Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,

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1 R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel , IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice questions. Ech question hs four choices A, B, C nd D out of which ONLY ONE is correct. x cos, x 0. Let f x = x, x R, then f is 0, x = 0 A differentible both t x = 0 nd t x = B differentible t x = 0 but not differentible t x = C not differentible t x = 0 but differentible t x = D differentible neither t x = 0 nor t x =. B x f x = 0 x cos, x, cos, x x > 0 x = 0 x < 0 At x = 0 : At x = : h f 0 h f 0 LHD = Lim = Lim h 0 h h 0 differentible t x = 0. f h f RHD = Lim h 0 h = Lim h 0 cos h h h = 0 = RHD cos 0 h h = h cos f h f LHD = Lim = h h 0 h h LHD RHD. = IIT JEE 0 08 Apr Question & Solutions per I www. prernclsses.com

2 . The function f : [0, ] [, 9] defined by f x = x 5x 6x, is A one one nd onto. B onto but not one one. C one one but not onto. D neither one one nor onto.. B f x = x 5x 6x f ' x = 6x 0x 6x = 6 x x f ' x = 0 t x =, f x is not monotonic hence not one one. f '' x = x 0 < 0 t x =, f = 9 f 0 =, f = 8, hence onto.. The ellipse E : x y = is inscribed in rectngle R whose sides re prllel to the coordinte 9 xes. Another ellipse E pssing through the point 0, circumscribes the rectngle R. The eccentricity of the ellipse E is A B C D. C 0, 5, 0, 0 Rectngle hving dimension = 6 x y Another ellipse be = b which psses through 0, nd, =, b = 6 e = /. IIT JEE 0 08 Apr Question & Solutions per I www. prernclsses.com

3 . The point is the intersection of the stright line joining the points Q,, 5 nd R,, with the plne 5x y z =. If S is the foot of the perpendiculr drwn from the point T,, to QR, then the length of the line segment S is A B C D. A Q,, 5 T,, S λ 5x y z = R,, λ λ λ 5 Q : R = λ : :,, lies on 5x y z = λ = λ λ λ /, /, / x y z 5 QR : = = = k D.R. of TS = k, k, k D.R. of QR =,, So, k k k = 0 k = /. S : /,, 9 / 9 S = =. 5. The locus of the mid point of the chord of contct of tngents drwn from points lying on the stright line x 5y = 0 to the circle x y = 9 is A 0 x y 6x 5y = 0 B 0 x y 6x 5y = 0 C 6 x y 0x 5y = 0 D 6 x y 0x 5y = 0 5. A h, k α, β Let h, k is ny point on the locus. Eq. of chord with middle point h, k is T = S xh yk = h k... i Also, from ny point α, β on the line x 5y = 0, eqution of the chord of contct is T = 0 i.e. xα yβ = 9... ii i nd ii re sme line α β 9 9 h 9 k = = α =, β = h k h k h k h k α, β is on the line x 5y = 0 Eq. of the locus : 0 x y 6x 5y = 0. IIT JEE 0 08 Apr Question & Solutions per I www. prernclsses.com

4 6. Let = [ ij ] be mtrix nd let Q = [b ij ], where b ij = i j ij for i, j. If the determinnt of is, then the determinnt of the mtrix Q is A 0 B C D 6. D Let = = = Q = = = =. 7. The integrl sec x sec x tn x 9 / dx equls for some rbitrry constnt K A B C D sec x tn x / sec x tn x 7 sec x tn x / sec x tn x 7 sec x tn x / sec x tn x 7 sec x tn x / sec x tn x 7 K K K K 9 / 7. C I = sec x sec x tn x sec x tn x dx sec x tn x sec x sec x tn x 7 / / = sec x tn x sec x tn x k 7 9 / sec x tn x dx = 7 / / 7 sec x tn x = x x x x sec tn / sec tn 7 k k IIT JEE 0 08 Apr Question & Solutions per I www. prernclsses.com

5 8. The totl number of wys in which 5 blls of different colours cn be distributed mong persons so tht ech person gets t lest one bll is A 75 B 50 C 0 D 8. B,, or,, Totl number of wys = 5 C C! /! 5 C C! /! = 50. x x 9. If Lim x b =, then x x A =, b = B =, b = C =, b = D =, b = 9. B Q = x x Lim x b x b x b Lim = x x x x = 0 nd b = =, b =. 50. Let zbe complex number such tht the imginry prt of zis non zero nd = z z is rel. Then cnnot tke the vlue A B C 50. D As is rel = z z = z z z z z z = 0 z z = [ since z z 0 ] Rez = Rez = / x = / Let z = / iy = iy iy = As y 0 cnnot be /. y D IIT JEE 0 08 Apr Question & Solutions per I 5 www. prernclsses.com

