Karnataka 2nd PUC Model Question Paper 2025 for Maths

ACADEMIC YEAR

2025

Karnataka
Board
Model Paper

BLUEPRINT FOR MODEL QUESTION PAPER-1
SUBJECT:MATHEMAMATICS(35)
CLASS : IIPUC :2024-2025
CHAPTER/ NO REMEMBER UNDERSTAND APPLY CREATE/EVALUATE
SL. DOMAIN/ OF M
MARKS
PART-A PART-B PART-C PART-D PART-E PART-A PART-B PART-C PART-D PART-E PART-A PART-B PART-C PART-D PART-E PART-A PART-B PART-C PART-D PART-E TOTAL
NO T. A
HOU R 1 1 2 3 5 4 1 1 2 3 5 4 1 1 2 3 5 6 4 1 1 2 3 5 6 4
RS K MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK MARK
MARK
MARK MARK MARK MARK MARK MARK M
MCQ FB SA SA LA LA MCQ FB SA SA LA LA MCQ FB SA SA LA LA MCQ FB SA SA LA LA AR
S LA K
LA

RELATIO 9 1 9
1 NS AND 9 1 1
FUNCTI
ONS

INVERSE
6 1 1 6
2 TRIGONO
METRIC
6 1
FUNCTIO
NS

9 1 1 1 9
3 MATRICES 9
12 1 1 1 1 12
4 DETERMINANTS 12
CONTINUITY 17 1 1 1 1 17
5 ANDDIF 20 1 1 1
FERENTIABIL
ITY

APLLICA 8 1 1 1 8
6 TION
10 1 1
OFDERIV
ATIVES
18 1 1 1 18
7 INTEGRALS 22 1 1 1
APPICA 5 1 5
8 TION 5
OFINTE
GRATIO
N
DIFFER 8 1 1 8
9 ENTIAL 10 1
EQUATI
ONS

8 1 1 1 8
10 VECTORALGEBRA 11 1 1
THREEDIMEN 6 1 1 6
11 SIONAL 8 1
GEOMETRY

6 6
12 LINEARPROGRAM
MING
7 1
8 1 1 1 1 8
13 PROBABILITY 11 1

TOTAL 140 120 5 2 0 0 6 0 8 1 6 6 1 1 0 2 3 2 0 1 1 3 0 0 1 0 1 120

DESIGN OF THE QUESTION PAPER
SECOND PUC: MATHEMATICS(35): 2024-25

Pattern of the Question Paper:
Part Type of Number of Number of TOTAL Remarks
questions questions questions MARKS
to be set to be
answered

A 1 mark 15+5=20 15+5=20 20(20) Compulsory
Questions part(MCQ+FB)

B 2 marks 9 6 9× 2=18
Questions (6× 2=12)

C 3marks 9 6 9× 3=27
Questions (6× 3=18)

D 5marks 7 4 7× 5=35 Questions must
Questions (4× 5=20) be asked from
7 specific topics

E 6 and 4 2+2 1+1 6× 2=12 Questions must
marks (6× 1=6) be asked from
Questions 4× 2=8 4 specific topics
(Both (4× 1=4)
Internal
choice)
The weightage marks distribution across different
dimensions shall be as follows:

A. Weightage to Objectives.

Objective Weightage Marks

Remember 40% 36/120

Understand 30% 48/120

Apply 20% 24/120
Analyse/Create/Evaluate 10% 12/120

B.Weightage/marks across difficulty level:
Level Weightage Marks
Easy 40% 48/120
Average 40% 48/120
Difficult 20% 24/120
C: Weightage Framework:
Chapter No. Chapter No. of Marks
teaching
Hours

1 RELATIONS AND FUNCTIONS 9 9

2 INVERSE TRIGONOMETRIC FUNCTIONS 6 6

3 MATRICES 9 9

4 DETERMINANTS 12 12

5 CONTINUITY AND DIFFERENTIABILITY 20 17

6 APPLICATION OF DERIVATIVES 10 8

7 INTEGRALS 22 18

8 APPLICATION OF INTEGRALS 5 5

9 DIFFERENTIAL EQUATIONS 10 8

10 VECTORS 11 8

11 THREE D GEOMETRY 8 6

12 LINEAR PROGRAMMING 7 6

13 PROBABILITY 11 8

TOTAL 140 120

Quality & Length of the Question Paper
1.While framing a question, the time required to solve it should be decided

properly and the marks should be awarded accordingly.

2.Marks allotted for each question should be properly mentioned in the question

paper.

General Instructions:
1.Question paper should be prepared by preparing separate blueprint by keeping the

weightage of marks allotted to each chapter in mind.

