MATHEMATICS QUESTION PAPER CODE 65/1/1 SECTION A

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1 MATHEMATICS Time allowed : hours Maimum Marks : General Instructions : (i) (ii) The question paper consists of three sections A, B and C. Section A is compulsory for all students. In addition to Section A, every student has to attempt either Section B Section C. For Section A Question numbers to 8 are of marks each. Question numbers 9 to 5 are of marks each. Question numbers 6 to 8 are of 6 marks each. (iii) For Section B/Section C Question numbers 9 to are of marks each. Question numbers to 5 are of marks each. Question number 6 is of 6 marks. (iv) All questions are compulsory. (v) Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions. (vi) Use of calculator is not permitted. However, you may ask for logarithmic and statistical tables, if required. QUESTION PAPER CODE 65// SECTION A. If A, prove that A A A.. Show that a b c, where a, b, c are in A.P.. In a single throw of three dice, determine the probability of getting (a) a total of 5, (b) a total of at most 5. 7

2 . A class consists of boys and 8 girls. Three students are selected at random. Find the probability that the selected group has (i) all boys, (ii) all girls, (iii) boys and girl. 5. Evaluate : sin a sin b cos d 6. Evaluate : 6 (log ) d 7. Form the differential equation representing the family of curves y ay a, where a is an arbitrary constant. 8. Solve the following differential equation : ( ) dy y ( d )( ) S : p q Solve the following differential equation : dy d 9. Prove that : y p q (p q) (q p) Test the validity of the following argument : y S : ~ p S : q. Evaluate : ( y) sec( y) sec lim y y 7

3 Evaluate : lim [ ) (sin. Differentiate sin w.r.t. from first principles. t. If t dy a and y, find. t t d ]. The surface area of a spherical bubble is increasing at the rate of cm /sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.. Evaluate : 5. Evaluate : d ( )( )( ) π d 5 cos y z z 9 A( ), 6. Using matrices, solve the following system of linear 7 equations 5 d : y z If find and y such that A I ya. Hence find A. 7. A wire of length 6 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum. 8. Find the area bounded by the curve y and the straight line y Evaluate the following as limit of sums : 7

4 Section B 9. Epress the vactor ^ a 5i j 5k ^ ^ as sum of two vactors such that one is parallel to the vector ^ ^ b i k and the other is perpendicular to b.. If the vectors c ab. ^ ^ ^ a i a j c k, i k and ^ ^ ^ ^ ^ c i c j b k be coplanar, show that. A car, travelling with a uniform acceleration, has a velocity of 8 km/hour at a certain time and 5 km/hour after covering a distance of 5 m. How much further will it travel to attain a velocity of 7 km/hour? 5. A body falls freely from the top of a tower. It covers th of the whole distance 9 in the last second. Find the height of the tower and the total time taken by the body to fall down. A cricket ball is projected with a velocity of 9. m/sec. Find (i) the greatest range on the horizontal plane; and (ii) the angle of projection to give a range of. m.. C (,, ) Find the co-ordinates of the foot of the perpendicular Q PS drawn from the point A (, 8, ) to the line joining the points B (,, ) and.. The resultant of two forces and acting at a point is at right angles to force P, while the resultant of forces and, acting at the same angle, is at right angles to force S. Prove that P SQ. 5. P and are two unlike parallel forces. When the magnitude of P is doubled, it is found that the line of action of is midway between the lines of action of the new and the original resultants. Find the ratio of P and Q. Three forces, acting on a particle, are in equilibrium. If the angle between the first force and the second force be and that between the second force and the third force be 5, find the ratio of their magnitudes. 6. Find the cartesian as well as the vector equation of the planes passing through ^ ^ ^ ^ ^ the intersection of the planes r.(i 6 j ) and r.(i j k ) which are at unit distance from the origin. 75

