IIT-JEE 2012 : PAPER-2

Transcription

1 IIT-JEE : PAPER- Date : 8-4- Duration : Hours Ma. Marks : 98 Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose. INSTRUCTIONS : A. General. This booklet is your Question Paper. Do not break the seals of this booklet before being instructed to do so by the invigilators.. The question paper CODE is printed on the right hand top corner of this page and on the back page of this booklet.. Blank spaces and blank pages are provided in this booklet for your rough work. No additional sheets will be provided for rough work. 4. Blank papers, clipboards, log tables, slide rules, calculators, cameras, cellular phones, pagers, and electronic gadgets are NOT allowed inside the eamination hall. 5. Answers to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided separately. You should not separate these parts. The invigilator will separate them at the end of the eamination. The upper sheet is a rnachine-gradable Objective Response Sheet (ORS) which will be taken back by the invigilator. You will be allowed to take away the bottom sheet at the end of the eamination. 6. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficient pressure so that the impression is created on the bottom sheet. 7. DO NOT TAMPER WITH/MUTILATE THE ORS OR THE BOOKLET. 8. On breaking the seals of the booklet check that and all the 6 questions and corresponding answer choices are legible. Read carefully the instructions printed at the beginning of each section. B. Filling the Right Part of the ORS 9. The ORS has CODES printed on its left and right parts.. Check that the same CODE is printed on the ORS and on this booklet. IF IT IS NOT THEN ASK FOR A CHANGE OF THE BOOKLET. Sign at the place provided on the ORS affirming that you have verified that all the codes are same.. Write your Name, Registration Number and the name of eamination centre and sign with pen in the boes provided on the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubble UNDER each digit of your Registration Number in such a way that the impression is created on the bottom sheet. Also darken the paper CODE given on the right side of ORS (R4). C. Question Paper Format The question paper consists of parts (Physics, Chemistry and Mathematics). Each part consists of three sections.. Section I contains 8 multiple choice questions. Each question has four choices (A). (B), (C) and (D) out of which ONLY ONE is correct.. Section II contains paragraphs each describing theory, eperiment, data etc. There are 6 multiple choice questions relating to three paragraphs with questions on each paragraph. Each question of a particular paragraph has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 4. Section III contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY or MORE are correct. D. Marking Scheme 5. For each question in Section I and Section II, you will be awarded marks if you darken the bubble corresponding to the correct answer ONLY and zero () marks if no bubbles are darkened. In all other cases, minus one ( ) mark will be awarded in these sections. 6. For each question in Section III, you will be awarded 4 marks If you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. No negative marks will be awarded for incorrect answer(s) in this section. Write your Name, Registration Number and sign in the space provided on the back page of this booklet

2 PART - I : PHYSICS PHYSICS SECTION - I : Single Correct Answer Type This section contains 8 multiple choice questions, Each question has four choices, (A), (B), (C) and (D) out of which ONLY ONE is correct.. A loop carrying current lies in the -y plane as shown in the figure. the unit vector kˆ is coming out of the plane of the paper. the magnetic moment of the current loop is : (A) (B) a kˆ (B) a kˆ (C) a kˆ (D) ( )a kˆ Area a + a 4 a + a A a kˆ. A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If c is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is (A) more than half filled if c is less than.5 (B) more than half filled if c is less than. (C) half filled if c is less than.5 (D) less than half filled if c is less than.5 (A) Let outer volume of shell is V Let inner volume of shell is V i Let volume of water inside the shall is v. V Vg + C (V V i )g g [Equlibrium] V + C (V V i ) V Page #

3 PHYSICS C (V V i ) C V V V V i V V if C < V V V Vi < V V V < V i V < V i V > V i so (A). An infinitely long hollow conducting cylinder with inner radius R/ and outer radius R carries a uniform current density along is length. The magnitude of the magnetic field, B as a function of the radial distance r from the ais is best represented by : (A) (B) (C) (D) (D) Case-I < R B Case-II B.d I B R R R J B J R 4 Page #

4 Case-III B. d I > R PHYSICS B R R J B J R B so JR 8 4. Consider a disc rotating in the horizontal plane with a constant angular speed about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc completed 8 rotation. (ii) their range is less than half disc radius, and (iii) remains constant throughout. Then (A) P lands in the shaded region and Q in the unshaded region (B) P lands in the unshaded region and Q in the shaded region (C) Both P and Q land in the unshaded region (D) Both P and Q land in the shaded region (C) IIT answer C or D. Page # 4

5 PHYSICS Since distance of particle P from point O is initially decreasing then increasing so, its angular velocity will initially increase then decrease. So, angle swept by P is more than angle swept by disc. So it will fall in unshaded portion. Since distance of particle Q from O is continuously increasing so its is continuously decreasing. So angle swept by Q is less than angle swept by disc. So it will fall in unshaded portion. 5. A student is performing the eperiment of Resonance Column. The diameter of the column tube is 4cm. The distance frequency of the tuning for k is 5 Hz. The air temperature is 8 C in which the speed of sound is 6 m/s. The zero of the meter scale coincides with the top and of the Resonance column. When first resonance occurs, the reading of the water level in the column is (A) 4. (B) 5. (C) 6.4 (D) 7.6 (B) V 4( e) f + e V 4f here so V e 4f e (.6)r (.6) (). cm cm In the given circuit, a charge of +8 C is given to the upper plate of the 4F capacitor. Then in the steady state, the charge on the upper plate of the F capacitor is : (A) + C (B) +4 C (C) +48 C (D) +8 C (C) Page # 5

6 PHYSICS q C C C.Q q C 7. Two identical discs of same radius R are rotating about their aes in opposite directions with the same constant angular speed. The disc are in the same horizontal plane.at time t, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is v r. as function of times best represented by (A) (B) (C) (D) (A) v r v sin ) v sin t) Page # 6

7 PHYSICS 8. Two moles of ideal helium gas are in a rubber balloon at C.The balloon is fully epandable and can be assumed to require no energy in its epansion. The temperature of the gas in the balloon is slowly changed to 5 C.The amount of heat required in raising the temperature is nearly (take R 8. J/mol.K) (A) 6J (B) 4 J (C) 4 J (D) 8 J (D) Q nc P T f R R R R T J SECTION II : Paragraph Type This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B) (C) and (D) out of which ONLY ONE is correct. Paragraph for Questions 9 and The - decay process, discovered around 9, is basically the decay of a neutron (n), In the laboratory, a proton (p) and an electron (e ) are observed as the decay products of the neutron. therefore, considering the decay of a neutron as a tro-body dcay process, it was predicted theoretically that thekinetic energy of the electron should be a constant. But eperimentally, it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process, i.e. n p + e + e, around 9, Pauli eplained the observed electron energy spectrum. Assuming the anti-neutrino ( e ) to be massless and possessing negligible energy, and neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maimum kinetic energy of the lectron is.8 6 ev. The kinetic energy carried by the proton is only the recoil energy. 9. What is the maimum energy of the anti-neutrino? (A) Zero (B) Much less than.8 6 ev (C) Nearly.8 6 ev (C) KE ma of Q.8 6 ev KE P KE KE Q KE P is almost zero When KE then KE Q KE P Q (D) Much larger than.8 6 ev Page # 7

8 . If the anti-neutrino had a mass of ev/c (where c is the speed of light) instead of zero mass, what should be the range of the kinetic energy, K, of the electron? (A) K.8 6 ev (B). ev K.8 6 ev (C). ev K <.8 6 ev (D) K <.8 6 ev (D) KE Q KE P KE KE Q Paragraph for Question and Most materials have therefractive inde, n >. So, when a light ray from air enters a naturally occurring material, then by Snells' law, sin sin n n, it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the c refractive inde of the medium is given by the relation, n rr where c is the speed of electromagnetic waves in vacuum, v its speed in the medium, r and r are negative, one must choose the negative root of n. Such negative refractive inde materials can now be artificially prepared and are called meta-materials. They ehibit significantly different optical behavior, without violating any physical laws. Since n is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.. Choose the correct statement. (A) The speed of light in the meta-material is v c n (B) The speed of light in the meta-material is v c n (C) The speed of light in the meta-material is v c. (D) The wavelength of the light in the meta-material ( m ) is given by m air n, where air is the wavelength of the light in air. (B) n v c for metamaterials v c n. For light incident from air on a meta-material, the appropriate ray diagram is : PHYSICS (A) (B) (C) (D) (C) Page # 8

