. The metal sheet is heated uniformly, by a small temperature $\\Delta T$, so that its new temperature is $T+\\Delta T$. Calculate the increase in the volume of the metal box.","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$4\\pi a^3\\alpha\\Delta T$","is_correct":false},{"key":"B","content":"$4a^3\\alpha\\Delta T$","is_correct":false},{"key":"C","content":"$\\frac43\\pi a^3\\alpha\\Delta T$","is_correct":false},{"key":"D","content":"$3a^3\\alpha\\Delta T$","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":7,"question":"Y, K and $\\eta$ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$Y=\\frac{9K\\eta}{2\\eta+3K}$ N/m$^2$","is_correct":false},{"key":"B","content":"$Y=\\frac{9K\\eta}{3K-\\eta}$ N/m$^2$","is_correct":false},{"key":"C","content":"$K=\\frac{Y\\eta}{9\\eta-3Y}$ N/m$^2$","is_correct":true},{"key":"D","content":"$\\eta=\\frac{3YK}{9K+Y}$ N/m$^2$","is_correct":false}],"correct_answer":"C","explanation":"From $Y=\\frac{9K\\eta}{3K+\\eta}$: $K(9\\eta-3Y)=Y\\eta$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":8,"question":"If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph?","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q8.png","type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"a constant negative, then jumps to a constant positive value (single step)","is_correct":true},{"key":"B","content":"a sinusoid-like curve","is_correct":false},{"key":"C","content":"a square wave alternating repeatedly between positive and negative values","is_correct":false},{"key":"D","content":"a triangular (rising then falling) curve","is_correct":false}],"correct_answer":"A","explanation":"Constant negative slope then constant positive slope.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":9,"question":"1 mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes. $A\\to B$: Isothermal expansion at temperature T so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$. $B\\to C$: Isobaric compression at pressure $P_2$ to initial volume $V_1$. $C\\to A$: Isochoric change leading to change of pressure from $P_2$ to $P_1$. Total workdone in the complete cycle ABCA is:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q9.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"0","is_correct":false},{"key":"B","content":"$nRT\\ln2$","is_correct":false},{"key":"C","content":"$nRT\\left(\\ln2+\\frac12\\right)$","is_correct":false},{"key":"D","content":"$nRT\\left(\\ln2-\\frac12\\right)$","is_correct":true}],"correct_answer":"D","explanation":"$nRT\\ln2-P_2V_1=nRT\\ln2-\\frac{nRT}{2}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":10,"question":"Match List I with List II. List I: (a) Isothermal (b) Isochoric (c) Adiabatic (d) Isobaric. List II: (i) Pressure constant (ii) Temperature constant (iii) Volume constant (iv) Heat content is constant. Choose the correct answer from the options given below:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"(a)$\\to$(i), (b)$\\to$(iii), (c)$\\to$(ii), (d)$\\to$(iv)","is_correct":false},{"key":"B","content":"(a)$\\to$(iii), (b)$\\to$(ii), (c)$\\to$(i), (d)$\\to$(iv)","is_correct":false},{"key":"C","content":"(a)$\\to$(ii), (b)$\\to$(iv), (c)$\\to$(iii), (d)$\\to$(i)","is_correct":false},{"key":"D","content":"(a)$\\to$(ii), (b)$\\to$(iii), (c)$\\to$(iv), (d)$\\to$(i)","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":11,"question":"In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q11.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$A\\sqrt{\\frac{M+m}{M}}$","is_correct":false},{"key":"B","content":"$A\\sqrt{\\frac{M}{M+m}}$","is_correct":true},{"key":"C","content":"$A\\sqrt{\\frac{M-m}{M}}$","is_correct":false},{"key":"D","content":"$A\\sqrt{\\frac{M}{M-m}}$","is_correct":false}],"correct_answer":"B","explanation":"Momentum conserved at the mean position: $A\\'=\\frac{Mv}{(M+m)\\omega\\'}=A\\sqrt{\\frac{M}{M+m}}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":12,"question":"A cell $E_1$ of emf 6 V and internal resistance 2 $\\Omega$ is connected with another cell $E_2$ of emf 4 V and internal resistance 8 $\\Omega$ (as shown in the figure). The potential difference across points X and Y is:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q12.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"2.0 V","is_correct":false},{"key":"B","content":"3.6 V","is_correct":false},{"key":"C","content":"5.6 V","is_correct":true},{"key":"D","content":"10.0 V","is_correct":false}],"correct_answer":"C","explanation":"Net emf 2 V, $I=0.2$ A; $V_{XY}=4+0.2\\times8=5.6$ V (E2 is being charged).","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":13,"question":"A current through a wire depends on time as $i=\\alpha_0t+\\beta t^2$ where $\\alpha_0=20$ A/s and $\\beta=8$ As$^{-2}$. Find the charge crossed through a section of the wire in 15 s.","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"260 C","is_correct":false},{"key":"B","content":"2100 C","is_correct":false},{"key":"C","content":"11250 C","is_correct":true},{"key":"D","content":"2250 C","is_correct":false}],"correct_answer":"C","explanation":"$q=10(225)+\\frac83(3375)=11250$ C.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":14,"question":"Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1 : 2","is_correct":false},{"key":"B","content":"2 : 1","is_correct":false},{"key":"C","content":"4 : 1","is_correct":false},{"key":"D","content":"1 : 4","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":15,"question":"A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is –Q. The electric field at the centre of cube is:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q15.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac{-Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$","is_correct":false},{"key":"B","content":"$\\frac{Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$","is_correct":false},{"key":"C","content":"$\\frac{-2Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$","is_correct":true},{"key":"D","content":"$\\frac{2Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$","is_correct":false}],"correct_answer":"C","explanation":"Equivalent to $-2Q$ at the origin: $E=\\frac{2Q}{4\\pi\\varepsilon_0(3a^2/4)}$ pointing to the origin.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":16,"question":"When an emitter current is changed by 4 mA, the collector current changes by 3.5 mA. The value of $\\beta$ will be:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"7","is_correct":true},{"key":"B","content":"0.875","is_correct":false},{"key":"C","content":"0.5","is_correct":false},{"key":"D","content":"3.5","is_correct":false}],"correct_answer":"A","explanation":"$I_B$ change 0.5 mA: $\\beta=7$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":17,"question":"In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E. The transitions A, B and C respectively represent:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q17.