GATE Mathematics Foundation: Calculus – Previous Year Questions

29 GATE previous year questions on Calculus (Mathematics Foundation, Environmental Science and Engineering) with answers and explanations, from every paper.

1. GATE 2021 ES Q11 (1 mark, Multiple choice) – Find slope of the tangent to y=x² at(x, y)=(3,9).
2. GATE 2021 ES Q12 (1 mark, Multiple choice) – Evaluate x0x2/ x.
3. GATE 2021 ES Q30 (1 mark, Multiple select) – For dependent y and independent x, classify dy/dx=x2y.
4. GATE 2021 ES Q31 (1 mark, Multiple select) – Which listed values solve x3-10x2+31x-30=0?
5. GATE 2021 ES Q51 (2 marks, Numerical answer) – f(x)+3f(g(x))=x-2,where g(x)=(3x+1)/(x-3). Find f(5)/f(8) as an integer.
6. GATE 2021 ES Q52 (2 marks, Numerical answer) – Function y satisfies y-y=0,y→1 as x→−∞,and y(0)=2. Find y(0) as an integer.
7. GATE 2022 ES Q13 (1 mark, Multiple choice) – For s>0, find the Laplace transform of f(x)=(ax).
8. GATE 2022 ES Q34 (1 mark, Numerical answer) – Find x0(1+x-1)/x, to one decimal.
9. GATE 2022 ES Q36 (2 marks, Multiple choice) – For y+4y=0, y(0)=0 and y(0)=1, identify problem and solution.
10. GATE 2022 ES Q52 (2 marks, Numerical answer) – Find area enclosed by y=x, y=3x and x=0,x=1, to one decimal.
11. GATE 2023 ES Q11 (1 mark, Multiple choice) – Classify P:y+x=x y and Q:y+xy=ex\,y.
12. GATE 2023 ES Q13 (1 mark, Multiple choice) – Classify the infinite series P= n=1(n2+1)/n2 and Q= n=1(1+1/n)-n.
13. GATE 2023 ES Q33 (1 mark, Numerical answer) – Evaluate (4/) 0/22x\,dx,to two decimals.
14. GATE 2023 ES Q47 (2 marks, Numerical answer) – Solve y-y-2y=0 with y(0)=2,y′(0)=1 and find y(1),to three decimals.
15. GATE 2023 ES Q49 (2 marks, Numerical answer) – Find the maximum of f(x)=x4-x2 on[−2,2],to two decimals.
16. GATE 2024 ES Q32 (1 mark, Numerical answer) – f(x)=x3-4.5x2-12x has a local maximum at what integer x in[−2,2]?
17. GATE 2024 ES Q33 (1 mark, Numerical answer) – Solve dy/dx-x2+ex=0 with y(0)=1. Find y(1),taking e=2.7,to two decimals.
18. GATE 2024 ES Q37 (2 marks, Multiple choice) – Solve y-3y+2y=2e-x with y(0)=2 and y′(0)=1.
19. GATE 2024 ES Q42 (2 marks, Multiple select) – For f(x)=( x),which Taylor-expansion statements for f(x+h) are correct?
20. GATE 2024 ES Q49 (2 marks, Numerical answer) – Find area under y=x2+x above the x-axis between x=1 and x=2,to two decimals.
21. GATE 2025 ES Q11 (1 mark, Multiple choice) – For s> a ,find the Laplace transform of f(x)= ax.
22. GATE 2025 ES Q12 (1 mark, Multiple choice) – Find the general solution of d2y/dx2+4y=0.
23. GATE 2025 ES Q13 (1 mark, Multiple choice) – Classify P= n=11/n and Q= n=11/n2.
24. GATE 2025 ES Q34 (1 mark, Numerical answer) – Evaluate x0(1+x)/(2 x),to two decimal places.
25. GATE 2025 ES Q49 (2 marks, Numerical answer) – Evaluate 0(4x)/( x)\,dx,to two decimals.
26. GATE 2026 ES Q12 (1 mark, Multiple choice) – Evaluate x/1+ x\,dx,where c is an integration constant.
27. GATE 2026 ES Q13 (1 mark, Multiple choice) – The global minimum of f(x)=3x4+4x3-12x2+6 for real x is
28. GATE 2026 ES Q36 (2 marks, Multiple choice) – Classify the differential equation dy/dx+x2y=0.
29. GATE 2026 ES Q64 (2 marks, Numerical answer) – Find the first-quadrant area bounded by y=8-x and the coordinate axes, in integer square units.