9 (C) Meta material has a negative refractive inde PHYSICS n (C) sin sin n n is negative negative Paragraph for Q. No. -4 The general motion of a rigid body can be considered to be a combination of (i) a motioon --- centre of mass about an ais, and (ii) its motion about an instantanneous ais passing through center of mass. These aes need not be stationary. Consider, for eample, a thin uniform welded (rigidly fied) horizontally at its rim to a massless stick, as shown in the figure. Where disc-stick system is rotated about the origin ona horizontal frictionless plane with angular sp---, the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass the disc about the z-ais, and (ii) a rotation of the disc through an instantaneous vertical ais pass through its centre of mass (as is seen from the changed orientation of points P and Q). Both the motions have the same angular speed in the case. Z Q P P Q Y X Now consider two similar systems as shown in the figure: case (a) the disc with its face ver--- and parallel to -z plane; Case (b) the disc with its face making an angle of 45 with -y plane its horizontal diameter parallel to -ais. In both the cases, the disc is weleded at point P, and systems are rotated with constant angular speed about the z-ais. Z Z Q Q P Y P 45 Y X Case (a) X Case (b). Which of the following statement regarding the angular speed about the istantaneous ais (passing through the centre of mass) is correct? (A) It is for boht the cases (B) it is for case (a); and w for case (b). (C) It is for case (a); and (D) for case (b) (D) It is for both the cases Page # 9

10 PHYSICS Angular Velocity of rigid body about any aes which are parallel to each other is same. So angular velocity is. 4. Which of the following statements about the instantaneous ais (passing through the centre of mass) is correct? (A) It is vertical for both the cases (a) and (b). (B) It is verticle for case (a); and is at 45 to the -z plane and lies in the plane of the disc for case (b) (C) It is horizontal ofr case (a); and is at 45 to the - z plane and is normal to the plane of the disc for case (b). (D) It is vertical of case (a); and is at 45 to the - z plane and is normal to the plane of the disc for case (b). (A) Since z- coordinate of any particle is not changing with time so ais must be parellel to z ais. SECTION III : Multiple Correct Answer(s) Type This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 5. Two solid cylinders P and Q of same mass and same radius start rolling down a fied inclined plane form the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the ais. Which statement (s) is (are) correct? (A) Both cylinders P and Q reach the ground at the same time (B) Cylinder P has larger linear acceleration than cylinder Q. (C) Both cylinder Q reaches the ground with same translational kinetic energy. (D) Cylinder Q reaches the ground with larger angular speed. (D) I P > I Q gsin a P I P mr gsin a Q I Q mr a P < a Q V u + at t a t P > t Q V u + as v a V P < V Q Translational K.E. mv TR KE P < TR KE Q V R V P < Q 6. A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement (s) is (are) : (A) the emf induced in the loop is zero if the current is constant. (B) The emf induced in the loop is finite if the current is constant. (C) The emf induced in the loop is zero if the current decreases at a steady rate. (D) Theemf induced in the loop is finite if the current decreases at a steady rate. (A,C) Page #

11 PHYSICS () loop for all cases so induced emf 7. In the given circuit, the AC source has rad/s. considering the inductor and capacitor to be ideal, the correct choice (s) is(are) (A) The current through the circuit, is. A (B) The current through the circuit, is. A. (C) The voltage across resistor V (D) The voltage across 5 resistor V (A,C or C) C F, C 6 ()( ) X C, X L L () (.5) 5 Z C Z L sin t i sin (t + /4) i 5 sin (t + /4) I 5 sin (t /4) I (. ) (.4) Page #

12 PHYSICS (.) (I) rms 5. A (V ) rms (I ) rms ). V V 5 ) rms.4 5 V Since rms. A so A may or may not be correct. 8. Si point charges are kept at the vertices of a regular heagon of side L q and centre O, as shown in the figure. Given that K 4 following statement (s) is (are) correct? L, which of the (A) the elecric field at O is 6K along OD (B) The potential at O is zero (C) The potential at all points on the line PR is same (D) The potential at all points on the line ST is same. (A,B,C) E 6 K (along OD) V Potential on line PR is zero (A), (B), (C) 9. Two spherical planets P and Q have the same unfirom density, masses M P and M Q, an surface areas A and 4A, respectively. A spherical planet R also has unfirom density and its mass is (M P + M Q ). The escape velocities from the planets P, Q and R, are V P, V Q and V respectivley. Then (A) V Q > V R > V P (B) V R > V Q > V P (C) V R /V P (D) V P /V Q (B,D) V es 4 GM.G. R R R V es R Sarface area of P A 4R P Surface area of Q 4A 4 R Q 4G R R Q R p mass R is M R M P + M Q 4 R R 4 R P + 4 R Q Page #

13 R R R P + R Q 9R P PHYSICS R R 9 / R P R R > R Q > R P Therefore V R > V Q > V P V V R P 9 / and V P V Q. The figure shows a system consisting of (i) a ring of outer radius R rolling clockwise without slipping on a horizontal surface with angular speed and (ii) an inner disc of radius R rotating anti-clockwise with angular speed /. The ring and disc are separated b frictionaless ball bearings. The system is in the -z plane. The point P on the inner disc is at distance R from the origin, where OP makes an angle of º with the horizontal. Then with respect to the horizontal surface, (A) the point O has linear velocity R î. (B) the point P has a linear velocity 4 R î + Rkˆ 4 (C) the point P has linear velocity Rw 4 î Rkˆ 4 (D) The point P has a linear velocity 4 Rw î + Rwkˆ. 4 (A,B) V R î V P (R R cos 6º) î + R sin 6 ĵ R R î î 4 4 Page #

14 PART - II : CHEMISTRY CHEMISTRY SECTION - I : Single Correct Answer Type This section contains 8 multiple choice questions, Each question has four choices, (A), (B), (C) and (D) out of which ONLY ONE is correct.. NiCl {P (C H 5 ) (C 6 H 5 )} ehibits temperature dependent magnetic behaviour (paramagnetic/ diamagnetic). the coordination geometries of Ni + in the paramagnetic and diamagnetic states are respectively (A) tetrahedral and tetrahedral (B) square planar and square planar (C) tetrahedral and square planar (D) square planar and tetrahedral (C) [NiCl {PEt Ph}] contains Ni + with electronic configuration Ni + [Ar] d 8 4s In high spin state, it is paramagnetic, sp hybridised, tetrahedral. In low spin state, it is diamagnetic, dsp, square planar.. The reaction of white phosphorous with aqueous NaOH gives phosphine along with another phosphorus containing compound. The reaction type; the oidation states of phosphorous in phosphine and the other product are respectively (A) redo reaction; and 5 (B) redo reaction ; and + 5 (C) disproportionation reaction ; and + 5 (D) disproportionation reaction; and + (C) Oidation states of P in Na PO 4 & PH are +5 & respectively. It is a disproportionation reaction.. In the cyanide etraction process of silver from argentite ore, the oidizing and reducing agents used are (A) O and CO respectively (C) HNO and Zn dust respectively. (B) (B) O and Zn dust respectively (D) HNO and CO respectively In etraction of silver, Ag S is leached with KCN in presence of air : Ag S + NaCN + O Na [Ag(CN) ] + Na S O Thus, O is oidant. Ag(CN) + Zn [Zn(CN) 4 ] + Ag Page # 4

15 4. The compound that undergoes decarbolylation most readily under mild condition is CHEMISTRY (A) (B) (C) (D) (B) In decarboylation, -carbon acquires charge. Whenever charge is stabilized, decarboylation becomes simple. In (B), it is stabilized by m & of C O, which is best amongst the options offered, 5. Using the data provided, calculate the multiple bond energy (kj mol ) of a CC bond C H. That energy is (take the bond energy of a C H bond as 5 kj mol ) C(s) + H (g) C H (g) H 5 kj mol C(s) C (g) H 4 kj mol H (g) H(g) H kj mol (A) 65 (B) 87 (C) 865 (D) 85 (D) H +4 + (5 ) CC + 5 CC KJ/mol. 6. The shape of XeO F molecule is (A) trigonal bipyramidal (B) square plannar (C) tetrahedral (D) see-saw (D) XeO F has trigonal bipyramidal geometry. Due to presence of lone pair on equitorial position, the shape is see-saw. Page # 5

16 7. The major product H in the given reaction sequence is CHEMISTRY CH CH CO CH G CN 95% H SO4 Heat H (A) (B) (C) (D) (A) CN HCN 95% H SO 4 8. For a dilute solution containing.5 g of a non- volatile non- electrolyte solute in g of water, the elevation in boiling point at atm pressure is ºC. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure (mm of Hg) of the solution is (take K b.76 K kg mol ) (A) 74 (B) 74 (C) 76 (D)78 (A) T b ºC ; m a.5 g T b K b m m solvent g K b.76 K. kg. mol P solution?.76 m m.76 P P P m MM 76 P P P P P 76 P torr 75.6 torr 74 torr SECTION II : Paragraph Type This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B) (C) and (D) out of which ONLY ONE is correct. Page # 6

