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 between h = ht 0 k = t 0 vectors b nd c, then vlue of sin is : () Ans. () ().c b (b.c) b c. c 0 b. c b c () cos = sin = (). Let O be the vertex nd Q be ny point on the prbol, x = 8y. If the point P divides the line segment OQ internlly in the rtio :, then the locus of P is :- () y = x () x = y () x = y () y = x Ans. () Let P(h, k) divides segment OQ in the rtio : O P Q(t, t ) k = h locus of P is x = y. If the ngles of elevtion of the top of tower from three colliner points A, B nd C, on line leding to the foot of the tower, re 0, 5 nd 60 respectively, then the rtio, AB : BC, is : () : () : () : () : Ans. () Let height of tower is 'h' Let AB = x, BC = y, CD = z then, in AED, tn 0º = ED h AD x y z x + y + z = h in BED, tn 5º = ED BD y + z = h in CED, tn 60º = ED CD z = h x = h, y = h, z = h AB x AB : BC : BC y JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only.
JEE(MAIN)-05. The number of points, hving both co-ordintes s integers, tht lie in the interior of the tringle with vertices (0, 0), (0, ) nd (, 0) is : () 80 () 780 () 0 () 86 Ans. () Let P be (x, y) x y y + y < x + y 0 x + y + t = 0, t 0 (x ) + (y ) + t = 8 No. of points = 8+ C = 0 C = 780 5. The eqution of the plne contining the line x 5y + z = ; x + y + z = 5, nd prllel to the plne, x + y + 6z =, is : () x + y + 6z = 7 () x + 6y + z = () x + 6y + z = () x + y + 6z = 7 Let eqution of plne prllel to x + y + 6z = is x + y + 6z = point on line of intersection is (,, 0), it lso lie on plne x + y + 6z = so required plne is x + y + 6z = 7 6. Let A nd B be two sets contining four nd two elements respectively. Then the number of subsets of the set A B, ech hving t lest three elements is : () 75 () 50 () () 56 Ans. () n(a) =, n(b) = n(a B) = 8 hence No. of subsets hving t lest '' elements is 8C + 8 C + 8 C 5 + 8 C 6 + 8 C 7 + 8 C 8 = 8 7 = 7. Locus of the imge of the point (, ) in the line (x y + ) + k (x y + ) = 0, k R, is () circle of rdius () circle of rdius () stright line prllel to x-xis () stright line prllel to y-xis 8. P (, ) A (, ) B (, ) P is the fixed point for given fmily of line PB = PA ( ) + ( ) = + (x ) + (y ) = locus is circle of rdius ( cosx) ( cosx) lim is equl to : xtn x x 0 () () () () ( cosx)( cos x) lim x0 (x)(tn x) sin x (x )() = lim. x 0 = x tn x (x) x x JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only.
CODE-D. The distnce of the point (, 0, ) from the point of intersection of the line x y z nd the plne x y + z = 6, is : () () () () 8 Ans. (). The sum of first terms of the series 5... is : () () () 7 () 6 Ans. () Sum = r (r )... r 5... r r r Point of intersection of line x y z (let) & plne x y + z = 6 ( + ) ( ) + ( + ) = 6 = Q(5,, ) & given P(, 0, ) PQ = 0. The sum of coefficients of integrl powers of x in the binomil expnsion of x 50 is : Ans. () () 50 () 50 () 50 () 50 x 50 = 50 C 0 50 C x + 50C x +...+ 50 C 50 50 x consider x 50 = 50 C 0 + 50 C x +...+ 50 C 50 50 x dd both equtions nd put x = sum of coefficients of integrl powers of x 50 JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only. 5 = r (r ) 0 r r = 6. The re (in sq. units) of the region described by {(x, y) : y x nd y x } is : () 5 6 () Ans. () 7 Required re = () () 5 6 / / y y (y ) y 8 6 dy
JEE(MAIN)-05. The set of ll vlues of for which the system of liner equtions : x x + x = x x x + x = x x + x = x hs non-trivil solution () contins two elements () contins more thn two elements () is n empty set () is singleton For nontrivil solution = 0 = 0 =, -. A complex number z is sid to be unimodulr if z =. Suppose z nd z re complex numbers such tht z z z z is unimodulr nd z is not unimodulr. Then the point z lies on : () circle of rdius () circle of rdius () stright line prllel to x-xis () stright line prllel to y-xis z z z z = z z = z z z z z z = z z z z z + z z z = 0 ( z ) z ( z ) = 0 z = 0 z = is circle of rdius nd centre t origin. 5. The number of common tngents to the circle x + y x 6y = 0 nd x + y + 6x + 8y + 6 = 0, is : () () () () r r c (, ) c (, ) distnce between centres (c c ) = r = 5 & r = 8 6 8 c c = r + r is cse of externl touching (s shown in bove figure), so three common tngents cn be drwn. 6. The number of integers greter thn 6000 tht cn be formed, using the digits,5,6,7 nd 8 without repetition, is : () 0 () 7 () 6 () Ans. () Number of digit numbers greter thn 6000 is = 7 Number of 5 digit numbers greter thn 6000 is 5! = 0 So totl number of numbers = 7 + 0 = 7. Let y(x) be the solution of the differentil eqution (x log x) dy + y = x log x, (x ). Then y(e) is equl to : () () e () e () 0 JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only. 6
