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PRELIMINARY EXAMINATION 1 (2024-25) Mathematics Grade : XII Maximum Marks: 80 ---------------------------------------------------------------------------------------------------------------Time Allotted: Three Hours Reading Time: Additional Fifteen minutes Instructions to Candidates -------------------------------------------------------------------------------------------------------------- You are allowed an additional 15 minutes for only reading the paper. You must NOT start writing during reading time. The question paper has 12 printed pages. The Question Paper is divided into three sections and has 22 questions in all. Section A is compulsory and has fourteen questions. You are required to attempt all questions either from Section B or Section C. Section B and Section C have four questions each. Internal choices have been provided in two questions of 2 marks, two questions of 4 marks and two questions of 6 marks in Section A. Internal choices have been provided in one question of 2 marks and one question of 4 marks each in Section B and Section C. While attempting Multiple Choice Questions in Section A, B and C, you are required to write only ONE option as the answer. The intended marks for questions or parts of questions are given in the brackets [ ]. All workings, including rough work, should be done on the same page as, and adjacent to, the rest of the answer. Mathematical tables and graph papers are provided. -------------------------------------------------------------------------------------------------------Instruction to Supervising Examiner Kindly read aloud the instructions given above to all the candidates present in the examination hall. ISC This paper consist of 12 printed pages Page 1 SECTION A 65 MARKS Question 1 In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), [15x 1] answer the questions as instructed. i) If A = [ ], then the value of ( ) is: a) b) c) 0 d) A ii) If, ( ( ) ) then f(x) will be: a) b) c) d) iii) The trigonometric equation a) | | has solution for b) | | c) | | d) All real value of a. iv) Assertion (A): Degree of the differential equation ( ) cannot be determined. Reason (R): If each term involving derivatives of a differential equation is a polynomial (or can be expressed as a polynomial) then the highest exponent of the highest order derivative is called the degree of the differential equation. a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). c) Assertion (A) is true, but Reason (R) is false. d) Assertion (A) is false, but Reason (R) is true. 2 v) Let A and B be independent events with P(A)= 1/4, and P(AUB) = 2P(B) p(A). Find P(B). a) b) 3/5 c) 2/3 d) 2/5 vi) If | | a) b) c) d) vii) then what is the value of | | 6k 3k 2k k/6 Statement I: ( ) ( ) { is continuous at x= 0 but f1(x) is not continuous at x = 0. Statement II: The derivative of a continuous function need not be a continuous function. Which one of the following is correct? a) b) c) d) viii) Both (I) and (II) are correct and (II) is the correct explanation of (I). Both (I) and (II) are correct and (II) is not the correct explanation of (I). (I) is correct but (II) is incorrect. (II) is correct but (I) is incorrect. In which of the following intervals is the function ( ) increasing? a) (-2,2) b) (- ) ( c) ( ) d) (ix) ) ) Statement 1: The intersection of two equivalence relations is always an equivalence relation. Statement 2: The Union of two equivalence relations is always an equivalence relation. Which one of the following is correct? a) Statement 1 implies Statement 2. b) Statement 2 implies Statement 1. c) Statement 1 is true only if Statement 2 is true. d) Statement 1 and 2 are independent of each other. 