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GOKULDHAM HIGH SCHOOL & JUNIOR COLLEGE MATHEMATICS [ II PRELIMS 13-01-2020 ] (Maximum Marks : 100) (Time allowed : Three hours) (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.) ----------------------------------------------------------------------------------------------The Question Paper consists of three sections A, B and C Candidates are required to attempt all the questions from Section A and all the questions EITHER from Section B OR Section C. Section A : Internal choice has been provided in three questions of four mark each and two questions of six marks each. Section B : Internal choice has been provided in two questions of four mark each. Section C : Internal choice has been provided in two questions of four mark each. All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables and squared paper are provided. Slide rule may be used. -----------------------------------------------------------------------------------------------SECTION A [80 Marks] Question 1 [10x2] i) If the binary operation * on the set of R of real numbers defined by 3 a * b = , write the identity element in R for * 7 ii) Solve cos 1 ( cos 1 ) = iii) For what value of x the following matrix is singular? 5 +1 [ ] 2 4 Using properties of determinants, solve for x + | + | + Find the differential equation of the family of curves y = A + B , where A and B are arbitrary constants. Using L hospital s rule , Evaluate :lim [ ] iv) v) vi) 6 vii) viii) ix) x) 2 2 Evaluate : 02 log dx Given that two numbers appearing on throwing two dice are different. find the probability of the event the sum of the numbers on the dice is 4. Given that the events E and F are such that P(E) = P (E F) and P(F) = p Find p if E and F are independent events. Write the equation of the normal to the curve y = 2 2 + 3 Sin x at x= 0 ---------------------------------------------------------------------------------------------------This paper consists of 5 printed pages. 1220 860 ISC XII Question 2 2 4 +3 Show that the function f in A = R { } defined as f(x) = is one-one and 3 6 4 onto. Hence find 1 [4] Question 3 Solve for x tan 1 ( + 1) + tan 1 ( 1 ) =tan 1 [4] 8 31 . Question 4 Prove, using properties of determinants: 2 + | 2 + |= (x + y + z ) (x - y)( y - z )(z x ) 2 + [4] Question 5 [4] a) Check the continuity and differentiability of f(x) = | 2 | at x = 2 OR b) It is given that Rolle s theorem holds good for the function 4 f(x) = x 2 + a 2 + . [1,2]at the point x = . . 3. Question 6 [4] + If y = + , then show that = Question 7 Integrate : [4] cos 1 1 2 dx Question 8 [4] a) A closed circular cylinder has height 16 cm and radius r cm . The total surface area is A 2 , prove that = 4 ( r + 8 ) Hence calculate an approximate increase in area if the radius increases from 4 to 4.02cm, the height remaining constant. OR b) Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole of the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the vertical angle of the conical vessel is 6 Question 9 [4] + a) Find the particular solution of differential equation = , 1+ given that y = 1, when x = 0 OR b) Solve the differential equation y dx (x + 2 2 ) dy = 0 ---------------------------------------------------------------------------------------------------1220 860 ISC XII Question 10 a) b) [6] 4 2 3 Find = [1 1 1 ] 3 1 2 Hence, solve the system of equations, 4x+ 2y + 3z = 2 , x + y + z = 1,3x + y - 2z = 5 OR Using elementary transformation find the inverse of the matrix, 8 4 3 [ 2 1 1] 1 2 2 1 Question 11 [4] On dialing certain telephone numbers, assume that on an average one telephone number out of five is busy. Ten telephone numbers are randomly selected and dialed. Find the probability that at least three of them will be busy Question 12 [6] ) ; b) Evaluate : ( + + + ) dx OR dx Question 13 [6] A window is in the form of a rectangle surmounted by a semicircle. The total perimeter of the window is 10 m . Find the dimensions of the rectangular part of the window to admit maximum light through it . Question 14 [6] In a bolts factory, three machines A, B and C manufactures 25% , 35% and 40% of the total production respectively . Of their respective outputs , 5% , 4% and 2 % are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C. SECTION B [20 Marks] Question 15 a) Find the angle between the line 2x + 3y 5z = 4 b) c) [3 x 2] +1 2 = 3 = 3 6 and the plane The Cartesian equation of a line is 2x -3 = 3y +1 = 5 - 6z , Find the vector equation of a line passing through ( 7, -5, 0) and parallel to the given line . Find if the scalar projection of ---------------------------------------------------------------------------------------------------1220 860 ISC XII = + + 4 = 2 + 6 + 3 4 Question 16 [4] a) Find the Cartesian equation of the plane passing through the intersection of the planes . (2 + 6 ) + 12 = 0 and 3x y + 4 z = 0 and at a unit distance from origin. OR b) Find the coordinates of the foot of perpendicular drawn from the point A ( -1, 8 , 4) to the line joining the points B (0 , -1, 3) and C (2,-3,-1). Hence find the image of the point A in the line BC. Question 17 a) If = 2 + + 3 and b = + 2 + 4 , then find a unit vector which is perpendicular to both the vectors ( + ) and ( - ) . OR b) If , , are perpendicular to each other , then prove that ( a . ( b x c ))2 = 2 2 2 [4] Question 18 Find the area of the region in the first quadrant enclosed by x axis, the line x = 3y and the circle 2 + 2 = 4 [6] SECTION C [20 Marks] Question 19 [3 x 2] a) Find the cost of increasing from 100 to 200 units if the marginal cost in Rs per unit is given by the function MC = 0.003 2 0.01 + 2.5. b) A monopolist demand function is p = 250 7x. Find the marginal revenue function. c) If the regression equation of x and y is given by n. d) x y +10 = 0 and the equation of y on x is given by -2x + 5y 14 = 0, Find the value of n if the coefficient of correlation between x and y 1 is 20 Question 20 [4] a) The data for marks in English and Biology obtained by six students are given below : English 8 7 8 12 7 15 Biology 11 13 17 25 17 10 Using this data , find the line of regression in which Biology is taken as in dependent variable. b) In a distribution given it was found x=3 ,the regression line of y on x is ---------------------------------------------------------------------------------------------------1220 860 ISC XII 8 10 + 66 = 0 while regression line of x and y is 40 18 214 = 0. Calculate , , (x,y) and y . Question 21 a) A firm has the cost function c (x) = 3 x 2 + 15x + 27 and demand function x = 50 p i) Write the total revenue function in terms of x. ii) Find the profit. [4] OR a) x3 The cost function of a product is given by C(x) = 90x 2 1800 x, 3 where x is the number of units produced. In order to minimize the marginal cost, how many units of the product must be produced? Question 22 [6] The postmaster of the local post office wishes to hire extra helper during the Christmas season because of large increase in volume of mail handling and delivery. Because of limited office space and budgetary condition, the number of temporary helper must not exceed 10. According to the past experience, a man can handle 300 letter and 80 packages per day , on an average and a woman can handle 400 letters and 50 packets per day .The postmaster believe that the daily volume of extra mail and packages will not less than 3400 and 680 respectively. A man receives Rs 225 a day and woman receive Rs 200 per day. How many man and woman should be hired to keep a payroll at the minimum? Formulate L.P.P and solve it graphically. -x-x-x-x-x- ---------------------------------------------------------------------------------------------------1220 860 ISC XII
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