Mathematics · Class 12 Science

Assam Ahsec 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: 245
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

—2026Paper not yet available

39 Q2025complete

29 Q2024complete

29 Q2023complete

29 Q2022complete

—2021Exam cancelled (COVID-19)

29 Q2020complete

29 Q2019complete

29 Q2018complete

2026 — Paper not yet available: This year’s exam was held, but neither AHSEC’s official site nor any source we check has published the question paper yet. 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): AHSEC cancelled the HS Final (Class-12) examination in 2021 due to COVID-19; no annual question paper was conducted or printed that year, so none exists to publish. Results were declared using a formula based on HSLC marks, institutional practicals, and internal assessment.

AHSEC Higher Secondary (HS) Final 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: 29
Duration: 180 min
Sections: 3

The marks / questions / duration above are the official exam pattern. We currently have 32 of this paper’s questions (110% of the full paper), with 32 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	subjective	10	1	10
BSection B	subjective	12	4	48
CSection C	subjective	7	6	42
Total		29		100

The question paper

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

Board Examination

Mathematics

AHSEC Higher Secondary (HS) Final Examination 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 100

General Instructions

This question paper contains 29 questions divided into 3 sections — A, B, C.
Section A comprises 10 questions of 1 mark each (subjective).
Section B comprises 12 questions of 4 marks each (subjective).
Section C comprises 7 questions of 6 marks each (subjective).

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

Section A

subjective · 1 mark each · 13 of 10 shown

Q1.

A=\{a, b, c\}

and

B=\{1, 2, 3\}

, then find the total number of relations from the set

A

B

[1]

Q2.

The value of

\hat{i}\cdot(\hat{j}\times\hat{k})+\hat{j}\cdot(\hat{i}\times\hat{k})+\hat{k}\cdot(\hat{i}\times\hat{j})

(i) $0$
(ii) $-1$
(iii) $1$
(iv) $3$

[1]

Q3.

What are the direction cosines of

x

-axis?

[1]

Q4.

Find the principal value of

\sin^{-1}\left(\sin\frac{3\pi}{5}\right)

[1]

Q5.

Let

A

be a square matrix of order

3\times 3

, and

\det A=7

(

|A|=7

). Find the value of

\det(\text{adj } A)

[1]

Q6.

The function

f(x)=|x-a|

(i) continuous at $x=a$ , but not differentiable at $x=a$
(ii) both continuous and differentiable at $x=a$
(iii) neither continuous nor differentiable at $x=a$
(iv) differentiable at $x=a$ , but not continuous at $x=a$

[1]

Page 1 of 6

Q7.

If A= cosα -sinα ; sinα cosα and A+AT=I, 0<α<90^°, then find the value of α.

[1]

Q8.

Evaluate : ∫(sin⁻¹x+cos⁻¹x)dx

[1]

Q9.

If f(x)=sin²√(x), then f'(x)=?

[1]

Q10.

The Signum function f:mathbbR→mathbbR is given by f(x)= 1, if x>0 \0, if x=0 \-1, if x<0 then (i) f(x) is both one-one and onto (ii) f(x) is neither one-one nor onto (iii) f(x) is one-one but not onto (iv) f(x) is onto but not one-one

[1]

Q11.

The value of ∫₁√(3)fracdx1+x² is (i) (π)/(3) (ii) (2π)/(3) (iii) (π)/(6) (iv) (π)/(12)

[1]

Q12.

If A and B are two events such that A⊂ B and P(B)≠ 0, then which of the following is correct? (i) P(A|B)=(P(B))/(P(A)) (ii) P(A|B)<P(A) (iii) P(A|B)≥ P(A) (iv) None of the above

[1]

Q13.

The integrating factor of the differential equation x(dy)/(dx)-y=2x² is (i) e-x (ii) -(1)/(x) (iii) (1)/(x) (iv) x

[1]

Section B

subjective · 4 marks each · 12 of 12 shown

Q1.

(a) If -(3π)/(2)<x<(π)/(2), then prove that tan⁻¹(cos x)/(1-sin x)=(π)/(4)+(x)/(2).

[4]

Page 2 of 6

Q2.

Answer both (a) and (b) : (a) [2 marks] If A= 3 1 \7 5 , I= 1 0 \0 1 , then find x and y such that A²+xI=yA. (b) [2 marks] If A and B are invertible matrices of the same order, then prove that (AB)⁻¹=B⁻¹A⁻¹.

