JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART- : JEE(Advaced) 08/Paper- SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse trigoometric fuctio ta assume values i,. ) The, which of the followig statemet(s) is (are) TRUE? (A) As. (D) Sol. 5 j ta ƒ 0 55 0 ' (B) j j j j ƒ (0) sec ƒ (0) 0 (C) For ay fied positive iteger, lim ta ƒ (D) For ay fied positive iteger, limsec ƒ f () ( j) ( j) ta j ( j)( j) f () [ta ( j) ta ( j)] j f () = ta ( + ) ta ta(f ()) = ta[ta ( + ) ta ] ta(f ()) = ( ) ( ) ta(f ()) = JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
sec (f ()) = + ta (f ()) sec (f ()) = JEE(Advaced) 08/Paper- lim sec (f ()) lim =. Let T be the lie passig through the poits P(, 7) ad Q(, 5). Let F be the set of all pairs of circles (S, S ) such that T is tagets to S at P ad taget to S at Q, ad also such that S ad S touch each other at a poit, say, M. Let E be the set represetig the locus of M as the pair (S, S ) varies i F. Let the set of all straight lie segmets joiig a pair of distict poits of E ad passig through the poit R(, ) be F. Let E be the set of the mid-poits of the lie segmets i the set F. The, which of the followig statemet(s) is (are) TRUE? (A) The poit (, 7) lies i E 7 (B) The poit, 5 5 does NOT lie i E (C) The poit, lies i E (D) The poit 0, does NOT lie i E As. (D) P Sol. A M Q commo taget (radical ais) AP = AQ = AM Locus of M is a circle havig PQ as its diameter Hece, E : ( ) ( + ) + (y 7)(y + 5) = 0 ad Locus of B (midpoit) is a circle havig RC as its diameter E : ( ) + (y ) = 0 Now, after checkig the optios, we get (D) (, ) B R C(0, ) JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper-. Let S be the of all colum matrices variables) b b b such that b, b, b ad the system of equatios (i real + y + 5z = b y + z = b y + z = b has at least oe solutio. The, which of the followig system(s) (i real variables) has (have) at least oe solutio of each b b S? b (A) + y + z = b, y + 5z = b ad + y + 6z = b (B) + y + z = b, 5 + y + 6z = b ad y z = b (C) + y 5z = b, y + 0z = b ad y + 5z = b (D) + y + 5z = b, + z = b ad + y 5z = b As. (A,C,D) Sol. We fid D = 0 & sice o pair of plaes are parallel, so there are ifiite umber of solutios. Let P + P = P P + 7P = P b + 7b = b (A) D 0 uique solutio for ay b, b, b (B) D = 0 but P + 7P P (C) D = 0 Also b = b, b = b Satisfies b + 7b = b (Actually all three plaes are co-icidet) (D) D 0. Cosider two straight lies, each of which is taget to both the circle + y = ad the parabola y =. Let these lies itersect at the poit Q. Cosider the ellipse whose ceter is at the origi O(0, 0) ad whose semi-major ais is OQ. If the legth of the mior ais of this ellipse is which of the followig statemet(s) is (are) TRUE? (A) For the ellipse, the eccetricity is ad the legth of the latus rectum is, the the (B) For the ellipse, the eccetricity is ad the legth of the latus rectum is (C) The area of the regio bouded by the ellipse betwee the lies ad = is (D) The area of the regio bouded by the ellipse betwee the lies JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 ( ) ad = is ( ) 6
As. (A,C) Sol. Q / (0,0) JEE(Advaced) 08/Paper- Let equatio of commo taget is y = m + m 0 0 m m m + m = 0 m = ± Equatio of commo tagets are y = + ad y = poit Q is (, 0) y Equatio of ellipse is / b (A) e = ad LR = a (C) (,0) y =/ (,0) Area.. d si / / = 8 8 correct aswer are (A) ad (D) JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- 5. Let s, t, r be the o-zero comple umbers ad L be the set of solutios z = + iy, y,i of the equatio sz tz r 0, where z iy. The, which of the followig statemet(s) is (are) TRUE? (A) If L has eactly oe elemet, the s t (B) If s = t, the L has ifiitely may elemets (C) The umber of elemets i L z : z i 5 is at most (D) If L has more tha oe elemet, the L has ifiitely may elemets As. (A,C,D) Sol. Give sz tz r 0... () o takig cojugate s z tz r 0... () from () ad () ellimiatig z z s t rt rs (A) If s t the z has uique value (B) If s = t the r t rs may or may ot be zero so L may be empty set (C) locus of z is oll set or sigleto set or a lie i all cases it will itersect give circle at most two poits. (D) I this case locus of z is a lie so L has ifiite elemets 6. Let ƒ : (0, ) be a twice differetiable fuctio such that ƒ()si t ƒ(t)si lim t t si for all (0, ). If ƒ, the which of the followig statemet(s) is (are) TRUE? 6 (A) ƒ (B) ƒ for all (0, ) 6 (C) There eists (0, ) such that ƒ'() = 0 (D) ƒ" ƒ 0 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 5
