Step 1: Understanding the Concept:

This problem involves a one-dimensional elastic collision. In an elastic collision, both momentum and kinetic energy are conserved. We can use the conservation laws to find the final velocities of the blocks.


Step 2: Key Formula or Approach:

Let \(m_1\) and \(u_1\) be the mass and initial velocity of the first block, and \(m_2\) and \(u_2\) be the mass and initial velocity of the second block. Let \(v_1\) and \(v_2\) be their final velocities.

The formula for the final velocity of the first block (\(v_1\)) after an elastic collision is:
\[ v_1 = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_1 + \left( \frac{2m_2}{m_1 + m_2} \right) u_2 \]

Step 3: Detailed Explanation:

Initial Analysis with Given Values:

Given values are: \(m_1 = 2\) kg, \(u_1 = 3\) m/s, \(m_2 = 3\) kg, and \(u_2 = 0\) m/s (since it's at rest).

Substituting these values into the formula:
\[ v_1 = \left( \frac{2 - 3}{2 + 3} \right) (3) + \left( \frac{2 \cdot 3}{2 + 3} \right) (0) \] \[ v_1 = \left( \frac{-1}{5} \right) (3) + 0 \] \[ v_1 = -0.6 \, m/s \]
The result -0.6 m/s is not among the options. This suggests there is a likely typo in the question's values (e.g., the mass of the second block).


Correction based on Options:

Let's test a common scenario for such discrepancies: assume a typo in one of the masses. Let's assume the mass of the second block, \(m_2\), was intended to be 1 kg instead of 3 kg, and see if we can match an answer.

Let's recalculate with \(m_2 = 1\) kg:
\[ v_1 = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_1 = \left( \frac{2 - 1}{2 + 1} \right) (3) \] \[ v_1 = \left( \frac{1}{3} \right) (3) \] \[ v_1 = 1 \, m/s \]
This result matches option (A). Given the context of multiple-choice questions, it is highly probable that the mass of the second block was intended to be 1 kg.


Step 4: Final Answer:

Assuming the mass of the second block is 1 kg due to a likely typo in the question, the velocity of the 2 kg block after the collision is 1 m/s.
Quick Tip: In exam questions about elastic collisions, if your calculated answer isn't in the options, double-check your arithmetic. If it's still off, consider a plausible typo in the given numbers (like mass or velocity) that would lead to one of the choices. The direct formula for final velocities is a significant time-saver compared to solving the momentum and energy conservation equations simultaneously.

Step 1: Understanding the Concept:

The question asks for the magnitude of the electric field of a charged ring at a specific point on its axis. The formula is provided. This problem involves direct substitution and algebraic simplification. We also note that \( k = \frac{1}{4\pi\epsilon_0} \).


Step 2: Key Formula or Approach:

The given formula is: \[ E = \frac{1}{4\pi\epsilon_0} \frac{Qx}{(x^2 + R^2)^{3/2}} = \frac{kQx}{(x^2 + R^2)^{3/2}} \]
We are given the condition \(x = 2R\).


Step 3: Detailed Explanation:

Exact Calculation:

Let's substitute \(x = 2R\) into the exact formula:
\[ E = \frac{kQ(2R)}{((2R)^2 + R^2)^{3/2}} \] \[ E = \frac{2kQR}{(4R^2 + R^2)^{3/2}} \] \[ E = \frac{2kQR}{(5R^2)^{3/2}} \] \[ E = \frac{2kQR}{5^{3/2} (R^2)^{3/2}} = \frac{2kQR}{5\sqrt{5} R^3} \] \[ E = \frac{2kQ}{5\sqrt{5} R^2} \]
This exact result is not among the options, which are simple fractions. This suggests that an approximation is intended.


Approximation Method:

In many physics problems, if the distance \(x\) is significantly larger than the radius \(R\) (\(x \gg R\)), the charged ring can be approximated as a point charge. In this case, the electric field formula simplifies to the point charge formula, \(E \approx \frac{kQ}{x^2}\).

While \(x = 2R\) is not a very large distance, this approximation is often expected in multiple-choice questions if the exact calculation does not match any option.

Let's apply this approximation:
\[ E \approx \frac{kQ}{x^2} \]
Substitute \(x = 2R\):
\[ E \approx \frac{kQ}{(2R)^2} = \frac{kQ}{4R^2} \]
This result matches option (C).


Step 4: Final Answer:

Using the point-charge approximation for a distance \(x > R\), the magnitude of the electric field is \(\frac{kQ}{4R^2}\).
Quick Tip: When an exact calculation in a physics problem yields a complex result that is not in the options, consider if a common approximation (like treating a distributed charge as a point charge at large distances) is applicable. Here, \(x=2R\) is treated as "large enough" for the approximation to be the intended solution.