3 x) In a third order matrix A= denotes the element of the row and column. { Assertion: Matrix A is not invertible. Reason: Determinant A = 0 Which of the following is correct? a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. c) Assertion is true and Reason is false. d) Assertion is false and Reason is true. xi) Teena is practising for an upcoming Rifle Shooting Tournament. The probability of her shooting the target in the 1st , 2nd , 3rd and 4th shots are 0.4 , 0.3, 0.2 and 0.1 respectively. Find the probability of at least one shot of Teena hitting the target. xii) Evaluate: xiii) If A = {1,2,3}, then the number find the number of equivalence relations on A containing (1,3). xiv) Show that the modulus function xv) There are three machines and 2 of them are faulty. They are tested one by one in a random , defined by f(x) = | | is not onto. order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines? Question2 (A) Differentiate ( ) [2] with respect to x. OR (B) Prove that the function ( ) is increasing on R. Question 3 Evaluate : [ ] [2] 4 Question 4 At what point on the circle [2] the tangents are parallel to x axis? Question 5 (A) [2] Evaluate: OR (B) Evaluate: [2] ( ) Question 6 Find the value of : ( [2] ) Question 7 ( Solve for x: ). [4] Question 8 [4] Evaluate : Question 9 (A) A man of height 180cm is moving away from a lamp post at the rate of [4] 1.2metres per sec. If the height of the lamp post is 4.5m, find the rate at which i) his shadow is lengthening, ii) the tip of his shadow is moving. OR (B) If ( ) . [4] 5 Question 10 (A) In a Kabaddi league, two matches are being played between Jaipur and [4] Delhi. It is assumed that the outcomes of two games are independent . The probability of Jaipur winning, drawing and losing the game against Delhi are , 3/10 and 1/5 respectively. Each team gets 5 points for win, 3 points for draw and 0 point for losing a game. After two games, find the probability that: a) Jaipur has more points than Delhi b) Jaipur and Delhi have equal points. OR (B) In an office three employees James, Sophia and Oliver process incoming [4] copies of a certain form. James processes 50% of the forms, Sophia processes 20% and Oliver the remaining 30% of the forms. James has an error rate of 0.06, Sophia has an error rate of 0.04 and Oliver has an error rate of 0.03. Based on the above information, solve the following questions: a) Find the probability that Sophia processed the form and committed an error. b) Find the probability of committing an error in processing the form. c) The manager of the company wants to do a quality check. During inspection, he selects a form at random from the days output of processed form. If the form selected at random has an error, find the probability that the form is not processed by James. Question 11 Using matrices, solve the following system of equations: x + 2y = 5 y + 2z = 8 2x + z = 5 Question 12 (A) [6] A running track of 440 m is to be laid out enclosing a football field. The [6] football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then find the length of its sides. Also calculate the area of the football field. OR 6 (B) Rohit has bought a closed cylindrical dustbin. The radius of the dustbin is r cm and height is h cm. It has a volume of 20 cm3. a) Express h in terms of r , using the given volume. b) Prove that the total surface area of the dustbin is c) Rohit wants to paint the dustbin. The cost of painting the base and top of the dustbin is Rs 2 per cm2 and the cost of painting the curved side Rs 25 per cm2. Find the total cost in terms of r , for painting the outer surface of the dustbin including the base and top. d) Calculate the minimum cost for painting the dustbin. Question 13 (A) [6] Evaluate the following integral: ( ) OR (B) Consider the following differential Equation and answer the questions. [ ( ) ( )] [ ( ) a) Transform the above equation in the form [6] ( )] ( ) b) Use appropriate substitution to transform it into variable separable form. c) Write the differential equation in variable separable form. d) Prove that the solution of the differential equation is ( ) e) Find the solution if x = 1, y = 1. 7 Question 14 Jacob has four coins, one of the coins is biased such that when it is thrown [6] the probability of obtaining a head is 7/10. The other three coins are fair. Jacob throws all four coins once. The number of heads that he obtains is denoted by the random variable X. The probability distribution table for X is as follows: X 0 1 2 3 4 P(X=x) 3/80 a b c 7/80 a) Show that a = 1 /5 b) Find the value of b and c c) Find P(X ) d) Find the expected value of Probability distribution. e) Section B- 15 marks Question 15 i) [5] The angle between the vectors - and is A) D) B) C) ii) Consider the following statements and choose the correct option: Statement 1: If and represents two adjacent sides of a parallelogram then the diagonals are represented by + and . Statement 2: If and represents two diagonals of a parallelogram then the adjacent sides are represented by 2( + ) and 2( ). Which of the following is correct? a) Only Statement 1 b) 1 Only Statement c) Both Statements 1 and 2 d) Neither Statement 1 nor Statement 2 8 iii) If and , find the projection of on . iv) Find a vector of magnitude 20 units parallel to vector v) . Shown below is a cuboid. Find Question 16 (A) If the vectors = 2 - + , = + 2 - 3 and = 3 + + 5 are [2] coplanar, then find the value of . OR (B) Find | - |, if two vectors and are such that | | = 2, | |= 3 and . = [2] 4. Question 17 (A) Find the equation of the plane passing through the point (7,14,5) and [4] perpendicular to the planes 3x + 2y 3z = 1 and 5x 4y + z = 5. OR (B) Find the length and foot of the perperdicular drawn from the point (2, -1, 5) to the line [4] . Question18 If = + + , = - 2 + 3 and = + + , then find a vector of magnitude 6 units, which is parallel to the vector - + 3 . [4] 9 Section C- 15 marks Question 19 i) [5] A company sells hand towels at Rs. 100 per unit. The fixed cost for the company to manufacture hand towels is Rs.35,000 and the variable cost is estimated to be 30% of total revenue. What will be the total cost function for manufacturing hand towels? a) 35000 + 3x b) 35000 + 30x c) 35000 + 100x d) 35000 + 10x ii) Read the following statements and choose the correct option: (I) If r = 0, then regression lines are not defined. (II) If r = 0, then regression lines are Parallel. (III) If r = 0, then regression lines are Perpendicular. (IV) If r = , then regression lines coincoid. Which of the following is correct? a) Only IV is coreect. b) Only I and II are correct. c) Only I and IV are correct. d) Only III and IV are correct. iii) The total revenue received from the sale of x unit of a product is given by ( ) Find the marginal revenue when x = 5. iv) If the covariance of x and y is 58.08, the variance of x is 121 and the standard deviation of y is 8 then, find the regression coefficient of x and y. v) The total cost of producing and marketing x units of bulls by a whole seller is given by ( ) . What will be the average cost of producing 3 units of bulbs? 10 Question 20 (A) A company has fixed cost of Rs. 10,000 and cost of producing one unit of [2] its product is Rs 50. If each unit sells for Rs. 75, find the break- even value . Also, find the values of x for which the company results in profit. OR (B) The demand function of a monopoly is given by x = 100 4p. Find the [2] quantity at which the MR will be zero. Question 21 (A) The line of regression of marks in Statistics (X) and marks in Accountancy [4] (Y) for a class of 50 students is 3y 5x + 180 = 0. The average score in Accountancy is 44 and the variance of marks in Statistics is ( ) of variance of marks in Accountancy. a) Find the average score in Statistics. b) Find the coefficient of correlation between marks in Statistics and marks in Accountancy. OR (B) A movie cinema is considering significantly reducing the price of their [4] popcorn as they believe their customers spend more on drinks when they buy popcorn. They recorded the following data of the daily revenue from popcorn, x , and the daily revenue from drinks, y over 8 randomly selected days. Popcorn revenue ( x ) Drinks revenue ( y ) 14 22 12 23 13 17 14 24 16 18 10 25 13 23 12 24 11 a) Find . b) Using , find regression coefficient of y on x. c) The equation of the regression line y on x is in the form y = a + bx. Calculate the values of a and b. Question22 A manufacturing company produces two types of cell phones, Android and [4] iOS. The company has resources to make at the most 300 sets a week. It takes 1800 to make an Android set and 2700 to make an iOS set. The company cannot spend more than 648000 a week to make cell phones. The company makes a profit of 510 per Android and 675 per iOS set. If x and y denote, respectively, the number of Android sets and iOS sets made each week, then formulate this problem as a Linear Programming Problem (LPP) given that the objective is to maximize the profit. Based on it, answer the questions that follow. a) What will be the maximum profit function on x and y sets? (write your objective based on the above data). b) What will be the values of your objective function in the feasible region? (at corner points) c) At what point the maximum profit will occur? d) What s the weekly cost (in ) of manufacturing the sets? 12
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