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CBSE Class 12 Sample / Model Paper 2021 : Mathematics (with Marking Scheme / Solutions)

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Class: XII Session: 2020-21 Subject: Mathematics Sample Question Paper (Theory) 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 case-based 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 SectionIV and 3 questions of Section-V. You have to attempt only one of the alternatives in all such questions. Part A Sr. No. Mark s Section I All questions are compulsory. In case of internal choices attempt any one. 1 Check whether the function : defined as OR = is one-one or not. 1 Page 1 of 10 = . How many reflexive relations are possible in a set A whose 1 2 A relation R in = { , , } is defined as = { , , , , , , , }. Which element(s) of relation R be removed to make R an equivalence relation? 3 A relation R in the set of real numbers R defined as function or not. Justify ={ , : = } is a 1 1 OR An equivalence relation R in A divides it into equivalence classes What is the value of and , , . 1 4 If A and B are matrices of order and respectively, then find the order of matrix 5A 3B, given that it is defined. 1 5 Find the value of , where A is a 2 2 matrix whose elements are given by ={ = 1 OR 1 Given that A is a square matrix of order 3 3 and |A| = - 4. Find |adj A| 6 Let A = [ where 7 ] be a square matrix of order 3 3 and |A|= -7. Find the value of + + is the cofactor of element Find OR Evaluate 8 9 cot + 2 2 sin Find the area bounded by = and = . 1 1 1 = , axis and the lines 1 How many arbitrary constants are there in the particular solution of the differential equation = ; y (0) = 1 1 OR 10 11 For what value of n is the following a homogeneous differential equation: = + 1 Find the area of the triangle whose two sides are represented by the vectors 2 . 1 Find a unit vector in the direction opposite to 1 Page 2 of 10 12 13 Find the angle between the unit vectors , given that | + | = 1 Find the direction cosines of the normal to YZ plane? + 1 14 Find the coordinates of the point where the line plane. 15 The probabilities of A and B solving a problem independently are respectively. If both of them try to solve the problem independently, what is the probability that the problem is solved? 1 16 The probability that it will rain on any particular day is 50%. Find the probability that it rains only on first 4 days of the week. 1 = = cuts the XY 1 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 mark 17 An architect designs a building for a multi-national company. The floor consists of a rectangular region with semicircular ends having a perimeter of 200m as shown below: Design of Floor Building Based on the above information answer the following: (i) If x and y represents the length and breadth of the rectangular region, then the relation between the variables is a) b) c) d) x + y = 100 2x + y = 200 x + y = 50 x + y = 100 Page 3 of 10 (ii)The area of the rectangular region A expressed as a function of x is a) b) c) d) 1 + (iii) The maximum value of area A is a) b) c) d) 1 (iv) The CEO of the multi-national company is interested in maximizing the area of the whole floor including the semi-circular ends. For this to happen the valve of x should be 1 a) 0 m b) 30 m c) 50 m d) 80 m (v) The extra area generated if the area of the whole floor is maximized is : a) b) c) 1 d) No change Both areas are equal Page 4 of 10 18 In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03 Based on the above information answer the following: (i) The conditional probability that an error is committed in processing given that Sonia processed the form is : 1 a) 0.0210 b) 0.04 c) 0.47 d) 0.06 (ii)The probability that Sonia processed the form and committed an error is : 1 a) 0.005 b) 0.006 c) 0.008 d) 0.68 (iii)The total probability of committing an error in processing the form is 1 a) 0 b) 0.047 c) 0.234 Page 5 of 10 d) 1 (iv)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 forms. If the form selected at random has an error, the probability that the form is NOT processed by Vinay is : 1 a) 1 b) 30/47 c) 20/47 d) 17/47 (v)Let A be the event of committing an error in processing the form and let E1, 1 E2 and E3 be the events that Vinay, Sonia and Iqbal processed the form. The | A is value of = a) 0 b) 0.03 c) 0.06 d) 1 Part B Section III 19 Express 20 ( ), < < 2 in the simplest form. If A is a square matrix of order 3 such that = , then find the value of |A|. 