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CBSE Class 12 Pre Board 2021 : Mathematics (Kendriya Vidyalaya (KV), Lucknow Region)

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Kendriya Vidyalaya Sangathan(Lucknow Region) Class: XII Session: 2020-21 Subject: Mathematics SECOND PRE BOARD EXAM. Time Allowed: 3 Hours Maximum Marks: 80 General Instructions: 1. This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks 2. Part-A has Objective Type Questions and Part -B has Descriptive Type Questions 3. Both Part A and Part B have choices. Part A: 1. It consists of two sections- I and II. 2. Section I comprises of 16 very short answer type questions. 3. Section II contains 2 case studies. Each case study comprises of 5 casebased MCQs. An examinee is to attempt any 4 out of 5 MCQs. Part B: 1. It consists of three sections- III, IV and V. 2. Section III comprises of 10 questions of 2 marks each. 3. Section IV comprises of 7 questions of 3 marks each. 4. Section V comprises of 3 questions of 5 marks each. 5. Internal choice is provided in 3 questions of Section III, 2 questions of Section- IV and 3 questions of Section-V. You have to attempt only one of the alternatives in all such questions. Sr. No 1 Part A Section I All questions are compulsory. In case of internal choices attempt any one. If n(A) = p, then Write number of bijective functions from set A to set A. Marks 1 OR 2 If set A has 3 elements and set B has 4 elements, then write the number of injective functions that can be defined from set A to set B. For the set A = {1. 2, 3}, define a relation R in the set A as follows: R = {(1, 1),(2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation. 3 What is the range of function f(x)= | 1| ( 1) 1 1 ? OR State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive relation. 0 If A = [ ], then find 16 . 0 0 cos sin If A =[ ] and A + / = I, then what is value of ? sin cos OR 1 6 Given that A is a square matrix of order 2 2 and |A| = - 1. Find |adj A| 2 4 2 4 Find value of x if | |= | | 2 1 3 1 7 Find (sin x + cos x) 1 4 5 8 9 OR Find cos x Find the area bounded by the curve y = sin and the x axis between x = 0 and x = 2 . 3 2 1 1 1 Write the degree of differential equation (1 + ) =( ) . OR For what value of n is the following a homogeneous differential equation: 3 = 2 + 2 10 11 12 Find the projection of the vector on the vector + . Find| |, for a unit vector , ( ) .( + )=12. , where P and Q are the Find the unit vector in the direction of vector points (1, 2, 3) and (4, 5, 6), respectively 13 If direction cosines of a line are 14 15 16 1 1 1 1 , , , then find value of k. 3 3 3 +3 1 5 Find the coordinates of the point where the line = = cuts the XY 3 1 5 plane. A speak truth in 70% cases and B speak truth in 85% cases, then find probability that they speak the same fact. 1 3 Given that the events A and B are such that P(A)= , P(AUB)= and P(B)=p. 2 5 Find p if they are independent. Section II Both the Case study based questions are compulsory. Attempt any 4 sub parts from each question (17-21) and (22-26). Each question carries 1 1 1 17 1 mark Mr Shashi, who is an architect, designs a building for a small company. The design of window on the ground floor is proposed to be different than other floors. The window is in the shape of a rectangle which is surmounted by a semi-circular opening. This window is having a perimeter of 10 m as shown below Based on the above information answer the following (i) If 2x and 2y represents the length and breadth of the rectangular portion of the windows, then the relation between the variables is a) 4y 2 = 10 b)4y=10 (2 ) c) 4y=10 (2 + ) d) 4y 2 = 10 + (ii) The combined area (A) of the rectangular region and semi-circular region of the window expressed as a function of x is a) A = 10x + (2 + ) x2 2 ) x2 2 ) x2 2 1 1 b) A = 10x (2 + c) A = 10x (2 d) A = 4xy + 2 5 2 1 4 x2 where y= + (2+ )x (iii) The maximum value of area A, of the whole window is 50 a) A = m2 b) A c) A 1 4 50 = m2 +4 100 = m2 +4 50 2 d) A = 4 m (iv) The owner of this small company is interested in maximizing the area of the whole window so that maximum light input is possible. For this to happen, the length of rectangular portion of the window should be 1 20 m +4 10 b) m +4 4 c) m +10 100 d) m +4 a) (v) In order to get the maximum light input through the whole window, the 1 area (in sq. m) of only semi-circular opening of the window is a) b) c) 18 100 (4+ )2 50 +4 50 (4+ )2 d) same as the area of rectangle portion of window The members of a consulting firm rent cars from three rental agencies : 50% from agency X, 30% from agency Y and 20% from agency Z. From past experience, it is known that 9% of the cars from agency X need a service and tuning before renting, 12% of cars from agency Y need a service and tuning before renting and 10% of the cars from agency Z need a service and tuning before renting. Assume that the rental car delivered to the firm needs service and tuning. Based on the above information answer the following: (i) The probability that the cars need service and tuning, if it came from agency Y, is a) 9% b) 12% c) 10% d) 30% (ii) The probability that the cars need service and tuning, if it came from agency Z, is a) 50% b) 12% c) 10% d) 20% 1 (iii) What is the probability that the car needs service and tuning? 1 a) 10.1% b) 11.0% c) 1.01% d) 10.01% (iv) If the rental car delivered to the firm need service and tuning, then the probability that agency X is to be blamed, is 20 a) 1 1 101 b) c) d) 45 101 25 101 81 101 (v) If the rental car delivered to the firm need service and tuning, then the probability that agency Z is not to be blamed, is a) 20 101 b) 65 101 c) d) 81 101 25 101 1 Part-B Section-III 19 1 1 Prove that : 3sin 1 = sin 1 (3 4 3 ), x [ , ] 2 1 Find the value of x , if 2[ 0 2 2 20 3 ]+[ 1 2 0 5 6 ]=[ ] 2 1 8 OR 3 1 If A=[ ], Show that A2 5A+7 I = 0 1 2 21 Find the value(s) of k so that the following function is continuous at = 5 2 + 1 5 F(x)={ 3 5 > 5 22 23 Prove that the curves x = y2 and x y = k cut at right angles if 8k2 = 1. 4 Evaluate 0 log(1 + tan ) dx OR Evaluate 3 1+ 6 1 tan 2 2 dx Find the area of the region bounded by the parabola 2 = 8 and the line = 2. Solve the following differential equation: + y = 1,(y 1) 2 26 Find a vector in the direction of vector =5 +2 which has magnitude 8 units. 2 27 Find the vector and Cartesian equation of the lines that passes through the origin and (5,-2,3). 2 28 Find the probability distribution of number of heads in three tosses of a fair coin. OR 2 24 25 2 4 29 Probability that A speaks truth is . A coin is tossed. A reports that a head 5 appears. What is the probability that actually there was head? Section IV All questions are compulsory. In case of internal choices attempt any one. Show that relation in the set A={ x , 0 12}, = {( , ), | | is multiple of 4} is an equivalence relation. 30 If cos = cos( + ), cos 1, 31 Find Find of function y=sin 1 ( 2 +1 ). 1+4 2 = ( + ) . sin 3 3 OR of function 3 sin y=tan 1 ( ). 1+cos 32 3 Prove that y = 4 sin (2+ cos ) sec +tan x is an increasing function of x in [0, 33 Evaluate 0 34 Find the area enclosed by the ellipse 2 ]. 3 dx 2 9 + 2 4 = 1. 3 OR Find the area bounded by the curve y= cos x between x=0 and x= 2 35 Find general solution of differential equation x = + Section V All questions are compulsory. In case of internal choices attempt any one. 3 36 1 1 0 2 2 4 Given A=[2 3 4] and B=[ 4 2 4] Find the product of matrices AB 0 1 2 2 1 5 and hence solve the system of equations x-y=3 2x +3y + 4z = 17 y +2z =7 OR 1 1 1 For the matrix A=[1 2 3], Show that A3 6A2+5A+11 I=0.Hence, find A-1. 2 1 3 5 37 + )+ ( + ) and Find the shortest distance between the lines = ( +2 = (2 )+ (2 + +2 ) OR Find equation of the plane passing through the intersection of the planes x+ y+ z = 6 and 2x+ 3y+ 4z +5= 0, and the point(1, 1, 1 ). Solve linear programming problems graphically: Maximise Z=3x+2y Subject to x+2y 10, 3x+y 15, x, y 0. OR Minimise Z= 3x+ 4y Subject to x+ 2y 8, 3x+2y 12, , x, y 0. 5 38 5

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