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Class 12 Math Practice Test Online

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About this Class 12 Math practice test

Class 12 Math demands a mature grasp of matrices and determinants, integral calculus, 3D coordinate geometry, and multi-variable probability. This diagnostic engine supports senior students preparing for school-leaving boards and engineering pathways.

Class 12 Math Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

  1. 1. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,3)} be a relation defined on A. Check whether R is reflexive, symmetric, and transitive.

  2. 2. Show that the function f: R → R defined by f(x) = 2x + 3 is both one-one and onto.

  3. 3. Find the value of cos^-1(cos(7π/6)).

  4. 4. Using the formula tan^-1 x + tan^-1 y = tan^-1((x + y)/(1 - xy)), find the value of tan^-1(1/2) + tan^-1(1/3).

  5. 5. If A = [[2, -1], [3, 4]] and B = [[1, 2], [0, -3]], find the matrix 2A - B.

  6. 6. If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find the matrix product AB.

  7. 7. If A = [[1, 2, 3], [4, 5, 6]], find A^T (the transpose of

    • A) and state its order.

  8. 8. Find the determinant of the matrix [[5, 3], [2, 4]].

  9. 9. Using determinants, find the area of the triangle whose vertices are (1, 0), (6, 0), and (4, 3).

  10. 10. Solve the following system of equations using the matrix method: x + 2y = 5 and 3x + y = 5.

  11. 11. Find dy/dx if y = 3x^4 - 5x^3 + 2x - 7.

  12. 12. Find dy/dx if y = sin(3x^2 + 1), using the chain rule.

  13. 13. The radius of a circle is increasing at a rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

  14. 14. Find the interval in which the function f(x) = x^2 - 4x + 6 is increasing.

  15. 15. Find the local maximum and local minimum values of f(x) = x^3 - 3x^2 - 9x + 5.

  16. 16. Evaluate the integral ∫ (2/x + 3 sec^2 x) dx.

  17. 17. Evaluate the definite integral ∫ from 1 to 2 of (2x + 3) dx.

  18. 18. Find the area bounded by the curve y = x^2, the x-axis, and the lines x = 0 and x = 3.

  19. 19. Find the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 + y = 0.

  20. 20. Solve the differential equation dy/dx = x/y, given that y = 2 when x = 0.

  21. 21. If vector a = 2i + 3j - k and vector b = i - j + 4k, find a + b.

  22. 22. Find the dot product of a = i + 2j + 3k and b = 3i - j + 2k, and state whether a and b are perpendicular.

  23. 23. Find a × b (the cross product) if a = i + j + k and b = 2i - j + 3k.

  24. 24. Find the magnitude of the vector a = 3i - 4j + 12k.

  25. 25. Find the direction cosines of the vector 2i - 3j + 6k.

  26. 26. Find the vector equation and the Cartesian equation of the line passing through the point (1, 2, 3) and parallel to the vector 2i - j + 2k.

  27. 27. Find the distance between the points P(1, -2, 3) and Q(4, 2, -1).

  28. 28. The corner points of the feasible region for a linear programming problem are (0, 0), (5, 0), (3, 4), and (0, 5). If Z = 4x + 3y, find the maximum value of Z.

  29. 29. Bag A contains 4 red and 2 black balls, and Bag B contains 3 red and 3 black balls. A bag is chosen at random and a ball is drawn from it, which turns out to be red. Using Bayes' theorem, find the probability that the ball was drawn from Bag A.

  30. 30. A random variable X has the probability distribution: P(X = 0) = k, P(X = 1) = 2k, P(X = 2) = 3k, P(X = 3) = 4k. Find the value of k and the expected value E(X).

Syllabus & Core Topics

Relations and functions, reflexive symmetric transitive propertiesInverse trigonometric functions, principal value branches, and equationsMatrix algebra, matrix multiplication, transpose, and inverse matricesDeterminants, cofactor expansions, and solving linear systemsVectors, dot products, cross products, and 3D coordinate geometry

Advanced math connects the abstract logic of variable functions with the physical reality of spatial structures. Finding your confidence with vector products and calculus derivatives prepares you to tackle the mathematical rigor of science and tech majors.

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Why this practice page is useful

  • Working through differential equations, limit rules, and integrals now builds the calculus fluency university engineering courses assume.

  • Guided steps for determinants and inverse matrices — using the inverse rather than a matrix 'division' that doesn't actually exist — clear up one of the most common board-exam mix-ups.

  • Practice with vector cross-products, line equations, and 3D coordinates builds the spatial thinking this section is actually tested on.

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

  1. 1. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,3)} be a relation defined on A. Check whether R is reflexive, symmetric, and transitive.

    Reflexive: yes; Symmetric: no; Transitive: no.

    R is reflexive because (1,1), (2,2), (3,3) are all in R. It is not symmetric because (1,2) is in R but (2,1) is not. It is not transitive because (1,2) and (2,3) are in R but (1,3) is not.

  2. 2. Show that the function f: R → R defined by f(x) = 2x + 3 is both one-one and onto.

    f is one-one and onto (bijective).

    One-one: if f(x1) = f(x2), then 2x1 + 3 = 2x2 + 3, giving x1 = x2. Onto: for any y in R, choosing x = (y - 3)/2 gives f(x) = y, so every real y has a pre-image. Hence f is bijective.

  3. 3. Find the value of cos^-1(cos(7π/6)).

    5π/6

    Since 7π/6 lies outside the principal range [0, π] of cos^-1, compute cos(7π/6) = cos(π + π/6) = -cos(π/6) = -√3/2. Then cos^-1(-√3/2) = 5π/6, since 5π/6 is in [0, π] and cos(5π/6) = -√3/2.

  4. 4. Using the formula tan^-1 x + tan^-1 y = tan^-1((x + y)/(1 - xy)), find the value of tan^-1(1/2) + tan^-1(1/3).

    π/4

    Here xy = (1/2)(1/3) = 1/6 < 1, so the formula applies directly: (x + y)/(1 - xy) = (5/6)/(5/6) = 1. So the sum equals tan^-1(1) = π/4.

  5. 5. If A = [[2, -1], [3, 4]] and B = [[1, 2], [0, -3]], find the matrix 2A - B.

    [[3, -4], [6, 11]]

    2A = [[4, -2], [6, 8]]. Subtracting B entrywise: [[4-1, -2-2], [6-0, 8-(-3)]] = [[3, -4], [6, 11]].

  6. 6. If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find the matrix product AB.

    [[4, 6], [10, 12]]

    Row 1 of AB: (1*2+2*1, 1*0+2*3) = (4, 6). Row 2 of AB: (3*2+4*1, 3*0+4*3) = (10, 12). So AB = [[4, 6], [10, 12]].

  7. 7. If A = [[1, 2, 3], [4, 5, 6]], find A^T (the transpose of A) and state its order.

    A^T = [[1, 4], [2, 5], [3, 6]], order 3×2.

    The transpose is formed by converting rows of A into columns. A is 2×3, so A^T is 3×2, with rows (1,4), (2,5), (3,6).

  8. 8. Find the determinant of the matrix [[5, 3], [2, 4]].

    14

    For a 2x2 matrix [[a,b],[c,d]], det = ad - bc = 5*4 - 3*2 = 20 - 6 = 14.

  9. 9. Using determinants, find the area of the triangle whose vertices are (1, 0), (6, 0), and (4, 3).

    7.5 square units

    Area = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| = (1/2)|1(0-3) + 6(3-0) + 4(0-0)| = (1/2)|-3+18+0| = (1/2)(15) = 7.5.

  10. 10. Solve the following system of equations using the matrix method: x + 2y = 5 and 3x + y = 5.

    x = 1, y = 2

    Writing as AX = B with A = [[1,2],[3,1]], det(A) = 1*1 - 2*3 = -5. A^-1 = (1/-5)[[1,-2],[-3,1]]. Then X = A^-1 B = [[-1+2],[3-1]] = [[1],[2]], so x=1, y=2, which checks: 1+2(2)=5 and 3(1)+2=5.

