Class 12 Maths Olympiad Previous Paper PDF and 40 Solved Questions

Prepare for Class 12 Maths Olympiad with the downloadable School Connect Olympiad maths question paper below and 40 original practice questions with fully worked answers on this page. Revise relations, matrices, determinants, calculus, vectors, three-dimensional geometry and probability while learning how to approach challenging problems.

Start with the SCO Class 12 Maths subject guide and its 2026 syllabus to plan your preparation. The downloadable paper is a previously published practice resource, while the 40 explained questions below are new exercises, not claimed as past examination questions. Check the instructions for your own exam session rather than assuming a past paper shows the latest format.

Download the Class 12 Maths Olympiad question paper

Use the PDF preview or Download control below for additional question practice. Try each question on your own and check your working with a teacher or matching paper-specific answer key where available. For problems you want to learn step by step, continue to the solved practice questions after the PDF.

40 Class 12 Maths Olympiad Practice Questions with Worked Answers

The questions below are original practice examples. Each has four choices and a clearly written solution immediately after it. They begin with functions and algebra and progress through calculus, geometry and probability. Work through each step before checking the answer.

Relations and functions

1. Question 1. If f(x)=2x+1 and g(x)=x², find f(g(3)).

   A. 10   |   B. 19   |   C. 37   |   D. 81

   Answer: B. 19. First g(3)=9. Then f(9)=2×9+1=19.

2. Question 2. For f(x)=3x−4 on the real numbers, what is its inverse?

   A. (x−4)/3   |   B. (x+4)/3   |   C. 3x+4   |   D. 1/(3x−4)

   Answer: B. (x+4)/3. Write y=3x−4 and solve for x: x=(y+4)/3. Thus the inverse maps x to (x+4)/3.

3. Question 3. A set contains 3 elements. How many different relations can be defined from this set to itself?

   A. 64   |   B. 128   |   C. 256   |   D. 512

   Answer: D. 512. Its Cartesian product with itself has 3×3=9 ordered pairs. Every subset is a relation, giving 2⁹=512 relations.

4. Question 4. If f(x)=x² on x≥0, which rule gives its inverse on y≥0?

   A. f⁻¹(y)=y²   |   B. f⁻¹(y)=−√y   |   C. f⁻¹(y)=√y   |   D. f⁻¹(y)=2y

   Answer: C. f⁻¹(y)=√y. On the nonnegative domain x² is one-to-one, so solving y=x² gives x=√y.

Inverse trigonometric functions

5. Question 5. What is the principal value of arcsin(1/2)?

   A. π/3   |   B. π/6   |   C. 2π/3   |   D. −π/6

   Answer: B. π/6. The principal range of arcsin is [−π/2,π/2], and sin(π/6)=1/2.

6. Question 6. What is the principal value of arccos(−1/2)?

   A. π/3   |   B. π/2   |   C. 2π/3   |   D. −2π/3

   Answer: C. 2π/3. Arccos has principal range [0,π]. The angle in that range with cosine −1/2 is 2π/3.

7. Question 7. Simplify arctan(1)+arctan(1/√3), taking principal values.

   A. π/3   |   B. 5π/12   |   C. 7π/12   |   D. π/2

   Answer: B. 5π/12. The two values are π/4 and π/6. Their sum is (3π+2π)/12=5π/12.

8. Question 8. Find sin(arccos(3/5)).

   A. 3/5   |   B. −4/5   |   C. 4/5   |   D. 5/3

   Answer: C. 4/5. Let θ=arccos(3/5). Since θ is in the first quadrant, sin θ=√(1−9/25)=4/5.

Matrices and determinants

9. Question 9. What is the determinant of the matrix [[1,2],[3,4]]?

   A. −2   |   B. 2   |   C. 10   |   D. −10

   Answer: A. −2. For a 2×2 matrix, determinant=ad−bc. Here 1×4−2×3=−2.

10. Question 10. If A is a diagonal matrix with diagonal entries 2 and 3, what is det(A²)?

    A. 6   |   B. 12   |   C. 18   |   D. 36

    Answer: D. 36. det(A)=2×3=6. The determinant of A² is (det A)²=36.

11. Question 11. Find the inverse of the matrix [[2,1],[5,3]].

    A. [[3,−1],[−5,2]]   |   B. [[3,1],[5,2]]   |   C. [[2,−1],[−5,3]]   |   D. [[−3,1],[5,−2]]

    Answer: A. [[3,−1],[−5,2]]. The determinant is 2×3−1×5=1. Swap diagonal entries and negate the off-diagonals, giving [[3,−1],[−5,2]].

12. Question 12. Find the determinant of the upper-triangular matrix [[1,2,0],[0,3,4],[0,0,5]].

    A. 8   |   B. 12   |   C. 15   |   D. 30

    Answer: C. 15. For a triangular matrix, the determinant is the product of diagonal entries: 1×3×5=15.

13. Question 13. A is a 2×3 matrix and B is a 3×4 matrix. What is the order of AB?

    A. 3×3   |   B. 2×4   |   C. 4×2   |   D. 2×3

    Answer: B. 2×4. The inner dimensions 3 and 3 match; the outer dimensions are 2 and 4, so AB is 2×4.

14. Question 14. Solve 2x+y=7 and x−y=2.

    A. x=2, y=3   |   B. x=4, y=−1   |   C. x=3, y=1   |   D. x=1, y=5

    Answer: C. x=3, y=1. Add the equations to get 3x=9, so x=3. Substitute into x−y=2 to get y=1.

Continuity and derivatives

15. Question 15. Differentiate x³ sin x.

    A. 3x²sin x+x³cos x   |   B. 3x²cos x   |   C. x³cos x   |   D. 3x²sin x−x³cos x

    Answer: A. 3x²sin x+x³cos x. Apply the product rule: derivative of x³ is 3x², and derivative of sin x is cos x.

16. Question 16. For f(x)=ln(x²+1), find f'(x).

    A. 1/(x²+1)   |   B. 2x/(x²+1)   |   C. 2/(x²+1)   |   D. ln(2x)

    Answer: B. 2x/(x²+1). By the chain rule, derivative of ln u is u’/u. Here u=x²+1 and u’=2x.

17. Question 17. Differentiate e^(2x).

    A. e^(2x)   |   B. 2e^x   |   C. 2e^(2x)   |   D. e^(x²)

    Answer: C. 2e^(2x). The chain rule multiplies e^(2x) by the derivative of 2x, which is 2.

18. Question 18. At which x-values does f(x)=x³−3x have stationary points?

    A. Only x=0   |   B. x=±1   |   C. x=±3   |   D. Only x=1

    Answer: B. x=±1. Stationary points satisfy f'(x)=3x²−3=0. Therefore x²=1 and x=−1 or +1.

19. Question 19. What is the minimum value of f(x)=x²−6x+8?

    A. −1   |   B. 0   |   C. 1   |   D. 8

    Answer: A. −1. Complete the square: f(x)=(x−3)²−1. The squared term is at least zero, so the minimum is −1.

20. Question 20. Differentiate sin(2x) with respect to x.

    A. cos(2x)   |   B. 2sin(2x)   |   C. 2cos(2x)   |   D. −2cos(2x)

    Answer: C. 2cos(2x). By the chain rule, d/dx sin u=cos u × u’. For u=2x the derivative is 2cos(2x).

Continue with calculus, geometry and probability

Integrals and applications

21. Question 21. Evaluate ∫2x dx.

    A. 2x²+C   |   B. x²+C   |   C. x+C   |   D. x³/3+C

    Answer: B. x²+C. An antiderivative of 2x is x² because d(x²)/dx=2x. Include the integration constant C.

22. Question 22. Evaluate the definite integral of 3x² from x=0 to x=2.

    A. 4   |   B. 6   |   C. 8   |   D. 12

    Answer: C. 8. An antiderivative is x³. Evaluate x³ between 0 and 2: 8−0=8.

23. Question 23. Find ∫x e^x dx.

    A. xe^x+C   |   B. e^x(x−1)+C   |   C. e^x(x+1)+C   |   D. x²e^x/2+C

    Answer: B. e^x(x−1)+C. Integration by parts gives x e^x − ∫e^x dx = e^x(x−1)+C.

24. Question 24. Find ∫[1/(1+x²)] dx.

    A. ln|x|+C   |   B. arctan x+C   |   C. arcsin x+C   |   D. x/(1+x²)+C

    Answer: B. arctan x+C. Since the derivative of arctan x is 1/(1+x²), the integral is arctan x+C.

