ANSWERSHEET (TOPIC = ALGEBRA) COLLECTION #2

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1 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) ANSWERSHEET (TOPIC ALGEBRA) COLLECTION # Questio Type A.Sigle Correct Type Q. (B) Sol ( 5 7 ) ( )!!!! C + C + C + C C + C + C + C + C + C + C. ] Q. (A) Sol k + k k () Ck+ ly C k+ ( + ) ( ) k+ ( ) ( + ) ( ) C k! k! k!. C k! k!! ( + ) ( )! k! k!. k! k!! k + k k + 6 k 5k 8 () From () ad () ad k + k 8 As. ] or k + k Q. (C) Sol Number of terms i ( + x) () + additio terms i () + additio terms i x x + x x x x x... x... x () (commo to ad ) x + x x ) 5 Hece total As. Alteratively : A B C A + B + C A B + B C + C A + A B C ( ) As. ] ( ) A B B C C A Q. (C) Sol Let z a + ib z a ib Hece we have 008 z z z

2 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) z z 0 z 0 or z ; if if z 0 z 0 ( 0, 0) 009 z z zz z 009 value of z Total 00 As.] Q. 5 (B) Sol If z + i + z i 8, PF + PF 8 z B max Q. 6 (D) Sol π π π π + i cos + i si cos + i si π f + i real part of z cos ( + ) π π 6a 6a a log cos log cos ( 0 0) a such term a ( 6a + ) a 8a a As.] Q. 7 (C) Sol Usig cosie rule

3 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) z z + z z + z z cos0 ad z z z + z z z cos z + z 9 z + z 7 7 N As. ] Q. 8 (C) Sol z + z i + 6z i + zi + i + i /8 z + i + i z + z + i /8 z + i d z + i Area /8 /8 5/.. As] Q. 9 (A) Sol W z cos isi cos / isi / cos / ( θ) θ ( θ ) ( θ ) ( θ ) cos ( θ / ) isi ( θ / ) θ cot i isi ( θ / ) + isi θ / cos θ / + isi θ / Hece Re( w) w moved o the lie x 0 parallel to y-axis. ] Q. 0 (D) Sol Give z z z z + + ( z z + )( z z + ) ( z + z )( z + z ) zz z z + z z + + z + zz + z z + z z + z z + z z + z z z +

4 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) ( z + z) ( z + z) z + + z ( z + z) z + + z 0 z + z + z z z + z z + z + zz + z + z + zz z z + z + z z + z + either z + z z ix purely imagiary z lies o y axis x 0 or z + + z z lie o the segmet joiig (, 0) ad (, 0 ) Q. (B) Sol z + 6z z + z z + z ( z + )( z + ) zz zz z z 0 or zz z z 0 Ceter - coefficiet of z, 0 Radius αα r Hece cetre, 0 & radius + + x + x + + x + x + x B ] Q. (B) Sol ( x + x x 6 + x 8...) + ( x x + x 5 x 7...) or ( + ) + ( + ) ( ) x x + 0 x ( ) x + ( ) 0 0 x x + x + (divide by + ) x As.] D ] Q. (C) Sol x px + 0 0

5 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) x 0x + p 0 If p 0 the x px + 0 x 0x + p 0 p x + 0 p 0 x ad p Hece there are values of x i.e. { 0 + 5, 0 5, } ] 5 Q. (A) Sol i.e. a c D b a c < 0 D b a c < 0 > b..() hece a c > b.() multiplyig () ad () a a c c > b b < b b b b Now cosider for f ( x ) D b b a a c c b b D < 0 g( x) > 0 x R ( A ) ] Q. 5 (C) Sol Product will be divisible by if atleast oe digit is 0,, 6, 9 Hece total digit umbers 9.0 Number of digit umbers without 0,, 6 or Number of umbers As. ] Q. 6 (B) Sol Sum of sigle digit umber S S + 0 S + 0S S Sum of two digit umber Sum of three digit umber Sum of four digit umber Total 80S ] Q. 7 (B) Sol ; P( r ) C. 8 Q. 8 (A) Sol S S S S S 666S P H T or TH p q ad r P (success) As. ]