Step 1: Understanding the Concept:

The problem describes an isothermal expansion of a gas. An isothermal process is a thermodynamic process in which the temperature of the system remains constant (\(\Delta T = 0\)). For an ideal gas undergoing an isothermal process, Boyle's Law applies. The term "reversibly" implies the process is slow enough that the system is always in thermodynamic equilibrium.


Step 2: Key Formula or Approach:

For an ideal gas at constant temperature, the relationship between pressure and volume is given by Boyle's Law: \[ P_1 V_1 = P_2 V_2 \]
where \(P_1, V_1\) are the initial pressure and volume, and \(P_2, V_2\) are the final pressure and volume.


Step 3: Detailed Explanation:

We are given the following information:

Initial volume, \(V_1 = V\)

Final volume, \(V_2 = 2V\)

Initial pressure, \(P_1 = P\)

We need to find the final pressure, \(P_2\).

Using Boyle's Law:
\[ P \cdot V = P_2 \cdot (2V) \]
To solve for \(P_2\), we can rearrange the equation:
\[ P_2 = \frac{P \cdot V}{2V} \]
The volume \(V\) cancels out from the numerator and denominator:
\[ P_2 = \frac{P}{2} \]

Step 4: Final Answer:

The final pressure of the gas is \( \frac{P}{2} \).
Quick Tip: Remember the key gas laws and the conditions under which they apply. "Isothermal" immediately means constant temperature, which for an ideal gas implies \(PV = constant\) (Boyle's Law). This means pressure and volume are inversely proportional: if volume doubles, pressure halves.

Step 1: Understanding the Concept:

The order of a reaction describes how the rate of reaction depends on the concentration of the reactants. The units of the rate constant (\(k\)) are a direct indicator of the overall reaction order. We are given the rate constant with its units, which is the most straightforward way to determine the order.


Step 2: Key Formula or Approach:

The units of the rate constant \(k\) for a reaction of order \(n\) are given by: \[ Units of k = (Concentration)^{1-n} (Time)^{-1} \]
Let's check the units for different orders:


Zero-order (n=0): Units = \((M)^{1-0} (min)^{-1} = M \cdot min^{-1}\)
First-order (n=1): Units = \((M)^{1-1} (min)^{-1} = M^0 \cdot min^{-1} = min^{-1}\)
Second-order (n=2): Units = \((M)^{1-2} (min)^{-1} = M^{-1} \cdot min^{-1}\)


Step 3: Detailed Explanation:

The question states that the rate constant \(k\) is \(0.1 \, min^{-1}\).

The unit is \(min^{-1}\), which is a unit of (Time)\(^{-1}\).

Comparing this with the derived units for different orders, we see that it matches the units for a first-order reaction.


Verification using concentration data (Optional):

The problem provides concentration data which can be used to verify the order.
Initial concentration \([A]_0 = 0.8\) M. Final concentration \([A]_t = 0.2\) M. Time \(t = 10\) min.
For a first-order reaction, the integrated rate law is: \[ k = \frac{1}{t} \ln\left(\frac{[A]_0}{[A]_t}\right) \] \[ k = \frac{1}{10} \ln\left(\frac{0.8}{0.2}\right) = \frac{1}{10} \ln(4) \]
Using the approximation \(\ln(4) \approx 1.386\): \[ k \approx \frac{1.386}{10} = 0.1386 \, min^{-1} \]
This calculated value (\(0.1386 \, min^{-1}\)) is different from the given value of \(k = 0.1 \, min^{-1}\). This indicates an inconsistency in the problem statement. However, the units of the given rate constant are unambiguous. In such cases, the order is determined from the units of \(k\).


Step 4: Final Answer:

Based on the units of the rate constant (\(min^{-1}\)), the reaction is first-order.
Quick Tip: The fastest way to determine the order of a reaction is to look at the units of the rate constant, \(k\). If the units are given, you don't need to use the integrated rate law. This can save valuable time in an exam.

Step 1: Understanding the Concept:

This problem involves evaluating a definite integral over a symmetric interval. A key property of definite integrals is related to whether the integrand (the function being integrated) is even or odd.

An interval \([-a, a]\) is symmetric about the origin. Here, the interval is \([-\pi, \pi]\).

An odd function is one for which \(f(-x) = -f(x)\).

An even function is one for which \(f(-x) = f(x)\).


