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LA MARTINIERE FOR BOYS, KOLKATA REHEARSAL EXAMINATION 2023-24 1 SUBJECT: MATHEMATICS Time: 3 hours + 15 mlns. Reading time CLASS:XII Full marks: 80 (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.) This Question Paper consists ofthree sections A, B & C Candidates are required to attempt all questions from Section A and all questions EITHER from Section B QB Section C. Sectio'! A: biternal choice has been provided in two questions o/lHIO marks each, two questions of/our marlcs and two questions ofsix marks each. Section B: Internal choice has been provided in one question oftwo marks and one question offour marks. Section C: Internal choice has been provided in one question oftwo marks and one question offour marks. All workings, including rough works, should be done on the same sheel as, and adjacent to the rest ofthe answer. The intended marks for questions or parts ofquestions are given in braclcels [ J. Matl,ematical tables and grap!, papers are provided. SECTION 65MARKS Q uestiool /. a " ,1 h . h b In subparts (i) to (x), choose_the co"ect options and ,n t e su ,parts ,x,/ to 1xv/. answer I e questions as instructed. (i). If [- 16 -2~ (a) (-2, 0) J -r1 (b )(2, O) +a then a is equal to (ii).lf xsinxdx = -xcosx ' \ (a) sinx + c (b) cosx + c (iii). . - 1 (tan(-6)) is The principal value of tan (a)-6 (1) is a skew-symmetric matrix, then (x, y) = 6 (c)(-2, 1) (c) c (c)6 - 2rr (b)2rr I 2-!'thlllflllllln -Ill (UI B) (d)(2,-1). (1) (d) none of these. (l I (d) 6. (iv). 0 rd ~; a (cty) er and degree of the differ~tial equation ~~+sin = 0 are respectively 2 ~)I; (c} 2, not defined {d)not de~ed, 2. llJ 1 (v).Associated to a d . ran om expenment,two events A and B are such that p (B) = ! , 1 P(A/8) = d P(A ) s ~ U B = 5 Then the value of P(A) is {a)l. (c) .!. 10 10 (vi). 22 The value of 23 (a) 2 lf f(x) 2 1 = Ix - i 25 is 24 25 . 26 (b) 26 9 (vii). 23 24 (d) (1) (c) 22 91, x e R, then at x (d) 0. =9, Ul (a) f(x) is not continuous (b) f(x) is continuous but not differentiable (c) f(x) is continuous and differentiable (d) f(x) is neither continuous nor differentiable (viii). If A= {l,2,3} andB= {a,b}, thenthenumberoffunctionsfromAtoBis (a) 3 (ix). (b) 6 (c) 8 (d) 12. The mapping/: Z Z given by f(x) = 2x + 1 is (a)injective but not surjective (b1surjective but not injective (c)bijective (d)neither injective nor swjective. (x).Let A be a square matrix. Assertion (A): Every square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. Reason(R): a) b) c) d) (xi). (xii). (1] flJ (1) (A + A') is a symmetric matrix and (A - A') is a skew-symmetric matrix. Both A and R are true and R is the correct explanation for A. Both A and R arc true but R is not the correct explanation for A. A is true and R is false. A is false and R is true. Writ.e the smallest equivalence relation on the set A = {l, 2, 3}. 1x-11 1 ? What is the range of the function ( x ) = -;:i-,x * ll-l'thtllciulla 211 (I.MB) (1) (1) (xiii). A ~ag contain s 3 red and S black balls and 11 second bag contain s 6 red 4 black balls. A ball is dra wn from eac h bag. Fin d the probability that both ball s are red. Ill (xiv). Find the val ue of sln - 1 c05 - 1 Ill (xv). Find the slop e of normal to the curve y = ~ at x = o. Ill s (-if-)+ (- ). Qu esti on 2 (21 . ._ jf (x) = ::;is not defined at x= 3. What value should be assigned to / (3) for contin~ity r- {f /(x ) at x = 3? Justify your answer. (ii V ~ o: per ime ter of a squa,,, ;,, dea eas ing decreasing when the side is 10 an? the rate of 2 cm /s . At what rate ;, ;ts side Qu est ion 3 lll z,;;f) dx. f_l1 log (2-J Eva lua te: Qu est ion 4 (21 Th e equation of the tangent at (2,3) on the curve y 2 val ues of a and b. Qu~ (i) S = ax 3 + bis y = 4x - 5. Fin d the lll Eva lua te: J[sln (lo gx) + cos (log x)]d x OR ' (ii) (ii) ./ii ni Eva lua te: f sin x cou dx Qu est ion 6 Find the dom ain of the function stn (11 -t x - 1. Queitlon 7 (41 Find the principal value of sin- 1 (cos {s1n- 1 ( - ~)}). Question 8 Evaluate: (4) f x 2 tan- 1 xdx. Question 9 X y Q f4J x 2 - 2x + 3, when x < 1 Given f (x) = 2, when x = 1 { 2 2x - Sx + 5, whe nx > 1. Examine whether f (x) is (a) continuous and (b) differentiable at x / IIfr> = e -1 then prove that d% ~ = I. OR 101 ' (1+lo 1x) 1 don 10 [4) unbiased coin is to~. If the result is a head, a pair of unbiased dice is rolled and the of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ... , 12 is picked and the number on the card is no~ . What is the probability that the noted num ber is either 7 or 8? / V OR (1i).l n a test, an examinee either guesses or copi es or knows the answer to a multiple choice ques tion with four choices. The probability that he makes a guess is and the probability that be copies the answer The probability that his answer is correct, give n that he copied it, isi. Find the probability