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CBSE XII Sample / Mock 2009 : MATHEMATICS with Answers (3 Sample Papers)

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Design of Question Paper Mathematics - Class XII Time : 3 hours Max. Marks : 100 Weightage of marks over different dimensions of the question paper shall be as follows : A. Weightage to different topics/content units S.No. 1. 2. 3. 4. 5. 6. B. Topics Relations and functions Algebra Calculus Vectors & three-dimensional Geometry Linear programming Probability Total Marks 10 13 44 17 06 10 100 Weightage to different forms of questions S.No. Forms of Questions 1. 2. 3. Very Short Answer questions (VSA) Short answer questions (SA) Long answer questions (LA) Total Marks for each question 01 04 06 No. of Questions 10 12 07 29 Total Marks 10 48 42 100 C. Scheme of Options There will be no overall choice. However, an internal choice in any four questions of four marks each and any two questions of six marks each has been provided. D. Difficulty level of questions S.No. 1. 2. 3. Estimated difficulty level Easy Average Difficult Percentage of marks 15 70 15 Based on the above design, separate sample papers along with their blue prints and Marking schemes have been included in this document. About 20% weightage has been assigned to questions testing higher order thinking skills of learners. (1) (2) Class XII MATHEMATICS Blue-Print I Sample Question Paper - I MATHEMATICS Class XII Time : 3 Hours Max. Marks : 100 General Instructions 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each. 3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, Internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION - A 1. Which one of the following graphs represent the function of x ? Why ? y y x x (a) 2. (b) What is the principal value of 2 2 1 cos 1 cos + sin sin ? 3 3 3. A matrix A of order 3 3 has determinant 5. What is the value of | 3A | ? 4. For what value of x, the following matrix is singular ? 5 x 2 (3) x + 1 4 5. Find the point on the curve y = x 2 2 x + 3 , where the tangent is parallel to x-axis. 6. What is the angle between vectors a & b with magnitude 3 and 2 respectively ? Given a . b = 3 . 7. Cartesian equations of a line AB are. 2x 1 4 y z + 1 = = 2 7 2 Write the direction ratios of a line parallel to AB. e 3 log x (x )dx 4 8. Write a value of 9. Write the position vector of a point dividing the line segment joining points A and B with position vectors a & b externally in the ratio and b = i + 3 + 1 : 4, where a = 2 i j + 4k j+ k 10. 3 1 2 1 4 and B = 2 2 If A = 4 1 5 1 3 Write the order of AB and BA. SECTION - B 11. Show that the function f : R R defined by f (x ) = 2x 1 , x R is one-one and onto function. Also find the 3 inverse of the function f. OR Examine which of the following is a binary operation (i) a*b= a+b , a, b N 2 (ii) a*b = a+b , a, b Q 2 for binary operation check the commutative and associative property. 12. Prove that 63 5 3 tan 1 = sin 1 + cos 1 16 13 5 (4) 13. Using elementary transformations, find the inverse of 2 6 1 2 OR Using properties of determinants, prove that bc a 2 + ac b 2 + bc c 2 + bc ac a 2 + ab b 2 + ab 14. c 2 + ac = (ab + bc + ca )3 ab Find all the points of discontinuity of the function f defined by x + 2, x 1 f (x) = x 2, 1 < x < 2 0, x 2 15. p q If x y = (x + y ) p +q , prove that dy y = dx x OR 1 + x2 + 1 x2 dy 1 , 0 < | x | <1 Find , if y = tan dx 1 + x2 1 x2 (x + 1)(x (x + 3)(x 2 2 2 2 ) dx 5) +4 16. Evaluate 17. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi- 1 2 1 vertical angle is tan . Water is poured into it at a constant rate of 5 cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m. 18. Evaluate the following integral as limit of sum 19. Evaluate 20. Find the vector equation of the line parallel to the line (5) x 1 3 y z + 1 = = and passing through (3, 0, 4). Also 5 2 4 find the distance between these two lines. 21. In a regular hexagon ABC DEF, if AB = a and BC = b , then express CD, DE, EF, FA, AC, AD, AE and CE in terms of a and b . 