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Ryan International School Bangalore I Pre board Examination 2019 2020 Subject: Mathematics Class: XII Date: .0 . 2019 Maximum Marks : 100 Duration : 3 Hrs ROLL No. : The Question Paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions Either from Section B OR Section C. Section A : Internal choice has been provided in the three questions of four marks each and two questions of six marks each. Section B : Internal choice has been provided in two questions of four marks each. Section C :Internal choice has been provided in two questions of four marks 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 graph papers are provided. SECTION A ( 80 MARKS ) Question 1 (i) [10X2=20] The binary operation if ( ) is defined on the of all positive integers by find . Also find identity element in (ii) Find the principal value of : (iii) For the matrix * ( + find the numbers where I is the identity matrix of order 2. . ( Relations and Functions ) ) ( Relations and Functions ) and such that ( Algebra ) (iv) Find the value of if the area of the triangle whose vertices are ( ) is 15 sq units. ( Algebra ) and ( (v) Find the value of so that ( ) { Page 1 of 9 is continuous at );( ) (Calculus) (vi) Find the interval on which the function ( ) ( Calculus ) (vii) Evaluate : ( on is strictly decreasing. ( Calculus ) ) (viii) Form the differential equation representing the family of parabolas having vertex at the origin and axis along the positive direction of axis. ( Calculus ) (ix) Cards numbered 1 to 12 are placed in a box mixed up thoroughly and then a card is drawn at random from the box. It is known that the number on the card drawn is more than 3 , find the probability that it is an even number. ( Probability ) (x) A and B are independent events. The probability that both A and B occurs is and the probability neither of them occurs is . Find the probability of occurrence of A. ( Probability ) Question 2 [4] Let is a one one function defined by ( ) the range of , Show that is invertible. Hence find inverse of ( Relations and Functions ) where S is equal to ( ) and Question 3 [4] a) If ( Relations and Functions ) then prove that OR b) Solve for : ( ) ( ) () for ( Relations and Functions ) Question 4 Prove that : | [4] | ( Page 2 of 9 ) ( Algebra ) Question 5 Find [4] and if the function given by ( ) is differentiable at { ( Calculus ) Question 6 If [4] ) then show that ( ( ) ( ( Calculus ) ) Question 7 a) [4] Find the equation of the normal to the curve at ( Calculus ) and OR b) Verify Lagrange s Mean value theorem for the function ( ) ( Calculus ) in [ Question 8 Evaluate : [4] ( Calculus ) Question 9 a) ] [4] Find the particular solution of the following differential equation ( ) , given that when ( Calculus ) OR b) Find the particular solution of the following differential equation ( )( ) ,given that when Page 3 of 9 ( Calculus ) Question 10 [4] a) There are three categories of students in a class of 60 students: A : Very hard working students , B : regular but not so hard working students C : Careless and irregular. 10 students are in category A , 30 are in category B and the rest in C. It is found that probability of category A, unable to get good marks in the final year examination is 0.002 , of category B is 0.02 and of category C is 0.2. A student selected at random was found to be the one who couldn t get good marks in the examination . Find the probability that the student is of category C. ( Probability ) OR b) There are two bags. One bag contains 4 white and 2 black balls. Second bag contains 5 white and 4 black balls. Two balls are transferred from first bag to second bag . Then one ball is taken from the second bag. Find the probability that it is white. ( Probability ) [6] Question 11 a) If [ ] and ] Find [ . Hence solve the system of equations ( Algebra ) OR b) If [ ] , find . Hence solve the system of equations ( Algebra ) Question 12 [6] Show that the semi vertical angle of a right circular cone of maximum volume and given slant height is ( Calculus ) Page 4 of 9 Question 13 [6] Find the particular solution of the differential equation ( ) ( ) given that when ( Calculus ) Question 14 [6] a) Three cards are drawn at random without replacement from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find mean and variance of the distribution. ( Probability ) OR b) An experiment succeeds thrice as often it fails. Find the (i) distribution (ii) probability that in the next five trials , there will be atleast 3 successes. (iii) find mean and variance for 10 trials. ( Probability ) SECTION B ( 20 MARKS) Question 15 a) For what value of ( Vectors ) are the vectors and are collinear. [2] b) Find the equation of the plane through the intersection of planes ) ( Three Dimensional Geometry ) and the point ( [2] c) Show that the lines [2] and are coplanar. ( Three Dimensional Geometry ) Question 16 [4] and a) Find a unit vector perpendicular to each of the vectors and ( Vectors ) where OR b) Find such that the four points A ( 1 , 4 , ),B( and D ( 3 , 2 , ) are coplanar. ( Vectors ) Page 5 of 9 , , ),C( 3,8, ) Question 17 [4] a) Find the image of the point ( , , ) in the line = = . Also find the equation of the line joining the given point and its image. ( Three Dimensional Geometry ) OR b) Find the equation of the plane passing through the point ( 1 , 2 , ) and perpendicular to the line joining the points A ( 3 , 2 , ) and B ( 1 , 4 , ) ( Three Dimensional Geometry ) Question 18 [6] Find the area of the region in the first quadrant enclosed by the axis , the line and the circle using integration. ( Application of Integrals ) SECTION C ( 20 MARKS) Question 19 a) The demand function is where is the number of units demanded and is the price per unit. Find (i) Revenue function ( ii) Marginal Revenue in terms of ( Application of Calculus ) [2] b) A company paid `16100 towards rent of a building and interest on loan. The cost of producing one unit of item is `20.If each unit is sold for ` 27, find the break even point . ( Application of Calculus ) [2] c) The coefficient of correlation between and is . If the variance of the variance of is 144 find the regression coefficients. ( Linear Regression ) Page 6 of 9 is 16 and [2] Question 20 [4] a) The manufacturing cost of an item consists of `900 as overhead charges, the material cost is `3 per item and the labour cost is ` for items produced. How many items should be produced to have average cost minimum. ( Application of Calculus ) OR b) The average cost function associated with producing and marketing an item is given by units of find (i) The total cost function and marginal cost function (ii) The range of values of the output , for which is increasing. ( Application of Calculus ) Question 21 [4] a) Given that the observations are ( ); ( ); ( ); ( ) ( ) ( ) ( ) ( ) Find the line of regression on and hence estimate the value of when . ( Linear Regression ) OR b) If and are two lines of regression find (i) The coefficient correlation (ii) The variance of , if the variance of is 9 ( Linear Regression ) Question 22 [6] A fruit grower can use two types of fertilizer in his garden , brand P and brand Q. The amounts ( in kg) of nitrogen , phosphoric acid, potash and chlorine in a bag of each brand are given the table. Tests indicates that the garden needs atleast 240kg of phosphoric acid , atleast 270kg of potash and atmost 310kg of chlorine. Page 7 of 9 Kg per bag Brand P Brand Q Nitrogen 3 3.5 Phosphoric acid 1 2 Potash 3 1.5 Chlorine 1.5 2 If the grower wants to minimize the amount of nitrogen added to the garden , how many bags of each brand should be used ? What is the minimum amount of nitrogen added in the garden ? ( Linear programming ) ************************************** Page 8 of 9 Blue Print for I Preboard Examination 2019-2020 Section A: Sl No. 1 Chapters 2 Marks 4 Marks 6Marks Total Relations and Functions 2 2+1 - 12 2 Algebra 2 1 1+1 14 3 Calculus 4 5+2 2 40 4 Probability 2 1+1 1+1 14 2X10=20 4X9=36 6X4=24 80 Total Section B: Sl no. 1 Chapters 2 Marks 1 4 Marks 1+1 6 Marks - Total Vectors 2 Three Dimensional Geometry 2 1+1 - 8 3 Application of Integrals - - 1 6 2X3=6 4X2=8 6X1=6 20 Total 6 Section C: Sl Chapters no. 1 Application of Calculus 2 Marks 2 4 Marks 1+1 6 Marks - Total 8 2 Linear Regression 1 1+1 - 6 3 Linear programming - - 1 6 2X3=6 4X2=8 6X1=6 20 Total Page 9 of 9
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