MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J + + d Thn, for an arbitrary constant C, th valu of J I quals (A) log + + + C + (B) log + + + C + (C) log + + + C + (D) log + + + C + Sol. Ans [C] J I d + + + + ( ) d + + t ( t ) + t + / t t + + t dz, z whr z t + t z log + c z + log + + + + c. Lt g() log f () whr f () is a twic diffrntiabl positiv function on (, ) such that f ( + ) f (). Thn, for N,,,... g N + g (A) + 9 + +... + (N ) (B) + 9 + +... + (N ) (C) + + +... + 9 (N + ) (D) + +... + 9 (N + ) + Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 Sol: Ans [A] g() log f () g( + ) log f ( + ) log ( f ()) log + g() g ( + ) g () Putting N,,..., and adding, w gt g N + g + + +... + 9 (N ). Lt two non-collinar unit vctor â and bˆ form an acut angl. A point P movs so that at any tim t th position vctor OP (whr O is th origin) is givn by aˆ cos t + b ˆ sin t. Whn P is farthst from origin O, lt M b th lngth of OP and û b th vctor along OP. Thn, (A) uˆ aˆ + bˆ and M ( + aˆ + bˆ aˆ b ˆ ) / (B) uˆ aˆ bˆ and M ( + aˆ bˆ aˆ b ˆ ) / (C) uˆ aˆ + bˆ and M ( + aˆ + bˆ aˆ b ˆ ) / (D) uˆ aˆ bˆ and M ( + aˆ bˆ aˆ b ˆ ) / Sol: Ans [A] OP ( ( a ˆ cos t + bˆ sin t)( aˆ cos t + bˆ sin t) + aˆ bˆ sin(t) For maimum valu ndd sin t M ( + aˆ b ˆ ) / for maimum valu t π/ a ˆ + bˆ OP aˆ + bˆ uˆ aˆ + bˆ π π. Lt th function g : (, ), b givn by g(u) tan ( u ) π. Thn, g is (A) vn and is strictly incrasing in (, ) (B) odd and is strictly dcrasing in (, ) (C) odd and is strictly incrasing in (, ) (D) nithr vn nor odd, but is strictly incrasing in (, ) Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 Sol. Ans [D] g(u) tan ( u ) π/ u g ( u) > u + [g(u) is strictly incrasing in, ] Clarly it is nithr vn nor odd.. Considr a branch of th hyprbola y y 6 with vrt at th point A. Lt B b on of th nd points of its latus rctum. If C is th focus of th hyprbola narst to th point A, thn th ara of th triangl ABC is (A) (B) (C) + (D) + Sol. Ans [B] y y 6 ( ) ( y ) + a, b 6 For point A, ( ) a ( y ) So, A( +, ), + For point B, b a, a So, B( 6+, ) Similarly focus C ( + 6, ) Hnc, ara of th triangl is sq.units Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 6. A particl P starts from th point z + i, whr i. It movs first horizontally away from origin by units and thn vrtically away from origin by units to rach a point z. From z th particl movs units in th dirction of th vctor i + j and thn it movs through an angl π in anticlockwis dirction on a circl with cntr at origin, to rach a point z. Th point z is givn by (A) 6 + 7i (B) 7 + 6i (C) 7 + 6i (D) 6 + 7i Sol. Ans [D] Z is + i and M is (6, ), Q is (6, ) Coordinats of P ar 7, y 6 Lt R ( + iy) 6 y cosº sinº So, + iy 7+ 6i iπ / + iy 6 + 7i 7. Th ara of th rgion btwn th curvs y + sin cos and y sin cos boundd by th lins and π is (A) t ( + t ) t (B) ( + t t ) t (C) + ( + t t ) t (D) + t ( + t ) t Sol: Ans [B] y + sin cos + cos(( π / ) ) sin(( π / ) ) π cot y sin π tan cos Now π/ π π π 8 Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 π tan π tan π cot So π cot π tan So Ara π / π cot π tan d π / sin( / ) d cos( / ) tan ( / ) Putting tan (/) t (/)sc (/) d (tan 8 π ) π tan 8 ( + t t ) t ( + t t ) t 8. An primnt has qually likly outcoms. Lt A and B b two non-mpty vnts of th primnt. If A consists of outcoms, th numbr of outcoms that B must hav so that A and B ar indpndnt, is (A), or 8 (B), 6 or 9 (C) or 8 (D) or Sol. Ans [D] No. of possibl outcoms n(a) Lt n(b) k and n(a B) whr lis btwn and Bcaus A and B ar indpndnt, so P(A B) P(A). P(B) So, So, K. K can b and only as K is an intgr. So, K or. Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 9. Considr thr points P ( sin(β α), cos β), Q (cos(β α), sin β) and R (cos(β α + θ), sin (β θ)), whr < α, β, θ < π. Thn, (A) P lis on th sgmnt RQ (C) R lis on th sgmnt QP (B) Q lis on th lin sgmnt PR (D) P, Q, R ar non-collinar Sol: Ans [D] cos( β α) sin( β α) cos( β α θ) sin β cos β sin( β θ) R R (cos θ R + sin θ R ) cos( β α) sin( β α) sin β cos β (cos θ + sin θ) [ (cos θ + sin θ)] cos(β α) β < π/ and α > (β α) < π/ cos(β α) Also sinc < θ < π/ cos θ + sin θ > P, Q and R ar not collinar. SECTION- II ASSERTION REASON TYPE This sction contains multipl choic qustions numbrd to. Each qustion contains STATEMENT- (Assrtion) and STATEMENT- (Rason). Each qustion has choics (A), (B), (C) and (D) out of which ONLY ONE is corrct.. Suppos four distinct positiv numbrs, a, a, a, a ar in G.P. Lt b a, b b + a, b b + a and b b + a. STATEMENT-: Th numbr b, b, b, b ar nithr in A.P. nor in G.P. and STATEMENT-: Th numbrs b, b, b, b ar in H.P. (A) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is a corrct planation for Statmnt- (B) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is NOT a corrct planation for Statmnt- (C) Statmnt- is Tru, Statmnt- is Fals (D) Statmnt- is Fals, Statmnt- is Tru Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - 6 -
IIT-JEE-8 Sol: Ans [C] a a a ar a ar a ar b a b a( + r) b a( + r + r ) b a( + r + r + r ) b a b a( r ) r b a( r ) r b a( r ) r b a r b a( r ) b r a( r ) r b a( r ) (D) is th corrct answr.. Considr L : + y + p L : + y + p +, whr p is a ral numbr, and C : + y + 6 y +. STATEMENT-: If lin L is a chord of circl C, thn lin L is not always a diamtr of circl C. and STATEMENT-: If lin L is a diamtr of circl C, thn lin L is not a chord of circl C. (A) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is a corrct planation for Statmnt- (B) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is NOT a corrct planation for Statmnt- (C) Statmnt- is Tru, Statmnt- is Fals (D) Statmnt- is Fals, Statmnt- is Tru Sol: Ans [D] Givn circl is + y + 6 y + ( + ) + (y ) cntr is (, ) 6 + + p If L is chord thn < + 9 p + 6 < < p + 6 < 6 < p < 6 For L to b diamtr, 6 + + p + p Statmnt is fals (D) is answr.. Lt a solution y y() of th diffrntial quation dy y y d satisfy y(). Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - 7 -
IIT-JEE-8 STATEMENT-: y() and sc sc π 6 STATEMENT-: y() is givn by. y (A) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is a corrct planation for Statmnt- (B) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is NOT a corrct planation for Statmnt- (C) Statmnt- is Tru, Statmnt- is Fals (D) Statmnt- is Fals, Statmnt- is Tru Sol: Ans [C] dy d y y y sc(sc π/6) sct is tru. y Again, cos( sc π / 6) cos( sc ) cos π / 6 + sin( sc ) sin π / 6 y + (Sct. is fals). Lt a, b, c, p, q b ral numbrs. Suppos α, β ar th roots of th quation + p + q and α, β ar th roots of th quation a + b + c, whr β {,, }. STATEMENT-: (p q)(b ac) and STATEMENT-: b pa or c qa. (A) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is a corrct planation for Statmnt- (B) Statmnt- is Tru, Statmnt- is Tru; Statmnt- is NOT a corrct planation for Statmnt- (C) Statmnt- is Tru, Statmnt- is Fals (D) Statmnt- is Fals, Statmnt- is Tru Sol: Ans [B] Sinc aα + bα + c...() α + pα + q... () Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - 8 -
