Maharashtra Board Class 12 Mathematics and Statistics Important Questions 2027 (Repeated in Board Exams)
74 question patterns that appeared in two or more of the analysed papers (8 past board papers (2022–2026)). The numbers change; the question does not.
Mathematical Logic
1. Using a truth table, examine whether a statement pattern is a tautology/contradi
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Using truth table, determine whether statement pattern [(p v q) ^ ~p] ^ ~q is a tautology or a contradiction or a contingency.
2. Find the dual of a given logical statement
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Write the dual of p ^ t.
3. Using a truth table, verify a logical equivalence of two statement patterns
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Using truth table, prove that ~p ^ q = (p v q) ^ ~p.
4. Determine which quantified statement over a set is not true
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If A = {1, 2, 3, 4, 5} then which of the following is not true? (a) there exists x in A such that x + 3 = 8 (b) there exists x in A such that x + 2 < 9 (c) for all x in A, x + 6 >= 9 (d) for all x in A, x + 5 <= 10
5. Simplify a switching circuit to an equivalent one with minimum switches
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Give an alternative equivalent simple circuit for the following switching circuit. (Switch S1 in series with a parallel block of switches S1' and S2', with lamp L.)
6. Express a switching circuit symbolically and construct its switching table
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Express the following switching circuit in the symbolic form of logic. Construct the switching table and interpret it. (Parallel switches S1 and S2 in series with switches S1' and S2', with lamp L.)
Matrices
7. Find the adjoint of a given matrix
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If A = [[2, -4], [3, 1]], then the adjoint of matrix A is ____. (a) [[1, 3], [4, -2]] (b) [[-1, 3], [-4, 1]] (c) [[-1, -3], [-4, 2]] (d) [[1, 4], [-3, 2]]
8. Verify the property A(adjA) = (adjA)A = |A|I for a given matrix
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If A = [[1, 2], [3, 4]], prove that A.(adj A) = (adj A).A = |A|.I.
9. Find the cofactors of the elements of a given matrix
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Find the cofactors of the elements of the matrix [[2, -3], [3, 5]].
10. Find the inverse of a matrix by elementary row transformations
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Find the inverse of matrix A by elementary row transformations, where A = [[2, -3], [-1, 2]].
Trigonometric Functions
11. Find the general solution of a given trigonometric equation
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Find the general solution of equation cos 5(theta) = 1/2.
12. Prove an identity involving inverse trigonometric functions
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Prove that: sin-1(3/5) + cos-1(12/13) = sin-1(56/65).
13. Evaluate an expression involving inverse trigonometric functions
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The value of sin[sin-1(-1/sqrt(2)) - 3 sin-1(sqrt(3)/2)] = ____. (a) -1/sqrt(2) (b) 1/sqrt(2) (c) sqrt(3)/2 (d) -sqrt(3)/2
14. Solve an equation involving inverse trigonometric functions
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If tan-1(2x) + tan-1(3x) = pi/4, then x = ____. (a) -1 (b) 1/6 (c) 1/3 (d) 3/2
15. Prove a given identity for a triangle using its properties
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In triangle ABC, prove that a(b cos C - c cos B) = b2 - c2.
16. Find the principal solutions of a given trigonometric equation
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The principle solutions of the equation cos(theta) = 1/2 are ____. (a) pi/6, 5pi/6 (b) pi/3, 5pi/3 (c) pi/6, 7pi/6 (d) pi/3, 2pi/3
17. Find an angle of a triangle using the cosine rule
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In triangle ABC, if c2 + a2 - b2 = ac, then angle B = ____. (a) pi/4 (b) pi/3 (c) pi/2 (d) pi/6
Pair of Straight Lines
18. Find the unknown constant using a relation between sum and product of slopes of
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Find k, if the sum of the slopes of the lines represented by x2 + kxy - 3y2 = 0 is twice their product.
19. Find the unknown constant given that a line is represented by a pair-of-lines eq
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Find the value of k, if 2x + y = 0 is one of the lines represented by 3x2 + kxy + 2y2 = 0.
20. Derive the formula for the angle between a pair of lines
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Show that the acute angle theta between the lines represented by ax2 + 2hxy + by2 = 0 is given by tan(theta) = |2 sqrt(h2 - ab)/(a + b)|.
21. Write the combined (joint) equation of two given lines
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Find the combined equation of the pair of lines 2x + y = 0 and 3x - y = 0.
Vectors
22. Find the ratio in which a point divides a segment using position vectors
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If a, b, c are the position vectors of points A, B, C respectively and 10a = 7b + 3c then find the ratio in which the point C divides the line segment AB.
23. Determine whether four given points are coplanar using scalar triple product
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Are the four points A(1, 2, 1), B(2, -3, 4), C(3, 4, -5), D(2, 3, -2) coplanar? Justify your answer.