17 Paragraph for Questions Nos. 9 to CHEMISTRY The electrochemical cell shown below is a concentration cell. M M + (saturated solution of a sparingly soluble salt,mx ) M + (. mol dm ) M The emf of the cell depends on the difference in concetration of M + ions at the two electrodes. The emf of the cell at 98 is.59 V 9. The solubility product (K sp ; mol dm 9 ) of MX at 98 based on the information available the given concentration cell is (take. R 98/F.59 V ) (A) 5 (B) 4 5 (C) (D) 4 (B) M M + (aq) M + (aq) M. M Anode : M M + (aq) + e Cathode : M + (aq) + e M M + (aq) c M + (aq) a E cell.59 M (aq) log a M (aq) log a M (aq) log a M + (aq) a solubility s K sp 4s 4 ( 5 ) 4 5. The value of G (kj mol ) for the given cell is (take F 965 C mol ) (A) 5.7 (B) 5.7 (C).4 (D).4 (D) G nfe cell kj/mole.4 kj/mole Paragraph for Questions Nos. to Bleaching powder and bleach solution are produced on a large scale and used in several house hold products. The effectiveness of bleach solution is often measured by iodometry.. 5 ml of household bleach solution was mied with ml of.5 M KI and ml of 4N acetic acid. In the titration of the liberated iodine, 48 ml of.5 N Na S O was used to reach the end point. The molarity of the household bleach solution is (A).48 M (B).96 M (C).4 M (D).4 M (C) milli mole of Hypo.5 48 milli mole of Cl Page # 7

18 CHEMISTRY milli mole of Cl milli mole milli mole of Cl milli mole of CaOCl So, molarity 6 M 5.4 M. Bleaching powder contains a salt of an ooacid as one of its components. The anhydride of that ooacid is (A) Cl O (B) Cl O 7 (C) ClO (D) Cl O 6 (A) CaOCl Ca(OCl)Cl OCl Hypochlorite ion which is anion of HOCl Anhydride of HOCl Cl O Paragraph for Questions Nos. to 4 In the following reactions sequence, the compound J is an intermediate. (CHCO) O CH COONa J (i) H,Pd / C (ii) SOCl K (ii) anhyd. AlCl J (C 9 H 8 O ) gives effervescence on treatment with NaHCO and positive Baeyer's test. The compound K is (A) (B) (C) (D) (C) 4. The compound is (A) (B) (C) (D) (A) ( to 4) (CH CO) O CHCOONa H Pd / C SOCl AlCl Page # 8

19 CHEMISTRY SECTION III : Multiple Correct Answer(s) Type This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 5. With respect to graphite and diamond, which of the statement(s) given below is (are) correct? (A) Graphite is harder than diamond. (B) Graphite has higher electrical conductivity than diamond (C) Graphite has higher thermal conductivity than diamond (D) Graphite has higher C C bond order than diamond (BD) (A) Diamond is harder than graphite. (B) Graphite is better conductor of electricity than diamond. (C) Diamond is better conductor of heat than graphite. (D) Bond order of graphite ( ~.5) > Bond order of diamond ( ) 6. The given graph / data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and pressure. Which of the following choice (s) about I, II, III and IV is (are) correct (i) (ii) (iii) (iv) (A) I is physisorption and II is chemisorption (C) IV is chemisorption and II is chemisorption (B) I is physisorption and III is chemisorption (D) IV is chemisorption and III is chemisorption Page # 9

20 CHEMISTRY (AC) In physisorption on increasing temperature at constant pressure, adsorption decreases while in chemical adsorption on increasing temperature due to requirement of activation energy adsorption will increase at same pressure. So, I is physisorption while II is chemisorption. III is physical adsorption as on increasing temperature, etent of adsorption is decreasing. IV is representing enthalpy change (which is high) during chemical adsorption (due to bond formation) So, is valid for chemical adsorption. So, answer is (A) and (C) 7. The reversible epansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct? (A) T T (B) T > T (C) w isothermal > w adiabatic (AD) (D) U isothermal > U adiabatic (A) T T (due to isothermal) (B) T > T (incorrect) cooling will take place in adiabatic epansion) (C) W isothermal > W adiabatic { with sign, this is incorrect} (D) U isothermal > U adiabatic ve So, answer is (A) and (D) Page #

