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Ryan International School Bangalore I PREBOARD 2018 2019 Subject : Mathematics Class: XII Date : .0 . 2018 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 [10X2] (i) If f : R (ii) If R defined by ( ) , (iii) For what value of - , then show that , find . , the matrix [ ] is a skew symmetric matrix. (iv) Find the equation of the line passing through ( determinant. (v) Using L Hospitals rule evaluate : is neither one-one nor onto. * ) and ( ) using the + (vi) The radius of a cylinder is increasing at the rate of 2 cm/sec and its height is decreasing at the rate of 8 cm/sec. Find the rate of change of its volume when radius is 3cm and height is 6cm. RIS 12th STD Mathematics I Preboard 2018-2019 [Page 1 of 7] (vii) Prove that ( ) (viii) Find the general solution of the following differential equation (ix) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75. What is the probability of passing the Hindi examination. (x) Two integers are selected at random from integers 1 through 11. If the sum is even , find the probability that both the numbers are odd. Question 2 [4] Show that the function f in A = onto. Hence find , - defined as ( ) is one - one and ( ) Question 3 [4] Prove that : ( ) ( ) ( ) Question 4 [4] Prove using properties of determinants : ( | ) ( ) | ( RIS 12th STD Mathematics I Preboard 2018-2019 ( ) ) [Page 2 of 7] Question 5 If [4] ( ) { ( ) is continuous at . Find the value of Question 6 [4] a) If ) , then show that ( ( ) ( ) OR b) Verify Rolle s theorem for the function f , given by ( ) * ) on ( + Question 7 [4] a) Find the intervals in which the function ( ) is (i) Strictly increasing (ii) Strictly decreasing OR b) If the tangent to the curve line , find and at ( ) is perpendicular to the . Question 8 [4] Evaluate: RIS 12th ( ) STD Mathematics I Preboard 2018-2019 [Page 3 of 7] Question 9 [4] Find the particular solution of the following differential equation given that when ( ) Question 10 [4] a) Of the students in a school , it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year , one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? OR b) A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets the total of 10. If A starts the game, then find the probability that B wins the game. [6] Question 11 If A = [ ] and B = [ , ] , Find AB. Hence solve the system of equations , Question 12 [6] Find the maximum volume of the cylinder which can be inscribed in a sphere of radius cm. (Leave your answer in terms of ) Question 13 [6] a) Evaluate : OR b) Evaluate : RIS 12th ( ) by the method of limit of sums. STD Mathematics I Preboard 2018-2019 [Page 4 of 7] Question 14 [6] a) Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement , then find the probability distribution of number of bad oranges drawn. Hence find the mean and standard deviation of the distribution. OR b) A box contains 4 red and 5 black marbles . Find the probability distribution of the red marbles in a random draw of three marbles. Also find the mean and standard deviation of the distribution. SECTION B ( 20 MARKS) Question 15 a) Write a unit vector in the direction of the sum of the vectors b) Find the angle between the two planes ( ) and [2] and ( ) [2] c) Show that the lines and are coplanar. Question 16 [2] [4] a) If the volume of the parallelopiped is 45 cubic units whose coterminus , edges are represented by the vectors and . Find the value of . OR b) If and , then find such that and Question 17 [4] a) Find the vector equation of the plane which contains the line of intersection of ) the planes ( and ( ) and which is ) perpendicular to the plane ( RIS 12th STD Mathematics I Preboard 2018-2019 [Page 5 of 7] OR b) Find the distance between the point ( of the line ) and the point of intersection and the plane Question 18 [6] Find the area of the region in the first quadrant enclosed by the and the circle using integration. axis , the line SECTION C ( 20 MARKS) Question 19 a) The demand for a certain product is represented by the function where is the number of units demanded and is the price per unit ( in ` ). Find the marginal revenue when 10 units are sold. [2] b) Determine the cost of increasing output from 100 to 200 units if the marginal cost MC ( in ` per unit) is given by MC = [2] c) Find the regression coefficient bxy for the following data : , , , , , [2] Question 20 a) A radio manufacturer produces [4] sets per week at a total cost of ` ( ) He is a monopolist and the demand of his product is , where is the price in rupees per set . Show that the maximum net revenue is obtained when about 30 sets are produced per week. What is the monopoly price. OR b) The cost of manufacturing of certain items consists of `1600 as overhead charges, `30 per item as the cost of material and the labour cost ` for items produced. How many items must be produced to have a minimum average cost? RIS 12th STD Mathematics I Preboard 2018-2019 [Page 6 of 7] Question 21 [4] a) Given the data x 1 2 3 4 5 y 3 7 9 10 11 Find the regression line x on y and hence predict the value of x , if y = 20 OR b) If the two regression lines of bivariate distribution are , calculate : (i) and , the arithmetic means of x and y respectively. (ii) Estimate the value of x when y = 7 (iii) Find the variance of y when x = 3 and Question 22 [6] A fruit grower can use two types of fertilizers in his garden , brand P and brand Q. The amount (in kg) of nitrogen , phosphoric acid, potash and chlorine in a bag of each brand are given in adjoining table. Test indicate that the garden needs atleast 240 kg of phosphoric acid , atleast 270 kg of potash and atmost 310 kg of chlorine. 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? Nitrogen Phosphoric acid Potash Chlorine Kg per bag Brand P Brand Q 3 1 3 1.5 3.5 2 1.5 2 ************************************** RIS 12th STD Mathematics I Preboard 2018-2019 [Page 7 of 7]
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