that he knew the answer to the question, given that he correctly answered it. is i Quesdon 11 [6) A shopkeeper has 3 varieties of pens : A, B and C. Aman purchased I pen of each variety for a total of t 21 . Akbar purchased 4 pens of varie ty A, 3 pens of B and 2 pens of c for ~ 60. Anthony purchased 6 pens o( A, 2 pens of B and 3 pens of C for t 70. Using matrix method, find the cost of each pen of A, B and C. l2 M lhuna lln ~ (LMB) 2 vc the diffcrcnti al \. y dx- (x + (6) . . equation = O,glven that y = 1 whenx = 2Y 2 )dy 1. OR Evaluate: f xz-x+1 dx xz+x+1 Qu~ 3 (6) ' _y("~poster is to contain 72 cm of printed matter with borders of 4 cm each at the top and 2 ttom and 2 cm on each side. Find the dimensions of the poster if the total area of the poster isminimum. 'I ~ ind the height of the cone ofmaximwn :::.,., which can be inscribed in a spbcrc of dius 12cm. Question 14 (6) Two numbers are selected at random (without replacement) from the positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. a) Constru ct the probability distribution for X. b) Find the probability that the larger number is less than S. c) Find the mean of X SECTIONB IS MARKS QuestJo n 15 . (5) Jn subparts (I) and (ii), choose the correct options and in the subparts (iii) to (v), answer the questions as instructed. (i) Vectors aand b represent diagonals of a rhombus. Then: (a) ii. b (ii) =0 x b= 0 (c) ldl = lbl (d) ii= b. If 'i,J, k are three mutually perpendicular vectors, then (a)i-j= l (iii) (b) ii (b)FxJ =O (c) t ~=O (d) l X ~ = 0. Find a vector of magnitude of 1Ounits and parallel to the vector 2r + 3J t l-~fa1h,nu 1ln .511 (LMB) I<. . Find the direction cosine s of the hne (iv) X+2 2)1- 1 7 =5 3-Z =, (v) If the intercepts made by the plane 2x 3y + 4z = 12 on the coordinate axe s are a, b and c respectively, find the val ue of a + b + c. Question 16 (i) (ii) If (d + b) 2 = (2) ca_- b) 2 then find the angle between the vectors d and OR b. Two adjacent sides of a parallelogram are given by t + le and 2 + J +le .Fi nd the area of the parallelogram. Question 17 (i) (4) Find the coordinates of the point where the line through the points A(3,4, 1) and B(S,1,6) crosses the XZ plane. OR (ii) Fin d the equation of the pla ne passing through the poi nts (1, 1, 2) and (2, 4, 3) and perpendicular to the plane x - 3y + 7z + S = 0. Quesdon 18 (4) Findthe area of the region bounded by the curve 2 y = 2x + 1 and the line x - y - 1 = 0. SECTIONC 15M AR KS Quesdon 19 (51 In subparts (i) and (ii), choose the co" ect options and in the subparts (iii) to (v). ans wer the questions as instructed. (i). If the . cost function of a certain commodity is C(x ) average cost of producing (a)!t 451 (b)~ 450 (c) ' 449 = 2000 + S0x S - x 2 , units then the is: (d) l 224 5. (ii) . lf the objective function is z 3x + Sy and the com er points of the feasible region are (1, 5), ( 4, 2) and (3, 3), wh ere docs the minimum value of z = (a) (1, 5) (b) {4, 2) (c) (3, 3) U-M atlit mat ka., . O,MB) {d)Noneofthese. occur? (iii). lfth e two lines ofrc ~cs s 0" 10n arc 2x _ " _ _ the mean of y. . ~ 4 - 0 and 8x - 3y - 38 = 0, then find (iv). The demand function of a m r . which MR (maroina} o)nopo 1st is given by x = 100 - 4p. State the quantity at oa revenue = o. (v).The average revenue run~ . "'"on f an item is given by AR = x - 3x2 Find the maroin111l revenue funct.ion. - I tion lO (l) A company produces a commodity with t 24,000 fixed cost. The variable cost is . a~e<l to be 25% of the total revenue re<:orded on sellin g the product at a rate oft 8 per umt. Fm c cost function and the revenue function. OR The total cost function of producing x units of a commodity is given by C == 16 - I2x + 2x 2 Find the level of output at which it is minimwn . Question 21 (4) Find the line of regression of y on x from the following data: I I % 1 y 9 \ 2 3 4 I I 10 12 8 Hence, estimate the value of y when x = 5. Q stion 22 (4) An aero plane can carr y n maximum 200 pass engers. A profit of Rs 1,000 is made on ach first-class ticket and n profit of Rs 600 is made on each economy class ticket. The airline reserves at least ,20 seats for first class. How ever, at least 4 times as many passengers prefer to travel by economy class to the first class. Determine how many tickets of each type must be sold in order to maximiz e the profit for the airline. What is the maximum profit? Formulate the above LPP mathematically and then solve it graphically. OR . A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food I 12,Ma tbrmatln -7/1 (lMB) contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while food ll contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. ll costs Rs 500 per kg to purchase food 1 and Rs700 per.kg lo purchase food n. Detennine the minimum cost of such a mixture. ll,.M11bemalln -lfl Cl.MB)
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