JEE(Advaced) 08/Paper- As. (B,C,D) f ()si t f (t)si Sol. lim t t by usig L'Hopital si f ()cos t f '(t)si lim t f()cos f'()si = si si f '()si f () cos si f () d si f () c si Put = & f 6 6 c = 0 f() = si (A) f (B) f() = si as si >, si < + 6 6 f() < + (C) f'() = si cos (0, ) 6 f'() = 0 ta = there eist (0, ) for which f'() = 0 (D) f"() = cos + si f ", f f " f 0 6 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
7. The value of the itegral As. () is. 0 6 SECTION d JEE(Advaced) 08/Paper- Sol. 0 d 0 Put I = 6 d 6 6 / / = t d ( ) / = dt dt / ( ) 6 / t t 8. Let P be a matri of order such that all the etries i P are from the set {, 0, }. The, the maimum possible value of the determiat of P is. As. () Sol. = a a a b b b (abc abc abc ) (abc abc abc ) c c c y Now if ad y the ca be maimum 6 But it is ot possible as = each term of = ad y = each term of y = i a b c i i i = ad which is cotradictio aibici = i JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 7
so ow et possibility is which is obtaied as ( ) ( ) ( ) = JEE(Advaced) 08/Paper- 9. Let X be a set with eactly 5 elemets ad Y be a set with eactly 7 elemets. If is the umber of oe- oe fuctios from X to Y ad is the umber of oto fuctios from Y to X, the the value of 5! is. As. (9) Sol. (X) = 5 (Y) = 7 Number of oe-oe fuctio = 7 C 5 5! Number of oto fuctio Y to X a a.. a 7 b b.. b 5,,,,,,,, 7! 7! 7 7 7 5! 5! C. C 5! C 5!!! (!)! 7 7 C C 5 5 9 5! 0. Let ƒ : be a differetiable fuctio with ƒ(0) = 0. If y = ƒ() satisfies the differetial equatio the the value of lim ƒ() is. dy 5y5y d, As. (0.) Sol. dy d 5y dy So, 5y d Itegratig, y l 5 c 5 y 5 5 8 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- 5y l 0( c) 5y Now, c = 0 as f(0) = 0 Hece 5y 5y e (0) let 5f () 5f () let e (0) Now, RHS = 0 (5f () ) 0 let let f () 5. Let ƒ : be a differetiable fuctio with ƒ(0) = ad satisfyig the equatio As. () Sol. The, the value of log e (ƒ()) is. ƒ( + y) = ƒ()ƒ'(y) + ƒ'()ƒ(y) for all, y. P(, y) : f( + y) = f()f(y) + f() f(y), y R P(0, 0) : f(0) = f(0)f(0) + f(0) f(0) = f'(0) f'(0) = P(, 0) : f() = f(). f(0) + f'().f(0) f() = f () f '() f'() = f () f() = e l(f()) =. Let P be a poit i the first octat, whose image Q i the plae + y = (that is, the lie segmet PQ is perpedicular to the plae + y = ad the mid-poit of PQ lies i the plae + y = ) lies o the z-ais. Let the distace of P from the -ais be 5. If R is the image of P i the y-plae, the the legth of PR is. JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 9
As. (8) Sol. Let P(,, ) Q(0, 0, ) & R(,, ) Now, PQ ˆi ˆj ˆ i ˆ j ˆ i ˆ j JEE(Advaced) 08/Paper- = Also, mid poit of PQ lies o the plae + = 6 = Now, distace of poit P from X-ais is 5 5 6 as = = as = Hece, PR = = 8. Cosider the cube i the first octat with sides OP, OQ ad OR of legth, alog the -ais, y-ais ad z-ais, respectively, where O(0, 0, 0) is the origi. Let S,, be the cetre of the cube ad T be the verte of the cube opposite to the origi O such that S lies o the diagoal OT. If p SP, q SQ, r SR ad t ST, the the value of p q r t is. As. (0.5) Sol. z R S T O P Q y p SP,, ˆ i ˆ j k ˆ 0 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- q SQ,, ˆ i ˆ j k ˆ r SR,, ˆ i ˆ j k ˆ t ST,, ˆ i ˆ j k ˆ ˆi ˆj kˆ ˆi ˆj kˆ (pq) r t i j i j ˆ ˆ ˆ ˆ = ˆk 6 0 0 0 0 0. Let X = C C C... 0 C, where 0 C r, r {,,..., 0} deote biomial coefficiets. The, the value of X is. 0 As. (66) Sol. r X r. C ; 0 r0 r r r0 X. C. C r r r X. C. C X. C ; 0 9 X 0. C 9 X. 9 C9 0 = 66 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
5. Let E = : ad 0 SECTION JEE(Advaced) 08/Paper- ad E : si log e is a real umber. Here, the iverse trigoometric fuctio si assumes values i,. Let ƒ : E be the fuctio defied by ad g : E be the fuctio defied by LIST-I P. The rage of ƒ is. ƒ() loge. g() si loge LIST-II Q. The rage of g cotais. (0, ) R. The domai of ƒ cotais. e,, e e, S. The domai of g is. (,0) (0, ) 5. e, e 6. e (,0), e The correct optio is : (A) P ; Q ; R ; S (B) P ; Q ; R 6; S 5 (C) P ; Q ; R ; S 6 (D) P ; Q ; R 6; S 5 As. (A) JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- Sol. E : 0 + 0 E : (, 0) (, ) + E : Now e e 0 e (e ) 0 e( ) also + /(e ) +, (, ) e (e ) e + e 0 0 e/(e ) + e (, ), e So E : e,, e e as Rage of is R + {} Rage of f is R {0} or (, 0) (0, ) JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- Rage of g is, \ {0} or,0 0, Now P, Q, R, S Hece A is correct 6. I a high school, a committee has to be formed from a group of 6 boys M, M, M, M, M 5, M 6 ad 5 girls G, G, G, G, G 5. (i) Let be the total umber of ways i which the committee ca be formed such that the committee has 5 members, havig eactly body ad girls. (ii) Let be the total umber of ways i which the committe ca be formed such that the committee has at least members, ad havig a equal umber of boys ad girls. (iii) Let be the total umber of ways i which the committe ca be formed such that the committee has 5 members, at least of them beig girls. (iv) Let be the total umber of ways i which the committee ca be formed such that the commitee has members, havig at least girls ad such that both M ad G are NOT i the committee together. LIST-I LIST-II P. The value of is. 6 Q. The value of is. 89 R. The value of is. 9 S. The value of is. 00 The correct optio is :- 5. 8 6. 6 (A) P ; Q 6, R ; S (B) P ; Q ; R ; S (C) P ; Q 6, R 5; S (D) P ; Q ; R ; S As. (C) 65 Sol. () 00 So P JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- 65 65 65 65 65 () 5 5 = 5 = 6! So Q 6 5 6 56 5 6 56 () 5 0 56 56 5 0 5 = 8 So R 5 5 6 5 56 5 () 89 So S 7. Let H : a y b, where a > b > 0, be a hyperbola i the y-plae whose cojugate ais LM subteds a agle of 60 at oe of its vertices N. Let the area of the triagle LMN be. LIST-I LIST-II P. The legth of the cojugate ais of H is. 8 Q. The eccetricity of H is. R. The distace betwee the foci of H is. S. The legth of the latus rectum of H is. The correct optio is : (A) P ; Q, R ; S (B) P ; Q ; R ; S (C) P ; Q, R ; S (D) P ; Q ; R ; S JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 5
JEE(Advaced) 08/Paper- As. (B) Sol. L b O b 0 a 0 N M b ta 0 a a b Now area of LMN =.b.b b b = & a b e a P. Legth of cojugate ais = b = So P Q. Eccetricity e So Q R. Distace betwee foci = ae = 8 So R S. Legth of latus rectum = b () a So S 6 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08
JEE(Advaced) 08/Paper- 8. Let f :, f :,, f :, e by ad f : be fuctios defied (i) f () = si e (ii) f () = i,, si if 0 ta, where the iverse trigoometric fuctio ta assumes values if 0 (iii) f () = [si(log e ( + )], where for t, [t] deotes the greatest iteger less tha or equal to t, (iv) f () = si if 0 0 if 0 List-I List-II P. the fuctio f is. NOT cotiuous at = 0 Q. The fuctio f is. cotiuous at = 0 ad NOT differetiable at = 0 R. The fuctio f is. differetiable at = 0 ad its derivative is NOT cotiuous at = 0 S. The fuctio f is. differetiable at = 0 ad its derivative is cotiuous at = 0 The correct optio is : (A) P ; Q, R ; S (B) P ; Q ; R ; S (C) P ; Q, R ; S (D) P ; Q ; R ; S As. (D) JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08 7
JEE(Advaced) 08/Paper- Sol. (i) f() = si e ' f () cos e. 0 e.( ) e at = 0 ' f () does ot eist So. P si, 0 (ii) f () = ta 0 0 si lim ta 0 f () does ot cotiuous at = 0 So Q (iii) f () = si ( ) = 0 < + < e / 0 < ( + ) < si(( + ) < f () = 0 So R (iv) f () = So S si, 0 0, 0 8 JEE(Advaced) 08/Paper-/Held o Suday 0 th May, 08