To identify the first and second members of the ketone homologous series, we need to understand the general structure of ketones and the naming conventions (IUPAC nomenclature) for organic compounds. Ketones are organic compounds containing a carbonyl group (\(C=O\)) bonded to two alkyl or aryl groups. The smallest ketone must have at least three carbon atoms.
Therefore, the first and second members are Propanone and Butanone, respectively.
Correct Option: A) Propanone, Butanone
To find the approximate value, we will round off each number to its nearest whole number or to one decimal place if necessary, and then perform the calculations according to the order of operations (BODMAS/PEMDAS).
The equation becomes approximately:
\[ 27 \times 4 - 8 \times 2 + 27 \]The equation now is:
\[ 108 - 16 + 27 \]So, the approximate value is \(119\).
C) 119 — By rounding off the numbers to their nearest whole numbers and following the order of operations, the approximate value is calculated as 119.
To find the profit percentage, we first need to calculate the cost price (CP) and selling price (SP) of one article. Then, we can determine the total profit and use the profit percentage formula.
Calculate the Cost Price (CP) of one article:
The man buys 15 identical articles for a total of Rs. 15.
\[ \text{CP of 1 article} = \frac{\text{Total Cost}}{\text{Number of articles}} = \frac{15}{15} = \text{Rs. } 1 \]Identify the Selling Price (SP) of one article:
He sells each article for Rs. 1.23.
\[ \text{SP of 1 article} = \text{Rs. } 1.23 \]Calculate the Profit per article:
Profit is the difference between the selling price and the cost price.
\[ \text{Profit per article} = \text{SP} - \text{CP} = 1.23 - 1 = \text{Rs. } 0.23 \]Calculate the Profit Percentage:
The profit percentage is calculated using the formula:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \% \]Substitute the values:
\[ \text{Profit Percentage} = \left( \frac{0.23}{1} \right) \times 100 \% = 0.23 \times 100 \% = 23 \% \]A) 23 % is the correct profit percentage.
The question asks to identify the organization established by Raja Rammohan Roy that served as an early precursor to socio-religious reforms in Bengal. We need to recall the key organizations founded by Raja Rammohan Roy and their timelines.
Correct Option: C) Atmiya Sabha was established by Raja Rammohan Roy in 1815. It was a pioneering association that held discussions on religious and moral subjects, promoting monotheism and rational thought, thus acting as a precursor to the larger socio-religious reform movements in Bengal.
Find the roots of \( \frac{6}{x}-\frac{2}{x-1}-\frac{1}{x-2} = 0 \)
\( \frac{4}{5} \) and 3
\( \frac{4}{3} \) and \( \frac{3}{2} \)
\( \frac{4}{5} \) and \( \frac{3}{2} \)
\( \frac{4}{3} \) and 3
To find the roots of the given equation, we need to combine the fractions by finding a common denominator, simplify the resulting expression, and then solve the quadratic equation that arises.
The given equation is:
\[ \frac{6}{x}-\frac{2}{x-1}-\frac{1}{x-2} = 0 \]First, combine the terms on the left-hand side by finding a common denominator, which is \(x(x-1)(x-2)\). Note that \(x \neq 0, 1, 2\).
\[ \frac{6(x-1)(x-2) - 2x(x-2) - 1x(x-1)}{x(x-1)(x-2)} = 0 \]For the fraction to be zero, the numerator must be zero (provided the denominator is not zero). So, we set the numerator equal to zero:
\[ 6(x-1)(x-2) - 2x(x-2) - x(x-1) = 0 \]Expand the terms:
Substitute these expanded terms back into the equation:
\[ (6x^2 - 18x + 12) + (-2x^2 + 4x) + (-x^2 + x) = 0 \]Combine like terms:
\[ (6x^2 - 2x^2 - x^2) + (-18x + 4x + x) + 12 = 0 \]\[ 3x^2 - 13x + 12 = 0 \]Now, we have a quadratic equation. We can solve this using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) or by factoring. Let's try factoring. We need two numbers that multiply to \(3 \times 12 = 36\) and add up to \(-13\). These numbers are \(-4\) and \(-9\).
\[ 3x^2 - 9x - 4x + 12 = 0 \]\[ 3x(x - 3) - 4(x - 3) = 0 \]\[ (3x - 4)(x - 3) = 0 \]Set each factor to zero to find the roots:
Both roots \(x = \frac{4}{3}\) and \(x = 3\) are valid as they do not make the original denominators zero.
