PART I : PHYSICS (C) of middle plate remains constant. Hence heat received by second plate = heat emitted by it

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2 PART I : PHYSICS SECTION I : Single Correct Answ er Type This section contains m ultiple choice questions. Each question has four choices (A), (B), (C ) and (D) out of which ONLY ON E is correct.. Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high theermal conductivity. The first and third plates are maintained at temperatures T and T respectively. The temperature of the middle (i.e. second) plate under steady state condition is (A) (C) 65 4 T (B) T (C) Under steady state condition, temperature T of middle plate remains constant. Hence heat received by second plate = heat emitted by it 4 7 T T T T T T (D) 97 4 T 97 T 4 T. Consider a thin spherical shell of radius r which its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field and the electric potential V r E r with the distance r from the centre, is best represented by which graph? (A) (B) (C) (D) (D) Inside a shell of uniform charge, the electric field is zero. Outside it varies as E. r Inside the shell of charge, potential is constant and outside V. r Hence (D is the correct answer.. Young s double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are G, R and, respectively. Then, B (D) (A) G B R (B) B G R (C) R B G (D) R G B

3 D Since R G B R G B d 4. Two large vertical and parallel metal plates having a separation of cm are connected to a DC voltage source of potential difference X. A proton is released at rest midway between the two plates. It is found to move at 45 to the vertical JUST after release. Then X is nearly (A) 5 V (B) 7 V (C) 9 V (D) V (C) There are two forces acting on the particle () Electrical force = F e and () Gravitational force = mg As it moves at 45 hence Fe mg X. e mg d d.6 X mg e volts. 5. A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index n of the first lens is.5 and that of the second lens is.. Both the curved surfaces are of the same radius of curvature R = 4 cm. For this bi-convex lens, for an object distance of 4 cm, the image distance will be (A) 8. cm (B) 4.cm (C).5cm (D).cm Key B Say the focal length of first half lens is f and that of second half is f.5 f 4 f 8cm Similarly f 7cm F f f 8 7 F = cm V u F V 4 V = 4 cm

4 SRIGAYATRI EUDCATIONAL INSTITUTIONS 6. A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undeergoing circular motion in the x y plane with centre at O and constant angular speed. If the angular momentum of the system, calculated about O and P are denoted by L and LP respectively, then (A) LO and LP do not vary with time. (B) LO varies with time while LP remains constant (C) LO remains constant while LP varies with time (D) LO and LP both vary with time (C) The resultant force (of tension and wieght) passes through the point O. Therefore torque about O is zero. d L L constant vector dt The torque of resultant force about P varies in direction as the mass rotates in a circle. Hence LP is not a constant vector 7. A mixture of moles of helium gas (atomic mass = 4 amu) and mole of argon gas (atomic mass = 4 v helium rms amu) is kept at K in a container. The ratio of the rms speeeds vrms argon is (A). (B).45 (C).4 (D).6 (D) V V rms rms Helium Argon M M V rms RT M 8. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with cosntant angular speed, as shown in the figure. At time t, a small insect starts from O and moves with constant speed v with respect to the rod towards the other end. It reaches the end of the rod at t T and stops. The angular speed of the system remains throughout. The magnitude of the torque on the system about O, as a function of time is best represented by which plot?

5 (A) (B) (C) (D) (B) rod J I Irod Iinsect I mv t dj mv t dt t When insect stops, I becomes constant, hence J. Hence B is the correct answer 4MLg 9. In the determination of Young s modulus Y ld by using Searle s method, a wire of length L m and diameter d.5 mm is used. For a load M =.5 kg, an extension l.5 mm in the length of the wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively. They have the same pitch of.5 mm. The number of divisions on their circular scale is. The contributions to the maximum probable error of the Y measurement (A) Due to the error in the measurements of d and l are the same (B) Due to the error in the measurement of d is twice that due to the error in the measurement of l. (C) Due to the error in the measurement of l is twice that due to the error in the measurement of d. (D) Due to the error in the measurement of d is four times that due to the error in the measurement of l. (A) Y l d Y l d M, L, g have fixed values pitch l no. of divisions an circular scale d l 5 mm Now l relative error in l 5 mm l.5mm.5 l 5 mm l d Similarly l 5 d 5 mn 6.5 mn

6 . A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by. m and released from rest at t. It then executes simple harmonic motion with angular frequency / rad s. Simultaneously at t, a small pebble is projected with speed v from point P at an angle of 45 as shown in the figure. Point P is at a horizontal distance of m from O. If the pebble hits the block at t s, the value of v is (take g m / s ) (A) 5 m / s (B) 5 m / s (C) 5 m / s (D) 5 m / s (A) T 6s For the block, w T 6.5 s 4 4 Therefore in second it travels a distance less than A. Let x Acos wt for the block x. cos..m Therefore when the pebble hits the block, the block is at a ditance of 4.9m.m.m 5m from. Now time of flight = T = sec. V cos 45 Range = R 5 m 5m V 5 5 m / s

7 S ECTION II : Multiple Correct Answer(s) Type This section contains 5 m ultiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. For the resistance network shown in the figure, choose the correct option(s). (A) The current through PQ is zero (B) I A (C) The potential at S is less than that at Q (D) I A (A,B,C,D) Due to symmetry in the circuit P and Q are at same potential also S and T are at same potential. There is no current in PQ. Hence A is correct. RAB 4 I A 4 also, 6I I I I I I I I A I Q P A V V & Kirchhoff s rule V 4 V V 4 V P VP VQ VS S P S. A person below into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe (A) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is open (B) a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open (C) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed (D) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed

8 (B,D) Since at open end pressure node is formed hence high pressure pulse is relfected as low pressure pulse B is correct & A is wrong At closed end pressure antinode is formed. Hence high pressure pulse is relfected as high pressure pulse D is correct & C is wrong. Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields E E ˆ j and B B ˆ j. At time t, this charge has velocity v in the x-y plane, making an angle with the x-axis. Which of the following option(s) is (are) correct for time t? (A) If, the charge moves in a circular path in the x-z plane. (B) If, the charge undergoes helical motion with constant pitch along the y-axis (C) If, the charge undergoes helical motion with its pitch increasing with time, along the y-axis (D) If 9, the charge undergoes lilnear but accelerated motion along the y-axis. (A, C, D) If, V Viˆ, =, Fel ; Fmag If V V ˆ ˆ xi Vy j, V y is varying due to Ê and V x gives circular path in zx-plane. helical path with increasing pitch At 9, Fmag ; but Fel accelerated linear motion. 4. A cubical region of side a has its centre at the origin. It encloses three fixed point charges, q at, a / 4,, q at,, and q at, a / 4,. Choose the correct option(s) (A) The net electric flux crossing the plane x a / is equal to the net electric flux crossing the plane x a / (B) The net electric flux crossing the plane y a / is more than the net electric flux crossing the plane y a /. q (C) The net electric flux crossing the entire region is (D) The net electric flux crossing the plane z a / is equal to the net electric flux crossing the plane x a / (A,C,D) As q is placed symmetrically w.r.t. origin flux through y z planes or x a / equal to flux through x a /. Hence A is correct

9 As the Qin q Net q. Hence C is correct As the planes at z a / and z a / are symmetric w.r.t. charge distribution. Flux at planes Z a / Flux at plane z a /. Hence D is correct. 5. A small block of mass of. kg lies on a fixed inclined plane PQ which makes an angle with the horizontal. A horizontal force of N acts on the block through its centre of mass as shown in the figure. The block remains stationary if take g = m/s (A) =45 (B) >45 and a frictional force acts on the block towards P. (C) >45 and a frictional force acts on the block towards Q. (D) <45 and a frictional force acts on the block towards Q. (A,C) From the figure, N mg cos sin or N sin cos For 45 : sin cos Block is stationary. Hence option (A) is correct For 45 : sin cos Block will be stationary if frictional force acts towards Q. Hence (C) is correct If 45,cos sin Block will be stationary if frictional force acts towards P. Hence option (D) is wrong SECTION III : Integer Answer Type This section contains 5 questions. The question to each question is a single digit integer, ranging from to 9 (both inclusive) 6. A cylindrical cavity of diameter a exists inside a cylinder of diameter a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude N of the magnetic field at the point P is given by aj, then the value of N is

(5) Magnetic field at P (B) = B B B Magnetic field due to whole cylinder without cavity B Magnetic field due to cavity assuming carrying current in opposite direction B Ja and B 5 Ja B Ja N 5 7.

10 (5) Magnetic field at P (B) = B B B Magnetic field due to whole cylinder without cavity B Magnetic field due to cavity assuming carrying current in opposite direction B Ja and B 5 Ja B Ja N 5 7. A proton is fired from very far away towards a nucleus with charge Q e, where e is the electronic charge. It makes a closest approach of fm to the nucleus. the de Broglie wavelength (in units of fm) of 7 the proton at its start is : (take the proton mass, 9 9 m / F; 4 5 fm m ) (7) KE of proton = Potential energy of system at distance of closest approach e p e m 4 r p me 4 r h h de-broglie wavelength me p 4 r m p 5 5/ kg; h / e 4. J. s / C; Substituting the values 5 7 m 7 8. A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side a a R having two turns is placed with its center at z R along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45 with respect to the z-axis. If the mutual inductance between the loops is given by /, then the value of p is ar p

11 (7) i r B 4 8r i B 6 i. a = M.i. 6r a M 8 r a 7/ r Hence P 7 9. An infinitely long solid cylinder of radius R has a uniform volume charge density. It has a spherical cavity of radius R/ with its centre on the axis of the cylinder. as shown in the figure. The magnitude of the electric field at the point P, which is at a distance r from the axis of the cylinder, is givenby the expression R 6k. The value of k is (6) E P q. R 4 4R R and R Q R EP K

12 . A lamina is made by removing a small disc of diameter R from a bigger disc of uniform mass density and radius R, as shown in the figure. The moment of intertia of this lamina about axes passing through O and P is I O and I, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio P to the nearest integer is I I P O () I I P R 7 4 R 4 7 I P / I PART - II : CHEMISTRY SECTION - I : Single Correct Answer Type This section contains multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.. For one mole of a van der Waals gas when b = and T = K, the PV vs V value of the van der Waals constant a (atm. liter mol ) is plot is shown below. The A). B) 4.5 C).5 D). Key: C For mole vandewaal s euqations is a P V b RT V a P V RT V b = a PV RT V

13 PV RT a. V Slope = y x y x a.5 a.5. The number of optically active products obtained from the complete ozonolysis of the given compound is A) B) C) D) 4 Key : A Ozonolysis products CH CH CHO OCH C CHO OCH C CHO CH CHO H CH H None of the products are asymmetric hence ans = zero. The colour of light absorbed by an aqueous solution of CuSO 4 is A) orange - red B) blue - green C) yellow D) violet Key: A Sol: Orange - Red 4. Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen? A) HNO, NO, NH4C l, N B) HNO, NO, N, NH 4Cl C) HNO, NH4C l, NO, N D) NO, HNO, NH4C l, N Key: B 5 HNO NO N N H Cl 4 5. As per IUPAC nomeclature, the name of the complex Co H O NH C 4 l is A) Tetraaquadiaminecobalt (III) chloride B) Tetraaquadiamminecobalt (III) chloride C) Diaminetetraaquacobalt (III) chloride D) Diamminetetraaquacobalt (III) chloride Key : D Diamminetetraaquacobalt (III) chloride 6. The carboxyl functional group (-COOH) is present in A) picric acid B) barbituric acid C) ascorbic acid D) aspirin Key: D O C CH O COOH OH Barbituric acid O=C O NH-C CH NH-C O NO NO Picric acid NO

14 A) B) C) D) 4 Key : C Aldol condensations as there are hydrogens. Choose the correct reason(s) for the stability of the lyophobic colloidal particles. A) preferential adsorption of ions on their surface from the solution B) preferential adsorption of solvent on their surface from the solution C) attraction between different particles having opposite charges on their surface D) potential diffrence between the fixed layer and the diffused layer of ooposite charges around the colloidal particles. Key : A & D [Conceptual] 7. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a is Bohr radius] A) Key: C K. E h 4 ma mv and also nh mv, m v r h B) 6 ma mvr nh n h 4 r, mv n h 4 mr C) h ma D) h 64 ma n h 4 h h KE 4 mr 4 m4a ma ma 8. In allene (C H 4 ), the type(s) of hybridisation of the carbon atoms is (are) A) sp and sp B) sp and sp C) only sp D) sp and sp Key : B CH = C= CH sp sp sp 9. A compound M p X q has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The empirical formula of the compound is A) MX B) MX C) M X D) M 5 X 4 Key: B M is at M edge centers = 4 ¼ = M M is at body center = = M Total M X at 8 corners and 6 face centers = 8 6 4X 8 Formuale = M X 4 = MX. The number of aldol reaction(s) that occurs in the given transformation is

. For an ideal gas, consider of P - V work in going from an initial state X to the final state Z. The final state Z can be reached by either of the two paths shown in the figure.

[Take S as change in entropy and w as work done] S S S B) w xz w xy w yz A) xz xy yz w C) xyz xy Key : A & C w D) Sxyz Sxy Entropy changes from x y and y z s sxy syz Work done from y z w xyz wxy.

15 . For an ideal gas, consider of P - V work in going from an initial state X to the final state Z. The final state Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct? [Take S as change in entropy and w as work done] S S S B) w xz w xy w yz A) xz xy yz w C) xyz xy Key : A & C w D) Sxyz Sxy Entropy changes from x y and y z s sxy syz Work done from y z w xyz wxy. Which of the following hydrogen halides react(s) with AgNO (aq) to give a precipitate that dissolves in Na S O (aq)? A) HCl B) HF C) HBr D) HI Key : AgF is soluble in H O hence A,C,D 4. Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as shown in the given scheme. A) C6H5OH and C6H5COOH B) C6H5COOH and C6H5CHOH C) C6H5CHOH and C6H5OH D) C6H5CHOH and C6H5CHCOOH Key : B,D 5. Which of the following molecules, in pure form, is (are) unstable at room temperature? A) B) C) D) Key : B & C are anti aromatic and 6. The periodic table consists of 8 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic table? 6 9 Cu H 6 n H X Key : 6 Cu H 6 n 4 He H 5 Fe = 8 6 9

16 7. An organic compound undergoes first - order decomposition. The time taken for its decomposition to / [t / 8] 8 and / of its initial concentration are t /8 and t / respectively. What is the value of? [t ] (take log =.) Key : t/8 log 8 t/8 log() 9 t/ log t/ 8. 9.% (w/w) HCl stock solution has a density of.5 g ml. The molecular weight of HCl is 6.5 g mol. The volume (ml) of stock solution required to prepare a ml solution of.4 M HCl is dx.5 9. Key : M GMW 6.5 V M = V M V.4 V = 8ml 9. The substituents R and R for nine peptides are listed in the table given below. How many of these peptides are positively charged at ph = 7.? H N CH CO NH CH CO NH CH CO NH CH CO O H R R H / Key : 4 [IV, VI, VIII, IX] 4. When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is CHO CH CHOH CHOH CHOH CH OH

17 Key : 8 H * OH H OH H H * * HOH C H * O H OH Total no.of stereoisomers = 4 = 6 No.of stereoisomers of D - Pyranose = 8 PART - II : MATHEMATICS SINGLE ANSWER TYPE QUESTIONS : 4. Let Sol : x cos, x f x x, x,, x then f is (A) differentiable both at x and at x (B) differentiable at x but not differentiable at x (C) not differentiable at x but differentiable at x (D) differentiable neither at x nor at x f ' lim x f x f Similarly at x. x x cos x cos x lim finite x x x lim finite x x. 4. The function f :,, 9, defined by f x x 5x 6x, is (A) one-one and onto (B) onto but not one-one (C) one-one but not onto (D) neither one-one nor onto f x x 5x 6x Sol : f ' x 6x x 6 6 x x neither increasing nor decreasing in, f is not one - one Range codomain onto. 4. The ellipse : x y E is inscribed in a rectangle R whose sides are parallel to the coordinate axes. 9 4 Another ellipse E passing through the point, 4 circumscribes the rectangle R. The eccentricity of the ellipse E is. (A) (B) (C) (D) 4

18 Sol: x y E : a. 6 It passes,4 9 4 a a 6 E : x y e b 6 a e The point P is the intersection of the straight line joining the points Q,,5 and,, 4 If S is the foot of the perpendicular drawn from the point,,4 plane 5x 4y z. the length of the line segment PS is (A) Sol: Any point on QR is t, 4 t,5 t it lies on plane t. it is perpendicular to line joining T to this 4 9 P,, S,, (B) (C) (D) 9 9,, R with the T to QR, then PS 45. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x 5y to the circle x y 9 is (A) x y 6x 45y (B) (C) 6 x y x 45y (D) Sol: Let, Then 4 5 Chord of contact of w.r.t circle x y 9 Equation of chord in terms of midpoints is xx yy x y & same 9x y x 9y y x x y 6x 45y 6 x y x 45y

