Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Absolutely, let’s delve into solving the problems step by step:
### Problem 74
The question requires us to find the length of the perpendicular from the point \((0,0)\) to the line \(3x + 4y - 10 = 0\).
We can use the formula for the distance \(d\) from a point \((x_0, y_0)\) to a line \(ax + by + c = 0\), which is given by:
[tex]\[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \][/tex]
1. Substitute \((x_0, y_0) = (0,0)\) into the formula:
[tex]\[ d = \frac{|3 \cdot 0 + 4 \cdot 0 - 10|}{\sqrt{3^2 + 4^2}} \][/tex]
2. Simplify the expression in the numerator:
[tex]\[ d = \frac{|0 + 0 - 10|}{\sqrt{9 + 16}} = \frac{|-10|}{\sqrt{25}} = \frac{10}{5} = 2 \][/tex]
So, the length of the perpendicular from \((0,0)\) to the line \(3x + 4y - 10 = 0\) is \(2\).
The correct answer is \(\boxed{2}\).
### Problem 75
We need to find the value of \(x\) for which the matrix
[tex]\[ A = \begin{pmatrix} 6 & x-2 \\ 3 & x \end{pmatrix} \][/tex]
has no inverse.
A matrix has no inverse if its determinant is zero. The determinant of matrix \(A\) can be found by:
[tex]\[ \text{det}(A) = 6 \cdot x - 3 \cdot (x - 2) \][/tex]
1. Calculate the determinant:
[tex]\[ \text{det}(A) = 6x - 3(x - 2) \][/tex]
Simplify the expression:
[tex]\[ \text{det}(A) = 6x - 3x + 6 = 3x + 6 \][/tex]
2. Set the determinant equal to zero:
[tex]\[ 3x + 6 = 0 \][/tex]
3. Solve for \(x\):
[tex]\[ 3x = -6 \implies x = -2 \][/tex]
So, the value of \(x\) for which the matrix has no inverse is \( -2 \).
The correct answer is \(\boxed{-2}\).
Each problem has been carefully broken down and solved with clear steps to arrive at the correct answers.
### Problem 74
The question requires us to find the length of the perpendicular from the point \((0,0)\) to the line \(3x + 4y - 10 = 0\).
We can use the formula for the distance \(d\) from a point \((x_0, y_0)\) to a line \(ax + by + c = 0\), which is given by:
[tex]\[ d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \][/tex]
1. Substitute \((x_0, y_0) = (0,0)\) into the formula:
[tex]\[ d = \frac{|3 \cdot 0 + 4 \cdot 0 - 10|}{\sqrt{3^2 + 4^2}} \][/tex]
2. Simplify the expression in the numerator:
[tex]\[ d = \frac{|0 + 0 - 10|}{\sqrt{9 + 16}} = \frac{|-10|}{\sqrt{25}} = \frac{10}{5} = 2 \][/tex]
So, the length of the perpendicular from \((0,0)\) to the line \(3x + 4y - 10 = 0\) is \(2\).
The correct answer is \(\boxed{2}\).
### Problem 75
We need to find the value of \(x\) for which the matrix
[tex]\[ A = \begin{pmatrix} 6 & x-2 \\ 3 & x \end{pmatrix} \][/tex]
has no inverse.
A matrix has no inverse if its determinant is zero. The determinant of matrix \(A\) can be found by:
[tex]\[ \text{det}(A) = 6 \cdot x - 3 \cdot (x - 2) \][/tex]
1. Calculate the determinant:
[tex]\[ \text{det}(A) = 6x - 3(x - 2) \][/tex]
Simplify the expression:
[tex]\[ \text{det}(A) = 6x - 3x + 6 = 3x + 6 \][/tex]
2. Set the determinant equal to zero:
[tex]\[ 3x + 6 = 0 \][/tex]
3. Solve for \(x\):
[tex]\[ 3x = -6 \implies x = -2 \][/tex]
So, the value of \(x\) for which the matrix has no inverse is \( -2 \).
The correct answer is \(\boxed{-2}\).
Each problem has been carefully broken down and solved with clear steps to arrive at the correct answers.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.