The question asks for the population threshold that defines a city as a "metropolitan city" under Indian law for development purposes. This is a factual question related to urban classification in India.
B) 10 lakh (1,000,000) is the correct population threshold for a city to be considered a metropolitan city under Indian law for development purposes. This definition is consistent across various government classifications and policies.
The question asks about the migration of ions during the chlor-alkali process, specifically towards the cathode. The chlor-alkali process involves the electrolysis of an aqueous solution of sodium chloride (brine). Understanding the principles of electrolysis, including the nature of electrodes and the movement of ions, is key to answering this question.
B) Na⁺ — Sodium ions (Na⁺) are positively charged cations. During electrolysis, cations are attracted to and migrate towards the negatively charged electrode, which is the cathode. While water is reduced at the cathode, Na⁺ ions still move towards it, contributing to the formation of NaOH in the solution.
To find the product of 597 and 603, we can use the algebraic identity \((a-b)(a+b) = a^2 - b^2\). This method simplifies the multiplication of numbers that are equidistant from a common number.
The numbers are 597 and 603. We can observe that both numbers are close to 600.
Comparing this with the identity \((a-b)(a+b)\), we can set:
Substitute the values of \(a\) and \(b\) into the identity \(a^2 - b^2\):
\[ 597 \times 603 = (600 - 3)(600 + 3) \]\[ = 600^2 - 3^2 \]Now, subtract the value of \(b^2\) from \(a^2\):
\[ 360000 - 9 = 359991 \]Thus, \(597 \times 603 = 359991\).
C) 359991 is the correct product obtained by applying the algebraic identity \((a-b)(a+b) = a^2 - b^2\).
To determine which organic compound does not belong to the same homologous series, we need to identify the general formula for each given compound and classify them into their respective homologous series (alkanes, alkenes, alkynes, etc.). A homologous series is a series of organic compounds with the same functional group but with each successive member differing by a \(-CH_2\) group.
Correct Option: C) C₄H₈
The mean proportional between
8 + 4\(\sqrt{2}\) and 6 - 3\(\sqrt{2}\) is:
3\(\sqrt{6}\)
\(\sqrt{6}\)
2\(\sqrt{6}\)
4\(\sqrt{6}\)
The mean proportional (or geometric mean) between two numbers, say \(a\) and \(b\), is given by the formula \(\sqrt{ab}\). We need to calculate this value for the given expressions \(8 + 4\sqrt{2}\) and \(6 - 3\sqrt{2}\).
Let the two given numbers be \(a = 8 + 4\sqrt{2}\) and \(b = 6 - 3\sqrt{2}\).
The mean proportional, let's call it \(M\), is given by:
\[ M = \sqrt{ab} \]Substitute the values of \(a\) and \(b\) into the formula:
\[ M = \sqrt{(8 + 4\sqrt{2})(6 - 3\sqrt{2})} \]Factor out common terms from each expression:
Substitute these factored forms back into the expression for \(M\):
\[ M = \sqrt{4(2 + \sqrt{2}) \cdot 3(2 - \sqrt{2})} \]Rearrange the terms:
\[ M = \sqrt{4 \cdot 3 \cdot (2 + \sqrt{2})(2 - \sqrt{2})} \]Multiply the constants and use the difference of squares formula \((x+y)(x-y) = x^2 - y^2\) for the terms in the parentheses:
\[ M = \sqrt{12 \cdot (2^2 - (\sqrt{2})^2)} \] \[ M = \sqrt{12 \cdot (4 - 2)} \] \[ M = \sqrt{12 \cdot 2} \] \[ M = \sqrt{24} \]Simplify \(\sqrt{24}\):
\[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \]C) The calculated mean proportional is \(2\sqrt{6}\), which matches option C.
The question asks what needs to be managed for sustainability. Sustainability refers to meeting the needs of the present without compromising the ability of future generations to meet their own needs. This concept is fundamentally linked to the responsible use and preservation of resources that are essential for life and economic activity.
B) Natural resources — Sustainability is inherently about the responsible use and conservation of natural resources (like water, air, land, forests, minerals, and biodiversity) to ensure their availability and quality for present and future generations. Without proper management of these fundamental resources, long-term sustainability is impossible.
Find the value of \(\frac{\mathrm{tan}^{2}60^\circ+\mathrm{cot}^{2}30^\circ}{\mathrm{tan}^{2}20^\circ-\mathrm{sec}^{2}20^\circ}-\frac{\mathrm{sin}90^\circ+\mathrm{cos}0^\circ}{\mathrm{cosec}^{2}20^\circ-\mathrm{cot}^{2}20^\circ}\)
The problem requires us to evaluate a trigonometric expression. We will use standard trigonometric identities and values for specific angles to simplify the expression. The key identities to recall are \(\mathrm{sec}^2\theta - \mathrm{tan}^2\theta = 1\) and \(\mathrm{cosec}^2\theta - \mathrm{cot}^2\theta = 1\).
