WBJEEM - 05 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. μ β γ δ 0 B A A D 0 B A C A 0 B C A * 04 C B B C 05 D D B A 06 A A B C 07 A * C A 08 D C D A 09 C C A * 0 C B D D B C A A D A A B A C A B 4 * A C B 5 C A B C 6 B D A B 7 B A D B 8 D * B D 9 D D D D 0 B A D A A B C B B C C A C A B B 4 A B D A 5 A B C D 6 B B C A 7 A C D D 8 A B A B 9 C D * A 0 A D C A D B A A A B * C A D D B 4 A C A C 5 C C B B 6 B D B C 7 B C D A 8 D D B D 9 D A A B 40 D C B C 4 C C A A 4 C B D B 4 A B B D 44 * A A D 45 D B C C 46 A A B C 47 B C C D 48 A B B C 49 B A C C 50 A D A C 5 C A C B 5 B A C C 5 C B D B 54 C A C A 55 B D A D 56 A B B C 57 C A A C 58 C D C A 59 D C B B 60 C C A A 6 C D A C 6 D A A A 6 B C C A 64 D C D A 65 A A B D 66 C C C A 67 A D A C 68 A B D C 69 C A A D 70 A A C B 7 C A,B,C A,D B,C 7 B,D B,C A,C B,C 7 C,D B,C C A,D 74 A,D B,C B,D A,C 75 A,D C,D B,C A,B,C 76 A,C A,D B,C B,C 77 A,B,C A,D C,D C 78 B,C A,C A,D B,D 79 B,C C A,B,C C,D 80 B,C B,D B,C A,D * None of the options is correct
WBJEEM - 05 (Answers & Hint) Code-. The value of CATEGORY - I (Q to Q60) Each question has one correct option and carries mark, for each wrong answer /4 mark will be deducted. t lim dt is ANSWERS & HINT for WBJEEM - 05 SUB : MATHEMATICS (A) 0 (B) (C) 8 (D) 6 d t dt d Lt 4 d ( ) d. If cot tan cos ec k, then the value of k is (A) (B) (C) (D) cot + tan cosec k. If,, then the value of 4 4cos sin 4cot cos 4 is (A) cot (B) cot (C) cos (D) sin cos + cot + cos cot, 4. The number of real solutions of the equation (sin ) (cos ) 0 is (A) (B) (C) (D) 4 sin only 0 one solution cos - / / solutions - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 5. Which of the following is not always true? (A) a b a b if a and b are perpendicular to each other (B) a b a for all R if a and b are perpendicular to each other (C) a b a b a b (D) a b a for all R if a is parallel to b 6. If the four points with position vectors i ˆ ˆj k, ˆi ˆj k, ˆ ˆj kˆ and ˆj kˆ are coplanar, then (A) (B) (C) - (D) 0 Let ˆ ˆ ˆ ˆ ˆ ˆ a i j k, b i j k, c ˆj k, ˆ d ˆj kˆ ˆ b a i, c b ˆi k, ˆ d c ( )j ˆ kˆ 0 0 0 0 0 a a a 7. For all real values of a 0, a, a, a satisfying a0 0, the equation a 4 0 + a + a + a 0 has a real root in the interval (A) [0, ] (B) [, 0] (C) [, ] (D) [, ] f(0) f() 0 4 a a a f () a0 4 f () 0 has at least one real root in [0, ] (Rolle s theorem) 8. Let f : R R be defined as 0, is irrational f () sin, is rational Then which of the following is true? (A) f is discontinuous for all (B) f is continuous for all (C) f is discontinuous at k, where k is an integer (D) f is continuous at k, where k is an integer sin 0 k - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 9. If 0, / f : R is defined as tan f ( ) tan tan. Then the range of f is tan (A) (, ) (B) (, ] (C) [, ) (D) (, ] f () sec Range of f [,) 0. If A and B are two matrices such that AB B and BA A, then A + B equals (A) AB (B) BA (C) A + B (D) AB A.A + B.B A (BA) + B (AB) BA + AB A + B. If is an imaginary cube root of unity, then the value of the determinant (A) (B) (C) (D) 0 (zero) is 0 0 (c c + c ) ( + ) ( 4 + ) ( ) ( ) +. Let a, b, c, d be any four real numbers. Then a n + b n c n + d n holds for any natural number n if (A) a + b c + d (B) a b c d (C) a + b c + d, a + b c + d (D) a b c d, a b c d Put n, a + b c + d... () Put n a + b c + d... () from () and () ab cd Consider a quadratic with roots (a, b ) (a + b ) + a b 0... () Consider another quadratic with roots (c, d ) (c + d ) + (cd) 0... (4) Since a + b c + d and (cd) (ab) Both quadratic are same and quadratic cannot have more than roots. Here a c and b d or a d, b c - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint). If are the roots of p + 0 and is a root of + p + 0, then () () is (A) 0 (zero) (B) (C) (D) p or ( + ) ( + ) 0 /5 / 4. Number of irrational terms in the binomial epansion of 7 00 (A) 90 (B) 88 (C) 9 (D) 95 Ans : (No option is correct) 00r r 00 r 5 is Tr C () (7) r 0, 5, 0, 45, 60, 75, 90 7 rational terms No. of irrational terms 0 7 94 (not in the option). 