# PCM- PART TEST-1 13th.pdf

29 Mar 2023

### PCM- PART TEST-1 13th.pdf

• 2. ROUGHWORK PART - A Select the correct alternative(s) (choose one or more than one) Q.1 AdiscofradiusR rollsonahorizontalsurfacewithlinearvelocityVof centre'O'and angularvelocity about 'O'. There is a point P on circumference of the disc at angle , which has a vertical velocity.Here  isequal to (A)  + sin–1  R V (B)     R V sin 2 1 (C)     R V cos 1 (D)     R V cos 1 Q.2 Ahorizontalturntableintheformofadiscofradiusrcarriesafixedsmallgunat G and rotates with a angular velocityw0 about a vertical axis passing through thecentreO.Theincrease inangularvelocityof thesystem if the gunfires a bullet of mass m with a tangential velocity v, w.r.t. the gun is (where I0 is moment ofinertiaofgunexcludingbullet +table, about O) (A) 0 I 2 mvr (B) r 2 v (C) 0 I mvr 2 (D) 2 0 mr I mvr  Q.3 Apoint object P of mass m is slipping down on asmooth hemispherical body of mass M & radius R.The point object is tied to a wallwith light inextensible string as shown.At a certain instant the speed of hemisphere isV& its acceleration a (as shownin figure). Then speed Vp &acceleration ap of the particlehasvalue(neglectfriction) (A) Vp = V sin 30 (B) Vp = V (C) ap = a (D) ap = 2 / 1 2 2 2 2 a 2 3 a R V                      
• 3. ROUGHWORK For Prob. 4 to 5 (3 Questions) Considertheshownarrangement.Theblockofmass misreleasedasshown on thesmoothcurvedtrack of wedge of mass M (=2m). One end of the ideal spring is fixed to the stand 'S' fixed to the ground. There is no friction between wedge & floor & between block & floor. Q.4 Ifspringconstant isK, maximumcompressioninthespringwould be (A) K mgh (B) K 3 mgh 4 (C) K 2 mgh 3 (D) none of these Q.5 Astheblockrebounces from thesprings& climbsup thewedgeagain, themaximum heightattainedby the block on the wedge would be (A) 9 h (B) 3 h 2 (C) 9 h 7 (D) none of these Q.6 Inyoung'sdoubleslitexperiment,thefringesaredisplacedbyadistancex whenaglassplateofrefractive index 1.5is introducedinthepathofoneoftheinterferingwave. Whenthis plateis replacedbyanother plate of thesame thickness, theshift of fringes is (3/2)x. Therefractive index ofthesecond plate is (A) 1.75 (B) 1.50 (C) 1.25 (D) 1.00 Q.7 Calculatetotalmaximummass(kg)whichcanbeliftedby10identicalballoon(eachhavingvolume82.1 lit. and mass of balloon & gas = 3 kg) at a height 83.14 m at Mars where g = 5 m/s2 & atmosphere containsonlyAr(At. wt.40).AtMars temperature is 10K and densityofatmosphereat groundlevel is 821 . 0 2 k gm/lit. [Given : e–0.1 = 0.9] (Assume dH = d0 RT Mgh e to be applicable). (A) 2000 × 0.81–30 (B) 1970 (C) 2000 × 0.9 –30 (D) 2000 × 0.81 – 3
• 4. ROUGHWORK Q.8 How manymg of quick lime is required to remove hardness of 1 kg of hard water having 366 ppm of HCO3 – and contains Ca2+ as the only cation (A) 72 mg (B) 84 mg (C) 168 mg (D) 170 mg Q.9 A 10 litre box contains O3 and O2 at equilibrium at 2000K. Kp = 4.17 × 1014 for 2O3 3O2. Assume that 3 2 O O P P  and if total pressureis 7.33 atm, then partial pressure of O3 will be (A) 9.71 × 10–5 atm (B) 9.71 × 10–7 atm (C) 9.71 × 10–6 atm (D) 9.71 × 10–2 atm Q.10      KOH Br2 If the reactant is (d) (dextrorotatory) then in final product  (A)inversionofconfigurationtakes place (B)racemisationtakes place (C)retentionofconfigurationtakes place (D) none of the above Q.11 Select the correct statement(s) If S = f (E, V, n1, n2) &A= f (T, V, n1, n2) (A) S is equal to f (E, V, n1, n2) (B) S is equal to f (E, V, n1, n2) (C) Ais equal to f (T, V, n1, n2) (D) Ais equal to f (E, V, n1, n2)
• 5. ROUGHWORK Q.12 Compound Bhas a neutralisationequivalent112. G is adichloro alkane. (A) A = , B = , C = , E = ; F = & G = ClCH2CH2CH2CH2Cl (B) A = , B = , C = , E = & G = ClCH2CH2CH2CH2CH2Cl (C) A = , B = , C = , E = , F = & G = ClCH2CH2CH2CH2CH2Cl (D) A = , B = , C = ; E = ; F = & G = ClCH2CH2CH2CH2CH2Cl
