GATE 2025 MA – Question 29

Calculus · Multivariable Differentiation and Optimization · 1 mark · Numerical answer

At P=(1,1,√2), a unit vector a is tangent to the intersection 2x²+3y²−z²=3 and x²+y²=z². For f=x²+2y²−2√11 z, find the absolute directional derivative at P along a (integer).

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Correct answer: 2 to 2

Explanation

Surface normals are (4,6,−2√2) and (2,2,−2√2). Their cross product is (−8√2,4√2,−4), so a may be $(-2\sqrt2,\sqrt2,-1)/\sqrt{11}$. Gradient of f is (2,4,−2√11). Dot product is [−4√2+4√2+2√11]/√11=2. Reversing tangent changes only sign; absolute value **2**.

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