5 SECTION C 9. Solve the following linear programming problem graphically : Maimise z 6 5y subject to constraints y 5. Two tailors A and B earn Rs. 5 and Rs. per day respectively. A can stitch 6 shirts and pants per day while B can stitch shirts and pants per day. Form a linear programming problem to minimise the labour cost to produce at least 6 shirts and pants.. A company has two plants to manufacture motor cycles. 7% motor cycles are manufactured at the first plant, while % are manufactured at the second plant. At the first plant, 8% motor cycles are rated of the standard quality while at the second plant, 9% are rated of standard quality. A motor cycle, randomly picked up, is found to be of standard quality. Find the probability that it has come out from the second plant.. The probability that a student entering a university [Use, y y : will e 9 graduate.679] is.. Find the probability that out of students of the university : (i) none will graduate, (ii) only one will graduate, (iii) all will graduate In a book of pages, misprints are randomly distributed. Using Poisson's distribution calculate the probability that a randomly observed page of the book will be found to have at least errors.. A, B and C are engaged in a printing business. A being the working partner, receives % of the net profit as salary. The remaining profit is divided among themselves in the ratio : 5 : 9. If A gets in total Rs.,,, find the total profit in the business and the shares of B and C in it. A and B are partners in a business sharing profits and losses equally. They admit a new partner C and it is agreed that now the profits and losses will be shared amongst A, B and C in the ratio 9 : 8 : 7 respectively. If C paid Rs.. lakh as premium for the goodwill, find the shares of A and B in the premium. 76

6 . Find the present worth of an ordinary annuity of Rs. per annum for years at % per annum, compounded annually. [Use : (.).] 5. A calculator manufacturing company finds that the daily cost of producing calculators is given by C() 75. (i) (ii) If each calculator is sold for Rs. 5, find the minimum number of calculators that must be produced daily and sold to ensure no loss. If the selling price is increased by Rs. 5, what would be the break-even point? 6. A bill was drawn on April, at 8 months after date and was discounted on July, at 5% per annum. If the Banker's gain is Rs., find the face value of the bill. QUESTION PAPER CODE 65/ SECTION A. If A, f (), show that f (A).. Using properties of determinants, solve for a : a a a a a a a a. An integer is chosen at random from the first positive integers. Find the probability that it is divisible by 6 or 8.. X is taking up subjects Mathematics, Physics and Chemistry in the eamination. His probabilities of getting Grade A in these subjects are.,. and.5 respectively. Find the probability that he gets. (i) (ii) (iii) Grade A in all subjects; Grade A in no subject; Grade A in two subjects. 5. Evaluate : sin (a b cos ) d 77

7 6. Evaluate : log (log ) d 7. Solve the following differential equation : dy y cot cot d 8. Solve the following differential equation : ( y) dy ( y ) d Solve the following differential equation : d y dy e cos, given that y, when. d d 9. Test the validity of the following argument : S : p q; S :~ p; S:~ q If B is a Boolean Algebra and ( y) ('. y'). Evaluate :, y B, then show the following : ( y) sec( y) sec lim y y. Differentiate tan w.r.t. from first principles.. If y { a }, prove that dy d ny a n. Find the intervals in which the function f () 5 6 is strictly increasing or decreasing. Also find the points on which the tangents are parallel to the -ais.. Evaluate : d 6 78

8 5. Evaluate : 5 f () d, where f () Show that the height of the cone of maimum volume that can be inscribed in a sphere of radius cm is 6 cm. Prove that the curves y and y k cut at right angles if 7. Using matri method solve the following system of linear equations : y z 8. Find the area of the region bounded by the curve and the line y. Find the area enclosed by the parabola SECTION B y and line y. 8 k y y y. z z 9. If a, b and c are three mutually perpendicular vectors of equal magnitude, find the angle between a and (a b c ).. Show that the four points A, B, C and D, whose position vectors are 6i 7 j, ^ ^ ^ 6i 9 j k, j 6k and ^ ^ ^ ^ ^ ^ i 5 j k respectively, are coplanar.. A body moves for seconds with a uniform acceleration and describes a distance of 8 m. At that point the acceleration ceases and the body covers a distance of 6 m in the net seconds. Find the initial velocity and acceleration of the body.. A body is projected with a velocity of m/sec at an angle of 6 with the horizontal. Find (i) the equation of its path; (ii) its time of flight; and (iii) the maimum height attained by it. A particle is projected so as to graze the top of two walls, each of height m, at 5 m and 5 m, respectively from the point of projection. Find the angle of projection. ^ 79