2. Weightage allotted to each topic cannot be changed but the question setter has the

liberty to choose the question type as per instructions given.

4. Miscellaneous worked examples and exercise problems can also be included in the

Question paper.

5. Question order in the question paper need not to be in accordance with the chapters

in the textbook.

6. The problems that are based on the concepts discussed in the book [prescribed by

the Department of School Education (Pre-university)] can be asked. However, this

does not mean that problems should be given as they appear in the textbook.

7.No question should be asked from the historical notes and appendices given in the

textbook.

8.Questions should not be split into subdivisions.

9.Questions should be clear, unambiguous, understandable and all unwanted data in

the questions should be avoided.

10.Instructions to use graph sheet in linear programming problem should be given in

the question paper.

11.Repetition of the same concepts, laws, facts etc. which generate the

same concept in different parts of the question paper should be avoided.

12. In MCQ section, stimulus questions, comprehensive, identifying the true or false
statements should be asked but then should not exceed three questions.

13. Six numerical options should be given for the five Fill in the blanks questions.

14.Questions for Part D should be given from the following 7 specific topics only.

i). RELATIONS AND FUNCTIONS
ii). MATRICES
iii). DETERMINANTS
iv). CONTINUITY AND DIFFERENTIABILITY
v). INTEGRALS
vi). APPLICATION OF INTEGRALS
vii). DIFFERENTIAL EQUATIONS

15.Questions for Part E should be given from the following 4 specific topics only.

(Both Six marks and Four marks questions should have internal choice.)

i). INTEGRALS
ii). LINEAR PROGRAMMING
iii). CONTINUITY AND DIFFERENTIABILITY
iv). DETERMINANTS
*************

GOVERNMENT OF KARNATAKA
KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD
WEIGHTAGE FRAMEWORK FOR MQP 1: II PUC MATHEMATICS(35):2024-25
Chapter

Number PART PART PART
of PART A B C D
CONTENT PART E Total
Teaching 1 mark 2 3 5
hours mark mark mark
6 4
MCQ FB
mark mark
RELATIONS AND 9 9
1 FUNCTIONS 1 1 1

INVERSE
2 TRIGONOMETRIC 1 1 1
FUNCTIONS 6 6

MATRICES 9 9
3 1 1 1
DETERMINANTS 12 12
4 1 1 1 1
CONTINUITY AND 20 17
5 DIFFERENTIABILITY 2 1 1 1 1 1
APPLICATION OF 10 8
6 2 1 1 1
DERIVATIVES
INTEGRALS 22 18
7 2 1 1 1 1
APPLICATION OF 5 5
8 INTEGRALS 1
DIFFERENTIAL 10 8
9 1 1 1
EQUATIONS
VECTOR ALGEBRA 11 8
10 2 1 1 1
THREE D 8 6
11 1 1 1
GEOMETRY
LINEAR 7 6
12 ROGRAMMING 1
PROBABILITY 11 8
13 2 1 1 1

TOTAL 140 120
15 5 9 9 7 2 2

GOVERNMENT OF KARNATAKA
KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD
Model Question Paper -1
II P.U.C : MATHEMATICS (35): 2024-25
Time : 3 hours Max. Marks : 80

Instructions :

1) The question paper has five parts namely A, B, C, D and E. Answer all the
parts.

2) PART A has 15 MCQ’s ,5 Fill in the blanks of 1 mark each.

3) Use the graph sheet for question on linear programming in PART E.

4) For questions having figure/graph, alternate questions are given at the end of
question paper in separate section for visually challenged students.

PART A

I. Answer ALL the Multiple Choice Questions 151 = 15

1. Let the relation R in the set A = { x ∈ Z: 0 ≤ x ≤ 12}, given by
R={(a, b):|a-b| is multiple of 4}, then [3], the equivalence class containing 3 is
A) {1,5,9} B) 𝜙 C)A D) {3, 7,11}
2. If cot −1 𝑥= y, then
𝜋 𝜋 𝜋 𝜋
(A) 0 ≤ 𝑦 ≤ 𝜋 (B) 0 < 𝑦 < 𝜋 (C) − 2 ≤ 𝑦 ≤ 2 (D ) - 2 < 𝑦 < 2 .
3. If 𝐴 = [𝑎𝑖𝑗 ] is a symmetric matrix of order 𝑚 × 𝑛 then
(A) m=n and 𝑎𝑖𝑗 =0 for i=j B) m=n and 𝑎𝑖𝑗 =𝑎𝑗𝑖 for all i,j
(C) 𝑎𝑖𝑗 =𝑎𝑗𝑖 for all i,j D) m=n and 𝑎𝑖𝑗 =−𝑎𝑗𝑖 for all i,j
3 𝑥 3 2
4. If | |=| | then the value of x is equal to
𝑥 1 4 1
A) 2 B) 4 C) 8 D) ±2√2.

