CET MATHEMATICS 2013

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Bryan Summers
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1 CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The mimum re of rectngle tht cn be inscribed in circle of rdius units is () 8 sq. units () sq. units () sq. units () 8 sq. units r ; mimum rectngle is squre with ech side r therefore re 8. If the length of the sub-tngent t n point to the curve n is proportionl to the bsciss, then n is () n non-zero rel number () () - () Ans: () Differentiting n we get n non-zero rel number. ST - n since it is proportionl to n cn be n n cos. n+ sin n cot () n Ans: () d, n is () n cot n Given integrl cn be epressed s ( )e. ( + ) () Ans: () e + ( + )e ( + ) d d () e ( + ) cot sin n () d e d ( + ) ( + ) () cot n n cot n n e ( + ) e ( + ) () () n cot n. e ( + )

2 / / 6. If I. sin d nd I. cos d, then which one of the following is true? () I I () I + I () I.I () I + I / I. sin d -cos + sin / I cos d sin - I + I sin / - 7. The vlue of d is () () () () Ans: () d d + d + d 8. cos cos + sin d () () () 8 () Ans: () cos cos + sin cos d cos + sin d 9. The re bounded b the curve sin, -is nd lines nd is () 9 () () 6 () Ans: () Put t given integrl sint dt () 6. The generl solution of the differentil eqution () sin () + c () sin () + c () sin ( + c) () sin () + c.d. d +. d is

3 Ans: () Put z Diff. eqution is sin - z + c z sin ( + c) sin ( + c) z d dz dz z d integrl. If m nd n re the order nd degree of the differentil eqution ( ) +. ( ) + sin, then () m, n () m, n () m, n () m, n Multipl b Order degree. If ( + ) () + A B + C +, then sin - A + tn - B + sec - C + () 6 Multipl b + ( + ) A ( + ) + (B + C) Compre coefficient A B C sin - + tn - + sec - 6 () () 6. The sum of the series, to n terms is... n+ () + n + n + () n + Ans: () Checking with options Putting n S + stisfies onl n + + () n + n + () n +. If the roots of the eqution + + b + c re in A.P., then 9b () 9c () 8c () 7c () -7c + + b + c Let α - β γ nd ( + ) ( ) ( - ) + - b - nd c Substitute in options 9b -7c stisfies

4 . If the vlue of C o +. C +. C + + (n + ). C n 76, then n is () 7 () () 6 () 9 Ans: () C o + ( + d) C + ( + d) C +. + ( + nd) C n ( + nd) n d ( + n) n 76 n 7 6. The inverse of the proposition (p ~ q) r is () (~r) (~p) q () (~p) q (~r) () r p (~q) () (~p) (~q) r Ans: () Inverse is, ~ [p ~ q] ~ r (~p) q ~ r 7. The rnge of the function f () sin [], - < < where [] denotes the gretest integer, is () {} () {, -} () {, ± sin } () {, -sin } Clerl sin.,, sin [] sin,, [] - sin [] sin (-) -sin 8. If the line λ ( + ) is prllel to is, then λ () -7 () - () 7 () Ans: () [6 + λ] m - λ -7 7 λ 9. The ngle between the lines sin α.. cos α + (cos α - ) is () 9 o () α () α Ans: () Clerl + b sin α + (cos α - ) θ 9 o () α. The minimum re of the tringle formed b the vrible line cos θ. + sin θ. nd the co-ordinte es is () () GE Are cos. θ cos + sin θ. θ sin θ sin θ () 9 () When Are is minimum, sin θ is mimum A min

5 . log (sin o ). log (sin o ). log (sin o ) log (sin 79 o ) () is positive () is negtive () lies between nd 8 () is zero log sin. log sin.log sin 9..log sin79 log sin.log sin.log..log sin 79. If sin sin nd cos cos, then tn ( + ) () 8 () - 8 () () - Ans: () + cos sin + sin sin tn ( + ) tn + tn + tn + - ( ) ( ). In tringle ABC, if Ans: (). cos A cos B b cos C c () () () sin A We know cos A Given () () cos B b sin B b sin C c cos C c ; tn A tn B tn C ΔABC is equilterl Are lim e lim log ( + ). nd, then its re is.... ().. () () log e () () log e () Ans: () + n n log + log o e log (), if is irrtionl. Let f () then f is, if is rtionl () continuous everwhere () discontinuous everwhere () continuous onl t () continuous t ll rtionl numbers

6 Ans: () lim f () lim lim + f () lim f () f () is continuous t + Note: tht between ever two rtionls there eists one irrtionl number nd vicevers. 6. In regulr grph of vertices the sum of the degree of the vertices is 6. Then the degree of ech verte is.. () () () () Ans: () Let degree of ech verte be k Thus k 6 k 7. The reminder when,. ( + ) ( + ) is divided b 6 is () () () () 6 Ans: ().( + ) ( + ) is product of consecutive integers nd hence is divisible b! 6. Reminder 8. A vlue of stisfing (Mod ) is () () () () Ans: () (mod ) 7 (mod ) 7 clerl stisfies this reltion 9. The smllest positive divisor greter thn of composite number is.. () < () () > () Stndrd result (Propert). If A nd B re squre mtrices of order n such tht A B (A B) (A + B), then which of the following will be true? () Either of A or B is zero mtri () A B () AB BA () Either of A or B is n identit mtri Ans: () A B A BA + AB B -BA + AB AB BA Note: Even though (), () nd () stisf the given eqution none of those is necessr condition for A B (A B) (A + B) α. If A nd A, then α. α () ± () ± () ± () ± Ans: () A A A α α 9 α ± 6

7 . If A nd B da, then d () B + () B () -B () B Ans: () A ( ) ( ) + ( ) + + A + da (B ) B d. If the determinnt of the djoint of (rel) mtri of order is, then the determinnt of the inverse of the mtri is (). () ± () 6 dj A we hve dj A A n A A ± A ± A ±. () ±.. If the mtri A + B, where A is smmetric nd B is skew smmetric, then B. () () () () B (A A ). In group (G, ), for some element of G, if e, where e is the identit element, then () () e () Ans: () Direct since group is betin - () e 6. In the group (Z, ), if b + b n, b Z, where n is fied integer, then the inverse of (-n) is.. () n () n () -n () n * e + e n e n (identit) To find inverse : α * (-n) n α - n n n α n 7

8 7. If (,, ), b (, -, ), c (,, ) nd ( b c ) α + β b γ c, then () α, β, γ () α, β, γ - () α + β + γ 8 () α β γ Ans: (Question is wrong) Question would hve been (b c ) α + β b + γ c (. c) b (. b) c α + β b+ γ c α β. c + + γ (.b) ( + ) b b b 8. If nd ( + ) ( + m ), then m Ans: () () - () (). b (+ b).(+ mb). + m(. b) + (b. ) + m(b. b) m b m - b b () 9. If, b, c re unit vectors such tht + b + c, then. b + b. c + c. () Ans: () + b + c b c GE - () GE () (). If is vector perpendiculr to both nd, then (). ( b c ) () ( b c ) () ( b + c ) () + ( b + c ) Ans: () Tke i, b j, c k i (j k) i i Other options will be not correct b c 8

9 . A tngent is drwn to the circle + + t point A nd it meets the line + t B (, ), then AB. Ans: () ) ) ) ) A B(, ) + AB length of Tngent to the circle from B. AB units.. The re of the circle hving its centre t (, ) nd touching the line + is. ) 6 sq. units ) sq. units ) sq. units ) sq. units Ans: () C (, ) () + () r A r 6 units. The number of rel circles cutting orthogonll the circle is ) ) ) ) infinitel mn Ans: () r + 7, imginr Given circle is n imginr circle. Number of rel circles cutting orthogonll given imginr circle is zero.. The length of the chord of the circle intercepted b the -is is ) ) 8 ) 9 ) Intercept mde b -is f C () A (cos θ, sin θ), B (sin θ, - cos θ) re two points. The locus of the centroid of ΔOAB, where O is the origin is ) + ) ) + 9 ) + Ans: () Tke θ A,, B,, O (, ) Centroid onl eqution holds v v, v v v, 9

10 6. The sum of the squres of the eccentricities of the conics + nd - is. Ans: () ) ) + e e ) 7 ) e The eqution of the tngent to the prbol inclined t n ngle of to the +ve direction of -is is.. ) + ) + ) ) + m + c be tngent θ m. + c Condition is c m If the distnce between the foci nd the distnce between the directrices of the hperbol Ans: () - b re in the rtio :, then : b is. ) + ) : ) : ) : Given e / e e + b b b 9. If the re of the uillr circle of the ellipse + ellipse, then the eccentricit of the ellipse is.. ) ) Are of uilir circle + is re of ellipse b Given, b b e b ) b ( > b) is twice the re of the )

11 . cos cos + sin ) Ans: () cos cos + cos ) + sin 6 ) - cos cos + cos + cos sin cos sin sin. The vlue of tn - - tn -,, > is + ) ) - ) ) 6 6 ) - Ans: () Tke, LHS tn - tn - (). The generl solution of sin cos, for n integer n is.. ) n + ) n ) (n + ) ) n Ans: () sin cos Method of Inspection For n () holds (), doesn t hold () doesn t hold () doesn t hold. The modulus nd mplitude of ) nd ) nd 6 + i ( i) re.. ) nd ) nd Ans: () + i ( i) + i i + i + + Modulus mplitude i + i. i tn

12 . If - + I, then the vlue of ( + + ) 6 ( + ) 6 ) ) 6 ) 6 ) + i + i ω LHS ( - ω + ω) 6 ( - ω + ω ) 6 (-ω ) 6 (-ω) If + tn - nd ) d d ) Ans: () + tn - + tn - d d + d d d f (), then f (). d - ) ) d d ( ) d d d d when < < 6. f () when Then which of the following is true? ) f () is not differentible t. ) f () is discontinuous t. ) f () is continuous for ll <. ) f () is differentible for ll. Ans: () f ( - ) - nd f ( + ) f ( - ) f ( + ) 7. Let f () cos - ( cos sin ). Then f (.) ). ) ) ) Ans: () f () cos - cos sin cos - [cos α. cos sin α. sin ] cos - [ cos ( + α)] + α f () f (.)

13 8. If f () is function such tht f () + f () nd g () [f ()] + [f ()] nd g () 8, then g (8).. ) ) ) ) 8 g () [f ()] + [f ()] g () f () f () + f () f () f () [f ()+ f ()] f () [] g () constnt Given tht g () 8 g () 8 9. If f () f () f () + f () + nd f (), then f (). Ans: () ) e ) e ) e ) e If f () e then f () e. + e. + e. f () e e nd f () e o + 6. The function f () + decreses in the intervl Ans: () ) (-, ) ) (-, ) ) (, ) ) (-9, 9) f () f () + < < < 9 (-, )

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).