Recall the values of trigonometric ratios for standard angles:
Recall the fundamental trigonometric identities:
Substitute these values and identities into the given expression:
The given expression is:
\[ \frac{\mathrm{tan}^{2}60^\circ+\mathrm{cot}^{2}30^\circ}{\mathrm{tan}^{2}20^\circ-\mathrm{sec}^{2}20^\circ}-\frac{\mathrm{sin}90^\circ+\mathrm{cos}0^\circ}{\mathrm{cosec}^{2}20^\circ-\mathrm{cot}^{2}20^\circ} \]Evaluate the first term:
The numerator of the first term is \(\mathrm{tan}^{2}60^\circ+\mathrm{cot}^{2}30^\circ\). Substituting the values:
\[ (\sqrt{3})^2 + (\sqrt{3})^2 = 3 + 3 = 6 \]The denominator of the first term is \(\mathrm{tan}^{2}20^\circ-\mathrm{sec}^{2}20^\circ\). Using the identity \(\mathrm{sec}^2\theta - \mathrm{tan}^2\theta = 1\), we have \(\mathrm{tan}^2\theta - \mathrm{sec}^2\theta = -1\). So, for \(\theta = 20^\circ\):
\[ \mathrm{tan}^{2}20^\circ-\mathrm{sec}^{2}20^\circ = -1 \]Thus, the first term is:
\[ \frac{6}{-1} = -6 \]Evaluate the second term:
The numerator of the second term is \(\mathrm{sin}90^\circ+\mathrm{cos}0^\circ\). Substituting the values:
\[ 1 + 1 = 2 \]The denominator of the second term is \(\mathrm{cosec}^{2}20^\circ-\mathrm{cot}^{2}20^\circ\). Using the identity \(\mathrm{cosec}^2\theta - \mathrm{cot}^2\theta = 1\), we have:
\[ \mathrm{cosec}^{2}20^\circ-\mathrm{cot}^{2}20^\circ = 1 \]Thus, the second term is:
\[ \frac{2}{1} = 2 \]Combine the results from step 4 and step 5:
The original expression becomes:
\[ -6 - 2 = -8 \]B) The value of the given expression is -8.
This problem involves the relationship between distance, speed, and time. We are given a distance, and two scenarios involving changes in speed and time. We can set up equations based on the formula \(\text{Distance} = \text{Speed} \times \text{Time}\) and solve for the usual time.
Let the usual speed of the vehicle be \(S\) km/h and the usual time taken be \(T\) hours.
The distance covered is 1800 km. Using the formula \(\text{Distance} = \text{Speed} \times \text{Time}\), we can write the first equation:
\[ 1800 = S \times T \quad \text{(Equation 1)} \]In the second scenario, the speed decreases by 10 km/h, so the new speed is \((S - 10)\) km/h.
The time taken increases by 2 hours, so the new time is \((T + 2)\) hours.
The distance remains the same (1800 km). So, we can write the second equation:
\[ 1800 = (S - 10) \times (T + 2) \quad \text{(Equation 2)} \]From Equation 1, we can express \(S\) in terms of \(T\):
\[ S = \frac{1800}{T} \]Substitute this expression for \(S\) into Equation 2:
\[ 1800 = \left(\frac{1800}{T} - 10\right) \times (T + 2) \]Expand the right side of the equation:
\[ 1800 = \frac{1800(T + 2)}{T} - 10(T + 2) \] \[ 1800 = 1800 + \frac{3600}{T} - 10T - 20 \]Simplify the equation:
\[ 0 = \frac{3600}{T} - 10T - 20 \]Multiply the entire equation by \(T\) to eliminate the denominator:
\[ 0 = 3600 - 10T^2 - 20T \]Rearrange the terms to form a quadratic equation:
\[ 10T^2 + 20T - 3600 = 0 \]Divide the entire equation by 10 to simplify:
\[ T^2 + 2T - 360 = 0 \]Solve the quadratic equation for \(T\). We can factorize it or use the quadratic formula. Let's factorize:
We need two numbers that multiply to -360 and add up to 2. These numbers are 20 and -18.
\[ (T + 20)(T - 18) = 0 \]This gives two possible values for \(T\): \(T = -20\) or \(T = 18\).
Since time cannot be negative, we discard \(T = -20\).
Therefore, the usual time taken is \(T = 18\) hours.
A) 18 hours is the correct answer. This is the positive value obtained for \(T\) from the quadratic equation, representing the usual time taken to cover the distance.
The question asks to identify the sportsperson who was awarded the Padma Bhushan in January 2022. This requires knowledge of the Padma Awards recipients for that specific year, particularly in the sports category.
Correct Option: D) Devendra Jhajharia was awarded the Padma Bhushan in January 2022. He is a celebrated Indian Paralympic javelin thrower, who has won multiple gold medals at the Paralympics.
Find the value of
222.22+22.22+12+0.22+0.02?
To find the sum of the given decimal numbers, we need to align them by their decimal points and then add them column by column, starting from the rightmost digit.
So, the hundredths digit of the sum is 8.
So, the tenths digit of the sum is 6.
So, the units digit of the sum is 6.
So, the tens digit of the sum is 5.
So, the hundreds digit of the sum is 2.
D) 256.68 is the correct sum obtained by aligning the decimal points and adding the numbers.