Mathematics · Class 12 Science

Maharashtra Msbshse Class 12 Mathematics — Real Previous-Year Papers

with complete answers

Real previous-year board papers, year by year — the official exam pattern, the full question paper, and every question solved the concept-first way. Distinct from the chapter-wise textbook bank.

2016–2026
Years of papers: 10
Total Papers: 10
Real Board Papers: 0
Sample papers: 414
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

44 Q2026complete

44 Q2025complete

44 Q2024complete

44 Q2023complete

44 Q2022complete

—2021Not available

44 Q2020complete

30 Q2019complete

40 Q2018complete

40 Q2017complete

40 Q2016complete

Maharashtra HSC (MSBSHSE) Board 2026 · Set ANNUAL

Real board examination

About this paper

The real Class-12 board examination held in 2026. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 80
Questions: 44
Duration: 180 min
Sections: 5

The marks / questions / duration above are the official exam pattern. We currently have 44 of this paper’s questions (100% of the full paper), with 44 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
ASection A	compulsory	8	2	16
ASection A	compulsory	4	1	4
BSection B	choice	12	2	24
CSection C	choice	12	3	36
DSection D	choice	8	4	32
Total		44		80

The question paper

The questions we hold for this paper, laid out by section. Solutions are on the Answers tab.

Board Examination

Mathematics

Maharashtra HSC (MSBSHSE) Board 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 44 questions divided into 5 sections — A, A, B, C, D.
Section A comprises 8 questions of 2 marks each (compulsory).
Section A comprises 4 questions of 1 mark each (compulsory).
Section B comprises 12 questions of 2 marks each (choice).
Section C comprises 12 questions of 3 marks each (choice).
Section D comprises 8 questions of 4 marks each (choice).

Above is the official exam pattern. The questions printed below are those we currently hold for this paper.

Section A

compulsory · 2 marks each · 12 of 8 shown

Q1.

Write the dual of

(p\vee q)\vee r \equiv p\vee(q\vee r)

[1]

Q2.

Evaluate:

\cos^{-1}\left(\dfrac12\right)+2\sin^{-1}\left(\dfrac12\right)

[1]

Q3.

Evaluate:

\displaystyle\int \dfrac{2x}{1+x^2}\,dx

[1]

Q4.

Write the degree of the differential equation

(y''')^2+3(y'')^3+3xy'+5y=0

[1]

Q5.

The converse of contrapositive of

\sim p \rightarrow q

is ____.

(a) $q\rightarrow p$
(b) $\sim q \rightarrow p$
(c) $p\rightarrow \sim q$
(d) $\sim q \rightarrow \sim p$

[2]

Q6.

A=\begin{bmatrix}2 & -4\\ 3 & 1\end{bmatrix}

, then the adjoint of matrix

A

is ____.

(a) $\begin{bmatrix}1 & 3\\ 4 & -2\end{bmatrix}$
(b) $\begin{bmatrix}-1 & 3\\ -4 & 1\end{bmatrix}$
(c) $\begin{bmatrix}-1 & -3\\ -4 & 2\end{bmatrix}$
(d) $\begin{bmatrix}1 & 4\\ -3 & 2\end{bmatrix}$

[2]

Page 1 of 8

Q7.

If tan⁻¹(2x)+tan⁻¹(3x)=(π)/(4), then x= __. (a) -1 (b) dfrac16 (c) dfrac13 (d) dfrac32

[2]

Q8.

The angle between the line bar r=(hat i+2hat j+hat k)+λ(hat i+hat j+hat k) and the plane bar r·(2hat i-hat j+hat k)=8 is __. (a) sin⁻¹((√2)/(3)) (b) sin⁻¹((√3)/(2)) (c) sin⁻¹(dfrac12) (d) sin⁻¹((1)/(√2))

[2]

Q9.

If y=sec(tan⁻¹x), then (dy)/(dx) at x=1 is __. (a) dfrac12 (b) 1 (c) (1)/(√2) (d) √2

[2]

Q10.

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

[2]

Q11.

∫₁² (1)/(x²)· e1/xdx= __. (a) √ e + 1 (b) √ e - 1 (c) √ e(√ e - 1) (d) (√ e - 1)/(e)

[2]

Q12.

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) dfrac19 (b) dfrac29 (c) (1)/(27) (d) (2)/(27)

[2]

Section A

compulsory · 2 marks each · 12 of 8 shown

Q1.

Write the dual of (pvee q)vee r ≡ pvee(qvee r)

[1]

Q2.

Evaluate: cos⁻¹(dfrac12)+2sin⁻¹(dfrac12)

[1]

Page 2 of 8

Q3.

Evaluate: ∫ (2x)/(1+x²)dx

[1]

Q4.

Write the degree of the differential equation (y''')²+3(y'')³+3xy'+5y=0

[1]

Q5.

The converse of contrapositive of sim p arrow q is __. (a) qarrow p (b) sim q arrow p (c) parrow sim q (d) sim q arrow sim p

[2]

Q6.

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

[2]

Q7.

