To find the kinetic energy of an object, we use the formula for kinetic energy, which depends on the object's mass and velocity. We will substitute the given values into the formula and calculate the result, ensuring the correct units are used.
The kinetic energy (\(KE\)) of an object is given by the formula:
\[ KE = \frac{1}{2}mv^2 \]The unit \(\text{kg m}^2\text{s}^{-2}\) is equivalent to Joules (\(\text{J}\)), which is the standard unit for energy.
\[ KE = 640 \text{ J} \]Correct Option: D) 640 J
\(\frac{\sqrt[3]{0.000027}}{\sqrt{0.0004}}\) =?
To solve this problem, we need to evaluate the cube root of the numerator and the square root of the denominator separately. Then, we will divide the result of the numerator by the result of the denominator. The key is to convert the decimal numbers into fractions or powers of 10 to simplify the root calculations.
The numerator is \(\sqrt[3]{0.000027}\).
We can write \(0.000027\) as a fraction:
\[ 0.000027 = \frac{27}{1000000} \]Now, take the cube root:
\[ \sqrt[3]{0.000027} = \sqrt[3]{\frac{27}{1000000}} = \frac{\sqrt[3]{27}}{\sqrt[3]{1000000}} \]We know that \(3^3 = 27\) and \(100^3 = 1000000\) (since \(10^2 \times 10^2 \times 10^2 = 10^6\), so \(100^3 = (10^2)^3 = 10^6\)).
So, the numerator becomes:
\[ \frac{3}{100} = 0.03 \]The denominator is \(\sqrt{0.0004}\).
We can write \(0.0004\) as a fraction:
\[ 0.0004 = \frac{4}{10000} \]Now, take the square root:
\[ \sqrt{0.0004} = \sqrt{\frac{4}{10000}} = \frac{\sqrt{4}}{\sqrt{10000}} \]We know that \(2^2 = 4\) and \(100^2 = 10000\).
So, the denominator becomes:
\[ \frac{2}{100} = 0.02 \]Now, divide the result from step 1 by the result from step 2:
\[ \frac{0.03}{0.02} \]To simplify this division, we can multiply both the numerator and the denominator by 100:
\[ \frac{0.03 \times 100}{0.02 \times 100} = \frac{3}{2} \]Finally, perform the division:
\[ \frac{3}{2} = 1.5 \]Correct Option: C) 1.5
The question asks for the term that describes the number of atoms constituting a molecule. We need to identify the correct chemical term among the given options.
Correct Option: B) Atomicity is the number of atoms constituting a molecule. For example, the atomicity of \(\text{O}_3\) (ozone) is 3, and for \(\text{P}_4\) (white phosphorus) it is 4.
To find the selling price of the toy, we need to calculate the profit amount based on the given profit percentage and the cost price. Then, add this profit to the cost price to get the selling price.
The cost price of the toy is given as Rs 132.
\[ \text{CP} = \text{Rs } 132 \]The profit percentage is given as 15%.
\[ \text{Profit Percentage} = 15\% \]The profit amount is 15% of the cost price.
\[ \text{Profit Amount} = \text{Profit Percentage} \times \text{CP} \]\[ \text{Profit Amount} = 15\% \times 132 \]\[ \text{Profit Amount} = \frac{15}{100} \times 132 \]\[ \text{Profit Amount} = 0.15 \times 132 \]\[ \text{Profit Amount} = \text{Rs } 19.80 \]The selling price is the sum of the cost price and the profit amount.
\[ \text{SP} = \text{CP} + \text{Profit Amount} \]\[ \text{SP} = 132 + 19.80 \]\[ \text{SP} = \text{Rs } 151.80 \]B) 151.80
The calculated selling price is Rs 151.80, which matches option B.
The question asks about the concentrations of ions in pure water. We need to recall the autoionization of water and the definition of a neutral solution.