9 . Find the equation of the line passing through the point P (,, ) and prependicular to lines and y z. 5. The resultant of forces R is doubled. If and acting at a particle is R. If is doubled, is reversed, R is again doubled. Prove that A and B are two fied points in a horizontal line at a distance 5 cm apart. Two fine strings AC and BC of length cm and cm respectively support a weight W at C. Show that the tensions in the strings CA and CB are in the ratio of :. 5. The resultant of two unlike parallel forces of 8 N and N acts along a line at a distance of cm from the line of action of the smallar force. Find the distance between the lines of action of the two given forces. 6. Find the equation of the sphere passing through the points (,, ), (, 5, ), and having its centre on the plane SECTION C [Use ( z Q P, : Q y, : er ) z..5] : :. 9. A speaks the truth 8 times out of times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?. A coin is tossed times. Find the mean and variance of the probability distribution of the number of heads. For a Poisson distribution, it is given that P (X ) P (X ). Find the value of the mean of the distribution. Hence find P (X ) and P (X ).. If the banker's gain on a bill be th of banker's discount, the rate of interest 9 being % per annum, find the unepired period of the bill.. A bill of Rs. 5, drawn on 6th January, for 8 months was discounted on th February, at % per annum. Find the banker's gain and discounted value of the bill.. In a business partnership, A invests half of the capital for half of the period, B invests one-third of the capital for one-third of the period, and C invests the rest of the capital for the whole period. Find the share of each in the total profit of Rs.,9,. 8

10 . A plans to buy a new flat after 5 years, which will cost him Rs. 5,5,. How much money should he deposit annually to accumulate this amount, if he gets interest 5% per annum compounded annually? [Use : (.5) 5.76] 5. The cost function of a firm is given by C(), where stands for the output. Calculate : (i) the output at which the marginal cost is minimum; (ii) the output at which the average cost is equal to the marginal cost. The total cost and the total revenue of a firm that produces and sells units of its product daily are epressed as C() 5 5 and R() 5. Calculate : (i) the break-even points, and (ii) the number of units the firm will produce which will result in loss. 6. A manufacturer produces two types of steel trunks. He has two machines A and B. The first type of the trunk requires hours on machine A and hours on machine B. The second type of trunk requires hours on machine A and hours on machine B. Machines A and B are run daily for 8 hours and 5 hours respectively. There is a profit of Rs. on the first type of the trunk and Rs. 5 on the second type of the trunk. How many trunks of each type should be produced and sold to make maimum profit? 8

11 Marking Scheme ---- Mathematics General Instructions :. The Marking Scheme provides general guidelines to reduce subjectivity in the marking. The answers given in the Marking Scheme are suggested answers. The content is thus indicative. If a student has given any other answer which is different from the one given in the Marking Scheme, but conveys the meaning, such answers should be given full weightage.. Evaluation is to be done as per instructions provided in the marking scheme. It should not be done according to one's own interpretation or any other consideration Marking Scheme should be strictly adhered to and religiously followed.. Alternative methods are accepted. Proportional marks are to be awarded.. Marks may not be deducted in questions on integration if constant of integration is not written. 5. In question(s) on differential equations, constant of integration has to be written. 6. If a candidate has attempted an etra question, marks obtained in the question attempted first should be retained and the other answer should be scored out. 7. A full scale of marks - to has to be used. Please do not hesitate to award full marks if the answer deserves it. 8

12 8 QUESTION PAPER CODE 65// EXPECTED ANSWERS/VALUE POINTS SECTION A. [ ] I A A A LHS Now I A A [ ] RHS I A A A. A c b a R R R R c b b c a A m c a b then A.P. in are c and b a, If c b A. (a) Let A be the event of getting a total of ),,,, (, (A) P

13 (b) Let B be the event of getting at most sum 5 This can be obtained as [,,), (,,), (,,), (,,), (,,), (,,), (,,), (,,), (,,), (,,) ] ( P (B) Total number of student 8 8 (i) P (All boys) (½½) m (ii) P (All girls) 8C 7 (½½) m 8C (iii) P (boys, girl) C 8C 5 6C (½½) m 5. Let a sin b cos t C 8C 5 a sin cos b sin cos dt d or (a b ). sin d dt dt I log t C (a b )t a b log a sin b cos C (a b ) d 6. Let log t dt I 6 t dt 8

14 t 6 t 8 log t 6 t C log 6 (log ) 8log log 6 (log ) C 7. y ay a...(i) dy dy y a d d or a dy y d dy d...(ii) Putting this value of a in (ii), we get y dy y d y dy d dy.y d dy y. d d d d or ( dy y ) d dy y d 8. The given differential equation can be written as dy. y d ( ) I.F. The solution is tan C 85

15 dy d y y This is a homogeneous differential equation dy dv Putting y v v d d dv v v d v or dv d v log c log v v or v v c or y y c 9. The statement results in following Truth Table q ( p pq) qq) q(q (p p ) q) (q p) p q p q T T T T T T Truth T F F T F F m Table F T T F F F F F T T T T S S S S p q ~ p q Truth Table T T F T T T F F T F F T T T T Identifying Critical Row F F T F F The given argument is valid 86

16 . [ cos cos ( y) ] lim sec y y cos cos ( y) lim y y y.sin.sin y cos.cos ( y). sec tan sec sec The given problem becomes or Let ( sin sec sin θy) sec( θ( tan cos y) θ) ) sec lim. θ y cos θ θy lim y {sec ( y) y sec } se lim θ θ.sin tan θ.lim θ θ θ... y sin sin y sin sin dy lim d cos.sin 87

17 sin lim cos sin lim cos. cos. cos. ( t ) a t y t ( t ) d dt dy dt dy d a. ( t ). t t( t ) at ( ) ( ) t ( t ). t.t t ( t ) ( t ) t ( t ) ( t ). ( t ) ( t ). ( t ) at t at. Let s be the surface area of the spherical bubble Now ds cm / sec. dt s πr or ds dt π. dr r. dt dr dr 8πr dt dt πr v Volume of bubble πr 88

18 dv πr dt dr dt πr r πr dv 6 cm /sec dt at r 6. I d ( ) ( ) ( ) Let ( ) ( ) ( ) A B C I 6 d d d log log log C 6 Getting A, B and C 6 π d 5. I 5 cos π d tan 5 tan m I π sec 9 tan d Let tan t sec d dt I dt t tan t π. π 89

19 9 6. Writing the given equation as B A X B AX or 9 z y A 5 8 A Adj. m 5 8 A z y z, y, A ya I A 5y 7y y y or 5y 7y y y m y 6 and 8 y 8 8 y 8, 8A 8I A (For every four correct co-factors, one mark may be given)

20 or A 8A 8I or A 8 [ 8I A] A or Let the length of two pieces be and 6 - (in cms) Let the piece of length be turned into a square and the other into an equilateral triangle. side of square, side of 6 an equilateral triangle Area of 6 square and Area of equilateral triangle 6 Let A be the sum of the areas of two da d 8 8 (6 ) ( ) A 6 6 ( 6 ) da d 8 8 (6 ) or 8 8 (6 ) 9 d A Showing > Area is minimum for this value of d Length of one piece cm 9 Length of second piece cm 9 9

21 8. Figure Calculating and as points of intersection Required Area m 8 9 sq.units 8 ( ) I d Here a, b, f (), h n I lim h [ f () f ( h) f ( h)... f ( n h) ] h d lim h [ (h h) (h h)... {(n ) h (n )h ] h (n ) n.(n ) h.n(n ) lim h h. h 6 8 lim (n )n.(n ) n 6n n n.(n ) 8 SECTION B 9. a 5î ĵ 5kˆ, b î kˆ a λ b c, where c b c a λ b 9

22 ( 5 λ) î ĵ (5 λ)kˆ c b b. c (5 λ) ( ). (5 λ) and c î ĵ kˆ. For the vectors a î aĵ ckˆ, î kˆ and cî cĵ bkˆ to be coplanar, a c a c c b c ab or c s λ λ vinitial 5 5km/hour m 5m/sec b u 6î as a velocity kˆ 5km/hour m/sec 5 5m/sec.. a. 5 ab 5. Let u 8km/hour 5 m/sec m. a. 5 Again final velocity 7km/hour m/sec. 9

23 Let s be the distance travelled 5 5 s or 75. s s 7.5m 5. Distance travelled in nth second g (n )...(i) Total distance covered where n is number of seconds. g. n...(ii) 5 It is given that ( i) of (ii) 9 g (n ) 5. g n 9 5n 8n 9 or n, neglecting n 5 5 Height of tower g () 9 g m. (i) u (9.) Maimum horizontal range 88. m g 9.8 (ii) u sin α Range. g. 9.8 sin α α or α 5 9