5. The quadratic epression (+) p + q 0 for any real if (A) p 6p 8q < 0 (B) p 8p + 6q < 0 (C) p 8p 6q < 0 (D) p 6p + 8q < 0 4 + (4 p) +q+ 0 D < 0 p 8p 6q < 0 64 64 i i 6. The value of i i (A) 0 (zero) (B) (C) (D) i i i 64 + 64 + is 7. Find the maimum value of z when z, z being a comple number. z (A) (B) (C) (D) z z z z 0 z ma - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (4)
WBJEEM - 05 (Answers & Hint) 8. Given that is a real number satisfying 5 6 5 0, then 0 (A) 5 (B) 5 (C) > 5 (D) 5 5 0 or < <5 5,,5 5 9. The least positive value of t so that the lines t +, y + 6 0 and y are concurrent is (A) (B) 4 (C) 6 (D) 8 6 6 t least positive value 8 0. If in a triangle ABC, a cos A b c 0, then (A) A 4 (B) A (C) A (D) A < 4 cos A b c a cos A < b +c a < 0cosA < 0 A,. { R : cos sin } 0, (A) 0,, 4 4 (B) 0,, 4 5 (C) 0,, 4 4 (D) 0, - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (5)
WBJEEM - 05 (Answers & Hint) y cos sin o 4 4 0,, 4 4. A particle starts moving from rest from a fied point in a fied direction. The distance s from the fied point at a time t is given by s t + at b + 7, where a, b are real numbers. If the particle comes to rest after 5 sec at a distance of s 5 units from the fied point, then values of a and b are respectively (A) 0, (B) 0, (C) 8, (D) 0, ds dt t 5 t5 a 0 t a 0 and 5 5 + 5a b+7 (s5, t5) b. lim n... n 0 n n (A) (B) (C) (D) 0(zero) lt n n r n lt n r n n n n 0 d 0 4. If 0 ae blog lim then the values of a, b are respectively (A), (B), (C), (D),0 - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (6)
WBJEEM - 05 (Answers & Hint) (LH Rule) lt 0 ae b ae () (form 0 0 ) a+0 b 0 a b () (Again LH Rule in ) a e e lt 0 a a a 5. Let P() be a polynomial, which when divided by and 5 leaves remainders 0 and 6 respectively. If the polynomial is divided by ( ) ( 5) then the remainder is (A) + 6 (B) 6 (C) 6 (D) 60 P() ( )( 5) q() + (a + b) P() 0; p(5) 6 a + b 0; 5a + b 6 a ; b 6 Remainder + 6 dy 6. The integrating factor of the differential equation ( d tan y )(+y )0 is (A) e (B) e (C) e (D) e tan y z dy dz y d d dz d +. z I. F. e d e - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (7)
WBJEEM - 05 (Answers & Hint) 7. If y e cos then which of the following differential equations is satisfied? d y dy (A) 5y 0 d d d y dy (B) 5 y 0 d d (C) d y d dy 5 y 0 d y y e sin () y y + e sin 4e cos y y y 4y y + y + 5y 0 (D) d y d dy 5y 0 d 8. In a certain town, 60% of the families own a car, 0% own a house and 0% own both a car and a house. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both? (A) 0.5 (B) 0.7 (C) 0. (D) 0.9 A Car ; B House P (AB) P(AB) P(AB) 70 0 0.5 00 9. The letters of the word COCHIN are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word COCHIN is (A) 60 (B) 9 (C) 96 (D) 48 Rank of COCHIN (4!)4 + 97 Number of words 97 96 0. Let f : RR a continuous function which satisfies t (A) 0 (zero) (B) (C) 5 (D) f () f() (Leibnitz theorem) f() ke and 0 f 0 f t dt 0 k 0 f() 0 f(log e 5) 0 0 f f dt. Then the value of f (log e 5) is. The value of, such that the following system of equations has no solution, is y z y + z 4 + y + z 4 (A) (B) (C) 0(zero) (D) 0 - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (8)
