Mathematics · Class 12 Science

Gujarat Gseb 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: 8
Total Papers: 8
Real Board Papers: 0
Sample papers: 532
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

77 Q2026complete

77 Q2025complete

77 Q2024complete

20 Q2023complete

—2022Paper not yet available

—2021Exam cancelled (COVID-19)

68 Q2020complete

68 Q2019complete

—2018Paper not yet available

2022 — Paper not yet available: This year’s exam was held, but no verified question paper for this subject has been published by any source we check — the official GSEB archive and the public past-paper archives. We publish only a paper we can verify against a real printed original — this one will appear here once it is.

2021 — Exam cancelled (COVID-19): GSEB cancelled the Class-12 Higher Secondary Certificate (HSC) Examination in 2021 due to COVID-19; students were promoted via internal/alternative assessment. No exam was administered that year, so there is no genuine previous-year paper to publish.

2018 — Paper not yet available: This year’s exam was held, but no verified question paper for this subject has been published by any source we check — the official GSEB archive and the public past-paper archives. We publish only a paper we can verify against a real printed original — this one will appear here once it is.

GSEB Higher Secondary Certificate (HSC) Examination 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: 100
Questions: 68
Duration: 180 min
Sections: 4

The marks / questions / duration above are the official exam pattern. We currently have 77 of this paper’s questions (113% of the full paper), with 77 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
Part ASection Part A	mcq	50	1	50
Part B — Section A (विभाग-A)Section Part B — Section A (विभाग-A)	subjective	8	2	16
Part B — Section B (विभाग-B)Section Part B — Section B (विभाग-B)	subjective	6	3	18
Part B — Section C (विभाग-C)Section Part B — Section C (विभाग-C)	subjective	4	4	16
Total		68		100

The question paper

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

Board Examination

Mathematics

GSEB Higher Secondary Certificate (HSC) Examination 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 100

General Instructions

This question paper contains 68 questions divided into 4 sections — Part A, Part B — Section A (विभाग-A), Part B — Section B (विभाग-B), Part B — Section C (विभाग-C).
Section Part A comprises 50 questions of 1 mark each (mcq).
Section Part B — Section A (विभाग-A) comprises 8 questions of 2 marks each (subjective).
Section Part B — Section B (विभाग-B) comprises 6 questions of 3 marks each (subjective).
Section Part B — Section C (विभाग-C) comprises 4 questions of 4 marks each (subjective).

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

Section A

Q1.

The function

f(x) = \dfrac{k\tan 2x}{x-\pi}

for

x \neq \pi

, and

f(x) = 2

for

x = \pi

. If

f

is continuous at

x = \pi

, then

k =

____.

(a) 1
(b) -1
(c) 2
(d) -2

[1]

Q2.

The derivative of

\cos 2x

with respect to

\sin x

is ____.

(a) $2\sin x$
(b) $-2\sin x$
(c) $4\sin x$
(d) $-4\sin x$

[1]

Q3.

e^x(y+1) = 1

, then

\dfrac{d^2y}{dx^2} =

____.

(a) $y$
(b) $y+1$
(c) $\dfrac{dy}{dx}$
(d) $\dfrac{dy}{dx}+1$

[1]

Q4.

The radius of an air bubble is increasing at the rate of

\dfrac{1}{4}

cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

(a) $\dfrac{\pi}{2}$ cm $^3$ /s
(b) $\pi$ cm $^3$ /s
(c) $2\pi$ cm $^3$ /s
(d) $4\pi$ cm $^3$ /s

[1]

Q5.

Wheat is being poured into a cylindrical tank of radius 10 m at the rate of 314 m

^3

/h. The rate of increase in the depth of the wheat poured is:

(a) 1 m/h
(b) 0.1 m/h
(c) 1.1 m/h
(d) 0.5 m/h

[1]

Q6.

In which of the following intervals is

y = x\cdot e^{-x}

increasing?

(a) $(-\infty, 1)$
(b) $(1, \infty)$
(c) $(-\infty, \infty)$
(d) $(-1, \infty)$

[1]

Page 1 of 10

Q7.

The point on the curve y² = 2x that lies at the minimum distance from (4, 0) is: (a) (0,0) (b) (1, ±√(2)) (c) (2, ± 2) (d) (3, ±√(6))

[1]

Q8.

[1]

Q9.

∫ dfracsec⁻¹x√(x⁴-x²)dx = __ + C. (a) -sec⁻¹x (b) sec⁻¹x (c) -(1)/(2)(sec⁻¹x)² (d) (1)/(2)(sec⁻¹x)²

[1]

Q10.

If ∫ sin³ xdx = Acos x + Bcos³ x + C, then B - A = __. (a) 4/3 (b) -2/3 (c) -4/3 (d) 2/3

[1]

Q11.

∫ dfrac1√(5x²-2x)dx = __ + C. (a) (1)/(√5)log|(5x-1)+√(25x²+10x)| (b) log|(5x-1)+√(25x²+10x)| (c) (1)/(√5)log|(5x-1)+√(25x²-10x)| (d) log|(5x-1)+√(25x²-10x)|

[1]

Q12.

∫-π/4π/4 (x²sin x + tan³ x - 1)dx = __. (a) 0 (b) -π/2 (c) π/2 (d) π/4

[1]

Q13.

If f(a+b-x) = f(x), then ∫ab x· f(x)dx = __. (a) (a+b)/(2)∫ab f(b-x)dx (b) (a+b)/(2)∫ab f(b+x)dx (c) (b-a)/(2)∫ab f(x)dx (d) (a+b)/(2)∫ab f(x)dx

[1]

Q14.

∫ ex((1-x)/(1+x²))² dx = __ + C. (a) (ex)/(1+x²) (b) -(ex)/(1+x²) (c) (ex)/((1+x²)²) (d) -(ex)/((1+x²)²)

[1]

Q15.

∫π/6π/3 dfrac11+√(tan x)dx = __. (a) π/6 (b) π/4 (c) π/12 (d) 0

[1]

Q16.

The area of the region bounded by the curve y = x|x|, the x-axis, and the ordinates x=-1 and x=1 is __. (a) 0 (b) 1/3 (c) 2/3 (d) 4/3

[1]

Page 2 of 10

Q17.

The area of the region bounded by the curve y=sin x between x=0 and x=π is __. (a) 1 (b) 2 (c) 3 (d) 4

[1]

Q18.

The area of the region bounded by the curve y=x² and the line y=16 is: (a) 128/3 (b) 256/3 (c) 64/3 (d) 512/3

[1]

Q19.

The degree of the differential equation (1+y₁²)3/2 = y₂ is __. (a) 4 (b) 3 (c) 2 (d) Not defined

[1]

Q20.

The general solution of the differential equation exdy + (yex+2x)dx = 0 is __. (a) y· ex + x² = C (b) y· ex - x² = C (c) x· ex + y² = C (d) x· ex - y² = C

[1]

Q21.

The integrating factor of the differential equation (tan⁻¹y - x)dy = (1+y²)dx is __. (a) e^tan⁻¹x (b) e^tan⁻¹y (c) e^-tan⁻¹x (d) e^-tan⁻¹y

[1]

Q22.

If x=0, y=1, then the particular solution of the differential equation (dy)/(dx) - y = 1 is __. (a) y=2ex+1 (b) y=ex+1 (c) y=ex-1 (d) y=2ex-1

[1]

Q23.

A vector of magnitude 5 units along the vector vec a = hat i - 2hat j + 3hat k is __. (a) dfrac1√(14)(5hat i-10hat j+15hat k) (b) -dfrac1√(14)(5hat i-10hat j+15hat k) (c) dfrac1√(14)(hat i-2hat j+3hat k) (d) -dfrac1√(14)(hat i-2hat j+3hat k)

[1]

Q24.

The projection of the vector hat i+3hat j+7hat k on the vector 7hat i-hat j+8hat k is __. (a) 60/114 (b) 60/√(114) (c) 66/114 (d) 66/√(114)

[1]

Q25.

[1]

Q26.

The angle between the vectors hat i-hat j-hat k and hat i-hat j+hat k is __. (a) sin⁻¹(2√2)/(3) (b) cos⁻¹(-dfrac13) (c) cos⁻¹(2√2)/(3) (d) sin⁻¹(-dfrac13)

[1]

Page 3 of 10

Q27.

If the position vectors of points A and B from the origin are vec a=2hat i-3hat j+2hat k and vec b=2hat i+3hat j+hat k respectively, then the area of triangle OAB = __. (a) √(229) (b) √(157) (c) frac12√(229) (d) frac12√(157)

[1]

Q28.

If the vectors vec a and vec b are in mutually opposite directions, then vec a·vec b = __. (a) |vec a||vec b| (b) |vec a×vec b| (c) -|vec a||vec b| (d) -|vec a×vec b|

[1]

Q29.

If a line makes angles 90^°, 60^°, 30^° with the x, y and z axes respectively, then its direction cosines are __. (a) 0, frac12, (√3)/(2) (b) 0, (√3)/(2), frac12 (c) 0, frac12, -(√3)/(2) (d) 0, -frac12, (√3)/(2)

[1]

Q30.

The equation of the line passing through the point (1,0,0) and parallel to the y-axis is __. (a) (x)/(1)=(y+1)/(0)=(z)/(0) (b) (x)/(1)=(y-1)/(0)=(z)/(0) (c) (x+1)/(0)=(y)/(1)=(z)/(0) (d) (x-1)/(0)=(y)/(1)=(z)/(0)

[1]

Q31.

The direction ratios of the line perpendicular to both the lines (x-5)/(7)=(y+2)/(-5)=(z)/(1) and (x)/(1)=(y)/(2)=(z)/(3) are __. (a) 17, 20, 19 (b) -17, 20, 19 (c) 17, -20, 19 (d) 17, 20, -19

[1]

Q32.

For a linear programming problem, the objective function Z=3x+9y has the corner points of the bounded feasible region (0,10), (5,5), (15,15) and (0,20). The maximum value of Z is __. (a) Only at (15,15) (b) Only at (0,20) (c) Any point on the line segment joining (15,15) and (0,20) (d) (5,5)

[1]

Q33.

For a linear programming problem, the objective function Z=510x+675y has the corner points of the bounded feasible region (0,0), (300,0), (180,120) and (0,240). The maximum value of Z is __. (a) 1,72,800 (b) 1,62,000 (c) 1,53,000 (d) 1,70,000

[1]

Q34.

If P(A)=3/5, P(B)=4/9, and A and B are independent events, then P(A'∩ B') = __. (a) 4/15 (b) 8/45 (c) 1/3 (d) 2/9

[1]

Q35.

If P(A/B) > P(A), then which of the following is true? (a) P(B|A) < P(B) (b) P(A∩ B) < P(A)· P(B) (c) P(B|A) > P(B) (d) P(B|A) = P(B)

[1]

Page 4 of 10

Q36.

The maximum number of equivalence relations on the set A=\1,2,3\ is __. (a) 2 (b) 3 (c) 4 (d) 5

[1]

Q37.

If n(A)=5 and n(B)=6, how many one-one and onto functions from A to B are there? (a) 0 (b) 120 (c) 720 (d) 30

[1]

Q38.

Which of the following functions from Z to Z is one-one and onto? (a) f(x)=x³ (b) f(x)=x+2 (c) f(x)=2x+1 (d) f(x)=x²+1

[1]

Q39.

sin[2cot⁻¹(-(5)/(12))] = __. (a) 24/169 (b) -120/169 (c) -24/169 (d) 120/169

[1]

Q40.

The principal branch of tan⁻¹ is __. (a) [-π/2,π/2] (b) (-π/2,π/2) - \0\ (c) (-π/2,π/2) (d) mathbb R

[1]

Q41.

The domain of cos⁻¹(2x-1) is __. (a) [-1,1] (b) [0,1] (c) (-1,1) (d) [0,π]

[1]

Q42.

If sin⁻¹(1-x) - 2sin⁻¹x = π/2, then the value of x is: (a) 0, frac12 (b) 1, frac12 (c) 0 (d) frac12

[1]

Q43.

How many 2×2 matrices are there each of whose entries is 0, 1, or 2? (a) 27 (b) 64 (c) 81 (d) 512

[1]

Q44.

Matrices A and B will be inverses of each other only if __. (a) AB=BA (b) AB=BA=0 (c) AB=0, BA=I (d) AB=BA=I

[1]

Q45.

If A and B are symmetric matrices of the same order, then AB+BA is a __. (a) skew-symmetric matrix (b) symmetric matrix (c) zero matrix (d) identity matrix

[1]

Q46.

If A= 12; 34 and I= 10; 01 , and A² = kA + 2I, then k = __. (a) 5 (b) -5 (c) 1 (d) -1

[1]

Page 5 of 10

Q47.

If x x; 2 x+1 = 0, then x = __. (a) -1 (b) 3 (c) -3 (d) 1

[1]

Q48.

If the area of the triangle with vertices (k,0), (4,0) and (0,2) is 4 square units, then the value of k is __. (a) 0, -8 (b) 8, -8 (c) only 0 (d) 0, 8

[1]

Q49.

If A is an invertible matrix of order two, then |A⁻¹| = __. (a) |A| (b) |A|⁻¹ (c) 1 (d) 0

[1]

Q50.

If A= 100; 020; 003 , then |3A| = __. (a) 18 (b) 54 (c) 162 (d) 182

[1]

Section B

Q1.

Express tan⁻¹((cos x)/(1-sin x)), for -(π)/(2) < x < (π)/(2), in its simplest form.

[2]

Q2.

Prove that: sin⁻¹(8)/(17) + sin⁻¹(3)/(5) = tan⁻¹(77)/(36).

[2]

Q3.

If y = tan⁻¹((3x-x³)/(1-3x²)), for -(1)/(√3) < x < (1)/(√3), find (dy)/(dx).

[2]

Q4.

Find ∫ x·tan⁻¹xdx.

[2]

Q5.

Find the area of the circle x²+y²=81.

[2]

Page 6 of 10

Q6.

Find the area of the region bounded by the line y=3x+2, the x-axis, and the ordinates x=0 and x=1.

[2]

Q7.

Find the solution of the differential equation y· eydx = (x· ey+y²)dy, (y≠0).

[2]

Q8.

Let P and Q be two points with position vectors overrightarrowOP=3vec a-2vec b and overrightarrowOQ=vec a+vec b. Find the position vector of a point R which divides the line joining P and Q in the ratio 2:1 (i) internally, (ii) externally.

[2]

Q9.

Show that the line through the points (1,-1,2) and (3,4,-2) is perpendicular to the line through the points (0,3,2) and (3,5,6).

[2]

Q10.

Find the shortest distance between the given lines l₁ and l₂: vec r=hat i+2hat j-4hat k+λ(2hat i+3hat j+6hat k) and vec r=3hat i+3hat j-5hat k+μ(2hat i+3hat j-6hat k).

[2]

Q11.

Find P(A∪ B) if 2P(A)=P(B)=(5)/(13) and P(A/B)=dfrac25.

[2]

Q12.

An urn contains 10 black and 5 white balls. Two balls are drawn one after another, and the first ball is not returned to the urn before the second is drawn. Assuming that each ball in the urn is equally likely to be drawn, find the probability that both balls drawn are black.

[2]

Page 7 of 10

Section C

Q1.

Prove that the relation R defined on A=\x∈ mathbb Z : 0≤ x≤ 12\ by R=\(a,b): |a-b| is a multiple of 4\ is an equivalence relation.

[3]

Q2.

If F(x)= cos x -sin x 0; sin x cos x 0\0 0 1 , then prove that F(x)· F(y)=F(x+y).

[3]

Q3.

If A= 133; 143; 134 , verify that AcdotadjA=|A|· I, and find A⁻¹.

[3]

Q4.

If x√(1+y)+y√(1+x)=0 for -1<x<1, prove that (dy)/(dx)=-(1)/((1+x)²).

[3]

Q5.

Find the intervals in which the function given by f(x)=4x³-6x²-72x+30 is (a) increasing, (b) decreasing.

[3]

Q6.

The scalar product of the vector hat i+hat j+hat k with the unit vector along the sum of the vectors 2hat i+4hat j-5hat k and λhat i+2hat j+3hat k is equal to 1. Find the value of λ.

[3]

Page 8 of 10

Q7.

Find the shortest distance between the lines vec r=6hat i+2hat j+2hat k+λ(hat i-2hat j+2hat k) and vec r=-4hat i-hat k+μ(3hat i-2hat j-2hat k).

[3]

Q8.

[For general students] Using the graphical method, maximize Z=3x+2y subject to the constraints: x+2y≤10, 3x+y≤15, x,y≥0. **OR** [For visually impaired students only] For a linear programming problem, the objective function Z=200x+500y has, over the bounded feasible region, the corner points (0,5), (4,3), and (0,6). Find the maximum and minimum value of Z.

[3]

Q9.

In a factory manufacturing bolts, machines A, B, and C manufacture 25%, 35%, and 40% of the total output, respectively. Of their outputs, 5%, 4%, and 2%, respectively, are defective. A bolt is drawn at random from the total output and is found to be defective. What is the probability that it was manufactured by machine B?

[3]

Section D

Q1.

Express the matrix B= 2-2-4; -134; 1-2-3 as the sum of a symmetric matrix and a skew-symmetric matrix.

[4]

Q2.

Solve the following system of equations by the matrix method: 3x-2y+3z=8; 2x+y-z=1; 4x-3y+2z=4.

[4]

Q3.

Differentiate with respect to x: (x+dfrac1x)x + x(1+frac1x).

[4]

Page 9 of 10

Q4.

[For general students] Prove that the height of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is (4r)/(3). **OR** [For visually impaired students only] Find the maximum and minimum values of f(x)=3x⁴-8x³+12x²-48x+25 on the interval [0,3].

[4]

Q5.

Find ∫(√(cot x)+√(tan x))dx.

[4]

Q6.

In a certain bank, the principal grows at the rate of r\% per annum. If Rs. 100 doubles itself in 10 years, find the value of r. (loge 2 = 0.6931)

[4]

Page 10 of 10