24 . Let ( α, β, γ) be the foot of perpendicular (as shown in figure) D.R.of BC are Equation of BC is...(i) As ( α, β, γ) lies on (i) α λ, β λ, γ λ...(ii) or As D,.RAA' α of y α, ( α AA' β BC ) are z γ γ ( α( 5 β ), 8) ( β ( 8), γ ) ( γ ) 5P Q9cos 5 From (i) and αs P, cos, α (ii), we get λ (say) λ Q P The foot of perpendicular is. Resultant of P and Q acting at angle α is perpendicular to P...(i) Similarly S P cos α...(ii) From (i) and (ii) P S P Q.S Q P or P QS 95

25 5. Figure In case (i), P BC Q AC R AB P BC. AB...(A) m Q P In case (ii), P BC' Q AC' R' AB P BC'. AB...(B) P Q P Q P P P Q P Q F F F sin 5 sin 5 sin ½ F F F F : F : F : : 6 6. Cartesian equation of one-plane is 6y or y 6...(i) Equation of second plane is - y z...(ii) Any plane passing through the intersection of (i) and (ii) is ( λ) y ( λ) λz 6 Its distance from (,, ) is unity. 96

26 6λ 6 λ ± The equation of planes are ρ r. (î ĵ kˆ ) SECTION C 9. Figure Maimise Z 6 5y 6 Their 6 When yy yy vector λ 5 9, ; 6 equation, y y are z z, y y r 5y 6. (î 8 y z ĵ kˆ ) Constraints : ( λ) ( λ) 6λ Z at O Z at A 6 8 Z at B Z at C Z is maimum when, y. Let the tailor A works for days and the tailor B works for y days We have to Minimise 5 y Z Subject to the constraints m 97

27 ., E and A are the events defined as follows E E : Plant I is selected to manufacture E : Plant II is selected to manufacture A : The motor cycle is of standard quality 7 (E), P(E ), P 8 9 P(A/E ), P(A/E ) By Baye's Theorem, P(E ) P(A/E ) P(E /A) 7 C 5 P(E 5 ) P(A/E) P(E ). P(A / E ) Let p be the probalility that a student entering a university will graduate. p., 5 q, n 5 5 (i) P (none will graduate) (ii) 5 P (only one will graduate) C (iii) P (All will graduate) 8 C

28 Here λ P (at least three misprints) Let the total profit be Rs A's Salary % of Rs Balance of Profit The ratio of profit sharing is : 5 : 9 9 P (X ) P (X ) P (X ) () e e Rs Rs e e 8! 5! Rs As' share in A's total share Total Profit B's share C's share Rs 5 99

29 Initially ratio of sharing profit : After C joins, the ratio of sharing profit 9 : 8 : 7 9 A has sacrificed 8 8 B has sacrifice 6 Premium sharing ratio of A and B is : B's share in premium Rs ( 9) Rs. We know that where P is the present value. Here R Rs, n, i. (.) P. 5 A's share 5 ( in i) n 75 premium Rs P R Rs 9 i 7 (.) Present value of annuity is Rs (i) Here C() 75 R () 5 For no loss, R() C() i.e. 5 calculators must be produced and sold daily for no loss.

30 (ii) Here C() 75 R () 5 For Break even points, R() C() or i.e. 5 calculators be produced for break even points. 6. Let the face value of bill be Rs S, Time 8 months m Date ( 75 Legal Face Number of discounting due 6 of 75 date days 7th for th which December, July, 7) the 6 bill days was discounted early Time Value Rs 5 Srt, r 5% 65 rt 5 B.G BD TD 5 S 5 5 S 5 5 S Note : If candidate has taken 66 days for the year and gets the answer as Rs 577 (app.), full credit may be given.

31 QUESTION PAPER CODE 65/ EXPECTED ANSWERS/VALUE POINTS SECTION A. f (A) A A I 5 5 (½½) m 5 5. Using R R R and R R R we get a a a Using C C C we get a (a ) a [ (a ) (a ) ] or (a ), a (Note : Using any two operations (ark each) and finding answer - mark ) Let A: the chosen integer is divisible by 6 B: the chosen integer is divisible by 8 P(A or B) P(A) P(B) P(A and B)

32 (½½½) m 5. P (getting grade A in all subjects) (.)(.)(.5). P (getting grade A in no subject) (.8)(.7)(.5).8 P (getting grade A in two subjects) (.8)(.)(.5)(.)(.7)(.5)(.)(.)(.5)..7.. cos.sin d 5. Writing I and putting (a b cos ) t, to get sin d dt. (a b cos ) b I t a a dt dt b t b t t 5 t 8 I e c c t log a a log t c log (a b cos ) C b t b a b cos t t 6. Put log t e and d e. dt I t e dt t t using [ f () f '()] e d f ().e c, we get (½) m dy cot d 7. cot.y cot, I.F e sin d y. sin (.cos sin ) d

33 y. sin.sin sin d sin d c y sin sin c or y c. cosec 8. Here, dy d y y dy dν, Put ν ν y d d dν ν d ν ν or dν d ν ν ν ν d ν dν dν d y ν log ν log c. y log log c (½ ½) m (½ ½) m d y d dy d e cos e sin C dy d when C dy d e sin y e cos C y when C y e cos 9. The truth table is Hypotheses Conclusion p q pvq ~ p ~ q T T T F F F TRUTH TABLE : T F T F T F T T T F Critical row Identification : F F F T T There is only one critical row in which the conclusion is false. Hence, the given argument is invalid.

34 LHS y ( y ) (y y ). [ cos cos ( y) ] Lim sec y y. cos. cos ( y) y y sin sin Lim sec y y. cos. cos ( y) ( y ( [ ( y) y)sec( ).(.y ) y y) y )] sec (y [..y( tan ) y ) ] [sec ( tan y) sec] Lim f () tan f () Lim Lim y y y. tan.sec sec or sec ( tan ) ½ m y. Let f () Lim tan [ ][ tan tan ] Lim tan [ ] [ ]. tan.tan [ ] tan Lim. sec. 5

35 n n dy. y [ a ] n. [ a ]. dy d n. d n [ ] ( a ) a. a a n [ a ]. n. a ny a. f () 5 6 f () 6 6 f () 6( 5 6) ( )( ),, Possible intervals are (, ), (, ), (, ) for f () increasing decreasing in ((, ), ) (, ) and Points at which the tangents are parallel to -ais are (, 8), (, 9) 6. I d d d d 6 d log 6 6 d ( ) ( ).log 6 6 tan C 6

36 5. f ()d d d 5 d ( )d ( )d ( )d ( 5)d m ( ) [ ( ) ] Figure Let O be the centre of sphere OC cm Let radius of cone cm and height h cm (h ) and h AD AOODOD OD (h - )cm Volume of cone [ (h ) ] π.h πh π V [ h h h h] [ h h ] π V dv 8h h, 6 h h 6 cm dh d v dh π [ 8 6h] π ( 8) i.e. negative For Maimum volume, height 6cm 7

37 Getting the point of intersection dy d y k y Slope (m ) / dy y y k Slope (m ) / d k m.m / k k / 8k 7. Writing the given equations as y z or A.X B X A.B / a( k /, k ) A () () () a a [Note : For every four a a a correct cofactors, one mark m a a a may be given] y z, y, z 8

38 8. Correct figure Point of intersection of two curves is at Required area () m 6 sq. units Correct figure Getting y -- and y as points of intersection Required area a d d Required area y y y SECTION B ( y)dy y dy m 9 sq. units 8 9. Given that a. b b. c c. a and a b c Let θ be the angle between and a b c 9

39 Cos θ a. a b c a a b c a a. a b c a.b a. c Cos θ a ρ a b c θ Cos. A, B, C, D are coplaner if AB, AC, AD or Scalar triple product of any three vectors through all four points AB î ĵ kˆ, AD î ĵ kˆ AC 6î ĵ 6kˆ AB, AC, AD 6 6 using νt ν6 u or at we ν get 6 u a ν 9 m/s using s ut at we get 8 u a () ( 8) ( ) 8. A 8 m B 6 m C t s. v t s. Let velocity at an initial point u m/s, and it covers a distance of 8m in sec. with acceleration a m/s to reach B with velocity v m/s. and in net sec. it covers 6m in seconds without any accelaration....(i) and...(ii) Solving (i) and (ii) we get u m/s and a m/s

40 . u m/s α 6 (i) Equation of path is g y tan α u cos α y g 576 or y g 88 (½ ½) m (ii) u sin α T. sec g 9.8 (½½) m (iii) 576 u sin α Maimum height. m (½½) m g 9.6 ( ) ( ) î ĵ kˆ î ĵ 5kˆ At A (5, ), we can write 5g 5 tan α...(i) u cos α At B (5, ) 5g 5 tan α...(ii) u cos α Multiplying (i) by 9 and subtracting from (ii) we get 8 9 tan α tan α 8 9 α tan 8 9. A vector m perpendicular to both given lines is î ĵ kˆ i.e. m î ĵ 8kˆ or î 7 ĵ kˆ 5

41 Equation of line passing through and parallel to is. If is the angle between then R P Q PQ cos α...(i) R P Q PQ cos α...(ii) and R P Q PQ cos α...(iii) Eliminating α between (i) and (ii) we get Eliminating α between (i) and (iii) we get P 6 Q 9 R 6 P : Q : R or : P Q : R α (,, y ) P z m P and Q Q Q R 5R 7 Correct figure Since () () (5) ACB 9 ACD (9 A) and BCD (9 B) Using Lami's theorem we get T W sin(9 B) sin(9 A) sin 9 T T T cos B cos A W T W cos B W. 5 5 W

42 T W cos A W. W 5 5 T : T : 5. Figure Let a force of 8N act at A and of N at B and the resultant at C. Let AB cm 8 BC AC or AB 8 () m Distance between the lines of action of two forces 6. Let the equation of sphere be y z Solving and ( or, u u the, 5, u 5 cm ) ) 6v centre vy ii, v vlies iii,iv 8w dw wz w ( & u, d v, d d v, 6 u w)...(i) 9 5 lies, v on, u w Equation on sphere is y z y 6v v z d and 6y w d z d (,, ) lies on sphere 9 6 u 6v 8w d...(ii)...(iii)...(iv)...(v) m

43 SECTION C 9. Let E : getting 5when a dieis tossed. E : Not getting a 5 H : reports that it was 5 (E) P(E ) 6 P ( H / E ) P( H / E ) P P ( E / H) P(E )P P(E )P(H/E ) ( H / E ) P(E )P( H / E ) Here n probability of success (p) probability of failure (q) Since distribution is binomial Mean np. Variance npq.. P(X ) e.5 P(X ) P(X ) λ e. λ! λ e. λ! (½½) m Mean of the distribution ( λ) P(X ) e ()! 6 (.5 ).9

44 . B.G. BD (B.D TD).BD 8B.D 9T. D 9 9 S.r.t 8.S.r.t 9. 8 rt rt 5 t.5 years. 8. S Rs 5, Legal due date is 9th Sept.. From th February to 9th September, number of days (6 9) 9 days. [ 5 8] Rs 98 Discounted value Rs 8 5 B.G BD TD B.D S.r.t Rs 5 8 Rs Let total investment be Rs and total time be 't' months. A invested Rs for a period of t months A ' s adjusted capital for onth Rs t. Similarly and t B 's adjusted capital for onth Rs. C 's adjusted capital for onth Rs. t 6 Ratio is : 9 : 6 or 9 : : 6 5

45 9 A 's share in Profit Rs,9, Rs 9, 9 B 's share in Profit Rs,9, Rs, 9 C 's share 6,. A Rs 5,5, n 5 i.5 Using Rs Rs Rs (.5) P 5 Rs,, 5. n [( i ), ] P A.i C A ().C P n i [ i] (i) M.C C'() d d (MC) units d and d (MC) i.e ve Output for minimum MC units. (ii) M.C AC 5 units Output at which MC AC is 5 units 6

46 (i) At Break even point C() R() or 5 5 units or 5 units (ii) or 5 5 >, ( 5 )( ) > > 5 or < So, there will be loss if firm produces less than units Or more than 5 units. 6. If number of trunks of Ist type & y number of trunks of second type be manufactured. Subject to : Maimise P 5y For L.P.P., loss y y y R() 8, 5, is. C() < < Correct graph m Etreme points of feasible region are A(, 6), B(, ) and C(5, ) Profit at A 5 6 Rs. 5 Profit at B 5 Rs. 65 Profit at C 5 5 Rs. 5 For maimum profit he should manufacture trunks of each type. m 7