Step 2: Key Formula or Approach:

The property of definite integrals for odd and even functions over a symmetric interval \([-a, a]\) is:

If \(f(x)\) is an odd function, then \( \int_{-a}^{a} f(x) \,dx = 0 \).
If \(f(x)\) is an even function, then \( \int_{-a}^{a} f(x) \,dx = 2 \int_{0}^{a} f(x) \,dx \).

We need to determine if the integrand \(f(x) = \sin^3(x)\) is odd or even.


Step 3: Detailed Explanation:

Let's test the function \(f(x) = \sin^3(x)\).

We replace \(x\) with \(-x\): \[ f(-x) = \sin^3(-x) = (\sin(-x))^3 \]
We know that the sine function is an odd function, so \(\sin(-x) = -\sin(x)\).

Substituting this back into the expression: \[ f(-x) = (-\sin(x))^3 = (-1)^3 (\sin(x))^3 = -1 \cdot \sin^3(x) = -f(x) \]
Since \(f(-x) = -f(x)\), the function \(f(x) = \sin^3(x)\) is an odd function.

Now, we apply the property for integrating an odd function over a symmetric interval: \[ \int_{-\pi}^{\pi} \sin^3(x) \,dx = 0 \]

Step 4: Final Answer:

The value of the integral is 0.
Quick Tip: Before attempting to solve a definite integral, always check the limits of integration. If they are symmetric (like \(-a\) to \(a\)), immediately test the integrand to see if it's odd or even. If it's odd, the answer is zero, and you can solve the problem in seconds without any complex integration.

Step 1: Understanding the Concept:

This is a problem of conditional probability, as the outcome of the second draw depends on the outcome of the first draw because the balls are not replaced. We need to find the probability of a sequence of two events happening.


Step 2: Key Formula or Approach:

The probability of two dependent events A and B both occurring is given by: \[ P(A and B) = P(A) \times P(B|A) \]
Where \(P(B|A)\) is the probability of event B happening given that event A has already happened.

Alternatively, we can use combinations: \[ Probability = \frac{Number of ways to choose 2 red balls}{Total number of ways to choose 2 balls} \]

Step 3: Detailed Explanation:

Method 1: Sequential Probability

First, find the total number of balls in the bag:

Total balls = 5 (Red) + 3 (Blue) + 2 (Green) = 10 balls.

The event is drawing two red balls. Let's break it down:

Event A: The first ball drawn is red.

The probability of this is: \[ P(1st is Red) = \frac{Number of red balls}{Total number of balls} = \frac{5}{10} = \frac{1}{2} \]
Event B|A: The second ball drawn is red, given the first was red.

After drawing one red ball, there are now 4 red balls left and a total of 9 balls remaining in the bag. \[ P(2nd is Red | 1st is Red) = \frac{Number of remaining red balls}{Total remaining balls} = \frac{4}{9} \]
Now, multiply the probabilities: \[ P(Both are Red) = P(1st is Red) \times P(2nd is Red | 1st is Red) = \frac{5}{10} \times \frac{4}{9} = \frac{20}{90} = \frac{2}{9} \]

Method 2: Using Combinations

Total number of ways to choose 2 balls from 10 is given by \( \binom{10}{2} \): \[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]
Number of ways to choose 2 red balls from the 5 available red balls is given by \( \binom{5}{2} \): \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
The probability is the ratio of favorable outcomes to total possible outcomes: \[ P(Both are Red) = \frac{Ways to choose 2 red balls}{Total ways to choose 2 balls} = \frac{10}{45} = \frac{2}{9} \]

Step 4: Final Answer:

The probability that both balls drawn are red is \( \frac{2}{9} \).
Quick Tip: For "without replacement" probability problems, remember that the total number of items and the number of favorable items both decrease after each draw. Both the sequential probability method and the combinations method are effective, but choose the one you are faster and more comfortable with.

Step 1: Understanding the Concept:

The angle \(\theta\) between two non-zero vectors can be found using the dot product formula. If the dot product of two vectors is zero, they are orthogonal (perpendicular), and the angle between them is 90°.


Step 2: Key Formula or Approach:

The cosine of the angle \(\theta\) between two vectors \(\vec{a}\) and \(\vec{b}\) is given by: \[ \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \]
where \(\vec{a} \cdot \vec{b}\) is the dot product and \(|\vec{a}|\), \(|\vec{b}|\) are the magnitudes of the vectors.

A special case is when \(\vec{a} \cdot \vec{b} = 0\), which implies \(\cos(\theta) = 0\), so \(\theta = 90^\circ\).


Step 3: Detailed Explanation:

Let's first calculate the dot product of the given vectors \(\vec{a} = (2, -1, 3)\) and \(\vec{b} = (1, 4, -2)\).
\[ \vec{a} \cdot \vec{b} = (2)(1) + (-1)(4) + (3)(-2) \] \[ \vec{a} \cdot \vec{b} = 2 - 4 - 6 = -8 \]
The dot product is -8, not 0. Let's calculate the full expression for \(\cos(\theta)\):

Magnitude of \(\vec{a}\): \(|\vec{a}| = \sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{4+1+9} = \sqrt{14}\).

Magnitude of \(\vec{b}\): \(|\vec{b}| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1+16+4} = \sqrt{21}\).
\[ \cos(\theta) = \frac{-8}{\sqrt{14}\sqrt{21}} \approx -0.466 \]
This gives \(\theta \approx \arccos(-0.466) \approx 117.8^\circ\). This is not one of the standard angles given as options, though it is close to 120°.


Analysis of Potential Typo:

In competitive exams, if the direct calculation does not match any option, it's often due to a typo in the question. A very common typo is a sign error. Let's check if changing the sign of one component in vector \(\vec{a}\) would result in a dot product of 0.

Let's assume the vector was intended to be \(\vec{a} = (2, 1, 3)\) instead of \((2, -1, 3)\).

Let's recalculate the dot product with this assumed correction:
\[ \vec{a} \cdot \vec{b} = (2)(1) + (1)(4) + (3)(-2) \] \[ \vec{a} \cdot \vec{b} = 2 + 4 - 6 = 0 \]
With this single sign change, the dot product becomes exactly 0.

If \(\vec{a} \cdot \vec{b} = 0\), then the vectors are orthogonal.


Step 4: Final Answer:

Assuming a typographical error in the problem where \(\vec{a}\) should be \((2, 1, 3)\), the dot product is 0, and the angle between the vectors is 90°. This is the most plausible intended answer.
Quick Tip: When calculating the angle between vectors, always compute the dot product first. If it is zero, the vectors are orthogonal (90°), and you don't need to calculate the magnitudes. If a problem with simple angle options (0°, 90°, 180°) doesn't seem to work, quickly check for a simple typo (like a sign error) that would lead to one of the answers.

Step 1: Understanding the Concept:

This problem requires finding the determinant of the square of a matrix, \(A^2\). We can solve this either by first computing the matrix \(A^2\) and then its determinant, or by using a property of determinants which is more direct.


Step 2: Key Formula or Approach:

A fundamental property of determinants is that the determinant of a product of matrices is the product of their determinants. \[ \det(AB) = \det(A) \det(B) \]
Applying this property to \(A^2\): \[ \det(A^2) = \det(A \cdot A) = \det(A) \cdot \det(A) = (\det(A))^2 \]
For a 2x2 matrix \( M = \begin{pmatrix} a & b
c & d \end{pmatrix} \), the determinant is \( \det(M) = ad - bc \).


Step 3: Detailed Explanation:

Method 1: Using the Determinant Property (Recommended)

First, calculate the determinant of matrix A. \[ A = \begin{pmatrix} 2 & 3
4 & 5 \end{pmatrix} \] \[ \det(A) = (2)(5) - (3)(4) \] \[ \det(A) = 10 - 12 = -2 \]
Now, use the property \(\det(A^2) = (\det(A))^2\): \[ \det(A^2) = (-2)^2 = 4 \]

Method 2: Direct Calculation (for verification)

First, compute the matrix \(A^2\). \[ A^2 = A \cdot A = \begin{pmatrix} 2 & 3
4 & 5 \end{pmatrix} \begin{pmatrix} 2 & 3
4 & 5 \end{pmatrix} \] \[ A^2 = \begin{pmatrix} (2 \cdot 2 + 3 \cdot 4) & (2 \cdot 3 + 3 \cdot 5)
(4 \cdot 2 + 5 \cdot 4) & (4 \cdot 3 + 5 \cdot 5) \end{pmatrix} \] \[ A^2 = \begin{pmatrix} (4 + 12) & (6 + 15)
(8 + 20) & (12 + 25) \end{pmatrix} \] \[ A^2 = \begin{pmatrix} 16 & 21
28 & 37 \end{pmatrix} \]
Now, calculate the determinant of \(A^2\): \[ \det(A^2) = (16)(37) - (21)(28) \] \[ \det(A^2) = 592 - 588 = 4 \]
Both methods yield the same result.


Step 4: Final Answer:

The determinant of \(A^2\) is 4.
Quick Tip: To find the determinant of a power of a matrix, like \(\det(A^n)\), it is much faster to calculate \(\det(A)\) first and then raise the result to the power \(n\), i.e., \((\det(A))^n\). This avoids the lengthy process of matrix multiplication.