EXAMINATION PAPER IIT-JEE 2010

Transcription

1 EXAMINATIN PAPER IIT-JEE 00 PAPER II DE - Part I (EMISTRY) SETIN I (Single orrect hoice Type) This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), () and (D), out of which NLY NE is correct.. Assuming that und's rule is violated, the bond order and magnetic nature of the diatomic molecule B is (A) and diamagnetic (B) 0 and diamagnetic () and paramagnetic (D) 0 and paramagnetic [A]. The compounds P, Q and S 3 P Q S were separately subjected to nitration using N 3 / S 4 mixture. The major product formed in each case respectively, is 3 (A) N (B) () N N N 3 N N 3 N N (D) N 3 N N [] N RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

2 3. The packing efficiency of the two-dimensional square unit cell shown below is L (A) 39.7 % (B) 68.0% () 74.05% (D) 78.54% [D] 4. The species having pyramidal shape is (A) S 3 (B) BrF 3 () Si 3 (D) SF 3 [D] 5. In the reaction (A) () Na/Br N () l (B) N T the structure of the product T is 3 () N (D) N [] 6. The complex showing a spin-only magnetic moment of.8 B.M. is (A) Ni() 4 (B) [Nil 4 ] () Ni((PPh 3 ) 4 (D) [Ni(N) 4 ] [B] SETIN II Integer Type This section contains a group of 5 questions. The answer to each questions is a single digit integer ranging from 0 to 9. The correct digit below the question number in the RS is to be bubbled. 7. Silver (atomic weight = 08 g mol ) has a density of 0.5 g cm 3. The number of silver atoms on a surface of area 0 m can be expressed in scientific notation as y 0 x. The value of x is [7] 8. Among the following, the number of elements showing only one non-zero oxidation state is, l, F, N. P, Sn, Tl, Na, TI [] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

3 9. ne mole of an ideal gas is taken from a to b along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is w s and that along the dotted line paths is w d, then the integer closest to the ratio w d \w s is P (atm) a V(lit) [] 0. The total number of diprotic acids among the following is 3 P 4 S 4 3 P 3 3 S 7 3 B 3 3 P r 4 S 3 [6]. Total number of geometrical isomers for the complex [Rhl()(PPh 3 )(N 3 )] is [3] SETIN III Paragraph Type This section contains paragraphs. Based upon each of the paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), () and (D) out of which NLY NE is correct. Paragraph for questions to 4 Two aliphatic aldehydes P and Q react in the presence of aqueous K 3 to give compound R, which upon treatment with N provides compound S. n acidification and heating, S gives the product shown below b. The compounds P and Q respectively are - 3 (A) and 3 (B) and RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

4 () and 3 (D) and 3 [B] 3. The compound R is (A) (B) 3 3 () (D) [A] 4. The compound S is 3 (A) (B) 3 3 N N N N () (D) 3 [D] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

5 Paragraph for Questions 5 to 7 The hydrogen like species Li + is in a spherically symmetric state S with one radial node. Upon absorbing light the ion undergoes transition to a state S. The state S has one radial node and its energy is equal to the ground state energy of the hydrogen atom. 5. The state S is (A) s (B) s () p (D) 3s [B] 6. Energy of the state S in units of the hydrogen atom ground state energy is (A) 0.75 (B).50 ().5 (D) 4.50 [] 7. The orbital angular momentum quantum number of the state S is (A) 0 (B) () (D) 3 [B] SETIN IV Matrix Type This Section contains questions. Each question has four statements (A, B, and D) given in olumn I and five statements (p, q, r, s and t) in olumn II. Any given statement in olumn I can have correct matching with one or more statement(s) given in olumn II. For example, if for a given question, statement B matches with the statements given in q and r, then for that particular question, against statement B, darken the bubbles corresponding to q and r in the RS. 8. Match the reactions in olumn I with appropriate options in olumn II. olumn I olumn II (A) Na / (p) Racemic mixture N l + 0 N = N (B) S (q) Addition reaction () 3. LiAl (r) Substitution reaction (D) S l Base [A r,s; B t; p,q; D r] S (s) oupling reaction (t) arbocation intermediate 9. All the compounds listed in olumn I react with water. Match the result of the respective reactions with the appropriate options listed in olumn II. olumn I olumn II (A) ( 3 ) Sil (p) ydrogen halide formation (B) XeF 4 (q) Redox reaction () l (r) Reacts with glass (D) Vl 5 (s) Polymerization (t) formation [A p,s; B p,q,r,t; p,q,t; D p] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