IIT-JEE-8 a b p c q b pa, c aq Again (p q) and (b ac) > (p q)(b ac) SECTION- III LINKED COMPREHENSION TYPE This sction contains Paragraphs P -6 and P 7-9. Basd upon ach paragraph, multipl choic qustions hav to b answrd. Each qustion has choics (A), (B), (C) and (D), out of which ONLY ONE is corrct. P -6 : Paragraph for Qustion Nos. to 6 Considr th function f : (, ) (, ) dfind by f () a +, < a < + a +. Which of th following is tru? (A) ( + a) f () + ( a) f ( ) (B) ( a) f () + ( + a) f ( ) (C) f () f ( ) ( a) (D) f () f ( ) ( + a) Sol: Ans [A] f () a + + a + a + a + [( + a + ) ( + a)] f () a ( + a + ) ( a( ) + a + ) a[ + + a] f () ( + a + ) a a f () ( + a), f ( ) ( a) ( + a) f () + ( a) f ( ). Which of th following is tru? (A) f () is dcrasing on (, ) and has a local minimum at (B) f () is incrasing on (, ) and has local maimum at Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - 9 -
IIT-JEE-8 (C) f () is incrasing on (, ) but has nithr a local maimum nor a local minimum at (D) f () is dcrasing on (, ) but has nithr a local maimum nor a local minimum at Sol: Ans [A] a( ) f () ( + a + ) f () < < (, ) So in (, ), f () is dcrasing a f () ( + a) > and f () So is point of local minima. f ( t) 6. Lt g () + t. Which of th following is tru? (A) g () is positiv on (, ) and ngativ on (, ) (B) g () is ngativ on (, ) and positiv on (, ) (C) g () changs sign on both (, ) and (, ) (D) g () dos not chang sign on (, ) Sol: Ans [B] f ( t) g() + t g () f ( ) + (by Libnitz rul of diffrntiation) Now + > So g () > f ( ) > > (sinc f () is incrasing in (, ) (, )) > So g () is positiv in (, ) g () < f ( ) < < < (Sinc f () is dcrasing in (, )) < So g () is ngativ in (, ) Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 P 7-9 : Paragraph for Qustion Nos. 7 to 9 Considr th lins L : L : + y + z + y + z 7. Th unit vctor prpndicular to both L and L is (A) i + 7j + 7k 99 (B) i 7j + k (C) i + 7j + k (D) 7i 7j k 99 Sol: Ans [B] dr s for lin prpndicular to L and L givn by (, 7, ) Unit vctor i 7j + k 8. Th shortst distanc btwn L and L is (A) (B) 7 (C) (D) 7 Sol: Ans [D] Shortst distanc btwn L and L b ( ) ( 7) ( + ) + ( ) + ( + ) 7 9. Th distanc of th point (,, ) from th plan passing through th point (,, ) and whos normal is prpndicular to both th lins L and L is (A) 7 (B) 7 7 (C) 7 (D) 7 Sol: Ans [C] Equation of plan will b ( + )( ) + (y + )( 7) + ( + )( ) + 7y + distanc + 7 + + 9 + 7 Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 SECTION- IV MATRIX-MATCH TYPE This sction contains qustions. Each qustion contains statmnts givn in two columns which hav to b matchd. Statmnt (A, B, C, D) in Column I hav to b matchd with statmnts (p, q, r, s) in Column II.. Considr th lins givn by L : + y L : ky L : + y Match th Statmnts/Eprssions in Column-I with th Statmnts/Eprssions