• 6. ROUGHWORK Q.13 Whichofthefollowingstatement(s)is/arefalse (A) If a, b and c are positive numbers not equal to 1 and a < b, then logac < logbc. (B) The equation x2 – b = 0 has a real solution for x for any real number b. (C) The sequence an defined byan = 3(0.2)–n is a geometric sequence. (D) cos(cos(x)) < 1/2,  x  R Q.14 Let f (x)= x – x 1 thenwhich one ofthe following statement are incorrect (A)Functionis invertible ifdefined from R – {0} R. (B) f (x1) > f (x2),  x1 > x2 and x1 ,x2  0. (C) Graphof the function has exactlyone asymptote. (D) Functionis one-oneineverycontinuous interval [a, b]defined on one sideof origin. Q.15 If f (x) =      0 x if 2 } 0 { ) 1 , 1 ( x , x 1 sec x tan x 1 1         , then f '(0) is (A) equal to – 1 (B) equal to 0 (C) equal to 1 (D)nonexistent Q.16 If a,b are positive realnumbers suchthat a– b =2,then the smallest valueof the constant Lfor which bx x ax x 2 2    < L for all x > 0, is (A) 1/2 (B) 2 1 (C) 1 (D) 2 Q.17 If K R0 then det. {adj (KIn)} is equal to (A) Kn – 1 (B) Kn(n – 1) (C) Kn (D) K Q.18 Let f(x)is 2 1 sin101         x x where [x] denotes step up function then f(x) is (A) both odd as well as even (B) neitherodd nor even (C)odd function (D)evenfunction
• 7. ROUGHWORK PART - B MATCH THE COLUMN INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Q.1 Inthefigureshowntwoidentical smallcharged balls havingmassmand charge 'q'aresuspendedwith thehelpoftwolight inextensiblesilk stringseach oflength 'l'.At equilibrium theangularseparationbetween thestrings is '' ColumnI Column II (A) If  is verysmall then charge'q'is (P) 3/2 proportional to (B)TensionTinstringisproportional to (Q) l (C) Ifsystem is taken ina satellite. Then (R)  tension T is proportional to (D)Angularseparationbetweenthe charges (S) l–2 (at equalibrium)in thesatellite is Q.2 In thefigure shown,upperblockis given a velocity6m/s & lower blocka velocity3m/s. Whenrelative motionbetweenthem stops (Here block 2 Kg is verylong) ColumnI ColumnII (A)Work done byfriction on 1 kg block (P) 3 Joule (B)Work done byfriction on 2 kg block (Q)negative (C) Net work done by friction (R)positive (D) Loss in K.E. of system (2kg + 1kg block) (S) 7 Joule
• 8. ROUGHWORK Q.3 Matchthefollowing ColumnI ColumnII (A) Canaryyellowprecipitate with (NH4)2MoO4 (P) NO3 – (B) Brownringtest (Q) NO2 – (C)Acidradical whichevolves gas with conc. HCl (R) AsO4 3– (D)Acidradical whichgives gas withdil. H2SO4 (S) PO4 3– Q.4 Matchthefollowing List I List II (A)    CuCl (P)Freeradical mechanism (B) CH3–CH=CH2 (Q) Non Classical Carbocation  Cl2/ Cl | CH CH CH 2 2   (C) CH3–CH=CH2       Cu / Zn / I CH 2 2 (R) Carbenoid (D) CH3–CH=CH2          OH / NaBH ) ii ( O H / ) OAc ( Hg ) i ( 4 2 2 (S)Through carbocation
• 9. ROUGHWORK Q.5. Column–I Column–II (A) Numberofrealsolution(s)oftheequation (P) 2    2 2 2 2 5 1 2 2 x x x     (B) The range of the function f(x) = tan–1 x x   1 1 – tan–1x (Q) 1 consists ofhowmanyelements (C) TheintegralvalueofxintheDomainofdefinitionof (R) 0 the function f (x) = log 1 9 3 · 10 1 x 2 x     + ) x 1 ( cos 1   (D) Value ofthe expression   0 0 0 0 4 1 78 sin 66 sin 42 sin 6 sin log (S) 3 is Q.6 Column–I Column–II (A) If 2 2 y x  = ) x / y ( tan 1 ae  , a > 0, assuming (P) 5 2  y > 0, then, y'' (0) = – 2 / e a K   where K is (B) If y = sin–1 2 x 1 x 2  then dx dy at x = –2 (Q) 2 (C) If 2 0 x x x cos B x sin A x 2 sin 4 Lim     (R) 5 exist thenA–B (D) If f(x) = [2 + 3sinx] 0 < x <  where [ ] denotes (S) –4 GIFthenno.ofpoints at whichfunction is non differentiable
• 10. PART - C Q.1 A force F acts on a uniform rectangular cabinet weighing 400 N as shown in figure.Thecabinetslides withconstant speedwhenaforceF=200Nisapplied at height h = 0.4 m at angle 370 from horizontal.At what distancex (in cm), from edgeA,the resultant normal reactionacts on cabinet (sin370 = 3/5)
• 11. Q.2 Asmall ball of mass m is suspendedbyastiff massless rodof length l.Asmall bulletofsamemassm strikes theballhorizontallywithvelocityV&emerges with velocity 2 V horizontally.Ifaftercollisionballis justabletoswingthrougha a complete circle, then find the force (in newton) applied byhinge on the rod immediatelyaftercollision(herecollisiontimeisnegligible)(Herem=1kg, g = 10m/s, l = 1m)
• 12. Q.3 Whenthegapbetweentwoidenticalconcavethinlens(=3/2,f=10cm)placedincontactis filledwith certain liquid, the image of an object placed at 15 cm from lens shifts away from the lens by5/4 cm. What is the R.I. of the liquid ?
• 13. Q.4 Considertheshownidentical arrangements inwhichauniform rod (length l, mass M)is suspended bymeans ofa ideal spring andastring.It is given thattherodis perfectlyhorizontal in the shownequilibriumposition.Assumethatthestringiscutin arrangement (i) & springis cut in arrangement (ii). If a1 & a2 be the values of acceleration of centreof mass of rod in (i) & (ii) respectively,immediatelyafterthestring/springis cut,thenfinda1/a2 .
• 14. Q.5 1.0gofamixturecontainingSb2O3 andSb2O5 andsomeinert impurities required 100ml0.01 Niodine fortitration.Theresultingsolutionis thenacidifiedandexcessofKIwasadded.Theliberated I2 required 100 ml 0.02 M Na2S2O3·5H2O for complete reaction. Calculate the % of Sb2O5 in the mixture. The reactions are [at. wt. Sb = 122] Sb2O3 + 2I2 + 2H2O  Sb2O5 + 4H+ + 4I– Sb2O5 + 4H+ + 4I–  Sb2O3 + 2I2 + 2H2O
• 15. Q.6 Aballooncontaining1moleairat 1 atminitiallyis filled furtherwithairtill pressure increasesto 3 atm. The initial diameter of the balloonis 1 m and the pressure at eachstate is proportion to diameter of the balloon.Ifballoonwillburst whenpressureincreasesto7 atm.Calculatethenumberofmolesofairthat must be addedafterinitial condition toburst theballoon.
• 16. Q.7 Thekeyreactioninthemanufactureofsyntheticcryolightforaluminiumelectrolysisis HF(g) + Al(OH)3 (s) + NaOH (aq)  Na3AlF6(aq) + H2O(l) Assuminga96%yieldofdried,crystallized product, whatmass (inkg)ofcryolitecanbeobtainedfrom the reaction of 351 kg ofAl(OH)3, 1.10 m3 of 50.0% by mass aqueous NaOH (d = 1.50 g/mL), and 225 m3 of gaseous HF at 312.08 kPa and 87oC? (assume that the ideal gas law holds) [Given: Al = 27, O = 16, H = 1, Na = 23, F = 19, R = 8.3 JK–1 mol–1]
• 17. Q.8 The vapour pressure of water at 300 K is 25 torr. If the standard state pressure is defined as 1 bar (750 torr) estimate the G° [inkJ/mol] for the process. H2O (l)  H2O (g) at 300 K (Neglect variation of H and S with pressure for liquid) Use [R = 8.314 JK–1 mol–1 ; X og X n l l = 2.3, log 3 = 0.48]
• 18. Q.9 Thenumberofpositiveintegral solutions oftheequation 1 z yz xz z y 1 y xy z x y x 1 x 3 2 2 2 3 2 2 2 3    =30 is:
• 19. Q.10 In a triangleABC with altitudeAD,  BAC = 450, DB = 3 and CD = 2. The area of the triangleABC is
• 20. Q.11 Asequenceofequilateraltriangleisdrawn.Thealtitude ofeachis 3 timesthealtitudeofthepreceding traingle,thedifferencebetweenthe areaofthefirst triangleandthesixthtriangleis 3 968 square unit. Theperimeterofthefirsttriangleis.
• 21. Q.12 Findtheintegralvalueofaforwhich theequation (x2 + x + 2)2 – (a – 3) (x2 + x + 2) (x2 + x + 1) + (a – 4) (x2 + x + 1)2 = 0 has at least one real root.