mediterranean region and bring non-monsoonal precipitation to Northwest India during the winter months. This precipitation is vital for the rabi crops grown in the northern plains.
To solve this problem, we will first calculate the individual work rates of A, B, and C. Then, we'll determine the amount of work completed by A, B, and C together in the first 2 days. After A leaves, we'll find the combined work rate of B and C and use it to calculate the time taken to complete the remaining work.
Calculate individual work rates:
Calculate combined work rate of A, B, and C:
Their combined 1-day work is:
\[ \text{Work}_{A+B+C} = \frac{1}{11} + \frac{1}{33} + \frac{1}{48} \]To add these fractions, find the Least Common Multiple (LCM) of 11, 33, and 48.
LCM \((11, 33, 48) = 2^4 \times 3 \times 11 = 16 \times 3 \times 11 = 48 \times 11 = 528\)
\[ \text{Work}_{A+B+C} = \frac{48}{528} + \frac{16}{528} + \frac{11}{528} = \frac{48 + 16 + 11}{528} = \frac{75}{528} \]Calculate work done in the first 2 days:
A, B, and C worked together for 2 days. So, the work done in 2 days is:
\[ \text{Work in 2 days} = 2 \times \frac{75}{528} = \frac{150}{528} \]Calculate remaining work:
Total work is 1. Remaining work is:
\[ \text{Remaining Work} = 1 - \frac{150}{528} = \frac{528 - 150}{528} = \frac{378}{528} \]Calculate combined work rate of B and C:
After 2 days, A left. So, B and C will complete the remaining work. Their combined 1-day work is:
\[ \text{Work}_{B+C} = \frac{1}{33} + \frac{1}{48} \]LCM \((33, 48) = 3 \times 11 \times 2^4 = 3 \times 11 \times 16 = 528\)
\[ \text{Work}_{B+C} = \frac{16}{528} + \frac{11}{528} = \frac{16 + 11}{528} = \frac{27}{528} \]Calculate time taken to complete the remaining work:
Time = \[\frac{\text{Remaining Work}}{\text{Combined Work Rate of B and C}}\]
\[ \text{Time} = \frac{\frac{378}{528}}{\frac{27}{528}} = \frac{378}{27} \] \[ \text{Time} = 14 \text{ days} \]Correct Option: C) 14
The remaining work was completed in 14 days.
The question asks about the duration for which the East India Company's charter was renewed by the Charter Act of 1793. This requires knowledge of the specific provisions of this historical act.
C) 20
The Charter Act of 1793 renewed the East India Company's charter for a period of 20 years, granting it continued exclusive trading privileges and administrative powers in India.
This problem involves calculating income based on savings and expenditure percentages. We know that Income = Savings + Expenditure. Given the percentage of savings and the absolute value of expenditure, we can determine the percentage of expenditure and then use this to find the total income.
Let the total income be \(I\).
The person saves 40% of his income. So, Savings \(= 40\% \text{ of } I\).
The remaining percentage of income is spent. Therefore, Expenditure percentage \(= 100\% - \text{Savings percentage}\).
Calculate the expenditure percentage:
\[ \text{Expenditure percentage} = 100\% - 40\% = 60\% \]
We are given that his expenditure is ₹360. This means that 60% of his income is ₹360.
\[ 60\% \text{ of } I = 360 \]
Convert the percentage to a decimal or fraction to solve for \(I\):
\[ \frac{60}{100} \times I = 360 \]
Solve for \(I\):
\[ I = \frac{360 \times 100}{60} \]
\[ I = \frac{36000}{60} \]
\[ I = 600 \]
So, his income is ₹600.
Correct Option: A) 600