in Column-II and indicat your answr by darkning th appropriat bubbls in th givn in th ORS. Column I Column II (A) L, L, L ar concurrnt, if (p) k 9 (B) On of L, L, L is paralll to at last on of th othr two, if (q) k 6/ (C) L, L, L form a triangl, if (r) k /6 (D) L, L, L do not form a triangl, if (s) k Sol: Ans [A-(s); B-(p),(q); C-(r); D-(p),(q),(s)] (A) Lins L, L and L ar concurrnt If k k (B) Th lins hav slops, So atlast two lins ar paralll if k or k k 9 or k, 6 k (C) Lins will form a triangl in all othr cass cpt A and B i.., for k 6 (D) Lins will not form a triangl in (A) and (B) i.., for k 9, 6, Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8. Considr all possibl prmutations of th lttrs of th word ENDEANOEL. Match th Statmnts/Eprssions in Column-I with th Statmnts/Eprssions in Column-II and indicat your answr by darkning th appropriat bubbls in th givn in th ORS. Column I Column II (A) Th numbr of prmutations containing th word ENDEA is (p)! (B) Th numbr of prmutations in which th lttr E occurs in th first (q)! and th last positions is (C) Th numbr of prmutations in which non of th lttrs D, L, N (r) 7! occurs in th last fiv positions is (D) Th numbr of prmutations in which th lttrs A, E, O occur only (s)! in odd positions is Sol: Ans [A-(p); B-(s); C-(q); D-(q)] (A) Considring ENDEA as on group, rmaining lttrs ar N, O, E, L So no. of prmutations! (B) E occurs in Ist and last positions. Rmaining lttrs ar N, N, D, A, O, E, L No. of prmutations 7! 7 6!!! (C) D, L, N should not occur in last fiv positions D, L, N should occur in Ist four positions, but w hav D, L, N, N So ways of arranging D, L, N, N in Ist four positions!! Ways of arranging rmaining E, E, A, O, E in last fiv positions Total no. of prmutations! (D) A, E, O occur in odd positions No of odd positions and lttrs ar E, E, E, A, O i.., Ways of arranging ths lttrs in odd positions!!!! Rmaining lttrs D, L, N, N can b arrangd in rmaining positions in Total no. of prmutations!!! ways Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8. Match th Statmnts/Eprssions in Column-I with th Statmnts/Eprssions in Column-II and indicat your answr by darkning th appropriat bubbls in th givn in th ORS. Column I Column II + + (A) Th minimum valu of is + (p) (B) Lt A and B b matrics of ral numbrs, whr A is symmtric, (q) B is skw-symmtric, and (A + B) (A B) (A B) (A + B). If (AB) t ( ) k AB, whr (AB) t s th transpos of th matri AB, thn th possibl valus of k ar (C) Lt a log log. An intgr k satisfying < ( k+ a ) <, (r) must b lss than (D) If sin θ cos ϕ, thn th possibl valus of π θ ± ϕ π ar (s) Sol: Ans [A-(r); B-(q),(s); C-(r),(s); D-(p),(r)] (A) Now, + + + + + + + + + + + + + + /( + ) + + + + + + ( + ) + (B) (A + B)(A B) (A B)(A + B) A + BA AB B A B + AB BA BA AB BA AB...(i) Now (AB) t B t A t ( B)A (sinc A is symmtric B is skw-symmtric) BA AB (Using (i)) k odd k, (C) a log log a log Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -
IIT-JEE-8 a a log log So k+ a k+log k log k So < k < < k < log < k < log log < k < log (/) log > k > log (/) k < log But < log < k < k < and k < (D) sin θ cos ϕ cos(9 θ) cos ϕ 9 θ nπ ± ϕ θ ± ϕ π nπ θ ± ϕ ( π / ) π n vn intgr, Amity Institut for Comptitiv Eaminations : Phons: 6/, 7///, 9-89/ - -