Autoionization of Water: Pure water undergoes a process called autoionization (or autodissociation), where water molecules react with each other to form hydronium ions (\(\text{H}_3\text{O}^+\), often simplified as \(\text{H}^+\)) and hydroxide ions (\(\text{OH}^-\)).
\[ \text{2H}_2\text{O} ightleftharpoons \text{H}_3\text{O}^+ + \text{OH}^- \]Or, more simply:
\[ \text{H}_2\text{O} ightleftharpoons \text{H}^+ + \text{OH}^- \]Neutrality of Pure Water: Pure water is by definition a neutral solution. A neutral solution is characterized by having an equal concentration of hydrogen ions (\(\text{H}^+\)) and hydroxide ions (\(\text{OH}^-\)).
Concentrations at 25°C: At standard temperature (\(25^\circ\text{C}\)), the ion product of water (\(K_w\)) is \(1.0 \times 10^{-14}\).
\[ K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \]Since \([\text{H}^+] = [\text{OH}^-]\) in pure water, we can write:
\[ [\text{H}^+]^2 = 1.0 \times 10^{-14} \]Taking the square root of both sides:
\[ [\text{H}^+] = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \text{ M} \]Therefore, in pure water at \(25^\circ\text{C}\), \([\text{H}^+] = 1.0 \times 10^{-7} \text{ M}\) and \([\text{OH}^-] = 1.0 \times 10^{-7} \text{ M}\).
Conclusion: The concentrations of \(\text{H}^+\) and \(\text{OH}^-\) ions are equal in pure water.
D) H⁺ and OH⁻ ions are equal
This is correct because pure water is a neutral solution, meaning the concentration of hydrogen ions (\(\text{H}^+\)) is equal to the concentration of hydroxide ions (\(\text{OH}^-\)).
A) H⁺ ions is equal to that of oxide ions
— This is incorrect. Oxide ions (\(\text{O}^{2-}\)) are not formed in significant concentrations in pure water. The ions formed are \(\text{H}^+\) (or \(\text{H}_3\text{O}^+\)) and \(\text{OH}^-\).
B) OH⁻ ions is less than that of H⁺ ions
— This is incorrect. If \([\text{OH}^-]\) were less than \([\text{H}^+]\), the solution would be acidic. Pure water is neutral, so these concentrations are equal.
C) OH⁻ ions is more than that of oxides
— This is incorrect. As explained for option A, oxide ions (\(\text{O}^{2-}\)) are not relevant in this context for pure water. Also, if \([\text{OH}^-]\) were significantly higher than \([\text{H}^+]\), the solution would be basic, which pure water is not.
The question relates to basic Mendelian genetics, specifically the outcome of a monohybrid cross in the F₁ generation. Understanding the concepts of dominant and recessive traits is key to answering this question.
C) Dominant. The dominant character is the one that predominates and is clearly seen in the F₁ generation because it masks the expression of the recessive character.
To subtract decimal numbers, align the decimal points of the numbers and then perform the subtraction column by column, starting from the rightmost digit. If one number has fewer decimal places than the other, you can add trailing zeros to make the number of decimal places equal, which helps in proper alignment and calculation.
So, the calculation is:
\[ \begin{array}{r} 1.004 \\ -0.400 \\ \hline 0.604 \\ \end{array} \]
The result is \(0.604\).
Correct Option: D) \(0.604\) is the correct result of subtracting \(0.4\) from \(1.004\).
The question asks to identify a consumption tax levied at each stage of the supply chain where value is added, from production to the point of sale. We need to evaluate each option based on this definition.
A) VAT — Value Added Tax (VAT) is a consumption tax levied on a product whenever value is added at each stage of the supply chain, from production to the point of sale. This definition precisely describes VAT.
This question tests logical reasoning, specifically the ability to identify implicit assumptions in a statement. An assumption is something taken for granted or accepted as true without proof. We need to analyze what must be true for Lalitha's statement to make sense.
Correct Option: D) Only II is implicit
Assumption II is implicit because for Lalitha to suggest having ice cream, she must assume that both she and her friend would like it. An invitation to do something generally implies that the activity is desirable to those involved.