24. Express a coplanar vector as a unique linear combination of two vectors
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Let a and b be non-collinear vectors. If vector r is coplanar with a and b then show that there exist unique scalars t1 and t2 such that r = t1 a + t2 b. For r = 2i + 7j + 9k, a = i + 2j, b = j + 3k, find t1, t2.
25. Evaluate a scalar triple product expression of unit vectors
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If p, q, r are mutually perpendicular unit vectors and form a right handed triplet then the value of p.(q x r) + q.(p x r) + r.(p x q) = ____. (a) -2 (b) 0 (c) 1 (d) 3
26. Prove the section formula for internal division using vectors
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If A(a) and B(b) are any two points in space and R(r) be a point on the line segment AB dividing internally in the ratio m : n then prove that r = (m b + n a)/(m + n).
Line and Plane
27. Find the angle between two lines given their vector equations
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Find the angle between the lines r = (i + 2j + 3k) + lambda(2i - j + k) and r = (i + 2j + 3k) + mu(i + 2j + k).
28. Find the shortest distance between two skew lines
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Find the shortest distance between the lines, (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5.
29. Find the vector equation of a plane through three given points
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Find the vector equation of the plane passing through the points A(2, 1, 1), B(0, 2, 3) and C(4, 5, 6).
30. Find the angle between a line and a plane
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The angle between the line r = (i + 2j + k) + lambda(i + j + k) and the plane r.(2i - j + k) = 8 is ____. (a) sin-1(sqrt(2)/3) (b) sin-1(sqrt(3)/2) (c) sin-1(1/2) (d) sin-1(1/sqrt(2))
31. Find the direction angle of a line given the other two direction angles
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A line makes angles of measure 45 deg and 60 deg with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.
32. Find the distance between two parallel lines in space
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Find the distance between parallel lines x/2 = y/-1 = z/2 and (x+1)/2 = (y-1)/-1 = (z+1)/2.
33. Find the equation of a plane given a point and its normal's direction ratios
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Find the cartesian equation of the plane passing through the point A(-1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
34. Find the vector equation of a plane through a point parallel to two vectors
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Find the vector equation of the plane passing through the point A(-1, 2, -5) and parallel to the vectors 4i - j + 3k and i + j - k.
Linear Programming
35. Solve a linear programming problem graphically
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Solve the linear programming problem (L.P.P.) by graphical method. Minimize z = 8x + 10y Subject to 2x + y >= 7, 2x + 3y >= 15, x >= 0, y >= 2.
Differentiation
36. Prove the parametric differentiation formula and apply it
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If x = f(t) and y = g(t) are differentiable functions of t, so that y is differentiable function of x and dx/dt != 0 then prove that dy/dx = (dy/dt)/(dx/dt). Hence find dy/dx, if x = sin t, y = cos t.
37. Prove a second-order differential relation for a given function
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If y = sin-1 x, then show that: (1 - x2) d2y/dx2 - x dy/dx = 0.
38. Differentiate an implicit function
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Find dy/dx, if sqrt(x) + sqrt(y) = sqrt(a).
39. Find the derivative of a composite function at a given point
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If y = sec(tan-1 x), then dy/dx at x = 1 is ____. (a) 1/2 (b) 1 (c) 1/sqrt(2) (d) sqrt(2)
40. Find the derivative of an implicit function at a given point
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If y is a function of x and log(x + y) = xy then the value of dy/dx at x = 0 is ____. (a) 1 (b) -1 (c) 2 (d) 0
Applications of Derivatives
41. Find the approximate value of a function/expression using differentials
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The approximate value of the function f(x) = x3 - 3x + 5 at x = 1.99 is ____. (a) 6.09 (b) 6.91 (c) 7.09 (d) 7.91
42. Verify Rolle's theorem for a given function on an interval
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Verify Rolle's theorem for the function f(x) = x2 - 5x + 9, x in [1, 4].
43. Find a related rate of change in a geometric context
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A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm?
44. Verify Lagrange's mean value theorem for a given function on an interval
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Verify LMVT for the function f(x) = log x, on [1, e].
45. Find dimensions that maximise/minimise area/volume (optimisation word problem)
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A wire of length 36 cm is bent to form a rectangle. Find its dimensions, if the area of the rectangle is maximum.
46. Show that a given function is increasing/decreasing on an interval
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Show that the function f(x) = x3 + 10x + 7, x in R is strictly increasing.
Indefinite Integration
47. Integrate a function using a suitable substitution
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Evaluate: integral ex(1 + x)/cos2(x ex) dx.