6 SECTION II Multiple Correct Answers Type This section contins 5 multiple choice questions. Ech question hs four choices A, B, C nd D out of which ONE OR MORE re correct. 5. Let S be the re of the region enclosed by y = e x, y = 0, x = 0 nd x =. Then A C S e S e B D S e S e 5. ABD 0, y = e x 0, 0 / = x S e dx, Since 0 x then 0...x x x x x x e e x e x.... x x e dx e dx e dx... dx S... e e S s well s e S. e As 0. but = Hence, C is not correct. e e For option : D Are of the two rectngle in the grph / e S e IIT JEE 0 08 Apr Question & Solutions per I 6 www. prernclsses.com

7 5. If yx stisfies the differentil eqution y' y tn x = x sec x nd y0 = 0, then A y = 8 B y = 8 C y = 9 D y = 5. AD y' y tn x = x sec x I.F. = e tn x dx = cos x Its solution is : y. I.F. = x sec x I.F. dx y cos x = x k At y0 = 0 then k = 0 y = x sec x Hence, y =, y =, y' =, y =. x y 5. Tngents re drwn to the hyperbol =, prllel to the stright line x y =. The 9 points of contct of the tngents on the hyperbol re A 9, B 9, C, D, x y 5. AB =, Tngent line : x y = 0, m = 9 Eq. of tngent : y = mx ± m b = x ±... i Eq. of tngent t point x, y is xx 9yy 6 = 0... ii Compring eq. i nd ii [Q they represent sme line] x y ± = 0 xx 9yy 6 = 0 x 9 y = = ± 6 9 x, y = ±,. IIT JEE 0 08 Apr Question & Solutions per I 7 www. prernclsses.com

8 θ θ 5. Let θ, φ [0, ] be such tht cos θ sin φ = sin θ tn cot cos φ, tn θ > 0 nd < sin θ <. Then θ cnnot stisfy A 0 < φ < B < φ < C < φ < D < φ < 5 5. ACD tn θ < 0 & sin θ, < θ < cos θ = sin θ φ LHS, / < sin θ φ < 5 < θ φ < < φ < 6 6 Hence, options ACD re correct. 55. A ship is fitted with three engines E, E nd E. The engines function independently of ech other with respective probbilities, nd. For the ship to be opertionl t lest two of its engines must function. Let denotes the event tht the ship is opertionl nd let, nd denote respectively the events tht the engines E, E nd E re functioning. Which of the following is re true? A C c 7 [ ] = B [Exctly two engines of the ship re functioning ] = 6 8 [ ] = 5 6 D [ ] = 55. BD = 7 6 = = A.. / = = 8 B Exctly two engines of the ship re functioning / = = 8 IIT JEE 0 08 Apr Question & Solutions per I 8 www. prernclsses.com

9 IIT JEE 0 08 Apr Question & Solutions per I 9 www. prernclsses.com C / = robbility tht occurs given tht engine E hs strted = 5 / 8 D / = robbility tht '' occurs given tht engine E hs strted = 7 / 6.

10 SECTION III Integer Answer Type This section contins 5 questions. The nswer of ech question is single digit integer, rnging from 0 to 9 both inclusive. 56. The vlue of 6 log... is 56.. Let S =... S = S S S = 0 S =, S = neglected 6 log / = 57. Let S be the focus of the prbol y = 8x nd let Q be the common chord of the circle x y x y = 0 nd the given prbol. The re of the tringle QS is 57.. Let ny point on y = 8x be t, t Since this point lies on x y x y = 0 t t 6 t = 0 t t t t = 0 t = 0, t = For t = 0, Q0, 0 For t =,, nd S, 0 Are = r 58. If b r, nd c r r r r re unit vectors stisfying r r r b b c c = 9 r r r, then 5 b 5 c is r r r r r r r r r r r r 58.. b c. b c b. b =. = 9 r r r r r r r r r r r b c = 0 = b c 5 b c = 5 = IIT JEE 0 08 Apr Question & Solutions per I 0 www. prernclsses.com

11 59. Let f : R R be defined s f x = x x. The totl number of points t which f ttins either locl mximum or locl minimum is f x = x x, x < = x x, x < 0 = x x, 0 x < = x x, x f' x = x, x < = x, < x < 0 = x, 0 < x < = x, < x For extremum f ' x = 0 x = /, / So, totl number of points of extremum = 5. / 0 / y = fx 60. Let px be rel polynomil of lest degree which hs locl mximum t x = nd locl minimum t x =. If p = 6 nd p =, then p' 0 is 'x = x x = x x x x = x x C = 6 / C = 6 = C = = ' 0 = = 9. IIT JEE 0 08 Apr Question & Solutions per I www. prernclsses.com

JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS

JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle