12th IIT 3rd June Test Paper-2 Solution

Transcription

1 th IIT 3rd June Test Paper- Solution Subject-PHYSICS Part : SECTION-A (NEXT FOUR QUESTIONS: SINGLE CHOICE ONLY ONE CORRECT ANSWER) Q The block shown in the figure is in equilibrium. Find the acceleration of the block just after the string burns: 3g 5 5g 5 4g 3 3g 4 (c) If T is the tension in the string and F is the force in the spring, then T F sin 53 and mg F cos 53 Q Two small bodies of masses 'm' and 'm' are placed in a fixed smooth horizontal circular hollow tube of mean radius 'r ' as shown. The mass 'm' is moving with speed 'u ' and the mass 'm' is stationary. After their first collision, the time elapsed for next collision is: [coefficient of restitution e / ] r u 4 r u 3 r u r u 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

2 3 th IIT 3rd June Test Paper- Solution (b) If just after collision, relative velocity v, then v u v u r r time between st and nd collision, t Q3 rel 4 r rel u The end of uniform wire length L and of weight W is attached rigidly to a point in the roof and a weight W is suspended from its lower end. If s is the area of cross-section of the wire, the stress in the wire at a height 3L 4 from its lower end is: W W s W 4 W s Q4 3W 4 W W s s (c) A transverse sinusoidal wave moves along a string in the positive x-direction at a speed at 0 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 0 cm. At a particular time t, the snapshot of the wave is shown in figure. The velocity of point P when its displacement is 0.05 m, is 3 ˆ jm / s 50 3 ˆ jm / s 50 3 ˆ im/ s 50 (a) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / ˆ im / s 50

3 4 th IIT 3rd June Test Paper- Solution SECTION-B (MULTIPLE CHOICE ONE OR MORE THAN ONE CORRECT ANSWER) Q A sphere is rolling without slipping on a fixed horizontal plane surface. In figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then vc v A (v B vc ) v v v v C A B C Q v B vb v A vc v A 4 vb v C (b, c) A one mol sample of an ideal gas is carried around the thermodynamics cycle shown in the figure. The cycle consists of three steps (i) an isothermal expansion (a b), (ii) an isobaric compression (b c), and (iii) a constant volume increases in pressure (c a). If Ta 400 K, Pa 4 atm and Pb Pc atm, the work done by the gas per cycle is 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

4 5 th IIT 3rd June Test Paper- Solution Q3.6 J 58 J J 6 J (d) A body takes 5 minutes for colling from 50 C to 40 C. Its temperature comes down to C in next 5 minutes. Now, choose the correct statement(s) from the following? The temperature of surrounding is 0 C The temperature of surrounding is 5 C The temperature of the body in further 5 minutes becomes 8.88 C The temperature of the body in further 5 minutes becomes 8.88 C (a, d) Q4 For the situation shown in Fig., mark out the correct statement(s) Potential of the conductor is q /[4 0 (d R )] Potential of the conductor is q / 4 0 d Potential of the conductor cannot be determined as nature of distribution of inducted charges is not known 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

5 6 th IIT 3rd June Test Paper- Solution Potential at point B due to induced charges is qr /[4 0 (d R ) d ]. (a, d) As no charge is present inside the conductor, potential at any point inside the conductor is same as that of the potential of conductor. So, potential of the conductor = Potential at the centre Vq Vinduced charges i.e., Vconductor q (d R) As the total induced charge at conductor s surface is equal to zero, hence the potential at centre due to the induced charges is zero. For point B Vconductor Vat point B Vq Vinduced charges Vinduced charges q q qr 4 0 (d R) 4 0 d 4 0 d (d R) The second of these is zero, so the work that must be done is equal to U. The distances between the charges are r3 a and r3 a, so If q3 is brought in from infinity along the x -axis, it is attracted by q but is repelled more strongly by q ; hence, positive work must be done to push q3 to the position at x a. The total potential energy of the assemblage of three charges is Q5 Two balls, having linear momenta P Piˆ and P Piˆ, undergo a collision in free space. There is no external force acting on the balls. Let P ' and P ' be their final momenta. The following option(s) is (are) not allowed for any non-zero value of P, a, a, b, b c and c. P ' aiˆ b ˆj ckˆ; P ' a iˆ b ˆj P ' aiˆ b ˆj ckˆ; P ' aiˆ b ˆj ckˆ P ' ckˆ; P ' c kˆ P ' aiˆ b ˆj; P ' aiˆ b ˆj (a, d) Momentum before collision, 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

