Karnataka II PUC Mathematics (35) 2024-25 Past Paper PDF

GOVERNMENT OF KARNATAKA KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD WEIGHTAGE FRAMEWORK FOR MQP 3: II PU MATHEMATICS (35):2024-25 Chapter Number PART PART PART...

GOVERNMENT OF KARNATAKA KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD WEIGHTAGE FRAMEWORK FOR MQP 3: II PU 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 1 1 1 FUNCTIONS INVERSE 2 6 2 2 TRIGONOMETRIC 6 FUNCTIONS 9 1 9 3 MATRICES 1 1 1 12 12 4 DETERMINANTS 1 1 1 1 CONTINUITY AND 20 17 5 2 1 1 1 1 1 DIFFERENTIABILITY APPLICATION OF 10 8 6 2 1 1 1 DERIVATIVES 22 18 7 INTEGRALS 1 1 1 1 1 1 APPLICATION OF 5 5 8 1 INTEGRALS DIFFERENTIAL 10 8 9 1 1 1 EQUATIONS 11 8 10 VECTOR ALGEBRA 2 2 THREE D 8 6 11 1 1 1 GEOMETRY LINEAR 7 6 12 1 ROGRAMMING 11 8 13 PROBABILITY 1 1 2 120 TOTAL 140 15 5 9 9 7 2 2  GOVERNMENT OF KARNATAKA KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD Model Question Paper -3 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. PART A I. Answer ALL the Multiple Choice Questions 151 = 15 1. The element needed to be added to the relation R={(1,1), (1,3), (2,2),(3,3) } on A = {1, 2, 3} so that the relation is neither symmetric nor transitive A) (2, 3) B) (3, 1) C) (1, 2) D) (3, 2) 2. The graph of the function 𝑦 = cos−1 𝑥 is the mirror image of the graph of the function y = cosx along the line A) x = 0 B) y = x C) y = 1 D) y = 0 3. The value of tan−1 (√3) + sec −1 (−2) is equal to 2π π π 𝐴) π B) C) − 3 D) 3 3 4. If A and B are matrices of order 3 × 2 and 2 × 2 respectively, then which of the following are defined A) AB B) BA C) A2 D) A + B 5. A square matrix A is invertible if A is A) Null matrix B) Singular matrix C) skew symmetric matrix of order 3 D) Non-Singular matrix 𝑑𝑦 6. If 𝑦 = sin−1(𝑥√𝑥), then = 𝑑𝑥 1 2√𝑥 3√𝑥 −3√𝑥 A) B) C) D). √1−𝑥 3 3√1−𝑥 3 2√1−𝑥 3 2√1−𝑥 3 𝑑𝑦 7. If y = 𝑥 a + 𝑎x + 𝑎a for some fixed a > 0 and x > 0, then = 𝑑𝑥 (A) 𝑎𝑥 a−1 + 𝑎x loga + 𝑎𝑎a−1 B) 𝑎𝑥 a−1 + 𝑎x loga C) 𝑎𝑥 a−1 + 𝑥𝑎x−1 + 𝑎𝑎a−1 D) 𝑎𝑥 a−1 + 𝑎x loga + 𝑎a. 8. Consider the following statements for the given function y=f(x) defined on an interval I and c∈ I, at x = c I. 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) < 0 ⟹ f attains local maxima II. 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) > 0 ⟹ f attains local minima III. 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) = 0 ⟹ f attains both maxima and minima A) I and II are true B) I and III are true C) II and III are true D) all are false 9. If each side of a cube is x units, then the rate of change of its surface area with respect to side is A) 12x B) 6x C) 6x2 D) 3x2 1 10. Statement 1: The anti-derivative of (√1+𝑥 2 ) with respect to x is 𝑥 1 2 √1 + 𝑥 2 + 2 log|𝑥 + √1 + 𝑥 2 | + 𝐶. 𝑥 1 Statement 2: The derivative of √1 + 𝑥 2 + 2 log|𝑥 + √1 + 𝑥 2 | + 𝐶 2 1 with respect to x is √1+𝑥 2. 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) Both statements are false. 3 d2 y dy 2 dy 11. The degree of the differential equation (dx2 ) + (dx) + sin (dx) + 1 = 0 is A) 2 B) 3 C) 5 D) not defined 12. The position vector of a point which divides the join of points with position vectors 3𝑎⃗ − 2𝑏⃗⃗ and 𝑎⃗ + 𝑏⃗⃗ externally in the ratio 2 : 1 is 5𝑎⃗⃗ A) 3 B) 4𝑎⃗ − 𝑏⃗⃗ C) 4𝑏⃗⃗ − 𝑎⃗ D) 2𝑎⃗ +𝑏⃗⃗ 𝜋 𝜋 13. If a vector 𝑎⃗ makes angles with with 𝑖̂ and with 𝑗̂ and an acute 3 4 angle 𝜃 with 𝑘̂, then θ is 𝜋 𝜋 𝜋 𝜋 A) 6 B) 4 C) D) 3 2 14. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b is A) 450 B) 300 C) 600 D) 900  1 1 15. If A and B are two independent events such that 𝑃(𝐴) = 4 and 𝑃(𝐵) = 2 then P(neither A nor B) 1 3 7 1 A) B) C) D). 