2 OR If 21 =[ 2 ], show that A A + I = O. Hence find A . Find the value(s) of k so that the following function is continuous at = 2 Page 6 of 10 c s 22 = Find the equation of the normal to the curve y= 23 si ={ + Find , > 0 perpendicular to the line 2 2 = . 2 2 OR Evaluate 24 25 Find the area of the region bounded by the parabola . 27 28 = = and the line Solve the following differential equation: = 26 2 , 2 2 = . Find the area of the parallelogram whose one side and a diagonal are represented by coinitial vectors - + and 4 + 5 respectively 2 A refrigerator box contains 2 milk chocolates and 4 dark chocolates. Two chocolates are drawn at random. Find the probability distribution of the number of milk chocolates. What is the most likely outcome? 2 Find the vector equation of the plane that passes through the point (1,0,0) and contains the line = . 2 OR Given that E and F are events such that P(E) = 0.8, P(F) = 0.7, P (E F) = 0.6. Find P ( E | F ) 2 Section IV All questions are compulsory. In case of internal choices attempt any one. 29 30 31 Check whether the relation R in the set Z of integers defined as R = { , + is "divisible by "} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [ ]. If y = 2 + sin , find . Prove that the greatest integer function defined by differentiable at = = [ ], < < is not 3 3 3 Page 7 of 10 OR = If 32 , = Find the intervals in which the function = tan , a) strictly increasing 33 Find 34 2 2+ 2+ , 2 given by = 3 3 is b) strictly decreasing 3 2+ . Find the area of the region bounded by the curves + = ,y = 3 OR Find the area of the ellipse 35 + = 3 using integration Find the general solution of the following differential equation: = + 3 Section V All questions are compulsory. In case of internal choices attempt any one. 36 = [ If ], find 5 . Hence Solve the system of equations; = = + = OR 5 Evaluate the product AB, where =[ = ] = [ ] Page 8 of 10 37 + + = + = Find the shortest distance between the lines = + + ( + + ) = + ( + + ) If the lines intersect find their point of intersection 5 OR 38 Find the foot of the perpendicular drawn from the point (-1, 3, -6) to the plane + + = . Also find the equation and length of the perpendicular. 5 Solve the following linear programming problem (L.P.P) graphically. Maximize = + subject to constraints ; + + , 5 OR The corner points of the feasible region determined by the system of linear constraints are as shown below: 5 Answer each of the following: (i) Let = be the objective function. Find the maximum and minimum value of Z and also the corresponding points at which the maximum and minimum value occurs. Page 9 of 10 (ii) Let = + , where , > be the objective function. Find the condition on and so that the maximum value of occurs at B , C , . Also mention the number of optimal solutions in this case. Page 10 of 10 ` Class: XII Session: 2020-21 Subject: Mathematics Marking Scheme (Theory) Sr.No. 1 = Let = Objective type Question Section I for , = , Hence Marks 1 is one one OR reflexive relations 1 2 (1,2) 1 3 Since is not defined for , . = 1 OR 4 3x5 5 = = ] =[ OR =[ ][ ]=[ 1 1 ] 1 |adj A|=(-4)3-1=16 6 0 1 cot ) + C 7 1 OR is an odd function sin 1 = 8 1 = = = [ ] Page 1 of 14 ` 9 0 1 OR 1 3 10 11 12 1 | |= 1 | | = | + | = + + . = . = | || | cos = = . = = 1 13 1,0,0 1 14 (0,0,0) 1 15 16 1 = 1 ( ) ( ) =( ) Section II 17(i) (b) 1 17(ii) (a) 1 17(iii) (c) 1 17(iv) (a) 1 17(v) (d) 1 18(i) (b) 1 18(ii) (c) 1 18(iii) (b) 1 18(iv) (d) 