5. Statement 1: Left hand derivative of f(x) = | x | at x = 0 is -1.
Statement 2: Left hand derivative of f(x) at x = a is lim f (a − h)
h →0

A) Statement 1 is true, and Statement 2 is false.
B) Statement 1 is true, and Statement 2 is true, Statement 2 is correct
Explanation for Statement 1
C) Statement 1 is true, and Statement 2 is true, Statement 2 is not a correct
Explanation for Statement 1
D) Statement 1 is false, and Statement 2 is false.

6. The derivative of log(secx+tanx) with respect to x is
1
A)secx B) tanx C) secx.tanx D) 𝑠𝑒𝑐𝑥+𝑡𝑎𝑛𝑥

7. The absolute maximum value of the function f given by
f (x) = 𝑥 3 , x ∈ [-2, 2] is
A)-2 B)2 C)0 D)8

8. The point of inflection for the following graph is

𝜋 𝜋
A)− 2 B) 2 C) 0 D) point of inflection does not exist

1 1
9. ∫ 𝑒 𝑥 (𝑥 − 𝑥 2 ) 𝑑𝑥 =

𝑒𝑥 𝑒𝑥 –𝑒 𝑥
A) 𝑒 𝑥 + 𝑐 B) +𝑐 (C) +𝑐 (D) +𝑐
𝑥2 𝑥 𝑥

10.  x sin xdx =

A) −𝑥𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 + 𝑐 B) 𝑥𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 + 𝑐

C) 𝑥𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 + 𝑐 D) −𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 + 𝑐
𝜋
11. The projection vector of the vector ⃗⃗⃗⃗⃗
AB on the directed line l, if angle 𝜃 = 2 will

be.

A) 𝑍𝑒𝑟𝑜 vector. ⃗⃗⃗⃗⃗
B) 𝐴𝐵 ⃗⃗⃗⃗⃗
C) 𝐵𝐴 D) 𝑈𝑛𝑖𝑡 vector.

12. For the given figure, 𝑃⃗ − 𝑄
⃗ 𝑖𝑠

⃗⃗⃗⃗⃗
A) 𝑂𝐶 ⃗⃗⃗⃗⃗
B) 𝐶𝑂 C) ⃗⃗⃗⃗⃗
𝐵𝐴 D) ⃗⃗⃗⃗⃗
𝐴𝐵

13. The direction cosines of negative z-axis.
(A) -1 , -1 , 0 (B) 0 , 0 , −1 (C) 0 , 0 , 1 (D) 1 , 1 , 0
1
14. If P(A) = 2, P (B) = 0, then P (A|B) is
1
A) 0 B) 2 C) 1 D) not defined

15. An urn contains 10 black and 5 white balls, 2 balls are drawn

one after the other without replacement, then the probability that both drawn

balls are black is
3 4 2 2
A) B) C) D)
7 9 3 9

II. Fill in the blanks by choosing the appropriate answer from those
given in the bracket (0, 1, 2, 3, 4, 5) 51=5

16. The number of points in R for which the function f(x) = |x|+ |x + 1| is
not differentiable, is_________
17. The value of 𝑖̂. (𝑗̂ × 𝑘̂) − 𝑗̂. (𝑘̂ × 𝑖̂) − 𝑘̂ . (𝑗̂ × 𝑖̂) is________
18. The sum of the order and degree of the differential equation.
𝑑2 𝑦 𝑑𝑦
2𝑥 2 (𝑑𝑥 2 ) − 3 (𝑑𝑥 ) + 𝑦 𝑖𝑠_______

19.The total revenue in rupees received from the sale of x unit of a
product is given by R(x)=2𝑥 2 − 4𝑥 + 5 ,The marginal revenue when x=2
is________
3 2 2
20. If P(A) = 𝑘, P(A∩ B) = 5 and P(B|A) = 3 , then k is _________

PART B

Answer any SIX questions: 6  2 = 12
1 1
21. Show that sin−1 (2x√1 − x 2 ) = 2sin−1 (x), − ≤ x ≤ .
√2 √2
22. Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear
using determinants.
dy
23. Find , if 2 x + 3 y = sin x .
dx
24. Find the local maximum value of the function g ( x) = x 3 − 3x 

25. Evaluate  sin 3x cos 4 x dx 

𝑦𝑑𝑥−𝑥𝑑𝑦
26. Find the general solution of the differential equation = 0.
𝑦

27. Find |𝑥|, if for a unit vector 𝑎, (𝑥 − 𝑎 ) ∙ (𝑥 + 𝑎) = 15.

28. Find the equation of the line in vector form that passes through
the point with position vector 2 𝑖̂ − 𝑗̂ + 4 𝑘̂ and is in the direction 𝑖̂ + 2 𝑗̂ − 𝑘̂.
29. Prove that if E and F are independent events, then so are the events E and 𝐹′.
PART C

Answer any SIX questions: 63 = 18.