22. A football match may be either won, drawn or lost by the host country s team. So there are three ways of forecasting the result of any one match, one correct and two incorrect. Find the probability of forecasting at least three correct results for four matches. OR A candidate has to reach the examination centre in time. Probability of him going by bus or scooter or by other 1 1 3 1 3 , , respectively. The probability that he will be late is and respectively, if he 4 10 10 5 3 travels by bus or scooter. But he reaches in time if he uses any other mode of transport. He reached late at the centre. Find the probability that he travelled by bus. means of transport are SECTION - C 23. Find the matrix P satisfying the matrix equation 2 1 3 2 1 2 3 2 P 5 3 = 2 1 24. Find all the local maximum values and local minimum values of the function f (x ) = sin 2 x x, <x< 2 2 OR A given quantity of metal is to be cast into a solid half circular cylinder (i.e., with rectangular base and semicircular ends). Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its circular ends is : ( + 2 ). 25. Sketch the graph of | x 2 | +2, x 2 f (x ) = 2 x>2 x 2, Evaluate 26. What does the value of this integral represent on the graph ? ( Solve the following differential equation 1 x 2 y )d xy = x , given y = 2 when x = 0 dx 2 (6) 27. Find the foot of the perpendicular from P(1, 2, 3) on the line x 6 y 7 z 7 = = 3 2 2 Also obtain the equation of the plane containing the line and the point (1, 2, 3) 28. Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that if x = 0 or 1 kx P(X = x ) = 2kx if x = 2 , k (5 x ) if x = 3 or 4 k is +ve constant (a) Find the value of k. (b) What is the probability that you will get admission in exactly two colleges? (c) Find the mean and variance of the probability distribution. OR Two bags A and B contain 4 white 3 black balls and 2 white and 2 black balls respectively. From bag A two balls are transferred to bag B. Find the probability of drawing (a) 2 white balls from bag B ? (b) 2 black balls from bag B ? (c) 1 white & 1 black ball from bag B ? 29. A catering agency has two kitchens to prepare food at two places A and B. From these places Mid-day Meal is to be supplied to three different schools situated at P, Q, R. The monthly requirements of the schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchens A and B are 60 and 70 packets per month respectively. The transportation cost per packet from the kitchens to schools is given below : How many packets from each kitchen should be transported to school so that the cost of transportation is minimum ? Also find the minimum cost. (7) MARKING SCHEME SAMPLE PAPER - I Mathematics - XII Q. No. Value Points Marks SECTION A 1. (a) for every value of x there is unique y 2. 1 3. 135 1 4. 3 1 5. (1, 2) 1 6. 7. 1 (1, - 7, 2) or their any multiple 1 8. 1 9. 1 order of AB is 2 x 2 10. order of BA is 3 x 3 (8) Q. No. Value Points Marks SECTION B 11. To show f is one-one Let f is one-one To show f is onto Let or 3y = 2x 1 or 1 for all y R (codomain of f), there exist (codomain of f), such that every element in codomain of f has its pre-image in the domain of f. f is onto. To find f 1 Let (9) Q. No. Value Points Marks f 1 : R R given by f 1(y) = 1 OR (i) v a, may or may not belong to N. a * b is not always natural no. * is not a binary operation on N (ii) v a, b Q a * b Q. * is a binary operation on Q (iii) For * is commutative (iv) v a, b, c, Q. (10) Q. No. Value Points Marks a * (b * c) (a * b) * c 12. v a, b, c, Q * is not associative, Let & (1 + 1) ( ) (1) ( ) (11) Q. No. Sol.13. Value Points Marks A= IA ( ) Let R1 R2 ( ) R2 R2 2R1 (1) R1 R1 R2 (1) R2 R2 ( ) ( ) OR Operate R1 aR1, R2 bR2, R3 cR3 1 (12) Q. No. Value Points Marks Take a, b, c common from C1, C2, C3 respectively 1 R1 R1 + R2 + R3 C1 C1 C3, C2 C2 C3 1 On expanding by R1 we get Sol.14. Being a polynomial function f(x) is continuous at all point for x < 1, 1 < x < 2 and x 2. Thus the possible points of discontinuity are x = 1 and x = 2. To check continuity at x = 1 1 1 f(1) = 3. since, f(x) is not continuous at x = 1. (13) Q. No. Value Points Marks To check continuity at x = 2 f(2) = 0 (1) since f (x) is continuous at x = 2. The only point of discontinuity is x = 1. (1) Sol.15. Take log on both sides (2) (1) or or or or (1) OR (14) Q. No. Value Points Put Marks x2 = cos ( ) (1) ( ) 1 y = + cos 1 x 2 4 2 (1) ( ) or dy 1 = dx 2 2x 4 1 x (1) Sol. 16. Consider (1) (15) Q. No. Value Points Marks Consider (1) (2) Sol.17. Let r = radius of cone formed by water at any time h = height of cone formed by water at any time Given Also h = 2r Volume of this cone (1) (16) Q. No. Value Points Marks (1) (1) But or when h = 10m or 18. (1) For a = 1, b = 2, h = ( ) f(x) = 3x2 1 ( ) (1) (1) (17) Q. No. Value Points Marks (1) = 6 19. Given (1) .............. (1) (1) Consider ............. (2) (18) (1) Q. No. Value Points Marks From (1) and (2) 20. (1) Given line ( ). ............... (i) or, is passing through (1, 3, 1) and has D.R. 5, 2, 4 . Equations of line passing through (3, 0, 4) and parallel to given line is ............... (ii) Vector equations of line (i) & (ii) (1) ( ) Also (19) Q. No. Value Points Marks (1) Distance between two parallel lines. ( ) ( ) 21. From fig. ( ) ( ) ( ) ( ) ( ) ( ) ( ) (20) Q. No. Value Points Marks ( ) 22. P (Correct forecast) (1) P (Incorrect forecast) P (At least three correct forecasts for four matches) = P(3 correct) + P(4 correct) ( ) (1+1) Ans. ( ) OR Let E : Candidate Reaches late A1 : Candidate travels by bus A2 : Candidate travels by scooter A3 : Candidate travels by other modes of transport ( ) (1 ) By Baye s Theorem (1) (1) (21) Q. No. Value Points Marks SECTION C 23. Given Let ( ) ( ) Since R and S are non-singular matrices R 1 and S 1 exist. (1) (1) Now given (1) (2) (22) Q. No. Value Points Marks 24. ( ) or (1) or ( ) (1) is point of local minima (1) is point of local maxima Local minimum value is (1) Local maximum value is (1) OR (23) Q. No. Value Points Let Marks h = length of cylinder r = radius of semi-circular ends of cylinder ( ) v= S = Total surface area of half circular cylinder = 2 (Area of semi circular ends) + Curved surface area of half circular cylinder + Area of rectangular base. (1) ( ) (1) (1) d 2s (1) dr 2 S is minimum when (1) Which is required result. (24) Q. No. Value Points Marks 25. or (1) To sketch the graph of above function following tables are required. For Also f(x) = x2 2 represent parabolic curve. x y 2 2 3 7 4 14 5 6 23 34 (2) (2) (25) Q. No. Value Points On the graph Marks represents the area bounded by x-axis the lines x = 0 ; x = 4 and the curve y = f (x). i.e. area of shaded region shown in fig. (1) 26. or ( ) I.F. (1) Solution of diff. equation is (1) (1) (1) When x = 0, 2=c Solution is y = 2 (1) ( ) (26) Q. No. 27. Value Points Marks The given line is .................. (i) Let N be the foot of the perpendicular from P(1, 2, 3) to the given line Coordinates of N = (3 + 6, 2 + 7 2 + 7) D.R. of NP 3 + 5 , 2 + 5 , 2 + 4 D.R. of AB 3 , 2 , 2 Since NP AB 3(3 + 5) + 2(2 + 5) 2( 2 + 4) = 0 or = 1 Coordinates of foot of perpendicular N are (3, 5, 9) Equation of plane containing line (i) and point (1, 2, 3) is Equation of plane containing point (6, 7, 7) & (1, 2, 3) and parallel to line with D.R. 3 , 2 , 2 is ( ) (1) (1) ( ) ( ) (1 ) 28. or, Given 18x 22y + 5z + 11 = 0. (1) (1) But pi = 1 (1) (27) Q. No. Value Points Marks Probability distribution is (1) Probability of getting admission in two colleges = (1) (1) (1) OR Three cases arise, when 2 balls from bag A are shifted to bag B. (28) Q. No. Value Points Marks Case 1 : If 2 white balls are transferred from bag A. (1) Case 2 : If 2 black balls are transferred from bag A (1) Case 3 : If 1 white and 1 black ball is transferred from bag (1) (a) Probability of drawing 2 white balls from bag B = P(WAWA). P(WBWB) + P(BABA). P(WBWB) + P(WABA).P(WB.WB) = (b) (1) Probability of drawing 2 black balls from bag B = P(WAWA). P(BBBB) + P(BA.BA). P(BBBB) + P(WABA).P(BB.BB) = (1) = (c) Probability of drawing 1 white and 1 black ball from bag B (1) 29. (29) Q. No. Value Points Marks Let x no. of packets from kitchen A are transported to P and y of packets from kitchen A to Q. Then only 60 x y packets can be transported to R from A. Similarily from B 40 x packets can be transported to P