Karnataka 2nd PUC Model Question Paper 2022 for Mathematics

DEPARTMENT OF PRE-UNIVERSITY EDUCATION

MODEL QUESTION PAPER FOR ANNUAL EXAMINATION APRIL-2022

II PUC

SUB: MATHEMATICS (35)

TIME: 3 Hours 15 MinutesMAX. MARKS: 100

Instructions :

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

(ii) Use the graph sheet for the question on Linear programming in PART E.

PART – A

Answer any TEN questions 10 X1=10

1. Give an example of a relation which is symmetric and transitive but not reflexive.

2. Define a binary operation.

3. Find the principal value of cos .

4. If sin sin + cos 𝑥 = 1 , then find the value of x.

5. Define a row matrix.

𝑥 2 6 2
6. Find the value of x if = .
18 𝑥 18 6

7. If 𝑦 = 𝑒 , 𝑓𝑖𝑛𝑑 .

8.If 𝑦 = 𝑠𝑖𝑛(𝑥 + 5), 𝑓𝑖𝑛𝑑 .

9.Find ∫(2𝑥 + 𝑒 )𝑑𝑥.

10.Evaluate ∫ 𝑑𝑥.

11.Find the unit vector in the direction of vector 𝑎⃗ = 2𝚤̂ + 3𝚥̂ + 𝑘.

12.Write two different vectors having same magnitude.

13.Write the direction cosines of x-axis.

14.Define feasible region of a linear programming problem.

15.Find P(A|B), if P(B) = 0.5 and P(A∩B) = 0.32.

PART- B

Answer any TEN questions 10 X2=20

1 𝑖𝑓 𝑥 > 0
16. Show that the signum function f:RR given by 𝑓(𝑥) = 0 𝑖𝑓 𝑥 = 0
−1 𝑖𝑓 𝑥 < 0

is neither one-one nor onto.

17.Find the value of tan √3 − sec (−2).

18.Write the domain and range of y =tan 𝑥.

4 3 𝑦 𝑧
19. Find the values of x, y and z from the equation = .
𝑥 5 1 5

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

21. If 𝑥 + 𝑥𝑦 + 𝑦 = 100, find .

22. If 𝑥 = 𝑎𝑡 , 𝑦 = 2𝑎𝑡, 𝑓𝑖𝑛𝑑 .

23. Differentiate sin 𝑐𝑜𝑠(𝑥 ) with respect to x.

24. Find the slope of tangent to curve 𝑦 = 𝑥 − 𝑥 + 1 at the point whose x-co-ordinate is 2.

( )
25.Find ∫ dx.

26.Find ∫ dx.

27.Evaluate ∫ 𝑐𝑜𝑠 𝑥 𝑑𝑥 .

28.Find the order and degree of the differential equation y1 + y = ex.

29.Find the projection of the vector 𝑎⃗ = 2𝚤̂ + 3𝚥̂ + 2𝑘 on the vector 𝑏⃗ = 𝚤̂ + 2𝚥̂ + 𝑘.

30.Find the area of the parallelogram whose adjacent sides are given by the vectors

𝑎⃗ = 𝚤̂ − 𝚥̂ + 3𝑘 and 𝑏⃗ = 2𝚤̂ - 7𝚥̂ + 𝑘.

31.Find the intercepts cut-off by the plane 2x+y-z=5.

32.Find the distance of the point (-6,0,0) from the plane 2x – 3y + 6z -2 = 0

33.The random variable X has a probability distribution P(X) of the following form where k is

some number. Find the value of k.

𝑘 , 𝑖𝑓 𝑥 = 0
2𝑘, 𝑖𝑓 𝑥 = 1
P(X) =
3𝑘 , 𝑖𝑓 𝑥 = 2
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

PART – C

Answer any TEN questions 10 X3=30

34. Show that the relation R in the set {1, 2, 3} given by R = { (1, 1), (2, 2), (3, 3), (1, 2), (2,

3)}

is reflexive but neither symmetric nor transitive.

35.Prove that 2 tan + tan = tan .

2 3
36. Find the inverse of the matrix 𝐴 = using elementary operations.
5 7

37. Verify that the value of the determinant remains unchanged if its rows and columns are

2 −3 5
interchanged by considering third order determinant 6 0 4 .
1 5 −7

38. If 𝑥𝑦 = 𝑒 𝑓𝑖𝑛𝑑 .

39. If 𝑥 = 𝑎(𝑐𝑜𝑠𝜃 + 𝜃𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(𝑠𝑖𝑛𝜃 − 𝜃𝑐𝑜𝑠𝜃) find .

40.Verify mean value theorem, if 𝑓(𝑥) = 𝑥 − 4𝑥 − 3 in the interval [a, b]

where a = 1 and b = 4.

41. Find the intervals in which the function f given by 𝑓(𝑥) = 2𝑥 − 3𝑥 − 36𝑥 + 7 is

(a) increasing (b) decreasing.

42.Find ∫ ( )
dx.

43.Evaluate ∫ ( )( )
𝑑𝑥.

44.Evaluate ∫ 𝑒 dx as the limit of a sum.

45.Find the area of the region bounded by y2 = 9x, x = 2, x= 4 and the x-axis

in the first quadrant.

46.Form the deferential equation representing the family of curves y=a.sin(x+b), where

a, b are arbitrary constants.

47.Find the general solution of the differential equation = .

48. If 𝑎⃗, 𝑏⃗ and 𝑐⃗ are unit vectors such that𝑎⃗+ 𝑏⃗ + 𝑐⃗ = 𝑜⃗, find the value of

𝑎⃗ .𝑏⃗ + 𝑏⃗ . 𝑐⃗ + 𝑐⃗.𝑎⃗.

49.Find x such that the four points A(3,2,1), B(4,x,5), C ( 4,2, -2) and D(6,5, -1) are co-planar.

50.Find the shortest distance between the lines = = and = = .

51.A man is known to speak truth 3 out of 4times. He throws a die and reports that it is a six.

Find the probability that it is actually a six.

PART –D

Answer any SIX questions 6X5=30

52. Let A = R  {3} and B = R{1}. Consider the function f: A  B defined 𝑓(𝑥) = .

Is f one-one and onto? Justify your answer.

53.Let f: NR be a function defined as 𝑓(𝑥) = 4𝑥 + 12𝑥 + 15 .

Show that f: NS where, S is the range of f is invertible. Find the inverse of f.

−1 2 3 −4 1 −5
54. If 𝐴 = 5 7 9 and 𝐵 = 1 2 0 , then verify that
−2 1 1 1 3 1

(i) (A+B)T= AT+BT (ii) (A-B)T= AT-BT.

55. Solve system of linear equation, using matrix method.

2𝑥 + 3𝑦 + 3𝑧 = 5
𝑥 − 2𝑦 + 𝑧 = −4
3𝑥 − 𝑦 − 2𝑧 = 3
56.If 𝑦 = 3 cos(𝑙𝑜𝑔𝑥) + 4sin (𝑙𝑜𝑔𝑥) Show that 𝑥 𝑦 + 𝑥𝑦 + 𝑦 = 0.

57. The length x of a rectangle is decreasing at the rate of 5cm/minute and the width y is

increasing at the rate of 4cm/minute.When x = 8 cm and y = 6cm find the rates of change of

a) the perimeter

b) the area of the rectangle.

58.Find the integral of with respect to x and hence evaluate ∫ dx.

59. Find the area enclosed by the circle 𝑥 + 𝑦 =𝑎 using integration.

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

61.Derive the equation of a plane perpendicular to a given vector and passing through a given

point both in vector and Cartesian form.

62.A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize

is . What is the probability that he will win a prize

a) atleast once

b) exactly once

63.Probability of solving specific problem independently by A and B are and respectively.

If both try to solve the problem independently, find the probability that

(i) the problem is solved (ii) exactly one of them solves the problem

PART – E

Answer any ONE question1 X10=10

64. (a) Maximise Z = 3x + 2y subject to the constraints x + 2y ≤ 10 , 3x + y ≤ 15, x,y ≥0

2 3
b) If the matrix 𝐴 = satisfies the equation 𝐴 − 4𝐴 + 𝐼 = 𝑂, when I is 2×2
1 2

identify matrix and O is 2×2 zero matrix. Using this equation find 𝐴 .

√
65. (a)Prove that ∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑓(𝑎 − 𝑥)𝑑𝑥 and hence evaluate ∫ 𝑑𝑥 .
√ √

𝐾𝑥 + 1 𝑖𝑓 𝑥 ≤ 5
(b) Find the value of K so that the function 𝑓(𝑥) =
3𝑥 − 5𝑖𝑓 𝑥 > 5

is continuous at x = 5.

66. (a) Prove that the volume of the largest cone that can be inscribed in a sphere of

radius R is of the volume of the sphere.

b) By using properties of determinants

𝑎−𝑏−𝑐 2𝑎 2𝑎
Show that 2𝑏 𝑏−𝑐−𝑎 2𝑏 = (𝑎 + 𝑏 + 𝑐) .
2𝑐 2𝑐 𝑐−𝑎−𝑏

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