30. Show that the relation R in the set of real numbers R defined as
R = ( a, b ) : a  b ,is reflexive and transitive but not symmetric.
4 12 33
31. Prove that cos−1 5 + cos −1 13 = cos−1 65 .

3 5 
32. Express   as the sum of a symmetric and a skew symmetric matrix.
1 −1
𝑑𝑦
33. Find 𝑑𝑥 if 𝑥 = 𝑎(cos 𝜃 + 𝜃 sin 𝜃) and y = 𝑎(sin 𝜃 − 𝜃 cos 𝜃).

34. Find the intervals in which the function f(x)=(x-2)3(x+4)3 is
a) increasing b)decreasing.
x
35. Find  dx 
( x + 1) ( x + 2)

36. If 𝑎, 𝑏⃗ & 𝑐 are three vectors such that |𝑎| = 3, |𝑏⃗| = 4, |𝑐 | = 5 and each
vector is orthogonal to sum of the other two vectors then find |𝑎 + 𝑏⃗ + 𝑐|.
37. Find the distance between the lines 𝑟 = 6𝑖̂ + 2 𝑗̂ + 2 𝑘̂ + 𝜆 (𝑖̂ - 2 𝑗̂ +2 𝑘̂)
and 𝑟 = -4𝑖̂ - 𝑘̂+ 𝜇 (3𝑖
̂ -2 𝑗̂ -2 𝑘̂).
38. Bag I contains 4 Red and 4 Black balls, Bag II contains 2 Red and 6 Black
balls .One bag is selected at random and a ball is drawn is found to be Red.
What is the probability that bag I is selected?
PART D
Answer any FOUR questions: 5 4 = 20.
39. State whether the function f: R →R defined by f (x) = 3 – 4x
is one-one, onto or bijective. Justify your answer.

1 0 2 
40. If A =  0 2 1  , prove that A3 − 6 A2 + 7 A + 2 I = O .
 2 0 3 
41. Solve the following system of equations by matrix method:
2x + y -z =1; x + y =z and 2x + 3y + z = 11.
𝑑2 𝑦 𝑑𝑦
42. If 𝑦 = 3 cos(𝑙𝑜𝑔𝑥) + 4 sin (𝑙𝑜𝑔𝑥), prove that 𝑥 2 𝑑𝑥 2 + 𝑥 𝑑𝑥 + 𝑦 = 0.
dx
with respect to x and evaluate 
1
43. Find the integral of .
a −x
2 2
7 − x2

𝑑𝑦
44. Solve the differential equation 𝑑𝑥 + y secx = tanx (0≤ 𝑥 ≤ 𝜋/2).

45. Find the area of the circle 𝑥 2 + 𝑦 2 = 𝑎2 by the method of integration.

PART E
Answer the following questions:
46. Maximize and Minimise ; z = 3x + 9 y subject to constraints
x + 3 y  60, x + y  10, x  y, x  0, y  0 by graphical method.
OR

b b 3
1
Prove that  f ( x) dx =  f (a + b − x) dx and hence evaluate  dx . 6
a a  1 + tan x
6

kx + 1, if x  5
47. Find the value of k so that the function f ( x ) =  , at x = 5 is a
3x − 5, if x  5

continuous function.
OR

2 3 1 −2
If A =[ ] and B =[ ] then verify that (𝐴𝐵)−1 = 𝐵 −1 𝐴−1.
1 −4 −1 3
4
PART F
(For Visually Challenged Students only)
8. The point of inflection of the function f(x)=sinx in the interval
  
 − 2 , 2  is
 
𝜋 𝜋
A)− 2 B) 2 C) 0 D) point of inflection does not exist

12. In a parallelogram OACB, ⃗⃗⃗⃗⃗
𝑂𝐴 = 𝑃⃗ ⃗⃗⃗⃗⃗ = 𝑄
and 𝑂𝐵 ⃗ , then 𝑃⃗ − 𝑄
⃗ is

⃗⃗⃗⃗⃗
A) 𝑂𝐶 ⃗⃗⃗⃗⃗
B) 𝐶𝑂 ⃗⃗⃗⃗⃗
C) 𝐵𝐴 ⃗⃗⃗⃗⃗
D) 𝐴𝐵

*******

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