(HELD ON 22nd MAY SUNDAY 2016) MATHEMATICS CODE - 2 [PAPER -2]
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![MATHEMATICS CODE - [PAPER -] b th img of th poit (,,7) with rspct to th pl - y + z. Th th qutio of th y z pl pssig through P d cotiig th stright li is 7.](/docs-images/93/113007992/images/1-4.jpg "(c) () + y - z (b) + z (c) - 4 y + 7z (d) - y Img of (,,7) is - y - z - 7 P : - lt th pl b ( + ) + b( y - 5) + c( z - ) -( - + 7 - ) + + (,,) lis o it Þ - + 5b + c...(i) y z prpdiculr to li Þ + b + c.")
1 QUESTION PAPER WITH SOLUTION OF JEE ADVANCED Lt P (HELD ON d MAY SUNDAY 6) FEEL THE POWER OF OUR KNOWLEDGE & EXPERIENCE Our Top clss IITi fculty tm promiss to giv you uthtic swr ky which will b fstst i th whol coutry. MATHEMATICS CODE - [PAPER -] b th img of th poit (,,7) with rspct to th pl - y + z. Th th qutio of th y z pl pssig through P d cotiig th stright li is 7.(c) () + y - z (b) + z (c) - 4 y + 7z (d) - y Img of (,,7) is - y - z - 7 P : - lt th pl b ( + ) + b( y - 5) + c( z - ) -( ) + + (,,) lis o it Þ - + 5b + c...(i) y z prpdiculr to li Þ + b + c...(ii) By (i) d (ii) - 4 y + 7z b c th qutio of th pl is Þ P (-,5,) 8. Lt é ù P 4 ë 6 4 d I b th idtity mtri of ordr. If Q [ q ij ] is mtri such tht 5 P - Q I, th 8.(b) q + q q quls () 5 (b) (c) (d) 5 P é ù é ù é ù ë 6 4 ë 6 4 ë 6 ( + ) 4 é ù é ù é ù é ù P P 4 4, ë 6 ( + ) 4 ë 6 4 ë 6 (+ + ) 4 ë 6 ( ) 4 4 H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg follow us o : momtumcdmy momtumcdmy
2 P 5 é ù é ù é ù q ë 6 ( ) (5 + ) ë q q q + q 6 5(5 + ) / + + q ë 9. Lt b i > for i,,...,. Suppos log b,log b,...,log b r i Arithmtic Progrssio (A.P.).) with th commo diffrc log If 5,,..., r i A.P. such tht b. Suppos t b + b b d s , th () s > t d 9.(b) > b (b) s t b logb - logb log Þ b > d < b (c) s t < d > b (d) s t d 5 b5. < b < d \ b, b, b,... b, b, b, b,..., b r i G.P.,..., 5,..., AP r i li. b,..., b,..., b GP r i potil curv. 5 by grph > b, > b,..., 6 > b6,..., 5 > b5 but b > > Þ s > t b b b... b... b b th Th vlu of p p - cos d + ò is qul to () p - (b) 4 p + (c) 4 p - p (d) p + p 4. () Lt I ò p / -p / cos d +...(i) Þ I / ò p -p / æ p p ö ç è ø æ p p ö æ p p ö ç cos ç è ø è ø d + Þ I ò p / cos d + -p / - / ò p -p / + ( cos ) d...(ii) H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg follow us o : momtumcdmy momtumcdmy
3 (i)+(ii) p / æ+ ö Þ I ò ç cos d -p / è+ ø Þ p / ò cos I d / I p p Þ (si ) - (- cos ) + ( - si ) Ar of th rgio y y y, Î : ³ +, is qul to () 6 (b) 4 (c) (d) 5 4.(c) As figur Ar -æ + 9 ö ò d -4 ç 5 è ø æ + 9 ö + ò - + d -ç è 5 ø æ ö ç + 9 / / - 9 æ 8 ö ç - - (- -) (8) 5-4 ç è 5 ø ç è ø ( + 8) 5 - (9) 4. Th vlu of æ p p ö æ p p ö siç + si ç + è 4 6 ø è 4 6 ø å is qul to ( k ) k k - () - (b) - (c) - (d) + 4.(c) ææ p kp ö æ p ( k -) p öö si ç ç + - ç è ø è ø è ø. æ p ( k -) p ö æ p kp å ö æ p ( k -) p ö æ p kp ö æ p ö si ç + si ç + si ç + si ç + si ç è 4 6 ø è 4 6 ø è 4 6 ø è 4 6 ø è 6 ø å k k H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg follow us o : momtumcdmy momtumcdmy
