WBJEE MATHEMATICS

Size: px

Start display at page:

Download "WBJEE MATHEMATICS"

Francine Cross
5 years ago
Views:

1 WBJEE - 06 MATHEMATICS Q.No. 0 A C B B 0 B B A B 0 C A C C 0 A B C C 05 A A B C 06 B C B C 07 B C A D 08 C C C A 09 D D C C 0 A C A B B C B A A C A B D A A A B B D C 5 B C C C 6 C A B B 7 C A A B 8 C B B A 9 C B A C 0 D C C C A D C A C A C B B B D A A A C A 5 B D C D 6 A B C C 7 C B A B 8 C C B A 9 B C C B 0 B C A A A C A C C D B C C A B C A C C D 5 B B D C 6 A A A C 7 A B B C 8 D A A A 9 C C D B 0 B C B C A B B A B B C A A A C B C C C B 5 C C C C 6 C A D D 7 D B A A 8 C A C B 9 C A B A 50 C D A D 5 B A A D 5 D C A B 5 B B A A 5 C B C A 55 D B B D 56 B D B A 57 A B B A 58 A C D A 59 D D B C 60 A B C B 6 A A D B 6 A A B B 6 C D A D 6 B A A B 65 B A D C 66 B,D A,B B,D A,B 67 B,D A,C A,B B,C 68 A,B B,D A,C B,D 69 B,C B,D A,C A,B 70 B,D A,B A,B A,C 7 A,B B,C A,C A,C 7 A,C B,D B,D A,B 7 A,C A,B B,D A,C 7 A,B A,C A,B B,D 75 A,C A,C B,C B,D For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

2 WBJEE - 06 (Aswers & Hit) Code- CATEGORY - I (Q to Q50) Oly oe aswer is correct. Correct aswer will fetch full marks. Icorrect aswer or ay combiatio of more tha oe aswer will fetch ¼ marks.. Let A ad B two evets such that ANSWERS & HINT for WBJEE - 06 SUB : MATHEMATICS 7 P A B, P(A B) ad P the A ad B are idepedet A ad B are mutually eclusive As : A P B 6 Hit : PB 7 P P(A B) P P P(A B) P 9 5 P P P(A B) A, B are idepedet. The value of cos 5 cos 7 si7 is B P A As : si7 cos 7 cos Hit : cos5 cos 7 si7 = 8. The smallest positive root of the equatio ta = 0 lies i (0, /) (/, ),, As : For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

3 WBJEE - 06 (Aswers & Hit) Hit : ta = 0 ta = Solutios are abscisse of poits of itersectio of the curves y = ta ad y =. It is clearly visible that solutio lies i,. y = O = = =. If i a triagle ABC,AD, BE ad CF are the altitudes ad R is the circumradius, the the radius of the circumcircle of DEF is R As : R Hit : Let, circumradius of DEF be R. We kow, FDE = 80 A ad FE = R sia Now, by sie rule i DEF, R Noe of these EF R sia R R si FDE si 80 A R = 5. The poits ( a, b), (a, b), (0, 0) ad (a, ab), a 0, b 0 are always lie o this lie. Hece, colliear colliear vertices of a parallelogram vertices of a rectagle lie o a circle As : Hit : The straight lie through (a, b) ad ( a, b) is b = ay. Obviously, (0, 0) ad (a, ab) always lie o this lie 6. The lie AB cuts off equal itercepts a from the aes. From ay poit P o the lie AB perpediculars PR ad PS are draw o the aes. Locus of mid-poit of RS is a y + y = a + y = a y = a As : Hit : Equatio of AB is + y = a Let, co-ordiates of the mid-poit be (h, k). So, R ad S are (h, 0) ad (0, k). Therefore, P must be (h, k). B F A D E C S P Now P lies o B. (h,k) R +y=a For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