6 Part II (MATEMATIS) SETIN I (Single orrect hoice Type) This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), () and (D), out of which NLY NE is correct. 0. A signal which can be green or red with probability 5 4 and 5 respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is 4 3. If the signal received at station B is green, then the probability that the original signal was green is (A) (B) () (D) 0 []. If the distance of the point P (,, ) from the plane x + y z = α, where α > 0, is 5, then the foot of the perpendicular from P to the plane is (A),, (B),, (),, (D),, [A]. Two adjacent sides of a parallelogram ABD are given by AB = î + 0 ĵ + kˆ and AD = î + ĵ + kˆ The side AD is rotated by an acute angle α in the plane of the parallelogram so that AD becomes AD. If AD makes a right angle with the side AB, then the cosine of the angle α is given by (A) 9 8 [B] (B) 7 9 () 9 (D) Let S = {,, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to (A) 5 (B) 34 () 4 (D) 4 [D] 4. Let f be a real-valued function defined on the interval (, ) such that e x f(x) = + t + dt, for all x (, ), and let f be the inverse function of f. Then (f ) () is equal to (A) (B) () 3 (D) e x 0 4 [B] 5. For r = 0,,, 0, let A r, B r and r denote, respectively, the coefficient of x r in the expansions of [D] ( + x) 0, ( + x) 0 and (+ x) 30. Then A r (B0Br 0A r ) is equal to 0 r= (A) B 0 0 (B) A 0 ( ) B () 0 (D) 0 B 0 0 0A0 RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

7 SETIN II (Integer type) This section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9. The correct digit below the question no. in the RS is to be bubbled. 6. Let a, a, a 3,.., a be real numbers satisfying a = 5, 7 a > 0 and a k =a k a k for k = 3, 4,.,. a + a a If [0] = 90, then the value of a + a a is equal to 7. Let f be a function defined on R (the set of all real numbers) such that f (x) = 00 (x 009) (x 00) (x 0) 3 (x 0) 4, for all x R. If g is a function defined on R with values in the interval (0, ) such that f(x) = ln {g(x)}, for all x R, then the number of points in R at which g has a local maximum is [] 8. Let k be a positive real number and let k k k 0 k k A = k k and B = k 0 k k k k k 0 If det (adj A) + det(adj B) = 0 6, then [k] is equal to [Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k] [4] 9. Two parallel chords of a circle of radius are at a distance 3 + apart. If the chords subtend at the center, angles π π of and, where k > 0, then the value of [k] is k k [Note : [k] denotes the largest integer less than or equal to k] [3] 30. onsider a triangle AB and let a, b and c denote the lengths of the sides opposite to vertices A, B and respectively. Suppose a = 6, b = 0 and the area of the triangle is 5 radius of the incircle of the triangle, then r is equal to [3] 3. If AB is obtuse and if r denotes the SETIN III (Paragraph Type) This section contains paragraphs. Based upon each of the paragraphs 3 multiple choice questions have to be answered. Each of these question has four choices (A), (B), () and (D) out of which NLY NE is correct. RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

8 Paragraph for Questions 3 to 33 onsider the polynomial f(x) = + x + 3x + 4x 3. Let s be the sum of all distinct real roots of f(x) and let t = s. 3. The real number s lies in the interval 3 3 (A), 0 (B), (), [] (D) 0, 4 3. The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t, lies in the interval 3 (A), 3 (B), () (9, 0) (D) 0, [A] 33. The function f (x) is (A) increasing in t, and decreasing in, t 4 4 (B) decreasing in t, and increasing in, t 4 4 () increasing in ( t, t) (D) decreasing in ( t, t) [B] Paragraph for questions 34 to 36. Tangents are drawn from the point P(3, 4) to the ellipse 34. The coordinates of A and B are (A) (3, 0) and (0, ) x y + =, touching the ellipse at points A and B. 9 4 (B) 8 6, and 9 8, () 8 6, and (0, ) (D) (3, 0) and 9 8, [D] 35. The orthocentre of the triangle PAB is (A) 5, (B), [] 8 (), 5 5 (D) 8 5 7, The equation of the locus of the point whose distances from the point P and the line AB are equal, is (A) 9x + y 6xy 54 x 6 y + 4 = 0 (B) x + 9y + 6xy 54x + 6 y 4 = 0 () 9x + 9y 6xy 54 x 6 y 4 = 0 (D) x + y xy + 7x + 3y 0 = 0 [A] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

9 SETIN IV (Matrix Type) This section contains questions. Each question has four statements (A, B, and D)given in olumn I and five statements (p, q, r, s and t) in column-ii. Any given statement in olumn I can have correct matching with one or more statement(s) given in olumn-ii. For example, if for a given question, statement B matches with the statements given in q and r, then for that particular question, against statement B, darken the bubbles corresponding to q and r in the RS. 37. Match the statements in olumn-i with those in olumn-ii. [Note : ere z takes values in the complex plane and lm z and Re z denote, respectively, the imaginary part and the real part of z] LUMN-I LUMN-II (A) The set of points z satisfying 4 (p) an ellipse with eccentricity 5 z i z = z + i z is contained in or equal to (q) the set of points z satisfying Im z = 0 (B) The set of points z satisfying z z 4 = 0 (r) the set of points z satisfying Im z is contained in or equal to () If w =, then the set of points (s) the set of points z satisfying Re z z = w w is contained in or equal to (D) If w =, then the set of points (t) the set of points z satisfying z 3 z = w + w is contained in or equal to A q,r ; B p ; p, s, t ; D q, r, s, t 38. Match the statements in olumn-i with the values in olumn-ii. LUMN-I LUMN-II (A) A line from the origin meets the lines (p) 4 8 x x y z + = = and 3 y 3 z = = + at P and Q respectively. If length PQ = d, then d is (B) The values of x satisfying tan (x + 3) tan (x 3) = sin 3 are, (q) 0 5 () Non-zero vectors a r, b r and c r satisfy a r. b r = 0, (r) 4 ( b r a r ). ( b r + c r ) = 0 and b r + c r = b r a r. If a r = µ b r + 4 c r, then the possible values of µ are (D) Let f be the function on [ π, π] given by (s) 5 9x x f(0) = 9 and f(x) = sin sin for x 0 π The value of π f (x)dx is (t) 6 A t ; B p, r ; p, q ; D r π RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

10 Part III (PYSIS) SETIN I Single orrect hoice Type This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), () and (D), out of which NLY NE is correct. 39. A uniformly charged thin spherical shell of radius R carries uniform surface charge density of σ per unit area. It is made of two hemispherical shells, held together by pressing them with force F (see figure). F is proportional to - F F (A) [A] σ ε 0 R (B) ε 0 σ R () ε 0 σ R (D) σ ε 0 R 40. A block of mass kg is free to move along the x-axis. It is at rest and from t = 0 onwards it is subjected to a time dependent force F(t) in the x direction. The force F(t) varies with t as shown in the figure. The kinetic energy of the block after 4.5 seconds is - F(t) 4N 3s 4.5s t (A) 4.50 J (B) 7.50 J () 5.06 J (D) 4.06 J [] 4. A tiny spherical oil drop carrying a net charge q is balanced in still air with a vertical uniform electric field of 8 π 5 strength 0 Vm. When the field is switched off, the drop is observed to fall with terminal velocity ms. Given g = 9.8 m s, viscosity of the air = Nsm and the density of oil = 900 kg m 3, the magnitude of q is - (A) (B) () (D) [D] 4. A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 30 ms, the mass of the string is - (A) 5 grams (B) 0 grams () 0 grams (D) 40 grams [B] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

