Determinants

Taking common and from and respectively, and applying and

Now expanding along , we get

We have,
Since , exists
Multiplying by on both sides of , we get

Given: is a homogeneous equation representing pair of straight lines.
The angle bisectors of the pair of lines represented by are given by:

Here: , ,
So:

The given constraints are: 1. 2. 3. 4. We need to find the maximum value of . First, plot the constraints and identify the feasible region. The lines corresponding to the inequalities are: Now, find the intersection points of these lines: 1. Intersection of and : Subtract the first equation from the second: Substitute in : So the intersection point is . Now calculate at this point: Thus, the maximum value of is 36.

We are given the following inequalities: Step 1: Rewriting the inequalities - The first inequality can be rearranged as: This represents a line with slope and y-intercept . - The second inequality can be rearranged as: This represents a line with slope and y-intercept . Step 2: Graphing the inequalities
- The line intersects the y-axis at and the x-axis at .
- The line intersects the y-axis at and the x-axis at .
- The feasible region is bounded by these lines and the axes. Step 3: Analyzing the options - Option (B) shows the correct feasible region where both inequalities are satisfied along with the conditions . Thus, the correct answer is option B.

Step 1: Use the property of adjugates

For any square matrix , it is known that: where is the adjugate of and is the identity matrix. In this case, we are given: This is equivalent to: where is the identity matrix.

Step 2: Compare both equations

Using the property of adjugates, we can equate the two expressions: This implies:

Final Answer:

The value of is .

We are given the matrix , and we need to find the value of when this matrix is singular.

Step 1: Condition for a matrix to be singular

A matrix is singular if and only if its determinant is zero. Therefore, we first need to find the determinant of matrix . The determinant of a 2x2 matrix is given by: For the matrix , we have: - - - - The determinant of is:

Step 2: Simplify the determinant expression

First, expand the terms: Expanding : So, the determinant becomes:

Step 3: Set the determinant equal to zero

For the matrix to be singular, the determinant must be zero:

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: Simplifying: This gives two possible solutions for :

Step 5: Calculate

We now calculate for both values of : - For : - For :

Final Answer:

In both cases, . Therefore, the value of is .

We are given the matrix , and we are asked to find the value of , where denotes the determinant of and denotes the adjugate (or adjoint) of .

Step 1: Find the determinant of

Since is a diagonal matrix, the determinant is simply the product of the diagonal elements:

Step 2: Find the adjugate of

The adjugate (or adjoint) of a matrix is the transpose of its cofactor matrix. For a diagonal matrix, the cofactor of each diagonal element is the product of the other diagonal elements. Hence, the cofactor matrix of is: Taking the transpose of this matrix gives the adjugate of :

Step 3: Multiply with

Now, we multiply the determinant by the adjugate : This gives:

Final Answer:

The matrix is equal to .

We are given the following linear programming problem: Maximize subject to the constraints: Step 1: Find the intersection points First, we convert the inequalities into equalities to find the boundary lines. 1. 2. We solve the system of equations by substitution or elimination. - From equation , solve for : - Substitute into the first equation: Substitute into : So, the intersection point is . Step 2: Calculate the value of Substitute and into : Thus, the maximum value of is , and the feasible region formed by the constraints is bounded by this point. Thus, the correct answer is .

To find the Y-intercept, we set in the equation of the circle. Substituting into , we get: This is a quadratic equation in . Solving using the quadratic formula: Thus, the two values of are: The length of the intercept is the difference between these two values: Therefore, the correct answer is 1.

We are given two equations of circles: 1. , which represents a circle with center . 2. , which we simplify to find the equation of the second circle. After simplifying the second circle's equation, we find the radius , and based on the relationship, the final result gives .

For a matrix to be singular, its determinant must be zero. Thus, we need to compute the determinant of matrix and set it equal to zero. The determinant of matrix is given by: We expand the determinant along the first row: Now, simplify each 2x2 determinant: For to be singular, we set : This gives two possible solutions: 1. 2. , which simplifies to , so Thus, the possible values for are and . Since doesn't satisfy the problem conditions, we conclude that the correct value of is .