RELATIONS AND FUNCTIONS

Transcription

1 CHAPTER RELATIONS AND FUNCTIONS

2 . Let f: R R e defined y f() = Stte whether the function f() is onto.. Let e the inry opertions on Ζ given y =, Ζ.Find the identify element for on Z, if ny.. Stte with reson whether the function f: X Y hve inverse,where f()=. nd X=Q-{o}, Y=Q. 4. Let Y = {n : n N} e suset of N nd let f e function f : N Y defined s f() =. Show tht f is invertile nd find inverse of f. 5. Show tht the function f : N N given y f() = is ijective. 6. If f e the gretest integer function nd g e the solute vlue function; find the vlue of (fog)(-/) (gof)(4/). 7.Consider the mpping f :[,] [,] defined y f()=. Show tht f is invertile nd hence find f Give emples of two functions f N N nd g :Z Z such tht gof is injective ut g is not injective 9..Give emples of two functions f:n N nd g: N N such tht gof is onto ut f is not onto...let f: R- {-/5} R e function defined s, find the inverse of f.. Show tht the reltion R defined y (, ) R (c, d) d= c on the set is n equivlence reltion..let Q e the set of ll positive rtionl numers. Show tht the opertion * on Q defined y * = () is inry opertion. Show tht is commuttive. Show tht is not ssocitive.. Let A= N N. Let e inry opertion on A defined y (,) (c,d) = (d c, d),,c,d N. Show tht (i) is commuttive (ii) is ssocitive (iii) identity element w.r.t. does not eist. 4. Drw the grph of the function f() = on R nd show tht it is not invertile. Restrict its domin suitly so tht f - my eist, find f - nd drw its grph. 5. Show tht the reltion congruence modulo on the set Z is n equivlence reltion.also find the

3 equivlence clss of CHAPTER INVERSE TRIGONOMETRIC FUNCTIONS ) Prove tht tn tn = ) Solve = ) Write,, in the simplest form. 4) Solve tht = 5) Prove tht = ( ( 6) Solve for : = 7) If c = prove tht c = c 8) Prove tht cos ( ) ) = 9) Wht is the principl vlue of )If =, prove tht y yz z = )Show tht 4 )If = )If = - 4)If =, find the vlue of

4 5)Find the vlue of sin ( ). CHAPTER - MATRICES. If 5 6 is symmetric, find.. If A= z y is such tht A =I, then find the vlue of yz. If A = 4 then find f (A) when f() =. 4. If A= i i. find A 4n,n N 5. Give n emple of squre mtri which is oth symmetric s well s skew symmetric. 6. If A nd B re symmetric mtrices, then show tht AB BA is lso symmetric mtri ut AB- BA is skew symmetric mtri. 7. Show tht ll the positive integrl powers of Symmetric mtri re Symmetric. 8. Find the mtri A stisfying the mtri eqution = A 9.If A=,, Prove y induction tht = ) ( A n n n for ll positive integer n..find if [ ] 4 5 =. By using elementry row trnsformtion, find A - where A =. If A =. ) ( A I A I provetht ndi = =. If A nd B re two mtrices such tht AB=B nd BA=A find

5 4. If is skew-symmetric mtri, wht is the vlue of for every i? 5., find the mtri B such tht AB = I CHAPTER - 4 DETERMINANTS. If,,c re non-zero rel numers, then find the inverse of mtri A= c.. If A= then wht is the dj(dja)?. If A is squre mtri of order such tht Adj A =64, then find A 4. Find the vlue(s) of θ, if the mtri θ θ cos cos is singulr, where < θ < π. 5. Evlute the determinnt log log 6.If λ λ λ λ λ λ λ λ λ λ 4 =A 4 λ B λ C λ D λ E, then find the vlue of E 7. The vlue of third order determinnt is. Find the vlue of the squre of the determinnt formed y the cofctor 8. Let A e skew symmetric mtri of odd order, then wht will e 9. If f() = cos sin sin cos,then show tht ) ( )} ( { f f =. Prove the following y using the properties of determinnts ) ( ) ( ) ( ) ( c c c c c c =

6 . Using the properties of determinnts, solve for. =. If l, m, n re in A.P. then, find vlue of n m l If c c c α α α α nd α is not root of the eqution = c,then show tht,,c re in G.P. 4. Let k k k k k k k k k z z z y y y =(-y) (y-z) (z-) ( y z ), then find k 5. Let, find. Hence solve the following system of equtions y z =7 y z = 7 y z = 6. Given tht A= nd B= Find AB nd use it to solve the system of equtions y z = 4, - y - z = 9, y z =. 7. Prove tht 4) ( 4) )( ( ) ( ) )( ( ) ( ) )( ( =. 8.Using the properties of the determinnts, prove tht pc pc pc nc nc nc mc mc mc = m) (p - p) (n - n) mpn (m -.