WBJEEM - 05 (Answers & Hint) 0 9 0. If f Then f (00) is equal to (A) 0(zero) (B) (C) 00 (D) 0 f () ( ) ( ) ( )( ) ( ) C ~ C C ; C ~ C C 0 0 ( ) ( ) 0 f(00)0 4. If sin... 4 8 6 where < then the value of is (A) (B) (C) (D) sin 6 [ < ] - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (9)
WBJEEM - 05 (Answers & Hint) 4. The area of the region bounded by the curve y, its tangent at (, ) and -ais is (A) (B) 6 (C) 7 (D) 5 Y A(, ) O B(/,0) C(/0) X Equation of tangent at A is y. Area of shaded region d d 5 0 4 8 5. If log 0. ( )> log 0.04 (+5) then (A) < < 4 (B) < < (C) < < 4 (D) < < log 0. ( ) > log 5 0. log 0. ( ) > log 0. (+5) ( ) < (+5) 4 < 0 < < 4 (but > ) < < 4 6. The number of real roots of equation log e + e 0 (A) 0(zero) (B) (C) (D) ylog e y e Clearly one root is there - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (0)
WBJEEM - 05 (Answers & Hint) 7. If the verte of the conic y 4y 4 4a always lies between the straight lines + y and + y 0 then (A) < a < 4 (B) (C) 0 < a < (D) Verte of y 4y 4 4a is (a, ) So, (a + ) (a + 4 ) < 0 (a ) (a + ) < 0 a a a 8. Number of intersecting points of the conic 4 + 9y and 4 + y 4 is (A) (B) (C) (D) 0(zero) C y y : C : 4 4 9 C C Clearly no intersection 9. The value of for which the straight line y z may lie on the plane y 0 is (A) (B) 0 (C) (D) there is no such The line must be perpendicular to the normal to plane.. + ( + ) ( ) + ( ).0 0. Also the point (,, ) on the line, lies on the plane. 0,. So no value of is possible. 40. The value of cot 4 cot is (A) 8 (B) (C) 4 (D) - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) tan tan 4 4 tan tan 4 tan 4 tan 4 4. If the point ( cos, sin ), for (0, ) lies in the region between the lines + y and y containing the origin, then lies in (A) 0,, (B) 0, (C), (D), 4 Y +y X O X y Y ( cos, sin ) will lie on the circle, + y 4, for the marked region in the figure. So, 4. Number of points having distance 5 from the straight line y + 0 and a distance from the line + y 0 is (A) (B) (C) 4 (D) 5 Let the point be (a, b) So, a b 5 a b + ± 5... () 5 and a b a + b ±...() Any one of equations () combined with any one of equations () will yield a point 4 points - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 4 4. Let f : R R be defined as f (). Then the range of the function f() is 4 (A) 5, 5 (B) 5, 5 (C) 5,, 5 (D) 5, 5 Let, 4 y 4 (y ) + (y + ) + 4y 4 0 For to be real, discriminant of the above quadratic equation should be greater than or equal to zero 5 y 5 (y 0 y 0 which is valid) 44. The least value of + y + y + y + 8 for real numbers and y is (A) (B) 8 (C) (D) Ans : (No option is correct) + y + y + y + 8 (4 + y + 4y + 4 6y + 6) (y 4) ( y ) so, least value is at Correct option is not there. 5,y 4. 45. Let f : [, ] R be a continuous function such that f () assumes only irrational values. If f (A) f (0) 0 (B) f (C) f (D) f A continuous function assuming only irrational value must be constant function So, f () 46. The minimum value of cos sin sin for (0, /) is (A) (B) (C) (D) sin cos sin.cos, then As the epression remain unchanged by interchanging sin and cos so minimum is achieved for sin cos so in 0, 4 so, minimum value - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 47. Let n...,n. 6 0 n(n ) Then the value of lim n n is (A) / (B) /9 (C) /8 (D) 0 (zero) n n n n n(n ) n n so comparing n n A., n A is real constant, 4 now,. So 9 A. 9 so lim A n n 9 48. The variance of first 0 natural numbers is (A) /4 (B) 79/ (C) / (D) 99/4 Variance of first n natural numbers is n so, for n 0 0 4 49. A fair coin is tossed a fied number of times. If the probability of getting eactly heads equals the probability of getting eactly 5 heads, then the probability of getting eactly one head is (A) /64 (B) / (C) /6 (D) /8 Let, the fied number is n. So, n c c n 8 n 5 8 8 Thus, required probability is c 50. If the letters of the word PROBABILITY are written down at random in a row, the probability that two B-s are together is (A) / (B) 0/ (C) / (D) 6/ 0!!!!! - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (4)
WBJEEM - 05 (Answers & Hint) 5. The number of distinct real roots of sin cos cos cos sin cos cos cos sin 0 in the interval 4 4 is (A) 0 (zero) (B) (C) (D) > Determinant (sin + cos ) (sin cos ) 0 so, either tan or tan only one solution 5. Let,,..., 5 be 5 distinct numbers chosen from,,,..., 5. Then the value of ( ) ( ), ( )... ( 5 ) is (A) always 0 (B) 0 (zero) (C) always even (D) always odd Among,,,..., 5 ; one of them, say, k is. k 0 the product given is 0 5. Let [] denote the greatest integer less than or equal to Then the value of for which the function f sin, 0, 0 is continuous at 0 is (A) 0 (B) sin( ) (C) sin() (D) sin lim 0 sin () if f () is continuous at 0, then lim f() f(0) sin () 0 54. Let f() denote the fractional part of a real number. Then the value of 0 f ( )d (A) (B) 0 (zero) (C) (D) f ( )d d d 0 0 0 d d d d 0 0 0.d.d.d 0 - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (5)
WBJEEM - 05 (Answers & Hint) 55. Let S {(a, b, c) N N N : a + b + c, a b c} and T {(a, b, c) N N N : a, b, c are in A.P.}, where N is the set of all natural numbers. Then the number of elements in the set S T is (A) 6 (B) 7 (C) (D) 4 a + b + c and b a c a + c 4 and b 7 So, a can take values from to 6, when c ranges from to 8, or a b c 7 So, 7 triplets 56. Let y e and y e sin be two given curves. Then the angle between the tangents to the curves at any point of their intersection is (A) 0 (zero) (B) (C) (D) 4 e e sin sin at point of intersection. dy y e e d Again, y e dy sin e sin e cos d dy e d (when sin i.e. cos 0) So, slopes of tangents are equal at point of intersection 57. Area of the region bounded by y and y + is (A) 4 sq. units (B) sq. units (C) sq. units (D) sq. units A B C y O y + In figure, B (0, ) OB and OABC is square. So, side is of length units area OABC sq. units. - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (6)
WBJEEM - 05 (Answers & Hint) 58. Let d(n) denote the number of divisors of n including and itself. Then d(5), d(5) and d(640) are (A) in AP (B) in HP (C) in GP (D) consecutive integers 5 5 d (5) 9 5 5 d (5) 4 640 7 5 d (640) 8 6 9,, 6 are in G.P. 59. The trigonometric equation sin sin a has a real solution if (A) a > (B) < a < (C) a > (D) a / sin / / sin a / /4 sin a /4 a a a 60. If + i and 5 i are the roots of the equation ( +a + b) ( + c + d) 0, where a, b, c, d are real constants, then product of all roots of the equation is (A) 40 (B) 9 5 (C) 45 (D) 5 i and 5 i are other roots. So, Product is ( + i) ( i) 5 9 45 5 i 5 i CATEGORY - II (Q6 to Q70) Each question has one correct option and carries marks, for each wrong answer / mark will be deducted. 6. In a triangle ABC, C 90, r and R are the in-radius and circum-radius of the triangle ABC respectively, then (r + R) is equal to (A) b + c (B) c + a (C) a + b (D) a + b + c B a r b r a r r r A C b r a r + b r R (r + R) a + b - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (7)