3 8 8 2 II. Fill in the blanks by choosing the appropriate answer from those 𝟓 𝟑 given in the bracket (-2, 𝟐, 0, 1, 2, 𝟐 ) 51 = 5 16. The number of all possible orders of matrices with 13 elements is ____ 𝑑2 𝑦 17. If y = 5 cos x – 3 sin x, then + 𝑦 =_____ 𝑑𝑥 2 18. If the function f given by f (x) = 𝑥 2 + ax + 1 is increasing on [1, 2], then the value of ‘a’ is greater than ____________ 2  19. | x | dx =_________ 1 20. If A and B are any two events such that P(A) + P(B) – P(A and B) =P(A), then P(A|B) is________ PART B Answer any SIX questions 6  2=12 1−𝑐𝑜𝑠𝑥 21. Write the simplest form of tan−1 (√1+𝑐𝑜𝑠𝑥) , 0 < 𝑥 < 𝜋. 3 24 22. Prove that 2 sin−1 5 = tan−1. 7 𝑐𝑜𝑠𝑥 −𝑠𝑖𝑛𝑥 0 23. If F(x)= [ 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 0], then show that F(x) F(y) = F(x + y). 0 0 1 24. Find the equation of line joining (1, 2) and (3, 6) using determinants. 25. Differentiate xsin x , x  0 with respect to x. 26. Find the intervals in which the function 𝑓 given by 𝑓(𝑥) = 𝑥 2 𝑒 −𝑥 is increasing. 27. Find  ( x 2 + 1) log x dx.. 𝑑𝑦 28.Verify the function 𝑦 =mx is the solution of − 𝑦 = 0, x ≠ 0. 𝑑𝑥 29. Find the distance between the lines 𝑟⃗ = 𝑖̂ + 2 𝑗̂ - 4 𝑘̂ + 𝜆 (2𝑖̂ + 3 𝑗̂ + 6 𝑘̂) and 𝑟⃗ = 3𝑖̂ + 3𝑗̂ - 5 𝑘̂ + 𝜇(2𝑖̂ + 3 𝑗̂ + 6 𝑘̂).  PART C Answer any SIX questions 6  3= 18 30. Let f : X →Y be a function. Define a relation R in X given by R = {(a, b): f(a) = f(b)}. Examine whether R is an equivalence relation or not. 𝑑𝑦 31. If 𝑥 3 + 𝑥 2 𝑦 + 𝑥𝑦 2 + 𝑦 3 = 81, then find. 𝑑𝑥 32. The length 𝑥 of a rectangle is decreasing at the rate of 3 cm/min and the width 𝑦 is increasing at the rate of 2 cm/min. When 𝑥 = 10 cm and 𝑦 = 6 cm, find the rate of change of the perimeter of the rectangle. 1 33. Find the integral of 2 with respect to x. a + x2 34. If the vertices A, B and C of a triangle are (1, 2, 3), (−1, 0, 0) and (0, 1, 2) respectively, then find the angle ∠𝐴𝐵𝐶.  1   1  35. Find the area of the rectangle, whose vertices are A  -i + j + 4k  , B  i + j + 4k  , 2  2     1   1  C  i − j + 4k  and D  −i − j + 4k .  2   2  36. Find the vector equation of the line passing through the point (1, 2, – 4) and x−8 y+19 z−10 x−15 y – 29 z−5 perpendicular to the two lines: = = and = =. 3 −16 7 3 8 −5 37. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red. 38. Three coins are tossed simultaneously. Consider the Event E ‘three heads or three tails’, F ‘at least two heads and G ‘at most two heads’. Of the pairs (E,F), (E, G) and (F, G), which are independent? Which are dependent? PART D Answer any FOUR questions 4  5 = 20 39. 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 40. If A =  −6 0 8  , B = 1 0 2  and C =  −2  ,      7 −8 0  1 2 0   3  calculate AB, AC and A(B + C). Verify that A ( B + C ) = AB + AC. 41. Solve the following system of linear equations by matrix method: 3 2x + y + z = 1, x – 2y – z = and 3y – 5z = 9. 2 𝒅𝟐 𝒚 42. If x = a (cos t + t sin t) and y = a (sin t – t cos t), find. 𝒅𝒙𝟐 x4 43. Find  ( x −1)( x2 +1) dx. 44. Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = –1 and x = 1 by integration method. 45. Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the ordinates of the point. PART E Answer the following questions:  a 2 f ( x ) dx, 46. (a) Prove that  f ( x ) dx =  0 a if f(x) is even −a 0, if f(x) is odd   2  sin 7 and evaluate x dx − 2 OR Solve the following linear programming problem graphically: Minimize and maximize 𝑍 = 𝑥 + 2𝑦, subject to constraints 𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0, 2𝑥 + 𝑦 ≤ 200 𝑎𝑛𝑑 𝑥, 𝑦 ≥ 0. 6 2 −1 1 47. If matrix A= [−1 2 −1] satisfying A3-6A2+9A-4I=O, then evaluate 𝐴−1. 1 −1 2 OR 𝑘 cos 𝑥 𝜋 , 𝐼𝑓 𝑥 ≠ 𝜋 𝜋−2𝑥 2 If 𝑓(𝑥) = { 𝜋 is continuous at 𝑥 = 2 , find k. 4 3, 𝑖𝑓 𝑥 = 2 *************