D) \( \frac{4}{3} \) and 3 — The roots of the equation are \( \frac{4}{3} \) and 3, which matches this option.
The question asks to identify the guest nation at the Hyderabad Literary Festival (HLF) 2022. This is a factual question related to current affairs and cultural events. To answer this, one needs to recall or know the specific details of the HLF 2022.
Correct Option: C) The United Kingdom was indeed the guest nation at the Hyderabad Literary Festival 2022. This was a significant cultural exchange, highlighting literary ties between India and the UK.
This problem involves analyzing inequalities to determine the truthfulness of given conclusions. We will combine the given statements and then check each conclusion against the combined statement.
The given statements are:
We can combine these into a single chain of inequalities:
\[ B \ge M > G \ge Y = L \]From the combined statement, we have \(B \ge M > G \ge Y\). Let's break this down:
Since \(M > G\) and \(G \ge Y\), it implies \(M > Y\). (If M is strictly greater than G, and G is greater than or equal to Y, then M must be strictly greater than Y).
Now, we have \(B \ge M\) and \(M > Y\). This implies \(B > Y\). (If B is greater than or equal to M, and M is strictly greater than Y, then B must be strictly greater than Y).
Therefore, Conclusion 1 is true.
From the combined statement, we have \(M > G \ge Y = L\). This means:
Combining these, we get \(M > G \ge L\). This clearly shows that \(M\) is strictly greater than \(G\), and \(G\) is greater than or equal to \(L\). Therefore, \(M\) must be strictly greater than \(L\) (i.e., \(M > L\)).
Thus, Conclusion 2, which states \(M = L\), is false.
A) Only conclusion 1 is true. As determined in the step-by-step analysis, Conclusion 1 (\(B > Y\)) is true, and Conclusion 2 (\(M = L\)) is false.
The question asks for the district in Madhya Pradesh where India's first commercial-scale biomass-based hydrogen plant is being established. This is a factual question related to current affairs and infrastructure development in the renewable energy sector.
Correct Option: A) The country's first commercial-scale biomass-based hydrogen plant is indeed coming up in the Khandwa district of Madhya Pradesh. This plant aims to produce 24 tonnes of hydrogen per day from 30 tonnes of biomass feedstock.
The question asks about the primary economic sector that supported the majority of India's population during the colonial period. This requires knowledge of the economic structure of colonial India.
Correct Option: D) Agriculture was the mainstay of the Indian economy during the colonial period, supporting approximately 85% of the population. This high dependence was a defining characteristic of the colonial economy.
The resistance of a wire depends on its material, length, and cross-sectional area. We will use the formula for resistance and the given relationship between the resistances of wires A and B to find the ratio of their radii.
Recall the formula for resistance:
The resistance \(R\) of a wire is given by the formula:
where \(\rho\) is the resistivity of the material, \(L\) is the length of the wire, and \(A\) is the cross-sectional area.
Express the cross-sectional area in terms of radius:
For a circular wire, the cross-sectional area \(A\) is given by:
where \(r\) is the radius of the wire. Substituting this into the resistance formula:
\[ R = \rho \frac{L}{\pi r^2} \]Apply the formula to wires A and B:
For wire A:
For wire B:
\[ R_B = \rho \frac{L_B}{\pi r_B^2} \]We are given that both wires are made of the same material, so their resistivities (\(\rho\)) are the same. They also have the same length, so \(L_A = L_B = L\).
Thus, the equations become:
\[ R_A = \rho \frac{L}{\pi r_A^2} \]\[ R_B = \rho \frac{L}{\pi r_B^2} \]Use the given relationship between \(R_A\) and \(R_B\):
We are given that \(R_A = 9 R_B\). Substitute the expressions for \(R_A\) and \(R_B\):
Simplify the equation and find the ratio of radii:
Cancel out the common terms \(\rho\), \(L\), and \(\pi\) from both sides:
Rearrange to find the ratio \(\frac{r_A}{r_B}\):
\[ \frac{r_B^2}{r_A^2} = 9 \]\[ \left( \frac{r_B}{r_A} \right)^2 = 9 \]Taking the square root of both sides:
\[ \frac{r_B}{r_A} = \sqrt{9} \]\[ \frac{r_B}{r_A} = 3 \]Therefore, the ratio of the radius of wire A to that of wire B is:
\[ \frac{r_A}{r_B} = \frac{1}{3} \]This can be written as 1 : 3.
C) 1 : 3