(x log x) dy = dy + I.F. = + y = x log x, (x ) y x log x = x log x e = loglogx e = logx = y log x = logx y log x = x(logx ) + C...() Put x = y.0 = + C C =...() Put x = e in () y log e = e (log e ) + C y(e) = C =... from () 8. If A = is mtrix stisfying the b eqution AA T = I, where I is identity mtrix, then the ordered pir (, b) is equl to : () (, ) () (, ) () (, ) () (, ) Ans. () T A A A I is orthogonl mtrix A b Three rows re corresponding to three mutully perpendiculr unit vectors b, +b+ = 0 & b+ =0 = & b = CODE-D. If m is the A.M. of two distinct rel numbers l nd n(l, n > ) nd G, G nd G re three geometric mens between l nd n, then equls. () lmn () l m n () l mn () lm n Ans. (), G, G, G, n in G.P. Let r be the common rtio r = n G G G = Here G G G r r r = n[ + n + n ] = n(+ n) = nm = m n (m = n + ) 50. The negtion of ~ s ~ r s is equivlent to : () s r ~ s () s r () s ~ r () s r ~ s Ans. () s ( r s) ( s ~ r) ( s s) ( s ~ r) t ( s ~ r) ~ ( s ~ r) s r 5. The integrl x (x ) equls : () x x c () x x () c x Ans. () I x (x ) / () x c JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only. 7 Put x. x x z x / z dz I z C z / C x c
JEE(MAIN)-05 5. The norml to the curve, x + xy y = 0, t (, ) : () meets the curve gin in the third qudrnt () meets the curve gin in the fourth qudrnt () does not meet the curve gin () meets the curve gin in the second qudrnt Ans. () x + xy xy y = 0 x(x + y) y(x + y) = 0 (x y) (x + y) = 0 Norml t (, ) will be x + y = Now, x + y = x + y = 0 y =, y = & x = (, ) which is in fourth qudrnt. 5. Let x tn y = tn x + tn x, where x <. Then vlue of y is : x x x x () () x x () Ans. () x x x () x x x x x tn y = tn x = tn x x x y x 5. If the function. g(x) = k x, 0 x mx, x 5 is differentible, then vlue of k + m is - () 0 () Ans. () k x 0 x g(x) mx x 5 If g(x) is differentible then it must be continuous t x = k = m. + k = m + Now, slope must be equl k m x x Now, 8m = m + m & 5 8 k 5 0 k m 5 55. The men of the dt set comprising of 6 observtions is 6. If one of the observtion vlued 6 is deleted nd three new observtions vlued, nd 5 re dded to the dt, then the men of the resultnt dt, is : () 5.8 ().0 () 6.8 () 6.0 Ans. () x i = 56 Let new observtions X i X i = (56 6) + ( + + 5) = 5 & number of observtions 8 () () 6 5 New men = Xi 8 = JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only. 8
CODE-D 56. The integrl logx log x log(6 x x ) is equl to : () () 6 () () I = I = (using I = log x log x log(x 6) log(6 x) log x log(x 6) b b f(x) f( b x) ) = x log x log(6 x) log x log(6 x) I = 57. Let nd be the roots of eqution x 6x = 0. If n = n n, for n, then the vlue of 0 8 is equl to : () () () 6 () 6 Given x 6x = 0 n+ 6 n+ n = 0 n n n = Now, put n = 8 0 8 = 58. Let f (x) be polynomil of degree four hving extreme vlues t x = nd x =. f(x) If lim x0 x =, then f () is equl to : () 0 () () 8 () Clerly f(x) = x + bx + cx + + e Now, x bx cx e lim x 0 x Clerly d = e = 0 Now, lim( x bx c) x0 c = hence, f(x) = x + bx + x f(x) = x + bx + x = x(x + bx + ) Now, x = nd x = re lso solutions b nd = x f(x) = x x nd b = f() = 8 6 + 8 = 0 5. The re (in sq. units) of the qudrilterl formed by the tngents t the end points of the lter rect to the ellipse () 7 () 7 Ans. () x y is : 5 () 7 () 8 JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only.
JEE(MAIN)-05 x y 5 =, b = 5 5 = ( e ) e = Eqution of tngent t (, 5 ) x y 5 5 x y (0, ) nd, 0 re the points of intersection of bove line with co-ordinte xes. 7 60. If identicl blls re to be plced in identicl boxes, then the probbility tht one of the boxes contins exctly blls is : () () 0 55 () () 55 0 Ans. (Bonus) The ctul solution corresponding to the lnguge given in question will be Totl number of wys = n(s) = Number of elements in smple spce = If E corresponds to the event tht one of the boxes contins exctly blls. Then n(e) = C C [ C ] + C 6 C 6 C Required probbility n(e) 55 07 n(s) Hence none of the nswer mtches from the given options so it seems tht the question is ill-frmed. If we ssume the event s one of the prticulr box contins exctly blls then n(e) = C. n(e) C. Probbility n(s) 55 JEE (Min + Advnced) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) nd JEE (Min) Leder Course (Trget-06) for XII Pssed / Appered students Strt on 5th April 05 (English / Hindi Medium) t Kot Centre only. 0