4 æ æ p ( k -) p ö æ p kp öö åç cotç + - cot + k 4 6 ç 4 6 è è ø è øø æ p æp p ö æp p ö æp p ö æp p ö æp p ö æp p öö ç cot - cot ç + + cot ç + - cot ç + + cot ç cot ç + - cot ç è è ø è ø è ø è ø è ø è øø æ p æ p p öö ç cot - cot ç è è øø u i + u j + u k b uit vctor i d wˆ ( iˆ + ˆj + kˆ ). Giv tht thr ists vctor 6 v r i such tht u ˆ v ˆ d wˆ.( uˆ v r ). Which of th followig sttmt(s) is (r) corrct? () Thr is ctly o choic for such v r (b) Thr r ifiitly my choics for such v r 4. Lt ˆ ˆ ˆ (c) If lis i th y - pl th u u (d) If lis i th z - pl th u u r r 4.(bc) Lt ui + u j + uk.giv uˆ v Þ v siq whr q is gl btw uˆ & v r. r wˆ.( uˆ vˆ ) wˆ. uˆ. v.siq cos Þ whr is gl btw uˆ v r & wˆ. th v r is vctor such tht uˆ v r & wˆ r prrlll. Thr r ifiitly my choics for such v r. r Now uˆ v lwˆ whr l >. iˆ ˆj kˆ l ˆ ˆ ˆ ˆ ˆ ˆ l u u u ˆ i + j + k Þ u ˆ ˆ v - uv i + uv - uv j + uv - uv k i + j + k 6 6 v v v u Þ u v l l & - uv Þ u u 6 6 (C) u th u u (D) 44. Lt æ æ ö æ ö ö ç ( + ) ç +... ç + f ( ) limç è ø è ø ç æ ö æ ö ç!( + ) ç ç è è ø è ø ø for ll >. Th () æ ö f ç ³ è ø f () (b) æ ö æ ö f ç f ç è ø è ø (c) f '() (d) f '() f '() ³ f () f () H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 4 follow us o : momtumcdmy momtumcdmy
5 44.(bc) æ ö ç æ öæ ö æ ö æ æ öæ ö æ ö ö ç ç + ç + ç ç + ç + ç + f ( ) limç è øè ø è ø limç è øè ø è ø ç æ öæ ö æ ö ç æ. öæ. ö æ. ö ç! ç ç ç ç ç ç. ç + è è øè ø è ø ø è øè ø è è ø ø th l ì og f og og l î è ø l æ öü ç ý è øþ æ ö r ( ) lim íå ç + - å + r r r r ì æ æ r ö ö æ æ r ö öü lim íå og ogr og og r r ç l ç + -l - ç ç + - ý å l l î è è ø ø r è è ø øþ ò ( ) ( ) ( ) ò l og f l og y dy l og y dy.lt y t th æ + t ö f '( ) æ + ö log f ( ) ò log( + t) dt - og( t ) dt og dt f ( ), og ò l + ò l ç Þ > l ç è + t ø f ( ) è+ ø + Now < Þ ³ th + f '( ) f ( ) ³ + d ³ Þ + th f ( ) is icrsig i (,) & dcrsig i [, ) (s f ( ) is positiv) f '( ) f ( ) hc æ ö f ç è ø f () d æ ö æ ö f ç f ç è ø è ø f '() æ ö log ç Þ f '() Q f ( ) > f () è 5 ø f '() æ 4 ö æ ö æ ö f '() f '() log ç log ç < log ç Þ < f () è ø è 5 ø è 5 ø f () f () 45. Lt, l, m Î. Cosidr th systm of lir qutios + y l - y m Which of th followig sttmt(s) is (r) corrct () If - th th systm hs ifiitly my solutios for ll vlus of l d m (b) ¹ -, th th systm hs uiqu solutio for ll vlus of l d m (c) If l + m, th th systm hs ifiitly my solutios for - (d) If l + m ¹, th th systm hs o solutio for - H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 5 follow us o : momtumcdmy momtumcdmy
6 45.(bcd) + y l - y m (A) If - th - + y l, y m (B) If ¹ - th - will hv ifiit my solutios if l - m othrwis ot ¹ i.. o - prlll lis i.. uiqu solutio for ll - l & m (C) If l - m th for - qutios - + y l, y - + -m will hv ifiit my solutios (D) If l ¹ -m th for - qutios - + y l, - + y -m will hv o solutio. 46. Lt P b th poit o th prbol circl Th y y y 4 which is t th shortst distc from th ctr S of th Lt Q b th poit o th circl dividig th li sgmt SP itrlly. () SP 5 (b) SQ : QP 5 + : (c) th - itrcpt of th orml to th prbol t P is 6 46.(cd) (d) th slop of th tgt to th circl t Q is S : + y - 4-6y + 64 i..ctr S (,8),rdius. Lt P b ( t, ) must pss through S th t th orml t P is.so orml t P is y + y + t t + t 8 + t t + t Þ t Þ P (4,4) i.. th - itrcpt of th orml to th prbol t P is 6. SP (4 - ) + (8-4) Lt Slop of SP is tq - th th slop of th tgt to th circl t Q is. Now QP SP - SQ 5 - th é ù f : -, R ë d SQ QP é ù g : -, R ë ¹ 5-4 b fuctios dfid by f ( ) [ ] - d g( ) f ( ) f ( ), whr [ y ] dots th grtst itgr lss th or qul to y for y Î R. Th () f is discotiuous ctly t thr poits i é ù -, ë (b) f is discotiuous ctly t four poits i é ù -, ë H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 6 follow us o : momtumcdmy momtumcdmy
7 æ ö ç -, (c) g is NOT diffrtibl ctly t four poits i è ø æ ö (d) g is NOT diffrtibl ctly t fiv poits i ç -, è ø ì -, - < -, < 47. (bc) f ( ) [ - ] [ ]- í-, <, <, î is discotiuous t,,, i..ctly 4 poits i é ù -, ë ì ( - - (4-7))( - ), - < ì 5 -,- < ( - (4-7))( - ), < ( (4 7)( )), 9 -, < < g( ) f ( ) í( - (4-7))( - ), < í 6-4, < ( + (4-7))(), < 7 / 4-7, < ( + (4-7))(),7 / 4 <, < ( + (4-7))(), î5-7, î th g is cotiuous t but f - 5, f + 9 i.. ot diffrtibl t d discotiuous t,,.so g is ot diffrtibl t 4 poit i æ ö ç -, è ø. 48. Lt f : R (, ) d g : R R b twic diffrtibl fuctios such tht f '' d g '' r cotiuous f ( ) g( ) fuctios o R. Suppos f '() g(), f ''() ¹ d g '() ¹. If lim, th f '( ) g '( ) () f hs locl miimum t (b) f hs locl mimum t (c) f ''() > f () (d) f ( ) - f ''( ) for t lst o Î R H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 7 follow us o : momtumcdmy momtumcdmy
8 f ( ) g( ) 48. (d) As f '(), g() th lim f '( ) g '( ) æ ö ç è ø form f '( ) g( ) + g '( ) f ( ) f '() g() + g '() f () g '() f () f ( ) g( ) Þ lim sic lim f ''( ) g '( ) + g ''( ) f '( ) f ''() g '() + g ''() f '() f ''() g '() f '( ) g '( ) Þ f ''() f () sic g '() ¹...(i) Þ f ''() > sic rg of f ( ) Î (, ) As f '() & f ''() > th f ( ) hs poit of miim t By (i), f () - f ''() th f ( ) - f ''( ) hv tlst o solutio i Î R 49. Lt, b Î R d f : R R b dfid by 49. (b) If f ( ) cos( - + b si( + ). Th f is () diffrtibl t if d b (b) diffrtibl t if d b (c) NOT diffrtibl t if d b (d) NOT diffrtibl t if d b - ³ Þ - - cos cos( ) - < Þcos - cos( - ) Q cos( - q) cosq is cotiuous d diffrtibl t Î R similrly si + si + si( + ) for ll Î R, is cotiuous d diffrtibl 5. Lt, b Î R d + b ¹. Suppos If z + iy d z Î S, th (, y ) lis o æ ö () th circl with rdius d ctr ç, for, è ø > b ¹ (b) th circl with rdius - d ctr ì ü S íz Î C : z, t Î R, t ¹ ý î + ibt þ, whr i -. æ ö ç -, for, è ø < b ¹ (c) th - is for ¹, b (d) th y - is for, b ¹ - ibt 5.(cd) z + iy + ibt + ibt - ibt + b t + y Þ ( + b t ) + b t - ibt + b t + y -bt Þ, y + b t + b t æ ö æ ö æ ö Þ ç - + ( y - ) ç è ø è ø is circl with ctr ç, è ø d rdius æ ö b Þ, y d ç, lis o -is. è ø, b ¹ Þ, y - will li o y-is. bt H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 8 follow us o : momtumcdmy momtumcdmy
9 PASSAGE : Footbll tms T d T hv to ply two gms gist ch othr. It is ssumd tht th outcom of th two gms r idpdt. Th probbilitis of T wiig, drwig d losig gm gist T r, 6 d, rspctivly. Ech tm gts poits for wi, poit for drw d poit for loss i gm. Lt X d Y dot th totl poits scord by tms T d T, rspctivly, ftr two gms. 5. P( X > Y ) is () 4 (b) 5 (c) (d) 7 5. P( X Y ) is () 6 (b) P( X > Y ) WW + DW + W D (whr 5. (b) P( X Y ) DD + WW + W W (whr 5. (c) is corrct (c) 6 W rprpsts W rprpsts T wis ) (d) T wis d D rprsts drw) PASSAGE - Lt F (,) d F (,) for < d >, b th foci of th llips y +. Suppos 9 8 prbol hvig vrt t th origi d focus t F itrscts th llips t poit M i th first qudrt d t poit N i th fourth qudrt 5. Th orthoctr of th trigl F MN is () æ 9 ö ç -, è ø (b) æ ö ç, è ø (c) æ 9 ö ç, è ø (d) æ ö ç, 6 è ø 54. If th tgts to th llips t M d N mt t R d th orml to th prbol t M mts th - is t Q, th th rtio of r of th trigl MQR to r of th qudriltrl MF NF is () : 4 (b) 4 :5 (c) 5 :8 (d) : H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg 9 follow us o : momtumcdmy momtumcdmy
10 5.() y + hs th 9, F (-,) y.equtio of prbol is 4 itrscts th llipsth 4 + Þ Þ Q ¹ -6 æ ö æ ö \ M ç, 6, N ç,- 6 è ø è ø. Slop of 6-6 F N -- 5 Altitud from F o MN is y, Altitud from M o F N is Puttig y w gt Þ (c) For M, t 6 Þ t.norml t M, Put y y + t t + t 5 æ ö y - 6 ç - 6 è ø 9 - th Orthoctr is æ 9 ö ç -, è ø. Þ y Þ + 7 th Q æ ö., ç è ø.tgt to llips t M is y Put y Þ 6 th R (6,).So r of th trigl MQR æ 7 ö. ç è ø 4 Ar of MF NF 6 th th rtio of r of th trigl MQR to r of th qudriltrl MF NF 5 6 is : 6 5:8. 4 H.O : JABALPUR : 55, Wright Tow, Ph. (76) 486, 4, , RAMPUR : Rmpur Chowk, Abov PNB RANJHI : Opp. Ji Mdir, Mi Rod, Rjhi Pg follow us o : momtumcdmy momtumcdmy
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MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
KCET 2016 TEST PAPER WITH ANSWER KEY (HELD ON WEDNESDAY 4 th MAY, 2016)
. Th maimum valu of Ë Ë c /. Th contraositiv of th convrs of th statmnt If a rim numbr thn odd If not a rim numbr thn not an odd If a rim numbr thn it not odd. If not an odd numbr thn not a rim numbr.