and 40 y to Q. Remaining requirement of R i.e. 50 (60 x y) can be transported from B to Q. Constraints are (1) (1) Objective function is: Minimise. z = L.P.P. is To Minimise. z = 3x + 4y + 370 subject to constraints (1) (1) Feasible Region is A B C D E F A with corner points A (0, 10) z = 3(0) + 4(10) + 370 = 410 B (0, 40) z = 3(10) + 4(40) + 370 =530 C (20, 40) z = 3(20) + 4(40) + 370 = 590 D (40, 20) z = 3(40) + 4(20) + 370 = 570 E (40, 0) z = 3(40) + 4(0) + 370 = 490 F (10, 0) z = 3(10) + 4(0) + 370 = 400 x = 10, y = 0 gives minimum cost of transportion. Thus No. of packets can be transported as follows (1) Minimum cost of transportation is Rs. 400. (30) (1) (31) Applications of Integrals Differential equations (d) (e) 2 (2) 10 (10) Probability Totals 6. - - Linear - Programming 3 - dimensional geometry 1 (1) 1 (1) 1 (1) 1 (1) 5. (b) Vectors Integrals (c) 4. (a) Applications of Derivatives (b) 1 (1) Continuity and Differentiability 3. (a) 2 (2) 1 (1) Matrices Determinants 48 (12) 8 (2) - - 4 (1) 4 (1) 12 (3) 4 (1) 8 (2) 4 (1) 4 (1) (4 Marks) (1 Mark) __________ SA VSA (b) 2. (a) Functions Inverse trigonometric Relations and Functions 1. (a) (b) Topic S.No. BLUE PRINT - II MATHEMATICS CLASS - XII 42 (7) - 6 (1) 12 (2) - 6 (1) 6 (1) - 6 (1) 6 (1) (6 Marks) LA 13 (5) 100 (29) 10 (4) 6 (1) 17 (4) 5 (2) } 19 (5) 44 (13) } 20 (6) 5 (2) 8 (3) 10 (2) TOTAL Sample Question Paper - II Mathematics - Class XII Time : 3 Hours Max. Marks : 100 General Instructions 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each. 3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION - A Q.1. If A is a square matrix of order 3 such that |adj A| = 64, find |A|. Q.2. If A, B, C are three non zero square matrices of same order, find the condition on A such that AB = AC B = C Q.3. Give an example of two non zero 2 2 matrices A, B such that AB = 0. Q.4. If f (1) = 4; f (1) = 2, find the value of the derivative of log f(ex) w.r.t x at the point x = 0. Q.5. Find a, for which f(x) = a(x + sin x) + a is increasing. Q.6. Evaluate, Q.7. dy dy Write the order and degree of the differential equation, y = x + a 1 + dx dx (where [x] is greatest integer function) (32) 2 Q.8. ; c =k +i + , find a unit vector in the direction of a + b + c j; b = j+ k If a = i Q.9. A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Find the probability that the number is divisible by 5. Q.10. The probability that an event happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. Find the probability that the event happens at least once. SECTION - B Q.11. Q.12. 5 2 + 2 tan 1 1 7 8 1 5 1 1 Find the value of 2 tan + sec 0 1 n n n 1 If A = , prove that (aI + bA ) = a .I + na bA where I is a unit matrix of order 2 and n is a 0 0 positive integer OR Using properties of determinants, prove that a + b + 2c a c b + c + 2a c a b b = 2 (a + b + c )3 c + a + 2b d2 y Q.13. If x = a sin pt and y = b cos pt, find the value of Q.14. Find the equations of tangent lines to the curve y = 4 x 3 3x + 5 which are perpendicular to the line dx 2 at t = 0. 9y + x + 3 = 0 . Q.15. Show that the function f (x ) = x + 2 is continuous at every x R but fails to be differentiable at x = 2. Q.16. Evaluate Q.17. x x2 + 4 dx Evaluate 4 + x 2 + 16 (33) OR Evaluate Q.18. If a , b , c are the position vectors of the vertices A, B, C of a ABC respectivily. Find an expression for the area of ABC and hence deduce the condition for the points A, B, C to be collinear. OR , i and 3i respectively, are 3 4 Show that the points A, B, C with position vectors 2i j+ k j 5k j 4k the vertices of a right triangle. Also find the remaining angles of the triangle. Q.19. Evaluate, Q.20. Solve the differential equation, dy + y sec 2 x = tan x sec 2 x ; y (0) = 1 dx OR 2 xy + y 2 2 x 2 Q.21. dy = 0 ; y (1) = 2 dx In a bolt factory machines, A, B and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their output 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. (i) What is the probability that the bolt drawn is defective ? (ii) If the bolt is found to be defective find the probability that it is a product of machine B. Q.22. Two dice are thrown