19 substitute in we get locus as x y 6x 45y 46. Let P a ij be a matrix and let Q b ij, where i b j ij aij for i, j. If the determinant of P is, then the determinant of the matrix Q is. (A) (B) (C) (D) Sol: P a a a Q a a a a a a Q P 4 sec 47. The integral x dx sec x tan x equals (for some arbitrary constant K) 9 (A) sec x tan x K sec x tan x 7 (B) sec x tan x K sec x tan x 7 (C) sec x tan x K sec x tan x 7 (D) sec x tan x K sec x tan x 7 Sol: sec x tan x t sec x tan x t sec x t t dt sec x dx t 48. The total number of ways in which 5 balls of different colours can be distributed among persons so that each person gets at least one ball is (A) 75 (B) 5 (C) (D) 4 Sol: Number of onto functions from set having 5 elements to set having elements c c

20 x 49. If lim x Sol: x ax b 4, x then (A) a, b 4 (B) a, b 4 (C) a, b (D) a, b x x ax ax bx b lim 4 x x Equating the coeff of x and coeff x 4 a a a b 4 b Let z be a complex number such that the imaginary part of z is non zero and a z z is real. Then a cannot take the value (A) (B) Sol: If Z x iy then a x y x i xy y xy y x y. (C). real (D) 4 a y y a a 4 and y 4 4 MULTIPLE TYPE QUESTIONS : x 5. Let S be the area of the region enclosed by, y e y, x, and x. Then (A) S (B) S e e (C) S (D) S 4 e e dy Sol: y tan x xsec x dx G. S. is y cos x x c y c y x cos x 4 8 y A cos. y ' cos x x x sin x 4 y ' x D

21 5. If y x satisfies the differential equation y y tan x x sec x and y, then (A) y 4 8 (B) y 4 8 (C) y 9 ( D ) y 4 Sol: (A) P X C X P E E E P E E E P E E E P E E E P E E E 8 8. Exactly two engines of the ship are functioning (B) P X P E E E PE E E PE E E P E E E P E E E P E E E P E E E (C) P X X (D) P X X 7 8 P E E E P E E E PE E E P E E E P E E E P E E E P E E E P E E E P E E E PE E E P E E E P E E E P E E E P E E E x y 5. Tangents are drawn to the hyperbola, parallel to the straight line x y. The 9 4 points of contact of the tangents on the hyperbola are 9 (A ). Sol: cos sin sin cos co s sin c o s s i n sin, tan 9 (B). (C), (D),

22 5x o cos. cos sin 5 n n, n,, Let,, be such that cos sin sin tan cot cos, Sol: tan and sin. Then cannot satisfy. 4 (A) (B) (C) 4 (D) x S e dx f x e x. is decreasing function A S b am B x e x x x x e x S e dx e x D f x 4e x x x e x S e 55. A ship is fitted with three engines E, E and E. The engines function independently of each other with respective probabilities, 4 and. For the ship to be operational at least two of 4 its engines must function. Let X denote the event that the ship is operational and let X, X and X denote respectively the events that the engines E, E and E are functioning. Which of the following is (are) true? c (A) P X X (B) 6 P [Exactly two engines of the ship are functioning (C) P X X (D) P X X X 8

23 Sol: Any line Parallel to x y is x y k k 4 x y 4 Point of contact 9 9,, INTEGER TYPE QUESTIONS 56. The value of 6 log Sol: x y x 4y, y 8x solving 4 y y y 4y 64 4 y y y, 48 56,, 4, P Q S Area 8 4 sq. u. 57. Let S be the focus of the parabola y x y x y is 8x and let PQ be the common chord of the circle 4 and the given parabola. The area of the triangle PQS is Sol: Let P x k x x k x 9x x Then P x K x x C P K C 6 4K C 6 P K C 4K 4 K P x x x P 58. If a, b 9. and c are unit vectors satisfying a b b c c a 9, then a 5b 5c is Sol: f x x x if x x x x x x x x x x

IIT-JEE 2012 PAPER - 1 PART - I : PHYSICS. Section I : Single Correct Answer Type

IIT-JEE 2012 PAPER - 1 PART - I : PHYSICS. Section I : Single Correct Answer Type IIT-JEE PAPER - PART - I : PHYSICS Section I : Single Correct Answer Type This section contains multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is