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ISC Class XII Board Exam 2016 : Mathematics

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Akshat Goyal
Kendriya Vidyalaya (KV) No. 1, Vasco-Da-Gama, Goa
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ISC 2016 MATHEMATICS x 62/87,21 2) x &200(176 2) &281&,/ (;$0,1(56 x 68**(67,216 )25 7($&+(56 'HGLFDWHG WR DOO P\ ORYHO\ VWXGHQWV 0D\ *RG KHOS \RX DOZD\V 7KLV VPDOO ERRNOHW FRQWDLQV VROXWLRQ RI ,6& 0DWKHPDWLFV 7KH FRPPHQWV IURP WKH FRXQFLO H[DPLQHUV XQGHU VROXWLRQ RI HYHU\ TXHVWLRQ PDNHV WKLV D YHU\ KDQG\ JXLGH IRU VWXGHQWV WR XQGHUVWDQG ZKDW WKH FRXQFLO H[SHFWV DV DQVZHU IURP WKH VWXGHQWV , KRSH WKDW WKH VWXGHQWV ZLOO ILQG WKLV WR EH XVHIXO Md. Zeeshan Akhtar VW -DQXDU\ %/$1. 3$*( MATHEMATICS SECTION A [10 u 3] Question 1 (i) Find the matrix X for which: (ii) 1 2 5 4 X= 1 3 1 1 Solve for x, if: tan(cos-1x) = (iii) Prove that the line 2x 3y = 9 touches the conics y2 = 8x. Also, find the point of contact. (iv) Using L Hospital s Rule, evaluate: v) (v) Evaluate: (vi) Using properties of definite integrals, evaluat evaluate: GFS G (vii) dx regression are x + 2y 2 5 = 0 and 2x + 3y 8 = 0 and the variance The two lines of regressions varian of y and the coefficient coe of x is 12. Find the variance of correlation. (viii) Express ( )( ) in the form of a + ib. Find its modulus and argument. (ix) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total otal of 8? (x) Solve the differential equation: x + = 3 2 www.guideforschool.com Comments of Examiners (i) Errors were made by candidates while finding inverse and correct = form. (ii) Some errors were made by candidates in converting inverse trigonometric functions (one to another) and also algebraic calculations. (iii) Some of the candidates made mistakes while finding the point of contact and also proving the condition of tangency. (iv) Mistakes were made while converting in the form, also applied L Hospital s Rule without observing the indefinite form . (v) Errors were made while using formula to convert o 1. A few candidates attempted it to by integration by parts and made the steps complicated - they could not proceed further. Suggestions for teachers Basic operations with Matrices need to be explained. Practice should be given in Inverse and Multiplication of Matrix by a Matrix. Explain clearly the conversion of inverse circular functions (one to another form). More practice is required in conversion through diagram and by using formulae. Tangency condition must be taught more clearly. Ample practice needs to be given for computing point of contact with and without formulae. GF FS i) Several candidate (vi) candidates made mistakes in applying tthe property ( = ) ( ) . Som Some attempted by other methods of integration and could not reach thee proper result. (vii) candidates solved the equations vii) Many unnecessarily and tried to identify byx arbitrari arbitrarily. represent The condition for the two equations to represe regression lines and the tests for identifying the them were not used by some. es failed to simplify in the fform (viii) Many candidates + also made mistakes while d careless l computing Modulus and Argument. (ix) A number of candidates made mistakes in getting all favorable cases. Several candidates had difficulty in calculating the values of ( ) , ( ) ( & ) . (x) Many candidates made errors while identifying the differential equation and attempted to solve it by variable-separable form. Applications of L Hospital s Rule for calculating Limits of Indeterminate minate Forms should be taught well. In the rule the numerator and denominator need to differentiate separately till form is removed. Varieties of questions of integration by substitution titution may be given for practice. Teach properties operties of ddefinite integrals well and see that the candidates learn to apply them appropriately. Properties when correctly used reduce cumbersome calculations into simple ones. Emphasize on mutually exclusive events and independent events. Plenty of practice is required to understand the application of ( ) ( = ) + ( ) ( ) . Give ample practice in various types of differential equations. www.guideforschool.com MARKING SCHEME Question 1. (i) 1 2 5 4 X= 1 3 1 1 1 4 5 4 = 1 5 1 1 = , = = 1 4 1 2 X= 1 5 1 3 3 14 X= 4 17 (ii) cos = tan GF FS S tan tan-1 = = 1 = x 5 5x = 4x2 2 9x2 = 5 x= or o cos = tan = cos x= (iii) Line 3y = 2x 9 y = 3 m = 2 3 F y2 = 8x a = 2 The condition: a = mc 2 = 5 5 2 1 Squaring on both sides 1 3 www.guideforschool.com 2 = 2 the line touches the parabola 3 = -8x or (2x 9)2 = 72 Point of contact= , 4x2 36x + 81 = 72x Substituting values of a & m 4x2 + 36x + 81 = 0 (2x + 9)2 = 0 -3 x = 9 2, y = -3 =-6 point of contact is , 6 1 cot GFS G FS (iv) = = = = = 3 = (v) I = . = tan tan = tan ( 1) = tan tan I1 (- log| )| I1 = tan tan x = t sec2 x dx = dt I1 = = = www.guideforschool.com I = + log |cos | + (vi) I = I= (i) dx I = (ii) Add (i) + (ii), 2I = 0 I = 0 (vii) bxy = and byx = -1 d r d 1 r = = = 0 866 = and = 4 GFS G F FS (viii) viii) = = = (2 + ) (1 + )(1 2 ) 2) (( ) ( ) ( )( ) )( ) = + i | = | + = Argument of z = tan-1 / z = tan-1 (1) = (ix) ^( 2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6) 6,1),(6,2),(6,3),(6,4),(6,5),(6,6),(3,5),(5,3)` P(E) = = Or P(A)+P(B)-P (A B) = + = . (x) x + = 3 2 + = 3 If = = e logx \[ [ G[ =x www.guideforschool.com yx = [ F y = Question 2 (a) Using properties of determinants, prove that: + (b) + [5] = 4abc + Solve the following system of linear equations using matrix method: [5] 3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2 Comments of Examiners (a) Many candidates ndidates were not able to attempt this part orrectly. Several candidates expanded directly. At correctly. times, correct cofactors were not used. ed. Suggestions for teachers Explain every property of the determinants with proper examples. This type of questions should be taught in the class by discussion method. Logical and rea reasoning skills should be used to apply the correct property. GF F FS (b)) Several candidates made errors in finding tthe cofactors of elements of matrix. For finding the unknown matrix X, some candidates used postmultiplication with inverse of A. A few candidates solved using Cramer s Rule. The basics withh regards to cofactors ng adjoint and of elements, obtaining inverse and meaning of prepre-and pre-and postpostcation of matrices must be multiplication covered thoroughly in class class;; e.g e.g. if hen X=A-1B and not BA-1 as AX=B then such multiplication of matrices is not commutative. MARKING SCHEME Question 2. (a) + , = + + R1 R 1 R 2 R 3 0 2 = + 2 + www.guideforschool.com c2 c2 c1, c3 c3 c1 0 2 = + 0 2 0 + = 2c (ab + b2 bc) -2b ( 0 ca c2+ bc) = 2abc + 2b2c 2bc2 + 2abc + 2bc2 2 b2c = 4 abc (b) 3 = 2 5 3 1 | = | 2 0 5 1 2 A-1 = 6 2 Hence Proved. 1 1 0 2 1 2 1 2 = 3(0 2) 2 1 2 1 4 2 2 GFS G X = A-1B 2 1 2 1 X = 6 1 4 0 2 2 2 2 2 = 2 2 1 = 1 = 1, = 1, = 1 1 1 Question 3 (a) If sin-1x + tan-1x = , prove that: [5] 2x2 + 1 = 5 (b) Write the Boolean function corresponding to the switching circuit given below: A A A A B B C % www.guideforschool.com [5] A, B and C represent switches in on position and A , % DQG & UHSUHVHQW WKHP LQ off position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit. Comments of Examiners (a) Many candidates attempted this question but errors were made while squaring, simplifying and solving higher degree algebraic equations. Most of the candidates failed to prove the result. (b) Many candidates were unable to find correct simplification leads to required result (B+C). A few candidates made errors in sketching the simplified circuit. Suggestions for teachers Inverse Circular Functions laws needs to be taught thoroughly. The applications need to be illustrated with examples. Inter conversion of functions should also be done. It helps the simplification process. Laws of Boolean algebra need to be explained properly and enough practice must be given on simplification and drawing drawing the different types of Boolean expressions. GFS GFS MARKING SCHEME Question 3. (a) sin-1x + tan-1x = 2 tan-1x = 2 sin-1 x = co cos-1 x tan-1x = tan-1 x= x 2 = 1 x4 = 1 x2 x4 + x2 1 = 0 x2 = But x2 (b) cannot be negative. x2 = x 2 + 1 = 5 ) $ % & $ $ % $ % $ % & $$ $% $ % $& % & 2 % $ $ $& % & % % & $& % % % & $& = B + C + AC = B + C www.guideforschool.com B C Question 4 (a) Verify the conditions of Rolle s Theorem for the following function: f(x) = log(x2 + 2) log 3 on [-1,1] [5] Find a point in the interval, where the tangent to the curve is parallel to x-axis. (b) Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is 10. Also, find its eccentricity. [5] GFS G FS ments of Examiners Comments (a) Several candidates made errors because of confusion continuity between the closed and open interval for continu and differentiability of the function. Most of the candidates did not find the point of contact of the tangent and the curve. b) Many candidates made errors while calculating values (b) of a and b due to application of wrong formula or simplification mistakes. Most of the candidates andidates forg forgot to find eccentricity and complete the equation of Ellipse. Suggestions for teachers The Th different erent criteria for Rolle s Rolle s and ge s Mean Value Va Lagrange s theorems need to be und understood and differentiated. Difference between closed and open intervals and its significance should be taught by c. sketching the curve, etc. Conics and nd their equations need to be thoroughly hly explained. explained. Basics such as axis, directrix, latus ectrix, eccentricity and la rectum, etc. etc. of a conic need to be explained with the help of good number of examples. MARKING SCHEME Question 4. (a) f(x) = log(x2+2) - log 3 is continuous [-1, 1] f ( = [ ) I [ H[LVWV LQ -1, 1) f( -1) = f (1) = 0 All the conditions of Rolle s theorem are satisfied then there exists c in ( - VXFK WKDW I F www.guideforschool.com 2 =0 (1 + ) c = 0 lies between -1 and 1. Hence, Rolle s theorem is verified. The point where the tangent is parallel to x axis is (0, log ) (b) Let the equation of the ellipse be + a > b. Given, 2b = 2ae - (i) b = ae =1 and = 10 - (ii) b2 = 5a We also know, b2 = a2 (1 e2) (iii). Substituting (i) and (ii) in (iii), we get 5a = a2 b2 = a2 5a 10a = a2 2 = 50 2 = 100 GFS G F FS equation of ellipse is + = 1 n x2 + 2y2 = 100 or complete equation Also, b2 = a2 (1 e2) A 0 = 100 (1 e2) 50 e2 = 1 - = e= Question 5 (a) If log y = tan-1x, prove that: tha 2 (1 + x ) (b) + (2 (2x 1) [5] =0 A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area. www.guideforschool.com [5] Comments of Examiners (a) First differentiation was done by many candidates and errors were made in finding the second derivative and framing the required result. Use of the rule for composite function as well as chain rule was forgotten by several candidates. (b) In most of the cases, candidates did not read the question attentively and drew wrong sketch. Due to this, area was not correct in variable. Candidates also made mistakes in calculating the dimensions of rectangle. Suggestions for teachers Derivatives of all forms of functions require continuous practice and review from time to time. Sufficient time needs to be spent on this topic. Students need to familiarize themselves with the area, perimeter, surface and volume of 2-Dimensional and 3-Dimensional figures in the syllabus. The function to be optimized needs to be expressed in terms of a single