PAPER ¼isij½- 1. SECTION-1 (Maximum Marks : 24)

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1 JEE (Advanced) (ADVANCED) PAPER ¼isij½ DATE: This question paper has three (0) parts: PART-I: Physics, PART-II: Chemistry and PART-III: Mathematics. Each part has total of eighteen (8) questions divided into three (0) sections (Section-, Section- and Section-). Total number of questions in Paper- : Fifty four (54). Paper- Maimum Marks : One Hundred Eighty (80). Instructions for Section- : Questions and Marking Scheme SECTION- (Maimum Marks : 4) This section contains SIX (06) questions. Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. Answer to each question will be evaluated according to the following marking chosen. Full Marks : +4 If only (all) the correct option(s) is (are) chosen. Partial Marks : Partial Marks : Partial Marks : Zero Marks : Negative Marks : + If all the four options are correct but ONLY three options are chosen. + If three or more options are correct but ONLY two options are chosen, both of which are correct options. + If two or more options are correct but ONLY one option is chosen and it is a correct option. 0 If none of the options is chosen (i.e. the question is unanswered). In all other cases. For Eample : If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in + marks. Selecting only one of the three correct options (either first or third or fourth option), without selecting any incorrect option (second option in this case), will result in + marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in marks. Answering Section- Questions : To select the option(s), using the mouse click on the corresponding button(s) of the option(s). To deselect chosen option(s), click on the button(s) of the chosen option(s) again or click on the Clear Response button to clear all the chosen options. To change the option(s) of a previously answered question, if required, first click on the Clear Response button to clear all the chosen options and then select the new option(s). To mark a question ONLY for review (i.e. without answering it), click on the Mark for Review & Net button. To mark a question for review (after answering it), click on Mark for Review & Net button answered question which is also marked for review will be evaluated. To save the answer, click on the Save & Net button the answered question will be evaluated. Registered & Corporate Office: CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.)-4005 Tel.No.: , 000, 0, Toll Free: Fa: Website: contact@resonance.ac.in CIN: U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal

2 Instructions for Section- : Questions and Marking Scheme SECTION- (Maimum Marks : 4) This section contains EIGHT (08) questions. The answer to each question is NUMERICAL VALUE. For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.5, 7.00, 0., 0.0,,0.7, 7.0) using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. Answer to each question will be evaluated according to the following marking scheme : Full Marks : + If ONLY the correct numerical value is entered as answer. Zero Marks : 0 In all other cases. Answering Section- Questions : Using the attached computer mouse, click on numbers (and/or symbols) on the on-screen virtual numeric keypad to enter the numerical value as answer in the space provided for answer. To change the answer, if required, first click on the Clear Response button to clear the entered answer and then enter the new numerical value. To mark a question ONLY for review (i.e. answering it), click on Mark for Review & Net button the answered question which is also marked for review will be evaluated. To mark a question for review (after answering it), click Mark for Review & Net button the answered question which is also marked for review will be evaluated. To save the answer, click on the Save & Net button the answered question will be evaluated. Instructions for Section- : Questions and Marking Scheme SECTION- (Maimum Marks : ) This section contains TWO (0) paragraphs. Based on each paragraph, there are TWO (0) questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme : Full Marks : + If ONLY the correct option is chosen. Zero Marks : Negative Marks : 0 If none of the options is chosen (i.e. the question is unanswered). In all other cases. Answering Section- Questions : To select an option, using the mouse click on the corresponding button of the option. To deselect the chosen answer, click on the button of the chosen option again or click on the Clear Response button. To change the chosen answer, click on the button of another option. To mark a question ONLY for review (i.e. without answering it), click on Mark for Review & Net button. To mark a question for review (after answering it), click on Mark for Review & Net button the answered which is also marked for review will be evaluated. To save the answer, click on the Save & Net button the answered question will be evaluated. Registered & Corporate Office: CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.)-4005 Tel.No.: , 000, 0, Toll Free: Fa: Website: contact@resonance.ac.in CIN: U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal

3 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS PART : III MATHEMATICS SECTION : (Maimum Marks : 4) This section contains SIX (06) questions. Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. Answer to each question will be evaluated according to the following marking chosen. Full Marks : Partial Marks : Partial Marks : Partial Marks : Zero Marks : +4 If only (all) the correct option(s) is (are) chosen. + If all the four options are correct but ONLY three options are chosen. + If three or more options are correct but ONLY two options are chosen, both of which are correct options. + If two or more options are correct but ONLY one option is chosen and it is a correct option. 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : In all other cases. For Eample : If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in + marks. Selecting only one of the three correct options (either first or third or fourth option), without selecting any incorrect option (second option in this case), will result in + marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in marks.. For a non-zero comple number z, let arg(z) denote the principal argument with < arg(z). Then, which of the following statement(s) is (are) FALSE? (A) Arg( i) = 4, where i = (B) The function f : R (, ], defined by f(t) = arg( + it) for all t R, is continuous at all points of R, where i = z (C) For any two non-zero comple numbers z and z, arg arg(z ) arg(z ) z is an integer multiple of (D) For any three given distinct comple numbers z, z and z, the locus of the point z satisfying (z z )(z z ) the condition arg (z z )(z z ) =, lies on a straight line. (ABD) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE #

4 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS Sol. (A) Arg ( i) = 4 (B) f(t) = Arg( + it) tan t t 0 ( tan t) t 0 If is discontinuous at t = 0 z (C) Arg z Arg z + Argz Arg z z = Arg z Arg z + n so the epression becomes n z (D) (z z )(z z ) Arg (z z )(z z = If is circle z z z. In a triangle PQR, let PQR = 0º and the sides PQ and QR have lengths 0 and 0, respectively. Then, which of the following statement(s) is (are) TRUE? (A) QPR = 45º (B) The area of the triangle PQR is 5 and QRP = 0º (C) The radius of the incircle of the triangle PQR is 0 5 (D) The area of the circumcircle of the triangle PQR is 00 (BCD) (PR) Sol. cosq = = 400 (PR) PR = (PR).0.0 = (PQ)(QR) sinq = 0.0 = r = 5( ) 0 5 s (0 0 ) by sine rule R = 0º sinr sinq (circumradius) = PR 0 circumradius = 0 sinq / Hence area of circumcircle = R = 00 REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE #

5 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS. Let P : + y z = and P : + y + z = be two planes. Then, which of the following statement(s) is (are) TRUE? (A) The line of intersection of P and P has direction ratios,, (B) The line 4 9 = y 9 = z is perpendicular to the line of intersection of P and P (C) The acute angle between P and P is 60º (D) If P is the plane passing through the point (4,, ) and perpendicular to the line of intersection of P and P, then the distance of the point (,, ) from the plane P is (CD) Sol. Direction ratio of common line is n n ˆi ˆj kˆ = ˆ i() ˆ j() k() ˆ = (i ˆ ˆ j k) ˆ (B) 4 / = y / = z This is to line of intersection (C) cos = i = 6 6 = 6 = = (D) P : y + z = satisfy (4,, ) 4 = y + z = 0 (,,) 4. For every twice differentiable function f : R [, ] with (f(0)) + (f(0)) = 85, which of the following statement(s) is (are) TRUE? (A) There eist r, s R, where r < s, such that f is one-one on the open interval (r, s) (B) There eists 0 ( 4, 0) such that f( 0 ) (C) lim f() (D) There eists ( 4, 4) such that f() + f() = 0 and f() 0 (ABD) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE #

6 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS Sol. f (0) + (f(0)) = 85 f : R [, ] (A) This is true of every continuous function f( 4) f(0) (B) f(c) = 4 0 f( 4) f(c) = f(0) 4 (C) f( 4) f(0) 4 f(-4) f(0) 4 This f(c) f() = lim Note f() should have a bound which can be concluded by considering 85 f() = sin 85 f() = 85 cos f (0) + (f(0) ) = 85 and lim f() does not eist (D) Consider H() = f () + (f() H(0) = 85 By (B) choice there eists some 0 such that (f( 0 )) for some 0 in (-4, 0) hence H( 0 ) = f ( 0 ) + (f( 0 )) 4 + H( 0 ) 5 Hence let p (-4, 0) for which H(p) = 5 (note that we have considered p as largest such negative number) similarly let q be smallest positive number (0, 4) such that H(q) = 5 Hence By Rolle's theorem is (p, q) H(c) = 0 for some c ( 4, 4) and since H() is greater than 5 as we move from = p to = q and f () 4 (f()) in (p, q) Thus H(c) = 0 ff + ff = 0 so f + f = 0 and f 0 5. Let f : R R and g : R R be two non-constant differentiable functions. If f() = (e (f() g() ) g() for all R, and f() = g() =, then which of the following statement(s) is (are) TRUE? (A) f() < log e (B) f() > log e (C) g() > log e (D) g() < log e (BC) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 4

7 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS Sol. f ' () = e f() g() g'() : f() = g() = e f() = e g() + c e f(). f '() = e g(). g'() f() g() d(e ) d(e ) e f() = e g() + c = e = e g() + c = e f() = c e g() > n e f() = e e g() e f() = e e g() f() > n e e f() = e g() e e g() + e f() = e e g() < e g() < n 6. Let f : [0, ) R be a continuous function such that f() = + 0 e t f(t) dt for all [0, ). Then, which of the following statement(s) is (are)) TRUE? (A) The curve y = f() passes through the point (, ) (B) The curve y = f() passes through the point (, ) (C) The area of the region {(, y) [0, ] R : f() y (D) The area of the region {(, y) [0, ] R : f() y is is 4 4 (BC) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 5

8 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS Sol. f() = + t e f(t) dt f(). e = ( ).e + 0 t e f(t) dt 0 f '()e e. f() =. e ( ). e + e. f() f '() f() = ( ) I.F. = e y. e = ( ).e d I II y. e = ( ). e e d y. e ( )e e = c y.e = ( ) c.e y = ( ) + c.e y = ( ) + c. e put = 0 = + c c = 0 y = which passes through point (, ) y= y=f() Now required area =. () SECTION : (Maimum Marks : 4) This section contains EIGHT (08) questions. The answer to each question is NUMERICAL VALUE. For each question, enter the correct numerical value (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.5, 7.00, 0., 0.0,,0.7, 7.0) using the mouse and the onscreen virtual numeric keypad in the place designated to enter the answer. Answer to each question will be evaluated according to the following marking scheme : Full Marks : + If ONLY the correct numerical value is entered as answer. Zero Marks : 0 In all other cases. REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 6

9 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS log (log 9) log The value of ((log 9) ) ( 7 ) is. (8) Sol. 7 (log 9) log (log 9) ( 7) log(log 9 ) log 4 (log 9).() = 4. = 8 8. The number of 5 digit numbers which are divisible by 4, with digits from the set {,,, 4, 5} and the (65) repetition of digits is allowed, is. Sol. Last two digits are,, 4, 5, 44 Number of numbers = = Let X be the set consisting of the first 08 terms of the arithmetic progression, 6,,., and Y be the set consisting of the first 08 terms of the arithmetic progression 9, 6,,.. Then, the number of elements in the set X Y is. (748) Sol. P = {, 6,,...} Q = {9, 6,,...} Common terms : 6, 5, 86 t p = 6 + (p )5 = 5p p 88.7 n (P Q) = n(p) + n (Q) n (P Q) = = The number of real solutions of the equation sin i i = cos i i i i ( ) i i lying in the interval, is. (Here, the inverse trigonometric functions sin and cos assume values in, and [0, ], () respectively). REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 7

10 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS i Sol. sin i i i = cos i i ( ) i i = ( ) ( ) = = 0 Let f() = f '() > 0 or = 0 f(0) = and f(/) = 9/8 so one root in 0, roots. For each positive integer n, let y n = n ((n + ) (n + ) (n + n))/n. () For R, let [] be the greatest integer less than or equal to. If. n n n n Sol. y n =... n n n log L = lim n n r n r log n = log( ) d = log d = log = log = log 4 e 0 lim y n n = L, then the value of [L] is L = 4 e [L] = REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 8

11 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS. Let a and b be two unit vectors such that a. b 0. For some, y R, let c a yb (a b). If c () = and the vector c is inclined at the same angle to both a and b, then the value of 8 cos is. Sol. c a yb a b & a.b 0 a c b c c.a c.b cos = y = cos c y a b = (4cos ) = 8cos + 8cos =. Let a, b, c be three non-zero real numbers such that the equation a cos + b sin = c,,, has two distinct real roots and with + = b. Then, the value of a is. (0.5) Sol. a cos + b sin = c, t a t + b t t a( t ) + 4bt = c ( + t ) t (c + a) 4bt + c a = 0 = 6 tan = b a = t t t t 4b c a c a = = c, where t = tan REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 9

