Maharashtra Board Class 12 Mathematics and Statistics Guess Paper 2027
A full practice paper on the official Mathematics and Statistics pattern, built from the most-likely questions by repetition analysis. Not an official or leaked paper.
Maharashtra Board Class 12 Mathematics and Statistics — Guess Paper 2027
Time: 3 hours · Maximum marks: 80 · Practice paper
Section AQ1 eight MCQs (2 marks each) and Q2 four very short answer questions (1 mark each)
Q1.2 marks
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
Why this question: Applications of Derivatives → Find the approximate value of a function/expression using differentials — appeared 5× Board 2023Board 2024Board 2026
Q2.2 marks
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
Why this question: Indefinite Integration → Integrate a product of functions using integration by parts — appeared 4× Board 2023Board 2024Board 2026
Q3.2 marks
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
Why this question: Trigonometric Functions → Evaluate an expression involving inverse trigonometric functions — appeared 3× Board 2024Board 2026
Q4.2 marks
Write the dual of p ^ t.
Why this question: Mathematical Logic → Find the dual of a given logical statement — appeared 3× Board 2024Board 2026
Q5.2 marks
Find the angle between the lines r = (i + 2j + 3k) + lambda(2i - j + k) and r = (i + 2j + 3k) + mu(i + 2j + k).
Why this question: Line and Plane → Find the angle between two lines given their vector equations — appeared 3× Board 2024Board 2025Board 2026
Q6.2 marks
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
Why this question: Binomial Distribution → Find n of a binomial distribution given its mean and variance — appeared 3× Board 2024Board 2025Board 2026
Q7.2 marks
If tan-1(2x) + tan-1(3x) = pi/4, then x = ____. (a) -1 (b) 1/6 (c) 1/3 (d) 3/2
Why this question: Trigonometric Functions → Solve an equation involving inverse trigonometric functions — appeared 3× Board 2022Board 2024Board 2026
Q8.2 marks
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
Why this question: Probability Distributions → Find a probability from a given probability density function — appeared 2× Board 2025Board 2026
Q9.1 mark
Evaluate: integral ex(1 + x)/cos2(x ex) dx.
Why this question: Indefinite Integration → Integrate a function using a suitable substitution — appeared 5× Board 2024Board 2025Board 2026
Q10.1 mark
Write the order of differential equation (d2y/dx2)^(5/2) = cube root of (dy/dx).
Why this question: Differential Equations → Find the order/degree of a given differential equation — appeared 5× Board 2022Board 2023Board 2025Board 2026
Q11.1 mark
Find the combined equation of the pair of lines 2x + y = 0 and 3x - y = 0.
Why this question: Pair of Straight Lines → Write the combined (joint) equation of two given lines — appeared 2× Board 2023Board 2024
Q12.1 mark
What is the distance of point (3, -5, 6) from XZ plane?
Why this question: Line and Plane → Find the distance of a point from a coordinate plane — appeared 1× Board 2026
Section BShort answer questions, 2 marks each (attempt any 8 of 12)
Q13.2 marks
Find the general solution of equation cos 5(theta) = 1/2.
Why this question: Trigonometric Functions → Find the general solution of a given trigonometric equation — appeared 5× Board 2023Board 2025Board 2026
Q14.2 marks
Evaluate: integral from 0 to -1 of e^(-x) dx.
Why this question: Definite Integration → Evaluate a given definite integral — appeared 4× Board 2022Board 2024Board 2025
Q15.2 marks
Using truth table, determine whether statement pattern [(p v q) ^ ~p] ^ ~q is a tautology or a contradiction or a contingency.
Why this question: Mathematical Logic → Using a truth table, examine whether a statement pattern is a tautology/contradi — appeared 3× Board 2024Board 2026
Q16.2 marks
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.
Why this question: Vectors → Find the ratio in which a point divides a segment using position vectors — appeared 3× Board 2023Board 2024Board 2026
Q17.2 marks
Find the area bounded by curve x2 + y = 0, X-axis and lines x = 1 and x = 4.
Why this question: Application of Definite Integration → Find the area under a curve bounded by given ordinates — appeared 3× Board 2023Board 2024Board 2026
Q18.2 marks
Using truth table, prove that ~p ^ q = (p v q) ^ ~p.
Why this question: Mathematical Logic → Using a truth table, verify a logical equivalence of two statement patterns — appeared 3× Board 2022Board 2025
Q19.2 marks
Solve the D.E. 3ex tan y dx + (1 + ex) sec2 y dy = 0.
Why this question: Differential Equations → Solve a differential equation by variable separable method — appeared 3× Board 2022Board 2023Board 2026
Q20.2 marks
Test whether the function f(x) = x - 1/x, x in R, x != 0 is increasing or decreasing.
Why this question: Applications of Derivatives → Test whether a given function is increasing or decreasing — appeared 2× Board 2026
Section CShort answer questions, 3 marks each (attempt any 8 of 12)
Q21.3 marks
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.
Why this question: Linear Programming → Solve a linear programming problem graphically — appeared 6× Board 2022Board 2024Board 2025Board 2026
Q22.3 marks
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.
Why this question: Differentiation → Prove the parametric differentiation formula and apply it — appeared 5× Board 2022Board 2024Board 2025Board 2026
Q23.3 marks
A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.
Why this question: Binomial Distribution → Find a binomial probability for a given number of successes — appeared 5× Board 2022Board 2023Board 2024Board 2025Board 2026
Q24.3 marks
Solve the differential equation x dy/dx - sqrt(x2 + y2) = y.
Why this question: Differential Equations → Solve a homogeneous differential equation — appeared 4× Board 2024Board 2025Board 2026
Q25.3 marks
Prove that: sin-1(3/5) + cos-1(12/13) = sin-1(56/65).
Why this question: Trigonometric Functions → Prove an identity involving inverse trigonometric functions — appeared 4× Board 2023Board 2024Board 2025Board 2026
Q26.3 marks
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.
Why this question: Line and Plane → Find the shortest distance between two skew lines — appeared 3× Board 2023Board 2024Board 2025
Q27.3 marks
Prove that the homogeneous equation of degree two in x and y, ax2 + 2hxy + by2 = 0 represents a pair of lines passing through the origin if h2 - ab >= 0.
Why this question: Pair of Straight Lines → Prove a homogeneous second-degree equation represents a pair of lines through th — appeared 2× Board 2026
Q28.3 marks
If y = sin-1 x, then show that: (1 - x2) d2y/dx2 - x dy/dx = 0.
Why this question: Differentiation → Prove a second-order differential relation for a given function — appeared 3× Board 2022Board 2023Board 2024
Section DLong answer questions, 4 marks each (attempt any 5 of 8)
Q29.4 marks
Verify Rolle's theorem for the function f(x) = x2 - 5x + 9, x in [1, 4].
Why this question: Applications of Derivatives → Verify Rolle's theorem for a given function on an interval — appeared 2× Board 2025Board 2026
Q30.4 marks
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?
Why this question: Applications of Derivatives → Find a related rate of change in a geometric context — appeared 2× Board 2025Board 2026
Q31.4 marks
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
Why this question: Definite Integration → Evaluate a definite integral using a suitable substitution — appeared 2× Board 2025Board 2026
Q32.4 marks
Find the vector equation of the plane passing through the points A(2, 1, 1), B(0, 2, 3) and C(4, 5, 6).
Why this question: Line and Plane → Find the vector equation of a plane through three given points — appeared 2× Board 2025Board 2026
Q33.4 marks
Find E(X) and V(X), where X is the number obtained on uppermost face, when a fair die is thrown.
Why this question: Probability Distributions → Find the expected value and variance of a discrete random variable from an exper — appeared 2× Board 2025Board 2026