Mathematical Methods

To find the moment of inertia of the square laminar sheet about the y-axis, we need to integrate over the entire area of the sheet considering the given mass per unit area .

• The mass per unit area is given as .
• The small strip at a distance from the origin along the x-axis has a width and height , making its area .
• The mass of the strip, , is given by multiplying the area of the strip by the mass per unit area: .

The moment of inertia of this strip about the y-axis is:

To find the total moment of inertia, integrate over the width of the sheet from to :

Now, let's calculate the integral:

• After evaluating the definite integral:

According to the problem, the total mass of the sheet is:

Solving for , we get .

Substitute back to get :

Therefore, the moment of inertia of the sheet about the y-axis is

To solve this problem, we need to find the decimal equivalent of a binary number with digits, where is even. The given binary number alternates between 1's and 0's, with digit 1 at the highest place value.

Let's analyze how the binary number looks:

• With digits, alternating 1's and 0's, starting with 1, the pattern of the binary digits will be: 1010...10. Since is even, half of the digits will be 1's, and half will be 0's.

Let's calculate the decimal value of this binary number:

The binary number can be generalized as:

• If , for an integer , the binary sequence can be expressed as:
• 1 at positions:
• 0 at positions:

In decimal, this number is calculated by summing up the powers of 2 corresponding to each 1 in the binary sequence:

The decimal value of this number can be expressed as:

• This series is a geometric progression with the first term and common ratio .

Using the formula for the sum of a geometric series , where is the number of terms, we can find the sum:

• The number of terms is (since there are ones).
• So, .

Substitute , and simplify the expression:

Notice that , so , hence:

• Thus, .

To analyze the given 2 x 2 matrix , with , we need to check each of the statements provided in the options:

1. M is a real symmetric matrix:
   A matrix is symmetric if it is equal to its transpose. Let's compute the transpose of matrix : . Since the transpose of is equal to itself, matrix is indeed a real symmetric matrix.
2. One of the eigenvalues of M is greater than b:
   To find the eigenvalues of the matrix , we solve the characteristic equation: , where is the identity matrix.

1. .
   The determinant is calculated as: . Solving this quadratic equation: . The larger root: is clearly greater than , since .
2. One of the eigenvalues of M is negative:
   Using the solutions from before, the smaller eigenvalue is: . Since , the second term is greater than , hence is negative.
3. Product of eigenvalues of M is b:
   The product of the eigenvalues of a matrix is equal to its determinant. Calculating the determinant of : . Therefore, the product of the eigenvalues is , not .

Therefore, the correct statements about the matrix are:

• M is a real symmetric matrix.
• One of the eigenvalues of M is greater than b.
• One of the eigenvalues of M is negative.

To solve the integral over the area of a disk of radius 2 in the xy-plane, we use polar coordinates which simplify the integration over circular regions. The transformation is given by and . The Jacobian of the transformation from Cartesian to polar coordinates is , thus . The limits for will range from 0 to 2, and for from 0 to .
The integral becomes:
.
First, evaluate the inner integral with respect to :
.
Next, integrate with respect to :
.
This result, , confirms that the integral of over the disk is indeed .

1. Understand the Vector Field and Geometry

Curve : A unit circle in the -plane ( ) centered at the origin.

Vector Field: The expression given is . In standard vector notation, the first term without a specific unit vector corresponds to the (or ) component. Thus, the vector field is:

$ \oint_C \vec{F} \cdot d\vec{r} C \theta 0 2\pi d\vec{r} \vec{F} z=0 \vec{F} \cdot d\vec{r} \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS S xy C \hat{n} C xy \hat{n} = \hat{k} \vec{F} \hat{k} 2\pi$.

The blank should be filled with 2.

To solve for the integer in the given problem, we analyze the reflection coefficient as the energy becomes much greater than the potential .

In region 2 ( ), the momentum of the electron is given by:


The reflection coefficient is expressed as:


Substituting and , we have:


Simplifying the terms:


For , the approximation is:


Substitute back:


As , , confirming that the integer is determined by how rapidly approaches 0, characterized by the factor .

Comparing the exponents, .

To solve this problem, we need to utilize the equation of continuity given by:
$ p(1,1,1,1) \nabla \cdot \overrightarrow{j} j x p(x,y,z,t) p(x,y,z,t) = f(t) t p(x, y, z, 0) = 1 (1, 1, 1, 0) t=1, x=1, y=1, z=1 $ (rounded to 2 decimal places).

To solve the problem, we need to interpret the given equation in the context of the complex plane. The given equation is:

Let where are real numbers and is the complex conjugate, i.e., .

Substituting these into the equation:

Adding these:

Thus, the equation simplifies to:

Dividing through by 2 gives:

This is the standard form of a hyperbola in the complex plane. Hence, the correct interpretation of the equation is a hyperbola.

Correct Answer: Hyperbola

To solve this problem, we will utilize the Lorentz transformation equations which are used to relate the space and time coordinates of the same event in different inertial frames of reference moving relative to each other at a constant velocity.

The general form of the Lorentz transformation equations when moving with velocity along the x-axis is:

In the problem, the rocket moves with a speed . We will substitute this value into the Lorentz transformation equations.

Substituting :

1. The Lorentz factor, , is computed as follows:

1. Now, substitute and into the Lorentz transformation equations:

This correctly matches the first option provided in the question:

To solve the binary subtraction problem , follow these steps:

1. Write the numbers in alignment for binary subtraction:

1. Subtract from right to left. Note that binary subtraction is similar to decimal subtraction where 1-0 = 1, 0-0 = 0, 1-1 = 0, and if 0 is to be subtracted from 1, borrowing from the left is required:
2. Borrow where necessary, starting from the rightmost digit (LSB):
   • requires borrowing, change to .
   • Continue borrowing from left so that subtraction is possible.
3. Continue this process till the leftmost digit (MSB).
4. The final result of the subtraction is .
5. Cross-verify the subtraction for accuracy, tally with given options, and identify that the correct answer is 1010111.

Thus, the resultant of the binary subtraction is 1010111.

To solve the integral over the unit circle centered at the origin and traversed anticlockwise, we can employ Green's Theorem. This theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by .

Green's Theorem is expressed as:

where and . We can now calculate the partial derivatives:

Substituting these into Green's Theorem, we get:

The region is the unit circle, which has an area of , where . Therefore, the area of the unit circle is .

Thus, the double integral over becomes:

Therefore, the value of the given integral is 2π.

To solve this problem, we need to find the curl of each given vector field and identify which field has a non-zero curl. The curl of a vector field is given by:

1. Option 1:
   
   
   The curl is zero.
2. Option 2:
   
   
   The curl is zero.
3. Option 3:
   
   
   The curl is potentially non-zero, but let's verify the specific computations.
   Calculating further, , as the non-zero terms proportionate cancel out.
4. Option 4:
   
   
   Thus, the curl is non-zero.

Therefore, the vector field with a non-zero curl is Option 4: .

To analyze the given function , let's break it down step by step:

1. The inner absolute function reflects the input so that all outputs are non-negative. This means is a V-shaped graph centered at the origin.
2. The operation translates this graph downwards by 1 unit. Therefore, the expression takes the following form:
   • equals when , since at , .
   • It equals when (as for ).
   • It equals when (as for ).
3. The outer absolute function again makes any resulting negative values positive. Let's see what this transformation does to :
   • For , the result is . Thus, for , .
   • For , , and since is positive, , which linearly increases.
   • For , , which is negative, so , which also linearly increases as a reflection in the positive y-axis direction.

Now that we have broken down the transformations, the graph of is characterized by a constant value of 1 between and , and it increases linearly outside this interval.

With these observations, the correct plot for the function is the one that shows these characteristics. Based on the given options, the correct image corresponds to the description above:

Given:
Angle

Taylor series of sin θ:
Using only first two terms: Step 1: Compute approximation
Step 2: True value Step 3: Absolute error
Final Answer:

Given:
Angular speed:
Angle:

Unit vector time derivatives:
Unit vectors in Cartesian form:
Thus, x-components:
Sum: Evaluate at :
Compute: Multiply by : Final Answer:

Given:


Step 1: Find the gradient
Step 2: Evaluate at (1,2,1)
Step 3: Find magnitude
Step 4: Unit normal vector
Thus, Final Answer:

Given:
Width of incident pulse:
Width of transmitted pulse:
Tension in both strings is the same.

Concept:
Time duration of the pulse remains the same at the boundary. So, Wave speed on a string: Thus, Therefore, Calculation:
Squaring, Final Answer:

We need to find the sum of the eigenvalues and of the matrix , where the matrix is:

1. **Calculate **:

First, we compute , which is the square of the matrix : Performing matrix multiplication:

2. **Calculate matrix **:

Now, we compute , where is the identity matrix:

Simplifying:

3. **Find the eigenvalues of **:

To find the eigenvalues of , we use the characteristic equation:

The characteristic equation is: Expanding the determinant: Simplifying: Using the quadratic formula: The eigenvalues are:

4. **Sum of the Eigenvalues**:

The sum of the eigenvalues is:

Thus, the sum of the eigenvalues is approximately , which rounds to to .

Area of the Triangle OAB

The area of the triangle OAB is given by:

Area = (1/2) |OA × OB|

where OA × OB is the cross product of the vectors OA and OB.

Step 1: Compute the Cross Product

Given vectors:

The cross product is:

Step 2: Expand the Determinant

Computing each determinant:

Step 3: Compute the Magnitude

The magnitude of the cross product is:

Step 4: Compute the Area

Final Answer

The area of the triangle OAB is 4.33.

1. Matrix:
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: Yes (Inverse equals conjugate transpose)
   Result: This matrix is Hermitian and Unitary.
2. Matrix:
   Check:
   - Hermitian: No (Transpose + conjugate does not give the original matrix)
   - Unitary: Not applicable
   Result: Not Hermitian and Unitary.
3. Matrix:
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: No (Inverse does not equal conjugate transpose)
   Result: Not both Hermitian and Unitary.
4. Matrix:
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: No (Inverse does not equal conjugate transpose)
   Result: Not both Hermitian and Unitary.

Conclusion:

The matrix is Hermitian and Unitary.

The vector field is given in terms of its components along the , , and directions. The path forms a closed loop, and the line integral needs to be evaluated along this path.

Key Observations:

• The path is a circle with a unit radius.
• The components of the vector field depend on , , and .
• The integration will involve parametric equations representing the motion of the circle in 3D space.

Solution:

By applying the line integral along the closed loop and performing the necessary calculations, the result is:

Conclusion:

Thus, the correct answer is (A).

The vector field is , where and . Using the divergence formula:

after computing partial derivatives and simplifying, we find:

Final Answer: The divergence of is Zero.

The flux of the curl of a vector field through a surface is given by:

Using Stokes’ Theorem, this flux can be related to the circulation around the boundary of the surface, which in this case is a circular loop of radius 2 centered at the origin.

Since is a vector field in the x-y plane, the curl of in the z-direction can be evaluated using the given components.

Upon solving for the flux, we find that the magnitude of the flux through the circular loop is:

Conclusion:

Thus, the correct answer is (B).

Line Integral of the Electric Field and Electric Potential

The line integral of the electric field along a path is related to the change in electric potential between the start and end points of the path. Mathematically:

where:

• and are the electric potentials at points C and M, respectively.

1. Electric Potential at a Point

The electric potential due to a point charge at a distance is given by:

where is Coulomb’s constant.

2. Potential at Points C and M

Both points and are at the same distance from the charges (at A) and (at B), since is a semicircle of radius . Therefore, the electric potential at both and is the same.

For any point on the semicircle:

Thus,

3. Line Integral

Since the potentials at points and are equal ( ):

Final Answer

The line integral of the electric field along the given path is zero.

Taylor Expansion of

The Taylor expansion of can be derived by multiplying the Taylor expansions of and .

Step 1: Taylor Expansion of

Step 2: Taylor Expansion of

Step 3: Multiply the Two Series

Multiply the two series and collect terms up to :

Step 4: Identify Contributions to

• From :
• From :
• From :
• From :
• From :
• From :

Step 5: Compute the Coefficient of

Step 6: Simplify the Expression

Final Answer

The coefficient of in the Taylor expansion of is approximately .

By Vieta’s formulas for a polynomial, the sum and product of the roots of the polynomial are related to the coefficients of the polynomial.

For the given polynomial:

Step 1: Sum of the Roots

The sum of the roots, , is equal to the coefficient of (with the opposite sign):

Step 2: Product of the Roots

The product of the roots, , is equal to the constant term (with the opposite sign):

Step 3: Compute the Required Ratio

The required ratio is:

Conclusion:

Thus, the correct answer is option (B).

Step 1: Convert Binary Numbers to Decimal

1. : Convert to decimal:
2. : Convert to decimal:
3. : Convert to decimal:

Step 2: Add the Decimal Numbers

Now, add the converted decimal numbers:

Conclusion:

Therefore, the sum of the binary numbers in the decimal system is: