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JEE Main Calculus: practice questions with solutions

15 questions on Application of Derivatives, Area Under Curves, Continuity & Differentiability, Definite Integration, Differential Equations, Differentiation, Indefinite Integration, Limits. Try each one first, then open the solution.

Tip: solve on paper before opening a solution. Once you open it, that question is counted as practised and won't appear in your HeyGyan tests.
LimitsEasy

Q1. What is lim(x→0) (1 − cos 2x) / (x tan x)?

  1. 0
  2. 1
  3. 2
  4. 4
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Answer: (C) 2

Solution1 − cos 2x = 2sin²x. So the expression is 2 · (sin x/x) · (sin x/tan x), and both ratios tend to 1. The limit is 2.
Common trapUsing 1 − cos 2x ≈ x² instead of 2x².
Continuity & DifferentiabilityHardNumerical

Q2. How many points are there where f(x) = |x² − 3x + 2| + |x − 3| is not differentiable?

Numerical value question: type the answer, no options.

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Answer: 3

SolutionEach modulus creates a corner where its inside crosses zero with a non-zero slope. x² − 3x + 2 = 0 at x = 1 and 2; x − 3 = 0 at x = 3. None of these corners cancel, so there are 3 points.
Common trapCounting only the roots of the quadratic.
Definite IntegrationMedium

Q3. What is ∫₀^(π/2) sin x / (sin x + cos x) dx?

  1. π/2
  2. π/4
  3. 1
  4. 0
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Answer: (B) π/4

SolutionReplacing x with π/2 − x turns the integrand into cos x/(cos x + sin x). Adding the two forms gives 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.
Common trapTrying direct substitution, which gets messy.
Area Under CurvesEasy

Q4. What is the area of the region bounded by y = x² and y = x?

  1. 1/2
  2. 1/3
  3. 1/6
  4. 1/12
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Answer: (C) 1/6

SolutionThe curves meet at x = 0 and 1, and y = x is above. Area = ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.
Common trapIntegrating only one curve.
Differential EquationsMedium

Q5. If dy/dx + y/x = x and y(1) = 1, what is y(2)?

  1. 4/3
  2. 5/3
  3. 2
  4. 7/3
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Answer: (B) 5/3

SolutionIntegrating factor = x, so d(xy)/dx = x², giving xy = x³/3 + C. From y(1) = 1, C = 2/3. At x = 2: 2y = 8/3 + 2/3 = 10/3, so y = 5/3.
Common trapForgetting to divide by x at the end.
Application of DerivativesMedium

Q6. What is the largest area of a rectangle inscribed in a semicircle of radius r, with one side on the diameter?

  1. r²/2
  2. r²
  3. 2r²
  4. πr²/4
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Answer: (B) r²

SolutionWith base 2x and height √(r² − x²), area A = 2x√(r² − x²). It is maximum at x = r/√2, giving A = 2 × (r/√2) × (r/√2) = r².
Common trapUsing the full circle's result (2r²) for a semicircle.
Indefinite IntegrationHard

Q7. What is ∫ dx / (x(xⁿ + 1))?

  1. (1/n) ln|xⁿ/(xⁿ + 1)| + C
  2. ln|xⁿ/(xⁿ + 1)| + C
  3. (1/n) ln|(xⁿ + 1)/xⁿ| + C
  4. n ln|xⁿ/(xⁿ + 1)| + C
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Answer: (A) (1/n) ln|xⁿ/(xⁿ + 1)| + C

SolutionMultiply top and bottom by xⁿ⁻¹ and set t = xⁿ (dt = n xⁿ⁻¹ dx). The integral becomes (1/n)∫ dt/(t(t + 1)) = (1/n) ln|t/(t + 1)| + C.
Common trapLosing the 1/n from the substitution.
LimitsMedium

Q8. What is lim(x→∞) (1 + 2/x)³ˣ?

  1. e²
  2. e³
  3. e⁶
  4. e^(2/3)
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Answer: (C) e⁶

SolutionFor a 1^∞ form, the limit is e^(lim of (base − 1) × exponent) = e^((2/x)(3x)) = e⁶.
Common trapAnswering 1 because the base tends to 1. The exponent grows at the same time.
LimitsMedium

Q9. What is lim(x→0) (eˣ − 1 − x)/x²?

  1. 0
  2. 1/2
  3. 1
  4. 2
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Answer: (B) 1/2

Solutioneˣ = 1 + x + x²/2 + … So eˣ − 1 − x ≈ x²/2, and the limit is 1/2.
Common trapStopping the expansion at the x term gives 0/0 again.
DifferentiationMedium

Q10. For |x| < 1, if y = sin⁻¹(2x/(1 + x²)), what is dy/dx?

  1. 2/(1 + x²)
  2. 1/(1 + x²)
  3. −2/(1 + x²)
  4. 2/√(1 − x²)
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Answer: (A) 2/(1 + x²)

SolutionPut x = tan θ. Then 2x/(1 + x²) = sin 2θ, and for |x| < 1, y = 2θ = 2 tan⁻¹x. So dy/dx = 2/(1 + x²).
Common trapDifferentiating directly with the chain rule is long and error-prone.
Definite IntegrationMedium

Q11. What is ∫₀^π x sin x dx?

  1. 0
  2. 2
  3. π/2
  4. π
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Answer: (D) π

SolutionIntegration by parts: ∫x sin x dx = −x cos x + sin x. From 0 to π: (π + 0) − (0 + 0) = π.
Common trapTreating x as a constant and integrating only sin x gives 2.
Application of DerivativesEasy

Q12. What is the local maximum value of f(x) = x³ − 3x?

  1. −2
  2. 0
  3. 1
  4. 2
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Answer: (D) 2

Solutionf′(x) = 3x² − 3 = 0 at x = ±1. f″(−1) = −6 < 0, so x = −1 is a local maximum: f(−1) = −1 + 3 = 2.
Common trapGiving the point (x = −1) instead of the value (2), or using x = 1, which is the local minimum.
Continuity & DifferentiabilityEasy

Q13. f(x) = (sin 2x)/x for x ≠ 0, and f(0) = k. For what value of k is f continuous at x = 0?

  1. 0
  2. 1
  3. 2
  4. 1/2
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Answer: (C) 2

Solutionlim(x→0) sin 2x/x = 2 × lim sin 2x/(2x) = 2. For continuity, k must equal this limit: k = 2.
Common trapUsing lim sin x/x = 1 directly without the factor 2.
Indefinite IntegrationMedium

Q14. What is ∫ eˣ (sin x + cos x) dx?

  1. eˣ cos x + C
  2. eˣ sin x + C
  3. −eˣ sin x + C
  4. eˣ (sin x − cos x) + C
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Answer: (B) eˣ sin x + C

SolutionUse ∫ eˣ (f(x) + f′(x)) dx = eˣ f(x) + C. With f(x) = sin x, f′(x) = cos x, so the answer is eˣ sin x + C.
Common trapPicking f(x) = cos x, whose derivative is −sin x, not +sin x.
Differential EquationsEasy

Q15. What are the order and degree of the differential equation (d²y/dx²)³ + (dy/dx)² + y = 0?

  1. Order 2, degree 3
  2. Order 3, degree 2
  3. Order 2, degree 2
  4. Order 1, degree 3
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Answer: (A) Order 2, degree 3

SolutionOrder is the highest derivative present: d²y/dx², so order 2. Degree is the power of that highest derivative (once the equation is polynomial in derivatives): 3.
Common trapSwapping order and degree.

Frequently asked questions

Which Calculus chapters are covered here?

This page covers Application of Derivatives, Area Under Curves, Continuity & Differentiability, Definite Integration, Differential Equations, Differentiation, Indefinite Integration, Limits. Questions follow the JEE Main pattern, with multiple-choice and numerical value types.

Are these previous year JEE questions?

No. These are original JEE Main-level practice questions written on the latest pattern, with solutions and the common trap for each.