The IIT-JEE s premier institute VelocIITy..

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1 The IIT-JEE s premier institute VelocIITy.. Corporate ffice: # -/NR, Plot No., Janaki Towers, Road pp. Madhapur P.S, Kavuri Hills,Hyderabad 50008, Ph: 040- /55, Cell: , , Web: ur Branches: Madhapur Dilsukhnagar Secunderabad Ramanthapur Nallakunta Kukatpally R.K Puram (DSNR) S.R Nagar Madinaguda Visakhapatnam MATHEMATICS

2 MATHEMATICS PHYSICS. The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively..5% and % the maximum error in determining the densityis : ).5 % ).5% ) 4.5 % 4) 6% Ans: m d d m d m. All the graphs below are intended to represent the same motion. ne of them does it incorrectly. Pick it up. ) ) ) 4) Ans: Since slope should decrease for some time and then increase but the reverse is given. Two masses m 5 kg and m 0 kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.5. The minimum weight m that should be put on top of m to stop the motion is: ) 8. kg ) 7. kg ) 4. kg 4) 0. kg >><<

3 Ans: 0 m m 5 m. 4. A particle is moving in a circular path of radius a under the action of an k attractive potential U. Its total energy is: r k k k ) ) ) zero 4) 4a a a Ans: k mv F r r k mv r k k T.E mv P.E 0 r r 5. In a collinear collision, a particle with an initial speed v0 strikes a stationary particle of the same mass. If the final total kinetic energy is 50% greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is : v0 v0 v0 ) ) v0 ) 4) 4 Ans: mv0 mv mv v v v () 0 given, kf ki v v v0.() >><<

4 From () & (), we get v v v v v v 4v v 0 v0 v v 0 v v v0 0 v 6. Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is ) 9 MR ) 55 MR ) 7 MR 4) 8 MR Ans: 4 I I MR com IP 6 MR M R MR 7M R 9MR MR 6MR 55MR 6MR 55 MR 6MR >>4<< 8 MR 7. From a uniform circular disc of radius R and mass 9 M, a small disc of radius R is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is

5 ) 4MR ) 40 MR 9 ) 0 MR 4) 7 MR 9 Ans: R R R / R / 4 4 R R R R 9 9M 4 8 R 4MR R 9 R /R / 8. A particle is moving with a uniform speed in a circular orbit of radius R in a th central force inversely proportional to the n power of R. If the period of rotation of the particle is T, then : / ) T R for any n. ) T n / ) T R 4) T R n R n/ Ans: n F R n mv R v R R T v n R R n R n >>5<< R n R n

6 9. A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, dr r, is ) Ka mg Ans: 4 v r ) Ka mg ) mg Ka 4) mg Ka dv 4 r dr mg v / k a v mg dr ak r dv v dr r 0. Two moles of an ideal monoatomic gas occupies a volume V at7 C. The gas expands adiabatically to a volume V. Calculate (a) the final temperature of the gas and (b) change in its internal energy. ) (a) 89 K (b).7 kj ) (a) 95 K (b).7 kj ) (a) 89 K (b).7 kj 4) (a) 95 K (b).7 kj Ans: T.v Tv T ' v 0 T v T v T v 00 T f T 89 f n U c T r R K.J >>6<<

7 . The mass of a hydrogen molecule is molecules strike, per second, a fixed wall of area 7. 0 kg. If0 hydrogen cm at an angle of 45 to the normal, and rebound elastically with a speed of 0 m /s, then the pressure on the wall is nearly: ).5 0 N / m ) N / m ) Ans:.5 0 N / m 4) N / m F n mv cos 45 A t A 7 F P 4 A 0. P A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 0 /sec.what is the force constant of the bonds connecting one atom with the other? ( Mole wt. of silver = 08 and Avagadro number gm mole ) ) 6.4 N/m ) 7. N/m ). N/m 4) 5.5 N/m Ans: M m A M m A K m f m k K K 7.6 N / m >>7<<

8 . A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is.7 0 kg /m and its Ans: 0 Young s modulus is Pa. What will be the fundamental frequency of the longitudinal vibrations? ) 5 khz ).5 khz ) 0 khz 4) 7.5 khz f v 4L khz Three concentric metal shells A, B and C of respective radii a, b and c a b c have surface charge densities, and respectively. The potential of shell B is: a b ) c ) a ) 0 b c a 4) b 0 4 a 0 b 0 b b c c c a Ans: V B 4n K 4c K 4a K b b c a o b b c 5. A parallel plate capacitor of capacitance 90 pf is connected to a battery of 5 emf 0 V. If a dielectric material of dielectric constant K is inserted between the plates, the magnitude of the induced charge will be: ). nc ) 0. n C ).4 n C 4) 0.9 n C >>8<<

9 Ans: Qp Q k 5 5 kcv k pc 5 = nc 6. In an a.c. circuit, the instantaneous e.m.f and current are given by e 00sin 0t ; i 0sin0t 4 In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively: ) 50,0 ) 000,0 Ans: P i v cos av m m 0 00 cos 45 Wattles current irms sin 0 ) 50,0 4) 50, 0 7. Two batteries with e.m.f V and V are connected in parallel across a load resistor of 0. The internal resistances of the two batteries ate and respectively. The voltage across the load lies between: ).6V and.7v ).5V and.6v ).4V and.5v 4).7V and.8v Ans: >>9<<

10 An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii r e,r p,r respectively in a uniform magnetic field B. The relation between r e,r p,r is : ) r e r p r ) r e r p r ) r e r p r 4) r e r r p Ans: P K.E. m r r mk.e. qb m q m m q q m m m q q q p e p 9. The dipole moment of a circular loop carrying a current I, is m and the magnetic field at the centre of the loop is B. When the dipole moment is doubled by keeping the current constant, the magnetic field is the centre of B the loop is B. The ratio B is ) ) ) 4) >>0<<

11 Ans: m I R 0I B R m' m I R I R R R B B R B R 0 I R 0. For an RLC circuit driven with voltage of amplitude m an frequency the current exhibits resonance. The quality factor, Q is given by: 0 0 ) L R LC 0 ) R L R ) C 0 4) CR 0 Ans: 0 Q L R. An EM wave from air enters a medium. The electric fields are z E E0x cos t c in air and E E0 x cos k z ct in medium, where the wave number k and frequency refer to their values in air. He medium is non-magnetic. If and refer to relative permittivities of air Ans: r and medium respectively, which of the following options is correct? r r ) r 4 ) r ) 4) 4 r 0 r r r r >><<

12 C C r r C C r r r r 4. Unpolarized light of intensity I passes through an ideal polarizer A. Another identical polarizer B is placed behind A. The intensity of light beyond B is found to be I. Now another identical polarizer C is placed between A and B. The intensity beyond B is now found to be I. The angle between polarizer A 8 and C is: ) 0 ) 0 ) 45 4) 60 Ans: I I cos sin 8 cos sin 4 sin The angular width of the central maximum is a single slit diffraction pattern is 60. The width of the slit is m. The slit is illuminated by monochromatic plane waves. If another slit of same width is made near it, Young s fringes can be observed on a screen placed at a distance 50cm from the slits. If the observed fringe width is cm, what is slit separation distance?(i.e., distance between the centres of each slit.) ) 5 m ) 50 m ) 75 m 4) 00 m >><<

13 Ans: sin a a m D d d d An electron from various states of hydrogen atom emit radiation to come to the ground state. Let n, g be the de Broglie wavelength of the electron in th the n state and the ground state respectively. Let n be the wavelength of th the emitted photon in the transition from the n state to the ground state. For large n, (A, B are constants) B A ) n A B n ) n A B n 4) n ) n n Ans:.6 En n hc hc Eg.6eV En Eg n.6 n.6 n hc Using Binomial approximation, we get.6 n B n A since n ng n >><< n

14 5. If the series limit frequency of the Lyman series is L, then the series limit frequency of the Pfund series is: L ) 5 L ) 6L ) 6 Ans: 4 RC n n L RC P RC 5 L P 5 L RC...() RC..() 5 L 4) 5 6. It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is p d ; while for its similar collision with carbon nucleus at rest, fractional loss of energy is p c. The values of p d and p c are respectively: 89, 8 8, 89 0, Ans: ) ) ) 0,0 4) Fraction of energy lost by 4 8 P A P C m m m m m 7. The reading of the ammeter for a silicon diode in the given circuit is: ) 0 ) 5mA ).5mA 4).5mA >>4<<

15 Ans: 0.7 I I.5 ma A telephonic communication services is working at carrier frequency of 0GHz. nly 0% of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of 5kHz? ) 0 ) 4 0 ) 5 0 4) 6 0 Ans: No. of channels In a potentiometer experiment, it is found that no current passes through the galvanometer when the terminals of the cell are connected across 5cm of the potentiometer wire. If the cell is shunted by a resistance of 5, a balance is found when the cell is connected across 40cm of the wire. Find the internal resistance of the cell ) ).5 ) 4).5 Ans: P.d length 5.() 40 dividing () & () 5 5 r r 0 5 r r r.() >>5<<

16 0. n interchanging the resistances, the balance point of a meter bridge shifts to the left by 0cm. The resistance of their series combination is k. How much was the resistance on the left slot before interchanging the resistances? ) 990 ) 505 ) 550 4) 90 Ans: givenr R k Before interchange: x 00 x 00 x R 00 R.() R R x R x R After interchange: x 0 0 x R x 0.() R R R 0 x 00 x x 0 from () & () x 0 x 0 on simplifying x R from () 55 R 9 9 R R, R R 000 R 550 CHEMISTRY C H is. The ratio of mass percent of C and H of an organic compound X Y Z 6 :. If one molecule of the above compound CXHY Z contains half as much oxygen as required to burn one molecule of compound CXH Y completely to C and H.The empirical formula of compound CXHY Z is ) CH6 ) CH4 ) CH4 4) CH4 Ans: 4 X 6 Y X Y Y Y CXHY X XC H 4 Y X 4 Y X X Z Z X X 4 X Z CXHYZ CXHXX CH4 >>6<<

17 . Which type of defect has the presence of cations in the interstitial sites? ) Schottky defect ) Vacancy defect ) Frenkel defect 4) Metal deficiency defect Ans: Frenkel defect is shown by ionic solid. The smaller ion (usually cation) is dislocated from its normal site to an interstitial site. It creates a vacancy defect at its original site and an interstitial defect at its new location.. According to molecular orbital theory, which of the following will not be a viable molecule? ) He ) He ) H 4) H Ans: 4 Bond order of He.5 H.5 H 0 He 4. Which of the following lines correctly show the temperature dependence of equilibrium constant, K, for an exothermic reaction? ) A and B ) B and C ) C and D 4) A and D Ans: K H ln K R T T +ve slope for exothermic reaction. >>7<<

18 5. The combustion of benzene l gives C g and H l. Given that heat of combustion of benzene at constant volume is 6.9 kj mol at 5 C; heat of combustion in kj mol of benzene at constant pressure will be: R 8.4 JK mol ) 45.6 ) ) 60 4) Ans: 4 5 C6H6l g 6C g H l 5 n H U nrt kj / mol For molal aqueous solution of the following compounds, which one will show the highest freezing point? ) CoH Cl 6 ) CoH Cl Cl 5.H ) CoH Cl 4 Cl.H 4) CoH Cl.H Ans: 4 CoH Cl 6 4 ions Co H Cl Cl 5.H ions Co H Cl 4 Cl.H ions Co H Cl.H particle Depression in freezing point is less for highest freezing point. Co H Cl.H, hence, 7. An aqueous solution contains 0.0 M HS and 0.0 M HCl. If the equilibrium constants for the formation of HS from HS is. 0 then the concentration of that of S from HS ions is in aqueous solution is ) ) 0 0 ) 6 0 4) and S ions >>8<<

19 Ans: 0 a x H S H S K x. 0 x An aqueous solution contains an unknown concentration of Ba. When 50 ml of a M solution of NaS 4 is added, BaS 4 just begins to precipitate. Ans: The final volume is 500 ml. The solubility product of BaS 4 is 0. What is the original concentration of Ba? 9 9 ) 5 0 M ) 0 M ). 0 M 4).0 0 M Let Ba conc. be x and volume of this 0 Ba S x x Ba solution is 450 ml. 9. At 58 C, the rate of decomposition of a sample of gaseous acetaldehyde, Ans: initially at a pressure of 6 Torr, was.00 Torr when 5% had reacted and 0.5 Torrs when % had reacted. The order of the reaction is ) ) ) 4) k C n n n n 67 n k c log nlog 67 n 40. How long (approximate) should water be electrolysed by passing through 00 amperes current so that the oxygen released can completely burn 7.66 g of diborane? Atomic weight of B 0.8 u s ) 6.4 hours ) 0.8 hours ). hours 4).6 hours 0 0 >>9<<

20 Ans: BH6 B H zit w t t 580 sec t. hours 4. The recommended concentration of fluoride ion in drinking water is up to ppm as fluoride ion is required to make teeth enamel harder by converting Ca P 4.Ca H to: ) CaF ) CaF.Ca H ) Ca P 4.CaF 4) {Ca H }.CaF Ans: F ions make the enamel on teeth much harder by connecting Ca P 4.Ca H to Ca P 4.CaF. 4. Which of the following compounds contain(s) no covalent bond(s)? KCl, PH,, BH 6, HS 4 ) KCl, BH 6, PH ) KCl, HS4 ) KCl 4) KCl, BH6 Ans: KCl K Cl PH H P H H BH6 H H H B B H H H HS4 H S H 4. Which of the following are Lewis acids? ) PH and BCl ) AlCl and SiCl 4 ) PH and SiCl 4 4) BCl and AlCl >>0<<

21 Ans: 4 BCl and AlCl have incomplete octet and hence act as Lewis acids 44. Total number of lone pair of electrons in I ion is: ) ) 6 ) 9 4) Ans: I ion is I I I Total lone pairs = Which of the following salts is the most basic in aqueous solution? Al CN ) CHCK Ans: ) FeCl 4) PbCH C ) CHCK only anionic hydrolysis, hence most basic 46. Hydrogen peroxide oxidises FeCN but reduces FeCN 6 Fe CN products formed are, respectively: ) H and H Ans: Acidic medium H H e H 4 FeCN Fe 6 CN e 6 4 H Fe CN H Fe 6 CN H 6 Alkaline medium H H H e 4 e Fe CN Fe 6 CN 6 4 H FeCN H Fe 6 CN 6 H >><< 6 to to FeCN 6 in acidic medium in alkaline medium. The other ) H and H H 4) H and H H ) H and H

22 47. The oxidation states of Cr in Cr H 6 Cl, Cr C 6 H 6 and K Cr CN NH are: ) +, +4 and +6 ) +, + and +4 ) +, 0 and +6 4) +, 0 and +4 respectively Ans: Cr H Cl 6 Cr x 0 0 Cr C6H6 Cr X x 4 K Cr CN NH Cr x The compound that does not produce nitrogen gas by the thermal decomposition is: Ba N ) NH Cr ) 4 NH Ans: 4 ) Ba N Ba N NH Cr Cr N 4H NH N N H NH S NH HS NH NH N 4) S When metal M is treated with NaH, a white gelatinous precipitate X is obtained, which is soluble in excess of NaH. Compound X when heated strongly gives an oxide which is used in chromatography as an absorbent. The metal M is ) Zn ) Ca ) Al 4) Fe Ans: M X excess NaH Al NaH Al H NaAl Al H X Al Alumina >><<

23 50. Consider the following reaction and statements: Co NH Br 4 Br CoNH Br NH I) Two isomers are produced if the reactant complex ion is a cis-isomer. II) Two isomers are produced if the reactant complex ion is a trans-isomer. III) nly one isomer is produced if the reactant complex ion is a trans-isomer. IV) nly one isomer is produced if the reactant complex ion is a cis-isomer. The correct statements are: ) (I) and (II) ) (I) and (III) ) (III) and (IV) 4) (II) and (IV) Ans: b a b b M a b trans b a a M b b b cis M Co a Br b NH b b a a M a b only one Isomer b a a M b a b b a a M a b b 5. Glucose on prolonged heating with HI gives ) n-hexane ) -Hexene ) Hexanoic acid 4) 6-iodohexanal Ans: CH H H H H H H H H CH H HI/ 5. The trans alkenes are formed by the reduction of alkynes with ) H Pd /C,BaS4 ) NaBH 4 ) Na / liq.nh 4) Sn HCl >><<

24 Ans: Na/liq. NH R H R C C R C C H R 5. Which of the following compounds will be suitable for Kjeldahl s method for nitrogen estimation? ) ) NH ) N 4) + - N Cl N Ans: Diazonium salts, nitro compounds and aromatic compounds in which N present ring are not respond for Kjeldhal s method. 54. Phenol on treatment with C in the presence of NaH followed by acidification produces compound X as the major product. X on treatment Ans: with CH C in the presence of catalytic amount of HS 4 produces. ) H C H CH + C NaH H + ) H (X) C H CH ) CH (CH C) H + CH C H 4) (Aspirin) C CH CH 55. An alkali is titrated against an acid with methyl orange as indicator, which of the following is a correct combination? Base Acid End point ) Weak Strong Colourless to pink ) Strong Strong Pinkish red to yellow ) Weak Strong Yellow to pinkish red 4) Strong Strong Pink to colourless Ans: Acidic medium Basic medium Methyl orange Pink Yellow C H C H CH >>4<<

25 56. The predominant form of histamine present in human blood is pk Histidine 6.0 ) ) Ans: 4 H a N a, H N N H N N b NH c NH NH ) NH 4) H N NH H N N NH NH Lone pair on a is participating in resonance, hence least basic. Among b and c, B is sp and c is sp hence c is most basic and accepts a proton from blood. Hence the most predominant form is H N NH N 57. Phenol reacts with methyl chloroformate in the presence of NaH to form product A. A reacts with Br to form product B. A and B are respectively ) H CH and H Br ) CH and ) and Br 4) H CH and Br H CH Ans: H C CH C Br CH + Cl C CH NaH Br (A) Br (B) >>5<<

26 58. The increasing order of basicity of the following compounds is NH NH NH a) b) c) NH ) (a) < (b) < (c) < (d) ) (b) < (a) < (c) < (d) ) (b) < (a) < (d) < (c) 4) (d) < (b) < (a) < (c) d) NHCH Ans: a) b) NH H C amine with NH sp 'N' sp hybridization. c) NH NH NH NH d) NHCH amine with sp hybridization. 59. The major product formed in the following reaction is HI Heat ) H ) I ) H 4) I H I I H Ans: 4 HI heat H I + C H 5 I + CH H 60. The major product of the following reaction is: Br NaMe MeH ) Me ) ) 4) Me >>6<<

27 Ans: Br NaMe MeH 6. Two sets A and B are as under: MATHEMATICS A a,b R R : a 5 and b 5 ; B a,b R R : 4 a 6 9 b 5 6. Then: ) B A ) A B ) A B (an empty set) 4) neither A B nor B A Ans: A represents set of all points lies inside the square formed by x 4, x 6, y 4 and y 6. B represents set of all points on or inside the ellipse x 6 y , 4, 4, 6, 6, 4, 6, 6 lies inside the ellipse. Clearly the points 6. Let Ans: S x R : x 0 and x x x Then S: ) is an empty set ) contains exactly one element. ) Contains exactly two elements 4) contains exactly four elements. If x 9, then x 8 x 0 x x 4 If x, then x 4 x 0 x 4 x 6 6. If, C are the set distinct roots, of the equation x x 0, then 0 07 is equal to: ) ) 0 ) 4) Ans:, 0 07 >>7<<

28 x 4 x x 64. If to: x x 4 x A Bx x A, x x x 4 then the ordered pair (A, B) is equal ) 4, 5 ) 4, ) 4,5 4) 4,5 Ans: x 4 x x x x 4 x x x x 4 C C C, C C C x 4 0 x x 4 x 4 x 0 x 4 x 4 0 x x 4 x 0 x 4 x 4 5x If the system of linear equations x ky z 0 x ky z 0 x 4y z 0 has a non-zero solution x,y,z, then xz y is equal to: ) 0 ) 0 ) 0 4) 0 Ans: x ky z 0 x ky z 0 x 5z 0 x z (say) 5 5 4y 0 y 0 zx 5 0 y >>8<<

29 66. From 6 different novels and different dictionaries, 4 novels and dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is: ) at least 000 ) less than 500 ) at least 500 but less than 750 4) at least 750 but less than 000 Ans: Number of ways of selecting four novels from six novels is 6 C 4. Number of ways of selecting one dictionary from three dictionary is C. So, number of ways of arranging them is 6 C4 C 4! The sum of the co-efficient of all odd degree terms in the expansion of 5 5 x x x x, x is: ) ) 0 ) 4) Ans: C 0x Cx x C4x x x 0 x x 5x x x 0 0 Sum of odd degree terms 68. Leta,a,a,...,a 49 be in A.P. such that >>9<< a4k 46 and 9 4 k0 If a a... a7 40m, then m is equal to: ) 66 ) 68 ) 4 4) Ans: Let d be the common difference of A.P. a 46 7 a k0 4k 5 a 4d 46 a 4d a9 a4 a 50d 66 a 5d a 8, d 7 7 r r r a r r r r r a a 66.

30 69. Let A be the sum of the first 0 terms and B be the sum of the first 40 terms of the series IfB A 00, then is equal to: ) ) 48 ) 464 4) 496 Ans: A B B A For each t R, let [t] be the greatest integer less than or equal to t. Then 5 lim x... x0 x x x ) is equal to 0 ) is equal to 5 ) is equal to 0 4) does not exist (in R) Ans: We know that x x x r r r x x x r r r x x x r r r 5 0 r 0 5 x x x r r lim 0 5x lim x lim 0 x 5 x0 x0 r x0 5 lim x... 0 x0 x x x 7. Let x S t R : f x x. e sin x is not differentiable at t}. Then the set S is equal to: ) (an empty set) ) 0 ) 4) 0, >>0<<

31 Ans: f ' x e sin x lim x x x x lim e sin x 0 f is differentiable at. f ' 0 lim x0 x0 x x lim sin x 0 x e sin x f is differentiable everywhere. x 7. If the curves y 6x,9x by 6 intersect each other at right angles, then the value of b is: ) 6 ) 7 ) 4 4) 9 Ans: 4 Let p x, y is the point of intersection of 9x 6bx 6 (i) Slope of tangent of Slope of tangent of y 6x at px, y 9x by 6 If they intersects orthogonally y at px, y 9x 7x 6 (ii) Comparing (i) and (ii) 6b 7 9 b y 6x, 9x by 6 9x by 7x by 6 9x 7. Let f x x and g x x, x R,0,. If x local minimum value of hx is: x f x h x, then the g x ) ) ) 4) Ans: 4 hx x x x >><<

32 74. The integral ) tan x C sin x cos x dx is equal to: 5 5 sin x cos x sin x sin x cos x cos x ) cot x C (where C is a constant of integration) ) tan x 4) cot x C C Ans: sin x cos x 5 5 sin x cos x sin x sin x cos x cos x sin x cos x dx sin x cos x tan x sec x dx tan x tan x C dx 75. The value of ) 8 sin x x dx is: ) ) 4 4) 4 Ans: 4 I I sin x dx x sin x x x dx sin x x dx >><<

33 I sin x dx 4I cos x dx I Let g x cos x, f x x, and, be the roots of the quadratic equation 8x 9x 0. Then the area (in sq. units) bounded by the y g f x and the lines x, x and y 0, is curve ) ) ) 4) Ans: y gof x cos x Required area sin sin cos x cos sin cos sin Let y y x be the solutions of the differential equation dy sin x y cos x 4x, x 0, dx. If y 0, then y is equal to 6 4 ) 9 ) ) 4) >><<

