R Pena Towe, Road No, Contactos Aea, Bistupu, Jamshedpu 8, Tel (657)89, www.penaclasses.com IIT JEE Mathematics Pape II PART III MATHEMATICS SECTION I Single Coect Answe Type This section contains 8 multiple choice questions. Each question has fou choices, (B), (C) and (D) out of which ONLY ONE is coect.. The equation of a plane passing though the line of intesection of the planes x + y + z and x y + z and at a distance fom the point (,, ) is 5x y + z 7 (B) x + y (C) x + y + z (D) x y. Let equation of plane be x + y + z + λ (x y + z ) ( + λ) x + y ( λ) + z ( + λ) λ... () distance of () fom (,, ) is ( + λ ) + ( λ ) ( + λ ) λ ( + λ ) + ( λ ) + ( + λ ) Substituting value of λ in () gives 5x y + z 7. λ 7 /. If a and b ae vectos such that a + b 9 and a ( i ˆ + j ˆ + k ˆ ) ( i ˆ + j ˆ + k ˆ ) b, then a possible value of ( a + b ). ( 7 i ˆ + ˆ j + k ˆ ) is. (B) (C) (D) 8. (C) a + b x i + y j + z k x + y + z 9 ( a + b ) ( i + j + k ) i ( y z ) j ( z x ) + k ( x y ) x : y : z : : a + b λ i + λ j + λ k x + y + z 9 9λ 9 λ ± ( a + b ).( 7 i + j + k ) if λ. IIT JEE (8 Ap ) Question & Solutions Pape II www. penaclasses.com
. Let PQR be a tiangle of aea with a, 7 b 5 and c, whee a, b and c ae the lengths of the sides of the tiangle opposite to the angles at P, Q and R espectively. Then equals MATHEMATICS sin sin P sin P P + sin P (B) 5 (C) (D) 5. (C) sin P sin P sin P + sin P sin sin P ( cos P ) cos P P ( + cos P ) + cos P ( 5 / ) + ( 9 / ) 5 + 9 6 9 cos P. ( 5 / ) ( 7 / ) 7 5 cos P + cos P ( )( / )( / ) ( + ( 9 / 5 ) 9 / 5 ) 6.. Fou fai dice D, D, D and D, each having six faces numbeed,,,, 5 and 6, ae olled simultaneously. The pobability that D shows a numbe appeaing on one of D, D and D is 9 6 (B) 8 6 (C) 5 6 (D) 7 6. Case I When D, D, D all show diffeent numbe and one of the numbe is shown by D 6 C! 6 P(E ) 6 6 6 Case II When D, D, D all show same numbe and that numbe is shown by D P(E ) 6 ( / 6) / 6 Case III When two numbes shown by D, D, D ae same and one is diffeent and one of the numbe is shown by D P(E ) 6 C 5 C 6!! 6 6 9 Requied pob. P(E ) + P(E ) + P(E ). 6 IIT JEE (8 Ap ) Question & Solutions Pape II www. penaclasses.com
MATHEMATICS π / 5. The value of the integal x π / π + x + ln cos x dx is π x π (B) π (C) + (D) π π / 5. (B) x π / π + x + ln cos x dx π x π / x π / cos xdx π + x Q ln is an odd function π x [ ] / π / π π x cos xdx ( x ) sin x + x cos x. 6. If P is a matix such that P T P + I, whee P T is the tanspose of P and I is the identity matix, then thee exists a column matix X x y such that z PX (B) PX X (C) PX X (D) PX X 6. (D) Given P T P + I P P T + I P (P + I) + I P I PX IX X. 7. Let a, a, a,... be a hamonic pogession with a 5 and a 5. The least positive intege n fo which a n < is (B) (C) (D) 5 7. (D) If d is the common diffeence of an A.P., then ( / a ) ( / a ) ( / 5 ) ( d 9 9 Now, / a n ( / a ) + (n ) d / 5 ) 5 9 ( n ) / a n ( n ) < > 5 5 9 95 Least value of n is 5. n > 99 / IIT JEE (8 Ap ) Question & Solutions Pape II www. penaclasses.com
MATHEMATICS 6 8. Let α(a) and β(a) be the oots of the equation ( + a ) x + ( + a ) x + ( + a ) a >. Then Lim α ( a ) and Lim β ( a ) ae + a 5 and (B) + a and (C) 8. (B) Let t 6 + a (t ) x + (t ) x + (t ) (t + ) x + (t + t + ) x +, as t If a then t x + x + x, /. 7 and (D) 9 and whee IIT JEE (8 Ap ) Question & Solutions Pape II www. penaclasses.com
SECTION II Paagaph Type MATHEMATICS This section contains 6 multiple choice questions elating to thee paagaphs with two questions on each paagaph. Each question has fou choices, (B), (C) and (D) out of which ONLY ONE is coect. Paagaph fo Questions 9 to 5 Let f (x) ( x ) sin x + x ( t ) fo all x R, and let g ( x ) ln t f ( t ) dt fo all x (, ). t + 9. Which of the following is tue? g is inceasing on (, ) (B) g is deceasing on (, ) (C) g is inceasing on (, ) and deceasing on (, ) (D) g is deceasing on (, ) and inceasing on (, ) 5. Conside the statements : P : Thee exists some x IR such that f (x) + x ( + x ) Q : Thee exists some x IR such that f (x) + x ( + x) Then both P and Q ae tue (B) P is tue and Q is false (C) P is false and Q is tue (D) both P and Q ae false x Sol. 9. (B) 5. (C) 9. f (x) ( x) sin x + x ( x ) and g' ( x ) ln x f ( x ) x + ( x ) Let h ( x ) ln x ln x ( x + ) x + h' ( x ) ( x + ) x ( x ) h' ( x ) < x (, ) h(x) < h(), as h(x) is deceasing. x ( x + ) Hence, h(x) < g' (x) < x (, ). 5. P : ( x) sin x + x + x x + ( x ) (sin x ) which has no solution as LHS, x. Hence, P is false. Q : x + ( x) sin x + x + x x sin x ( x ) P ( x ) x, Q(x) sin x ) x ( IIT JEE (8 Ap ) Question & Solutions Pape II 5 www. penaclasses.com
MATHEMATICS / (, /).57 x Hence, thee exist some solutions. Hence, Q is tue. Paagaph fo Questions 5 and 5 Let a n denote the numbe of all n digit positive integes fomed by the digits, o both such that no consecutive digits in them ae. Let b n the numbe of such n digit integes ending with digit and c n the numbe of such n digit integes ending with digit. 5. Which of the following is coect? a 7 a 6 + a 5 (B) c 7 c 6 + c 5 (C) b 7 b 6 + c 6 (D) a 7 c 7 + b 6 5. Q a n b n + c n a n b n + c n b n b n + c n a 7 b 6 + c 6 Also, b 6 b 5 + c 5 a 5 + a 6 b 6 + c 6 + b 6 a 7. 5. The value of b 6 is 7 (B) 8 (C) 9 (D) 5. (B) b 6 contains 6 digits beginning and ending with I. If second digit fom beginning is zeo, thid place can be filled up with I only, then fouth and fifth places can be filled up in two ways, o,. If second digit fom beginning is, then if addition to it thid digit is zeo, then we have cases, (, ) if thid digit is, we have cases (,, ) so, total numbe of cases 8 IIT JEE (8 Ap ) Question & Solutions Pape II 6 www. penaclasses.com
Paagaph fo Questions 5 and 5 A tangent PT is dawn to the cicle x + y at the point P (, ) to PT is a tangent to the cicle (x ) + y 5. A common tangent to the two cicles is MATHEMATICS. A staight line L, pependicula x (B) y (C) x + y (D) x + y 6 5. (D) Equation of the tangent to cicle x + y is y + Also geneal equation of tangent to (x ) + y is mx + + m y mx + m... (i) y m ( x ) + + m y mx + + m m... (ii) (i) and (ii) ae same line + m m ± + m m Taking 5. A possible equation of L is m, equation of common tangent fom (i) is x + y 6 x y (B) x + y (C) x y (D) x + y 5 5. Equation of tangent to x + y at point (, ) is T x + y equation of line L x y + k... (i).. + k Q k o 5 Equation of L is x y + ( ) IIT JEE (8 Ap ) Question & Solutions Pape II 7 www. penaclasses.com
SECTION III Multiple Coect Answe Type MATHEMATICS This section contains 6 multiple choice questions. Each question has fou choices, (B), (C) and (D) out of which ONE o MORE ae coect. 55. Let X and Y be two events such that P ( X Y ), P ( Y X ) and P ( X Y ) following is (ae) coect? 55. (AB) P ( X Y ) (B) X and Y ae independent (C) X and Y ae not independent P ( X Y ) P ( X Y ) P ( Y ) P ( Y ) P ( X Y ) P ( Y X ) P ( X ) P ( X ) (D) P ( X Y ) P ( X ) + P ( Y ) P ( X Y ) / c P ( X Y ) Q P ( X Y ) P ( X ). P ( Y ) they ae independent. Also. X c & Y will be independent. c P ( X Y ) 6 6. Which of the x t 56. If f ( x ) e ( t )( t ) dt fo all x (, ) then f has a local maximum at x (B) f is deceasing on (, ) (C) thee exists some c (, ) such that f ''(c) (D) f has a local minimum at x 56. (ABCD) f ' (x) e x (x ) (x ) f ' (x) x, + + f (x) has local maximum at x and local minimum at x. Also f ' (x) < on (,). Since, f ' () f ' () and f ' (x) is continuous and diffeentiable, theefoe fom Rolle's Theoem thee must exist some c (, ) such that f '' (c). IIT JEE (8 Ap ) Question & Solutions Pape II 8 www. penaclasses.com
MATHEMATICS 57. Fo evey intege n, let a n and b n eal numbes. Let function f : IR IR be given by a n + sin π x, fo x [ n, n + ] f ( x ), fo all intege n. b n + cos π x, fo x [ n, n ] If f is continuous, then which of the following hold(s) fo all n? a n b n (B) a n b n (C) a n b n + (D) a n b n 57. (BD) a n f ( x ) b n + sin π x + cos π x x [( n ), n ] x [ n, ( n )] a n + + sin π x x [( n + ), n + ] f ( x ) b n + + cos π x x [ n +, ( n + )] Fo option : f ( (n ) + ) a n f ( (n ) ) b n + a n b n. Fo option (B) : f (n ) a n f (n + ) b n + a n b n. Fo option (C) : f (n + ) ) a n f (n + ) + ) b n + a n b n +. Fo option (D) : f (n ) ) a n f (n ) + ) b n a n b n. x y + z x + y + z 58. If the staight lines and ae coplana, then the plane(s) containing k 5 k these two lines is (ae) y + z (B) y + z (C) y z (D) y z 58. (BC) k (k ) k ± 5 k If Lines ae coplana x 5 y + ± z y z o y + z. ± IIT JEE (8 Ap ) Question & Solutions Pape II 9 www. penaclasses.com
59. If the adjoint of a matix P is MATHEMATICS 7, then the possible value(s) of the deteminant of P is (ae) (B) (C) (D) 59. (AD) Adj P 7 + + Adj P P P ± π π π 6. Let f : (, ) R be such that f (cos θ) fo θ,, sec θ of f is (ae). Then the value(s) (B) + (C) (D) + 6. (AB) f ( cos θ ) f (x ) + x cos θ + cos θ cos θ cos θ Put, x t x ± + t f ( t ) ± + t f ( / ) ± ±. + / IIT JEE (8 Ap ) Question & Solutions Pape II www. penaclasses.com