Vectors

This is a fundamental property of determinants. If a matrix has two identical rows or two identical columns, its determinant is zero. This can be understood by considering that swapping two rows negates the determinant. If the two rows are identical, swapping them changes nothing about the matrix, but the determinant must be negated. The only number that is equal to its own negative is zero. Therefore, the value of the determinant is 0.

Step 1: Recall the property of eigenvalues for a polynomial of a matrix. If is an eigenvalue of a matrix , then for any polynomial , the corresponding eigenvalue of the matrix is .
Step 2: Define the polynomial and list the eigenvalues of A. The matrix polynomial is . The corresponding scalar polynomial is . The eigenvalues of A are , , and .
Step 3: Calculate the new eigenvalues by applying the polynomial to each eigenvalue of A. - For : . - For : . - For : .
Step 4: Match the calculated eigenvalues with the given statements. The eigenvalues of are 2, 6, and 11. These correspond to statements A, B, and D. Statement C (8) is not an eigenvalue.

Step 1: Recall the cyclic property of powers of . , , , . The pattern repeats every 4 powers. To find , we can evaluate where is the remainder of divided by 4.
Step 2: Evaluate each expression. - A. : The remainder of 49 divided by 4 is 1. So, . This matches III. - B. : The remainder of 38 divided by 4 is 2. So, . This matches IV. - C. : The remainder of 103 divided by 4 is 3. So, . This matches II. - D. : 92 is perfectly divisible by 4 (remainder is 0). We can treat this as . So, . This matches I.
Step 3: Formulate the correct sequence. The matches are: A III, B IV, C II, D I. This corresponds to option (3).

Step 1: Express the complex number in polar form. In polar coordinates, , where the modulus is and the argument is .
Step 2: Apply the complex logarithm. The natural logarithm of is: Using the property , we get: Using the property , we get:
Step 3: Identify the real and imaginary parts. - The real part is . - The imaginary part is . This matches the expressions in option (4). Note that is also equal to .

Step 1: Use vector calculus identities. There are two fundamental vector identities that can solve this problem without direct calculation: 1. The curl of the gradient of any scalar field is always zero: . 2. The divergence of the gradient is the Laplacian: .
Step 2: Apply the identities to the given problem. - Curl of : We are given . Using the first identity, we immediately know that . Therefore, statement C is true and D is false. A vector field that is the gradient of a scalar potential is called a conservative or irrotational field. - Divergence of : Using the second identity, we need to calculate the Laplacian of . First derivatives are: , , . Second derivatives are: Therefore, . Statement A is true and B is false.
Step 3: Conclude the correct statements. Both statements A and C are true.

Step 1: Recall the formula for the scalar projection of vector onto vector . The projection is given by:
Step 2: Calculate the dot product .
Step 3: Calculate the magnitude of vector .
Step 4: Compute the projection.

Step 1: Apply the Divergence Theorem (Gauss's Theorem). The Divergence Theorem relates a surface integral over a closed surface S to a volume integral over the volume V enclosed by that surface:
Step 2: Substitute the given expression for . We are given that . Substituting this into the volume integral:
Step 3: Use the vector calculus identity for the divergence of a curl. A fundamental identity in vector calculus is that the divergence of the curl of any vector field is always identically zero:
Step 4: Evaluate the integral. Since the integrand is zero, the volume integral is zero. Therefore, the surface integral is also zero.

Step 1: Square both sides of the given equation. The magnitude of a vector is always non-negative, so we can square both sides without loss of information.
Step 2: Express the square of the magnitude using the dot product. Recall that for any vector , .
Step 3: Expand the dot products. Since and the dot product is commutative ( ):
Step 4: Simplify the equation to find the condition on the dot product. Canceling the and terms from both sides:
Step 5: Interpret the result. The dot product of two non-zero vectors is zero if and only if they are orthogonal (perpendicular). The angle between them is 90 degrees, or radians.

Step 1: Recall the definition of a conservative force. A force field is conservative if its curl is equal to the zero vector.
Step 2: Write out the components of the curl. The curl of is given by the determinant:
Step 3: Set the curl to zero and analyze the components. For the curl to be zero, each of its vector components must be zero. - component: . This is statement B. - component: . This is statement C. - component: . This is statement A. Therefore, for a force to be conservative, all three relations A, B, and C must be satisfied. Statement D describes a non-conservative force.