[4]

Q3.

Let A=mathbbR-\3\ and B=mathbbR-\1\. Consider the function f:A→ B defined by f(x)=(x-2)/(x-3). Is f one-one and onto? Justify your answer.

[4]

Q4.

Find (dy)/(dx) of the following (any two) : (i) ((x)/(a))m+((y)/(b))ⁿ=1 (ii) y=xx (iii) y=sin⁻¹(frac1-x²1+x²),0<x<1

[4]

Q5.

The volume of a cube is increasing at the rate of 9 cubic centimetres per second. How fast is the surface area increasing, when the length of an edge is 10 centimetres?

[4]

Q6.

Evaluate (any two) : (i) ∫fracdx√(x)+x (ii) ∫tan⁴xdx (iii) ∫ ex((1)/(x)-frac1x²)dx

[4]

Q7.

Using integration, find the area of the ellipse fracx²a²+fracy²b²=1.

[4]

Page 3 of 6

Q8.

Answer both (a) and (b) : (a) [2 marks] If veca=hati+2hatj-hatk and vecb=3hati+hatj-5hatk, then find a unit vector in the direction of veca-vecb. (b) [2 marks] For any two vectors veca and vecb, show that |veca·vecb|≤|veca||vecb|.

[4]

Q9.

If θ is the angle between two unit vectors hata and hatb, and hata+hatb is a unit vector, then find θ. Also find |hata-hatb|.

[4]

Q10.

Answer both (a) and (b) : (a) [3 marks] Find the angle between any two diagonals of a cube. (b) [1 mark] If a line makes angles 90^°, 135^°, 45^° with the x-axis, y-axis and z-axis respectively, find its direction cosines.

[4]

Q11.

Answer both (a) and (b) : (a) [2 marks] If E and F be any events of a sample space S, prove that P(E'|F)=1-P(E|F), where E' is the complementary event of the event E. (b) [2 marks] If P(A)=(6)/(11), P(B)=(5)/(11) and P(A∩ B)=(7)/(11), then find P(A|B).

[4]

Q12.

There are three coins. One is two-headed coin (having head on both faces), another is a biased coin that comes up head 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head. What is the probability that it is the two-headed coin?

[4]

Page 4 of 6

Section C

subjective · 6 marks each · 7 of 7 shown

Q1.

Answer both (a) and (b) : (a) [4 marks] If A= 0 -tan(α)/(2) ; tan(α)/(2) 0 and I is the identity matrix of order 2, then show that I+A=(I-A) cosα -sinα ; sinα cosα . (b) [2 marks] If 6i -3i 1 \4 3i -1 \20 3 i =x+iy, then find the values of x and y.

[6]

Q2.

Answer both (a) and (b) : (a) [3 marks] Find the relationship between a and b so that the function f defined by f(x)= ax+1, if x≤ 3 bx+3, if x>3 is continuous at x=3. (b) [3 marks] If x=√a^sin⁻¹t, y=√a^cos⁻¹t, then show that (dy)/(dx)=-(y)/(x).

[6]

Q3.

Find the local maximum and local minimum values of the function f given by f(x)=3x⁴+4x³-12x²+12.

[6]

Q4.

Evaluate (any two) : (i) ∫ x³e^x²dx (ii) ∫fracdx√7-6x-x² (iii) ∫sin 2x·cos 3xdx

[6]

Page 5 of 6

Q5.

Solve the following differential equations (any two) : (i) y(dy)/(dx)=xy+x+y+1 (ii) xdy-ydx=√x²+y²dx (iii) (dy)/(dx)+2ytan x=sin x, given y=0, when x=(π)/(3)

[6]

Q6.

Answer both (a) and (b) : (a) [3 marks] If hata and hatb be two unit vectors inclined at an angle θ, prove that sin(θ)/(2)=(1)/(2)|hata-hatb|. (b) [3 marks] In a bank, principal increases continuously at the rate r\% per year. Find the value of r if rupee100 doubles itself in 10 years. (loge2=0.6931)

[6]

Q7.

Solve the following linear programming problem graphically : Minimize Z=10(x-7y+190) subject to the constraints x+y≤ 8, x≤ 5, y≤ 5, x+y≥ 4, x≥ 0, y≥ 0.

[6]

Page 6 of 6