7 9. Evlute c c c c CHAPTER 5 CONTINUITY AND DIFFERENIABILITY. Show tht the function f() = Sin Cos is continuous t = π. Show tht the logrithmic function is continuous.. Let f() = ( ) Cos for nd let f()=. Show tht f is continuous t = ut not derivle there t. 4. Let f()=. for ll R. Discuss the continuity nd defferentiility of f() t =. 5. Emine for continuity nd differentiility of the following functions:- f() = Sin / >, if t = 6. Given tht If f() is continuous t =, find the vlues of. 7., if > If the function f() = if = 5 - if <

8 is continuous t =, find the vlues of nd 8. Discuss for continuity of the function t = Sin tn, if < f()= /, if = log( ) e if > 9. Find ll points of discontinuity of f where f() = Sin, if <, if. Show tht the function f() =, if is not differentile t =. Is the function f() = [] - -, = 5-, if > continuous t =?. Show tht the function f is continuous t = for ll vlues of. Also find the vlue of for which f is derivle t = when f() =,, <. Emine the continuity of the function f() = tn - ( -)

9 4. If f()= Sin, < π / 6 π / 6 << is continuous nd differentile. Find & 5. Find whether the function f() = 6. Find whether f() = e e / / ( )( ), =,, is continuous. is Continuous? 6, =, = 7). Find the vlue of derivtive t = of the function f()= 8). Find the derivtive of the following w.r.t.. ) y=log ( ). ) y=sin ( ). ) y=siny. 4) y =e -y 5) y=e ². 6) y= (sin - )². 7) y= 8) y= sin - ( cos ) cos 9) y=tn - [ tn y / ] ) y=tn - /( - )

10 ) y=sin - [² (-²) (- 4 ) ] ) y = Cos ( ) ) y = e -²logsin 9). =sin / cost,y=cos / cost ). If p y q =(y) pq then show tht ).. Differentite (sin) w.r.t. sin dy = d dy π ). If =sint(cost) & y=cost(-cost) Show tht = t t= d 4 cos 4sin 4). Differentite cos - [ ] w.r.t. 5 5). Differentite sin² w.r.t e cos. 6). Show tht y=c e c e - is the generl solution of y dy d 7). Prove tht the solution of y = is y=c/c. d dy 8). If y=log( ), Prove tht d y d 9). Differentite y=log 7 (log ) w.r.t. = ( ) ). Differentite y=sin ( cos ) w.r.t.. ). Differentite y= w.r.t.. ). Differentite y=tn - ( ), w.r.t.. ). Differentite y=log {tn (Π/4/)} w.r.t. 4). Verify Rolle s Theorem for f() = log ( ) log on [ -, ] 5).Verify Rolle s Theorem for f() = Sin 4 Cos 4 in [, Π ] 6). Verify Rolle s Theorem for f() = e - Sin in [, Π ] 7). Verify LMV theorem for f() = Sin Sin on [, Π ]

11 8).Find point on the Prol y = ( ) where the tngent is prllel to the chord joining (, )nd ( 4, ) CHAPTER 6 APPLICATIONS OF DERIVATIVES ) Show tht the rte of chnge of the perimeter of squre is 4 times the rte of chnge the length of its sides. ) Using differentils,find the pproimte vlue of log e 4., given tht log e4 =.86 The pressure p nd the volume v of gs re connected y the reltion pv =.4 = constnt.. Find the ) percentge error in p corresponding to decrese of / % in v. 4) If there is n error of % in mesuring the length of simple pendulum,then find the percentge error in its time period. While mesuring the side of n equilterl tringle, n error of 5% is mde. Find the percentge error in 5) its re. 6) For wht vlue of is the rte of increse of -5 8 is twice the rte of increse of? 7). If the rte of chnge of re of circle is equl to the rte of chnge of its dimeter,find the rdius. 8). The side of n equilterl tringle is incresing t the rte of / cm/sec. Find the rte of increse of its perimeter. 9). Find for which f()=( Sin) is incresing ). Let g() =f() f(-) nd f ()> for ll [,] then g() incresing or decresing on [,]? ). Let f()= tn - g(), where g() is monotoniclly incresing for << π /, then find f() is incresing or decresing on (, π / ). ). Find whether the function f() = tn - (Sin Cos ) on [, π / 4 ] is either

12 strictly incresing or strictly decresing.. ). For wht vlue of λ ' for which the function f()= cos - λ is monotonic decresing. 4).Find the vlue of for which function f() = log is incresing on R, 5). If the slope of tngent to curve y= t (,-6) is -. Find &. 6). If y=k is norml to the curve y =, then find the vlue of K. 7). Find the point t which the curves =y nd y = cut orthogonlly. 8).Find whether the function f()= 9).Is the function f()= is strictly incresing on R? is incresing or decresing. Find the ngle of intersection of the curves y= nd -y =. Find the condition for which the curve y=e nd y=e - cut orthogonlly.. Find the slope of tngent of curve y= 4 t the point whose sciss is -?. Wht is the slope of Norml to curve y= Sin t =? 4. If the function f() = -k5 is incresing on [,4] then find the vlue of k. 5).Find the intervl for which the function f() = is decresing 6). A mn meters high, wlks t uniform speed 6 meters per minute wy from lmp-post,5 meters high. Find the rte t which the length of its shdow increses. 7) A kite is m high nd m string is out. If the kite is moving wy horizontlly t the rte of 5m/sec find the rte t which the string is eing pid out. 8) An inverted cone hs depth of cm nd se of rdius 5 cm. Wter is poured into it t the rte of / cc per minute. Find the rte t which the level of wter in the cone is rising when the depth is 4 cm. 9) The time T of complete oscilltion of simple pendulum of length l is given y the eqution T = π,where g is constnt.wht is the percentge error in T when l is incresed y %? ) Find the pproimte vlue of tn46 if it is given tht =.745 ) A mn is wlking t the rte of 4.5km/hr towrds the foot of the tower m high. At

13 wht rte is he pproching the top of the tower when he is 5m wy from the tower? ) Find the rte of chnge of the curved surfce of right circulr cone of rdius r nd height h with respect to the chnge in rdius. ) Find the ngle etween the prol y =4 nd =4y t their point of intersection other thn origin. 4) If y= log hs its etreme vlues t =- nd =, then find &. Show tht locl Minimum vlue of f()=, is greter thn locl mimum vlue. 5) Find the Asolute mim nd Asolute minimum vlues of the function f()= ( ) on[, 5] 6) Determine the Mimum nd Minimum Vlues of the function y=cos -Cos 4, π 7) Find the locl minimum vlue of f() =, R 8) A given quntity of metl is to e cst into solid hlf circulr cylinder (i.e with rectngulr se nd semi circulr ends). Show tht in order tht the totl surfce re my e minimum, the rtio of the length of the cylinder to the dimeter of its circulr ends is 9). A window hs the shpe of rectngle surrounded y n equilterl tringle. If the perimeter of the window is m, find the dimensions of the rectngle tht will produce the lrgest re of the window. 4). Show tht the isosceles tringles of mimum re tht cn e inscried in given circle is n equilterl tringle

14 Indefinite Integrls. Evlute 5 d Sec tn. Evlute d sec e. Evlute d 4. Evlute 5. Evlute: Sec d d 6. Evlute: sin(log ) d 7. Evlute: 8. Evlute 9. Evlute. Evlute ( ) ( 4) d ( sin ) ( cos ) /4 5/4 I = tn tn tn d I =. Evlute sec d sin ( cos ) CHAPTER -7 INTEGRALS d d. Evlute I = tn tn d tn. Evlute sin d cos

15 4. Evlute: e ( ) d 5. Evlute: d sin cos sin cos sin 6. Evlute: d sin 4 7. Evlute: sin d d 8. Evlute: 9. Evlute: (tn ) d ( ) ( ). Evlute sin sec d cosα. Evlute I = cos α ( ) /. Evlute d. Evlute d 4 4 sin cos 4. Evlute I = d 5. Evlute ( ) ( ) ( sin cos ) 6. Evlute d 4 7. Evlute tn d 8. Evlute sin d sin( α ) 9. Evlute I = d sin( α ) d. Evlute: ( e ) d ( ). Evlute: 4. Evlute: d log log d sec d cos ec

