Mathematics · Class 12 Science

Goa Gbshse 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.

2018–2026
Years of papers: 5
Total Papers: 5
Real Board Papers: 0
Sample papers: 168
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

36 Q2026complete

36 Q2025complete

36 Q2024complete

—2023Not available

—2022Not available

—2021Not available

—2020Not available

30 Q2019complete

30 Q2018complete

GBSHSE Class 12 Board Exam 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: 36
Duration: 180 min
Sections: 6

The marks / questions / duration above are the official exam pattern. We currently have 36 of this paper’s questions (100% of the full paper), with 36 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
MCQ (1 mark)Section MCQ (1 mark)	compulsory	8	1	8
VSA (1 mark)Section VSA (1 mark)	compulsory	8	1	8
SA-I (2 marks)Section SA-I (2 marks)	compulsory	6	2	12
SA-II (3 marks)Section SA-II (3 marks)	compulsory	6	3	18
LA-I (4 marks)Section LA-I (4 marks)	compulsory	6	4	24
LA-II (5 marks)Section LA-II (5 marks)	compulsory	2	5	10
Total		36		80

The question paper

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

Board Examination

Mathematics

GBSHSE Class 12 Board Exam 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 36 questions divided into 6 sections — MCQ (1 mark), VSA (1 mark), SA-I (2 marks), SA-II (3 marks), LA-I (4 marks), LA-II (5 marks).
Section MCQ (1 mark) comprises 8 questions of 1 mark each (compulsory).
Section VSA (1 mark) comprises 8 questions of 1 mark each (compulsory).
Section SA-I (2 marks) comprises 6 questions of 2 marks each (compulsory).
Section SA-II (3 marks) comprises 6 questions of 3 marks each (compulsory).
Section LA-I (4 marks) comprises 6 questions of 4 marks each (compulsory).
Section LA-II (5 marks) comprises 2 questions of 5 marks each (compulsory).

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

Section A

Q1.

A matrix of order 2×2 whose elements are given by a_ij = (i+j)²/3 is ................. .

(a) [[4/3, 3], [3, 16/3]]
(b) [[4/3, 2/3], [2/3, 16/3]]
(c) [[2/3, 3], [3, 13/3]]
(d) [[4/3, 1/3], [2/3, 14/3]]

[1]

Q2.

The principal value of cot⁻¹(-1/√3) is ................. .

(a) π/3
(b) π/4
(c) 2π/3
(d) 4π/3

[1]

Q3.

Let R = {(4, 4), (6, 6), (7, 7), (4, 6), (6, 4), (4, 7), (6, 7)} be a relation defined on A = {4, 6, 7}, then this relation R is ................. .

(a) Reflexive, not symmetric and not transitive
(b) Reflexive, symmetric and transitive
(c) Reflexive and transitive but not symmetric
(d) Neither Reflexive, nor symmetric and nor transitive

[1]

Q4.

If y = 3^(log x), then dy/dx = ................. .

(a) (3^(log x) · log 3)/x
(b) 3^x/log 3
(c) 3^x/(x log x)
(d) (3^x log x)/x

[1]

Q5.

If a⃗ and b⃗ are two vectors such that |a⃗| = 2 and |b⃗| = 5, then the value of λ for which a⃗ + λb⃗ and a⃗ − λb⃗ will be perpendicular is ................. .

(a) 1/2
(b) 2/3
(c) 2/5
(d) 3/4

[1]

Q6.

∫ dx/(x² − 2x + 2) is ................. .

(a) tan⁻¹(x−1) + c
(b) tan⁻¹(x+1) + c
(c) tan⁻¹(x+2) + c
(d) tan⁻¹(x−2) + c

[1]

Page 1 of 6

Q7.

The value of k for the random variable y with the given probability distribution is ................. . [Table: y = 0, 1, 2, 3, 4 ; P(y) = k, 2k, 3k, 4k, 5k] (a) 1/2 (b) 1/3 (c) 1/15 (d) 1/20

[1]

Q8.

The equation of the plane having intercepts 2, −3, 1 on X, Y and Z axis respectively is ................. . (a) 2x − 3y + z = 0 (b) 3x − 2y + 6z = 6 (c) 3x + 2y + z = 3 (d) x + 2y + 3z = 2

[1]

Q9.

Find the vector equation of the line passing through the point (2, 3, 4) and parallel to the vector 2î + 5ĵ − 3k̂.

[1]

Q10.

If a⃗ = 2î + 3ĵ + 5k̂ and b⃗ = 3î + ĵ + k̂, then find the projection of a⃗ on b⃗.

[1]

Q11.

If A and B are two independent events with P(A) = 1/2 and P(B) = 1/3, then find P(A ∪ B).

[1]

Q12.

