If tangent to the curve f ( x ) = x 3 − α x 2 − x + β at point ( 1 , 3 ) on the curve, cut equals non zero intercepts on co-ordinate axes, then

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∵ ( 1 , 3 ) lies on the curve. Then, f ( 1 ) = 3 ⇒ 1 − α − 1 + β = 3 ⇒ β − α = 3 ∵ Tangent cuts equal intercepts that means, slope = − 1 ⇒ ⇒ ⇒ ⇒ ⇒ ​ f ′ ( 1 ) = − 1 3 ( 1 ) 2 − 2 ( 1 ) α − 1 = − 1 − 2 α = − 3 2 α = 3 α = 2 3 ​ ​ Then, β = 3 + 2 3 ​ = 2 9 ​ Therefore, α + β = 2 3 ​ + 2 9 ​ = 6

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