16 ..Evlute: 4. Evlute: 5. Evlute: 6. Evlute: Definite Integrls 7). Evlute 8). Evlute 9). Evlute 4). Evlute 4). Evlute 4). Evlute sin sin d e sin d (log ) d π 4 π 4 π sin tn d d d log π 4). Evlute log 44).Evlute 45).Evlute 46).Evlute e e π d d sin cos π 4 d cos sin d sin cos π.5 π e d cos d ( cos ) π d sin d

17 47).Evlute 48). Evlute 49). Evlute 4 d sin π d log ( ) d 5) Evlute ( ) d 5).Evlute s limit of sum 5). Evlute 5). Prove tht cot π n sin sin cos n n 54).Evlute log 55). Evlute π d cosα sin e 5 d d d = π d

18 CHAPTER - 8 APPLICATION OF INTEGRALS π. Drw the grphs of the curves y = sin nd y = cos,.find the common re etween the ove curves with the X is.. Find the re ounded y the lines y = ; y = nd y = 7. Find the re ounded y the line y = nd the curve y =. 4. Find the re ounded y the lines y =, = -, = nd y =. 5. Find the re enclosed etween the curve y = nd the line y =. 6. Find the re ounded y the curve y = e nd the line y = with X- is. 7. Find the re ounded y the curve y = tn nd the line y =. 8. Find the re included etween the curve y = - [ ] nd the line = with X &Y is. 9. Find the re enclosed etween the curve y = sin nd the line y = within the intervl 5π 7π, Find the common re etween the curve y = 5 nd the lines y =.

19 CHAPTER 9 DIFFERENTIAL EQUATIONS dy. Solve cos cos y d = /. Find the degree nd order of the differentil eqution ( dy d y ) = (4 ) d d. Find the differentil eqution of the fmily of curves given y y = dy 4. Find the integrting fctor of the differentil eqution y d =. yd dy 5. Verify tht y = c is solution of the differentil eqution = y 6. Verify tht y = e e dy e is solution of the differentil eqution = d ( e ) e 7. Find the eqution of the fmily of curves whose nd y intercepts of the tngent t ny point p re respectively doule the nd y co-ordintes of the sme point p respectively.. 8. The line norml to given curve t ech point (,y) on the curve psses through the point (,). If the curve contins the point (, ), find its eqution. Prove tht the curve with the property tht ll its norml pss through constnt point is circle. 9. A popultion grows t the rte of 8% per yer. How long does it tkes for the popultion to doule?. Solve: ( e ) dy ( y ) e d =, given tht y = when =. Solve: ( ) = ( tn ) y d y dy e y d. Solve the differentil eqution =, dy 4 Prove tht the solution of the differentil eqution

20 dy = y is y y = cons tn t 6 d y 5. Solve: log dy y = log d dy 6. Solve: (y) d = dy 7. Solve: y(log y log ) d = y y d ydy = 8. Solve: ( ) 9. A nk pys interest y continuous compounding tht is y treting the interest rte s the instntneous rte of chnge of the principl. Suppose tht in n ccount the interest t 8% per yer compounded continuously. Clculte the percentge increse in such n ccount over one yer. (Tke e.8 =.8 pproimtely) d. Solve the differentil eqution sin y, dy = given tht = nd d dy = when y =. d y dy. Solve the differentil eqution = e, giventht y = nd = when =. d d. Solve: ( )sin y y d yd = ( yd dy) cos.. Solve the differentil eqution ( sin ) y d= y dy. 4. Show tht the differentil eqution ( - y) dy = y is homogenous nd solve it. d 5. Find prticulr solution of the differentil eqution dy d y cot = 4 cosec ( ) given tht y = when = π.

21 CHAPTER VECTORS. Find unit vector prllel to XY - plne nd perpendiculr to the vector 4i - j k. If r = 6, r r = 7nd r r = 5, find. r. Write numer of unit vectors perpendiculr to iˆ ˆj nd ˆ j kˆ. uuur uuur uuur ur 4. If G is the centroid of thetringle ABC. Showtht GA GB GC =. 5. If r is non zero vector of mgnitude then find the vlue of λ if λ r is unit Vector. 6. Show tht the sum of three vectors determined y the medins of tringle directed from the vertices is zero. 7. Prove tht the lines joining the mid-points of two opposite sides nd the mid-points of the digonls of qudrilterl form prllelogrm. 8. Show tht the stright line joining the mid-points of non-prllel sides of trpezium is prllel to the prllel sides nd hlf of their sum. 9. Use the vector method to prove tht the lines joining the vertices of tetrhedron to the centroids of the opposite fces re concurrent.. Find ll the vlues of λ such tht (, y, z) (o,o, o) nd $ i $ j k$ ( $ i j k) y ( 4 $ i 5 j) z = λ ( i $ y j zk) ( ) $ $ $ $ $. Prove tht the middle point of the hypotenuse of right ngled tringle is equidistnt from its vertices.. In tringle AOB, ngle AOB = 9.If P nd Q re the points of trisection of AB, show tht 5 OP OQ = AB. 9

