Mathematics · 2nd Puc Science

Karnataka PUC 2nd Puc Science 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: 428
Real-paper Q & A: 0
Sample-paper Q & A

Real board-paper questions available, by year

47 Q2026complete

47 Q2025complete

52 Q2024complete

60 Q2023complete

66 Q2022complete

—2021Paper not available

52 Q2020complete

52 Q2019complete

52 Q2018complete

2021 — Paper not available: We publish a Karnataka II PUC board question paper only after verifying it against the official printed original from the Department of Pre-University Education (KSEAB / DPUE). No such verified paper is available for this year, so we show none rather than an unconfirmed copy.

Karnataka II PUC Board 2026 · Set V1

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: 47
Duration: 180 min
Sections: 5

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	15 MCQ + 5 Fill in the blanks — answer all	20	1	20
BSection B	Short answer — answer any 6	9	2	18
CSection C	Short answer — answer any 6	9	3	27
DSection D	Long answer — answer any 4	7	5	35
ESection E	Long answer with internal choice (one 6-mark + one 4-mark)	2	—	—
Total		47		80

The question paper

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

Board Examination

Mathematics

Karnataka II PUC Board 2026 · Set V1

Series/Set: V1Roll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 47 questions divided into 5 sections — A, B, C, D, E.
Section A comprises 20 questions of 1 mark each (15 MCQ + 5 Fill in the blanks — answer all).
Section B comprises 9 questions of 2 marks each (Short answer — answer any 6).
Section C comprises 9 questions of 3 marks each (Short answer — answer any 6).
Section D comprises 7 questions of 5 marks each (Long answer — answer any 4).
Section E comprises 2 questions (Long answer with internal choice (one 6-mark + one 4-mark)).

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

Section A

15 MCQ + 5 Fill in the blanks — answer all · 1 mark each · 20 of 20 shown

Q1.

If a relation

R

in the set

\{1, 2, 3\}

be defined by

R = \{(1, 1),(2, 2)\}

then

R

(a) symmetric but not transitive
(b) transitive but not symmetric
(c) symmetric and transitive
(d) neither symmetric nor transitive

[1]

Q2.

The domain of

\tan^{-1} x

(a) $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$
(b) $(0, \pi)$
(c) $[-1, 1]$
(d) $(-\infty, \infty)$

[1]

Q3.

A matrix has 13 elements. The number of possible different orders it can have

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

[1]

Q4.

For the matrix

A = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}

the value of

|adj\ A|

(a) 25
(b) 5
(c) 0
(d) 1

[1]

Q5.

The derivative of

\sin^{-1} x

exists in the interval

(a) $[-1, 1]$
(b) $(-1, 1)$
(c) $R$
(d) $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$

[1]

Q6.

x - y = \pi

then

\frac{dy}{dx}

(a) $\pi$
(b) $-\pi$
(c) $1$
(d) $-1$

[1]

Q7.

The minimum value of

f(x) = x

x \in R

(a) 0
(b) 1
(c) 2
(d) does not exist

[1]

Page 1 of 7

Q8.

Statement I : The function f(x) = x² is decreasing in the interval (0, ∞) Statement II : Any function y = f(x) is decreasing if (dy)/(dx) < 0. Which of the following is correct? (a) Both the Statements I and II are true (b) Both the Statements I and II are false (c) Statement I is true and Statement II is false (d) Statement I is false and Statement II is true

[1]

Q9.

The antiderivative of frac1x√(x²-1), x>1 with respect to x (a) sin⁻¹x+c (b) cos⁻¹x+c (c) -operatornamecosec⁻¹x+c (d) cot⁻¹x+c

[1]

Q10.

The value of ∫-(π)/(2)(π)/(2) sin⁷ x dx (a) 1 (b) 0 (c) -1 (d) 7

[1]

Q11.

If veca is a nonzero vector of magnitude a and λ, a nonzero scalar then λveca is a unit vector if (a) λ=1 (b) λ=-1 (c) a=|λ| (d) a=(1)/(|λ|)

[1]

Q12.

The position vector of the midpoint of the line joining the points P(2, 3, 4) and Q(4, 1, -2) (a) 3hati+2hatj+hatk (b) 3hati+2hatj-hatk (c) hati-hatj-3hatk (d) -hati+hatj+3hatk

[1]

Q13.

The direction ratios of x-axis are (a) 0, k, 0 (b) 0, 0, k (c) k, 0, 0 (d) k, k, k

[1]

Q14.

The probability of obtaining an even prime number on each die when a pair of dice is rolled (a) (1)/(36) (b) (1)/(6) (c) (1)/(18) (d) (1)/(4)

[1]

Q15.

If A and B are independent events with P(A)=0.3 and P(B)=0.4 then P(A∩ B) (a) 1.2 (b) 0.12 (c) 0.7 (d) (3)/(4)

[1]

Q16.

