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CBSE Class 12 Exam 2023 : Mathematics : cluster paper

6 pages, 54 questions, 0 questions with responses, 0 total responses,

Bhuvana Ramesh
The Velammal International School (TVIS), Panchetti, Chennai

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WEST CHENNAI SAHODAYA CLUSTER EXAMINATION 2022-23 CLASS XI MATHEMATICS (041) TIME: 3 HOURS MAXIMUM MARKS: 800 General Instructions: 1. This Question paper contains five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions. 2 Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each. Section B has 05 questions of 2 marks each. Section C has 06 questions of 3 marks each. Section D has 04 questions of 5 marks each. Section E has 03 Case-study / Source-based /Passage based question with sub- parts (4 marks each). There is no overall choice. However, internal choice has been provided in 0 2 Questions of Section B 03 Questions of Section C 0 2 Questions of Section D 0 2 Questions of Section E You have to attempt only one of the alternatives in all such questions. SECTION A (Question numbers 01 to 20 carry I mark each.) Followings are multiple choice questions. Select the correct option in each one of them. 1. If A is a symmetric matrix of order 3 symmetric matrix? (b) A. A (a) A A 02. L.et (c) A- AT P= ACB. If A=| B 5) 03. If - i+j-2k (b) 2x3 and. b= (a) 04. The value of k and C=5 1 -1) then, the order of matrix (b) 2i-j- 4k are two (d) 2x1 vectors, then +b (c) 3j function f(x) = equals (d) -2i+j+4k { x-I"*** is k, if x =1 at x=.is (a)2 (b) is not (d) A+A (c) 2x2 -i+2j+ 2k for which the following (2 (2 (a) 12 then, which of the (c) -1 (d) 1 continuous Pis 05. d x equals tan x- 1 (a)anX=)+C (b) +C 2 06. = 0 then, (a) I +C = (d) 4 (c) 3 The maximum value of Z =4x +3y subject to constraint x+ y s8. x.y >0 is (d) 24 (c) 65 (b) 32 The scalar projection of the vector 3i-xk on the vector i+ v2j+k is , (a) 0 09. (n") (b) 2 (a) 56 08. tan x - If m andn respectively, are the order and degree ofthe differential equation dx 07. (c) log tanx-1|+C (d) - tan x (d) -2 (c) 2 (b) 1 The shortest distance between the lines f = , +.b and f = , (a)bx (b) b 6x(,- 6,- (c) a,- then x = +ub is given by () (d) b b 3 10. 11. 12. If A = |4 -5 6 3 5 2 24* +A Ifa non-singular matrix A satistying f 2P(A) = a square matrix (b) 2 oforder 3 (b) and A|=4, (a) y 17. (d) 210 (d) is dx y (d) x-yC dy dx (b) ay (c) a (d) y(loga) LetA ={m, a, t, h}. If a relation R in the set A is given by R {(m, m), (a, ), (1, a), (h, t), (t, h)}, then R is (a) only reflexive (b) only symmetric 18. to 26 general solution ofthe differential equation (c) xy=C (b) x-y=C (a) y' x +C =a, (a>0) then, equal 15 The If y = P(AUB)== = 16. and (d) 2A - 1 then Jadj(2A)| is (c) 2 then P(B) =and P(A |B) =, (a)26 15. of third row -I=0 then, A (c) 1+2A (b)-2A-I Given that A is (a) 2 14. element 2nd column) is (d) 2 (c) 3 (b)-14 (a) 14 value at two points, then it has Ifa linear programming problem has same optimal (a) infinite solutions (b) two solutions (c) unique solution (d) three solutions (a) A+21 13. (the then, the cofactor of a only transitive (d) equivalence (C) and parallel to y-axis, 5 Vector equation of line passing through (1,0,-2) (a) f =j+2(i-2k) (6) f=i- 2k+(i) (c) f =i-2j+) (d) f i-2k 2 0 Pollowings are Assertion-keason based questions. In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). answer out of the following choices. Choose the correct (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true and R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. 19. Assertion (A): Unit vector along i+2j-k is t Reason (R): For two non-zero vectors and b, .b=0 implies aLb. 20. Assertion (A): |(x + sin x) dx = 0. Reason (R): if f-x) |fx)dx =2 f)dx, = f() 0 0, if f(-x) = -f(x) 21. SECTION B (Question numbers 21 to 25 carry 2 marks each.) A relation R in the set ofreal numbers R is given by R= {(a. b) : a > b, such that a, beR}. Check the transitivity of relation R. Is it symmetric? OR Find the value of 22. ( cosec|cosec The volume of a cube is increasing at the rate of 8 cm/s. How fast is the surface area increasing when the length of its edge is 12 cm? 23. Write the unit vectors which are perpendicular to both the vectors =i+j+k and b=i+j. OR Ifl + 5-60., -b-40 and al=22, then find 6. 24. If cosy Excos(a+ 25. If . b and .b+b.ei , y) then, prove cos (a+y) that dx sin a are unit vectors such that +b+ =0 then, find the value of SECTION C (Question numbers 26 to 31 carry 3 marks each.) 26. Find: 27. . Two integers are selected at random from the integers one to elcven. If their sum is even, find the probability that both the numbers are odd. OR A bag contains 5 red and 4 black balls, a second bag contains 3 red and 6 black balls. One of the two bags is selected at random and two balls are drawn at random (without replacement). What is the probability that both the balls drawn are red? 28. Evaluate: 4 Xdx 1+sinx OR Evaluate: [xl+x-2]]dx. 29. Solve the differential equation: (1+e*" jdx+e**|1-dy=. 29 OR Solve the differential equation: ydx - (x+2y )dy = 0. 30. Solve the following Linear Programming Problem graphically: Minimize Z=[x +8y subject to constraints x+yS5.xs4.y 2 2.x >0. 31. Also, write the coordinate of point at which Z Evaluate: |(Vtan x -Vcot x Jdx. is obtained. SECTION D (Question numbers 32 to 35 carry 5 marks each.) 32. Find the area enclosed by the parabola 4y = 3x* and the line 2y =3N - 12. 33. Determine the equations of a line passing through the point (1. 2. -4) and perpendicular to the two lines x-8 y+19-2-10 and; - -2 OR Find the foot of perpendicular drawn from the point (2, 3.-8) to the line 4 34. = Also, find the perpendicular distance from the given point to the line. Determine whether the relation R defined on the set R of all real numbers as R = {(a,b):a, b e R and a-b+V3 E S, where S is the set of all irrational numbers, IS reflexive, symmetric and transitive. OR Considr t: R- given by f(x)=* 3x+4 Show that fis one-one and onto. 35. Usingmatrices. solve the ix-2y7t =20. system of equations: 3x+4y+Sz = 18. 2x-y +8z =13, and SECTION E (Question numbers 36 to 38 carry 4 marks each.) This section contains three Case-study/ based Passage questions. First two questions have three and sub-parts (i), (i) (ii) of marks 1, 1 and 2 respectirely. Third question has two sub-parts of 2 marks each. 36. CASE STUDY I: Read the following passage and the answer the questions given below. X A factory makes an open cardboard box for a jewellery shop from a sheet of side 18 cm by cutting off squares from each corner and folding upsquare the flaps. Assume that x is the of each square cut from the side-length (i) Write the volume (V) of the open box as a function of corners. x, where 'x' is the sidelength of each square to be cut-off from the corners. (ii) Write the conditions on dV dx and dv S o that the volume (V) is maximum. (ii) Find the maximum volume (V) of the open box. Also find the total area of the removed squares. 37.CASE STUDY II: Read the following passage and answer the questions given below. 5 Reeta goes for walk in a Community Park daily. She notices two trees in AP = 16 n (1) Reeta and BQ = 22 m stands at a a line (as seen in the figure above). whse heights are respectively. are 20 m apart trom each other. point (sav, R) in between these trees such that AR = m. Obtain an expression for RP +RQ, in terms of X. (i) Let fi)= RP +RQ. Then, find r(). 11) Is the function f(x) differentiable in xe (0. 20)? Find the intervals in which. f(x) is strictly increasing / decreasing. OR ii) At what value of x, f'(x) =0? Can we say that f(x) is minimum? Also write the minimum value of f(x). Use second derivative test. 38. CASE STUDY III: Read the following passage and answer the questions given below. Raghunath and Haridas are playing a game using a biased die. This biased die is tossed and respective are listed in the following table: Face Probability 2 0.1 3 probabilities for various 45 0.24 0.19 0.18 0.15 faces to turn up 6 K (i) What is the value of K? (ii) If a face showing an even number has turned up, then what is the that it is the face with 4' or '6'? probability

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