WBJEEM - 05 (Answers & Hint) 6. Let be two distinct roots of a cos + b sin c, where a, b and c are three real constants and [0, ]. Then + is also a root of the same equation if (A) a + b c (B) b + c a (C) c + a b (D) c a tan b tan a c tan tan (c + a) tan / btan/ + (c a) 0 so, b a b (c + a) a so, c a 6. For a matri b c a b tan c a a c a is a root of the equation, b b c a 0 a A 0 0 0, if U, U and U are column matrices satisfying 0 AU, AU 0, AU 0 and U is matri whose columns are U, U and U Then sum of the elements of U is (A) 6 (B) 0 (zero) (C) (D) / ai b Let U i i i,, c i a a AU b 0, So a a b c 0, b, c a a AU b, a b c 0 So, a, b, c 4. Similarly, a, b, c So, U. So, sum of elements of U 4 is zero. - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (8)
WBJEEM - 05 (Answers & Hint) 64. Let f : N R be such that f() and f() + f() + f() +...... + nf(n) n(n + )f(n), for all nn, n, where N is the set of natural numbers and R is the set of real numbers. Then the value of f(500) is (A) 000 (B) 500 (C) /500 (D) /000 f() + f() + f() +... + nf(n) n(n + )f(n) f() + f() + f() +... + (n )f(n ) (n )nf(n ) Subtracting, nf(n) n(n + )f(n) n(n )f(n ) nf(n) (n )f(n ) Clearly, f(n) n. So, f(500) 000 65. If 5 distinct balls are placed at random into 5 cells, then the probability that eactly one cell remains empty is (A) 48/5 (B) /5 (C) 8/5 (D) /5 Required probability C 4 C C C 5 5 4 5 4 5 4 5 5 5 48 5 66. A survey of people in a given region showed that 0% were smokers. The probability of death due to lung cancer, given that a person smoked, was 0 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is 0.006, what is the probability of death due to lung cancer given that a person is a smoker? (A) /40 (B) /70 (C) /40 (D) /0 D S NS S person is smoker NS person is non smoker D death due to lung cancer P(D) P(S). P D S + P(NS).P D NS 0.006 0 D 80 D P P 00 S 00 0 S P D S 40. - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (9)
WBJEEM - 05 (Answers & Hint) 67. A person goes to office by a car or scooter or bus or train, probability of which are /7, /7, /7 and /7 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is /9, /9, 4/9, and /9 respectively. Given that he reached office in time, the probability that he travelled by a car is (A) /7 (B) /7 (C) /7 (D) 4/7 Required probability / d 68. The value of 7 is 7 7 9 7 8 5 8 7 7 9 7 9 7 9 7 9 (A) 0 4/ c (B) 0 /4 c (C) 5 4/ c d d I / 7/ 7/ (D) 0 5/ c Let, 5d t so, dt dt So, I c 7/ 4/ 5 t 0t 0 69. Let f : R R be differentiable at 0. If f(0) 0 and f(0), then the value of 4/ c lim () () ()... 05is 0 f f f f (A) 05 (B) 0 (zero) (C) 05 06 (D) 05 04 Applying L Hospital Rule, f () f () f ()... 05f (05) 05 06 lim 05 06 0 70. If and y are digits such that 7! 556y48096000, then + y equals (A) 5 (B) 6 (C) (D) Since 7! is divisible by 9 so sum of the digits 48 + + y must be divisible by 9. So, + y can be 5 or 6. Also 7! is divisible by so 0 + y must be multiple of or zero. The only possibility is y So, + y 5 - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (0)
WBJEEM - 05 (Answers & Hint) CATEGORY - III (Q7 to Q80) Each question has one or more correct option(s), choosing which will fetch maimum marks on pro rata basis. However, choice of any wrong option(s) will fetch zero mark for the question. 7. Which of the following is/are always false? (A) A quadratic equation with rational coefficients has zero or two irrational roots (B) (C) (D) A quadratic equation with real coefficients has zero or two non-real roots A quadratic equation with irrational coefficients has zero or two rational roots A quadratic equation with integer coefficients has zero or two irrational roots 7. If the straight line (a ) by + 4 0 is normal to the hyperbola y then which of the followings does not hold? (A) a >, b > 0 (B) a >, b < 0 (C) a <, b < 0 (D) a <, b > 0 Ans : (B,D) d Every normal to y must have positive slope as. So a 0 dy. b 7. Suppose a machine produces metal parts that contain some defective parts with probability 0.05. How many parts should be produced in order that probability of at least one part being defective is / or more? (Given log 0 95.977 and log 0 0.) (A) (B) (C) 5 (D) 4 Ans : (C,D) Probability of at least one part being defective probability of no part defective 0.95 n so n 0.95 n log0 95 log0 00 n so n 4, 5 74. Let : R R be such that ( ) () for all R. If is continuous at and (), then (A) () (B) () (C) is continuous only at (D) is continuous at all points Ans : (A,D) f f f f - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint)... f n n n f n n f n n Taking limit n, f() f(), as f() is continuous at f() is constant function d y dy 75. If cos and sin are solutions of the differential equation a0 a ay 0 d d, Where a 0, a, a are real constants then which of the followings is/are always true? (A) A cos + B sin is a solution, where A and B are real constants (B) (C) A cos 4 is a solution, where A is real constant A cos sin is a solution, where A is real constant (D) A cos Bsin 4 4 is a solution, where A and B are real constants Ans : (A,D) The general solution is y C cos + C sin where C and C are real constants 76. Which of the following statements is/are correct for 0? (A) cos / cos (B) cos /4 cos 4 (C) cos 5 cos 5/6 (D) 7 cos cos 7/8 6 Ans : (A,C) 8 n f( ) cos & g( ) cosn, 0 as 0 < n <, 0 n. So cosn ranges from to some positve value whereas cos n ranges from to 0. Also f ( ) g ( ) < 0. g( ) 0 f( ) / So, cos cos and cos 5 5 6 cos. 6 - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 77. Let 6 y y 44 represent a hyperbola. Then (A) length of the transverse ais is (B) length of each latus rectum is / (C) eccentricity is 9 / (D) equation of a directri is Ans : (A,B,C) 6( ) (y + ) 48 9 y so, 6 Length of transverse ais Length of L.R. eccentricity 9. equation of directri, 9 78. For the function f statements are true?, where [] denotes the greatest integer less than or equal to, which of the following (A) The domain is, (B) The range is 0 (C) The domain is,0, (D) The range is 0 Ans : (B,C) Domain is 0, Range is {,, 0} 79. Let f be any continuously differentiable function on [a,b] and twice differentiable on (a,b) such that f(a)f(a)0 and f(b) 0. Then (A) f(a) 0 (B) f() 0 for some a,b (C) f() 0 for some a,b (D) f() 0 for some a,b Ans : (B,C) Applying Rolle s Theorem on f() f(a) f(b) 0 so f() 0 for some c a,b again f(a) 0 f(c) so for some (a,c) i.e (a,b) f() 0. - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 ()
WBJEEM - 05 (Answers & Hint) 80. A relation ρ on the set of real number R is defined as follows: ρy if and only if y > 0. Then which of the followings is/are true? (A) ρ is refleive and symmetric (B) ρ is symmetric but not refleive (C) ρ is symmetric and transitive (D) ρ is an equivalence relation Ans : (B,C) : > 0 not true for 0 y y y so y > 0, y z so yz > 0. zy > 0, Hence z > 0 so z. - Regd. Office: Aakash Tower, Plot No.-4, Sector-, Dwarka, New Delhi-0075 Ph.: 0-476456 Fa : 0-47647 (4)