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NDA 2 2024 Maths, question 18

Topic: Matrices and determinants

Q18

If

Dn=∣n2030n24050n36070∣D_{n}=\left| \begin{matrix} n & 20 & 30 \\ n^{2} & 40 & 50 \\ n^{3} & 60 & 70 \end{matrix} \right|

then what is the value of ∑n=14Dn\sum _{n=1}^{4}D_{n}?

  1. (a) −10000-10000Correct answer
  2. (b) −10-10
  3. (c) 1010
  4. (d) 1000010000

Solution.

Answer: (a) −10000-10000

  1. Expansion along the first column gives Dₙ=−200n+400n²−200n³=−200n(n−1)².
  2. For n=1,2,3,4 the bracket values are 0,2,12,36.
  3. Their sum is 50, so ΣDₙ=−200×50=−10000.