Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.