CAT 2025 Slot 1 · Quant · Question 7
CAT 2025 Slot 1 previous year question on Quadratic Equations, with the answer and a step-by-step solution.
The number of non-negative integer values of k for which the quadratic equation {x}^{2}-5x+k=0 has only integer roots, is
Answer & solution
Answer: 3
If the roots are p and q, then p+q=5 and pq=k.
We need both roots to be integers and k\ge 0, so pq\ge 0. Since the two integers add to a positive number, they cannot both be negative, so both must be non-negative.
The non-negative integer pairs adding to 5 are \left( 0,5 \right), \left( 1,4 \right) and \left( 2,3 \right), giving k=0, k=4 and k=6.
(Pairs such as \left( 6,-1 \right) do add to 5, but give k=-6, which is not allowed.)
So there are 3 such values of k.
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