WBJEE - 7 Aswer Keys by, Kolkata Cetre MATHEMATICS Q.No. B A C B A C A B 3 D C B B 4 B C D D 5 D A B B 6 C D B B 7 B C C A 8 B B A A 9 A * B D C C B B D A A D B B C B 3 A D D D 4 C B A A 5 C B B B 6 C C B A 7 A A B D 8 D B B B 9 C B D D B A B C * C B B C D A B 3 A A A A 4 B B D C 5 D B B D 6 B B D B 7 B B B A 8 C D D C 9 A B A C 3 B B B C 3 B A A A 3 A A D D 33 C D B C 34 D B D B 35 A D C * 36 B B B C 37 B D B A 38 B A A B 39 B B C D 4 D A D B 4 B D B B 4 B B A C 43 A D C A 44 A C C B 45 D B C B 46 B B A A 47 D A D C 48 B C C D 49 D D B A 5 A B * B 5 C C B B 5 A B C C 53 D B A C 54 C A C A 55 B A B D 56 B C C C 57 A B C B 58 A C A B 59 C A D A 6 B C C A 6 C B B C 6 A C B B 63 C C A C 64 B A A A 65 C D C C 66 B C B,C C,D 67 B,C B,D A,C A,C 68 C B,C C C 69 B,D A,C C,D B 7 B,C C A,C B,C 7 A,C C,D C C 7 C A,C B B,D 73 C,D C B,C B,C 74 A,C B C A,C 75 C B,C B,D C * Either B or D.
WBJEE - 7 (Aswers & Hit) Code - CATEGORY - I (Q to Q5) Oly oe aswer is correct. Correct aswer will fetch full marks. Icorrect aswer or ay combiatio of more tha oe aswer will fetch 4 marks. No aswer will fetch marks.. Trasformig to parallel aes through a poit (p, ), the euatio + 3y + 4y + + 8y + 5 = becomes + 3y + 4y =. The p =, = 3 p =, = 3 p = 3, = 4 p = 4, = 3 4p + 3 + = 3p + 8 + 8 = p =, = 3. Let A(, 3) ad B (, ) be two agular poits of ABC. If the cetroid of the triagle moves o the lie + 3y =, the the locus of the agular poit C is give by + 3y = 9 3y = 9 3 + y = 5 3 y = 3 t G t, 3, = 3t = 3 t, + 3y = 9 3. The poit P (3, 6) is first reflected o the lie y = ad the the image poit Q is agai reflected o the lie y = to get the image poit Q. The the circumcetre of the PQQ is (6, 3) (6, 3) (3, 6) (, ) As : P(3, 6) ANSWERS & HINT for WBJEE - 7 SUB : MATHEMATICS Q(6,3) ( 3, 6)Q - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 ()
WBJEE - 7 (Aswers & Hit) 4. Let d ad d be the legths of the perpediculars draw from ay poit of the lie 7 9y + = upo the lies 3 + 4y = 5 ad + 5y = 7 respectively. The d > d d = d d < d d = d 5. The commo chord of the circles + y 4 4y = ad + y = 3 subteds at the origi a agle eual to 3 As : 4 This commo chord is passig through the cetre of the st circle. Therefore it will form a agle of 9º at the circumferetial poit (, ). 6. The locus of the mid-poits of the chords of the circle + y + y = which make a agle of 9º at the cetre is + y y = + y + y = + y + y = + y + y = 6 si45º = OP OP =, Cetre : (, ) B (,) O 9º 45º 45º p(h,k) A y 7. Let P be the foot of the perpedicular from focus S of hyperbola o the lie b ay = ad let C be the a b cetre of hyperbola. The the area of the rectagle whose sides are eual to that of SP ad CP is ab ab (a b ) a b y b y P C S(ae,o) Area = SP.CP = a.b 8. B is a etremity of the mior ais of a ellipse whose foci are S ad S. If SBS is a right agle, the the eccetricity of the ellipse is 3 3 - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 ()
WBJEE - 7 (Aswers & Hit) y B(,b) S ( ae,) O S (ae,) b = ae, e 9. The ais of the parabola + y + y 5 + 5y 5 = is + y = + y = y + = y = ( + y) = 5 5y + 5 ( + y ) = 5( y + ) Ais is + y =. The lie segmet joiig the foci of the hyperbola y + = is oe of the diameters of a circle. The euatio of the circle is + y = 4 + y = + y = + y = y + =, Foci =,, Cetre = (, ), Radius = Euatio of circle + y =. The euatio of the plae through (,, 3) ad (,, ) ad parallel to X-ais is y z + = y z = y + z = y + z + = As : y z 3 3 y+z+=. Three lies are draw from the origi O with directio cosies proportioal to (,, ), ( 3, ) ad (,, 3). The three lies are ot coplaar coplaar perpedicular to each other coicidet = (Coplaar) - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (3)
WBJEE - 7 (Aswers & Hit) 3. Cosider the o-costat differetiable fuctio f of oe variable which obeys the relatio ad f(5)=, the f( 5) is f f y = f( y). If f() = p p p p f() = a k f() = ka k l a k l a = p, ka 5k l a = a 5k = p f( 5) = k.a 5k l a p 4. If f() = log 5 log 3, the f(e) is eual to e log e 5 e log e 3 e log 5 e e log 3 e f() = log 5 l + log 5 log 3 e f() =.. l 5 l f(e) = e l 5 5. Let F() = e, G() = e ad H() = G(F()), where is a real variable. The dh d at = is e e H() = e e H() = e e.e H () = e 6. If f() = k, k, the the value of lim f 3f f 4 k k 3k 4k By L Hospital Rule is - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (4)
WBJEE - 7 (Aswers & Hit) 7. If y = e m si, the ( ) d y dy ky, where k is eual to d d m m 8. The chord of the curve y = + a + b, joiig the poits where = ad =, is parallel to the taget to the curve at abscissa = a b a b 3 3 As : +a = (+) + a = 9. Let f() = 3 + + 9 + 7 + 5 + 3 + + 9. The f() = has 3 real roots oly oe positive ad oly two egative real roots ot more tha oe real root has two positive ad oe egative real root f() = has o real root. Let f() = si iterval [, ] p, if, (p,, ). The Lagrage s mea value theorem is applicable to f() i closed, if for all p, oly whe p > oly whe p < for o value of p, p > lim f. lim si ta is is is does ot eist As : Either B Or D lim si ta ta Not i the domai hece does ot eist, But if approached like ta lim si lim si cos log d F c, where c is a arbitrary costat. Here F() =. coslog silog coslog silog cos log si log si log cos log - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (5)
WBJEE - 7 (Aswers & Hit) cos log d F c, Let log = t, I e t cos t dt e t cos t e t si t I, t t e cost e sit I cos log si log 3. d( ) 4 is 3 ta c ta c log e c log e c dividig by, / d, Let / 3 t, dt ta c t 9 si 4. Let I 8 d. The I 9 I 7 I 5 I 7 9 9 si d si 8 8 d 9 d (as si ) 8 9 =9 8 < 7 8 8 8 d as for 9 5. Let I d ad I d, where [] ad {} are itegral ad fractioal parts of ad eual to As : 3 I d d d d... d, = +++3+...+(-)= I d d I, I /I = 6. The value of lim... is,. The I /I is 4 4 4 lim... = lim... = lim r r d =/4 - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (6)
WBJEE - 7 (Aswers & Hit) 7. The value of the itegral e d is less tha is greater tha is less tha or eual to lies i the closed iterval [,e] e d 8. e d e e e (e ) e e [] d = 3 99 e d e d e d... e d 99 = e d e d e d... e d = (e ) dy is d 9. Solutio of y a a beig a cos tat y y c ta,c is a arbitrary costat y = a ta c, c is a arbitrary costat a a y ta,c is a arbitrary costat y = ta(+c), c is a arbitrary costat a c dy ( y) a d dy dz [Put + y = z ] d d dz z a z a z d dz d z z dz d z a ta C z a a y y c ta, c is arbitrary costat a a - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (7)
WBJEE - 7 (Aswers & Hit) 3. The itegratig factor of the first order differetial euatio dy y is d e dy y, I.F = d e d, e d d e l, = e / 3. I a G.P. series cosistig of positive terms, each term is eual to the sum of et two terms. The the commo ratio of this G.P. series is 5 5 5 5 t r = t r+ + t r+ ar = ar + ar + = r + r r = 3. If (log 5 )(log 3)(log 3 y) = log 3, the y euals 5 5 5/3 43 5 log log3 logy 3 log5 log log3 logy = 3log5 y = 5 3 = 5 33. The epressio i i euals i + i + i + ( i) i ( i) i i ( i) = ) i( = i = i = i + 34. Let z = + iy, where ad y are real. The poits (, y) i the X-Y plae for which z i z i a straight lie a ellipse a hyperbola a circle As : Let z = + iy is purely imagiary lie o - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (8)
