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Find the area of a semi-circle whose radius is 7cm. [Take π = 22/7]

Mathematics

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here are the solutions to your questions:

  1. If P=(43)P = \begin{pmatrix} 4 \\ 3 \end{pmatrix} and Q=(2−1)Q = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, find 4P−2Q4P - 2Q.
  1. Calculate $4P$.

    $$
    4P = 4 \begin{pmatrix} 4 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \times 4 \\ 4 \times 3 \end{pmatrix} = \begin{pmatrix} 16 \\ 12 \end{pmatrix}
    $$
    
  2. Calculate $2Q$.

    $$
    2Q = 2 \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \times 2 \\ 2 \times -1 \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}
    $$
    
  3. Subtract $2Q$ from $4P$.

    $$
    4P - 2Q = \begin{pmatrix} 16 \\ 12 \end{pmatrix} - \begin{pmatrix} 4 \\ -2 \end{pmatrix}
    $$
    $$
    4P - 2Q = \begin{pmatrix} 16 - 4 \\ 12 - (-2) \end{pmatrix}
    $$
    $$
    4P - 2Q = \begin{pmatrix} 12 \\ 12 + 2 \end{pmatrix}
    $$
    $$
    4P - 2Q = \begin{pmatrix} 12 \\ 14 \end{pmatrix}
    $$
    The calculated result is $\begin{pmatrix} 12 \\ 14 \end{pmatrix}$. This answer is not among the given options.
    

    28. Given that vector a=(−512)a = \begin{pmatrix} -5 \\ 12 \end{pmatrix} and b=(10x12)b = \begin{pmatrix} 10x \\ 12 \end{pmatrix}, find the value of x if a=ba = b.

  4. If two vectors are equal, their corresponding components must be equal.

    $$
    \begin{pmatrix} -5 \\ 12 \end{pmatrix} = \begin{pmatrix} 10x \\ 12 \end{pmatrix}
    $$
    
  5. Equate the x-components.

    $$
    -5 = 10x
    $$
    
  6. Solve for x.

    $$
    x = \frac{-5}{10}
    $$
    $$
    x = -\frac{1}{2}
    $$
    The value of x is $\boxed{\text{B. } -\frac{1}{2}}$.
    
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