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b. Find the image of a point \( A(3,4) \) after reflection in \( x \)-axis. 5.a. What is the formula of mean deviation from median in discrete series. Group- \( B[8 \times 2=16] \) b. Define transpose matrix of \( A \). 6.a. Define Rectangular matrix with example. b. If \( A=\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] \) and \( B=\left[\begin{array}{ll}3 & 2 \\ 6 & 4\end{array}\right] \) then find \( A B \). 7.a. Find the distance between the points \( A(-2,1) \) and \( B(4,3) \). b. For what value of \( K \) will be the point \( (2,1) \) lie on the curve. \( k x^{2}-2 y^{2}-2 x+3 y-3=0 \) ? 8.a. Find the mid - point of the line joining the points \( (-3,6) \) and \( (1,-2) \) b. Prove that: \( \left(1-\cos ^{2} \theta\right) \). \( \left(1+\cot ^{2} \theta\right)=1 \) 9.a. If \( A(1,3) \) and \( B(6,3) \) are the two end points of \( \overrightarrow{A B} \). Find \( \overrightarrow{A B} \) and hence find \( |\overrightarrow{A B}| \).

Ask by Huang Hill. in Nepal
Mar 16,2025

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**b. Image of Point \( A(3,4) \) after Reflection in the \( x \)-axis:** \[ (3, -4) \] **5.a. Mean Deviation from the Median (Discrete Series):** \[ \text{Mean Deviation} = \frac{\sum_{i=1}^{n} f_i |x_i - \tilde{x}|}{\sum_{i=1}^{n} f_i} \] **5.b. Transpose of Matrix \( A \):** Interchange the rows and columns of \( A \). **6.a. Rectangular Matrix:** A matrix with unequal numbers of rows and columns, e.g., \[ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} \] **6.b. Product \( AB \):** \[ AB = \begin{bmatrix} 21 & 14 \\ 30 & 20 \end{bmatrix} \] **7.a. Distance Between \( A(-2,1) \) and \( B(4,3) \):** \[ 2\sqrt{10} \] **7.b. Value of \( k \):** \[ k = \frac{3}{2} \] **8.a. Midpoint of \( (-3,6) \) and \( (1,-2) \):** \[ (-1, 2) \] **8.b. Proof:** \[ (1 - \cos^2 \theta)(1 + \cot^2 \theta) = 1 \] **9.a. Vector \( \overrightarrow{AB} \) and Its Magnitude:** \[ \overrightarrow{AB} = (5, 0), \quad |\overrightarrow{AB}| = 5 \]

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Bonus Knowledge

To find the image of point \( A(3,4) \) after reflection in the \( x \)-axis, you simply need to change the sign of the \( y \)-coordinate. So, the reflected point will be \( A'(3, -4) \). It's like taking a selfie—just flipping it upside down! The mean deviation from the median in a discrete series is calculated using the formula: \[ \text{MD} = \frac{\sum |x_i - M|}{N} \] where \( x_i \) represents each data point, \( M \) is the median, and \( N \) is the number of data points. Just think of it as measuring how far each number is from the central trend (the median) and averaging those distances!

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