Answer :
Sure, let's go through each of the statements one by one, providing a detailed explanation for each.
### Part I: True or False Statements
1. The dot product of two vectors is a vector.
- Explanation: The dot product of two vectors results in a scalar quantity, not a vector.
- Answer: False
2. Any vectors whose magnitude is zero is called unit vector.
- Explanation: A unit vector is defined as a vector with a magnitude (length) of 1. A vector with zero magnitude is simply the zero vector.
- Answer: False
3. If \( u = 2i + j \) and \( v = -2i - j \), then vector \( u \) and \( v \) are parallel vectors.
- Explanation: Two vectors are parallel if one is a scalar multiple of the other. In this case, \( v = -1 \cdot u \), making \( u \) and \( v \) parallel.
- Answer: True
4. The radius of the circle \( (x+2)^2 + y^2 - 9 = 0 \) is 9.
- Explanation: The given circle equation can be written as \( (x + 2)^2 + y^2 = 9 \). This conforms to the standard circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius. Here, \( r^2 = 9 \) implies \( r = 3 \).
- Answer: False
5. If \( A \) is a square matrix that has two identical rows, then \(\operatorname{det}(A) = 0 \).
- Explanation: A square matrix with two identical rows has a determinant of zero by properties of determinants.
- Answer: True
6. Every square matrix is invertible.
- Explanation: A square matrix is invertible if and only if its determinant is non-zero. Not every square matrix meets this criterion.
- Answer: False
7. If the determinant of a square matrix is zero, then the matrix is singular.
- Explanation: By definition, a matrix is considered singular if its determinant is zero.
- Answer: True
8. If \( u = (3, 5) \) and \( v = (-2, 2) \), then \( u \cdot v = 6 \).
- Explanation: The dot product of \( u \) and \( v \) is calculated as \( 3 \cdot (-2) + 5 \cdot 2 = -6 + 10 = 4 \).
- Answer: False
9. The image of the point \( (2, -3) \) after being reflected about the line \( L: x = 0 \) is \( (-2, -3) \).
- Explanation: Reflecting a point \( (x, y) \) across the line \( x = 0 \) results in \( (-x, y) \). So, \( (2, -3) \) reflects to \( (-2, -3) \).
- Answer: True
10. Rigid motion is a motion that preserves distance.
- Explanation: By definition, rigid motion (or isometry) preserves distances between points.
- Answer: True
### Part II: Choose the Correct Answer from Given Alternatives
11. Which of the following is a scalar quantity?
- Explanation: Scalar quantities are those that are described by a magnitude (or numerical value) alone. Examples include mass, temperature, and speed, which do not depend on direction.
- Specific example: Mass
Thus, the complete and accurate answer summary is:
### Part I Summary
1. False
2. False
3. True
4. False
5. True
6. False
7. True
8. False
9. True
10. True
### Part II Summary
11. Mass
### Part I: True or False Statements
1. The dot product of two vectors is a vector.
- Explanation: The dot product of two vectors results in a scalar quantity, not a vector.
- Answer: False
2. Any vectors whose magnitude is zero is called unit vector.
- Explanation: A unit vector is defined as a vector with a magnitude (length) of 1. A vector with zero magnitude is simply the zero vector.
- Answer: False
3. If \( u = 2i + j \) and \( v = -2i - j \), then vector \( u \) and \( v \) are parallel vectors.
- Explanation: Two vectors are parallel if one is a scalar multiple of the other. In this case, \( v = -1 \cdot u \), making \( u \) and \( v \) parallel.
- Answer: True
4. The radius of the circle \( (x+2)^2 + y^2 - 9 = 0 \) is 9.
- Explanation: The given circle equation can be written as \( (x + 2)^2 + y^2 = 9 \). This conforms to the standard circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius. Here, \( r^2 = 9 \) implies \( r = 3 \).
- Answer: False
5. If \( A \) is a square matrix that has two identical rows, then \(\operatorname{det}(A) = 0 \).
- Explanation: A square matrix with two identical rows has a determinant of zero by properties of determinants.
- Answer: True
6. Every square matrix is invertible.
- Explanation: A square matrix is invertible if and only if its determinant is non-zero. Not every square matrix meets this criterion.
- Answer: False
7. If the determinant of a square matrix is zero, then the matrix is singular.
- Explanation: By definition, a matrix is considered singular if its determinant is zero.
- Answer: True
8. If \( u = (3, 5) \) and \( v = (-2, 2) \), then \( u \cdot v = 6 \).
- Explanation: The dot product of \( u \) and \( v \) is calculated as \( 3 \cdot (-2) + 5 \cdot 2 = -6 + 10 = 4 \).
- Answer: False
9. The image of the point \( (2, -3) \) after being reflected about the line \( L: x = 0 \) is \( (-2, -3) \).
- Explanation: Reflecting a point \( (x, y) \) across the line \( x = 0 \) results in \( (-x, y) \). So, \( (2, -3) \) reflects to \( (-2, -3) \).
- Answer: True
10. Rigid motion is a motion that preserves distance.
- Explanation: By definition, rigid motion (or isometry) preserves distances between points.
- Answer: True
### Part II: Choose the Correct Answer from Given Alternatives
11. Which of the following is a scalar quantity?
- Explanation: Scalar quantities are those that are described by a magnitude (or numerical value) alone. Examples include mass, temperature, and speed, which do not depend on direction.
- Specific example: Mass
Thus, the complete and accurate answer summary is:
### Part I Summary
1. False
2. False
3. True
4. False
5. True
6. False
7. True
8. False
9. True
10. True
### Part II Summary
11. Mass