variable by using the given data. For maximum value f'(x)=0 and f''(x) < 00.. This needs to be taught taught well. GFS GFS MARKING SCHEME Question 5. (a) log y = tan-1 x = = (1+x + 2) = (1+x2) = Differentiating again, (1 + x2) + (2 = ) ( 1 + ) + (2 1) =0 www.guideforschool.com (b) = cos QR = sin A( ) = Area of the Rectangle. A( ) = PQ QR = 2OQ QR = r2 sin 2 = 2r2 cos 2 = 0 S R r Q O P OR Differentiating area: A=2xy=A=2x 1 cos 2 = 0 2 = = is the critical point. U2 sin 2 = 4r2 < 0 a = Area is maximum at So, sides of rectangle ar are GFS G F FS 2 , and Area = r2 sq. units nits Question 6 (a) Evaluate: [5] [5 (b) curves y = 6x x2 and y = x2 2x. Find the area of the region bound by the curv www.guideforschool.com [5 [5] Comments of Examiners (a) Some candidates made mistakes in correct substitution. In most of the cases, candidates who used appropriate substitution did not give the final answer in terms of but left in terms of the new variable. (b) Many candidates got the correct solution for this question. A few candidates solved the question with incorrect limits. Some candidates made mistakes in sketching the curve and finding limits. Suggestions for teachers Practice needs to be given in Integration using substitution and special integrals. Teachers must instruct students not to leave their answers in terms of the new variable. Sketching of curves may not be necessary always but a rough sketch will always help the student to understand the requirements, the area required to be found, the points of intersection and the limits of the definite integral. Since the functions given was simple algebraic the integration of such functions will not n cause problems and the solution can ca be easily found. GFS GFS SC MARKING SCHEME Question 6. (a) Let I = dx I = P t u = sin x cosx Put du = (sin x + cos x)dx u2 = 1 sin 2x sin 2x = 1 u2 ( ) = = ( ) -1 = sin + = sin-1 (b) ( ) + The curve y = 6x x2 y = (x 2 + 9 represents a parabola with vertex at The curve y = x 2 2 x y = (x 1)2 1 represents a parabola with vertex at (1 - i at (4, 8) 0 www.guideforschool.com 2 6 (6 ) ( 2 ) = (8 = 64 = 2 ) = 8. 2. sq units Question 7 (a) Calculate Karl Pearson s coefficient of correlation between x and y for the following data and interpret the result: [5] (1, 6), (2, 5), ), (3,7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12) GFS G FS FS (b) The marks obtained by 10 candidates in English and Mathematics are given below: Marks in Engli English Marks in Mathematics 20 13 18 21 11 12 17 14 19 15 17 12 23 25 14 8 199 21 22 19 Estimate the probable score for Mathematics if Estim i the marks obtained in English are 24. Comments omments of Examiners (a) Some candidatess were not clear about the differe different formulae for the calculation coefficient. culation of correlation c Most of the candidates did not write the interpretation of the result. A few candidates solved the problem by Spearman s Rank correlation coefficient method. (b) Candidates were not clear about the formulae used for byx and bxy. In most of the cases, candidates applied wrong formula to evaluate byx. Some candidates used the equation for the regression line of x on y instead of y on x to find y when x is given. Suggestions for teachers teacher Plenty of practice is required in order to solve problems on correlation. Teachers should make sure that the students understand the need for the appropriate formula at appropriate situations. Accuracy levels should be maintained at highest level as such simplifications involve simple calculations only. www.guideforschool.com [5] MARKING SCHEME Question 7. (a) x Y Xy x2 y2 1 6 6 1 36 2 5 10 4 25 3 7 21 9 49 4 9 36 16 81 5 8 40 25 64 6 10 60 36 100 7 11 77 49 121 8 13 104 64 169 9 12 108 81 1 144 45 81 462 285 85 789 GFS GF G FS r= ( ) ] [ ( ) ][ ( = = ][ ( ) ) ][ ( [ [ ( ) = 0 95 There is a high +ve correlation between the two variables. v (b) Eng (x) Maths (y) dx=x- dy=y- (dx)2 (dy)2 dxdy 20 17 4 -1 16 1 -4 13 12 -3 -6 9 36 18 18 23 2 5 4 25 10 21 25 5 7 25 49 35 11 14 -5 -4 25 16 20 12 8 -4 -10 16 100 40 17 19 1 1 1 1 1 14 21 -2 3 4 9 -6 19 22 3 4 9 16 12 15 19 -1 1 1 1 -1 www.guideforschool.com = 160 = 180 = 125 = 110 = 254 byx = ( ) ( ) = = = 1 13 Regression equation y on x is y 18 = ( 16) Probable score y 18 = (24 16) y = 9.0909 + 18 = 27 0909 = 27 1 Question 8 GFS G FS S (a) (b) A committee of 4 persons perso has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the prob o more re girls than boys. probability that the committee consists of An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn ur and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball ball. [5] [ [5] Comments of Examiners (a) a) Very few candidates understood the question and a answered it correctly. Most of the candidates andidates cou could not write the complete combinations correctly. A few fe f red based on the definition of candidates answered probability and others approached on conditional probability. (b) Few candidates not considered the possibility of one white and one black when two balls are drawn from the first urn. Most of the candidates answered correctly. Suggestionss for teachers Explain the probability laws and it its applications ons thoroughly. Teachers T must explain lain the concepts with the help of addition and multiplication rule and its applications including conditional probability and condition based problems. Explain and make students to understand the problem and identify the cases that satisfy the situation and conditions. www.guideforschool.com MARKING SCHEME Question 8. (a) Number of ways the committee can be selected: = 14 8 = 1001 70 = 931 or 6 . 8 + 6 . 8 + 6 . 8 + 6 No. of Committees consists of more girls than boys= 6 + 6 8 P(E) = = (b) = = 0 188 P(transferring 2 white balls to urn 2 and drawing a white ball from urn 2) = 10 5 13 10 GFS G = p (transferring 2 black balls to urn 2 land drawing a white from urn 2) = = P(transferring1white and a black ball to urn 2 and drawing a white ball from urn 2) = = Required probability = [45 + 24 + ] = Question 9 (a) Find the locus of a complex number, z = x + iy, satisfying the relation 2 . Illustrate the locus of z in the Argand plane. [5] (b) Solve the following differential equation: x2 dy + (xy + y2) dx = 0, when x = 1 and y =1. www.guideforschool.com [5] Comments of Examiners (a) Most of the candidates did not sketch the locus in the Argand plane. Made mistakes while marking centre and with correct radius. (b) Candidates made errors while solving the integration part due to lack of clear understanding about the formula of special integrals and sufficient practice. Suggestions for teachers Sketching of straight lines and curves (circle, conics etc.) should be practiced regularly. Post differentiation and integration solving of differential equations needs a lot of practice. MARKING SCHEME Question 9. (a) z = x + iy 2 GFS G GF FS + ( 3) 2 + ( + 3) Squaring x2 + y2+ 6y + 9 2 (x2 + y2 + 6y + 9) 8 +9 0 x2 + y2 + 18y x 2 + (y + 9)2 81 + 9 0 x 2 + (y + 9)2 72 72 72 R Re(z) (0, 9) (b) = put y = vx = + v + = = 2 ( ) = www.guideforschool.com 1 = log + log +2 2 = ( + 2) = ( + 2 ) 3x2y = y + 2x (when x = 1, y = 1) Or = c= GFS G FS Or =c log =c SECTION CTION B (20 (2 Marks) Question 10 (a) For any three vectors , , , show that , , are coplanar. [5] (b) Find a unit vector perpendicular to each of the vectors + and where = 3 + 2 + 2 and = + 2 2 [5] www.guideforschool.com Comments of Examiners (a) Some of the candidates failed to apply condition of coplanarity. Many candidates made errors while simplifying the scalar triple product , , . A few candidates failed to write the correct order of scalar triple product. (b) Some candidates used cross product without evaluating + . Some candidates made mistakes while evaluating the unit vector in the final answer. Suggestions for teachers Vector notations, usage, dot and cross products in terms of the vectors or their components need to be taught well and in detail. Unit vector and its properties must be taught with a number of examples. Scalar triple product and its applications need to be taught with the help of practical examples. Students must be asked to read the instructions given in the question y carefully. GFS GFS ARKING SCHEME MARKING Question 10. (a) a) [ ] = ( ) . ( ) ( ) ) . [ + ] = ( = ( ). [ + + ] . ( ) = . + . ( ) + . ( . . ( = [ ] [ ] (b) = [ ] [ ] = 0, hence vectors are coplannar. OR If a+b+c=0 , one vector can be represented as liner combination of other two a,b,c are coplanar Hence proved that a-b, b-c, c-a are coplanar Since (a-b)+(b-c)+(c-a)=0 = 3 + 2 + 2 , = + 2 2 + = 4 + 4 = 2 + 4 = ( + ) ( ) = (4 + 4 ) (2 + 4 ) = 4 4 0 = 16 16 8 2 0 4 www.guideforschool.com = 8 (2 2 ) = | | = = ) ( Question 11 (a) Find the image of the point (2, 1, 5) in the line = = [5] Also, find the length of the perpendicular IURP WKH SRLQW WR WKH OLQH. (b) Find the Cartesian equation of the plane, passing through the line of intersection of the planes: . (2 + 3 4k ) + 5 = 0 and . ( 5 + 7k ) + 2 = 0 and intersecting y-axis at (0, 3). [5] ners Comments of Examiners GFS GFS FS (a) Candidates andidates made mistakes in calculating a point M on the line PQ and then applying the condition to prove that AM PQ ie + + = 0. Suggestions for teachers Need to emphasize on complete understanding anding of perpendicularity perpend condition on at different stages. Give more practice in solving prob problems based on the concept of Image of a given point with reference to a line. Some made errors while calculating the length of perpendicular. perpendicul (b) b) Most of the candidates were able to score marks in this part. MARKING SCHEME Question 11. (a) Let A (2, -1, 5) be the point and an PQ be the line = = . Let M be any point on the line PQ is 0 U U U 8) $ DR s of AM U U U 13 AM is perpendicular to PQ a1, a2 + b1b2 + c1c2 = 0 U U U 13) = 0 P M 100 r + 90 + 16r + 4 + 121r + 143 = 0 $ U = M is (1, 2, 3) www.guideforschool.com Q 0 LV WKH PLGSRLQW RI $ $ = 1, = 2, =3 &RRUGLQDWHV RI $ DUH 1) Length of the perpendicular = ( 1) + 3 + ( 2) = 14 units Question 12 (a) In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B and C carries out this task. A has 45% chance, B has 35% chance and C has 20% chance of doing the task. The probability that A, B and C will take more than the allotted time is , respectively. If it is found that the time taken is more than the allotted time, what is the probability y that A has done the task? [5] GFS (b)) The difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution. [5 [5] Comments omments of Examiners Ex (a) a) Many candidates used Baye s theorem correctly but took the probabilities of A, B, C, as 45, 35, 20, instead of percenta percentages. Some candidates did not implement the correctly. theorem co (b) equations b) Several candidates made errors in solving the equat 2 2 np-npq=1, (np) -(npq) =11. Most of the candidates were w not clear in writing the correct Binomial Distribution. Distributio Suggestions for teachers Teachers need need to emphasize tha that no probabilities are ratios and not onditional probability numbers. Conditional ght clearly which concept is to be taught helps the he students to understand the th ced concept Baye s theorem. advanced Revision on must be done on concepts concept mean and vvariance binomial i of binomia probability distribution. While solving for n, p, and q it must be noted that p+q=1 . More practice in solving equations in two /three unknowns is essential. MARKING SCHEME Question 12. (a) Let E1, E2, E3, denote the events of carrying out the task by A, B and C respectively. Let H denote the event of taking more time. Then P(E1) = 0.45 P(E2) = 0.35 P(E3) = 0.20 P(H/E1) = P(H/E2) = P(H/E3) = www.guideforschool.com ( ) = ( ). ( / ) ( ). ( ). ( ). ( / . = . . . = 0.625 = (b) np npq = 1, np(1 q) = 1 2 2 (i) 2 2 (np) (npq) = 11, (np) (1 -q ) = 11 (ii) GLYLGLQJ LL E\ L ZH JHW T 1 + q = 11 11q 12q = 10 q = 5/6, p = 1/6 we get n = 36 GFS G F FS The distribution is given bby ( + )36 or x ~ B (36, ) SECTION CTION C (20 (2 Marks) Question 13 (a) per DW annum,D compounded semi-annually A man borrows DW D annum semi-annually and agrees to pay it in 10 equal semi-annual installments. Find the value of each installment, nstallment, if the first payment is due at the end of two years. (b) products A and B. Each unit of A requires 3 grams [[5] A company manufactures two types of produ of nickel and 1 gram of chromium, while each eac unit of B requires es 1 gram of nickel and 2 grams of chromium. The firm can produce 9 grams of nickel and 8 grams of chromi chromium. The profit is RQ HDFK XQLW RI SURGXFW RI on Q HDFK each X unit of type B. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution. www.guideforschool.com [5] [5 Comments of Examiners (a) Many candidates made errors while identifying the total time and deferred time. Deferred annuity formula was noted wrongly in most of the cases. Some substitutions were also not correct. Very few candidates solved this part correctly. Suggestions for teachers Help students to distinguish between total time and deferred time, the concept of deferred annuity and its application in solving problems. (b) In some cases, all the constraints were not used and hence coordinates of all feasible points were not obtained. The optimum function and all possible constraints in the form of inequations have to be put down from what is stated in the problem. Students must be made to solve the constraints equations in pairs to obtain all feasible points leading to maximum or minimum value of desired function. MARKING KING SCHEME GFS G F FS S Question estion 13. (a)) P= n= L 10 L and m = 3 P = ( ) ) ( ) ( ) 20,000 = ) ( ) a = 3235 90 Size of the installment is (b) Let x units of product oduct A and y units of product B. B y Maximize Z = 40x + 50 50y Subject to constraints 9 3x + y 9 x + 2y 8 4 A x 0, y 0 C Solving, we get A(0, 4), B(3, 0), C(2, 3) x B3 O 8 At A, z = 40 0 + 50 4 = 200 B, z = 40 3 + 50 0 = 120 C, z = 40 2 + 50 3 = 230 Maximum profit is 230, when 2 units of type A and 3 units of type B are produced. www.guideforschool.com Question 14 (a) The demand function is x = the price per unit. Find: where x is the number of units demanded and p is (i) The revenue function R in terms of p. (ii) The price and the number of units demanded for which the revenue is maximum. A bill of 1,800 drawn on 10th September, 2010 at 6 months was discounted for 1,782 at a bank. If the rate of interest was 5% per annum, on what date was the bill discounted? (b) [5] [5] Comments of Examiners (a) Some me of the candidates made mistakes in obtaining Revenue function in terms oof p. Several everal candidates GFS GFS FS forgot to apply the condition for the maxima < 0 for R. (b) b) Some candidates used True Discount formula instead of Banker s Discount or relevant formula for calculating n .. Several candidates had no idea how to get the Date of Discounting. Suggestions for teachers Question should be answered as asked in the paper and Revenue function n must be found in terms of p. Application p ication of Maxima aand Minima should be taught by taking up more examples. All relevant terms ms and formulae need to be taught well for complete understanding. nding. MARKING SCHEME Question 14. (a) Total revenue function = px Revenue function = = ( ) dR/dp = (24 4p)/3 For maximum or minimum dR/dp = 0 (24 4p) / 3 = 0 p=6 d2R / dp2 = -4/3<0 R is maximum Number of units x = (24 12)/3 x=4 Hence, R is maximum when 4 units are demanded at the price of www.guideforschool.com SHU XQLW (b) A = 1800; i= 5% p.a. BD = 1800 1782 = 18 BD = Ani 18 = 1800 n n = year = 73 days Date of expiry: March 13, 2011 Date of discounting: December 30, 2010. Question 15 (a) The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year , find the values of x and y p y 2000 is in the he data given below: [5] GFS GF G FS Commodity Price in the year 20 2000 ( Price in th the year 2010 ( b) (b) A 50 60 B x 24 C 30 y D 70 80 E 116 120 12 F 20 28 From the details given below, below calculate culate the five yearly moving averages of the number of candidat candidates who have studied in a school. Also, plot these and original data on the same graph paper. Year 1993 1994 1995 1996 19 1997 1998 1999 2000 2001 2002 Number of Students 332 317 357 392 402 40 405 410 427 405 438 www.guideforschool.com [5] Comments of Examiners (a) Most of the candidates attempted this part correctly. Errors were committed by a few candidates while calculating value of y . (b) Moving averages were mostly correctly calculated but for plotting, centered moving averages were required. Some candidates did not use the centered averages. In some cases, the graphs were not neat. Suggestions for teachers Help the students to identify whether the question is based on aggregate or average method. Thorough knowledge of the formula is required. Moving Averages need to be calculated correct to two decimal places. For plotting, centered averages correct to one decimal place are sufficient. The axes should be labelled and the plotting and sketching should be as neat as possible; the graph should be given a caption. GFS GF G FS FS MARKING SCHEME SC Question 15. (a) Commodity Price in 2000 200 2010 50 60 x 24 30 y 70 80 116 120 20 28 286+x 312+y A B C D E F 286 + x = 300 x = 14 P01 = 100 116 = 100 36 = y (b) Year Number 5yrs moving total www.guideforschool.com 5yr moving average 1993 332 -- -- 1994 317 -- -- 1995 357 1800 360 1996 392 1873 374 6 1997 402 1966 393 2 1998 405 2036 407 2 1999 410 2049 409 8 2000 427 2085 417 2001 405 -- -- 2002 438 -- -- GFS www.guideforschool.com S Note: For questions having more than one correct solution, alternate correct solutions, apart from those given in the marking scheme, have also been accepted. www.guideforschool.com GENERAL COMMENTS: (a) Topics found difficult by candidates: Indefinite Integrals (use of substitution or integration by parts) Definite Integrals use of properties. Inverse Circular Functions (formulae and relations) Differential Equations (solving Homogeneous and Linear Differential Equations) Vectors in general Annuity (Deferred annuities) Conics in general Probability use of sum and product laws and identifying all cases. Maxima and Minima (b) Concepts in which candidates got confused: Regression lines: y on x and x on y Sum and product laws of probability 3 D: Image age of a given ppoint andd perpendicular distance Conditional probability property in Baye s theorem Price Index by aggregate and Price Relative methods Differences between and usage of formulae for BD, TD, BG, DV, etc. GFS (c)) Suggestions for candidates: Study the entire syllabus thoroughly and revise from time to time. Concepts of Class XI m must be revised and integrated with the Class XII syllabus. Develop logical and reasoning skills to have a clear understanding. Revise all topicss and formulae involved and make a chapter wise or topic-wise list of these. Make wise choices from the options available in the question paper and management time wisely. Be methodical and neat in working. www.guideforschool.com

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