  11. 11. Find dy/dx if y = 3x^4 - 5x^3 + 2x - 7.

    dy/dx = 12x^3 - 15x^2 + 2

    Differentiate term by term using the power rule: d/dx(3x^4)=12x^3, d/dx(-5x^3)=-15x^2, d/dx(2x)=2, d/dx(-7)=0. Adding gives 12x^3 - 15x^2 + 2.

  12. 12. Find dy/dx if y = sin(3x^2 + 1), using the chain rule.

    dy/dx = 6x cos(3x^2 + 1)

    Let u = 3x^2 + 1, so y = sin(u). By the chain rule, dy/dx = cos(u) * du/dx = cos(3x^2+1) * 6x = 6x cos(3x^2+1).

  13. 13. The radius of a circle is increasing at a rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

    60π cm^2/s

    Area A = πr^2, so dA/dt = 2πr (dr/dt). Substituting r=10 and dr/dt=3: dA/dt = 2π(10)(3) = 60π cm^2/s.

  14. 14. Find the interval in which the function f(x) = x^2 - 4x + 6 is increasing.

    (2, ∞)

    f'(x) = 2x - 4. f is increasing where f'(x) > 0, i.e., 2x - 4 > 0, giving x > 2, so f increases on the interval (2, ∞).

  15. 15. Find the local maximum and local minimum values of f(x) = x^3 - 3x^2 - 9x + 5.

    Local maximum value 10 at x = -1; local minimum value -22 at x = 3.

    f'(x) = 3x^2 - 6x - 9 = 3(x-3)(x+1), giving critical points x=-1 and x=3. f''(x)=6x-6: f''(-1)=-12<0 (local max), f''(3)=12>0 (local min). f(-1) = -1-3+9+5 = 10 and f(3) = 27-27-27+5 = -22.

  16. 16. Evaluate the integral ∫ (2/x + 3 sec^2 x) dx.

    2 ln|x| + 3 tan x + C

    Using the standard formulas ∫(1/x)dx = ln|x| + C and ∫sec^2 x dx = tan x + C term by term: ∫2/x dx = 2ln|x|, and ∫3sec^2x dx = 3tan x. Summing with a constant gives 2 ln|x| + 3 tan x + C.

  17. 17. Evaluate the definite integral ∫ from 1 to 2 of (2x + 3) dx.

    6

    The antiderivative of 2x+3 is x^2+3x. Evaluating from 1 to 2: (2^2+3*2) - (1^2+3*1) = (4+6) - (1+3) = 10 - 4 = 6.

  18. 18. Find the area bounded by the curve y = x^2, the x-axis, and the lines x = 0 and x = 3.

    9 square units

    Area = ∫ from 0 to 3 of x^2 dx = [x^3/3] from 0 to 3 = 27/3 - 0 = 9 square units.

  19. 19. Find the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 + y = 0.

    Order = 2, Degree = 3

    The order is the highest derivative present, which is d^2y/dx^2, giving order 2. The degree is the power to which this highest-order derivative is raised (after the equation is a polynomial in derivatives), which is 3.

  20. 20. Solve the differential equation dy/dx = x/y, given that y = 2 when x = 0.

    y^2 = x^2 + 4

    Separating variables: y dy = x dx. Integrating both sides: y^2/2 = x^2/2 + C. Using y=2 at x=0: 4/2 = 0 + C, so C=2. Thus y^2/2 = x^2/2 + 2, i.e., y^2 = x^2 + 4.

  21. 21. If vector a = 2i + 3j - k and vector b = i - j + 4k, find a + b.

    3i + 2j + 3k

    Add corresponding components: (2+1)i + (3-1)j + (-1+4)k = 3i + 2j + 3k.

  22. 22. Find the dot product of a = i + 2j + 3k and b = 3i - j + 2k, and state whether a and b are perpendicular.

    a·b = 7; they are not perpendicular.

    a·b = (1)(3) + (2)(-1) + (3)(2) = 3 - 2 + 6 = 7. Since the dot product is not zero, the vectors are not perpendicular.

  23. 23. Find a × b (the cross product) if a = i + j + k and b = 2i - j + 3k.

    a × b = 4i - j - 3k

    Using the determinant formula for cross product: i-component = (1)(3)-(1)(-1)=4, j-component = -[(1)(3)-(1)(2)]=-1, k-component = (1)(-1)-(1)(2)=-3. So a × b = 4i - j - 3k.