WBJEE - 7 (Aswers & Hit) z i ( i(y )) ( i(y )) z i ( i( y)) ( i( y)) Re z i z i (y ) ( y) + y = 35. If p, are odd itegers, the the roots of the euatio p + (p + ) + = are ratioal irratioal o-real eual D = (p + ) 8p = (p ) always a perfect suare 36. Out of 7 cosoats ad 4 vowels, words are formed each havig 3 cosoats ad vowels. The umber of such words that ca be formed is 5 5 34 7 C 3 4 C 5! = 5 37. The umber of all umbers havig 5 digits, with distict digits is 99999 9 9 P 4 P 5 9 P 4 9 9 P 4 38. The greatest iteger which divides (p + )(p + )(p + 3)...(p + ) for all p ad fied is p!! p This is product of cosecutive atural umbers, so it will always be divisible by! 39. Let (( + ) + ) 9 = a + a + a +... + a 8 8. The a + a +... + a 8 = a + a 3 +... + a 7 a + a +... + a 8 is eve a + a +... + a 8 is divisible by 9 a + a +... + a 8 is divisible by 3 but ot by 9 a + a + a 4 +... + a 8 = 9 3 eve 4. 8 3y 5z The liear system of euatios 5 8y 3z has 3 5y 8z oly zero solutio oly fiite umber of o-zero solutios o o-zero solutio ifiitely may o-zero solutios As : 8 3 5 D 5 8 3 3 5 8 D = D = D 3 = ifiite solutios - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (9)
WBJEE - 7 (Aswers & Hit) 4. Let P be the set of all o-sigular matrices of order 3 over ad Q be the set of all orthogoal matrices of order 3 over. The, P is proper subset of Q Q is proper subset of P Neither P is proper subset of Q or Q is proper subset of P P Q = the void set Q is the proper subset of P 3 4. Let A 3, B 5. The all solutios of the euatio det (AB) = is,,,, 4,,,, 4, 3, 4,, 3 7 3 6 det AB 8 ( + ) ( 4) ( ) = =,,, 4 cos 43. The value of det A, where A cos cos lies cos i the closed iterval [, ] i the closed iterval [, ] i the ope iterval (, ) i the ope iterval (, ) det = ( + cos ) A, 44. Let f : be such that f is ijective ad f() f(y) = f( + y) for, y. If f(), f(y), f(z) are i G.P, the, y, z, are i A.P always G.P always A.P depedig o the value of, y, z G.P depedig o the value of, y, z f() = a a, a y, a z G.P, y, z A.P 45. O the set of real umbers we defie Py if ad oly if y. The the relatio P is refleive but ot symmetric symmetric but ot refleive trasitive but ot refleive refleive ad symmetric but ot trasitive As : (, ), (, ) satisfies the relatio y but (, ) does t satisfy relatio y. - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 ()
WBJEE - 7 (Aswers & Hit) 46. O, the relatio be defied by y holds if ad oly if y is zero or irratioal. The is refleive ad trasitive but ot symmetric. is refleive ad symmetric but ot trasitive. is symmetric ad trasitive but ot refleive. is euivalece relatio 47. Mea of observatios,,..., is. If a observatio is replaced by the the ew mea is ( ) ( ) As : New Mea = i i 48. The probability that a o leap year selected at radom will have 53 Sudays is /7 /7 3/7 49. The euatio si (si + cos ) = k has real solutios, where k is a real umber. The k 3 k 3 k 3 As : si cos = k k k k 5. The possible values of, which satisfy the trigoometric euatio ta ta are 4. + + = 4 + = - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 ()
WBJEE - 7 (Aswers & Hit) CATEGORY - II (Q5 to Q65) Oly oe aswer is correct. Correct aswer will fetch full marks. Icorrect aswer or ay combiatio of more tha oe aswer will fetch marks. No aswer will fetch marks. 5. O set A = {,, 3}, relatios R ad S are give by R = {(, ), (, ), (3, 3), (, ), (, )} S = {(, ), (, ), (3, 3), (, 3), (3, )} The R S is a euivalece relatio R S is refleive ad trasitive but ot symmetric R S is refleive ad symmetric but ot trasitive R S is symmetric ad trasitive but ot refleive RUS = {(,), (,), (3,3), (,), (,), (,3), (3,)} 5. If oe of the diameters of the curve + y 4 6y + 9 = is a chord of a circle with cetre (, ), the radius of this circle is 3 r (,) 5 (,3) r = 5 4 3 53. Let A (, ) ad B (, ) be two poits. A poit M moves i the plae i such a way that MBA = MAB. The the poit M moves alog a straight lie a parabola a ellipse a hyperbola As : 54. If f() = t dt, the for ay, f() is eual to + f() = t dt, = ( + ) 55. Let for all >, f() = lim, the f() + f f(y)=f()+f(y) f(y)= f(y) + yf() f(y) = f() + y f(y) - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 ()
