Karnataka 2nd PUC Model Question Paper 2026 for Mathematics

SAMPLE PAPER

Karnataka
Board
Model Paper

GOVERNMENT OF KARNATAKA
KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD
Model Question Paper - 1
II P.U.C: MATHEMATICS (35): 2025-26
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. Consider the following equivalence relation R on Z , the set of integers
R = ( a,b ) |2 divides a - b . If [x] represents the equivalence class of x, then [0] is the set
A) 0, 2, 4, 6, 8,........ B) 1, 3, 5, 7 , 9,........
C) 0, 1, 2, 3, 4,........ D) ....... − 4, −1, 2,5,8.......
2. If sin x = θ ( the principal value branch of sin x where 0  x  1 , then the range in
-1 -1

which θ lies
    
A)  − ,  B) 0 ,  C)  0,  D) −1  x  1
 2 2  2
3. The product of matrices A and B is equal to a diagonal matrix. If the order of the
matrix B is 2x3, then order of the matrix A is
A) 3x3 B) 2x2 C) 3x2 D) 2x3
5 0 
4. Let A be a 2x2 square matrix such that A =   then value of adjA is
0 5 
A) 25 B) 5 C) 0 D) I
dy
5. For some fixed a>0 and x>0, y = a + x + a then find
x a a
=
dx
A) a x loge a + a x ( a − 1) + aa a −1
B) a x loge a + x a −1
C) a x loge a + ax a −1
D) a x loge a + x a log x + a a log a
6. Choose the statement that is not true from the options given below:
A) Every polynomial function is continuous.
B) Every rational function is continuous.
C) Every differentiable function is continuous.
D) Every continuous function is differentiable.

SUBJECT: 35 – MATHEMATICS Page 1 of 5

7. Let C be the circumference and A be the area of a circle
Statement 1: The rate of change of the area with respective to radius is equal to C
Statement 2: The rate of change of the area with respective to diameter is C/2
A) Only Statement 1 is true B) Only statement 2 is true
C) Both statements are true D) Both statements are false
8. Function f ( x ) = a is increasing on R, if
x

A) a>0 B) a<0 C)0<a<1 D) a>1
1
9. The anti-derivative of , x > 1 with respective to x
x x2 - 1
A) sin −1 x + C B) cos−1 x + C C) − cos ec −1 x + C D) cot −1 x + C
1
10. Find the value of  x99 dx = _______
−1

A) 2 B) 3 C) 0 D) 1

11. For the given figure, AC is
A


−2a

B  C
3b

       
A) 2a − 3b B) 3b − 2a C) a + b D) 2a + b
12. The direction ratios of the vectors joining the points P ( 2, 3, 0 ) & Q ( -1, -2, 4 ) directed
form P to Q are
A) ( −3, −5, 4 ) B) ( −3, −5, −4 ) C) ( −1, −2, −4 ) D) (1,1,1)
x-5 y +4 z-6
13. The cartesian equation of a line is = = then the vector equation is
3 7 2
 
( ) ( ) ( ) ( )
   
A) r = −5i + 4 j − 6k +  3i + 7 j + 2k B) r = 5i + 4 j − 6k +  3i + 7 j + 2k
 
( ) ( ) ( ) ( )
   
C) r = 5i − 4 j + 6k +  3i + 7 j + 2k D) r = 3i + 7 j + 2k +  5i − 4 j + 6k
14. Two cards are drawn at random and without replacement from a pack of 52 playing
cards. Find the probability that both the cards are black is
1 1 25 25
A) B) C) D)
26 4 102 104
1 3
15. A and B are two events such that P ( A ) = , P ( A  B ) = , P ( B ) = q then the value of q
2 5
if A and B are mutually exclusive events
3 1 1 7
A) B) C) D)
10 10 5 10

SUBJECT: 35 – MATHEMATICS Page 2 of 5

II. Fill in the blanks by choosing the appropriate answer from those given in the
bracket (-1, 6, 0, 1, 4, 3) 51=5
dy
16. If xy = 81 , then at x = 9 is _______
dx
17. The absolute maximum value of the function f given by f ( x ) = x 2 , x   0, 2 is _________
18. If m and n respectively are the order and degree of the differential equation
3 2
 dy   d2 y 
1 +   = 7  2  then m − n =_____
 dx   dx 
 
