CDS 2026 (I) Elementary Mathematics, question 10

Topic: Algebra

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Q10

For what integral value of kk, the system of equations kx−5y+6=0kx-5y+6=0 and 4(k−1)y−12x+3=04(k-1)y-12x+3=0 has no solution ?

  1. (a) 44
  2. (b) 33
  3. (c) 11
  4. (d) No such value existsCorrect answer

Solution.

Answer: (d) No such value exists

  1. The coefficient determinant is 4k(k−1)−604k(k - 1) - 60.
  2. A system with no solution must have determinant zero, requiring k(k−1)=15k(k - 1) = 15.
  3. But the product of two consecutive integers is even, whereas 15 is odd.
  4. Thus every integral k gives a nonzero determinant and a unique solution.