If y = 8e⁻³ˣ, find d²y/dx².

[1]

Q13.

Find the integrating factor of the differential equation x dy/dx + y = 10.

[1]

Q14.

Find dy/dx for y = tan²(2x + 3).

[1]

Q15.

Show that the function f(x) = x³ − 3x² + 3x + 10 is always increasing.

[1]

Q16.

If tan⁻¹(1/3) = x, then find sin x.

[1]

Page 2 of 6

Section B

Q1.

Prove that: cot⁻¹x + cot⁻¹y = cot⁻¹((xy−1)/(y+x)), where x, y ∈ R.

[2]

Q2.

If A = [[1, −2], [−2, 3]] and B = [[1, 3], [2, 4]], verify whether (AB)ᵀ = BᵀAᵀ, where 'T' is the transpose of a matrix.

[2]

Q3.

Using Integration, show that: ∫[a to b] f(x) dx = ∫[a to b] f(a + b − x) dx.

[2]

Q4.

Solve the differential equation: dy/dx = (5x + y − 7)².

[2]

Q5.

If y = log(x + √(x² + 1)), then prove that: (1 + x²) d²y/dx² + x dy/dx = 0.

[2]

Q6.

Find the angle between the diagonals of a parallelogram having adjacent sides a⃗ = î + 2ĵ + 3k̂ and b⃗ = 2î − 2ĵ + k̂.

[2]

Section C

Q1.

Using integration, prove that: ∫ 1/(a² − x²) dx = 1/(2a) log|(a+x)/(a−x)| + c.

[3]

Page 3 of 6

Q2.

Express the matrix A = [[2, 4, 6], [0, 6, 8], [10, 14, −8]] as a sum of symmetric and skew symmetric matrix.

[3]

Q3.

Evaluate: ∫ 1/((x−1)²(x+2)) dx.

[3]

Q4.

Solve the differential equation: (x² − y²) dx + 2xy dy = 0.

[3]

Q5.

Show that the function f : R − 3/5 → R − 2/5 given by f(x) = (2x+5)/(5x−3) is a bijective function.

[3]

Q6.

If y = (cos x)x + x^(cot x), find dy/dx.

[3]

Section D

Q1.

Solve the Linear programming problem graphically. Maximize Z = 200x + 30y Subject to the constraints: 3x + y ≤ 60 ; 6x + y ≥ 60 ; x + y ≥ 20 ; x, y ≥ 0.

[4]

Page 4 of 6

Q2.

If A = [[2, 1, −1], [1, 1, −3], [3, 2, 1]], find A⁻¹ by using adjoint method. Hence solve the system of linear equation: 2x + y − z = 3 ; x + y − 3z = −7 ; 3x + 2y + z = 16.

[4]

Q3.

By defining the continuity of a function at point x = a, show that the given function is continuous at x = 0, where f(x) = [sin x(1 − cos 2x)]/x³, for x < 0 ; x² + 5x + 2, for x = 0 ; (e^(2x) − 1)/x + 3x/2, for x > 0 .

[4]

Q4.

Show that the lines: (x−2)/3 = (y−1)/2 = (z−3)/3 and (x+1)/4 = (y+1)/(−2) = (z−2)/5 do not intersect each other.

[4]

Q5.

A clay pot manufacturer hires three potters for making clay pots. The first potter X makes 2% defective pots whereas the other two potters Y and Z make 5% and 6% defective pots respectively. Potter X works for 50% of the allotted time, Y works for 30% and Z works for 20% of the allotted time. A defective pot is produced, what is the probability that it is produced by potter X ? **OR** A multinational company announces incentives to its employees of three departments on achieving their monthly targets. Department P has 200 employees, Q has 500 employees and R has 300 employees. The probability of achieving the target by employees of departments P, Q and R are 0.5, 0.3 and 0.6 respectively. One of the employees who got an incentive is selected. What is the probability that he is from Department Q ?

[4]

Q6.

Using integration, find the area common to parabola y² = 6x and the circle x² + y² = 16 in the first quadrant only. **OR** Using integration, find the area of the region in the first quadrant bounded by the circle x² + y² = 3, line x = √2 y and the x axis.

[4]

Page 5 of 6

Section E

Q1.

Evaluate: ∫ e^(tan⁻¹x)/(1+x²)² dx. **OR** Evaluate: ∫[0 to π] (x tan x)/(sec x + tan x) dx.

[5]

Q2.

A wire of length 20 cm is cut into two pieces and one of the pieces is made into a square and the second piece into an equilateral triangle. What should be the lengths of two pieces so that the combined area is minimum ? **OR** Show that the volume of the largest cylinder that can be inscribed in a sphere of radius R is 1/√3 times the volume of the sphere.

[5]

Page 6 of 6