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. Consider the following statements in respect of the determinant 

I. Δ is positive if k>0.

II. Δ is negative if k<0.

III. Δ is zero if k=0.

How many of the statements given above are correct?

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NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)
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  1. None 
  2. One 
  3. Two 
  4. All three

Answer (Detailed Solution Below)

Option 2 : One 
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Detailed Solution

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Calculation:

Given,

Δ  = 

Simplify the determinant by the column operation  :

Expanding along the third row,

Thus  .

Sign analysis

  • : if , Δ < 0; if , Δ > 0  ⇒ Statement I is false.
  • :    ⇒ Statement II is true.
  • :    ⇒ Statement III is false.

∴ Only Statement II is correct  ⇒  exactly one statement is true.

Hence, the correct answer is Option 2.

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