6 7 th IIT 3rd June Test Paper- Solution P P P Piˆ Piˆ 0 Option (a) and (d), add together can not give zero momentum. Q6 A particle is projected from a point P with a velocity v at an angle with horizontal. At a certain point Q it moves at right angles to its initial direction. Then: Velocity of time Velocity of particle at Q is v sin of flight from P to Q is (v / g ) cos ec time particle at Q is v cot of flight from P to Q is (v / g ) sec (b, c) Initial velocity of particle in vector form can be written as SECTION-C Paragraph for next Questions In fig. a thin horizontal bar AB of negligible weight and length L is pinned to a vertical wall at A and supported at B by a thin wire BC that makes an angle with the horizontal. A weight W can be moved anywhere along the bar; its position is defined by the distance x from the wall of its centre of mass. As a function of x, find 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

7 8 th IIT 3rd June Test Paper- Solution Q The tension in the wire is W Q Wx L sin Wx / L None of these The vertical component of the force exerted on the bar by the pin at A is x W L W ( k x) / L W ( x / L) Wx / L (-b, -a) Let T is the tension in the wire and FH and FV are the components of force exerted by the pin in horizontal and vertical directions respectively. The FBD is shown in figure. (-b) (-a) Paragraph for next Questions Two cylinder A and B having pistons (massless) of cross sectional area 00 cm and 00 cm respectively. The pistons are connected by massless rod. The pistons can move freely without friction. The cylinder A contains 00 gm of an ideal gas (.5) at pressure 0 5 N/m and temperature T0. The cylinder B contains identical gas at same temperature T0 but has different mass. The pistons are held at the state such that volume of gas in cylinder A and cylinder B are same and is equal to 0 m 3. The walls and piston of cylinder A are thermally insulated where as gas in cylinder B is maintained at temperature T0. The whole system is in vacuum. Now the temperature T0. The whole system is in vacuum. Now the piston is slowly released and its moves towards left and mechanical equilibrium 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

8 9 th IIT 3rd June Test Paper- Solution is reached at the state when the volume of gas in cylinder A becomes m 3. Q3 The mass of gas in cylinder B is Q4 00 gm 600 gm 500 gm kg 500 J 3000 The change in internal energy of gas in cylinder A is 000 J (3-d, 4-a) (3-d) (4-a) 000 J 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 J

9 0 th IIT 3rd June Test Paper- Solution SECTION-D Integer Answer Type This section Contains 4 questions. The answer to each of the questions is a single digit integer, ranging from 00 to 99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened. Q A cubical body (side 0. m and mass 0.00 kg ) float in water. It is pressed and then released, so that it oscillates vertically. Find the time period. Q (8) An iron ball having a surface area of 00 cm and at a temperature of 57 C is placed in an enclosure at 7 C. If the surface emissivity of iron be 0.4, at what rate is heat being lost by radiation by the ball? Q3 (44) A column of air at 5 C and a tuning fork produce 4 beats/s when sounded together. As the temperature of air column is decreased the number of beats per second tends to decrease and when temperature is 6 C the two produce beat/s. Find the frequency of tuning fork. (50) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

10 th IIT 3rd June Test Paper- Solution Q4 A burglar s car had started with an acceleration of m/s. A police vigilant party came after 5 s and continued to close the burglar s car with a uniform velocity of 0 m/s. Find the time taken in which the police van will over take the burglar s car. (5) If t is the time taken, then (t 5) 0 t After solving, t 5 s 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

11 th IIT 3rd June Test Paper- Solution Subject-CHEMISTRY Part : SECTION-A (NEXT FOUR QUESTIONS: SINGLE CHOICE ONLY ONE CORRECT ANSWER) Q The two compounds shown below are Diastereomer Enantiomer Epimer Identical (b) Non super imposable mirror image Q Which of the following salts give different result by the action of heat? Hydrated Q3 Li CO 3 and CaCO 3 MgCl and hydrated Na SO 4 NaHCO 3 and KHCO 3 Li SO 4 and MgSO 4 (c) ml of NO at NTP was liquefied, the density of the liquid being.5 g ml. The volume of the liquid is: 0. ml 0. ml 8.8 ml.5 (b) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 ml

12 3 th IIT 3rd June Test Paper- Solution Q4 One g of a mixture of NaCO3 and NaHCO3 consumes y equivalent of HCl for complete neutralization. One g of the mixture is strongly heated, then cooled and the residue treated with HCl. How many equivalent of HCl would be required for complete neutralization? y equivalent y equivalent 3y / 4 equivalent 3y / equivalent (b) Δ NaHCO 3 Na CO 3 H O CO NaHCO 3 HCl NaCl H O CO or Na CO 3 HCl NaCl H O CO Amount of HCl remains same. SECTION-B (MULTIPLE CHOICE ONE OR MORE THAN ONE CORRECT ANSWER) Q Which of the following statement is/are true regarding a real gas? dz of the curve between compressibility factor vs P dp Boyle temperature is the temperature at which the slop is same as that for an ideal gas over a wide range of pressure. At larger molar volumes, a real gas always obeys ideal gas laws. At low pressure, attractive forces in the real gases dominates and the gas is more compressible. At critical state, the compressibility factor is 3 / 8. Q (a, c, d) Which of the following compounds will give same product in hydroboration oxidation and catalytic hydration? (a, b, d) Symmetrical alkenes give the same products on hydroboration oxidation and catalytic hydration. Q3 Which of the following compounds undergo hydrolysis? CCl 4 SiCl 4 BCl 3 SiH (b, c, d) Due to availability of empty orbitals 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 4

13 4 th IIT 3rd June Test Paper- Solution Q4 Which of the following statement(s) is/are correct regarding photoelectric effect? No electrons are ejected, regarding of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the metal The kinetic energy of the ejected electrons increases linearly with the frequency of incident radiation but is independent of the intensity of the radiation. Even at low intensities, electrons are ejected immediately if the wavelength is above threshold value. It is a direct evidence for the dual nature of light. (a, b) Frequency of incident radiation should be greater than threshold frequency. Kinetic energy will only depends on frequency of incident radiation. Q5 For the reaction NH 3 ( g ) 3 N ( g ) H ( g ). If K p of the above reaction is 78. at 400 C which is not correct: 3 3 P At pressure P degree of dissociation of NH 3 is 4 k p / K c.43 mol / If we increase pressure it will decrease K p If we increase P by bringing piston down, then degree of dissociation will decrease (a, b, d) K p K c ( RT ) n K p K c ( ) ' Approximately P 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

14 5 th IIT 3rd June Test Paper- Solution Q6 The correct statement/statements is/are: The distance between two octahedral voids of a cubic close packing of atoms with radius r is r The distance between two tetrahedral voids of a cubic close packing of atoms with radius r is 6 r at the body diagonal axes The distance between two tetrahedral and octahedral voids is 3 r, where is the radius of atom forming the close packing The distance between the surface of atoms fcc arrangement along an edge is ( ) r (b, c, d) The distance between two tetrahedral voids of a cubic packed arrangement is 0.5 r r 0.6 r 6 r Distance between tetrahedral and octahedral voids is 0.5 r r.5 r Edge length a 3/ r 4r r Between surface of two atoms distance is a r r r r ( ) SECTION-C Paragraph for next Questions Addition of non-volatile solute always lowers the vapour pressure of solvent, therefore, it will be in equilibrium with solid phase at a lower pressure and hence at a lower temperature, the difference between the freezing points of the pure solvent and its solution is called depression of freezing point. The depression in freezing point when mol of solute is dissolved in 000 gm of solvent is called molal freezing point depression constant ( K f ). The value of K f for H O is.86 K mol kg. To understand this, a solution of 5 % (w/w) MgCl containing impurity of the chloride of a trivalent metal M having atomic mass 43.5 gm (4 % by weight of MgCl ) has been prepared. In this solution MgCl is completely dissociated. on the other hand only 50 % dissociation of MCl 3 takes place and molality is approximately equal to molarity. Q Depression in freezing point of the solution is 7.36 K 3.0 K 4.88 K.4 (a) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 K

15 6 th IIT 3rd June Test Paper- Solution Q If the solution is cooled to.7 C, what weight of ice will be formed? 5 gm 60 gm 40 gm 5 gm (a) Paragraph for next Questions A more widely used method for locating the double bond of alkene involves the use of ozone. Ozone reacts with alkene to form unstable compounds called ozonide. Ozonides on reduction with Zn and water produces carbonyl compound that can be isolated and identified. Ozonolysis can be either of reductive type or oxidative type. Q3 An organic compound A (C 5 H 8 ) on hydrogenation gives compound B (C 5 H ). A on ozonolysis gives methanal and -ketopropanal. The compound (A) (a) CH O3 H 3C C CH CH HCHO H 3C C CHO HCHO Zn. H O O 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

16 7 th IIT 3rd June Test Paper- Solution Q4 The reactant (A) is (b) SECTION-D Integer Answer Type This section Contains 4 questions. The answer to each of the questions is a single digit integer, ranging from 00 to 99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened. Q The number of bicarbonates that do not exist in solid form among the following is LiHCO 3, NaHCO 3, Ca (HCO 3 ), KHCO 3, NH 4 HCO 3, Ba (HCO 3 ), Mg (HCO 3 ). (4) Based on fact Q A crystal is made up of atoms X, Y and Z. Atoms X are in FCC packing. Y occupies all octahedral voids and Z occupies all tetrahedral voids. If all the atoms along two body diagonals are removed, the ratio of sum of effective number of atoms of Y and Z to the effective number of atoms of X is a :, what is the value of a? () 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

17 8 th IIT 3rd June Test Paper- Solution Q3 g of a monobasic acid dissolved in 00 g of water lowers the freezing point by 0.86 C. On the other hand when g of the same acid is dissolved in water so as to make the solution 00 ml, this solution requires 5 K kg ml of 0. N NaOH for complete neutralization. Calculate α for the acid in percent. K f.86 mol (60) Let molar mass of acid be m. then 0.86 i Also, So i Q m 00 () so m 80 m α 50 5 α 60 % Calculate the ratio of the time required for an electron taking one round of nd orbit of H -atom and He ion. (4) For nd orbit of H-atom: If rn and r are radius of H-atom. u n and u are velocity of electron in nth and I orbit of H-atom. Time required to complete one round in nth orbit is rn where u n is velocity of electron in nth orbit and un rn is radius of nth orbit. TH rn un ( rn r n and u n r n n u r n 3 n e rn 3 h TH e u ) n e u h (i) For nd orbit of He : 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

18 9 th IIT 3rd June Test Paper- Solution THe [rn He rh n Z rn He u n He : u n He u n H Z and u n H THe u ] n r n n Z u Z THe rn 3 h rn 3 h Z e e. Z TH Z 4 THe ( Z For He ) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

19 0 th IIT 3rd June Test Paper- Solution Subject-Mathematics Part : 3 SECTION-A (NEXT FOUR QUESTIONS: SINGLE CHOICE ONLY ONE CORRECT ANSWER) Q If the system of equations x y z a x y z b x 3 y 3z c and has infinite solution, then a a b c 0 b c 0 a b c 0 a b c 0 (b) Hence for at least one solution, if D D D3 0. a b 0 c 3 3 a b c 0 From 0 and 3 0, we get the same condition. Q If A ( ) A, where is a complex cube root of unity, then adj A equals A A ( ) A (d) Cofactor matrix of A ( ) A Q3 Suppose A and B are two events with P ( A) 0.5 and P ( A B) 0.8. Let P ( B) p, if A and B are 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

20 th IIT 3rd June Test Paper- Solution mutually exclusive, and P ( B ) q, if A and B are independent, then p q p q p q d (c) Q4 Number of roots of the equation tan x 0 tan x, for x (0, 3 ) is 6 3 (a) SECTION-B 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 q

21 th IIT 3rd June Test Paper- Solution (MULTIPLE CHOICE ONE OR MORE THAN ONE CORRECT ANSWER) Q If A, B are two square matrices of same order such that A B AB and I is identity matrix of order same as that of A, B, then AB BA A I 0 B I 0 A B 0 (a, c) Q If A and B are two 3 3 matrices such that their product AB is a null matrix, then det A 0 B must be a null matrix det B 0 A must be a null matrix If neither of A and B is null matrix, then at least one of the two matrices must be singular If neither det A nor det B is zero, then the given statement is not possible Q3 (a, b, c, d) A solution set of the equations x y z, x 3 y 4 z k, x 5 y 0 z k is ( 5, 3, ) (5, 3, ) ( 6,, ) ( (a, b) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783 6,, )

22 3 th IIT 3rd June Test Paper- Solution Q4 If a parabola touches the lines y x and y x at P (3, 3) and Q (, ) respectively, then 30 6, 3 3 Focus is Equation of directrix is 5 x y 0 Equation of line through origin and focus is x 5 y 0 Equation of line through origin and parallel to axis is x 5 y 0 (a, b, c, d) 5 Midpoint of PQ, Slope of OP 5 Slope of axis 5 [ Line joining the point of intersection of tangents at A and B on parabola and midpoint of chord AB is always parallel to axis of parabola] Slope of directrix 5 Equation of directrix is y 5 x 5 x y 0. Circle with OP as diameter is x y 3 x 3 y 0. Circle with OQ as diameter is x y x y 0. 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

23 4 th IIT 3rd June Test Paper- Solution 30 6,, which we get by solving above two 3 3 These circles pass through fixed point focus which is focus equations. Hence, equation of line joining origin, focus is x 5 y 0. Q5 The equation of the chord of the circle x y 3x 4 y 4 0 which passes through the origin such that origin divides it in the ratio 4 : is Q6 4 x y 0 7y 0 7 x 4 y 0 7 x 4 y 0 (a, b) The number of ways of arranging the letters AAAAA, BBB, CCC, D, EE, and F in a row if the letters C are separated from one another is 3 C3.! 5! 3!! 4! 3! 3!! 3! 5! 3! 3!! 5! 3!! 3 C 5! (3!)! 5! 3!! 5! 3! (a, d) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

24 5 th IIT 3rd June Test Paper- Solution SECTION-C Paragraph for next Questions The st, nd and 3rd terms of an arithmetic series are a, b, and a, where a is negative. The st, nd and 3rd terms of a geometric series are a, a, and b. Q The sum of infinite geometric series is Q The sum of the 40 terms of the arithmetic series is (c), (a) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

25 6 th IIT 3rd June Test Paper- Solution Paragraph for next Questions Let A, B, C be angles of triangle with vertex A (4, ) and internal angular bisectors of angles B and C be x 0 and x y 0 respectively. D, E, F are the points of contact of sides BC, CA, AB with in-circle of triangle ABC. Q3 Slope of BC is Q4 3 (b) B C cot If A, B, C are angles of triangle at vertices, A, B, C respectively, then cot (d) SECTION-D Integer Answer Type 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

26 7 th IIT 3rd June Test Paper- Solution This section Contains 4 questions. The answer to each of the questions is a single digit integer, ranging from 00 to 99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened. Q Complex number z and z satisfy z and z 3. If the included angle of their corresponding z z vectors is 60 then the value of 9 z z Q is. (7) If a, a, a3... is an arithmetic progression with common difference and a a a3... a98 37, the value of a a 4 a6... a98 is Q3 (93) n If the constant term in the binomial expansion of x, n N is 5 then the value of n is equal to. x 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783

27 8 th IIT 3rd June Test Paper- Solution (6) For x x n I r n C r ( x ) n r ( ) r x r n C r x n 3r ( ) r Constant term n C r ( ) r if n 3r i.e. coefficient of x 0. Hence n C n / 3 ( ) n / C 4 Q4 n 6 n a x If , the value of (n a ) is x, then find the value of (0) 0, Blue Diamond Complex, Fatehgunj, Vadodara-0.Tel. No / / 783