  24. 24. Find the magnitude of the vector a = 3i - 4j + 12k.

    13

    Magnitude = sqrt(3^2 + (-4)^2 + 12^2) = sqrt(9+16+144) = sqrt(169) = 13.

  25. 25. Find the direction cosines of the vector 2i - 3j + 6k.

    (2/7, -3/7, 6/7)

    Magnitude = sqrt(2^2+(-3)^2+6^2) = sqrt(4+9+36) = sqrt(49) = 7. Direction cosines are the components divided by the magnitude: 2/7, -3/7, 6/7.

  26. 26. Find the vector equation and the Cartesian equation of the line passing through the point (1, 2, 3) and parallel to the vector 2i - j + 2k.

    Vector: r = (i + 2j + 3k) + λ(2i - j + 2k); Cartesian: (x-1)/2 = (y-2)/(-1) = (z-3)/2

    A line through point (x0,y0,z0) parallel to direction vector (a,b,c) has vector equation r = (x0 i + y0 j + z0 k) + λ(a i + b j + c k) and Cartesian form (x-x0)/a = (y-y0)/b = (z-z0)/c. Substituting the given point and direction gives the stated equations.

  27. 27. Find the distance between the points P(1, -2, 3) and Q(4, 2, -1).

    sqrt(41)

    Distance = sqrt((4-1)^2 + (2-(-2))^2 + (-1-3)^2) = sqrt(9 + 16 + 16) = sqrt(41).

  28. 28. The corner points of the feasible region for a linear programming problem are (0, 0), (5, 0), (3, 4), and (0, 5). If Z = 4x + 3y, find the maximum value of Z.

    Maximum Z = 24 at (3, 4)

    Evaluate Z at each corner point: (0,0) gives 0; (5,0) gives 20; (3,4) gives 4(3)+3(4)=12+12=24; (0,5) gives 15. The maximum value, 24, occurs at (3,4).

  29. 29. Bag A contains 4 red and 2 black balls, and Bag B contains 3 red and 3 black balls. A bag is chosen at random and a ball is drawn from it, which turns out to be red. Using Bayes' theorem, find the probability that the ball was drawn from Bag A.

    4/7

    P(A)=P(B)=1/2, P(Red|A)=4/6=2/3, P(Red|B)=3/6=1/2. P(Red) = (1/2)(2/3)+(1/2)(1/2) = 1/3+1/4 = 7/12. By Bayes' theorem, P(A|Red) = P(A)P(Red|A)/P(Red) = (1/3)/(7/12) = 4/7.

  30. 30. A random variable X has the probability distribution: P(X = 0) = k, P(X = 1) = 2k, P(X = 2) = 3k, P(X = 3) = 4k. Find the value of k and the expected value E(X).

    k = 0.1, E(X) = 2

    Since probabilities must sum to 1: k+2k+3k+4k=10k=1, so k=0.1. E(X) = 0(k)+1(2k)+2(3k)+3(4k) = 2k+6k+12k = 20k = 20(0.1) = 2.

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

Relations, Inverse Trigonometry & Matrices (Questions 1-10)

Tests relations and functions, inverse trigonometric values, matrix operations, determinants, and solving linear systems using matrices.

Calculus: Derivatives & Integrals (Questions 11-20)

Measures derivatives and their applications (rates of change, increasing/decreasing intervals, maxima/minima), integration and area under curves, and differential equations.

Vectors, 3D Geometry & Probability (Questions 21-30)

Covers vector algebra, three-dimensional geometry, linear programming, and conditional probability.

Class 12 Math chapters covered

  1. Chapter 1: Relations and Sets: Equivalence connections, mappings, and principal trigonometric values.
  2. Chapter 2: Matrices and Determinants: Matrix operations, transposes, inverses, and solving systems of equations.
  3. Chapter 3: Calculus: Differentiation rules, integration, and finding the area under curves.
  4. Chapter 4: Coordinate Spaces: Vector projections, dot and cross products on cartesian reference planes.
  5. Chapter 5: Optimization: Modeling constraints, inequalities, and linear programming regions.

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