Matrices & Determinants
Determinants
Grade Class 12
Question:
<div>22. D = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><msup><mn>10</mn><mn>4</mn></msup><mo>+</mo><mn>2</mn></mtd><mtd><msup><mn>10</mn><mn>7</mn></msup><mo>+</mo><mn>3</mn></mtd><mtd><msup><mn>10</mn><mn>8</mn></msup><mo>+</mo><mn>8</mn></mtd></mtr><mtr><mtd><msup><mn>10</mn><mn>9</mn></msup><mo>+</mo><mn>9</mn></mtd><mtd><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><mn>8</mn></mtd><mtd><msup><mn>10</mn><mn>3</mn></msup><mo>-</mo><mn>4</mn></mtd></mtr><mtr><mtd><msup><mn>10</mn><mn>3</mn></msup><mo>-</mo><mn>5</mn></mtd><mtd><msup><mn>10</mn><mn>8</mn></msup><mo>+</mo><mi>b</mi></mtd><mtd><msup><mn>10</mn><mn>6</mn></msup><mo>+</mo><mi>a</mi></mtd></mtr></mtable></mfenced></math> where a, b, both ∈ {1,2,3,4,5,6,7,8,9}<br>Number of ordered pairs (a, b) such that D = 2n + 1, n ∈ Z is</div>
(A) 36
(B) 20
(C) 16
(D) 45
Step-by-Step Solution
Key Concept: The determinant D is evaluated modulo 2. Since 10^k is even for k >= 1, the parity of the elements depends on the constants added. The condition D = 2n + 1 means D is odd.
<div>The determinant D is evaluated modulo 2. Since 10^k is even for k ≥ 1, the parity of the elements depends on the constants added. The condition D = 2n + 1 means D is odd. Evaluating the determinant modulo 2, we find the condition for D to be odd based on the values of a and b. Counting the pairs (a, b) satisfying this condition gives 45.</div>
Correct Answer: D