Mathematics · Class 12 Science

Puducherry Tnboard 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: 539
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

47 Q2026complete

47 Q2025complete

47 Q2024complete

47 Q2023complete

47 Q2022complete

—2021Not available

47 Q2020complete

47 Q2019complete

70 Q2018complete

70 Q2017complete

70 Q2016complete

Tamil Nadu HSC (DGE) 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: 90
Questions: 47
Duration: 180 min
Sections: 6

The marks / questions / duration above are the official exam pattern. We currently have 47 of this paper’s questions (100% of the full paper), with 47 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	20	1	20
BSection B	compulsory	1	2	2
BSection B	choice	9	2	18
CSection C	compulsory	1	3	3
CSection C	choice	9	3	27
DSection D	compulsory	7	5	35
Total		47		90

The question paper

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

Board Examination

Mathematics

Tamil Nadu HSC (DGE) Board 2026 · Set ANNUAL

Series/Set: ANNUALRoll No. ________

Time Allowed: 3 hoursMaximum Marks: 90

General Instructions

This question paper contains 47 questions divided into 6 sections — A, B, B, C, C, D.
Section A comprises 20 questions of 1 mark each (compulsory).
Section B comprises 1 question of 2 marks each (compulsory).
Section B comprises 9 questions of 2 marks each (choice).
Section C comprises 1 question of 3 marks each (compulsory).
Section C comprises 9 questions of 3 marks each (choice).
Section D comprises 7 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

compulsory · 1 mark each · 20 of 20 shown

Q1.

If A is a

3\times3

non-singular matrix such that

AA^T=A^TA

and

B=A^{-1}A^T

, then

BB^T=

(a) $I_3$
(b) $A$
(c) $B^T$
(d) $B$

[1]

Q2.

\rho(A)=\rho([A\mid B])

, then the system

AX=B

of linear equations is :

(a) consistent and has infinitely many solutions
(b) consistent and has a unique solution
(c) inconsistent
(d) consistent

[1]

Q3.

z

is a complex number such that

z\in C\setminus R

and

z+\dfrac1z\in R

, then

|z|

is :

(a) $2$
(b) $0$
(c) $3$
(d) $1$

[1]

Q4.

The product of all four values of

\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)^{3/4}

is :

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

[1]

Q5.

f

and

g

are polynomials of degrees

m

and

n

respectively and if

h(x)=(f\circ g)(x)

, then the degree of

h

is :

(a) $m^n$
(b) $mn$
(c) $n^m$
(d) $m+n$

[1]

Q6.

x<0

, then

\tan^{-1}\left(\dfrac1x\right)

is equal to :

(a) $-\pi+\cot^{-1}(x)$
(b) $\tan^{-1}(x)$
(c) $-\pi+\tan^{-1}x$
(d) $\cot^{-1}(x)$

[1]

Page 1 of 10

Q7.

The eccentricity of the circle is : (a) dfrac12 (b) 0 (c) 2 (d) 1

[1]

Q8.

If a vector vecα lies in the plane vecβ and vecγ, then (a) [vecα, vecβ, vecγ]=0 (b) [vecα, vecβ, vecγ]=1 (c) [vecα, vecβ, vecγ]=2 (d) [vecα, vecβ, vecγ]=-1

[1]

Q9.

If the image of the point A(1, 2, 3) with respect to the plane vecr·(hati+2hatj+4hatk)=38 is A'(3, 6, 11), then the foot of the perpendicular from the point A to the given plane is : (a) (2, 5, 7) (b) (2, 3, 7) (c) (2, -4, 7) (d) (2, 4, 7)

[1]

Q10.

One of the closest points on the curve x²-y²=4 to the point (6, 0) is : (a) (3, √5) (b) (2, 0) (c) (√(13), -√3) (d) (√5, 1)

[1]

Q11.

The value of 'c' satisfied by the Rolle's theorem for the function f(x)=x³-3x², x∈[0, 3] is : (a) dfrac32 (b) 1 (c) 2 (d) √2

[1]

Q12.

The percentage error of fifth root of 31 is approximately how many times the percentage error in 31 ? (a) 5 (b) (1)/(31) (c) 31 (d) dfrac15

[1]

Q13.

Let A=\(x, y)mid a<x<b,c<y<d\⊂ R². If the function u:A→ R² is harmonic in A, then : (a) (∂²u)/(∂ x²)+(∂²u)/(∂ y²)=0∀(x,y)∈ A (b) (∂²u)/(∂ x²)+(∂²u)/(∂ y²)=1∀(x,y)∈ A (c) (∂²u)/(∂ x²)-(∂²u)/(∂ y²)=0∀(x,y)∈ A (d) (∂²u)/(∂ x²)-(∂²u)/(∂ y²)=1∀(x,y)∈ A

[1]

Q14.

If f(x)=∫₀xtcos tdt, then (df)/(dx)= (a) xcos x (b) cos x-xsin x (c) xsin x (d) sin x+xcos x

[1]

Q15.

If f(x)=∫₁xdfracesin uudu, x>1 and ∫₁³dfracesin x^2xdx=dfrac12[f(a)-f(1)], then one of the possible value of a is : (a) 9 (b) 3 (c) 5 (d) 6

[1]

Q16.

The solution of the differential equation 2x(dy)/(dx)-y=3 represents : (a) Parabola (b) Straight lines (c) Ellipse (d) Circles

[1]

Page 2 of 10

Q17.

P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then : (a) P=Ckt (b) P=Cekt (c) Pt=C (d) P=Ce-kt

[1]

Q18.

If the function f(x)=(1)/(12) for a<x<b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b ? (a) 7 and 19 (b) 0 and 12 (c) 16 and 24 (d) 5 and 17

[1]

Q19.

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is f(x)= dfrac1l 0<x<l; 0 l≤ x<2l . The mean and variance of the shorter of the two pieces are respectively : (a) l,(l²)/(12) (b) (l)/(2),(l²)/(3) (c) (l)/(2),(l²)/(12) (d) (l)/(2),(l²)/(6)

[1]

Q20.

The dual of lnot(pvee q)vee[pvee(pwedgelnot r)] is : (a) lnot(pwedge q)wedge[pwedge(pwedge r)] (b) lnot(pwedge q)wedge[pvee(pwedgelnot r)] (c) lnot(pwedge q)wedge[pwedge(pveelnot r)] (d) (pwedge q)wedge[pwedge(pveelnot r)]

[1]

Section B

compulsory · 2 marks each · 10 of 1 shown

Q1.

If A is a non-singular matrix of odd order, prove that |adjA| is positive.

[2]

Q2.

Simplify : Σn=1¹²iⁿ

[2]

Q3.

If x²+2(k+2)x+9k=0 has equal roots, find k.

[2]

Q4.

Simplify : sin⁻¹[sin10]

[2]

Page 3 of 10

Q5.

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

[2]

Q6.

Prove that the function f(x)=x²-2x-3 is strictly increasing in the interval (2, ∞).

[2]

Q7.

If f(x, y)=cos⁻¹((x)/(y)), then show that fy=dfracxy√(y²-x²).

[2]

Q8.

Evaluate : ∫₀(π)/(2)sin¹⁰xdx

[2]

Q9.

Determine the order and degree (if exists) of the differential equation x²(d²y)/(dx²)+[1+((dy)/(dx))²]1/2=0

[2]

Q10.

If √Var(X)=dfrac12 then, find the value of Var(2X+3).

[2]

Section B

compulsory · 2 marks each · 10 of 1 shown

Q1.

If A is a non-singular matrix of odd order, prove that |adjA| is positive.

[2]

Q2.

Simplify : Σn=1¹²iⁿ

[2]

Page 4 of 10

Q3.

If x²+2(k+2)x+9k=0 has equal roots, find k.

[2]

Q4.

Simplify : sin⁻¹[sin10]

[2]

Q5.

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

[2]

Q6.

Prove that the function f(x)=x²-2x-3 is strictly increasing in the interval (2, ∞).

[2]

Q7.

If f(x, y)=cos⁻¹((x)/(y)), then show that fy=dfracxy√(y²-x²).

[2]

Q8.

Evaluate : ∫₀(π)/(2)sin¹⁰xdx

[2]

Q9.

Determine the order and degree (if exists) of the differential equation x²(d²y)/(dx²)+[1+((dy)/(dx))²]1/2=0

[2]

Q10.

If √Var(X)=dfrac12 then, find the value of Var(2X+3).

[2]

Page 5 of 10

Section C

compulsory · 3 marks each · 10 of 1 shown

Q1.

Find the rank of the matrix 2 -2 4 3; -3 4 -2 -1; 6 2 -1 7 by reducing it to an echelon form.

[3]

Q2.

Find all real numbers satisfying the equation : 4x-3(2x+2)+2⁵=0.

[3]

Q3.

Prove that 2tan⁻¹dfrac12+tan⁻¹dfrac17=tan⁻¹(31)/(17)

[3]

Q4.

Find the points where the straight line passes through (6, 7, 4) and (8, 4, 9) cuts the xz and yz planes.

[3]

Q5.

If limθ→0((1-cos mθ)/(1-cos nθ))=1, then prove that m=± n

[3]

Q6.

Use the linear approximation to find approximate value of √[4]15

[3]

Q7.

If ∫₀∞e-xxⁿdx=5!, then find the value of ∫₀∞e-xxⁿ⁻¹dx

[3]

Page 6 of 10

Q8.

If Xsim B(n, p) such that 4P(X=4)=P(X=2) and n=6, find the distribution, mean and Standard Deviation of X.

[3]

Q9.

Verify (i) Closure property (ii) Associative property and (iii) Existence of identity for the following operation on the given set : m*n=m+n-mn;m, n∈ Z

[3]

Q10.

If z₁=overline1+i and overlinez₂=1-i, find the inverse of ((z₁)/(z₂))²⁰²⁶

[3]

Section C

compulsory · 3 marks each · 10 of 1 shown

Q1.

Find the rank of the matrix 2 -2 4 3; -3 4 -2 -1; 6 2 -1 7 by reducing it to an echelon form.

[3]

Q2.

Find all real numbers satisfying the equation : 4x-3(2x+2)+2⁵=0.

[3]

Q3.

Prove that 2tan⁻¹dfrac12+tan⁻¹dfrac17=tan⁻¹(31)/(17)

[3]

Q4.

Find the points where the straight line passes through (6, 7, 4) and (8, 4, 9) cuts the xz and yz planes.

[3]

Page 7 of 10

Q5.

If limθ→0((1-cos mθ)/(1-cos nθ))=1, then prove that m=± n

[3]

Q6.

Use the linear approximation to find approximate value of √[4]15

[3]

Q7.

If ∫₀∞e-xxⁿdx=5!, then find the value of ∫₀∞e-xxⁿ⁻¹dx

[3]

Q8.

If Xsim B(n, p) such that 4P(X=4)=P(X=2) and n=6, find the distribution, mean and Standard Deviation of X.

[3]

Q9.

Verify (i) Closure property (ii) Associative property and (iii) Existence of identity for the following operation on the given set : m*n=m+n-mn;m, n∈ Z

[3]

Q10.

If z₁=overline1+i and overlinez₂=1-i, find the inverse of ((z₁)/(z₂))²⁰²⁶

[3]

Page 8 of 10

Section D

compulsory · 5 marks each · 7 of 7 shown

Q1.

(a) Solve the following system of equations, using matrix inversion method. 2x₁+3x₂+3x₃=5 x₁-2x₂+x₃=-4 3x₁-x₂-2x₃=3 **OR** (b) Show that the equation z³+2bar z=0 has five solutions.

[5]

Q2.

(a) Prove that p→(lnot qvee r)≡lnot pvee(lnot qvee r) using truth table. **OR** (b) Prove that ∫₀(π)/(4)log(1+tan x)dx=(π)/(8)log2

[5]

Q3.

(a) Solve the equation (x+1)(x+3)(x-2)(x-4)+21=0 **OR** (b) Sketch the curve y=log(1+x).

[5]

Q4.

(a) Show that the line x-y+4=0 is a tangent to the ellipse x²+3y²=12. Also find the co-ordinates of the point of contact. **OR** (b) Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years ?

[5]

Page 9 of 10

Q5.

(a) By Vector method prove that : cos(α+β)=cosαcosβ-sinαsinβ **OR** (b) (x²+y²)dy=xydx. It is given that y(1)=1 and y(x₀)=e. Find the value of x₀.

[5]

Q6.

(a) A random variable X has the following probability mass function. | x | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | f(x) | k | 2k | 6k | 5k | 6k | 10k | Find (i) P(2<X<6) (ii) P(2≤ X<5) (iii) P(X≤4) (iv) P(3<X) **OR** (b) At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

[5]

Q7.

(a) A particle moves along a line according to the law s(t)=2t³-9t²+12t-4, where t≥0. (i) At what times the particle changes direction ? (ii) Find the total distance travelled by the particle in the first 4 seconds. (iii) Find the particle's acceleration each time the velocity is zero. **OR** (b) Find the non-parametric form of Vector equation and Cartesian equation of the plane passing through the point (1, -2, 4) and perpendicular to the plane x+2y-3z=11 and parallel to the line (x+7)/(3)=(y+3)/(-1)=(z)/(1)

[5]

Page 10 of 10