SECTION C Question numbers 3 to 9 carr 6 marks each. 3. Find the equation of the plane passing through the line of intersection of the planes r ˆi 3j ˆ 6 0 r 3i ˆ ˆ j k ˆ 0, whose perpendicular distance from origin is and unit. OR Find the vector equation of the line passing through the point (,, 3) and parallel to the ˆ ˆ ˆ r i j k 5 and r 3i ˆ ˆj kˆ 6. planes The equation of the plane passing the intersection of the given planes is r iˆ 3 ˆj 6 r 3i ˆ ˆj kˆ 0 r 3 iˆ 3 ˆj kˆ 6 0... This plane is at a unit distance from the origin. Therefore, length of perpendicular from origin = 3 3 6 6 6 3 3 6 36 Putting the values of in (), we have ˆ ˆ ˆ r i j k 6 0 and r iˆ ˆj kˆ 6 0 ˆ ˆ ˆ r i j k r i ˆ ˆj kˆ Or 3 and 3 Thus, the required equations of the plane are ˆ ˆ ˆ r i j k 3 and r iˆ ˆj kˆ 3 The vector equation of the line passing through iˆˆj 3kˆ and parallel to the vector aiˆ bj ˆ ckˆ r iˆ ˆj 3 kˆ k aiˆ bj ˆ ckˆ... OR is The equations of the given planes are
ˆ ˆ ˆ ˆ ˆ ˆ r i j k 5... r 3i j k 6... 3 Since line () is parallel to the planes () and (3), so a b c 0 a b c 0... a3 b c 0 3a b c 0... 5 Solving () and (5), we have a b c 6 3 a b c 3 5 a 3, b 5, and c Substituting the values of a, b and c in (), we have r iˆ ˆj 3kˆ k 3iˆ 5 ˆj kˆ r i ˆ ˆj 3kˆ k 3i ˆ 5 ˆj kˆ r i ˆ ˆj 3kˆ 3i ˆ 5 ˆj kˆ, where k Thus, the vector equation of the line is r iˆ ˆj 3k 3i ˆ 5 ˆj kˆ. In a hocke match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternatel and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not. Given that the captain of team A was asked to start. Team A can be declared the winner, if the captain of team A gets a six in an one of the given throws: first, third, fifth, seventh ; in all such throws, the captain of team B will not get a six in each of his throws. Team B can be declared the winner, if the captain of team B gets a six in an one of the given throws: second, fourth, sixth, eighth ; in all such throws, the captain of team A will not get a six in each of his throws. Probabilit of getting a six =. 6 5 Probabilit of not getting a six =. 6 6 Let, the probabilit that teams A and B are the winner be P (A) and P (B) respectivel.
5 5 5 5 5 5 P A... 6 6 6 6 6 6 6 6 6 5 5 6 P A.... 6 6 6 6 5 6 5 5 5 5 5 5 5 5 5 PB... 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 PB.... 6 6 6 6 36 5 6 The decision of the referee was not fair as chances of team A to be declared as winners are more than that of team B. 5. A manufacturer considers that men and women workers are equall efficient and so he pas them at the same rate. He has 30 and 7 units of workers (male and female) and capital respectivel, which he uses to produce two tpes of goods A and B. To produce one unit of A, workers and 3 units of capital are required while 3 workers and unit of capital is required to produce one unit of B. If A and B are priced at Rs 00 and Rs 0 per unit respectivel, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphicall. Do ou agree with this view of the manufacturer that men and women workers are equall efficient and so should be paid at the same rate? Let, the number of units of A and B that are produced be x and respectivel. Therefore, total revenue R = 00x + 0. Total number of workers used in the production of given units of A and B = x + 3. Total capital used in the production of given units of A and B = 3x +. As per the information given in the question, the following must hold true: x 3 30 and 3x 7 The problem is to maximize the value of R such that x 3 30, 3x 7, x 0 and 0.
The coordinates of the corner points A, B, C, and O are (0, 0); (3, 8); (7/3, 0) and (0, 0) respectivel. The value of R at the points A, B, C, and O are Rs 00, Rs 60, Rs 566.67, and Rs 0 respectivel. Therefore, maximum revenue would be obtained when 3 units of A and 8 units of B are produced. In doing so, 30 workers and 7 units of capital must be used. x 6. Find the area of the greatest rectangle that can be inscribed in an ellipse a OR. b Find the equations of tangents to the curve 3x = 8, which pass through the point,0 3. A rectangle ABCD is inscribed in an ellipse as shown below. Let the co-ordinates of the points A, B, C, D be asin, bcos and asin, bcos respectivel. asin, bcos,asin, bcos Therefore, AB a sin and BC b cos. Area of the rectangle ABCD, S = AB BC ab sin cos absin S absin ds ab cos abcos d,
ds For, the value of S to be greatest, 0 cos 0 d ds Now, 8ab 0 d Smax ab Therefore, the area of the greatest rectangle that can be inscribed in an x ellipse is equal to ab. a b Let the tangent be at point x Therefore, 3x 8...(), OR on the curve. Also, the slope of the tangent will be equal to The equation of the curve is 3x = 8. d d 3x 6x 0 dx dx x, d dx x, 3x Equation of the line passing through x, having slope is given 3x by X x The point Therefore,,0 3 3x lies on Y X x 3 3x 0 x 3x x 0... Solving () and () we get that x 8 x. Therefore, 3x 8 Equations of the tangents are 3x 0and 3x 0.
7. The management committee of a residential colon decided to award some of its members (sa x) for honest, some (sa ) for helping others and some others (sa z) for supervising the workers to keep the colon neat and clean. The sum of all the awardees is. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honest is 33. If the sum of the number of awardees for honest and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each categor. Apart from these values, namel, honest, cooperation and supervision, suggest one more value which the management of the colon must include for awards. As per the information given in the question, the following equations hold true: x z, x 3 z 33 and x z 0. The above three equations can be represented in the form of a matrix as x 3 3 = 33 z 0 x Or AX = B, where, A 3 3, X and B 33 z 0 A 30Thus, A is non-singular. Therefore, its inverse exists. 9 3 0 Adj A is given b 0 9 3 0 A adja 0 A 3 7 3 7 3 9 3 0 X A B 0 33 3 7 3 0 x 9 3 0 3 0 33 3 z 7 3 0 5 x 3,, and z 5. Therefore, the number of awardees for Honest, Cooperation and Supervision are 3,, and 5 respectivel. One more value which the management of the colon must include for awards ma be Sincerit.
8. Find the area of the region {(x, ): 6ax and x + 6a } using method of integration. Coordinates of point C is (a, 0). Let the coordinates of point A be (x, ). x 6ax 6a x a x 8a 0 x a, x 8 a as x lies in the first quadrant. 6ax a 3a a Area of the shaded region = 6 3 3a a d d 6a 0 0 8 a 3 a = 3a a 3 3 3 3 = square units 9. Show that the differential equation x sin dx xd 0 x is homogeneous. Find π the particular solution of this differential equation, given that when x =. xsin dx xd 0 x d x xsin...() dx x d dv Let, vx v x...() dx dx d Puttting the value of from () in (), we get that : dx dv dv xv x vx xsin v x x sin v dx dx dv dx cot v log x c cot log x c sin v x x
Given that, when x. cot log c c. Therefore, particular solution is cot log x x