WBJEE - 7 (Aswers & Hit) f() = lim log 56. Let = cos d, the = = = = = cos d = si d = 57. The area of the figure bouded by the parabolas = y ad = 3y is 4 3 suare uits 3 suare uits 3 7 suare uits 6 7 suare uits Curves itersect at (, ±) Area = 4 y dy 3 3 58. Tagets are draw to the ellipse is 7 s. uits y at the eds of both latus rectum. The area of the uadrilateral so formed 9 5 3 s. uits 5 4 s. uits 45 s. uits Euatio of taget i uadrilateral : y 9 3 59. The value of K i order that f() = si cos K + 5 decreases for all positive real values of is give by K < K K > K < f() = cos + si k (<) k > cos + si ma. (cos + si) = k > 6. For ay vector, the value of i j k is eual to 3 4 - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (3)
WBJEE - 7 (Aswers & Hit) = si si si = 6. If the sum of two uit vectors is a uit vector, the the magitude of their differece is uits uits 3 uits 5 uits a b a b a.b 3 6. Let ad be the roots of + + =. If be positive iteger, the + is cos 3 si 3 cos 3 si 3 i i 3 3 e e cos 3 63. For real, the greatest value of 4 4 9 is 4 4 y 4 9 or, (y ) (7y 3) or, 3 y 7 64. Let A =. The for positive iteger, A is A = B + I A = - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (4)
WBJEE - 7 (Aswers & Hit) 65. Let a, b, c be such that b(a+c) If a a a a b c b b b a b c c c c ( ) a ( ) b ( ) c =, the the value of is ay iteger zero ay eve iteger ay odd iteger A = A T or, a a a ( ) a a a b b b ( ) b b b ( + ) is eve or is eve c c c ( ) c c c CATEGORY - III (Q66 to Q75) Oe or more aswer(s) is (are) correct. Correct aswer(s) will fetch full marks. Ay combiatio cotaiig oe or more icorrect aswer will fetch marks. Also o aswer will fetch marks. If all correct aswers are ot marked ad also o icorrect aswer is marked the score = umber of correct aswer marked actual umber of correct aswers. 66. Let f : be twice cotiuously differetiable. Let f() = f() = f() =. The f() for all f(c) for some c f() if f() > for all Applyig Rolle s theorem twice f() = for some [, ] 67. If f() =, beig a o-egative iteger, the the values of for which f(+) = f() + f() for all, > is 5 As : (B, C) 68. Le f be a o-costat cotiuous fuctio for all. Let f satisfy the relatio f() f(a ) = for some a +. The a d I = f() is eual to a a 4 a f(a) a a a d d f()d I f() f(a ) f() I= a d I= a 69. If the lie a + by + c =, ab, is a taget to the curve y =, the a >, b < a >, b > a <, b > a <, b < As : (B,D) dy d - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (5)
WBJEE - 7 (Aswers & Hit) 7. Two particles move i the same straight lie startig at the same momet from the same poit i the same directio. The first moves with costat velocity u ad the secod starts from rest with costat acceleratio f. The they will be at the greatest distace at the ed of time u f from the start they will be at the greatest distace at the ed of time u f from the start their greatest distace is u f their greatest distace is u f As : (B, C) S = ut ft 7. The comple umber z satisfyig the euatio z i = z+ = is + i + i i As : (A,C) Im (, ) (,) Re 7. O, the set of real umbers, a relatio is defied as ab if ad oly if + ab >. The is a euivalece relatio is refelive ad trasitive but ot symmetric is refleive ad symmetric but ot trasitive is oly symmetric 73. If a, b {,, 3} ad the euatio a + b + = has real roots, the a > b a b umber of possible ordered pairs (a, b) is 3 a < b As : (C, D) (, ) (, 3) (, 3) 74. If the taget to y = 4a at the poit (at, at) where t > is a ormal to y = a at the poit (a sec, a ta ), the t = cosec t = sec t = ta t = cot As : (A, C) y yt = at or, t = cosec or t = ta asec ata 75. The focus of the coic 6 + 4y + = is (, 3) (3, ) (3, ) (, 4) ( 3) = 4 (y ) - Regd. Office: Aakash Tower, 8, Pusa Road, New Delhi-5, Ph.: -4763456 Fa : -476347 (6)