simultaneously. Let X denote the number of sixes, find the probability distribution of X. Also find the mean and variance of X, using the probability distribution table. SECTION - C Q.23. Let X be a non-empty set. P(x) be its power set. Let * be an operation defined on elements of P(x) by, A*B = A B v Then, (i) Prove that * is a binary operation in P(X). (ii) Is * commutative ? (iii) Is * associative ? (iv) Find the identity element in P(X) w.r.t. * (v) Find all the invertible elements of P(X) (vi) If o is another binary operation defined on P(X) as A o B = , (34) then verify that o distributes itself over *. OR Consider f : R+ [ 5, ) given by f(x) = 9x2 + 6x 5. Show that f is invertible. Find the inverse of f. Q.24. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, find the dimensions of the rectangle that will produce the largest area of the window. Q.25. Make a rough sketch of the region given below and find it s area using integration Q.26. Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The corresponding values for rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs. 4 per kg and rice Rs. 6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gms and 200 gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost. Frame an L.P.P. and solve it graphically. 1 1 1 Q.27. font A = 2 1 3 , find A 1 and hence solve the system of equations 1 1 1 x + 2y + z = 4 x + y + z = 0 {(x, y); y x 3y + z = 2 2 } 4x, 4x 2 + 4 y 2 9 Q.28. Find the equation of the plane containing the lines, ( + 2 r = i + j+ i j k ) ( and + + r =i j+ i j 2k ) Find the distance of this plane from origin and also from the point (1, 1, 1) OR Find the equation of the plane passing through the intersection of the planes, 2 x + 3 y z + 1 = 0 ; x + y 2z + 3 = 0 and perpendicular the plane 3x y 2z 4 = 0 . Also find the inclination of this plane with the xy plane. Q.29. Prove that the image of the point (3, 2, 1) in the plane 3x y + 4z = 2 lies on the plane, x + y + z + 4 = 0 . (35) MARKING SCHEME SAMPLE PAPER - II Mathematics - XII Q. No. Value Points Marks SECTION A 1. 8 (1) 2. | A | 0 / A is non singular A 1 exists / A is invertible (1) 3. (1) 4. (1) 5. a>0 (1) 6. 0.5 (1) 7. order 1, degree 2 ( , ) 8. (1) .9. (1) 10. 1 (0.6)3 (1) SECTION - B 11. (36) Q. No. Value Points Marks (1) (1) (1) (1) 12. for n = 1 (37) Q. No. Value Points Marks LHS = aI + bA, RHS = aI + bA The result is true for n = 1 (1) Let (aI + bA)k = ak I + k a k-1 b A be true = (aI + bA)k . (aI + bA) Now, (aI + bA)k-1 = (kak 1 bA + akI) (aI + bA) = k ak bA + k ak 1 b2 A2 + ak + 1 I + ak bA (aI + bA)k + 1 = k ak bA+ ak + 1 I + ak bA (a + bA)k + 1 = ak b(k + 1) A + ak + 1 I = anI + nan 1 bA (aI + bA)n is true v n N by principle of mathametical induction. ( ) (2) ( ) OR c1 c1 + c2 + c3 (1) (1) R2 R2 R1, R3 R3 R1 (1) (1) 13. (38) Q. No. Value Points Marks (1) (1 ) ( ) (1) 14. Let P (x1, y1) be the point on the curve y = 4 x 3 3x + 5 where the tangent is perpendicular to the line 9 y + x + 3 = 0 ( ) ( ) Slope of a line perpendicular to is 9 (1) Two coresponding points on the curve are (1, 6) and ( 1, 4). ( ) Therefore, equations of tangents are (1 ) 15. Let f(x) = | x + 2 | ( ) When x > 2 or x < 2, f(x) being a polynomial function is continuous. We check continuity at x = 2 ( ) (39) Q. No. Value Points Also Now, LHD Marks f(-2) = 0 f(x) is continuous at x = 2 (1) f(x) is continuous v x R We check differentiability at x = 2 at x = 2 ( ) ( ) ( ) f(x) is not differentiable at x = 2 ( ) 16. ( ) (1) (1) (1) ( ) (40) Q. No. Value Points Marks ( ) 17. (1) (1 ) ( ) = ( ) OR (1) ( ) (1) (41) Q. No. Value Points Marks (1) ( ) 18. (1) area = 1 2 c a b a ( ) (1) If A, B, C are collinear then area ABC = 0 ( ) ( ) ( ) OR We have, (1) Since A, B, C form a triangle. ( ) Also, BC CA < ( ) i.e. ABC is a right triangle. (42) Q. No. Value Points Marks 19. (1) (1) Let ( ) (1) ( ) ( ) (43) Q. No. Value Points Marks (1) ( ) 20. (1) (1) =I+c ( ) Solution is Now y(0) = 1 1.e0 = (0 1)e0 + c c=2 (1) ( ) OR (44) Q. No. Value Points Marks Put ( ) y = vx ( ) (1) (1) c= 1 ( ) 21. Let ( ) E1 E2 E3 A = = = = the bolt is manufactured by machine A the bolt is manufactured by machine B the bolt is manufactured by machine C the bolt is defective Then ( ) (45) Q. No. Value Points (i) Marks Now defective bolt could be from machine A, B or C ( ) (1) (ii) = 0.0345 We require probability that the bolt is manufactured by machine B given that bolt drawn is defective i.e., P(E2/A), So by Baye s theorem (1) (1) 22. Here x = 0, 1 or 2 ( ) (1 ) (46) Q. No. Value Points Marks (1) Now, mean (x) = ( ) var (x) = = ( ) SECTION - C 23. (i) * is a binary operation on P(X) as it is a function from P(X) P(X) to P(X) * is a binary operation on P(X) as (1) A, B, 0 P(X) ; A * B = A B also belongs to P(X) (ii) * is commutative as, A * B = A B = B A = B * A v A, B P(X) (iii) * is associative as, A * (B * C) = A (B C) = (A B) C v A, B, C P(x) = (A * B) * C (iv) X is the identity element is P(x) as, X * A = X A = A = A X = A * X (v) Let A be an invertible element in P(x) w.r.t * (1) (1) v A P(X) (1) ................... (i) A B = B A= X X = A (as A X) X A Since no element of P(X) other than X satisfies (i), X is the only invertible element of P(X) w.r.t. * (1) (vi) o is another binary operation of P(x) with A o B = A B (47) Q. No. Value Points Marks (1) Now, A o (B * C) [or, (B * C) o A = (B o A) * (C o A)] OR Sol. f is 1 1 Let (2) f is onto Let y = [ 5, %] Suppose f(x) = y i.e. Solving for x to get (2 ) Since, f is onto (1 ) 24. (48) Q. No. Value Points Marks (1) Maximize (1) ( ) (1) (1) (1) Area is maximum and dimensions of the window are and ( ) 25. (1 ) (49) Q. No. Value Points Marks Point of intersection as (1) (1 ) Required area = (1) (1) 26. Suppose x gms of wheat and y gm of rice are mixed in the daily diet. As per the data, x gms of wheat and y grams of rice will provide 0.1 x + 0.5 y gms of proteins Similarly 0.25 x + 0.5 200 Hence L.P.P. is Minimize ( ) subject to the constraints (2) (50) Q. No. Value Points Marks 2 The feasible region is unbounded and has vertices A (0, 1000), B(800, 0) and P(400, 200) Point Value of objective function, Z (800, 0) (400, 200) (1) (0, 1000) Clearly Z is minimum for x = 400 i.e., Wheat = 400 gm and rice = 200 gm 27. ( ) We have Now, (1) A 1 exists matrix of cofactors (1 ) adj ( ) (51) Q. No. Value Points Marks ( ) Now the given system of equations is expressible as At X = B where X = ( ) and, ( ) ( ) ( ) 28. in the required solution. ( ) is normal to the plane containing ..................... (1) and ..................... (2) (52) Q. No. Value Points Marks (2) Both lines (1) and (2) pass through eqns. of the plane containing (1) & (2) is , (1) is, ( ) = 3 + 3 + 3 r 3i j + 3k (1) or, Since this plane contains the origin, its distance from origin is zero. Distance of the plane from (1, 1, 1) i.e (1) , is (1) = OR Equation of family of planes passing through the intersection of ................. (i) (1) as plane (i) is perpendicular to plane (1 ) ( ) (53) Q. No. Value Points Marks Substituting in (i), are obtain, 7 x + 13 y + 4z = 9 is the required plane. (1) Let be the inclination of this plane with xy plane (i.e. z= 0) (1) 29. ( ) ( ) Equations of the line perpendicular the plane ................... (i) and passing through (3, 2, 1) is ................... (ii) ( ) is a general point in (ii) Now, The foot of perpendicular from (3, 2, 1) to the plane (i) is the point of intersection of (ii) and (i) ................... (i) (1) (1) foot of perpendieular ( ) Let Image of (3, 2, 1) in the given plane be (x1, y1, z1) (1) x1 = 0, y1 = 1, z1 = 3 Image of (3, 2, 1) in the plane which lies on the plane (as 0 1 3 + 4 = 0) (54) (1) (1) (55) Differentiability Continuity and Determinants Matrices Applications of Integrals Differential equations (d) (e) 4 (1) 48 (12) 10 (10) Probability Total - 4 (1) 6. - - 4 (1) 8 (2) 4 (1) - 12 (3) 4 (1) - 4 (1) 4 (1) SA Linear - Programming 3 - dimensional geometry 3 (3) 2 (2) - - 1 (1) 2 (2) 1 (1) 1 (1) VSA 5. (b) Vectors Integrals (c) 4. (a) Applications of Derivatives (b) 3. (a) (b) 2. (a) Functions Inverse trigonometric Relations and Functions 1. (a) (b) Topic S.No. BLUE PRINT - III MATHEMATICS - XII 42 (7) 6 (1) 6 (1) 6 (1) - 6 (1) 6 (1) 6 (1) - - 6 (1) - - LA 100 (29) 10 (2) 