JEE (MAIN) Date : Duration : 3 Hours Max. Marks : 360 IMPORTANT INSTRUCTIONS

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1 JEE (MAIN)- Date : 7-4- Duration : Hours Max. Marks : 6 IMPORTANT INSTRUCTIONS. Immediately fill the particulars on this page of the Test Booklet with Blue / Black Ball Point Pen. Use of pencil is strictly prohibited.. The Answer Sheet is kept inside this Test Booklet. When you are directed to open the Test Booklet, take out the Answer Sheet and fill in the particulars carefully.. The test is of hours duration. 4. The Test Booklet consists of 9 questions. The maximum marks are There are three parts in the question paper A, B, C consisting of Physics, Chemistry and Mathematics having questions in each part of equal weightage. Each question is allotted 4 (four) marks for correct response. 6. Candidates will be awarded marks as stated above in Instructions No. 5 for correct response of each question. ¼ (one fourth) marks will be deducted for indicating incorrect response of each question. No deduction from the total score will be made if no response is indicated for an item in the answer sheet. 7. There is only one correct response for each question. Filling up more than one response in any question will be treated as wrong response and marks for wrong response will be deducted accordingly as per instructions 6 above. 8. Use Blue/Black Ball Point Pen only for writing particulars/marking responses on Side- and Side- of the Answer Sheet. Use of pencil is strictly prohibited. 9. No candidate is allowed to carry any textual material, printed or written, bits of papers, paper, mobile phone, any electronic device, etc., except the Admit Card inside the examination hall/room.. Rough work is to be done on the space provided for this purpose in the Test Booklet only. This space is given at the bottom of each page and in pages at the end of the booklet.. On completion of the test, the candiate must hand over the Answer Sheet to the Invigilator on duty in the Room/Hall. However, the candidates are allowed to take away this Test Booklet with them.. The CODE for this Booklet is P. Maken sure that the CODE printed on Side- of the Answer Sheet is the same as that on this booklet. In case of discrepancy, the condidate should immediately report the matter to the Invigilator for replacement of both the Test Booklet and the Answer Sheet.. Do not fold or make any stray marks on the Answer Sheet. Name of the Candiate (in Capital letters) : Roll Number : in figures : in words : Examination Centre Number : Name of Examination Centre (in Capital letters) : Candidate's Signature : Invigilator's Signature :

2 PART A PHYSICS. A uniform cylinder of length L and mass M having cross - sectional area A is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density at equilibrium position. The extension x of the spring when it is in equilibrium is : () k Mg Mg LA () k M Mg LA () k M Mg LA (4) k M kx + F B mg kx + L Ag Mg Mg LAg x k Mg LA k M Ans (). A metallic rod of length l is tied to a string of length l and made to rotate with angular speed on a horizontal table with one end of the string fixed. If there is a vertical magnetic field B in the region, the e.m.f. induced across the ends of the rod is: () Bl () Bl () 4Bl (4) 5Bl e ( x)bdx B [() () ] Ans. (4) 5B

3 . This question has statement and Statement. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement - : A point particle of mass m moving with speed collides with stationary point particle of mass M. If the maximum energy loss possible is given as m m ƒ then ƒ. M m Statement - : Maximum energy loss occurs when the particles get stuck together as a result of the collision. () Statement - is true, Statment - is true, Statement - is the correct explanation of Statement -. () Statement - is true, Statment - is true, Statement - is not the correct explanation of Statement -. () Statement - is true, Statment - is false. (4) Statement - is false, Statment - is true. Maximum energy loss P P m (m M) P M mv m (m M) M m M Hence Statement - is wrong and statement is correct Hence Ans (4) M f m M 4. Let [ ] denote the dimensional formula of the permittivity of vacuum. If M mass, L length, T time and A electric current, then : () [ ] [M L T A] () [ ] [M L T 4 A ] () [ ] [M L T A ] (4) [ ] [M L T A] qq F 4 R qq 4FR C Hence N. m [AT] MLT. L [M L T 4 A ] Ans. () 5. A projectile is given an initial velocity of î ĵ m/s, where î is along the ground and ĵ is along the vertical. If g m/s, the equation of its trajectory is : () y x 5x () y x 5x () 4y x 5x (4) 4y x 5x v î ĵ x t...(i) y t (t )...(ii) y x 5x Hence Ans () From (i) and (ii)