1 18(v) (d) 1 Section III 19 ( [ si )= c s 4 4 4 [ ] i c ] Page 2 of 14 ` [ [ + 20 ] = = | ] = |=| | | || | = | | | | | | | | = or ] | |= 1 + | = [tan | =| || | | | 1 OR =[ =[ 21 ]=[ ][ ], + = = [ cos sin ] + = =[ + = =[ [ =O = = = ] 1 ]=O O ] [ ] sin ] 1 sin sin = 1 Page 3 of 14 ` 22 = + = 23 = + = = = = = , ( = tangent is parallel to it 1 > ) = tan = So that, =+ = + = 1 tan Put, = = =4 = , = = + 1 = = = a + 1 OR = = = + = [ + = [ 24 + = = [ + + + ] = ]= ] + + 1 + 1 Page 4 of 14 ` = [ ] = ]= [ = 25 = ; = sin = cos = 26 = cos + = = 1 = , = Let = + = 4 + x + | |= + = = 27 = = 3 + + 4 +4 Area of parallelogram = | x | = = Let the normal vector to the plane be Equation of the plane passing through (1,0,0), i.e., is ( ) = 0 .(1) plane (1) contains the line = + = 0 and = 0 28 + + 1 = 1 Hence equation of the plane is ( ) = 0 i.e., = 0 1 Let x denote the number of milk chocolates drawn X P(x) Page 5 of 14 ` 0 1 = ( ) 2 x = 1 = Most likely outcome is getting one chocolate of each type OR E F F P E | F ) = P Now = F = -----------(1) = P (E) + P (F) - P (E 1 F) = 0.8+0.7-0.6=0.9 Substituting value of P E | F ) = 29 .9 . = . . in (1) = Section IV (i) Reflexive : Since, a+a=2a which is even (a,a) Hence R is reflexive Z (ii) Symmetric: If (a,b) R, then a+b = 2 b+a = 2 (b,a) R, Hence R is symmetric 1 (iii) Transitive: If (a,b) R and (b,c,) R then a+b = 2 ---(1) and b+c =2 ---- (2) Adding (1) and (2) we get a+2b+c=2( + a+c=2 ( + ) a+c=2k ,where + b = k Hence R is transitive [0] = {...-4, -2, 0, 2, 4...} 30 Let u = (a,c) R 1 and v = sin Page 6 of 14 ` so that y = u + v Now, u = = Also , v = sin log v = = + ----(1) , Differentiating both sides w.r.t. x, we get 1 [ + ] ----- ( 2) log (sin Differentiating both sides w.r.t. x, we get = cot + log = sin 1 [ + log [ + + = Substituting from = , ] in ------ (3) ] + sin we get [ + log ] 31 RHD = = LHD = = = = [ + ] [ ] 1 = [ ] [ ] = 1 Since, RHD LHD Therefore f(x) is not differentiable at x = 1 1 OR = tan = sec = = sec tan Page 7 of 14 ` = sec tan = = = . = ] = = = For > For < cos < < < < < 1 < < < to be strictly decreasing 1 > > ] [ to be strictly increasing b) ec a [cot ] = ( ) = . = tan 32 a) cot 1 . . . , cot = > cos < > > < [ , ] 1 Page 8 of 14 ` 33 = Put + + + = + = + + + + + + = + + + (1) Comparing coefficients of y and constant terms on both sides of (1) we get A+B = 1 and 3A + 2B = 1 1 Solving, we get A = , B = 2 = 34 + + + We get + = Solving = + = + = + + = + + 1 = Required Area = = = [ + ] +[ + [ + sin ] ] 1 OR Page 9 of 14 ` Required Area = Y x [ = = [ 35 + ]= ] 1 1 The given differential equation can be written as + = = , IF = = = = = l g = 1 The solutions is : 36 = ( = + = | |= + ) A is nonsingular, therefore =[ = | | ] =[ 1 = + exists ] = 1 Page 10 of 14 ` The given equations can be written as: [ ] [ ]=[ ] Which is of the form = [ ] = [ = = , = 1 ] = [ ] ][ = , 1 = OR =[ [ ] [ =[ ( = X= [ ]= [ = , 1 ] ] ][ = , = + 1 = =[ [ ] = [ ] We have ][ ] = [ = , 37 ] )= ] ] = [ ] 1 = 1 = + + Page 11 of 14 ` = = | = = = + |= = + ( ). = + + + 1 1 = 1 The lines are intersecting and the shortest distance between the lines is 0. Now for point of intersection + + ( + + = + + = + + = = Solving (1) ad (2) we get, + = + + ( + 1 = Substituting in equation of line we get = + ( + ) = Point of intersection is , , 1 OR Let P be the given point and Q be the foot of the perpendicular. Equation of PQ + = Since Q lies in the plane + = 1 P , , , + , Let coordinates of Q be = + + + + + + + = + + + = = Page 12 of 14 ` + = = Length of the perpendicular = + , + = , + + 1 1 1 38 Max = Subject to + + + ------------- (1) (2) (3) , 3 = Corner Points A (0, 50) 50 B (0, 200) 200 C (50, 100) 250 D (20, 40) 100 = = , + 1 = 1 Page 13 of 14 ` OR (i) Corner points O(0,0) A(0,8) B(4,10) C(6,8) D(6,5) E(4,0) (ii) = = 0 -32 -28 -14 -2 12 = , 1 , Since maximum value of Z occurs at B(4,10) and C(6, 8) + = + = = Number of optimal solution are infinite 1 2 Page 14 of 14

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