Karnataka 2nd PUC Model Question Paper 2024 for Maths

MODEL
QUESTION PAPER
II PUC

2024

SECOND PUC MODEL QUESTION PAPER 2023-2024
SUBJECT : MATHEMATICS ( 35 )
TIME : 3 Hours 15 Minutes [Total questions : 52 ] 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 multiple choice questions, 5 fill in the blank
questions.
3. Use the graph sheet for question on linear programming
problem in Part E.
PART -A
I. Answer all the multiple choice questions : 15 x 1 = 15

1. The relation R in the set { 1,2,3 } given by { (1,2) ,(2,1) } is
a) reflexive b) symmetric
c) transitive d) equivalence relation
2. If f : R → R be defined as f(x) = 𝑥 ,then the function f is
4

a) one-one and onto b) many-oneandonto
c) one-one but not onto d ) neither one-one nor onto
3. The principal value branch of cot 𝑥 is
−1
𝜋 𝜋 𝜋 𝜋
a) − 2 , 2 b) − 2 , 2
c) 0 , 𝜋 d) 0 , 𝜋
4. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is
a) 27 b) 18 c) 81 d) 512
5. Let A be a nonsigular matrix of order 3 x 3 and | adj A|= 25, thena possible value
of |A| is
a) 625 b) 25 c) 5 d) 125
6. Which of the following x belongs to domain of the greatest integer function f(x
) = 𝑥 , 0 < x < 3 is not differentiable
a) 2 and 3 b) 1 and 2 c) 0 and 2 d) 1 and 3
dy
7. If y = log 7 2𝑥, then dx is
1 1
1) 𝑥 𝑙𝑜𝑔 7 b) 7 𝑙𝑜𝑔𝑥
𝑙𝑜𝑔𝑥 7
c) 7 d)𝑙𝑜𝑔𝑥
8. The point of inflection of the function y = 𝑥 3 is
a) ( 2, 8 ) b) ( 1, 1) c) ( 0, 0 ) d) ( –3, -27 )
9. 𝑠𝑖𝑛2𝑥 dx is
𝑠𝑖𝑛 2𝑥 𝑐𝑜𝑠 2𝑥
a) – +c b) – +c
2 2
𝑐𝑜𝑠 2𝑥 𝑠𝑖𝑛 2𝑥
c) +c d) +c

2 2
1 1
10. 𝑒𝑥 – 𝑥 2 dx is
𝑥
1 1 1 1
a)𝑒 –𝑥 +c b) 𝑒 –𝑥 + c c) 𝑒 𝑥 +c d) 𝑒 𝑥 +c
𝑥 𝑥2 𝑥 𝑥2

11. If θ is the angle between any two vectors 𝑎and𝑏, then𝑎 . 𝑏 = 𝑎 𝑥 𝑏 ,
when tanθis equal to,
1
a) 1 b) 3 c) 3 d) 0
12. Unit vector in the direction of 𝑎 = 2 𝑖 + 3 𝑗 + 𝑘 is
2𝑖 +3𝑗 + 𝑘 2𝑖–3𝑗 + 𝑘
a) b)
14 14
2𝑖 +3𝑗 + 𝑘 2𝑖 +3𝑗 – 𝑘
c) d)
14 14
1 3
13. If the direction cosines l,m,n of a line are 0, 2 , then the angle made by the
2
line with the positive direction of y – axis is
a) 600 b) 300 c) 900 d) 450
14. In a Linear programming problem , the objective function is always
a) a cubic function b) a quadratic function
c) a linear function d) a constant function
15. If A and B are two non empty events such that P 𝐴/𝐵 = P 𝐵/𝐴 and P( A∩B ) ≠∅
then
a) A ⊂ B but A ≠ B b) A = B
c) B⊂ A but A≠ B d) P(A) = P(B)

II. Fill in the blanks by choosing the appropriate answer from those given in the
bracket 5 x 1= 5
1 1
0, 1, 4, , 7,
36 6
𝜋 1
16. The value of sin 3 – sin−1 – 2 is ––––––––––––––––––––––––––––––
17. A square matrix A is a singular matrix if |A| is ––––––––––––––––––––––––––
𝑑4𝑦
18. The order of the differential equation 𝑑𝑥 4 + sin 𝑦 𝐼𝐼𝐼 = 0 is –––––––––––––––
𝑥 –5 𝑦 +2 𝑧 𝑥 𝑦 𝑧
19. The lines = = 1 and 1 = 2 =3 are perpendicular, then k is ––––––––––
𝑘 –5
20. The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is ______________________________

PART –B
Answer any six questions 6 x 2 =12
3 24
21. Prove that 2 sin−1 5 = tan−1 7
22. Find the equation of line joining ( 1, 2 ) , ( 3, 6 ) using determinant method
dy
23. Find dx , if y + siny = cosx
24. Find the rate of change of the area of a circle with respect to its radius r
when r = 3 cm

25. Find the local minimum value of the function f given by f(x) = 3 + |x| , x ∈ R
𝑑𝑥
26. Find (𝑥 +1)(𝑥 +2)
𝜋
𝑥 𝑥
27. Evaluate 0
2 𝑠𝑖𝑛2 2 – 𝑐𝑜𝑠 2 2 𝑑𝑥
28. Find the projection of the vector 𝑎 = 2 𝑖 + 3 𝑗 + 2𝑘 on the vector 𝑏 =𝑖 +2 𝑗 + 𝑘

29. Find the angle between the pair of lines given by
𝑟 = 3 𝑖 + 2 𝑗 – 4𝑘+ ( 𝑖 + 2 𝑗 + 2𝑘) and 𝑟 = 5 𝑖 –2 𝑗 +𝜇 ( 3 𝑖 + 2 𝑗 + 6 𝑘 )
30. A fair die is rolled. Consider events E = {1, 3, 5 } , F = { 2, 3 } , find P (E/F )
1 1 1
31. If A and B two events such that P(A) = 4 , P(B) =2 and P (A ∩ B) = 8 ,
find P (not A and not B)

PART – C

Answer any six questions 6 x 3 =18

32. Show that the relation R in the set A = {1,2,3,4,5} given by
R = {(a, b): |a − b| is even } is an equivalence relation
1 + 𝑥2– 1
33. Write in the simplest form tan−1 , x≠0
𝑥
3 5
34. Express A = as the sum of a symmetric anda skew symmetric matrix.
1 −1
35. Differentiate 𝑠𝑖𝑛2 𝑥 with respect to 𝑒 𝑐𝑜𝑠𝑥
36. Differentiate 𝑥 𝑠𝑖𝑛𝑥 , x > 0 with respect to x
37. Find the interval in which the function f(x) = 10 – 6x – 2x 2 is strictly increasing
38. Find 𝑥 sin−1 𝑥 dx
39. Find the equation of curve passing through the point ( –2, 3) , given that the slope
2𝑥
of the tangent to the curve at any point ( x, y ) is𝑦 2
40. Show that the position vector of the point P, which divides the line joining the
𝑚 𝑏 + 𝑛𝑎
points A and B having position vectors 𝑎and𝑏internally in the ratio m: n is 𝑚 + 𝑛
41. Find a unit vector perpendicular to each of the vectors 𝑎 + 𝑏 and (𝑎 – 𝑏) ,
where 𝑎 = 3 𝑖 + 2 𝑗 + 2𝑘 and 𝑏 = 𝑖 + 2 𝑗 – 2𝑘
42. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black
balls. One of the two bags is selected at random and a ball is drawn at random
from the bag and it is found to be red .Find the probability that the ballis drawn
from first bag ?

PART – D

Answer any four questions 4 x 5 = 20

43. Let f : N→ Y be a function defined as f(x) = 4x + 3, where Y = {y ∈ N ∶ y = 4x + 3 for
some x ∈ N}.Show that f is invertible. Find the inverse of f.
0 6 7 0 1 1 2
44. If A = – 6 0 8 B= 1 0 2 and C = – 2 then calculateAC, BC and
7 –8 0 1 2 0 3
(A + B )C. Also verify (A + B) C = AC + B C
45. Solve the system of linear equations by matrix method

2x – 3y + 5z = 11 , 3x +2y –4z = –5 , x + y –2z = – 3
46. If y = 3 cos(logx) + 4 sin (logx) , show that x 2 y2 + xy1 + y = 0
1 dx
47. Find the integral of x 2 −a 2 with respect to x and hence evaluate x 2 −16
𝑥2 𝑦2
48. Find the area of the region bounded by the ellipse + = 1 using integration.
16 9

49. Find the general solution of the differential equation
dy
x dx + 2y = x 2 logx , ( x ≠ 0 )
50. Derive the equation of a line in space through a given point and
parallel to a vector both in the vector and Cartesian form

PART – E

Answer the following questions

a
a 2 0 f x dx , if f x is an even function
51. P.T. −a
f x dx =
0 if f x is an odd function
π
and hence evaluate 2
π sin7 xdx 6
−
2
OR

Solve the following linear programming problem graphically
Minimise Z = 200x + 500y ,
subject to the constraints : x + 2y ≥ 10, 3x + 4y ≤ 24, x ≥ 0, y≥0

52. Show that the matrix A =
2
1
3
2
satisfies the equation 𝐴2 – 4A + I = o,
where I is 2 x 2 identity matrix and o is 2 x 2 zero matrix.
Using this equation, find 𝐴–1 . 4

OR
𝑘 𝑐𝑜𝑠𝑥 π
if x ≠ 2
𝜋 – 2𝑥
Find the value of k so that the function f(x) = 𝜋
3 if x = 2
π
is continuous at x = 2

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