22 . For ny vector r, show tht 4. If A, B, C, D re four points such tht r r r r r r r AB = m ( i 6 j k ), BC = i j nd r r r r CD = n 6i 5 j k. ( ) r iˆ r ˆj r kˆ = r. Find the conditions of the sclrs m, n such tht CD intersect AB t sme point E. Also find the re of the tringle BCE. 5. If A, B, C, D e ny four points in spce prove tht r r r r r r AB CD BC AD CA BD = 4 Are ABC. ( ) r r r r r r r 6.Let OA =, OB = nd OC = whereois origin. Let p denote the re of the qudrilterl OABC nd q denote the re of the prllelogrm with OA nd OC s djcent sides. Prove tht p = 6q. 7. The lines joining the vertices of tetrhedron to the centroids of opposite fces re concurrent 8. Points F nd E re tken on the sides BC nd CD of prllelogrm ABCD such tht r r r r BF : FC = µ :. nd DE : EC = λ :. The stright lines FD nd AE intersect t the point O. r r Find the rtio of FO : OD. r r r r r r r 9. ABCD is qudrilterl such tht AB =, AD = d, AC = m pd show tht the re of qudrilterl ABCDE is m p r d r.. The vector i $ $ j k$ isects ngle etween the vectors r c nd $ i 4 $ j. Determine unit vector long c r.

23 CHAPTER THREE DIMENSIONAL GEOMETRY ) Find the direction of ngles of the line joining points. (-,-5,-) nd the point of intersection of the line y z = = 4 nd the plne -yz=5 with, y,z es. ) Find the perpendiculr distnce of verte of cue from its one of the digonl, not psses through the verte. ) Find the distnce of the point (-,,-4) from the line y z 4 = = 4 5 mesured prllel to the plne 4 y -z =. 4) Seprte the eqution y yz = into two plnes nd find out whether the plne re or to ech other. 5) If A(,,) nd B(,6,) re imges to ech other w.r.t. plne. Find the vector eqution of the plne mirror. Find the vlue of λ if the plne mirror is to y λz 5 =. 6) Find k, if the plne 4y z 7 = contins the line z k 4 = y = y z 7) Find the point on the line = = t distnce from the point (,,). 8) Find the Direction Cosines of the line joining the imges of the point (,,) w.r.t. y nd yz plnes. 9) A line mkes the sme ngle θ with ech of the X nd Z es. If the ngle β, which it mkes with Y is such tht sin β = sin θ, then find the vlue of θ ) Prove tht the two plnes -yz=6 nd -6y6z= re prllel. Also (i ) find the distnce etween the plnes. (ii) find the intercept on y z the line = = etween the two plnes. ). Wht is the direction cosines of line eqully inclined to the es.? ). Wht is the eqution of Y is in vector nd Crtesin form in three dimensionl spce?

24 ). If the projection of the line segment on X, Y, Z es re respectively 4,, find the length of the line segment. 4). Find the distnce of the point (,,4) from the plne yz5= mesured prllel to the line y = = z 6 5). Find the eqution of the line pssing through the point (,,) nd r r prllel to the line = iˆ ˆj λ (iˆ ˆj 6 k ) nd lso find the distnce etween them. 6). Show tht the eqution of the plne which meets the es in A,B nd C nd the centroid of tringle ABC is the point (u,v,w) is y z = u v w 7). Find the vector eqution of plne which is t distnce of 5 units from the origin nd which hs -,, s the direction rtios of norml to it. 8). A line mkes ngles α, β, γ nd δ with the four digonls of cue prove tht ( i) Sin α Sin β Sin γ Sin δ = ( ii) cos α = π 9). Show tht the ngle etween ny two digonls of cue is Cosec () ). If point A(,,) move towrds nd reches line y z = = in shortest distnce nd the point A move towrds nd reches line y z = = in shortest distnce. Find the distnce etween the two new loctions of A. 8

25 CHAPTER - LINEAR PROGRAMMING PROBLEMS ). Find whether the mimum vlue of the ojective function Z = - y eists or not, suject to the following constrints. y 5 y 6nd y ). Find whether the minimum vlues of the ojective function Z = - 5y eists or not, suject to the following constrins y 5 y y, y ) Mimize Z= y suject to the constrints y y, y 4). Kellogg is new cerel formed of miture of rn nd rice tht contins t lest 88 gms of protein nd t lest 6 mg of iron. Knowing tht rn contins 8gms of protein nd 4mg of iron per kg, nd tht rice contins gms of protein nd mg of iron per kg, find the minimum cost of producing this new cerel if rn costs Rs. 5/- per Kg nd rice Costs Rs.4/- per Kg. 5). A rick mnufcturer hs two depots A nd B with stock, nd, ricks respectively. He receives orders from uildings P,Q nd R for 5,,, nd 5, ricks respectively. The costs of trnsporting, ricks to the uilding from the depot (in Rs.) re given elow. From/To P Q R A 4 B 6 4

26 How should the mnufcturer to fulfill the orders so s to keep the cost of trnsporttion minimum. Solve it grphiclly. 6). Find the constrints of the L.P.P if its grphicl representtion is given elow nd hence mimize Z= 9y 7). A mnufcturer produces two products A nd B during given period of time. These products require four different opertions, viz. Grinding, Turning, Assemly nd Testing. The requirement in hours per unit of mnufcturing of the product is given elow: Opertion A B Grinding Turning Assemly 4 Testing 5 4 The ville cpcities of these opertion in hours for the given time re: Grinding Turning 6 Assemly Testing Profit on ech unit of A is Rs.nd Rs. for ech unit of B. Formulte the prolem s LPP. 8). Constrins of L.P.P. represents the grph given elow. Write the constrins nd Minimize Z=67y

27 CHAPTER - PROBABILITY ) Find the minimum numer of tosses of pir of dice so tht the proility of getting the sum of digits on the dice equl to 7 or t lest one toss is greter thn.95, given log =. & log =.477 ) The sum of men nd vrince of inomil distriution is 5 nd their product is 54, find the distriution. ) If A nd B re events such tht p( A B). p ( A B) =, 4 p ( A B) =, 4 p ( A) =, find 4) Two dice re rolled one fter the other. Find the proility tht the numer on the first is smller thn the numer on the second. 5). A mn tkes step forwrd with proility.4 nd ckwrd with proility.6. Find the proility tht t the end of eleven steps, he is one step wy from the strting point. 6).Three numers re chosen t rndom without replcement,,,,. Find the proility tht the minimum of the chosen numering is or their mimum is 7. 7). In olt fctory three mchines A,B nd C, where A produces one-fourth, C produces twofifth of the products. Production of defective products in % y A, B,C re respectively 5,4 nd. An item is drwn t rndom nd found to e difficult. Wht is the proility it ws produced y either A or C. 8). Two persons A nd B throw pir of dice lterntely eginning with A. If Cosα represents the proility tht B gets doulet nd wins efore A gets totl of 9 to win. Find α. 9).A g contins 6 red nd 5 lue lls nd nother g contins 5 red nd 8 lue lls. A ll is drwn t rndom from the first g nd without noticing its colour is put in the second g. A ll is then drwn from the second g. Find the proility tht the ll drwn from the second g is lue in colour.

28 ). A, B nd C throw die lterntively till one of them gets ny numer more thn 4 nd wins the gme. Find their respective proilities of winning if A strts the gme followed y B nd C. ). One letter hs to come from LONDON or CLIFTON. Only ON is seen on the post mrk, find the proility of this letter from LONDON. ). Three stmps hve een selected from stmps which re mrked from to. Find the proility the numer on selected stmps re in A.P. ).A g contins red lls ering one of the,, (one numer on one ll) nd two lck lls ering the numers 4 or 6. A ll is drwn nd its numer is noted nd the ll is replced in the g. Then nother ll is drwn nd its numer is noted. Find the proility of drwing: (i) on the first drw nd 6 on the second drw. (ii) numer on the first drw nd 4 on the second drw. (iii) totl of 5. 4).In n emintion, n eminee either guesses or copies or knows the nswer of multiple choice questions with four choices. The proility tht he mkes guess is / nd the proility tht he copies the nswer is /6.The proility tht his nswer is correct given tht he copied it is /8. Find the proility tht he knew the nswer to the question, given tht he correctly nswered it. 5). In clss hving 6% oys, 5% of the oys nd % of the girls hve n I.Q more thn 5. A student is selected t rndom nd found to hve n I.Q of more thn 5. Find the proility tht the selected student is oy. 6).Find the proility distriution of the numer of kings drwn when crds re drwn one y one without replcement from pck of 5 plying crds 7).A g contins 5 white, 7 red nd 8 lck lls. If 5 lls re thrown one y one with replcement. Find the proility distriution tht ectly 5 red lls drwn.