If tan⁻¹(2x)+tan⁻¹(3x)=(π)/(4), then x= __. (a) -1 (b) dfrac16 (c) dfrac13 (d) dfrac32

[2]

Q8.

[2]

Q9.

If y=sec(tan⁻¹x), then (dy)/(dx) at x=1 is __. (a) dfrac12 (b) 1 (c) (1)/(√2) (d) √2

[2]

Q10.

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

[2]

Q11.

∫₁² (1)/(x²)· e1/xdx= __. (a) √ e + 1 (b) √ e - 1 (c) √ e(√ e - 1) (d) (√ e - 1)/(e)

[2]

Page 3 of 8

Q12.

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) dfrac19 (b) dfrac29 (c) (1)/(27) (d) (2)/(27)

[2]

Section B

choice · 2 marks each · 12 of 12 shown

Q1.

Construct the switching circuit of the statement pattern (sim pwedge q)vee(pwedge sim r).

[2]

Q2.

In triangle ABC, prove that a(bcos C - ccos B)=b²-c².

[2]

Q3.

Find the general solution of 4cos²θ=3.

[2]

Q4.

Find k, if the sum of the slopes of the lines represented by x²+kxy-3y²=0 is twice their product.

[2]

Q5.

Find the value of p, for which the vectors bar a=3hat i+2hat j+9hat k and bar b=hat i+phat j+3hat k are perpendicular to each other.

[2]

Q6.

Find the vector equation of the line passing through the points A(1,2,3) and B(2,3,4).

[2]

Q7.

Find (dy)/(dx), if √ x + √ y = √ a.

[2]

Page 4 of 8

Q8.

Find (d²y)/(dx²), if y=x³+7x²-2x-9.

[2]

Q9.

Test whether the function f(x)=x³+6x²+12x-7 is increasing or decreasing for all x∈ R.

[2]

Q10.

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?

[2]

Q11.

Evaluate: ∫ √(1+sin 2x)· dx

[2]

Q12.

Given that, Xsim B(n,p), if n=10, E(X)=8 then find Var(X).

[2]

Section C

choice · 3 marks each · 12 of 12 shown

Q1.

Examine whether the statement pattern (pwedge q)wedge(sim p vee sim q) is a tautology or contradiction or contingency.

[3]

Q2.

In triangle ABC, if A=45°, B=60° then find the ratio of its sides.

[3]

Page 5 of 8

Q3.

If two vertices of a triangle are A(3,1,4) and B(-4,5,-3) and the centroid of the triangle is G(-1,2,1), then find the coordinates of the third vertex C of the triangle.

[3]

Q4.

If D, E, F are the mid-points of the sides BC, CA, AB respectively of triangle ABC, then prove that overlineAD+overlineBE+overlineCF=bar 0.

[3]

Q5.

Show that the lines bar r=(hat i+hat j-hat k)+λ(2hat i-2hat j+hat k) and bar r=(4hat i-3hat j+2hat k)+μ(hat i-2hat j+2hat k) intersect each other.

[3]

Q6.

Find the cartesian equation of the plane bar r=(hat i-hat j)+λ(hat i+hat j+hat k)+μ(hat i-2hat j+3hat k).

[3]

Q7.

If y=f(u) is a differentiable function of u and u=g(x) is a differentiable function of x then prove that y is a differentiable function of x and (dy)/(dx)=(dy)/(du)×(du)/(dx).

[3]

Q8.

Verify LMVT for the function f(x)=log x, on [1,e].

[3]

Q9.

Evaluate: ∫ (sin x)/(sin 3x)dx

[3]

Page 6 of 8

Q10.

Solve the D.E. 3extan ydx+(1+ex)sec² ydy=0.

[3]

Q11.

Find E(X) and V(X), where X is the number obtained on uppermost face, when a fair die is thrown.

[3]

Q12.

A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.

[3]

Section D

choice · 4 marks each · 8 of 8 shown

Q1.

If A= 1 2\3 4 , prove that A·(adj A)=(adj A)· A=|A|· I

[4]

Q2.

Show that every homogeneous equation of degree two in x and y i.e. ax²+2hxy+by²=0, represents a pair of lines passing through the origin, if h²-ab≥ 0

[4]

Q3.

If A(bar a) and B(bar b) are any two points in space and R(bar r) be a point on the line segment AB dividing internally in the ratio m:n then prove that bar r=(mbar b+nbar a)/(m+n).

[4]

Page 7 of 8

Q4.

Solve the L.P.P. graphically: Minimize: z=5x+2y, Subject to, 5x+y≥ 10, x+y≥ 6, x≥ 0, y≥ 0

[4]

Q5.

Evaluate: ∫ (3x²+4x-5)/((x²-1)(x+2))dx

[4]

Q6.

Prove that: ∫ab f(x)dx=∫ab f(a+b-x)dx. Hence, find ∫π/6π/3 sin² xdx

[4]

Q7.

Find the area enclosed between the circle x²+y²=1 and the line x+y=1 lying in the first quadrant.

[4]

Q8.

Solve the differential equation x²·(dy)/(dx)=x²+xy+y².

[4]

Page 8 of 8