25. Question 25. What is the area beneath the curve y=x above the x-axis between x=0 and x=2?

    A. 1 square unit   |   B. 2 square units   |   C. 3 square units   |   D. 4 square units

    Answer: B. 2 square units. The area is ∫₀² x dx = [x²/2]₀² = 4/2 = 2 square units.

Differential equations

26. Question 26. If dy/dx=3x² and y(0)=2, find y(x).

    A. 3x³+2   |   B. x³+2   |   C. x²+2   |   D. x³−2

    Answer: B. x³+2. Integrate 3x² to get x³+C. Using y(0)=2 gives C=2.

27. Question 27. Solve dy/dx=2y with initial condition y(0)=3.

    A. y=3e^x   |   B. y=2e^(3x)   |   C. y=3e^(2x)   |   D. y=3+2x

    Answer: C. y=3e^(2x). Separate variables: dy/y=2dx, so y=Ce^(2x). The initial value gives C=3.

28. Question 28. Find the order and degree of (d²y/dx²)³ + dy/dx = 0.

    A. Order 3, degree 2   |   B. Order 2, degree 3   |   C. Order 2, degree 1   |   D. Order 1, degree 3

    Answer: B. Order 2, degree 3. The highest derivative is the second derivative, so order=2. Its highest power is 3, so degree=3.

Vectors

29. Question 29. Given vectors a=(1,2,2) and b=(2,0,1), calculate a·b.

    A. 2   |   B. 3   |   C. 4   |   D. 5

    Answer: C. 4. Dot product =1×2+2×0+2×1 = 2+0+2 = 4.

30. Question 30. What is the magnitude of vector a=(2,−3,6)?

    A. 5   |   B. 6   |   C. 7   |   D. 11

    Answer: C. 7. Length =√(2²+(−3)²+6²)=√(4+9+36)=√49=7.

31. Question 31. Find (1,0,0) × (0,1,0).

    A. (0,0,1)   |   B. (0,0,−1)   |   C. (1,1,0)   |   D. (0,1,0)

    Answer: A. (0,0,1). The standard unit vectors satisfy i×j=k, so the cross product is (0,0,1).

32. Question 32. What is the angle between the vectors (1,0,0) and (0,0,1)?

    A. 0°   |   B. 45°   |   C. 90°   |   D. 180°

    Answer: C. 90°. Their dot product is zero. Nonzero perpendicular vectors form a 90° angle.

Three-dimensional geometry

33. Question 33. Find the distance between points (1,2,3) and (4,6,3).

    A. 4   |   B. 5   |   C. 6   |   D. 7

    Answer: B. 5. Distance =√[(4−1)²+(6−2)²+(3−3)²]=√(9+16)=5.

34. Question 34. What is the vector from A=(1,2,3) to B=(3,1,7)?

    A. (−2,1,−4)   |   B. (2,1,4)   |   C. (2,−1,4)   |   D. (4,3,10)

    Answer: C. (2,−1,4). Subtract A’s coordinates from B’s: B−A=(3−1,1−2,7−3)=(2,−1,4).

35. Question 35. Find the shortest distance from the point (0,0,5) to the xy-plane.

    A. 0   |   B. 2   |   C. 5   |   D. 10

    Answer: C. 5. The xy-plane has z=0. Perpendicular distance to this plane is |z|=5.

Linear programming

36. Question 36. Maximise Z=3x+2y subject to x≥0, y≥0 and x+y≤4.

    A. 8   |   B. 10   |   C. 12   |   D. 14

    Answer: C. 12. The feasible triangle has vertices (0,0), (4,0), and (0,4). Their objective values are 0, 12, and 8, so the maximum is 12.

37. Question 37. Which point satisfies x≥0, y≥0 and x+y≤5?

    A. (2,2)   |   B. (4,3)   |   C. (−1,3)   |   D. (6,0)

    Answer: A. (2,2). Only (2,2) has nonnegative coordinates and sum 4≤5. Each other option violates a constraint.

Probability

38. Question 38. Two events A and B are independent with P(A)=0.6 and P(B)=0.5. Find P(A∩B).

    A. 0.1   |   B. 0.2   |   C. 0.3   |   D. 0.6

    Answer: C. 0.3. For independent events, multiply probabilities: 0.6×0.5=0.3.

39. Question 39. A condition affects 10% of a population. A test detects 90% of cases and has an 80% true-negative rate. What is P(condition | positive test)?

    A. 1/5   |   B. 1/4   |   C. 1/3   |   D. 1/2

    Answer: C. 1/3. P(condition and positive)=0.10×0.90=0.09. False positives=0.90×0.20=0.18, so among positives the fraction with the condition is 0.09/(0.09+0.18)=1/3.

40. Question 40. A fair coin is tossed three times. What is the probability of exactly two heads?

    A. 1/8   |   B. 1/4   |   C. 3/8   |   D. 1/2

    Answer: C. 3/8. There are 2³=8 equally likely outcomes and C(3,2)=3 outcomes with exactly two heads, so the probability is 3/8.

Class 12 Maths revision guide

If you get a question wrong, identify its topic first and practise the rule using new numbers. This approach builds more useful understanding than memorising an option letter.

Question numbers	Skill tested	What to practise next
1–4	Relations, composite functions and inverses	Check domains and the order of function composition
5–8	Inverse trigonometric functions	Sketch the unit circle and remember the principal ranges
9–14	Matrices and determinants	Check products, determinant signs and inverse formulas
15–20	Derivatives and stationary points	Apply the product and chain rules and substitute into results
21–25	Integrals and area	Differentiate a proposed antiderivative to verify it
26–28	Differential equations	Integrate and use initial values to determine constants
29–35	Vectors and 3D coordinates	Sketch axes and calculate dot products and distances
36–37	Linear programming	Identify the feasible region and evaluate corner points
38–40	Probability	Check independence and carefully count outcomes

Four short worked learning strategies

Check an inverse function

For f(x)=3x−4, the inverse is (x+4)/3. Verify the result by composition: f((x+4)/3)=3((x+4)/3)−4=x. The inverse really reverses the original operation.

Check integration by differentiation

For ∫x e^x dx, a proposed answer is e^x(x−1)+C. Its derivative is e^x(x−1)+e^x=x e^x, so the integration is correct.

Use geometric meaning in linear programming

For x+y≤4 in the nonnegative quadrant, draw the line x+y=4 and identify the triangle beneath it. A linear objective reaches its maximum at one of the feasible vertices when the feasible region is bounded.

Distinguish conditional probability from a test’s accuracy

A test with 90% sensitivity does not necessarily mean that 90% of its positive results are true positives. The proportion also depends on how common the condition is and how often the test produces false positives.

Frequently asked questions

Where can I download the Class 12 Maths question paper?

The downloadable School Connect Olympiad practice paper is provided near the beginning of this page, before the 40 worked questions. You can use the embedded preview or its download button.

Are the 40 solved questions from the attached examination paper?

No. They are original study exercises prepared for this article. Use the separate PDF for additional practice and the worked examples here to understand the mathematical methods.

Do I have to download anything to see the worked solutions?

No. All 40 original questions and their answers are available directly on this page. You can print the article from your browser if you prefer paper-based study.

Which Class 12 Maths topics are important?

Relations and functions, inverse trigonometry, matrices, determinants, calculus, differential equations, vectors, 3D geometry, linear programming and probability are among the subjects in SCO’s Class 12 Maths syllabus.

Are SCO and the International Mathematical Olympiad the same competition?

No. School Connect Olympiad’s Maths Olympiad and the separate international competition run by the IMO organisation are not interchangeable. Follow the preparation materials and rules for the competition you plan to enter.

How should I practise if I find calculus difficult?

Start with derivative rules, then simple antiderivatives. Move on to applications of derivatives, definite integrals and differential equations as your understanding improves.

Is this a timed mock exam?

No. The 40 solved questions are intended for concept practice. Follow the current organiser’s official instructions if you want to simulate actual examination conditions.

Where can I revise Class 11 Maths first?

Review the Class 11 Maths practice questions if you need foundations in trigonometry, sequences, coordinate geometry and elementary calculus.

Next step: Work through one topic group, compare your reasoning with the solutions, and visit the Class 12 syllabus to plan your next revision session.