4 WBJEE - 06 (Aswers & Hit) h + k = a + y = a y = 0, 5 + y = 0, y + = 0 are the three sides of a triagle. The area of the triagle is 6 square uit 9 square uit square uit 7 square uit As : Hit : If AB deotes : +8y = 0 () BC deotes : 5+y = 0 () ad CA deotes : y+ =0 () The solvig equatios (), () ad (), we get A (,), B (6,) ad C (,7). Hece, area of ABc is 9 square uits 8. The lie through the poits (a, b) ad ( a, b) passes through the poit (, ) (a, b) (a, ab) (a, b) As : ** ** Note : The poit i Optio D is already i the questio. Hit : The lie through (a, b) ad ( a, b) has the equatio b = ay. Hece, (a, ab) is always o the lie. 9. The locus of the poit of itersectio of the straight lies y K ad y, where k is a o-zero real a b a b k variable, is give by a straight lie a ellipse a parabola a hyperbola As : Hit : Let the poit itersectio be ( ). so, k ad a b a b k a b Locus : a y which is equatio of a hyperbola. b 0. The equatio of a lie parallel to the lie + y = 0 ad touchig the circle + y = 9 i the first quadrat is + y = 5 + y = 5 + y = 9 + y = 7 As : Hit : Let, the equatio be + y = k the, k y. By coditio of tagecy + y = 5 touches i the first quadrat. k 9 k 5 +y=5 +y= 5 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

5 WBJEE - 06 (Aswers & Hit). A lie passig through the poit of itersectio of +y= ad y= makes a agle ta with the -ais. It itersects the parabola y =( ) at poits (, y ) ad (, y ) respectively. The is equal to 6 9 As : 9 Hit : Poit of itersectio of +y= ad y= is (, ) The lie though this makig a agle is y 5 5 y Puttig y i y =( ), we have ad 9 9 = 9 ta with the ais. The equatio of auiliary circle of the ellipse 6 +5y + 00y=8 is y y 0 0 y y 0 y 00 As : y 5 y Hit : Simplifyig the give equatio, we have the ellipse as : 5 6 So, the auilliary circle is y 5 y y 0 0. If PQ is a double ordiate of the hyperbola eccetricity e satisfies a y such that OPQ is equilateral, O beig the cetre. The the b e e e e As : Hit : OPQ is equilateral, OP = PQ P (asec, bta ) a sec b ta b ta O Q (asec, bta ) For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

6 WBJEE - 06 (Aswers & Hit) a sec b ta si Now, a b si a b b a b a e e. If the verte of the coic y y = a always lies betwee the straight lies; +y= ad +y =0 the <a< <a< 0<a< a As : y a Hit : y y a verte : (a, ) Clearly, a a 5. A straight lie joiig the poits (,,) ad (0,0,0) itersects the plae +y+z=0 at (,, 5) (,, ) (,, 5) (,, 6) As : Hit : D.R. of lie (,,) let poit be (k, k, k) k+k+k = 0 5k = 0 k = Hece poit : (,,) 6. Agle betwee the plaes +y+z=6 ad y+z=9 is 6 As : Hit : +y+z = 6; y + z = 9 Agle betwee the plaes = agle betwee the ormals : cos. = cos cos 6 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

7 WBJEE - 06 (Aswers & Hit) dy the the value of d at = 0 is 7. If y... 0 As : Hit : y... l y l l... l dy... y d dy y.... d dy. d 0 8. If f() is a odd differetiable fuctio defied o, such that f() =, the f( ) equal to 9. 0 As : Hit : Let f() = f( ) f() = f( ). ( ) = f( ) f ( ) = f () = lim is does ot eist is As : is/ Hit : lim lim 0. If e log log f ta ta loge 6log the the value of f() is 0 As : log log log 6log Hit : f ta ta For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

8 WBJEE - 06 (Aswers & Hit) let, log = ta f = ta ta cos ta t = f() = 0. log d is equal to log c log c log c log c As : Hit : log d Let log = z d dz I = z dz = zdz. = z. c. log c = f f log d is equal to f()+c log + c f() + c + c As : Hit : f f log d Let g() = f() g() = f() + f()log = = (f() + f() log) g d = g() + c = f()+c For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

9 WBJEE - 06 (Aswers & Hit). 0 log d 0 Noe of these As : Hit : Let = log d log d 0 0 = 0 log d = = 0 = 0 b a b f d f a b d a.... The value of lt is As :... Hit : lt = lt... = r lt r = d dy If the solutio of the differetial equatio y e d be, y = e () + c the () is equal to + As : d Hit : If = e e y = e d e c 6. The order of the differetial equatio of all parabolas whose ais of symmetry alog -ais is Noe of these As : Hit : y = a( b) For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

10 WBJEE - 06 (Aswers & Hit) 7. The lie y = + is taget to the ellipse + y =. The is 5 6 As : Hit : = The area eclosed by y = 5 ad y = is 5 5 sq. uits sq. uits 5 sq. uits 5 sq. uits As : Hit : 5 5 d 5 d Area = 5 9. Let S be the set of poits whose abscissas ad ordiates are atural umbers. Let P S such that the sum of the distace of P from (8,0) ad (0,) is miimum amog all elemets i S. The the umber of such poits P i S is 5 As : Hit : Sum of distaces will be miimum if P, (8,0) ad (0,) will colliear y y = 8 (,y) (,9), (,6), (6,) 0. Time peirod T of a simple pedulum of legth l is give by T = l If the legth is icreased by %, the a g approimate chage i the time period is % % % Noe of these As : For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

11 WBJEE - 06 (Aswers & Hit) Hit : dt. d g T = dt.. d g 00 = T. g T T 00 %. The cosie of the agle betwee ay two diagoals of a cube is As :. If is a positive real umber differet from such that log a, log b, log c are i A.P., the a c b b ac log a b c As : ac Noe of,, are correct Hit : log b = log a + log c = log a log c log ac log b log alog c a log b log c log ac log b.log ac log a c ac log a b. If a, are real umbers ad a <, <, the + (+a) + (+a+a ) +... is equal to a a a a a a As : a a Hit :... a a.... if log 0. ( ) < log 0 09 ( ), the lies i the iterval =. a (, ) (, ) (, ) Noe of these As : Hit : log 0 ( ) < log (0.) ( ) log 0 ( ) < log (0 ) ( ) ( ) > (0. < ) ( ) ( ) > 0 <, > > ( < ) 5. The value of i i,i, is i i 0 As : Hit : i i i For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

12 WBJEE - 06 (Aswers & Hit) 6. If z, z, z are imagiary umbers such that z = z = z = z z z the z +z +z is Equal to Less tha Greater tha Equal to As : Hit : z.z z z z z z z z z z z z z 7. If p, q are the roots of the equatio + p + q = 0, the p =, q = p = 0, q = p =, q = 0 p =, q = As : Hit : p + q=0 p=, q = q(p+q+)=0 8. The umber of values of k for which the equatio + k = 0 has two distict roots lyig i the iterval (0, ) are Three Two Ifiitely may No values of k satisfies the requiremet As : Hit : f(0) > 0 f() > 0 K > ad D > 0 K 9 so < K < 9 9. The umber of ways i which the letters of the word ARRANGE ca be permuted such that the R s occur together is As : Hit : A A RR N G E. Number of arragemet = 6 0. If, 5 6 C C C, the the value of r equals to r r r 5 As :. For +ve iteger, + is always divisible by As :. I the epasio of ( ) ( )... ( 8), the coefficiet of 7 is As : Hit : Coefficiet of 7 is: ( ) =. + C cos + C cos C cos equals 89 7 cos cos cos cos cos As : For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

13 WBJEE - 06 (Aswers & Hit) Hit : Re ( C 0 + C e i +...) = Re ( + e i ) = Re (cos + + isi ) = cos cos. If, y ad z be greater tha, the the value of log y log z log log z y log log y z z y is log. logy. log z log + logy + log z 0 {(log ).(logy).(logz)} As : Hit : log logy logz log log log log logy logz log y logy log y log logy logz logz logz logz Takig log, logy, logz commo from R, R, R all rows are idetical. So =0 5. Let A is a matri ad B is its adjoit matri. If B = 6, the A = ± ± ±8 ± As : Hit : Adj = A = 6 A = ± 8 6. Let cos si Q ad the Q si cos is equal to As : 0 cos si Hit : If Q si cos, Q Q, Q / 0 = cos / si / si / cos / / / / /, Q / 0 7. Let R be a relatio defied o the set Z of all itegers ad Ry whe + y is divisible by. The R is ot trasitive R is symmetric oly R is a equivalece relatio R is ot a equivalece relatio As : 8. If A 5 : N ad B 6 : N, the A = B A B A B B A As : Hit : 5 6k, k z, A is a set of some multiple of 6 while set B is the set of all cosecutive multiple of If the fuctio f : R is defied by f() = ( +) 5, the f is oe-oe but ot oto oto but ot oe-oe either oe-oe or oto both oe-oe ad oto As : For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

14 WBJEE - 06 (Aswers & Hit) Hit : f 5 Sice f() is eve fuctio hece ot oe oe ad f() > 0 R hece ot oto 50. Stadard Deviatio of observatios a, a, a...a is. The the stadard deviatio of the observatios a, a,... a is As : 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 ½ marks. 5. The locus of the midpoits of chords of the circle +y = which subteds a right agle at the origi is / (0,0) / M (h,k) As : y y y = 0 y = 0 h k Hit : si /, h + k = / 5. The locus of the midpoits of all chords of the parabola y = a through its verte is aother parabola with directri P(at,at) (0,0) M(h,k) = a = a = 0 As : a Hit : h = at, k = at t k / a, k h a, a y a, Equatio of its directri = a/ 5. If [] deotes the greatest iteger less tha or equal to, the the value of the itegral 0 d equals As : Hit : 0 0 d 0d d / 7 / For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

15 WBJEE - 06 (Aswers & Hit) 5. The umber of poits at which the fuctio f() = ma {a, a +, b},, 0 < a < b caot be differetiable y b -a a 0 As : Hit : Possible graph of f() is as show. There are to sharp tur, Hece f() caot be differetiable at two poit 55. For o-zero vectors a ad b if a b a b, the a ad b are Colliear Perpedicular to each other Iclied at a acute agle Iclied at a obtuse agle As : Hit : a b a b a b a b a b a b cos a b a b cos, (where is a agle betwee a ad b vector a b cos 0, cos 0, is a obtuse agle 56. Geeral solutio of As : dy y by acos, d 0 < < is y = a(b si + cos) + ce b (b + )y = a(si + bcos) + ce b (b + )y = a(si + bcos) + ce b y = a(bsi + cos) + ce b Here c is a arbitrary costat Hit : Let y = z dy dz y d d dz bz acos d IF = b d e = eb z.e b = b a cos.e.d For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

16 WBJEE - 06 (Aswers & Hit) y e b = a (si b bcos )e c b (b + )y = a(si + bcos) + ce b 57. The poits of the ellipse 6 + 9y = 00 at which the ordiate decreases at the same rate at which the abscissa icreases is/are give by 6 6, &, 6 6, &,, &, , &, As : Hit : y (5cos, 0 si) = 5cos, y = 0 si d 5si d, dy 0 cos d d dy d d 5si = 0 cos ta = / cos = /5 or /5 si = /5 or /5 6 Poits are, ad 6, 58. The letters of the word COCHIN are permuted ad all permutatio are arraged i a alphabetical order as i a Eglish dictioary. The umber of words that appear before the word COCHIN is As : Hit : COCHIN C + ways /ways =! = If the matri A = , the A 0 = a a 0, N where b 0 a a =, b = a =, b = a =, b = a =, b = As : A 0 0 A 0 0 Hit : 0 0 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

17 WBJEE - 06 (Aswers & Hit) 60. The sum of terms of the followig series; is ( ) ( ) As : Hit : t r = (r ) S = tr = 8 r r 6 r r = ( ) r r r r 6. If ad are roots of a + b + c = 0 the the equatio whose roots are ad is a (b ac) + c = 0 a + (b ac) + c = 0 a + (b + ac) + c = 0 a + (b + ac) + c = 0 As : Hit : Let y = = y puttig y i the give equatio ay + b y + c = 0 b y = ay c b y = a y + c + acy a y (b ac)y + c = 0 So the required quadratic equatio is a (b ac) + c = 0 6. If is a imagiary cube root of uity, the the value of ( )( ) + ( )( ) ( )( )( ) is ( ) As : ( ) ( ) ( ) Hit : (r )(r )(r ) = (r ) = ( ) ( ) r r = 6. If C r = 6, C r = 8 ad C r+ = 6 the the value of C 8 is As : ( ) Hit : r r 6...() r r 8...() r r 6...() () () gives r 6 r 8 8r = 6 6r + 6 or 0r = () () () gives r 8 6r + 6 = 8 8r or 0r = (5) r 6 Solvig () ad (5) = 9, r = So C 8 = 9 C 8 = 9 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

18 WBJEE - 06 (Aswers & Hit) 6. I a group males ad 6 females, 8 ad of the males ad females respectively are aged above 0 years. The probability that a perso selected at radom from the group is aged above 0 years, give that the selected perso is female, is 7 As : Hit : Here out of 6 females are aged above 0 ad are aged below 0. So probability of perso aged above give female perso = 65. The equatio y + y = 0 represets As : a hyperbola ad two straight lies a straight lie a parabola ad two straight lies a straight lie ad a circle Hit : y + y = 0 ( y) + ( y) = 0 ( + )( y) = 0 So oly possibility is = y as + 0 So it represets a straight lie. CATEGORY - III (Q66 to Q75) Oe or more aswer(s) is (are) correct. Correct aswer(s) will fetch marks. Ay combiatio cotaiig oe or more icorrect aswer will fetch 0 marks. If all correct aswers are ot marked ad also o icorrect aswer is marked the score = umber of correct aswers marked / actual umber of correct aswers. 66. If the first ad the (+) th t erms of a AP, GP ad HP are equal ad their th terms are respectively a, b, c the always a = b = c a b c a + c = b ac b = 0 As : (B, D) Hit : There seems to be a pritig mistake here If there are ( ) terms istead of ( + ) terms the th terms of the A.P., G.P. ad H.P. are the A.M., G.M. & H.M of the first ad the last terms. So, a b c & ac b (B, D) otherwise if there are ( + ) terms the the th terms should be i decreasig order of A.P., G.P. & H.P. i.e. a b c. 67. The coordiates of a poit o the lie + y + = 0 which is at a distace 5 uit from the lie + y + = 0 are (, ) (, ) (0, ) (, 0) As : (B, D) Hit : Let (t, t ) be a parametric poit of the lie + y+ = 0 Distace of (t, t ) from + y + = 0 is For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

19 WBJEE - 06 (Aswers & Hit) t ( t ) 5 t = t + =, so t = or t = possible co-ordiates are (, 0) & (, ) 68. If the parabola = ay makes a itercept of legth 0 uit o the lie y = the a is equal to As : (A, B) Hit : (, y ) (, y ) y = + Solvig = ay with y =, = a( + ) a a = 0 Let & are the roots so, a a = a(a+) also, y y = ( ) = 6a(a+ ) ow ( ) y y a(a ) 6a(a ) 0 0a(a + ) = 0 a + a = 0 a =, 69. if f() is a fuctio such that f() = ( ) ( ), the f(0) = 0 f() is icreasig i (0, ) = is a critical poit of f() f() is decreasig i (, 5) As : (B, C) Hit : f() = ( ) ( ) + + The sig scheme of f() so clearly f() is icreasig i (0, ) as f() 0. = is a critical poit as f() = 0. from f(), we ca t determie f() uiquely so f(0) ca t be predicted For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

20 WBJEE - 06 (Aswers & Hit) 70. O the ellipse + 9y =, the poits at which the tagets are parallel to the lie 8 = 9y are, 5 5, 5 5, 5 5 As : (B, D), 5 5 Hit : Let cos, si be a poit o + 9y =, so equatio of taget at cos + y si = equatig slope with 8 = 9y cos, si is cos 8 ta si 9 Hece either cos, si 5 5 or cos, si 5 5 so the poits are, 5 5 or, 5 5, for 0t 0 otherwise the 7. If (t) r0 (t r ) t 06 dt a real umber 0 does ot eist As : (A, B) Hit : (t 06)( (t 0) (t 05) (t 06)).dt dt.(0 0 ).dt 0.dt = If the equatio + y 0 + = 0 has real roots = a ad y = the 7 y 7 y As : (A, C) Hit : 0 + (y + ) = 0 for real roots of, D 0 00 (y + ) 0 y y also, y = + 0 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

21 WBJEE - 06 (Aswers & Hit) for real roots of y, ( 7) ( ) If z = si i cos the for ay iteger, z z cos z z si z isi z As : (A, C) Hit : z = si i cos cos isi i( ) = e z z icos so, i z e cos isi i e cos isi z, so z = z z isi z cos cos 7. Let f : X X be such that f(f()) = for all X ad X R, the f is oe-to-oe f is oto f is oe-to-oe but ot oto f is oto but ot oe-to-oe As : (A, B) Hit : f f(). X so, f() = f () i.e.f() is self ivertible Hece f() has to be oe-oe & oto 75. If A, B are two evets such that P(A B) ad P(A B) the 8 8 P P 8 P.P 8 7 P P Noe of these 8 As : (A, C) Hit : P(A B) = P + P P (AB) For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

22 WBJEE - 06 (Aswers & Hit) P + P = P(A B) + P (AB) P(A B) P(A B) 8 8 so, 7 P(A B) P(A B) so, P P 8 P P 8 For ay query related to admissio i Egieerig colleges of West Begal, visit or cotact

WBJEE Answer Keys by Aakash Institute, Kolkata Centre

WBJEE Answer Keys by Aakash Institute, Kolkata Centre 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