12 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS 4. A farmer F has a land in the shape of a triangle with vertices at P(0, 0), Q(, ) and R(, 0). From this land, a neighbouring farmer F takes away the region which lies between the side PQ and a curve of the form y = n (n > ). If the area of the region taken away by the farmer F is eactly 0% of the area of PQR, then the value of n is (4) y y= n Sol. Q(,) P(0,0) R(,0) n ( )d 0 0 n = 4 n 0 5 n n 0 0 n 0 SECTION : (Maimum Marks : ) This section contains TWO (0) paragraphs. Based on each paragraph, there are TWO (0) questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding to the correct answer. Answer to each question will be evaluated according to the following marking scheme : Full Marks : + If ONLY the correct option is chosen. Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : In all other cases. PARAGRAPH X Let S be the circle in the y-plane defined by the equation + y = 4. (There are two questions based on PARAGRAPH X, the question given below is one of them) 5. Let E E and F F be the chords of S passing through the point P 0 (, ) and parallel to the -ais and the y-ais, respectively. Let G G be the chord of S passing through P 0 and having slope. Let the tangents to S at E and E meet at E, the tangents to S at F and F meet at F, and the tangents to S at G and G meet at G. Then, then, the points E, F, and G lie on the curve (A) +y = 4 (B) ( 4) + (y 4) = 6 (C) ( 4)(y 4) = 4 (D) y = 4 (A) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE # 0

13 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS Sol. Tangent at E and E are y 4 and y 4 They intersect at E (0, 4) E (0, 4) (,) E E (, ) 0 F (, ), F (, ), F (4, 0) G (0, ), G (, 0), G (, ) E, F, G lie on line + y = 4 PARAGRAPH X Let S be the circle in the y-plane defined by the equation + y = 4. (There are two questions based on PARAGRAPH X, the question given below is one of them) 6. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate aes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve (A) ( + y) = y (B) / + y / = 4/ (C) + y = y (D) + y = y (D) Sol. Let P( cos, sin ) Tanget is cos + y sin = M, 0 cos, N 0, cos = cos and y = sin + y PARAGRAPH A = + y = y There are five students S, S, S, S 4 and S 5 in a music class and for them there are five seats R, R, R, R 4 and R 5 arranged in a row, where initially the seat R i is allotted to the student S i, i =,,, 4, 5. But, on the eamination day, the five students are randomly allotted the five seats. (There are two questions based on PARAGRAPH A, the question given below is one of them) 7. The probability that, on the eamination day, the student S gets the previously allotted seat R, and NONE of the remaining students gets the seat previously allotted to him/her, is (A) 40 (B) 8 (C) 7 40 (D) 5 (A) REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE #

14 JEE (ADVANCED) 08 DATE : PAPER- MATHEMATICS 4!!!! 4! 9 Sol. Probability = 5! 0 40 PARAGRAPH A There are five students S, S, S, S 4 and S 5 in a music class and for them there are five seats R, R, R, R 4 and R 5 arranged in a row, where initially the seat R i is allotted to the student S i, i =,,, 4, 5. But, on the eamination day, the five students are randomly allotted the five seats. (There are two questions based on PARAGRAPH A, the question given below is one of them) 8. For i =,,,4, let T i denote the event that the students S i and S i+ do NOT sit adjacent to each other on the day of the eamination. Then, the probability of the event T T T T 4 is (A) 5 (B) 0 (C) 7 60 (D) 5 (C) Sol. Total cases = 5! favorable ways = } = 4 Probability = 4 0 REG. & CORPORATE OFFICE : CG Tower, A-46 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC0409 This solution was download from Resonance JEE ADVANCED 08 Solution portal PAGE #

Memory Based Questions & Solutions

PAPER- (B.E./B. TECH.) JEE (Main) 09 COMPUTER BASED TEST (CBT) Memory Based Questions & Solutions Date: 09 April, 09 (SHIFT-) TIME : (9.0 a.m. to.0 p.m) Duration: Hours Ma. Marks: 60 SUBJECT : MATHEMATICS