TOPIC : Physics : Electrstatics (Diple, Gauss law & its applicatin) Chemistry : Ideal and nn ideal slutin, clligative prperties, van t Hff factr Maths : Inverse f a functin, range f elementary functins (withut using calculus)
PHYSICS Directins (Qs. -) : These uestins cntains Statement- (S) and Statement- (S). Of the fur chices given (), (), and belw, chse the ne that best describes the tw statements. () Bth S and S are crrect and S is crrect eplanatin f S () Bth S and S are crrect but S is nt crrect eplanatin f S S is crrect but S is wrng S is wrng but S is crrect. S: Ecess charge always resides n the uter surface f cnductr. S: Under electrstatic cnditin the ptential at every pint n r inside cnductr is always same.. S: The electric flu thrugh a Gaussian surface nly depends n charge enclsed inside Gaussian surface. S: The charge utside Gaussian surface des nt cntribute in the electric field strength at the surface f Gaussian surface.. S: There is electrstatic pressure n the surface having charge distributin. S: The electrstatic pressure at a pint n the surface is f value utward drawn nrmal. directed always alng 4. A pint charge causes an electric flu f. Nm cul t pass thrugh a spherical Gaussian surface f. cm radius centred n the charge. If the radius f the Gaussian surface were dubled. Hw much flu wuld pass thrugh the surface. () Nm cul () Nm cul Nne f these 5. A cube f side l is placed in an electric field given by E b î, where b is a cnstant. The electric flu thrugh the cube is given by y () bl / ( ) () bl 7/ ( ) bl / ( ) bl 5/ ( ) z O l l E Directins : Questins 6 and 7 are based n the fllwing paragraph. Read the paragraph carefully and answe the uestins There is electric lines emanating frm a given psitive pint charge and terminates int negative pint charge as shwn in the figure. The number f electric lines frm a pint charge is directly prprtinal t the charge. A B 6. What wuld be the number f electric lines due t charge emanating frm making an angle with the line jining f charges and () cs sin () cs Nne f these 7. Cnsider a particular line that emantes frm pint charge making angle with line that jins and in given figure and terminates with by making with the line jining f charges then is given by () sin sin () sin sin sin sin sin sin 8. The side f the suare surface shwn in figure is 5mm. It is immersed in a unifrm electric field with E = N/C. The field lines make an angle f 6º with the utward pinting nrmal as shwn. Then the flu thrugh the surface is in Nm cul 6º E ().5 ().5.4.4
9. Charges,, + are placed at the crners f euilateral triangle f side l, then magnitude f resultant electric diple mment will be () l () l l 4l. The three diple (charge, separatin a) are placed inside a spherical shell f radius R. The electric flu passing thrugh the shell is () Zer () 6 6 R. Tw identical diple are placed at r distance apart as shwn in figure. p and p are diple mment f I and II diple respectively. (a<<r). The ptential energy f II diple in the presence f eternal field f I diple is pp () 4 I II a + + p p r r pp () r pp 4 Nne f these r. Fr a given surface the Gauss s law is stated as E.ds. Frm this we can cnclude that () E is necessarily zer n the surface () E is perpendicular t the surface at every pint The ttal flu thrugh the surface is zer The flu is nly ging ut f the surface. If the flu f the electric field thrugh a clsed surface is zer () the electric field must be zer everywhere n the surface () the electric field may be zer everywhere n the surface the net charge enclsed is nn-zer the charge in the vicinity f the surface must be zer 4. An electric diple is placed alng the -ais at the rigin. A pint P is at a distance f cm frm this rigin such that OP makes an angle /, with the - ais. If the electric field at P makes an angle with the -ais, the value f wuld be : () () tan tan 5. A clsed cylinder f radius R and length L is placed in a unifrm electric field E that is parallel t the cylinder s ais. The ttal flu linked with the surface f the cylinder is given by () R E () R R E Paragraph:(Q 6-8) R E zer - As Gauss law states that the ttal electric flu thrugh a clsed lp is always eual t times the charge enclsed, E ds enclsed + Gaussian surface 6. The change in value f causes () Change in bth electric flu thrugh clsed surface and E () Change in electric flu nly Change in E nly Nne f these 7. The change in value f causes the change in () electric flu nly () E nly bth E and electric flu Nne f these
8. Gauss law is valid () fr any kind f clsed surface () fr symmetrical clsed surface fr assymetrical clsed surface nne f these 9. A charged particle is placed at the centre O f a cube f side L(ABCDEFGH). A secnd charge is placed at O, where OO passes thrugh the centre X, f the face BCFG and OO L. The ttal electric flu thrugh the face BCFG is () () 8 8 6 6 D A E O H C B X F ' O' G. A lng cylindrical vlume cnsists a unifrmly distributed charge f density. What is the electric field at a pint inside cylindrical vlume at a distance frm ais () () zer. The mathematical frm f Gauss s law is E ds In this reference which f fllwing is crrect () E depends n the charge which is enclsed within the Gaussian surface nly () E depends n the charge which is inside and utside the Gaussian surface E des nt depend n the magnitude f charge Nne f these. A diple cnsists f tw particles, ne with charge Q and mass m and the ther with charge Q and mass m, separated by distance L. Fr small scillatins abut its euilibrium psitin, the angular freuency when placed in unifrm electric field E is - () QE ml QL me L m Q cm () m Q E QE ml Qm LE. In a certain regin f space there eists a unifrm electric field f kˆ Vm. A rectangular cil f dimensins cm cm is placed in XY-plane. The electric flu thrugh the cil is : () zer () 4 5 4 4 4. An electric diple is placed at the rigin and is directed alng the X-ais. At a pint P far away frm the diple, the electric field is parallel t the Y-ais. OP makes and angle with X-ais : () tan () tan 45 tan / 5. If an electric diple f mment p is placed in electric field f strength E, then which f the fllwing gives the ptential energy f the diple? () p E () p E p E nne f these 6. If E a be the electric field strength f a shrt diple at a pint n its aial line and E e that n the euatrial line at the same distance, then () E e = E a () E a = E e E a = E e nne f the abve 7. A charge Q is placed at the crner f a cube. The electric flu thrugh all the si faces f the cube is () Q / () Q / 6 Q /8 Q /
8. Flu cming ut frm a unit psitive charge placed in air is () () 4 4 9. When an electric diple p is placed in a unifrm electric field E then at what angle between the value f true will be maimum? () 9 () 8 45. If the electric flu entering and leaving an enclsed surface respectively is and, the electric charge inside the surface will be : () () / /. The electric field in a regin is radially utward with magnitude E = A r. What is the charge enclsed in a sphere f radius a centred at rigin; A = vm ; a = cm () 8.89 C () 6.89 C.69 C nne f these. A lng cylinderical wire carries a psitive charge f linear density. An electrn revlves arund it in a circular path under the influence f attractive electrstatic frce. Then what wuld be kinetic energy f electrn. [e = electrnic charge] () e e () e 4 nne f these. A charge is distributed unifrmally n a ring f radius r. A sphere f eual radius r is cnstracted with its centre at the periphery f ring. What is electric flu thrugh surface f sphere () Q Q () Q Q 4. A metallic shell has a pint change kept inside its cavity. Which ne f the fllwing diagrams crrectly represents the electric lines f frces? () () 5. The inward and utward electric flu frm a clsed surface are respectively 8 and 4 units. Then the net charge inside the clsed surface is : () 4 4 culmb () 4 culmb culmb 4 culmb
CHEMISTRY 6. Van t Hff s factr (i) : () Is less than ne in case f dissciatin () Is mre than ne in case f assciatin nrmal mlecular mass i bserved mlecular mass bserved mlecular mass i nrmal mlecular mass 7. Which slutin will shw maimum elevatin in b.p.? (). M KCl (). M BaCl. M FeCl. M Fe (SO 4 ) 8. Which is nt a clligative prperty? () Lwering f vapur pressure () Freezing pint Osmtic pressure Elevatin f biling pint 9. If liuids A and B frm an ideal slutin: () The entrpy f miing is zer () The free energy f miing is zer The free energy as well as the entrpy f miing are each zer The enthalpy f miing is zer 4. The rati f the value f any clligative prperty fr KCl slutin t that f sugar slutin is : () ().5 4 4. In the case f smsis, slvent mlecules mve frm: () Higher vapur pressure t lwer vapur pressure () Higher cncentratin t lwer cncentratin Lwer vapur pressure t higher vapur pressure Higher smtic pressure t lwer smtic pressure 4..8 g f fructse (C 6 H O 6 ) is added t kg f water. The freezing pint f the slutin is (k f fr H O is.86 K mlality ). ().86 ().86 C.9 C.9 C 4. The freezing pint f mlal NaCl slutin assuming NaCl t be % dissciated in water is : (K f =.86 K mlality ) () +.86 C ().7 C.86 C +.7 C 44. The mlal elevatin cnstant fr water is.5 K mlality. The elevatin caused in the biling pint f water by disslving.5 mle f a nn-vlatile slute in 5 g f water will be : () 5 C () 5. C.5 C.5 C 45. At 4 C the vapur pressures in trr, f methyl alchl-ethyl alchl slutins is represented by the euatin. P = 9 X A + 5; where X A is mle-fractin PA f methyl alchl, then the value f lim X is : () 54 trr () 5 trr 9 trr 4 trr A 46. The values f bserved and calculated mlecular weights f silver nitrate are 9.64 and 7 respectively. The degree f dissciatin f silver nitrate is : () 6% () 8.5% 46.7% 6.% 47. The depressin in f.p. f. m aueus slutin f urea, sdium chlride and sdium sulphate is in the rati : () : : () : : : : 4 : : 48. The amunt f ice that will separate ut n cling a slutin cntaining 5 g f ethylene glycl in g water t 9. C is : (K f =.86 K mlality ) () 8.7 g () 8.7 mg 4 g 4 mg A
49. A. mlal slutin f [Pt(NH ) 4 Cl 4 ] in water had a freezing pint depressin f.54 C. If K f fr water is.8, the crrect frmulatin f the abve mlecule is : () [Pt(NH ) 4 Cl ]Cl () [Pt(NH ) 4 Cl ]Cl [Pt(NH ) 4 Cl]Cl [Pt(NH ) 4 Cl 4 ] 5. The latent heat f vaprisatin f water is 9.7 kcal/ mle and if the biling pints is 7 K, the ebulliscpic cnstant f water will be ().5 K kg mle ().6 K kg mle.6 K kg mle.8 K kg mle 5. Phenl assciates in benzene as C H OH. C6H5OH 6 5 If is the degree f assciatin f phenl, then the ttal number f mles f particles present at euilibrium is () () + 5. A slutin cntaining 4. g f PVC in litre f diane (industrial slvent) was fund t have an smtic pressure 6. 4 atmsphere at 7 C. The mlecular mass f the plymer will be ().6 4 ().6 5.6.6 5. If M nrmal is the nrmal mlecular mass and is the degree f dissciatin f K [Fe(CN) 6 ], then the abnrmal mlecular mass f the cmple in the slutin will be () M nrmal () M nrmal eual t M nrmal 54. On February 4,, the temperature in Manali was fund t be C. A car was used, whse radiatr was filled with 5 L f water. What uantity f antifreezing agent ethylene glycl were added t water f radiatr in rder t use the car fr travelling? (given K f f water =.86 K kg mle, d g ml ) () g () 67 g g g 55. The vapur pressures f tw slutins f slute A and B are 6 mm Hg and mm Hg respectively. If mles f each A and B are present in a slutin A-B. Then, slutin A-B () shws ideal behavir, if the vapur pressure f slutin is mm Hg () shws psitive deviatin frm Rault s law, if the vapur pressure f slutin is mm Hg shws negative deviatin frm Rault s law, if the vapur pressure f the slutin is mm Hg bth () and are crrect 56. The cryscpic cnstant f water is.86 K kg mle. A. mlal acetic acid slutin prduces a depressin f.9 C in the freezing pint. The appr degree f dissciatin f acetic acid is () Zer () 4% 8% 4% 57. Cnsider the fllwing statements :. smtic pressure methd is the mst useful fr determining the mlecular weight f macrmlecules. the mlecular weight f the macr-mlecules des nt change in presence f an electrlyte in the slutin. clligative prperties are helpful t determine average mlecular weight f macr-mlecules Which f the abve statement(s) is/are crrect? () alne () and and alne 58. The mlality f slutin btained by miing eual vlume f % H SO 4 by weight (d =.8 g/ml) and 7% by weight f H SO 4 (.6 g/ml) ().4 () 7.6 5.6 5.7 59. Calculate the eperimental mlal elevatin cnstant f water frm the fact that a slutin cntaining.45 g f urea in.5 g f water gives a biling pint elevatin f.7 C. ().5 ().5.8.8
6. A 5% slutin f cane sugar is istnic with.877% f X. The mlecular weight f substance X is () 59.98 () 9.96 95.58 6.98 6. If pk a = 4 at a particular temperature then van t Hff factr, i fr. M mnbasic acid is, (). ()... 6. A 5.8% slutin f NaCl has vapur pressure clsest t () 5.8% slutin f urea () m slutin f glucse m slutin f urea 5.8% slutin f glucse 6. Depressin f f.pt. f. m a. CH COOH slutin is.46 K. m a. urea slutin freezes at.86 C. Assuming mlarity eual t mlality, ph f CH COOH slutin is () (). 4. 64. Accrding t Rault s law, the relative lwering f vapur pressure f slutin is eual t the () Mle fractin f slute () Mle f slute Mle fractin f slvent Mles f slvent Directins (Qs. 65-67) : These uestins are based n the fllwing paragraph. Lwering f vapur pressure is determined by Ostwald and Walner dynamic methd. It is based n the principle, when air is allwed t pass thrugh a slvent r slutin, it takes up slvent vapur with it t get itself saturated with vapurs at that temperature B P A A P A dry air slutin slvent Anhy. CaCl air A and B are weighed separately befre and after passing dry air. Lss in weight f each set, gives the lwering f vapur pressure. The temperature f air, the slutin and the slvent is kept cnstant. 65. Lss in weight f slvent (W ) will be prprtinal t 66. () P A P A PA PA PA PA PA () () P P W W W W W W is eual t A A P P () (W = lss in wt. f slutin bulb) A A W W W W W W 67. Dry air was passed thrugh 5 g f slute in 8 g f water and then thrugh pure water. The lss in weight f slutin was.5 g and that f pure water was.4 g. The mlecular weight f slute is () 7.4 () 7.4 74. 74. Directins (Qs. 68-7) : These uestins cntains Statement- (S) and Statement- (S). Of the fur chices given (), (), and belw, chse the ne that best describes the tw statements. () Bth S and S are crrect and S is crrect eplanatin f S () Bth S and S are crrect but S is nt crrect eplanatin f S S is crrect but S is wrng S is wrng but S is crrect 68. S: Amng the euimlecular slutins f urea, KCl and K SO 4, the slutin f urea shws lwest smtic pressure. S: Urea is nn electrlyte while KCl and K SO 4 are electrlytes. 69. S: Dilute slutins are taken as ideal slutins. S: The slutins which fllws the Rault s law at any temperature and cncentratin are knwn as ideal slutins. Mst f the slutins shw the deviatin frm Rault s law. 7. S: The mlecular weight f benzic acid when disslved in benzene is fund t be higher than the theretically calculated mlecular weight frm frmula. S: Many rganic slutes when disslved in nnaueus slvents underg dissciatin.
MATHEMATICS 7. Let f : (, ] (, ] be a functin defined by f() = 4, Then f () is () 4 () 4 4 nne f these 7. Let f :[4, ) [4, ) be a functin defined by 4 f 5, then f () is () 4 lg5 () 4 lg5 4 5 nne f these 7. Let f : R R be given by f() = ( + ),. Then f () () () des nt eist because f is nt ne-ne des nt eist because f is nt nt 74. Let f : R R be given by f() = ( + ),. Then, the set f values f fr which f() = f () is given by () {} () {, } { } nne f these 75. The value f the parameter, fr which the functin f() = +, is the inverse f itself, is () () 76. The inverse f the functin f () lga lga a a is a a () lga nne f these 77. If the functin f :[, ) [, ) is defined by () f() = ( ), then f () is () 4lg 4lg nt defined 78. Let f : R R, g : R R be tw functins given by f() =, g() = + 5. Then (fg) () is eual t () 7 / / () 7 / / 7 7 79. Let f : (4, 6) (6, 8) be a functin defined by f (where [.] dentes the greatest integer functin), then f () is eual t () () 8. Suppse f() = (+) fr. If h() is the functin whse graph is the reflectin f the graph f f() with respect t the line y = and g() is the functin whse graph is btained by shifting units t the left the graph f h(), then g() euals (), (), ( 4),, 8. Let f() = +, /, then the slutin f the euatin f () = f() is () = () = = / Nne f these 8. If f : R R is defined by f() = +, then the value f f (7) and f ( ) respectively are (),{ 4,4} () {,},,{,} { 4,4},
8. Range f f ( ) (),, f = is - cs = is + 84. Range f ( ) (),, f 85. Range f ( ) () () 5 = is - (), () 5,, 86. Range f f ( ) = 4 - + 4 () (, ) (),,,, 5, nne f these is, [, ], 87. Range f f() = a cs + b sin is () (a, b) () [a, b] [a + b, a b] a b, a b - + = is - () R [, ] () R (, ) R {} R {} 88. Range f f ( ) 89. Range f f ( ) + = is + () (, ) () [, ] {, } R 9. Range f f ( ) (), - e - e = e + e () {} 9. Range f f ( ) is, + - cs = is - - sin () {} () {} (, ) [, ] 9. Range f ( ) is e = where [.] is the G.I., and + f, [ ] () [, ) (), (, ], 9. Range f f ( ) (), (),,, 4 = sin + cs is 94. If f() = lg e ( 4 + 5) then () Df R, R f R () Df R, Rf lg, Df,, Rf R D f,, Rf R 95. Range f f ( ) = - + 5 - is () ( ),, (), (, )
96. Range f (),, 97. Range f f the G.I., is f sin 9 (),, sin, {.} is the F.P. & [.] is (), (), R set f natural number 98. Range f f() = cs[], where [.] is the G.I., is, 4 4 () {, cs } () {} {cs } nne f these 99. Dmain & range f f ( ) Df,, Rf, () () Df R, Rf, D f,, Rf, nne f these [ ] [ ] - = is ([.] is G.I.) + -. Range f f ( ) + 4 C + = is () {, } () {,, } {, } {,, } Directins (Qs. -5) : These uestins cntains Statement- (S) and Statement- (S). Of the fur chices given (), (), and belw, chse the ne that best describes the tw statements. () Bth S and S are crrect and S is crrect eplanatin f S () Bth S and S are crrect but S is nt crrect eplanatin f S S is crrect but S is wrng S is wrng but S is crrect. S: If f() =, R has inverse defined n their ranges. S: f() is ne-ne nt. S: y = f() has inverse S: y = f() is strictly mntne & cntineus in the dmain sin cs. S: Range f lg is (, ) S: lg a is defined fr >, a > but a 4. S: f() = sin + cs, g() =, g(f()) is invertible then, 4 4 S: f() = sin + cs has its inverse 5. S: Range f f() = + + + sin is S: Cubic plynmial has range R b G b G b G b
. ().. () 4. () 5. 6. () 7. () 8. () 9.. (). ().. () 4. () 5. 6. () 7. () 8. () 9.. (). (). (). 4. () 5. () 6. () 7. 8. () 9. (). (). (). (). 4. 5. () TEST NO. : 4 ANSWERS 6. 7. 8. () 9. 4. 4. () 4. 4. () 44. 45. () 46. () 47. () 48. () 49. () 5. () 5. 5. () 5. () 54. () 55. () 56. () 57. () 58. () 59. () 6. () 6. 6. () 6. () 64. () 65. () 66. () 67. () 68. () 69. () 7. 7. () 7. () 7. () 74. () 75. () 76. () 77. () 78. () 79. 8. 8. () 8. 8. 84. 85. 86. () 87. 88. () 89. 9. () 9. () 9. () 9. () 94. () 95. 96. () 97. 98. () 99. ().. (). (). 4. 5.
TEST NO. : 4 HINTS & SOLUTIONS PHYSICS. Charge in the cnductr cmes utside the surface s as ptential remains cnstant thrughut the cnductr.. Charge utside the Gaussian surface will nt cntribute in flu while E at surface is due t all the charges inside as well as utside.. Charge n the surface causes electrstatic pressure f value 4. Since always alng utward nrmal. E ds radius f sphere. enclsed 5. As b l l b l l 5/ 5/ b l bl 5/ b l this is independent f 6. Number f electric lines frm charge at is the semivertical angle f cne cs 7. Since cs cs sin sin 9.. = ( 5 ). Nm cul p. Since R P p p p cs 6 6 p p = l p p l. enclsed p U p E p E p r pp r. E ds. E ds E 4. at every part r angle be- tween ds & E is. p r E P N net flu. sin sin. 8. E s = E s cs 6 6 E s +...() tan tan tan tan
5. 9.. E ds enclsed flu thrugh cube Flu thrugh BCFG = 6 l Gaussian cylinder () and () tan 5. U P E P 6. Ea 4 r p Ec 4 r Ea = E e. Q 7. 8 By taking cube fr symmetry. 8. E ds if = unit charge. 9. P E = 9 PEsin l E l E. Principle f superpsitin.. I E Lsin fr t be small d EL dt I EL EL 4 EL I ml ml ml 4 4 E ml. E ds E S s = 4 m E = vm. 4. E 9 Since tan tan...() 9...().. E ds cs Aa 4a Q inside = 4 Aa mv. e e mv r r 4. Charge enclsed in sphere = Q/ hence flu crsses thrugh sphere = 4. Electric lines meet cnductr at 9 and n net lines eist in cnductr. 5. 6. CHEMISTRY ep. clligative prperties i ; nrmal clligative prperties put clligative prperties ml. wt. 7. Fe (SO 4 ) furnishes mre n. f ins. 8. Elevatin in b.p., smtic pressure, depressin in f.p. and depressin in vapur pressure are clligative prperties i.e., the prperties which depend n number f particles present in slutin. Q
9. Fr ideal slutin Vmiing, Hmiing ve 4. van t Hff s factr i Ep. C.P. Cal. C.P. y ; Fr KCl it is eual t and fr sugar it is eual t. 4. Osmsis is eplained in terms f vapur pressure thery, i.e., mvement f slvent particles frm higher vapur pressure t lwer vapur pressure. Nte that a slutin f high smtic pressure is cncentrated in cmparisn t ther having lw smtic pressure. 4. Tf.86.5 =.9; T f =.9 =.9 C..5.5 44. Tf.5 C 5 45. P = 9 X A = 5; 46. i fr AgNO P lim X A A A = 9 + 5 = 54 trr. nrmal ml. wt., bserved ml. wt. 7 =.85 = 8.5%. 9.64 47. mle urea gives mle mle NaCl gives mle mle Na SO 4 gives mle T rati : : 48. ' f K w T W.m.86 5 9. 6 W W = 6.9 Ice separated = 6.9 = 7.7 g. 49. [Pt(NH ) 4 Cl 4 ] Gives n mles f ins n cmplete inisatin, i.e., T = K f mlality n.54 =.8. (n) n = Thus, [Pt(NH ) 4 Cl ]Cl [Pt(NH ) 4 Cl ]+ + Cl n =. 5. Ebulliscpic cnstant K r 8 7 K b.56 97.5 K kg mle MBRT b Hv 5. C H OH C H OH 5. 6 5 6 5 At t = after assciatin ( ) Ttal number f mles f particles = + WB R T 4.8 5 MB.64 V 4 6 5.6 6 6 5. K FeCN K FeCN At t = After dissciatin Ttal number f particles after inisatin i Since, M M Number f particles after dissciatin Number f particles befre dissciatin nrmal abnrmal M nrmal r M abnrmal = 54. T use car f f f T T T K Tf Kf m M nrmal Mabnrmal
=.86 m m 5.8 fr 5 L f water number f mles f ethylene glycl added t water f radiatr = 5.8 5 = 6.88 = 6.88 6 g (weight in grams = number f mles mlecular mass) = 666.67 67 g. 55. Given P A = 6 mm Hg, P B = mm Hg n A = n B = XA XB Accrding t Rault s law Psl PA XA PB XB P 6 sl = 8 + = 8 mm Hg The vapur pressure f slutin ( mm Hg) is mre than that f calculated frm Rault s law (8 mm Hg). Hence, the slutin shws psitive deviatin frm Rault s law. 56. Given K f =.86 K kg mle, m =., Tf i Kf m Tf =.9 C.9.86. r i =.76.4 Hence, degree f dissciatin =.4 = 4%. 57. The mlecular weight f the macr mlecules gets changed in the presence f an electrlyte in the slutin because clligative prperties depend nly n number f particles. 7. y = 4-4 y f 4 MATHEMATICS But fr,, - f = - 4 - ( ) 7. ( - 4 y = 5 ) lg5 y 4 4 lg5 y 4 lg5 y But fr 4, 4, - f ( = + 4 + lg ) 5 7. y = ( + ) - y f But - 74. f ( ) = f ( ) f 4 lg5 ( ) - + + = ( + ) -, 75. y 76. y f r a - a - a - y = ; a + a - a + By C & D + = lg y a - y 77. same as (7) 78. ( )( ) ( ) - f = - + + f lga ( ) ( ) ( ) fg = f g = f + 5 = + 5 - y = + 7 / y 7 / 7 f
79. f 4, 6 y = + = y f () = 8. Abut the line y =, reflectin means inverse. y = ( + ) = y - - f = - ( ) h( ) = - g( ) = + - ( shifted by units) 8. y = + ± 4y - = f - ( ) = ± 4 - - f f ( ) = ( ) y - y = 5 y 5 y y 86. 87. - a + b a cs + bsin a + b 88. - + y = - y - y = - + - y + + + y = D 89. ( ) ( ) y 4. y f ± 4 - - + = which is satisfied by = f 7 K 7 f K K K 4 8. f f 8. cs 84. y = + y + y = y - + y D 4y.y y 5 85. y = - 9. 9. e e y e e e = - (use =, > ) y y > i cs isin ( ( )) Im f = cs sin tan Put 4 in R 4 e (f())
9. e y = + [ ] = + y ( min) 9. A.M. G.M. & csec 94. f ( ) lg( 4 5) 4 5 and ceff. f is psitive minimum value = / R f lg e /, 95. D f [, 5) 96. ( ) = ( ) = ( ) f f min = = 5 f f ma f 9 sin D ve 97. sin sin 98. 4 sin is N [ ] = - f ( ) 4 f 4 f = cs 99. R, f y y X y y y y ( = r. same).in n C, n, r & n, r, Î +ve integer r.one-ne and nt functin is bijective. Strictly mntnic and ne-ne with nt always be bijective. y sin 4 sin 4 y,4 { } 4. g f ( ) 5. = sin 4 4 f ( ) = sin + cs = = cnstant functin s it has n inverse. f sin Plynmial R Range f f() is R plynmial functin always be real fr all real, if pwer f variables be +ve integer. b G b G b G b