6 (1) 17 (6) 8 (2) }18 (5) }18 (4) 13 (5) - 10 (4) TOTAL 44 (11) Sample Question Paper - III Time : 3 Hours Max. Marks : 100 General Instructions : 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into three sections A, B and C. Section A contains 10 questions of 1 mark each, section B is of 12 questions of 4 marks each and section C is of 7 questions of 6 marks each. 3. There is no overall choice. However, an internal choice has been provided in four questions of 4 marks each and two questions of six marks each. 4. Use of calculators is not permitted. However, you may ask for Mathematical tables. SECTION - A R be a function defined as f (x ) = 2x , find f 1 : Range of 5x + 3 1. Let 2. Write the range of one branch of sin 1x, other than the Principal Branch. 3. If 4. If B is a skew symmetric matrix, write whether the matrix (ABA ) is symmetric or skew symmetric. 5. On expanding by first row, the value of a third order determinant is a11 A11 + a12 A12 + a13 A13. Write the expression , find x, 0 < x < when A + A = I 2 for its value on expanding by 2nd column. Where A ij is the cofactor of element a ij . 6. Write a value of 7. Write the value of 8. Let a and b be two vectors such that a = 3 and b = between a and b ? 9. ( ) ( ) ( ) + i . . +k Write the value of i j k j. k j i (56) 2 and a x b is a unit vector. Then what is the angle 3 10. For two non zero vectors a and b write when a + b = a + b holds. SECTION - B 11. { } Show that the relation R in the set A = x x W, 0 x 12 given by R = {(a , b) : (a b) is a multiple of 4} is an equivalence relation. Also find the set of all elements related to 2. OR Let * be a binary operation defined on N N, by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative. Also find the identity element for * on N N, if any. 12. Solve for x : 13. If a, b and c are real numbers and Show that either a + b + c = 0 or a = b = c. 14. x 5 + a , if x < 5 If f ( x) = x 5 a + b, if x = 5 x 5 + b, if x < 5 x 5 b+c c+a a+b c+a a+b b+c = 0 a+b b+c c+a is a continuous function. Find a, b. 15. y x If x + y = log a , find dy . dx 16. Use lagrange s Mean Value theorem to determine a point P on the curve y = x 2 where the tangent is parallel to the chord joining (2, 0) and (3, 1). 17. Evaluate : OR Evaluate : (57) 18. If a and b are unit vectors and is the angle between them, then prove that OR If are the dia gonals of a parallelogram with sides , find the area of parallelogram in terms of and d = 3i 2 and hence find the area with d1 = i + 2 j + 3k j + k. 2 19. Find the shortest distance between the lines, whose equations are x 8 y + 9 10 z x 15 58 2 y z 5 = = = = and . 16 7 16 3 5 3 20. A bag contains 50 tickets numbered 1, 2, 3, ........, 50 of which five are drawn at random and arranged in ascending 21. Show that the differential equation order of the number appearing on the tickets (x1 < x 2 < x3 < x 4 < x5 ) . Find the probability that x3 = 30. homogeneous and find its particular solution given that x = 0 when y = 1. OR Find the particular solution of the differential equation d1 and d 2 given that y = 0, when x = 22. Form the differential equation representing the family of ellipses having foci on x-axis and centre at origin. SECTION - C 23. 24. 25. A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probabiity that the letter has come from (1) Tata nagar (ii) Calcutta OR Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement from a bag containing 4 white and 6 red balls. Also find the mean and variance of the distribution. Find the distance of the point (3, 4, 5) from the plane x + y + z = 2 measured parallel to the line 2x = y = z. Using integration, compute the area bounded by the lines x + 2y = 2, y x = 1 and 2x + y = 7 OR 2 Find the ratio of the areas into which curve y = 6x divides the region bounded by x2 + y2 = 16. (58) 26. Evaluate : 27. A point on the hypotenuse of a right triangle is at a distance a and b from the sides of the triangle. Show that the [ minimum length of the hypotenuse is a 28. 23 ]. 