35. MATHEMATICS MQP II PUC 2023-24.pdf

SECOND PUC MODEL QUESTION PAPER 2023-2024 SUBJECT : MATHEMATICS ( 35 ) TIME : 3 Hours 15 Minutes [Total questions : 52 ]...

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 is dx 1 1 1) b) 𝑥 𝑙𝑜𝑔 7 7 𝑙𝑜𝑔𝑥 𝑙𝑜𝑔𝑥 7 c) d)𝑙𝑜𝑔𝑥 7 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 – dx is 𝑥 𝑥2 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, , then the angle made by the 2 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 – sin−1 – 2 is –––––––––––––––––––––––––––––– 3 17. A square matrix A is a singular matrix if |A| is –––––––––––––––––––––––––– 𝑑4𝑦 18. The order of the differential equation + sin 𝑦 𝐼𝐼𝐼 = 0 is ––––––––––––––– 𝑑𝑥 4 𝑥 –5 𝑦 +2 𝑧 𝑥 𝑦 𝑧 19. The lines = = and = =3 are perpendicular, then k is –––––––––– 𝑘 –5 1 1 2 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 , if y + siny = cosx dx 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) = , P(B) =2 and P (A ∩ B) = , 4 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 with respect to x and hence evaluate x 2 −a 2 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 + 2y = x 2 logx , ( x ≠ 0 ) dx 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 f x dx , if f x is an even function 51. P.T. −a f x dx = 0 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 ≠ 𝜋 – 2𝑥 2 Find the value of k so that the function f(x) = 𝜋 3 if x = 2 π is continuous at x = 2 *******@*******