21 8. For the given aqueous reaction which of the statement(s) is (are) true? CHEMISTRY (A) The first reaction is a redo reaction (B) White precipitate is Zn [Fe(CN) 6 ] (C) Addition of filtrate to starch solution gives blue colour. (D) White precipitate is soluble in NaOH solution (ACD) (D) with NaOH K Zn [Fe(CN) 6 ] + NaOH [Zn(OH) 4 ] (aq) + [Fe(CN) 6 ] 4 (aq) 9. With reference to the scheme given, which of the given statments(s) about T, U, V and W is (are) correct? (A) T is soluble in hot aqueous NaOH (B) U is optically active (C) Molecular formula of W is C H 8 O 4 (D) V gives effervescence on treatment with aqueous NaHCO Page #

22 (ACD) CHEMISTRY 4. Which of the given statement(s) about N, O, P and Q with respect to M is (are) correct? (A) M and N are non-mirror image stereoisomers (B) M and O are identical (C) M and P are enantiomers (D) M and Q are identical (ABC) Page #

23 PART - III : MATHEMATICS MATHEMATICS Section I : Single Correct Answer Type This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 4. The equation of a plane passing through the line of intersection of the planes + y + z and y + z and at a distance from the point (,, ) is (A) 5 y + z 7 (B) y (C) + y + z (D) y (A) Equation of required plane ( + y + z ) + ( y + z ) ( + ) + ( )y + ( + )z ( + ) distance from point (,, ) ( ) ( ) ( ) equation of required plane 5 y + z 7 4. If a and b are vectors such that a b 9 and a (î ĵ 4kˆ ) ( î ĵ 4kˆ ) b, then a possible value of (a b) ( 7î ĵ kˆ ) is (A) (B) (C) 4 (D) 8 (C) Let c î ĵ 4kˆ a c c b (a b) c (a b) c Page #

24 Let (a b) c a b c MATHEMATICS 9. 9 a b ( î ĵ 4kˆ ) Now ( a b).( 7î ĵ kˆ ) ± ( ) ± 4 4. Let PQR be a triangle of area with a, b 7 and c 5, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then sinp sinp equals sinp sinp (A) 4 (B) 45 4 (C) 4 (D) 45 4 (C) a QR b 7 PR c 5 PQ s a b c sinp sinpcosp sinp sinpcosp sin P( cos P) sin P( cos P) cosp cosp P sin P cos P tan (s b)(s c) s(s a) (s b) (s c) Four fair dice D, D, D and D 4 each having si faces numbered,,,4,5 and 6 are rolled simultaneously. The probability that D 4 shows a number appearing on one of D, D and D is 9 (A) 6 8 (B) 6 5 (C) 6 7 (D) 6 (A) Favourable : D 4 shows a number and Page # 4

25 Page # 5 only of D D D shows same number or only of D D D shows same number or all of D D D shows same number Required Probability 6 6 ) C 5 C 5 5 C ( C ) 5 ( The value of the integral d cos n / / is (A) (B) 4 (C) 4 (D) (B) / / n cos d / cos d function anodd is n / / sin d sin / sin d 4 4 cos d cos 4 / / If P is a matri such that P T P + I, where P T is the transpose of P and I is the identity matri, then there eists a column matri X z y such that (A) PX (B) PX X (C) PX X (D) PX X MATHEMATICS

26 (D) P T P + I (P T ) T (P + I) T P P T + I P (P + I) + I P I P I PX IX X MATHEMATICS 47. Let a, a, a,... be in harmonic progression with a 5 and a 5. The least positive integer n for which a n < is (A) (B) (C) 4 (D) 5 (D) Corresponding A.P.,... ( th term) d d a n < 4 (n ) < n > 4.75 < n Let (a) and (a) be the roots of the equation 6 a a a Then lim a (a) and lim a (a) are where a >. 5 (A) and (B) and 7 (C) and 9 (D) and (B) (( + a) / ) + ((a+) / ) + ((a+) /6 ) let a + t 6 (t ) + (t ) + (t ) (t + ) + (t + t + ) + As a, t + + and Page # 6

27 MATHEMATICS Section II : Paragraph Type This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. Paragraph for Question Nos. 49 to 5 (t ) Let f() ( ) sin + for all R and let g() nt f(t) dt for all (, ). t 49. Which of the following is true? (A) g is increasing on (, ) (B) g is decreasing on (, ) (C) g is increasing on (, ) and decreasing on (, ) (D) g is decreasing on (, ) and increasing on (, ) (B) f() ( ) sin + : R (t ) g() ln t t f(t)dt ( ) g() ln f(). let ( ) () ln () [( ) ( ).] ( ) 4 ( ) ( ) ( ) ( ) () for (, ), () < g() < for (, ) 5. Consider the statements : P : There eists some R such that f() + ( + ) Q : There eists some R such that f() + ( + ) Then (A) both P and Q are true (B) P is true and Q is false (C) P is false and Q is true (D) both P and Q are false (C) f() + ( ) sin + + f() + ( + ) ( ) sin Page # 7

28 ( ) sin + + ( ) + ( ) cos which can never be possible P is not true Let H() f() + ( + ) H() f() + H() f() + 4 so H() has a solution so Q is true MATHEMATICS Paragraph for Question Nos. 5 to 5 Let a n denote the number of all n-digit positive integers formed by the digits, or both such that no consecutive digits in them are. Let b n the number of such n-digit integers ending with digit and c n the number of such n-digit integers ending with digit. 5. Which of the following is correct? (A) a 7 a 6 + a 5 (B) c 7 c 6 + c 5 (C) b 7 b 6 + c 6 (D) a 7 c 7 + b 6 (A) # a n #a n So A choice is correct consider B choice c 7 c 6 + c 5 c 5 c 4 + c is not true consider C choice b 7 b 6 + c 6 a 6 a 5 + a 4 is not true consider D choice a 7 c 7 + b 6 a 7 a 5 + a 5 which is not true Aliter using the Recursion formula a n a n + a n Similarly b n b n + b n and c n c n + c n n and a n b n + c n n so a, a, a, a 4 5, a b, b, b, b 4, b 5 5, b c, c, c, c 4, c 5, c using this b n c n n Page # 8

29 5. The value of b 6 is (A) 7 (B) 8 (C) 9 (D) (B) MATHEMATICS b 6 a 5 a C + C + + C Paragraph for Question Nos. 5 to 54 A tangent PT is drawn to the circle + y 4 at the point P(, ). A straight line L, perpendicular to PT is a tangent to the circle ( ) + y. 5. A common tangent of the two circles is (A) 4 (B) y (C) + y 4 (D) + y 6 (D) 54. A possible equation of L is (A) y (B) + y (C) y (D) + y 5 (A) Q.No. 5 to 54 Equation of tangent at y 4, 5. Page # 9

30 MATHEMATICS B divides C C in : eternally B(6, ) Hence let equation of common tangent is y m( 6) m y 6m length of r dropped from center (, ) radius 6m m m ± equation is + y 6 or y Equation of L is y + c length of perpendicular dropped from centre radius of circle C C, 5 y or y 5 Section III : Multiple Correct Answer(s) Type This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 55. Let X and Y be two events such that P(X Y), P(Y X) and P(X Y) 6. Which of the following is (are) correct? (A) P(X Y) (B) X and Y are independent (C) X and Y are not independent (D) P(X C Y) (AB) P(X/Y) P(X Y) P(Y) P(Y) P(Y/X) P(X Y) P(X) P(X) Page #

31 MATHEMATICS P(X Y) P(X) + P(Y) P(X Y) P(X Y) P(X) P(X) X and Y are independent A is correct B is correct P(X c Y) P(Y) P(X Y) 6 6 D is not correct t e 56. If f() (t )(t )dt for all (, ), then (A) f has a local maimum at (B) f is decreasing on (, ) (C) there eists some c (, ) such that f(c) (D) f has a local minimum at (ABCD) f() t e (t )(t )dt f() e ( ) ( ) (i) is local maima (ii) is local minima (iii) It is decreasing in (, ) (iv) f () e ( ) + e e [ + + ( )( )] f() f() e ( + 4 5) f() < and f() > so f(c) where c (, ) ( ) + e ( ) ( ) 57. For every integer n, let a n and b n be real numbers. Let function f : R R be given by f() a b n n sin, cos, for [n, n ] for (n, n), for all integers n. If f is continuous, then which of the following hold(s) for all n? (A) a n b n (B) a n b n (C) a n b n+ (D) a n b n (BD) f(n) an f(n ) an f(n ) bn a a n n b b n n So B is correct Page #

32 MATHEMATICS f(n ) an f((n ) ) an f((n ) ) bn a a a n n b b n n b n n So D is correct 58. If the straight lines y z and k 5 y z are coplanar, then the plane(s) containing k these two lines is(are) (A) y + z (B) y + z (C) y z (D) y z (BC) For co-planer lines [ a c b d] a (,, ), c (,, ) b î kĵ kˆ d 5î ĵ kkˆ Now 5 k k k ± 6 ĵ 6 kˆ for k n b d 4 ĵ 4 kˆ for k n b d so the equation of planes are r a.n so answer is (B,C) r a.n y z... () y + z... () 59. If the adjoint of a matri P is 4 4 7, then the possible value(s) of the determinant of P is (are) (A) (B) (C) (D) (AD) Let A [a ij ] adj A adj A ( 7) 4(6 7) + 4( ) 4 A 4 A 4 A ± Page #

33 6. Let f : (, ) R be such that f(cos 4) is (are) sec MATHEMATICS for,,. Then the value(s) of f 4 4 (A) (B) + (C) (D) + (AB) cos4 cos cos cos Now f(cos4) sec cos cos + cos f ± NOTE : Since a functional mapping can't have two images for pre-image /, so this is ambiguity in this question perhaps the answer can be A or B or AB or marks to all. Page #

34 ANSWER KEY IIT-JEE - : PAPER- CODE-,,,, 4, 5, 6, 7, 8 & 9 IIT-JEE ANSW ER KEY Paper- CODE Que B B D A C D C or D B A C or D A D A C D C or D B A B C D A C D C or D B A B D D 4 C or D C D C or D B A B D A B 5 B D C or D B A B D A C A 6 C C or D B A B D A C D A 7 A B A B D A C D C or D B 8 D A B D A C D C or D B D 9 C C D B C D B D A C D B A C D A C C D B B C D D A B C D D D C D C A D C D A C A D A C D B D A B C D 4 A D B C C C D C B C 5 D AC or C AC ABC D BD AB ABC AC or C AC 6 AC AC ABC D BD AB AC or C BD ABC AC or C 7 AC or C ABC D BD AB AC or C AC AB BD ABC 8 ABC D BD AB AC or C AC ABC D AB BD 9 BD BD AB AC or C AC ABC D AC D AB AB AB AC or C AC ABC D BD AC or C AC D C C B C D A B D B B C B C D A B D B C D B C D A B D B C B A 4 B D A B D B C B C C 5 D A B D B C B C D B 6 D B D B C B C D A C 7 B D B C B C D A B D 8 A B C B C D A B D B 9 B A C C B C C D A A D C A A D A A B C C C B D C A C B C D C A D B A C A D A B A C A A D C D A C A D 4 A C C B A B C A C B 5 BD AD AC ACD BD ACD ABC ACD AD AC 6 AC AC ACD BD ACD ABC AD ACD ACD AD 7 AD ACD BD ACD ABC AD AC ABC ACD ACD 8 ACD BD ACD ABC AD AC ACD BD ABC ACD 9 ACD ACD ABC AD AC ACD BD AC BD ABC 4 ABC ABC AD AC ACD BD ACD AD AC BD 4 A A C C D B A B D A 4 C C C D B A B D A D 4 C C D B A B D A C B 44 A D B A B D A C C A 45 B B A B D A C C D D 46 D A B D A C C D B C 47 D B D A C C D B A B 48 B D A C C D B A B C 49 B B D A B D A C A B 5 C A A B C A B B D A 5 A B C D A A B D C D 5 B C B A D B C A B A 5 D A B C A C A A B C 54 A D A B B B D B A B 55 AB BD ABCD BC AB AD AB BC BD ABCD 56 ABCD ABCD BC AB AD AB BD AD BC BD 57 BD BC AB AD AB BD ABCD AB AD BC 58 BC AB AD AB BD ABCD BC AB AB AD 59 AD AD AB BD ABCD BC AB ABCD AB AB Page # 4