3 23 2 +b Using elementary transformations, find the inverse of the matrix 1 3 2 3 0 5 2 5 0 29. A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B and C. Production of one chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as L.P.P. and solve it graphically. e tan 1 x (1 + x ) 2 2 (59) dx . Marking Scheme Sample Paper - III Q. No. Value Points Marks SECTION - A 1. 2. 4. Skew symmetric 5. 7. Zero. 8. (or any other equivalent) 3. 6. log | x + log sin x | + c 9.1 10. a and b are like parallel vectors. 1 10 = 10 SECTION - B 1. 11. 2. (or any other equivalent) 3. (i) since (a a) = 0 is a multiple of 4, v a A R is reflexive (ii) (a, b) R (a b) is a multiple of 4 (b a) is also a multiple of 4 R is Symmetric (b, a) R v a, b A (iii) (a, b) R and (b, c) R (a b) = 4k, k z (b c) = 4m, m z v a, b, c A (a c) = 4(k + m), (k + m) z (a, c) R R is transitive Set of all elements related to 2 are {2, 6, 10} OR (i) v a, b, c, d N, (a, b) * (c, d) = (a + c, b + d) = (c + a, d + b) = (c, d) * (a, b) * is commutative (ii) [(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f) = ((a + c) + e, (b + d) + f) = (a + c + e, b + d + f) = (a + (c + e), b + (d + f)) v a, b, c, d, e, f, N = (a, b) * [(c, d) * (e, f)] * is associative (60) 1 1 1 1 1 Q. No. Value Points Marks (iii) Let (e, f) be the identity element, then (a, b) * (e, f) = (a, b) (a + e, b + f) = (a, b) e = 0, f = 0 but (0, 0) N N So, identity element does not exist 12. 1 1 We have 1 1 13. 1 1 1 14. 1 f(5) = a + b 1 1 + a = a + b = 7 + b b = 1, a = 7 1 , (61) Q. No. Value Points 15. Marks Let ................. (i) logu = y log x 1 log v = x log y 1 16. (i) (ii) (i) Since (x 2) 0 in [2, 3] so is continuous 1 f(x) is differentiable in (2, 3) exists for all Thus lagrang s mean value theorem is applicable; There exists atleast one real number c in (2, 3) such that 1 LMV is verified and the req. point is (2.25, 0.5) 1 17. 1 1 (62) Q. No. Value Points Marks 1 or, OR 1 1 1 18. 1 1 1 1 OR (63) Q. No. Value Points Marks Let ABCD be the parallelogram with sides 1 Now 1 When 19. 1 area of 11 gm 1 The given equations can be written as 1 The shortest distance between two lines is given by Here (64) Q. No. Value Points Marks 1 20. 1 Since 1 Required Probability is 2 1 21. Here Let F(x, y) is a homogeneous function of degree zero, thus the given differential equation is a homogeneous differential equation Put 1 1 or, c=2 1 OR Here integrating factor = 1 the solution of differential equation is given by y. sin x 1 (65) Q. No. Value Points Marks .............................(1) Substituting y = o and x = 1 , we get or 22. Equation of the said family is 1 Differentiating w.r.t x, we get 1 y. sin x 1 SECTION - C 23. Letter has come from tatanagar Letter has come from calcutta A: E1 : E2 : Let Obtaining two consecutive letters TA P(A|E 1) {Total possiblities TA, AT, TA, AN, NA, AG, GA, AR = 8, favourable = 2} P(A|E2) 1+1 {Total possiblities CA, AL, LC, CU, UT, TT, TA = 87 favourable = 1} (66) 1 Q. No. Value Points Marks 1 1 OR Let x = Number of white balls. 4=2 Thus we have Mean = 1 1 (67) Q. No. Value Points Marks 24. AB is parallel to the line or or Equation of AB is 1 For some value of , B is ( + 3, 2 + 4, 2 + 5) B lies on plane + 3 + 2 + 4 + 2 + 5 2 = 0 1 1 =2 B is (1, 0, 1) 1 1 = 6 units. 25. Solving the equations in pairs to get the vertices of as (0, 1), (2, 3) and (4 1) For correct figure (68) 3 = 1 1 Q. No. Value Points Marks Required area 1 1 = 12 4 2 = 6 sq. U. OR Correct figure 1 Area 2 1 A2 = Area of circle shaded area 1 (69) Q. No. Value Points Marks 26. Put 1 .................. (i) 1 1 Putting in (i) we get 1 (70) Q. No. Value Points Marks For correct figure 1 c = . AC = AP + PC = S (say) S = a sec + b cosec 1 27. Let Cot 1 Which is +ve as a, b > 0 and is acute S is minimum when 1 Minimum 1 1 28. Let (71) Q. No. Value Points Writing A = IA Marks or 1 Operating R2 R2 + 3R1, we have R3 R3 2R1 8 =4 1 (72) Q. No. Value Points 29. Let Marks number of chairs = x number of tables = y LPP is Maximise P = 30 x + 60y Subject to 1 For correct graph P P(A) PB PC PD 3 = 30 (x + 2y) = 30 (60) = 30(65) = 30(50) = 30(35) For Max Profit (30 65) No. of chairs = 15 No. of tables = 25 1 (73)

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