11 43. A biconvex lens of focal length 5 cm is in front of a plane mirror. The distance between the lens and the mirror is 0 cm. A small object is kept at a distance of 30 cm from the lens. The final image is - (A) virtual and at a distance of 6 cm from the mirror (B) real and at a distance of 6 cm from the mirror () virtual and at a distance of 0 cm from the mirror (D) real and at a distance of 0 cm from the mirror [B] 44. A vernier calipers has mm marks on the main scale. It has 0 equal divisions on the vernier scale which match with 6 main scale divisions. For this vernier calipers, the least count is - (A) 0.0 mm (B) 0.05 mm () 0. mm (D) 0. mm [D] SETIN II Integer Type This section contains Five questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The correct digit below the question number in the RS is to be bubbled. 45. A large glass slab (µ = 5/3) of thickness 8 cm is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius R cm. What is the value of R? Image of an object approaching a convex mirror of radius of curvature 0 m along its optical axis is observed to 5 50 move from m to m in 30 seconds. What is the speed of the object in km per hour? dn(t) dn(t) 47. To determine the half life of a radioactive element, a student plots a graph of l n versus t. ere is dt dt the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.6 years, the value of p is - 6 dn(t) l n dt Years 48. A diatomic ideal gas is compressed adiabatically to of its initial volume. In the initial temperature of the gas is 3 T i (in Kelvin) and the final temperature is at i, the value of a is At time t = 0, a battery of 0 V is connected across points A and B in the given circuit. If the capacitors have no charge initially, at what time (in seconds) does the voltage across them become 4 V? (Take : ln 5 =.6, ln 3 =.] MΩ µf A MΩ µf RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.: B

12 SETIN III omprehension Type This section contains paragraphs.. Based upon each of paragraph 3 multiple choice question have to be answered. Each of these questions has four choices (A), (B), () and (D) for its answer, out of which NLY NE is correct. Paragraph for Questions 50 to 5 When liquid medicine of density ρ is to be put in the eye, it is done with the help of a dropper. As the bulb on the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop. We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension T when the radius of the drop is R. When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper. 50. If the radius of the opening of the dropper is r, the vertical force due to the surface tension on the drop of radius R (assuming r << R) is (A) πrt (B) πrt () [] πr T R πr T (D) r 5. If r = m, ρ = 0 3 kgm 3, g = 0ms, T = 0. Nm, the radius of the drop when it detaches from the dropper is approximately (A) m (B) m () m (D) m [A] 5. After the drop detaches, its surface energy is (A) J (B) J () J (D) J [B] Paragraph for Questions 53 to 55 The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition. 53. A diatomic molecule has moment of inertia I. By Bohr's quantization condition its rotational energy in the n th level (n = 0 is not allowed) is (A) h n 8π I [D] (B) h n 8π I () h n 8π I (D) n h 8π I 54. It is found that the excitation frequency from ground to the first excited state of rotation for the moelcule is close to π 4 0 z. Then the moment of inertia of molecule about its center of mass is close to (Take h = π 0 34 J s) (A) kg m (B) kg m () kg m (D) kg m [B] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

13 55. In a molecule, the distance between (mass = a.m.u.) and (mass = 6 a.m.u.) where a.m.u. = kg, is close to (A) m (B) m () m (D) m [] SETIN IV Matrix Type This Section contains questions. Each question has four statements (A, B, and D) given in olumn I and five statements (p, q, r, s and t) in olumn II. Any given statement in olumn I can have correct matching with one or more statement(s) given in olumn II. For example, if for a given question, statement B matches with the statements given in q and r, then for that particular question, against statement B, darken the bubbles corresponding to q and r in the RS. 56. Two transparent media of refractive indices µ and µ 3 have a solid lens shaped transparent material of refractive index µ between them as shown in figures in olumn II. A ray traversing these media is also shown in the figures. In olumn I different relationships between µ, µ and µ 3 are given. Match them to the ray diagrams shown in olumn II. olumn I olumn II (A) µ < µ (p) µ 3 µ µ (B) µ > µ (q) µ 3 µ µ () µ = µ 3 (r) µ 3 µ µ (D) µ > µ 3 (s) µ 3 µ µ (t) µ 3 µ µ [A p,r; B q,s,t; p,r,t D q,s] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.:

14 57. You are given many resistances, capacitors and inductors. These are connected to a variable D voltage source (the first two circuits) or an A voltage source of 50 z frequency (the next three circuits) in different ways as shown in olumn II. When a current I (steady state for D or rms for A) flows through the circuit, the corresponding voltage V and V. (indicated in circuits) are related as shown in olumn I. Match the two olumn I olumn II V V (A) I 0, V is proportional to I (p) 6m 3µF V V V (B) I 0, V > V (q) 6m Ω V V V () V = 0, V = V (r) 6m Ω ~ V V V (D) I 0, V is proportional to I (s) 6m 3µF ~ V (t) V kω V 3µF ~ V [A r,s,t; B q,r,s,t; p,q; D q,r,s,t] RPRATE FFIE, P Tower, Road No., IPIA, Kota (Raj.) Ph.: