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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Determinants

NDA 2024 · Mathematics · Determinants

For what value of n is determinant | array ccc C(9,4) & C(9,3) &...

For what value of nn is determinant \left|C(9,4)C(9,3)C(10,n−2)C(11,6)C(11,5)C(12,n)C(m,7)C(m,6)C(m+1,n+1)\begin{array}{ccc} \mathrm{C}(9,4) & \mathrm{C}(9,3) & \mathrm{C}(10, n-2) \\ \mathrm{C}(11,6) & \mathrm{C}(11,5) & \mathrm{C}(12, n) \\ \mathrm{C}(m, 7) & \mathrm{C}(m, 6) & \mathrm{C}(m+1, n+1) \end{array}\right|=0 for every m>nm \gt n ?

Options

  1. A4
  2. B5
  3. C6
  4. D7

Correct answer

C. 6

Step-by-step solution

\left|C(9,4)C(9,3)C(10,n−2)C(11,6)C(11,5)C(12,n)C(m,7)C(m,6)C(m+1,7)−C(m+1,n+1)\begin{array}{ccc}\mathrm{C}(9,4) & \mathrm{C}(9,3) & \mathrm{C}(10, n-2) \\ \mathrm{C}(11,6) & \mathrm{C}(11,5) & \mathrm{C}(12, n) \\ \mathrm{C}(m, 7) & \mathrm{C}(m, 6) & \mathrm{C}(m+1,7)-\mathrm{C}(m+1, n+1)\end{array}\right|=0 Applying C3→C1+C2−C3∣C(9,4)C(9,3)C(10,4)−C(10,n−2)C(11,6)C(11,5)C(12,6)−C(12,n)C(m,7)C(m,6)C(m+1,7)−C(m+1,n+1)∣=0∵ Value of  determinant =0∴C3=0⇒n−2=4⇒n=6.\begin{aligned} & \text{ Applying } \mathrm{C}_3 \rightarrow \mathrm{C}_1+\mathrm{C}_2-\mathrm{C}_3 \\ & \left|\begin{array}{lcc}\mathrm{C}(9,4) & \mathrm{C}(9,3) & \mathrm{C}(10,4)-\mathrm{C}(10, n-2) \\ \mathrm{C}(11,6) & \mathrm{C}(11,5) & \mathrm{C}(12,6)-\mathrm{C}(12, n) \\ \mathrm{C}(m, 7) & \mathrm{C}(m, 6) & \mathrm{C}(m+1,7)-\mathrm{C}(m+1, n+1)\end{array}\right|=0 \\ & \because \text{ Value\text{ of } determinant }=0 \\ & \therefore \mathrm{C}_3=0 \Rightarrow n-2=4 \Rightarrow n=6 .\end{aligned}$
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