Choose from [0, 3, -1, 2, -2, 1]. The left hand derivative of |x| with respect to x at x=0 is __.

[1]

Q17.

Choose from [0, 3, -1, 2, -2, 1]. The point of inflection of the function f(x)=x³ in the interval [-1, 1] is __.

[1]

Page 2 of 7

Q18.

Choose from [0, 3, -1, 2, -2, 1]. If m and n are respectively the order and degree of the differential equation 2x²(d²y)/(dx²)-3(dy)/(dx)+y=0 then m+n= __.

[1]

Q19.

Choose from [0, 3, -1, 2, -2, 1]. The value of hati·hati+hatj·hatj= __.

[1]

Q20.

Choose from [0, 3, -1, 2, -2, 1]. If F is an event of a sample space S then P(S|F)= __.

[1]

Section B

Short answer — answer any 6 · 2 marks each · 9 of 9 shown

Q1.

Write tan⁻¹√((1-cos x)/(1+cos x)), 0 < x < π in simplest form.

[2]

Q2.

Find the equation of line joining (1, 2) and (3, 6) using determinants.

[2]

Q3.

Find (dy)/(dx) if x = sin t and y = cos 2t.

[2]

Q4.

The radius of a circle is increasing at the rate of 0.7 cm/s. Find the rate of increase of its circumference.

[2]

Q5.

Find ∫ (1)/(x + xlog x)dx.

[2]

Page 3 of 7

Q6.

Find the general solution of the differential equation (dy)/(dx) = (1+y²)/(1+x²).

[2]

Q7.

Find the projection of the vector veca = 2hati + 3hatj + 2hatk on the vector vecb = hati + 2hatj + hatk.

[2]

Q8.

Find the angle between the lines (x-5)/(4) = (y-2)/(1) = (z-3)/(8) and (x)/(2) = (y)/(2) = (z)/(1).

[2]

Q9.

A family has two children. What is the probability that both the children are boys given that atleast one of them is a boy?

[2]

Section C

Short answer — answer any 6 · 3 marks each · 9 of 9 shown

Q1.

Let T be the set of all triangles with R a relation in T given by R = \(T₁, T₂) : T₁ is congruent to T₂\. Show that R is an equivalence relation.

[3]

Q2.

Prove that 3cos⁻¹ x = cos⁻¹(4x³ - 3x), x ∈ [(1)/(2), 1].

[3]

Q3.

Express the matrix 1 5 \-1 2 as the sum of a symmetric and a skewsymmetric matrix.

[3]

Page 4 of 7

Q4.

Differentiate xsin x, x > 0 with respect to x.

[3]

Q5.

Find the absolute maximum and minimum values of a function f given by f(x) = 2x³ - 15x² + 36x + 1 on the interval [1, 5].

[3]

Q6.

Find ∫ (1)/((x+1)(x+2))dx.

[3]

Q7.

Find the area of triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices.

[3]

Q8.

Derive the equation of a line in space passing through a given point A and parallel to a given vector vecb in vector form.

[3]

Q9.

Bag I contains 3 red and 4 black balls while another bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag II.

[3]

Page 5 of 7

Section D

Long answer — answer any 4 · 5 marks each · 7 of 7 shown

Q1.

Consider the function f : R arrow R given by f(x) = 4x + 3. Show that f is invertible and write the inverse of f.

[5]

Q2.

If A = 1 \-4 \3 and B = -1 2 1 , verify that (AB)' = B'A'.

[5]

Q3.

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

[5]

Q4.

If y = Aemx + Benx show that (d²y)/(dx²) - (m+n)(dy)/(dx) + mny = 0.

[5]

Q5.

Find the integral of (1)/(a² - x²) with respect to 'x' and hence find ∫ (1)/(25 - x²)dx.

[5]

Page 6 of 7

Q6.

Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

[5]

Q7.

Find the general solution of the differential equation x(dy)/(dx) + 2y = x² (x ≠ 0).

[5]

Section E

Long answer with internal choice (one 6-mark + one 4-mark) · 2 of 2 shown

Q1.

Show that the matrix A = 2 3 \1 2 satisfies the equation A² - 4A + I = O Where I is 2 × 2 identity matrix and O is 2 × 2 zero matrix and hence find A⁻¹. OR Find the values of a and b such that the function defined by f(x) = 5, x ≤ 2 ax + b, 2 < x < 10 \21, x ≥ 10 is continuous.

[4]

Q2.

Prove that ∫₀a f(x)dx = ∫₀a f(a-x)dx and hence evaluate ∫₀(π)/(2) dfrac√(sin x)√(sin x) + √(cos x)dx. OR Solve the following Linear Programming Problem graphically : Maximise Z = 250x + 75y Subject to the constraints x + y ≤ 60, 25x + 5y ≤ 500, x ≥ 0, y ≥ 0

[6]

Page 7 of 7