Algebra Answers

You are given the system of equations

"\\begin{array}{ccc}2x & +4y & =10\\\\ 3x & +2y & =7 \\end{array} ."

Which of the following is true about the system above?

1. The determinant of the coefficient matrix is -8
2. With Cramer's rule, the value of x is "x=-\\frac{\\left| \\begin{array}{cc}10 & 4\\\\ 7 & 2 \\end{array} \\right |}{8}.\u200b"
3. With Cramer's rule, the value of y is "y=-\\frac{\\left| \\begin{array}{cc}2 & 10\\\\ 3 & 7 \\end{array} \\right |}{8}."
4. The determinant of the coefficient matrix is 8
5. With Cramer's rule, the value of y is "y=\\frac{\\left| \\begin{array}{cc}2 & 10\\\\ 3 & 7 \\end{array} \\right |}{8}."

You are given the circle "x^{2}+y^{2}+ax+by+c=0", where a, b and c are some constants. Given that this circle passes through the points (4,2); (0,3) and (3,-2).

Which of the following are true?

1. The system of equations connecting a, b and c is "\\begin{array}{cccc} 4a& +2b& +c&= -20\\\\ & +3b & +c& =-9 \\\\ 3a& -2b&+c&=-13 \\end{array}"
2. The determinant of the coefficient matrix for the system connecting a, b and c is 17
3. With Cramer's rule, the value of a is "a=\\frac{\\left| \\begin{array}{cccc}-20 & 2& 1\\\\ -9 & 3& 1\\\\ -13 & -2& 1 \\end{array} \\right |}{17}."
4. With Cramer's rule, the value of b is "b=\\frac{\\left| \\begin{array}{cccc}-20 & 2& 1\\\\ -9 & 3& 1\\\\ -13 & -2& 1 \\end{array} \\right |}{17}."

You are given a matrix 

"A=\\left(\\begin{array}{cccc}1 & -2 & -1\\\\ 2 & -3 & 0 \\\\1& -1 & -3\\end{array} \\right)."

Which of the following are true about A?

1. "det (A)=-4"
2. "adj (A)=\\left(\\begin{array}{cccc}9 & -5 & -3\\\\ 6 & -2 & -2 \\\\1& -1 & 1\\end{array} \\right)"
3. "A^{-1}=\\left(\\begin{array}{cccc}-\\frac{9}{4} & \\frac{5}{4} & \\frac{3}{4}\\\\ -\\frac{3}{2} & \\frac{1}{2} & \\frac{1}{2} \\\\-\\frac{1}{4}& \\frac{1}{4} & -\\frac{1}{4}\\end{array} \\right)"
4. The system AX=0 has no solution, where 0 denotes a zero matrix of order 3×1.
5. "adj (A)=\\left(\\begin{array}{cccc}9 & -5 & -3\\\\ 6 & -2 & -2 \\\\1& -1 & 1\\end{array} \\right)^{T}"

Determine the numerical value of the following expression without the use of a calculator: log10 (1000100) 100 + X 100 n=1 sin(πn) + 1 (−1)n ! · vuut 1000 Y m=1 1 cos(πm) 2 

Consider the system of equations 

"\\begin{array}{ccc} +x&&+z&=2\\\\ +x&+y& &=0\\\\ & +y&+z&=2\\end{array}."

​.

Which of the following statement(s) is/are true?

1. "y= \n\\frac{\\textup{det} \\begin{bmatrix}2&0&1\\\\0&1&0\\\\2&1&1\\end{bmatrix}}{\\textup{det} \\begin{bmatrix}1&0&1\\\\1&1&0\\\\1&0&1\\end{bmatrix}}"
2. "x=\\frac{\\textup{det} \\begin{bmatrix}2&0&1\\\\0&1&0\\\\2&1&1\\end{bmatrix}}{\\textup{det} \\begin{bmatrix}1&0&1\\\\1&1&0\\\\1&0&1\\end{bmatrix}}"
3. "z=\\frac{\\begin{array}{cccc}&-\\textup{det} \\begin{bmatrix}0&2\\\\1&2\\end{bmatrix} &+ \\textup{det} \\begin{bmatrix}1&2\\\\0&2\\end{bmatrix}\\end{array}}{2}"
4. "z=\\frac{\\begin{array}{cccccc}&\\textup{det} \\begin{bmatrix}0&2\\\\1&2\\end{bmatrix} &+ \\textup{det} \\begin{bmatrix}1&2\\\\0&2\\end{bmatrix}\\end{array}}{2}"
5. "y=\\frac{\\textup{det} \\begin{bmatrix}1&2&1\\\\1&0&0\\\\0&2&1\\end{bmatrix}}{2}"

Given that "A^{-1}=\\frac{1}{\\textup{det}(A)}\\begin{bmatrix}d&-b\\\\-c&a\\end{bmatrix}"  with "\\textup{det}(A)\\neq 0", consider the following statements:

(i) "\\frac{1}{\\textup{det}(A)}\\begin{bmatrix}d&-b\\\\-c&a\\end{bmatrix}=\\frac{1}{\\textup{det}(A)}\\textup{adj}(A)" ;

(ii) "A=\\begin{bmatrix}a&c\\\\b&d\\end{bmatrix}^T" ;

(iii) "\\textup{adj}(A)=\\begin{bmatrix}d&-c\\\\-b&a\\end{bmatrix}^T"

Which of the above are true?

Let A and B be n×n matrices such that "\\textup{det}(A)\\neq 0" and "\\textup{det}(B)\\neq 0". Consider the following statements. Based on the properties of determinants which of the following is false? 

1. "\\textup{det}(AB)=\\textup{det}(BA)"
2. If "A=\u2212B then \\textup{det}(A)=(-1)^{n}\\textup{det}(B)"
3. "\\textup{det}(A^{T}B^{T})=\\textup{det}(AB)"
4. "\\textup{det}((AB)^{-1})=\\frac{1}{\\textup{det}(A)\\textup{det}(B)}"
5. "\\textup{det}(AB)=\\textup{det}(A)+\\textup{det}(B)"

Polynomial Inequalities:

10) 6 + 5x - 6x^ 2 > 0

show number line and answer in interval notation

Polynomial Inequalities:

9) 2x^2 > 5x + 3

show number line and answer in interval notation

Polynomial Inequalities:

8) x^2 + 7x < -12

show number line test and answer in interval notation