34 Ans: dy sin x y cos x 4x dx sin x.dy y cos x dx 4x dx d y sin x 4x dx y sin x x c Given y 0 c Now y 6 y. 6 8 y A straight line through a fixed point, intersects the coordinate axes at distinct points P and Q. If is the origin and the rectangle PRQ is Ans: completed, then the locus of R is ) x y 6 ) x y xy ) x y xy 4) x y 6xy a b h k,, a,b h,k Line equation a b x y y x xy x y PQ a b >>4<<

35 79. Let the orthocentre and centroid of a triangle be A, 5 and B, Ans: respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is 5 ) 0 ) 0 ) 4) 5 h k 5,, h 6 k AC Radius If the tangent at, 7 to the curve x y 6 touches the circle x y 6x y c 0 then the value of c is ) 95 ) 85 ) 85 4) 95 Ans: 4 Tangent of parabola x y 7 6 x y 7 x y C C 5 00 C C 95 >>5<<

36 8. Tangent and normal are drawn at P 6, 6 on the parabola y 6x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and CPB, then a value of tan is ) ) ) 4) 4 Ans: y Slope of PC 6 Slope of PB 8 4 tan Tangents are drawn to the hyperbola 4x y 6 at the points P and Q. If T 0, then the area (in sq. units) of these tangents intersect at the point PTQ is ) 45 5 ) 54 ) 60 4) 6 5 Ans: x y 9 6 4x.0 y. 6 y 4x x x 4 >>6<<

37 , ;, 0, If L is the line of intersection of the planes x y z 0, x y z 0 and L is the line of intersection of the planes x y z 0, x y z 0, then the distance of the origin from the plane, containing the lines L and L is ) ) ) 4) 4 Ans: :x y z x y z 0 is plane containing L Finding point in L Substituting y 0 in L x z x z 6 x z 5x 7 7 x 5 x z 7 z 5 8 y 0 5 P 7,0, since P lies in 5 The required plane is :7x 7y 8z 0 Perpendicular distance of 0,0,0 from is >>7<<

38 84. The length of the projection of the line segment joining the points 5,, 4 and 4,, on the plane, x y z 7 is ) ) Ans: 4 h 5 k h, k, 4 4 A,, h 4 k 0 h, k, 0 B,, AB ) 85. Let u be a vector coplanar with the vectors a i j k and b j k. If u is perpendicular to a and u b 4, then u is equal to ) 6 ) 5 ) 56 4) 84 4) Ans: u a a b a b a a a b i j k 4 j k 4i 8 j 6k u b 4 u A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that his drawn ball is red, is ) ) ) 4) >>8<<

39 Ans: 4R 6B C C C C 0 0 C C C C P E If xi 5 9 and i i ,, x 5 45, then the standard deviation of the 9 i items x, x,...,x 9 is ) 9 ) 4 ) 4) Ans: x...x 9 Has same variance with xi 5 xi 5 9 xi If sum of all the solutions of the equation 8 cos x cos x cos x 6 6 in0, is k, then k is equal to ) ) ) 8 4) Ans: 8cos x cos x 4cos x 4 8 cos x 6cos x cos x ;cos x x 60,00,40, ,, k 9 9 >>9<<

40 89. PQR is a triangular park with PQ PR 00m. A T.V. tower stands at the midpoint of QR. If the angles of elevation of the top of the tower at P, Q and o R are respectively 45, 0 and 0, then the height of the tower (in m) is Ans: ) 00 ) 50 ) 00 4) 50 Tower of the height =h From the triangle SPM h tan45 PM From the triangle PQM h 00 From the triangle SQM h tan 0 h h By (ii) h 00 4, RS=tower... (i) (ii) by (i) 90. The Boolean expression ~ p q ~ p q is equivalent to ) ~ p ) p ) q 4) ~ q Ans: ~ p q ~ p q ~ p ~ q ~ p q >>40<<