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Calculate the determinant of the given 3 x 3 matrix.

Mathematics

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Answer

A

Step-by-step solution

8 steps

24.

  1. Calculate the determinant of the given 3×33 \times 3 matrix.

    The determinant of a matrix ∣abcdefghi∣\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} can be found by expanding along the first column: a(ei−fh)−d(bi−ch)+g(bf−ce)a(ei - fh) - d(bi - ch) + g(bf - ce).
    For the given matrix ∣032178054∣\begin{vmatrix} 0 & 3 & 2 \\ 1 & 7 & 8 \\ 0 & 5 & 4 \end{vmatrix}, we expand along the first column:

    det⁡=0⋅∣7854∣−1⋅∣3254∣+0⋅∣3278∣\det = 0 \cdot \begin{vmatrix} 7 & 8 \\ 5 & 4 \end{vmatrix} - 1 \cdot \begin{vmatrix} 3 & 2 \\ 5 & 4 \end{vmatrix} + 0 \cdot \begin{vmatrix} 3 & 2 \\ 7 & 8 \end{vmatrix}
  2. Simplify the expression.

    det⁡=0−1⋅((3×4)−(2×5))+0\det = 0 - 1 \cdot ((3 \times 4) - (2 \times 5)) + 0 det⁡=−1⋅(12−10)\det = -1 \cdot (12 - 10) det⁡=−1⋅(2)\det = -1 \cdot (2) det⁡=−2\det = -2

    The correct option is A.

    The final answer is A. -2\boxed{\text{A. -2}}.

    25.

  3. Find the exterior angle of the regular polygon.

    The interior angle of the regular polygon is given as 135∘135^\circ.
    The sum of an interior angle and its corresponding exterior angle is 180∘180^\circ.

    Exterior Angle=180∘−Interior Angle\text{Exterior Angle} = 180^\circ - \text{Interior Angle} Exterior Angle=180∘−135∘\text{Exterior Angle} = 180^\circ - 135^\circ Exterior Angle=45∘\text{Exterior Angle} = 45^\circ
  4. Calculate the number of sides of the polygon.

    The sum of the exterior angles of any polygon is 360∘360^\circ. For a regular polygon with nn sides, all exterior angles are equal.

    n×Exterior Angle=360∘n \times \text{Exterior Angle} = 360^\circ n×45∘=360∘n \times 45^\circ = 360^\circ n=360∘45∘n = \frac{360^\circ}{45^\circ} n=8n = 8

    The correct option is A.

    The final answer is A. 8\boxed{\text{A. 8}}.

    26.

  5. Use the property of parallel lines to find ∠LNM\angle LNM.

    Given that KL∥NMKL \parallel NM and LNLN is a transversal, the alternate interior angles are equal.
    Therefore, ∠KLN=∠LNM\angle KLN = \angle LNM.
    Given ∠KLN=54∘\angle KLN = 54^\circ, so ∠LNM=54∘\angle LNM = 54^\circ.

  6. Use the angle bisector property to find ∠KNL\angle KNL.

    Given that LNLN bisects ∠KNM\angle KNM, it means LNLN divides ∠KNM\angle KNM into two equal angles.
    So, ∠KNL=∠LNM\angle KNL = \angle LNM.
    Since ∠LNM=54∘\angle LNM = 54^\circ, then ∠KNL=54∘\angle KNL = 54^\circ.

  7. Calculate the total angle ∠KNM\angle KNM.

    ∠KNM=∠KNL+∠LNM\angle KNM = \angle KNL + \angle LNM ∠KNM=54∘+54∘\angle KNM = 54^\circ + 54^\circ ∠KNM=108∘\angle KNM = 108^\circ
  8. Calculate ∠KMN\angle KMN using the sum of angles in △KMN\triangle KMN.

    The sum of angles in a triangle is 180∘180^\circ. In △KMN\triangle KMN:

    ∠KMN+∠MKN+∠KNM=180∘\angle KMN + \angle MKN + \angle KNM = 180^\circ

    Given ∠MKN=35∘\angle MKN = 35^\circ and we found ∠KNM=108∘\angle KNM = 108^\circ.

    ∠KMN+35∘+108∘=180∘\angle KMN + 35^\circ + 108^\circ = 180^\circ ∠KMN+143∘=180∘\angle KMN + 143^\circ = 180^\circ ∠KMN=180∘−143∘\angle KMN = 180^\circ - 143^\circ ∠KMN=37∘\angle KMN = 37^\circ

    The final answer is 37∘\boxed{37^\circ}.

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