6 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) P( A ) 5, P( B ) ; P( C) P( D) P( A ).P ( D / A) + P( B ).P ( D / B) + P( C ).P ( D / C) Q. 9 (A) Sol A: exactly oe child B: exactly two childre C: exactly childre P( A ) ; P( B ) ; P( C) E: couple has exactly gradchildre P( E) P( A ).P ( E / A) + P( B ).P ( E / B) + P( C ).P ( E / C) / (, ) As ly / deotes each child havig two childre

7 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) deotes each child havig ad or ad childre As.] Q. 0 (A) Sol ta α + ta β p ta α ta β q p p ta ( α + β) q q ta ( α + β ) + p ta ( α + β ) + q + ta α + β + ( q ) p p + + q ( q ) ( ) p q q + p + ( ) + ( ) p + q p + p q + q q p q + q q p q q q ] p q Q. (D) Sol a, a, b, ( a b 6) i A.P. a + a b 6 a + b b a a b a a b; a Hece the series is,,,, 5,... s As.] 00 Q. (A) Sol [ ] r T r r ( r + ) r ( r + )( r + ) r [ ] r r 6r ( r ) + 6 r 6 S T r ( 60 ) 5 [ ] 5 [ ] 60.6 As.] Questio Type B.Comprehesio or Paragraph Q. () Sol C A.P.with a, d, 60 7

8 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) B C [Sol. r r r 0 + x + x a x..() Replacig x by x i equatio () the r r + + a r or ( + x + x ) a rx.() x x x From equatio () ad (), we get r 0 r 0 r r a rx a rx r 0 r 0 r x Comparig coefficiet of a r a r.() o both sides, the we get (0) Put x ad x i equatio (), the 9 a + a + a + a + a a a 0 ad a0 a + a a + a... + a a addig ad subtractig, the we get 9 + a 0 + a + a a a + a.() 9 ad a + a + a a a.(5) Now, a r a r Put r 0,,,6,...,a,a a a a 0 a a a a :. :. a + a + a + a a a a + a + a Now from equatio () 9 + ( a 0 + a + a a ) + a 9 + a a 0 + a + a a 9 + a a r As. 0 r 0 () ar a r Put r,,5, 7,...,, 8

9 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) a a a a a5 a 5 : : a a a + a + a + a + a a a + a a + a Now from equatio (5) a + a + a a () a coefficiet of x i ( + x + x 9 a r As. r ) coefficiet of C C + + C As. ] Q. () Sol Q. C Q. B Q. D [Sol. R ( + x) put x to get sum of all the coefficiets (i) for x ;R ( + ) 8 { + C x + x + C x + x +...} x i cosider f + f ' 8 C +... eve iteger sice I is iteger f + f ' must be a iteger but 0 < f + f ' < f + f ' f ' f ow + R Rf + R ( f ) 8 + ( + ).( ) As. (ii) r 8 8 r 8 r T + i ow T r+ Tr + x C x C whe r x

10 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) T 8 r + T > Cr > < 8 < T r T > r+ r < ( ) ( 9 r) r 9 r Cr ( ) ( ) 8! r!; 9 r!. r! 8 r! 8! for r,,, this is true i.e. T5 > T but for r 5 T6 < T5 T 5 is the greatest term (B) (iii) agai T C..x ; T C..x 8 k k 8 k k k+ k k k 8 k k k Tk C k..x.x we wat to fid the term havig the greatest coefficiet k 8 k 8. C >. C () k k k 8 k 8 ad. Ck >. Ck () from () k k 8!..8! > > k! 9 k! k! 8 k! 9 k k k > 8 k k > 6 Agai. C >. C k 8 k 8 k k k k 8!..8! > > k! 9 k! k! 0 k! k 0 k 0 k > k > k k < 7 6 < k < 7 T 6 ad T 7 term has the greatest coefficiet k 6 or 7 sum As. ] Q. 5 () Sol Q. D Q. B Q. C [Sol. (i) (ii) square) x + y a Figure is a square As. Area of the circle πd (where d diameter of circle side of the

11 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) π 5000π As. (iii) x + y < 5 x 0, y 0 or x + y 50 give oe each to x ad y x + y 8 x + y + z 8 umber of solutios 50 C 50 9 Number of solutios i all the four quadrats , 0 Number of solutios except ( 0, 0 ) o x ad y axis from ( ) to ( 0, 5 ) to ( 0, 5) are 00 Total solutios As. ] Q. 6 () Sol Q. D Q. A Q. B [Sol. P( E) P P( F) P( E F) + P( E F) 5, 0 ad P( F) P( E) P( F/ E) + P( E) P( F / E) p.+ ( p ). p (i) if p 0.75 p( F) ( p + ) ( ) P ( E F) P ( E / F) P F As.

12 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) (ii) ow P( E / F) Equality holds for p 0 or p 5p p ( p + ) For all others value of p ( 0, ), LHS RHS, > hece (A) (iii) If each questio has alteratives tha ( ) p + P( F) p + ( p) p + p P( E / F) which icreases as icreases for a fixed p (B) ] p + Q. 7 () Sol Q. B Q. A Q. C 5R [Sol. Ur I< B R Ur I< B A : first two draws resulted i a blue ball. B : ur-i is used P( B ) B : ur-ii is used P( B ) P( A / B ) P( A / B ) As.(i). ( ) 6 P B / A 6 7 E P( B / A) E E: third ball draw is red P E P E E + P E E As. ii As. (iii)

13 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) Q. 8 () Sol Q. [C] Q. [A] Q. [B] [Sol. () A: balls draw foud to be oe each of differet colours. B : ( W) + ( G) + ( R ) are draw; P( B ) B : ( W) ( G) ( R) + + are draw; P( B ) B : ( W) ( G) ( R) + + are draw; P( B ) B : They are draw i groups of,, (WGR) - (6 cases); P( B ) B 5 : ( W) + ( G) + ( R) ; P( B ) P A / B P A / B P A / B C 6 C 0 W G R R R R C 6 C 0 W G G G G R C 6 C 0 W W W W G R C. C. C 6 P( A / B ) 6. W G G R R R, C 0 P A / B 6 C. C. C 8 W W G G R R C As P( B i ).P ( A / B i ) r (). P( B / A) As. 00 () 8. 8 P( B 5 / A) Hece P (bag had equals umber of W ad G balls/a) P( B / A) P( B 5 / A) + As. ] 6 0 Questio Type C.Assertio Reaso Type

14 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) Q. 9 (C) Sol Statemet- is False Take eg. z + i z i z i z + i the figure is rectagle ] Q. 0 (D) Sol Area Hece S- is false ad S- is true.] Q. (B) Sol z, z ad 0 are o the same side the oly S is the reaso of S ] Q. (C) Sol For For λ > Im z, the umber values of z λ Im z, the umber of values of z Q. (A) Sol Let z cos θ + i si θ where cos θ, si θ Q + θ + θ si θ + isi θcos θ si θ si θ i cos θ z cos isi z si θ Now P( ) : si θ, cos θ Q N ca be provided by iductio if si θ, cos θ Q]

15 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) a p Q. (A) Sol Let A b q A c e a + p ab + pq ac + pr T AA ab + pq b + q bc + qr ac + pr bc + qr c + r a b c p q r T a p 0 a b c T T AA b q 0 p q r 0 AA is sigular. ] c r Q. 5 (A) Sol Give AB + A + B 0 AB + A + B + I I A B + I + B + I I ( A + I)( B + I) I ( A + I) ad ( B + I) are iverse of each other ( A + I)( B + I) ( B + I)( A + I) AB BA ] Q. 6 (B) Sol Let x, x, x f ( x) ( x x)( x x )( x x ) f ( i) ( i x )( i x )( i x ) f ( i) x i x i x i x + x + x + R be the roots of f ( x) 0 This is possible oly if x x x 0 f x x a 0 b c a + b + c 0* Q. 7 (D) Sol ix + + i x + i 0 αβ Im αβ 0.] Questio Type D.More tha oe may corect type Q. 8 () Sol B, D [Sol. Required lie is passig through P( α ) ad parallel to the vector OQ

16 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) Hece z α + iλa, λ R z α purely imagiary α z α Re 0 α ( B ) ( ) Re z α α 0 Re zα α 0 Also z α z α + 0 α α α( z α ) + α( z α ) 0 α z + αz α 0 D ] Q. 9 () Sol [A, B, C, D] [Sol. AP + PB AB z α + β z β α A is true Now z α + t ( β α ) ( t) α + tβ where Agai z α β α is real z α z α β α β α t 0, B is true z α z α 0 β α β α Agai z z α α 0 β β ( z α) z α β α β α As. if ad oly if 0 As. z α z α 0 α α 0 β α β α 0 Q. 0 () Sol Q. A, B, C, D

17 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) z z α α α put z i 0 if k i if k + ( i α r ) r i + i if k + i if k + [Sol. ( z )( z )...( z ) Q. () Sol Q. B, C, D 0 [Hit. PQ ] 0 B,C,D Q. () Sol A, B t p t p r say t t [Sol. p p If we start from t p, the p is the ( p + ) th term t is the th p + term ad if we start from t p, the t p t t + pd () p p ad t t + pd ( d c.d ) p p t t + t t (from equatio ()) p p p p t t t t r p p t p t p t p r p p ( r )( r ) 0 r, As. Alterative solutio : PD R ; R PD A + p D A p D R A + p D A + p D Also if PD 0 D 0 TP Tp Tp R ] Q. () Sol B, C x [Sol. ( log x) log + 9 log log x < ( log x)

18 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) ( log x) ( log x ) + 5 5log x < ( log x) Let log x t t t + 5 8t < t t 9t + 9 8t 8t + 5 < t t t + 6 < 0 t t 9 < 0 < t < 9 t < 9 < t < ad t > t > or t < hece, t (, ) (, ) x, (, 8) B,C 8 Q. () Sol Q. B, D x x + cos ax + b [Sol. ( x ) + cos( ax + b) for above equatio to have atleast oe solutio let f ( x) ( x ) + ad f ( x) cos( ax + b) if x the L.H.S. cos a + b ad R.H.S. hece, cos( a + b) a + b π, π, 5π but 0 a + b 0 a + b π or π B, D] Q. 5 () Sol [A, B, C, D] S ABCD [Sol. [ ]

Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 00 000 www.tekoclasses.com Q. 6 () Sol Q. B, C, D [Sol. (A) False is should be 9 P5 (B) x.! 8! 8! 8 x C!

19 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) Q. 6 () Sol Q. B, C, D [Sol. (A) False is should be 9 P5 (B) x.! 8! 8! 8 x C! (C) Vowels E E E E select places i 9 C ways arrage cosoat alphabetically oly us oe ways. 9 0 C C5 (D) True D correct aswer are ( B ), ( C ) ad Q. 7 () Sol B, C, D [Sol. Let umber of blue marbles is b ad umber of gree marbles is g bg Hece b g + C ( b + g)( g + b ) bg ( b + g) ( b + g) bg b + g + bg b g bg g bg g + b b 0 D b + b b 8b + must a perfect square. Hece possible values of b are, 6, 0 [ B,C, D] ] Q. 8 () Sol B, C, D [Sol. Let the H.P. be A A + D A + D A + A + D + A + D +... Correspodig A.P. Tp of AP A ( p ) D q p q + ( + ) Tq of AP A ( q ) D p p q + ( + ).().()

20 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) Tp + q of AP A + P + q D Now solvig equatio () ad (), we get A D pq p + q Tp + q of AP A + ( p + q ) D ( p + q) D pq Ad Tpq of AP A + ( pq ) D pqd p + q Tp + q of HP pq ad Tpq of HP p + q also p >, q > pq > p + q i.e. T > T ] p+ q Questio Type E.Match the Colums Q. 9 () Sol (A) Q, R; (B) P, S; (C) Q, S; (D), P, R pq i i i [Sol. (A) z ± + or π π amp z or ampz Q, R i i i (B) z ± + or π π amp z or P,S i π π (C) z i z cos + i si mπ ( π / ) mπ ( π / ) z cos + isi π π m 0, z cos + i si π π π π m, z cos + i si amp z or Q,S (D) z + i 0 + i π π z cos + isi mπ + ( π / ) mπ + ( π / ) z cos + isi π π m 0, z cos + i si

21 Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) π π π π m, cos + i si or cos + i si P,R] Q. 50 () Sol (A) Q: (B) R; (C) S 6 [Sol: (A) No pair C As. (Q) (B) atleast oe pair exactly oe + both pair C. C. + C As. ( R ) (C) fewer tha pairs o pair + exactly oe pair C. + C. C As ( S )] Q. 5 () Sol (A)-R (B)-S (C)-P (D)-Q fog : f g x [Sol: (A) l g ( x) l ( x ) (B) x > 0,, R gof : g f x l x 0, S (C) fof : f l f x l x l x > 0 (D) gog : g g x g x x > (, ) P x x, Q ]

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms. [ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural