CBSE Class 12 Mathematics Determinants Case Studies

Answer: (b) x − y = 50, 2x + y = 550

Explanation:
1. (x − 50)(y + 50) = xy gives 50x − 50y − 2500 = 0, i.e. x − y = 50.
2. (x − 10)(y − 20) = xy − 5300 gives −20x − 10y + 200 = −5300, i.e. 20x + 10y = 5500.
3. Dividing by 10: 2x + y = 550. So the equations are x − y = 50, 2x + y = 550.

Teacher's Note:
1. The xy terms cancel on both sides, which is why the conditions give linear equations.
2. Watch the signs: decreasing both sides gives +200 from (−10)(−20), not −200.

Answer: (a) \( \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)

Explanation:
1. The system is 1·x + (−1)·y = 50 and 2·x + 1·y = 550.
2. Coefficient matrix A = \( \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} \), X = \( \begin{bmatrix} x \\ y \end{bmatrix} \), B = \( \begin{bmatrix} 50 \\ 550 \end{bmatrix} \)
3. AX = B is option (a).

Teacher's Note:
1. Each row of A holds the coefficients of one equation, in the order x, y.
2. Before solving, check |A| = 1(1) − (−1)(2) = 3 ≠ 0, so the system has a unique solution.

Answer: (c) 200 m

Explanation:
1. |A| = 1(1) − (−1)(2) = 3, adj A = \( \begin{bmatrix} 1 & 1 \\ -2 & 1 \end{bmatrix} \)
2. X = \( A^{-1}B = \frac{1}{3}\begin{bmatrix} 1 & 1 \\ -2 & 1 \end{bmatrix}\begin{bmatrix} 50 \\ 550 \end{bmatrix} = \frac{1}{3}\begin{bmatrix} 600 \\ 450 \end{bmatrix} \)
3. x = 200 m

Teacher's Note:
1. For a 2 × 2 matrix, adj A is found by swapping the diagonal entries and changing the signs of the other two.
2. Quick check by adding the equations: 3x = 600 gives x = 200 at once.

Answer: (a) 150 m

Explanation:
1. From X = \( \frac{1}{3}\begin{bmatrix} 600 \\ 450 \end{bmatrix} \), y = \( \frac{450}{3} \) = 150 m.
2. Check in x − y = 50: 200 − 150 = 50 ✓; in 2x + y = 550: 400 + 150 = 550 ✓
3. y = 150 m

Teacher's Note:
1. Always substitute both values back into both equations; it takes seconds and catches sign slips.
2. Option (b) 200 m is the length, not the breadth, so read which variable is asked.

Answer: (b) 30000 sq. m

Explanation:
1. Area = length × breadth = x × y
2. Area = 200 × 150 = 30000
3. Area = 30000 sq. m

Teacher's Note:
1. Area must carry square units, so options in plain metres can be rejected straight away.
2. Check with the first condition: (200 − 50)(150 + 50) = 150 × 200 = 30000, the same area.