4 6. The amplitude of a damped oscillator decreases to.9 times its original magnitude is 5s. In another s it will decrease to times its original magnitude, where equals. ().7 ().8 ().79 (4).6 e bt A A m after 5 second b(5) e m.9a A After more second...(i) A A e b(5) m...(ii) From (i) & (ii) A.79 A Hence Ans. () 7. Two capacitors C and C are charged to V and V respectively. It is found that by connecting them together the potential on each one can be made zero. Then : () 5C C () C 5C () C + 5C (4) 9C 4C For potential to be made zero, after connection C C C 5C Ans. () 8. A sonometer wire of length.5 m is made of steel. The tension in it produces an elastic strain of %. What is the fundamental frequency of steel if density and elasticity of steel are 7.7 kg/m and. N/m respectively? () 88.5 Hz () 78. Hz ().5 Hz (4) 77 Hz f v T T Ad Also Y T A T Y A f y d.5m, y. N/m After solving f Hz 7 f 78. Hz Ans. ()., d 7.7 kg/m

5 9. A circular loop of radius. cm lies parallel to a much bigger circular loop of radius cm. The centre of the small loop is on the axis of the bigger loop. The distance between their centres is 5 cm. If a current of. A flows through the smaller loop, then the flux linked with bigger loop is : () 9. weber () 6 weber (). weber (4) weber ()( (.) (.5) ) (. ) on solving weber Ans (). Diameter of a plano - convex lens is 6 cm and thickness at the centre is mm. If speed of light in material of lens is 8 m/s, the focal length of the lens is : () 5 cm () cm () cm (4) cm n + (R mm) R + R R(mm) + (mm) R R 5 cm f Ans () 5 f cm. What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of R? () E f 5GmM 6R mv GMm R GMm E i + K R () m GmM R GM R () GmM R GMm GMm GMm R R 6R (4) GmM R E i E f 5GMm K 6R Ans ()

6 . A diode detector is used to detect an amplitude modulated wave of 6% modulation by using a condenser of capacity 5 pico farad in parallel with a load resistance kilo ohm. Find the maximum modulated frequency which could be detected by it. ().6 MHz ().6 khz () 5. MHz (4) 5. khz D Signal C R RC 5 sec.5 7 sec.5 5 sec The higher frequency which can be detected with tolerable distortion is f m RC.6.5 a 4 Hz 5. 5 Hz 4 4. Hz.6 KHz This condition is obtained by applying the condition that rate of decay of capacitor voltage must be equal or less then the rate of decay modulated singnal voltage for proper detection of mdoulated signal. Ans (). A beam of unpolarised light of intensity is passed through a polaroid A and then through another polaroid B which is oriented so that its principal plane makes an angle of 45º relative to that of A. The intensity of the emergent light is : () () / () /4 (4) /8 Relation between intensities is R cos (45 ) Ans. () 4

7 4. The supply voltage to room is V. The resistance of the lead wires is 6. A 6 W bulb is already switched on. What is the decrease of voltage across the bulb, when a 4 W heater is switched on in parallel to the bulb? () zero Volt ().9 Volt (). Volt (4).4 Volt P V R 4 R 4 6 R eq V 4 V 7.7 volt 46 6 V V Volt 54 V V.4 Volt Ans (4) 48 Volt 5. The above p-v diagram represents the thermodynamic cycle of an engine, operating with an ideal monoatomic gas. The amount of heat, extracted from the source in a single cycle is : () p v () p v 5 P V P V P V Ans () () p v (4) 4p v

8 6. A hoop of radius r and mass m rotating with an angular velocity is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip? () r 4 () r () r (4) r V R V mr mvr + mr v Ans () r v r 7. An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V and its pressure is P. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency : () AP V M () V MP A () A P MV (4) MV AP Mg P A Mg P A...() P V Pv' P Ax PA(x x) x let piston is displaced by x P x P (x x) x P x Mg x x A F restoring P A x x x F restoring [ x x x ] P Ax F x f P A x M P A MV Ans ()

9 8. If a piece of metal is heated to temperature and then allowed to cool ina room which is at temperature, the graph between the temperature T of the metal and time t will be closest to : () () () (4) According to Newtons cooling law option () is correct Answer. 9. This questions has Statement and Statement. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement - : Higher the range, greater is the resistance of ammeter. Statement - : To increase the range of ammeter, additional shunt needs to be used across it. () Statement - is true, Statment - is true, Statement - is the correct explanation of Statement -. () Statement - is true, Statment - is true, Statement - is not the correct explanation of Statement -. () Statement - is true, Statment - is false. (4) Statement - is false, Statment - is true. Statements I is false and Statement II is true Ans (4). In an LCR circuit as shown below both switches are open initially. Now switch S is closed, S kept open. (q is charge on the capacitor and RC is Capacitive time constant). Which of the following statement is correct? () Work done by the battery is half of the energy dissipated in the resistor () At t, q CV/ () At t, q CV ( e ) (4) At t, q CV ( e )

10 q CV ( e t/ ) at t q CV ( e ) Ans (). Two coherent point sources S and S are separated by a small distance 'd' as shown. The fringes obtained on the screen will be : () points () straight lines () semi-circles (4) concentric circles It will be concentric circles Ans (4). The magnetic field in a travelling electromagnetic wave has a peak value of nt. The peak value of electric field strength is : () V/m () 6V/m () 9V/m (4) V/m E B C E B. C V/m. Ans (). The anode voltage of a photocellis kept fixed. The wavelength of the light falling on the cathode is gradually changed. The plate current of the photocell varies as follows : () () () (4) As is increased, there will be a value of above which photoelectrons will be cease to come out so photocurrent will become zero. Hance (4) is correct answer. Ans (4)

11 4. The V characteristic of an LED is : () () () (4) For same value of current higher value of voltage is required for higher frequency hance () should be correct answers. Ans () 5. Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is T, density of liquid is and L is its latent heat of vaporization. () L/T () T / L () T/L (4) T/L When radius is decrease by dr decrease in surface energy Heat required for vaporisation (4rdr) T 4r dr r Ans. (4) T L 6. Ina hydrogen like atom electron make transition from an energy level with quantum number n to another with quantum number (n ). If n>>, the frequency of radiation emitted is proportional to : () n E h () n E k h (n ) n kn n (n ) Ans. (4) k n n () n (4) n

12 7. The graph between angle of deviation () and angle of incidence (i) for a triangular prism is represented by : () () () (4) Ans. () 8. Two charges, each equal to q, are kept at x a and x a on the x-axis. A particle of mass m and charge q q is placed at the origin. If charge q is given a small displacement (y <<a) along the y-axis, the net force acting on the particle is proportional to : () y () y () y (4) y F net Fcos F net q kq y a y a. y F net q kq y (y a ) / kq y a y Ans. ()

13 9. Two short bar magnets of length cm each have magnetic moments. Am and. Am respectively. They are placed on a horizontal table parallel to each other with their N poles poining towards the South. They have a common magnetic equator and are separated by a distance of. cm. The value of the resultand horizontal magnetic induction at the mid - point O of the line joining their centres is close to (Horizontal component of earth s magnetic induction is.6 5 Wb/m ) ().6 5 Wb/m ().56 4 Wb/m ().5 4 Wb/m (4) Wb/m B net B + B + B H B net (M M ) B H 4 r 7 (. ) (.) wb/m Ans. (). A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at distance L from the end A is : O L A L B () Q 8 L () Q 4 L () Q 4 Lln (4) Qln 4 L V L L kdq x L L q dx L 4 x Ans. (4) q n() 4 L

14 PART B CHEMISTRY. Which of the following complex species is not expected to exhibit optical isomerism? () [Co(en) ] + () [Co(en) Cl ] + () [Co(NH ) Cl ] (4) [Co(en) (NH ) Cl ]+ Ans. () [Co(NH ) Cl ] show facial as well as meridional isomerism. But both contain plane of symmetry. So, the answer is ().. Which one of the following molecules is expected to exhibit diamagnetic behaviour? () C () N () O (4) S Ans. () and () N and C both are diamagnetic Ans is () and ().. A solution of ( ) chloro phenylethane in toluene racemises slowly in the presence of a small amount of SbCl 5, due to the formation of : Ans. () () carbanion () carbene () carbocation (4) free radical 4. Given : E Cr.74 V ; / Cr E MnO4 /Mn.5 V E CrO 7 / Cr. V ; Cl / Cl.6 V E Based on the data given above, strongest oxidising agent will be : () Cl () Cr + () Mn + (4) MnO 4 Ans. (4) Higher the SRP, better is oxidising agent Hence MnO 4 - is strongest oxidising agent. 5. A piston filled with.4 mol of an ideal gas expands reversibly from 5. ml to 75 ml at a constant temperature of 7.ºC. As it does so, it absorbs 8 J of heat. The values of q and w for the process will be: (R 8.4 J/mol K) (ln 7.5.) () q +8 J, w 8 J () q 8 J, w 8 J () q 8 J, w + 8 J (4) q +8 J, w + 8 J Ans. () The process is isothermal expansion Hence, q - w u q + 8 J w 8 J(expansion work)

15 6. The molarity of a solution obtained by mixing 75 ml of.5(m) HCl with 5 ml of (M)HCl will be : ().875 M (). M ().75 M (4).975 M Ans. () M f MV MV V V M. 7. Arrange the following compounds in order of decreasing acidity : ; ; ; () II > IV > I > III () I > II > III > IV () III > I > II > IV (4) IV > III > I > II Ans. () electron releasing group decreases and electron withdrawing group increases acidic strength. 8. For gaseous state, if most probable speed is denoted by C*, average speed by C and mean square speed by C, then for a large number of molecules the ratios of these speeds are : () C* : C : C.5 :.8 : () C* : C : C.8 :.5 : () C* : C : C :.8 :.5 (4) C* : C : C :.5 :.8 Ans. () C* most probable speed RT M C average speed 8RT M C Mean square speed corrected as rms C* < C <C RT M C* : C : C : 4 : :.8 :.5 Note : As no option correspond to mean square speed, it is understood as misprint. It should be root means square speed So, Ans is ()

16 9. The rate of a reaction doubles when its temperature changes from K to K. Activation energy of such a reaction will be : (R 8.4 JK mol and log.) () 5.6 kj mol () 48.6 kj mol () 58.5 kj mol (4) 6.5 kj mol Ans. () K log K Ea.R T T K K ; T K T K log Ea. 8.4 E a J/ mol Ans is () 5.6 KJ/mol 4. A compound with molecular mass 8 is acylated with CH COCl to get a compound with molecular mass Ans. () 9. The number of amino groups present per molecule of the former compound is : () () 5 () 4 (4) 6 By reaction with one mole of with one NH group the molecular mass increases with 4 unit. Since the mass increases by (9 8) hence the number of NH groups is 5. ( HCl) 4. Which of the following arrangements does not represent the correct order of the property stated against it? () V + < Cr + < Mn + < Fe + : paramagnetic behaviour () Ni + < Co + < Fe + < Mn + : ionic size () Co + < Fe + < Cr + < Sc + : stability in aqueous solution (4) Sc < Ti < Cr < Mn : number of oxidation states Ans. () () V + unpaired electrons Cr + 4 unpaired electrons Mn + 5 unpaired electrons Fe + 4 unpaired electrons Hence the order of paramagnetic behaviour should be V + < Cr + Fe + < Mn + () ionic size decrease from left to right in same period () As per data from NCERT. Co + /Co +.97 ; Fe + /Fe +.77 ; Cr + /Cr Sc + is highly stable (It does not show +) (4) The oxidation states increases as we go from group to group 7 in same period. Ans is ()

17 4. The order of stability of the following carbocations : ; ; is : () III > II > I () II > III > I () I > II > III (4) III > I > II Ans. (4) The order of stability of carbocation will be 4. Consider the following reaction : Ans. () xmno 4 + yc O 4 + zh + xmn + + yco + z H O The values of x, y and z in the reaction are, respectively : () 5, and 6 (), 5 and 8 (), 5 and 6 (4) 5, and 8 MnO 4 + C O 4 + H + Mn + CO + H O vf (7 ) vf ( ) 5 Balanced Equation : MnO 4 + 5C O H + So, x, y 5 & z 6. Mn + + CO + 8H O 44. Which of the following is the wrong statement? () ONCl and ONO are not isoelectronic. () O molecule is bent () Ozone is violet-black in solid state (4) Ozone is diamagnetic gas. Ans. (All statement are correct there is no answer). () ONCl e ONO e (correct) () Ȯ. Central atom O is sp hybridised with lone pair, so bent shape (correct) O O () Ozone is violet-black in solid state. (Ref. NCERT & shriver atkins) (4) O has no unpaired electrons, so diamagnetic (correct)

18 45. A gaseous hydrocarbon gives upon combustion.7 g of water and.8 g. of CO. The empirical formula of the hydrocarbon is : () C H 4 () C H 4 () C 6 H 5 (4) C 7 H 8 Ans. (4) 8g H O contains g H.7 g H O contains.8 gh. 44 g CO contains g C.8 g CO contains.84 g C C : H.84.8 :.7 :.8 7 : 8 Empirical formula C 7 H In which of the following pairs of molecules/ions, both the species are not likely to exist? Ans. () () H +, He- () H -, He- () H +, He (4) H -, He+ H : Bond order He : Bond order So, both H & He do not exist. 47. Which of the following exists as covalent crystals in the solid state? () Iodine () Silicon () Sulphur (4) Phosphorus Ans. () Silicon exists as covalent crystal in solid state. (Network like structure, like diamond). 48. Synthesis of each molecule of glucose in photosynthesis involves : () 8 molecules of ATP () molecules of ATP () 8 molecules of ATP (4) 6 molecules of ATP Ans. () Fact 6CO + NADPH + 8 ATP C 6 H O 6 + NADP + 8 ADP 49. The coagulating power of electrolytes having ions Na +, Al + and Ba + for arsenic sulphide sol increases in the order : () Al + < Ba + < Na + () Na + < Ba + < Al + () Ba + < Na + < Al + (4) Al + < Na + < Ba + Ans. () According to Hardy Schulze rule, greater the charge on cation, greater is its coagulating power for negatively charged solution. So, order of coagulating power : Na + < Ba + < Al +.

19 5. Which of the following represents the correct order of increasing first ionization enthalpy for Ca, Ba, S, Se and Ar? () Ca < S < Ba < Se < Ar () S < Se < Ca < Ba < Ar () Ba < Ca < Se < S < Ar (4) Ca < Ba < S < Se < Ar Ans. () Order of increasing H IE : Ba < Ca < Se < S < Ar Ba < Ca ; Se < S : On moving top to bottom in a group, size increases. So ionisation energy decreases. Ar : Maximum value of ionisation energy, since it is an inert gas. 5. Energy of an electron is given by E.78 8 J Z. Wavelength of light required to excite an electron n in an hydrogen atom from level n to n will be : (h Js and c. 8 ms ) ().4 7 m ().86 7 m () m (4) m Ans. () hc E m 5. Compound (A), C 8 H 9 Br, gives a white precipitate when warmed with alcoholic AgNO. Oxidation of (A) gives an acid (B), C 8 H 6 O 4. (B) easily forms anhydride on heating. Identify the compound (A). () CH Br () C H 5 Br () CH Br (4) CH Br CH CH CH Ans. (4)

20 5. Four successive members of the first row transition elements are listed below with atomic numbers. Which one of them is expected to have the highest E + + value? M /M Ans. (4) () Cr(Z 4) () Mn(Z 5) () Fe(Z 6) (4) Co(Z 7) Eº Cr + /Cr +.4 V ; Eº Mn + /Mn V ; Eº Fe + /Fe V ; Eº Co + /Co V SRP value normaly increases from left to right in the period of d-block elements. Some SRP value are exceptionally higher due to stability of product ion. For e.g. Eº Mn + /Mn V ; Eº Co + /Co V. 54. How many litres of water must be added to litre an aqueous solution of HCl with a ph of to create an aqueous solution with ph of? (). L ().9 L (). L (4) 9. L Ans. (4) ph [H + ]. M ph [H + ]. M for dilution of HCl M V M V.. V V lt Volume of water added 9 litre. 55. The first ionisation potential of Na is 5. ev. The value of electron gain enthalpy of Na + will be : ().55 ev () 5. ev (). ev (4) +.55 ev Ans. () Na Na + + e I st I.E. Na + + e Na Electron gain enthalpy of Na + Because reaction is reverse so then. H eg 5. ev. 56. An organic compound A upon reacting with NH gives B. On heating B gives C. C in presence of KOH reacts with Br to given CH CH NH. A is : () CH COOH () CH CH CH COOH () CH CH COOH (4) CH CH COOH CH Ans. (4)

21 57. Stability of the species Li, Li and Li + increases in the order of : () Li < Li + < Li () Li < Li + < Li () Li < Li < Li + (4) Li < Li < Li + Ans. () Li s s s Bond order + Li s s s Bond order.5 Li s s s s Bond order.5 + Stability order Li > Li > Li 58. An unknown alochol is treated with the Lucas reagent to determine whether the alcohol is primary, secondary or tertiary. Which alcohol reacts fastest and by what mechanism : () secondary alcohol by S N () tertiary alcohol by S N () secondary alcohol by S N (4) tertiary alcohol by S N Ans. () The reaction of alcohol with lucas reagent is mostly an S N reaction and the rate of reaction is directly proportional to the carbocation stability formed in the reaction, since R OH forms carbocation hence it will react fastest. 59. The gas leaked from a storage tank of the Union Carbide plant in Bhopal gas tragedy was : () Methylisocyanate () Methylamine () Ammonia (4) Phosgene Ans. () Methyl isocyanate CH N C O (MIC gas) (Fact) 6. Experimentally it was found that a metal oxide has formula M 98 O. Metal M, present as M + and M + in its oxide. Fraction of the metal which exists as M + would be : () 7.% () 4.8% () 6.5% (4) 5.8% Ans. () M.98 O consider one mole of the oxide. Moles of M.98, Moles of O Let moles of M + x Moles of M +.98 x Doing charge balance (.98 x) + x.96 x + x x.4.4 % of M + 4.8%.98

22 PART C MATHEMATICS 6. Distance between two parallel planes x + y + z 8 and 4x + y + 4z + 5 is () () 5 () 7 (4) 9 () x + y + z 8...(P ) x + y + z (P ) Distance between P and P At present, a firm is manufacturing items. It is estimated that the rate of change of production P w.r.t. dp additional number of workers x is given by x. If the firm employs 5 more workers, then the dx new level of production of items is () 5 () () 5 (4) 45 () dp ( x )dx By integrating dp dx P x 8x / + C When x then P C Now when x 5 then P is P 5 8 (5) / P 5 x 6. Let A and B two sets containing elements and 4 elements respectively. The number of subsets of A B having or more elements is () 56 () () 9 (4) () n(a) n(b) 4 n(a B) 8 8 C + 8 C C C 8 C 8 C If the lines x y z 4 k and x k y 4 z 5 are coplanar, then k can have () any value () exactly one value () exactly two values (4) exactly three values

23 () [a c, b, d] k k k k ( + k) + ( + k ) ( k) k + k + k k + k k, Note : If appears in the denominator, then the correct way of representing the equation of straight line is x y ; z If the vectors AB î 4kˆ and AC 5î ĵ 4kˆ are the sides of a triangle ABC, then the length of the () median through A is () 8 () 7 () (4) 45 AB BC CA BC AC AB AC AB BM AB BM MA AC AB AB AM AB AC AM 4î ĵ 4kˆ AM 66. The real number k for which the equation, x + x + k has two distinct real roots in [, ] () lies between and () lies between and () lies between and (4) does not exist. (4) f(x) x + x + k f(x) 6x + > x R f(x) is strictly increasing function f(x) has only one real root, so two roots are not possible 67. The sum of first terms of the sequence.7,.77,.777,..., is () (79 ) () (99 ) () (79 + ) (4) (99 + )

24 () up to terms 7...up to terms up to terms up to terms A ray of light along x + y gets reflected upon reaching x-axis, the equation of the reflected ray is () y x + () y x () y x (4) y x () Take any point B(, ) on given line Equation of AB y x y x x y y x 69. The number of values of k, for which the system of equations : (k + )x + 8y 4k kx + (k + )y k has no solution, is () infinite () () (4) () k 8 4k k k k k + 4k + 8k k 4k + k, If k 8 then And If k 4. False 8 4. then True 6 9 therefore k Hence only one value of k.

25 7. If the equations x + x + and ax + bx + c, a,b,c R, have a common root, then a : b : c is () : : () : : () : : (4) : : () x + x +...(i) ax + bx + c...(ii) Since equation (i) has imaginary roots So equation (ii) will also have both roots same as (i). Thus a b c a, b, c Hence : : 7. The circle passing through (, ) and touching the axis of x at (, ) also passes through the point () ( 5, ) () (, 5) () (5, ) (4) (, 5) () Let the equation of circle be (x ) + (y ) + y As it passes through (, ) ( ) + ( ) + ( ) 4 equation of circle is (x ) + y 8 so (5, ) satisfies equation of circle 7. If x, y, z are in A.P. and tan x, tan y and tan z are also in A.P., then A (, ) () x y z () x y 6z () 6x y z (4) 6x 4y z () y x + z tan y tan x + tan (z) y tan y x z y x z xz x z tan xz y xz or x + z x y z A(, ) 7. Consider Statement-I : ( p ~ q) (~ p q) is a fallacy. Statement-II : (p q) (~ q ~p) is a tautology. () Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I. () Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I. () Statement-I is true; Statement-II is false. (4) Statement-I is false; Statement-II is true. () Statement-II : (p q) (~ q ~p) (p q) (p q) which is always true so statement -II is true Statement-I : (p ~q) (~ p q) p ~ q ~ p q p ~ p ~ q q f f f so statement -I is true

26 Alternate Statement-II : (p q) (~ q ~p) ~ q ~ p is contrapositive of p q hence (p q) (p q) will be a tautology statement -II (p ~q) (~ p q) p q P ^ ~ q ~p ^ q (p ^ ~ q) ^ (~ p ^ q) T T F F F T F T F F F T F T F F F F F F It is a fallacy 74. If f (x)dx (x), then x f(x ) dx 5 is equal to () x (x ) x (x )dx C () x (x ) x (x )dx C () () x (x ) x (x )dx C f (x) dx (x) 5 I x f(x ) dx (4) x (x ) x (x )dx C put x t dt x dx tf(t) dt t(t) (t) dt x (x ) x (x )dx c x (x ) x (x )dx c 75. ( cos x)( cos x) lim x x tan 4x is equal to (4) () 4 I im x im x () ( cosx) ( cosx). x sin x cos x x.. x tan4x x tan4x () (4).4. 4

27 / 76. Statement-I : The value of the integral / 6 dx tan x is equal to /6. Statement-II : b a b f (x)dx f(a b x)dx. a () Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I. () Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I. () Statement-I is true; Statement-II is false. (4) Statement-I is false; Statement-II is true. (4) / I / 6 dx tanx / dx tan / 6 x / / 6 / / 6 tanx dx tanx tanx dx tanx / I dx / 6 I 6, statement - is false b a b f (x)dx f(a b x)dx it is property a 77. The equation of the circle passing through the foci of the ellipse x y, and having centre at (, ) is 6 9 () () x + y 6y 7 () x + y 6y + 7 () x + y 6y 5 (4) x + y 6y + 5 a 4, b, e 9 6 Focii is (± ae, ) (± 7, ) 7 4 r r ( ae) b Now equation of circle is (x ) + (y ) 6 x + y 6y 7

28 78. A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is : 7 () 5 () 5 () 5 (4) 5 () p, q 5 C C The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (, ) (, ) and (, ) is : () + () () + (4) () y.. x - coordinate of incentre C (, ) B (, ) (, ) (, ) (, ) A (, ) x 8. The term independent of x in expansion of x / x x / x x x / is : () () 4 () () (4) (x / ) x x (x / x / ) T r+ c r (x / ) r ( x / ) r r r r T 5 C 4 4 r r

29 8. The area (in square units) bounded by the curves y x, y x +, x-axis, and lying in the first () quadrant is : 7 () 9 () 6 () 8 (4) 4 y x...() and y x +...() On solving both y, Required area (y ) y dy y y y Let T n be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If T n+ T n, then the value of n is : () 7 () 5 () (4) 8 () T n n C T n + n + C T n + T n n + C n C n C n 5. z 8. If z is a complex number of unit modulus and argument, then arg z equals : () () () z, arg z z e i z z () (4) arg z arg (z). z 84. ABCD is a trapezium such that AB and CD are parallel and BC CD. If ADB, BC p and CD q, then AB is equal to : () (p q )sin pcos qsin () p q cos pcos qsin () p p q cos q sin (p q )sin (4) (pcos qsin)

30 () Let AB x p tan ( ) tan ( + ) x q q x p cot ( + ) x q p cot ( + ) cot cot q p cot cot p q x q p q p q p cot cot qcot p q p q pcot qcos psin q p qsin pcos Alternative x q sin pqcos pqcos p pcos qsin sin AB (p q )sin. pcos qsin From Sine Rule AB sin p q sin( ( )) AB p q sin sin cos cos sin (p q ) sin qsin pcos cos p q q (p q )sin. pcos qsin 85. If P is the adjoint of a matrix A and A 4, then is equal to : 4 4 () 4 () () 5 (4) () P ( ) (4 6) + (4 6) 6 P A The intercepts on x-axis made by tangents to the curve, y t dt, x R, which are parallel to the line y x, are equal to : () ± () ± () ± (4) ±4 () dy x dx x ± points y t dt ± equation of tangent is y (x ) or y + (x + ) x - intercept ±. x

31 87. Given : A circle, x + y 5 and a parabola, y 4 5 x. Statement-I : An equation of a common tangent to these curves is y x + 5. Statement-II : If the line, y mx + m 5 (m ) is their common tangent, then m satisfies m 4 m +. () Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I. () Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I. () Statement-I is true; Statement-II is false. (4) Statement-I is false; Statement-II is true. () Let common tangent y mx + m 5 5 m m 5 y ± (x + m m m ( + m ) m 4 + m (m + ) (m ) m ± 5 ), both statements are correct as m ± satisfies the given equation of statement-. dy 88. If y sec(tan x), then at x is equal to : dx () () () (4) () y sec (tan x) Let tan x x tan y sec y x dy dx at x x x dy. dx tan A cot A 89. The expression can be written as : cot A tan A () sina cosa + () seca coseca + () tana + cota (4) seca + coseca

32 () Given expression sin A cos A sin A sin A cos A cos A cos A sin A cos A sin A sin A cos A sin A cos A cos A sina sin A sin A cos A cos sin A cos A A + sec A cosec A 9. All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of to each of the students. Which of the following statistical measures will not change even after the grace marks were given? () mean () median () mode (4) variance (4) If initially all marks were x i then Now each is increased by (x x) (xi ) (x ) N So variance will not change whereas mean, median and mode will increase by. i N