19. The value of  for which the vectors 2i - 3j + 4k & - 4i + λj - 8k are collinear is ______
S
20. If F be an event of a sample space S, then P   = _____
F
PART B
III. Answer Any Six Questions: 6  2 = 12
  1 
21. Find the value of tan −1  2 cos  2 sin −1  
  2 
22. Find the area of the triangle whose, vertices are ( 3,8) ,( −4, 2 ) & ( 5,1) using determinants
dy  2x 
23. Find , if y = sin −1  2 
, -1 < x < 1.
dx  1+ x 
24. Find the interval in which of the function f given by f ( x ) = x 2 + 2 x − 5 is strictly
increasing
x sin −1 x
25. Find  dx
1 − x2
dy 2 x
26. Find the general solution of the differential equation =
dx y 2
27. Find the area of the parallelogram whose adjacent sides are given by the vectors
  
a = 3i + j + 4k & b = i − j + k
x + 3 y −1 z + 3 x + 1 y − 4 z − 5
28. Find the angle between the pair of lines given by = = & = =
3 5 4 1 1 2
29. A die is thrown. If E is the event the number appearing is a multiple of ‘3’ and F be the
event the number appearing is ‘even’ then find whether E & F are independent?

PART C
IV. Answer Any Six Questions: 6  3 = 18
30. Let L be the set of all lines in a plane and R be the relation in L defined as
R = ( L1 ,L2 ) : L1 is perpendicular to L2  . Show that R is symmetric but neither reflexive nor
transitive.
 1 + x2 −1 
−1
31. Find the simplest form of tan   ,x  0
 x 
 

SUBJECT: 35 – MATHEMATICS Page 3 of 5

1 4 
32. Express the matrix A =   as the sum of symmetric and skew symmetric matrix
6 7 
33. Differentiate, y = ( sin x ) + sin −1 x w.r.t.x
x

34. Find the positive numbers whose sum is 15 and the sum of whose squares is
minimum
x
35. Integrate with respect to x by partial fraction
( x + 1)( x + 2 )
  
36. If a,b & c are three vectors such that a = 3, b = 4 & c = 5 and each vector is orthogonal
 
to sum of the other two vectors then find a + b + c .
37. Derive the equation of the line in space passing through the point and parallel to the
vector in the vector form.
38. There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls.
Box-II contains 5 red and 5 black balls, one of the two boxes I selected at random and
a ball is drawn from the box which is found to be red. Find the probability that the red
ball comes out from the box-II

PART D
V. Answer Any Four Questions: 5  4 = 20
39. State weather the function f : R → R defined by f ( x ) = 3 − 4 x is one-one, onto or
bijective. Justify your answer
 0 6 7 0 1 1  2
   
40. If A = −6 0 8 , B = 1 0 2 , C =  −2
     
 7 −8 0  1 2 0   3 
Calculate AC, BC and (A + B) C. Also, verify that (A + B) C = AC + BC.
41. Solve the system of linear equations by matrix method
4 x + 3 y + 2 z = 60, 2 x + 4 y + 6 z = 90 & 6 x + 2 y + 3z = 70
d2y dy
42. If y = 3 e + 2 e , then prove that
2x 3x
2
− 5 + 6 y = 0.
dx dx
1 dx
43. Find the integral of with respect to x and hence evaluate 
a −x2 2
25 − x 2
x2 y 2
44. Find the area of the ellipse + = 1 by the method of integration
a 2 b2
dy
45. Find the general solution of the differential equation + 2 y = sin x
dx

SUBJECT: 35 – MATHEMATICS Page 4 of 5

PART E
VI. Answer The Following Questions:
a a 4
x
46. Prove that  f ( x ) dx =  f ( a − x ) dx and hence evaluate  dx 6M
0 0 0 x + 4− x
OR
Solve the following graphically, maximise Z = 250 x + 75 y subject to the constraints
x + y  60, 25 x + 5 y  500, x  0, y  0
5 6
47. If A =   , satisfies the equation A2 − 8 A − 9 I = O where I is 2 x 2 identity matrix and
 4 3
O is 2 x 2 zero matrix. Using this equation, find A−1 . 4M
OR
kx + 1, if x  5
Find the value of k so that the function f ( x ) =  is a continuous at x = 5
3x − 5, if x  5

PART F
(For Visually Challenged Students only)
  
11. In a ABC , BA = 2a , BC = 3b then AC is
       
A) 2a − 3b B) 3b − 2a C) a + b D) 2a + b

******

SUBJECT: 35 – MATHEMATICS Page 5 of 5

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