Mathematics Past Paper 208 A PDF

# Mathematics ## Series Q5QPS/5 ## Set-1 ## GENERAL INSTRUCTIONS: - This Question paper contains 38 questions. All questions are compulsory. - Question paper is divided into FIVE Sections – Section A, B, C, D and E. - In Section A – Questions Number 1 to 18 are Multiple Choice Questions (MCQs)...

# Mathematics ## Series Q5QPS/5 ## Set-1 ## GENERAL INSTRUCTIONS: - This Question paper contains 38 questions. All questions are compulsory. - Question paper is divided into FIVE Sections – Section A, B, C, D and E. - In Section A – Questions Number 1 to 18 are Multiple Choice Questions (MCQs) type and Questions Number 19 & 20 are Assertion-Reason based questions of 1 mark each. - In Section B – Questions Number 21 to 25 are Very Short Answer (VSA) type questions, carrying 2 marks each. - In Section C – Questions Number 26 to 31 are Short Answer (SA) type questions, carrying 3 marks each. - In Section D – Questions Number 32 to 35 are Long Answer (LA) type questions, carrying 5 marks each. - In Section E – Questions Number 36 to 38 are case study based questions, carrying 4 marks each. - There is no overall choice. However, an internal choice has been provided in 2 questions in Section – B, 3 questions in Section – C, 2 questions in Section – D and 2 questions in Section – E. - Use of calculators is NOT allowed. ## SECTION – A This section has 20 multiple choice questions of 1 mark each. 1. A function f: IR → IR defined as f(x) = x² - 4x + 5 is : - injective but not surjective. - surjective but not injective. - both injective and surjective. - neither injective nor surjective. 2. If A = - 0 - -10 3. If A is a square matrix of order 3 such that the value of | adj.A| = 8, then the value of |AT | is: - √2 - 8 4. If inverse of matrix - -4 - 3 5. If \[x 2 0]\^-1 = [3 1]\ [-2]\,then value of x is : - -1 - 1 ## SECTION – B This section has 5 Very Short Answer questions of 2 marks each. 21. Find value of k if - sin-1 (k tan (2cos-1 (√ 3/2)) ) = π/3 22. (a) Verify whether the function f defined by - f(x) = x sin (1/x), x ≠ 0 - 0, x = 0 is continuous at x = 0 or not. OR (b) Check for differentiability of the function f defined by f(x) = |x − 5|, at the point x = 5. 23. The area of the circle is increasing at a uniform rate of 2 cm²/sec. How fast is the circumference of the circle increasing when the radius r = 5 cm ? 24. (a) Find: ∫cos³ x e^{log sin x} dx OR (b) Find: ∫(5+ 4x-x²) dx 25. Find the vector equation of the line passing through the point (2, 3, −5) and making equal angles with the co-ordinate axes. ## SECTION – C There are 6 short answer questions in this section. Each is of 3 marks. 26. (a) Find dy/dx if (cos x) = (cos y) *. OR (b) If √1-x² + √1-y2 = a(x - y), prove that dy/dx = (1-y2)/(√1-x2). 27. If x = a sin³ 0, y = b cos³ 0, then find d²y/dx² at 0 = π/4. 28. (a) Evaluate: ∫^π₀ (e^cosx/ e^cosx + e^-cosx) dx OR (b) Find: ∫ (2x+1) / √(x+1)² (x-1) dx 29. (a) Find the particular solution of the differential equation dy/dx - 2xy = 3x² ex²; y(0) = 5. OR (b) Solve the following differential equation : x² dy + y(x + y) dx = 0 30. Find a vector of magnitude 4 units perpendicular to each of the vectors 2i - j + k and i + j - k and hence verify your answer. 31. The random variable X has the following probability distribution where a and b are some constants : X | 1 | 2 | 3 | 4 | 5 P(X) | 0.2 | a | a | 0.2 | b If the mean E(X) = 3, then find values of a and b and hence determine P(X ≥ 3). ## SECTION – D There are 4 long answer questions in this section. Each question is of 5 marks. 32. (a) If A = [1 2 -3; 2 0 -3; 1 2 0], then find A-1 and hence solve the following system of equations : - x + 2y - 3z = 1 - 2x -3z = 2 - x + 2y = 3 OR (b) Find the product of the matrices [1 2 -3; 2 3 2; 3 -3 -4] and [-6 17 13; 14 5 -8; -15 9 -1] and hence solve the system of linear equations : - x + 2y - 3z = -4 - 2x + 3y + 2z = 2 - 3x - 3y - 4z = 11 33. Find the area of the region bounded by the curve 4x² + y² = 36 using integration. 34. (a) Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, -8) to the line (4-x)/2 = y/6 = (1-z)/3. Also, find the perpendicular distance of the given point from the line. OR (b) Find the shortest distance between the lines L₁ & L₂ given below : - L₁: The line passing through (2, −1, 1) and parallel to x/1 = y/1 = z/3 - L₂: r = i + (2μ + 1)j - (μ + 2)k. 35. Solve the following L.P.P. graphically : - Maximise Z = 60x + 40y - Subject to - x + 2y ≤ 12 - 2x + y ≤ 12 - 4x + 5y ≥ 20 - x, y ≥ 0 ## SECTION – E In this section there are 3 case study questions of 4 marks each. 36. (a) Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by R = {(11, 12): l₁ is parallel to 12}. On the basis of the above information, answer the following questions: - Find whether the relation R is symmetric or not. - Find whether the relation R is transitive or not. - If one of the rail lines on the railway track is represented by the equation y = 3x + 2, then find the set of rail lines in R related to it. OR (b) Let S be the relation defined by S = {(11, 12): l₁ is perpendicular to 12} check whether the relation S is symmetric and transitive. 37. A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : - On the basis of the above information, answer the following questions: - Write the expression for the area of the visiting card in terms of x. - Obtain the dimensions of the card of minimum area. 38. A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. - Based on the above information, answer the following questions: - Let E₁ and E₂ respectively denote the event of customer paying or not paying the first month bill in time. Find P(E₁), P(E₂). - Let A denotes the event of customer paying second month's bill in time, then find P(A|E₁) and P(A|E₂). - Find the probability of customer paying second month's bill in time. OR - Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.