GATE Calculus: Multivariable Calculus – Previous Year Questions

10 GATE previous year questions on Multivariable Calculus (Calculus, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q55 (2 marks, Numerical answer) – Let f(x,y)=8x²−2y. If M,m are its maximum and minimum on x²+y²=1,find M−m,to two decimals.
2. GATE 2022 ST Q14 (1 mark, Multiple choice) – Define f(x,y)=x²y/(x²+y²) off (0,0),and f(0,0)=0. Which holds on ℝ²?
3. GATE 2022 ST Q63 (2 marks, Numerical answer) – A point P=(x,y,z) lies on z=xy+5 and x+y+z=1 and is nearest to the origin. Find its distance from the origin as an integer.
4. GATE 2023 ST Q29 (1 mark, Numerical answer) – Find the maximum of f(x,y)=xy on x²+2y²=1. Round to two decimals.
5. GATE 2024 ST Q36 (2 marks, Multiple choice) – For f(x,y)=108xy−2x²y−2xy², which is not true?
6. GATE 2024 ST Q37 (2 marks, Multiple choice) – Define f(x,y)=(x³−y³)/(x²+y²) off the origin and f(0,0)=0. Which is not true?
7. GATE 2025 ST Q37 (2 marks, Multiple choice) – On U=(x,y):x+y≤2, find the minimum of f(x,y)=(x−1)⁴+(y−2)⁴.
8. GATE 2025 ST Q55 (2 marks, Numerical answer) – Find the number of saddle points of f(x,y)=x²y²+8x−4y (integer).
9. GATE 2026 ST Q36 (2 marks, Multiple choice) – For f(x₁,x₂)=2x₁⁴+x₂²+x₂x₁², which holds?
10. GATE 2026 ST Q49 (2 marks, Multiple select) – Define f(x,y)=(x³+y³)/√(x²+2y²) away from the origin and zero at the origin. Which hold?