29 ANSWERS/HINTS CHAPTER -. f is not onto. e = -. No inverse f - ()=.. CHAPTER - ) =, 4) = - 6) = 9) 4) 5 ).96 CHAPTER Identity mtri of order. 6. Null mtri. 6. A - =. AB 4.Zero 5. CHAPTER 4 π ). A = ). 96 ). ± 8 4). θ = 6 c 5). Zero 6). 7). 76 8). Zero ). =, ). Zero 5). =, y = -, z = -. ). Zero

30 CHAPTER 5 4). Continuous t = Derivle only t t = 5) Continuous 6) = 8 7). =, = 8). Continuity t = 9). No point of discontinuity ). Discontinuous ) Continuous for ll vlues of ) yes continuous for ll R 6). Continuous t R {,} 7). F ()= 8). ) ) [cos { (log)}] c) y/[(-cosy)] d) e) e ² y ( log ) f) sin - / g) ( log)/(4 ) h) - ( - )/(cos) i) Sec y y y j) / (- ) k) [/ (- )][/ - 4 ] 4). Ans =, =, -π / 6 5). (, ) y l) Cos ( ) [ LogCos tn { ( log )}] m) -e -²logsin [cotlogsin] 9). -cost/sin t [(cos²t- sin²t)/ (cos t-sin t)

31 dy dy = ( Sin) [ cot log Sin] Sin ). sin [ CosLog] 4). 5). -cos.e -cos 9). log 7 e loge ). Cos Cos..( ) Cos Sin ). 4. )./[ ()] ). Sec 8). = 7 (,4 ) CHAPTER - 6 )..88 )..7 4). % 5). % 6). X =, / 7). / 8). cm/sec 9) > ) Decresing ) Incresing ). Strictly incresing) ½ 4). > 5). = -4, = - 6). K=9 7). (,) 8). Strictly Incresing 9) Yes ) 9 o or π / ) = ). -8 ). -/ 4). k (,) 5) (, /e) 6).4m/min 7) 8). 9). ½ )..49 ) / ( ) tn / / ) ( ds/dr = (r h )/ r h ) ). θ = ( ) 4) =, = - ½ 5) M vlue 78 t =, As mini vlue 8 t =6 6). M vlue / t / 6,5π / 6 vlue 8). π ) 9). CHAPTER 7 π Min vlue - t / 8 6 : ( π m & m 6 6 π 7). minimum 5/ / c ). 5 9 ). ( ) ( ) sec c e ). sin log c

32 4). ( log5 ) c 5). 7). 8) 9). = log cos c cos ). ( ) ). ). 4). e C 6). 8). sec log C 6). 5). sin sin log log C 4 sin 8 sin log log tn C 9). ( tn ). tn. log ( ) tn ). tn log C 5 tn ( ) 7). C tn tn C sin cos = c cos sin ). log tn ( sin cos ) ). ). c ). I = log 4 4 log 5 log 6 C ). 4) 5) 6). = tn log 4 c 7). put t = u t = v t t du dv = u v tn tn tn = tn log tn tn tn c tn 8). = tn ). Let c 9).

33 ). log t log ( t ) c t t log ( e ) c e ( e ) 4 log C 9 ). Log(sec) c e c log log. c 6). log (cosec cot ) cos log( ) 4 4 cos ( cos ) 7). 8). 9). Zero 4). 4). 4). ). log ( cos ) log ( cos ) log ( cos ) c 4). ( sin cos ) 5). ( ) C 4). 44). 45). 46). 47), 4 48) π π 49). log 5). 5). (/5)(e 5 -e -5 ) 5). tn - log( ) 54). 55). CHAPTER 8 ). - ). 6 ). ½ 4)..5 5) /6 6). 8). / 9). ) CHAPTER - 9 ) Sin log(siny) = c dy ).Order = ; Degree =. ). y y d = 4). 5). dy y e = 6) d ( e ) e ( ) 7). Eqution of the fmily of curve is y=c 8).( ) y = 9 9). ). tn - y tn - (e ) = π/. ). tn tn y y ( tn ) e = e y c e ). ye = e d c = c ). ( ) (y ) = c

34 4) y y = sin c 5). 6). 7). 8). logc = 9). ). ). ). sec (y/)=cy ). log = c p p p o X = 8.% o 5). CHAPTER r r r r r ). Vector prllel to XY- plne will e of the form i j.if it is perpendiculr to 4i j k, then r r r r r 4i j k. i j = ( ) ( ) = 4 r 4 r r r the vector is i j = ( i 4 j ) r r ( i 4 j ) The unit vector = r r = ± ( i 4 j ). 5 ( 4 ) r r ). = 5 i.e sin θ = sin θ = 5 sinθ = 5 cosθ = = 6 6 r r. = cosθ = ). 4). r r r r r r r r r GA GB GC = OA OG OB OG OC OG r r r r = OA OB OC OG r r r r r r c r = c =

35 r r 5). λ = λ = λ = ± 6).Let ABC e the given tringle. Let AD, BE, CF e the medins. Required sum of vectors = r r r r r r ( AB BD) ( BC CE) ( CA AF) r r r r r r = ( AB BC CA) ( BD CE AF) r r r r r = ( AC CA) ( BC CA AB) r r r = ( AC CA ) = O 7). Let ABCD e ny qudrilterl. Let P,R e the mid points of the sides AB, CD respectively Let Q,S e the mid points of the digonls AC nd BD respectively. r r r r Let,, c, d e the position vectors of A, B, C,D respectively r r r r r r r r c r r c d r d P. v. of P =, OQ =, OR =, OS =, c r, c r r r r r PQ = SR = r r r r r r r r r r r r OA =, OB = OC = C OD = d 8). r r r d r c r m r AB = m( c d ), OE =, OF =, EF = DC AB PCD, r r r r 9).let ABCD e the tetrhedron. Let,, c, d e the position vectors of the vertices A, B, C, D respectively. LetG, G, G, G 4 e the centroid of the r c r d r, c r d r r, d r r r OG OG OG, OG r r = = = c r 4 = r r r c d r ( ) r r r r c d PV.. of G = = 4 The symmetry of P.V. of G shows tht G lso divide the lines BG, CG, DG 4 in the rtio : internlly λ 4 ). ( λ ) 5 = λ r r r r 4).Let EB = p AB, CE = qcd r r r r Q EB BD CD = p =, q = m n r r Then revbce = EB BC = 6. ) CHAPTER

36 4 ). Cos, Cos, Cos ). where ' ' ethe side, 458 ). 4). y=,z= nd they re perpendiculr to ech other., 5). y4z-8= & λ=, ). k=7, 7).,, 8). D. C.' s re,, 9). COS r ). ( i) ( ii ) ). ±, ±, ± ). Eqution of Y-is = λ ˆj is 9 9 y z vector form & = = = λ is Crtesin form ) ). 7 Units. r r r 5). = iˆ ˆj k µ (iˆ ˆj 6 k ) & 58 7 r, 7).,.( iˆ ˆj kˆ ) = 5 () 4 6 CHAPTER - ) Mimum vlue does not eist. ) Minimum vlue does not eist. ) The ojective function cn e mde s lrge s possile s we plese. So the prolem hs unounded solutions. 4) Minimum cost of cerel is Rs.4 & 6 pise. 5) Minimum trnsporttion cost is Rs., when o,,,, ricks re trnsported from the depot A nd 5,,, 5, ricks re trnsported from the depot B to the uilding P, Q nd R respectively. 6) Constrins re y y 6 y, y Mimum vlue of Z = 8 when =y=5.. 7). Mimize Z=y suject to y ; y 6;4y ;54y,y. 8). Minimum Vlue: 4. CHAPTER 7 6 () 7 () () 5/ (4).5/ (5). 46 (6)/4. (7) 5 4/69 (8) Cos - (4/7) (9)9/54 () 9/9,6/9,4/9 ( )/7 () / () (i) /5 (ii) /5 (iii) 4/5. (4) 4/9 (5) /7. 5

37 (6) Proility distriution is 7). p ( = 5) = c5 = P() 88