48. Integrate a product of functions using integration by parts
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If f(x) = sin-1 x/sqrt(1 - x2) and g(x) = e^(sin-1 x) then integral f(x).g(x) dx = ____. (a) e^(sin-1 x)(1 - sin-1 x) + c (b) e^(sin-1 x)(sin-1 x - 1) + c (c) e^(sin-1 x)(sin-1 x + 1) + c (d) e^(cos-1 x)(cos-1 x - 1) + c
49. Integrate a rational function using partial fractions
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Evaluate: integral (3x2 + 4x - 5)/((x2 - 1)(x + 2)) . dx.
50. Integrate a trigonometric quotient by adjusting the angle
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Evaluate: integral sin(x + a)/cos(x - b) dx.
51. Integrate a special-form rational function (standard integral)
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Evaluate: integral 1/(25 - 9x2) dx.
Definite Integration
52. Evaluate a given definite integral
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Evaluate: integral from 0 to -1 of e^(-x) dx.
53. Evaluate a definite integral using a suitable substitution
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integral from 1 to 2 of (1/x2) . e^(1/x) dx = ____. (a) sqrt(e) + 1 (b) sqrt(e) - 1 (c) sqrt(e)(sqrt(e) - 1) (d) (sqrt(e) - 1)/e
54. Prove a property of definite integrals and apply it
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Prove that: integral from a to b of f(x) dx = integral from a to b of f(a + b - x) . dx. Hence, find integral from pi/6 to pi/3 of sin2 x . dx.
55. Evaluate a definite integral using a trigonometric identity
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Evaluate: integral from 0 to pi/4 of sqrt(1 + sin 2x) dx.
Application of Definite Integration
56. Find the area under a curve bounded by given ordinates
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Find the area bounded by curve x2 + y = 0, X-axis and lines x = 1 and x = 4.
57. Find the area bounded by a line and the X-axis between given ordinates
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The area bounded by the line y = 2x, X-axis and the lines x = -1, x = 4 is ____ sq. units. (a) 15 (b) 17 (c) 16 (d) 14
58. Find the area bounded between two curves
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Find the area of the region lying between the parabolas y2 = 4x and x2 = 4y.
Differential Equations
59. Find the order/degree of a given differential equation
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Write the order of differential equation (d2y/dx2)^(5/2) = cube root of (dy/dx).
60. Solve a homogeneous differential equation
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Solve the differential equation x dy/dx - sqrt(x2 + y2) = y.
61. Solve a differential equation by variable separable method
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Solve the D.E. 3ex tan y dx + (1 + ex) sec2 y dy = 0.
62. Solve an application of differential equations (Newton's law of cooling)
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A body cools according to Newton's law from 100 deg C to 60 deg C in 20 minutes. The temperature of the surrounding being 20 deg C. How long will it take to cool down to 40 deg C?
63. Solve a linear differential equation
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Solve the D.E. dy/dx + y/x = x3 - 3.
64. Identify the solution of a variable-separable differential equation
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The solution of the D.E. sec2 x . tan y dx + sec2 y . tan x dy = 0 is ____. (a) tan x . cot y = c (b) cot x - cot y = c (c) tan x . tan y = c (d) cot x - tan y = c
65. Find the particular solution of a variable-separable differential equation with
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Find the particular solution of: r dr/d(theta) + cos(theta) = 5 at r = sqrt(2) and theta = 0.
66. Solve a differential equation reducible to variable separable form by substituti
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Solve: 1 + dy/dx = cosec(x + y); put x + y = u.
Probability Distributions
67. Find a probability from a given probability density function
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If the p.d.f. of a continuous r.v. X is f(x) = (x + 2)/18, for -2 < x < 4, = 0 otherwise then P(|X| < 1) = ____. (a) 1/9 (b) 2/9 (c) 1/27 (d) 2/27
68. Find the expected value and variance of a discrete random variable from an exper
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Find E(X) and V(X), where X is the number obtained on uppermost face, when a fair die is thrown.
69. Find the unknown constant and probabilities from a discrete probability distribu
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The probability distribution of X is as follows: x: 0, 1, 2, 3, 4; P[X=x]: 0.1, k, 2k, 2k, k. Find (i) k, (ii) P(X < 2), (iii) P[1 <= X < 4].
70. Find probabilities from a given probability density function
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Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = 1/5, for 0 <= x <= 5, = 0 otherwise. Find the probability that: (i) waiting time is between 1 to 3 minutes. (ii) waiting time is more than 4 minutes.
Binomial Distribution
71. Find a binomial probability for a given number of successes
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A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.
72. Find n of a binomial distribution given its mean and variance
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In a Binomial Distribution, E(X) = 6, Var(X) = 4.2 then the value of n is ____. (a) 22 (b) 24 (c) 20 (d) 26
73. Find binomial probabilities for a given number of successes
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Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that: (i) all the five cards are spades. (ii) none is spade.
74. Find n and p of a binomial distribution given its mean and variance
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If X ~ B(n, p) and E(X) = 6, Var(X) = 4.2, then find n and p.