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"The first member of the Lyman series, third member of Balmer series and second member of Paschen series.","is_correct":false},{"key":"B","content":"The ionization potential of hydrogen, second member of Balmer series and third member of Paschen series.","is_correct":false},{"key":"C","content":"The series limit of Lyman series, second member of Balmer series and second member of Paschen series.","is_correct":false},{"key":"D","content":"The series limit of Lyman series, third member of Balmer series and second member of Paschen series.","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":18,"question":"Given below are two statements: Statement I: Two photons having equal linear momenta have equal wavelengths. Statement II: If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above statements, choose the correct answer from the options given below.","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Both Statement I and Statement II are true","is_correct":false},{"key":"B","content":"Both Statement I and Statement II are false","is_correct":false},{"key":"C","content":"Statement I is true but Statement II is false","is_correct":true},{"key":"D","content":"Statement I is false but Statement II is true","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":19,"question":"The focal length f is related to the radius of curvature r of the spherical convex mirror by:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$f=r$","is_correct":false},{"key":"B","content":"$f=-r$","is_correct":false},{"key":"C","content":"$f=-\\frac12r$","is_correct":false},{"key":"D","content":"$f=+\\frac12r$","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":20,"question":"In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"4 : 1","is_correct":true},{"key":"B","content":"2 : 1","is_correct":false},{"key":"C","content":"1 : 4","is_correct":false},{"key":"D","content":"3 : 1","is_correct":false}],"correct_answer":"A","explanation":"$\\left(\\frac{3+1}{3-1}\\right)^2=4$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":21,"question":"The coefficient of static friction between a wooden block of mass 0.5 kg and a vertical rough wall is 0.2. The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be ________ N. [g = 10 ms$^{-2}$]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25","explanation":"$0.2F=5\\Rightarrow F=25$ N.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":22,"question":"An unpolarized light beam is incident on the polarizer of a polarization experiment and the intensity of light beam emerging from the analyzer is measured as 100 Lumens. Now, if the analyzer is rotated around the horizontal axis (direction of light) by 30° in clockwise direction, the intensity of emerging light will be ________ Lumens.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"75","explanation":"$100\\cos^230^\\circ=75$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":23,"question":"A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. The ratio of velocities of the balls after collision is x : y, where x is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"1","explanation":"Transverse momentum balance with equal angles gives equal speeds: 1 : 1.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":24,"question":"A hydraulic press can lift 100 kg when a mass 'm' is placed on the smaller piston. It can lift ________ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass 'm' on the smaller piston.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25600","explanation":"Load $\\propto(D/d)^2$: $\\times256$, so 25600 kg.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":25,"question":"An inclined plane is bent in such a way that the vertical cross-section is given by $y=\\frac{x^2}{4}$ where y is in vertical and x in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\\mu=0.5$, the maximum height in cm at which a stationary block will not slip downward is ________ cm.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25","explanation":"$\\frac{dy}{dx}=\\frac x2\\le0.5\\Rightarrow x=1$, $y=0.25$ m $=25$ cm.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":26,"question":"A resonance circuit having inductance and resistance $2\\times10^{-4}$ H and 6.28 $\\Omega$ respectively oscillates at 10 MHz frequency. The value of quality factor of this resonator is ________. [$\\pi$ = 3.14]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"2000","explanation":"$Q=\\frac{\\omega L}{R}=\\frac{2\\pi\\times10^7\\times2\\times10^{-4}}{6.28}=2000$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":27,"question":"An audio signal $v_m=20\\sin2\\pi(1500t)$ amplitude modulates a carrier $v_c=80\\sin2\\pi(100{,}000t)$. The value of percent modulation is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25","explanation":"20/80 = 25%.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":28,"question":"In connection with the circuit drawn below, the value of current flowing through 2 k$\\Omega$ resistor is ________ $\\times10^{-4}$ A.","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q28.png","type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25","explanation":"Zener holds 5 V across 2 k$\\Omega$: $2.5\\times10^{-3}=25\\times10^{-4}$ A.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":29,"question":"An electromagnetic wave of frequency 5 GHz, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are 2. Its velocity in this medium is ________ $\\times10^7$ m/s.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"15","explanation":"$c/2=15\\times10^7$ m/s.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":30,"question":"A common transistor radio set requires 12 V (D.C.) for its operation. The D.C. source is constructed by using a transformer and a rectifier circuit, which are operated at 220 V (A.C.) on standard domestic A.C. supply. The number of turns of secondary coil are 24, then the number of turns of primary are ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"440","explanation":"$24\\times\\frac{220}{12}=440$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":31,"question":"Which of the following are isostructural pairs? A. SO$_4^{2-}$ and CrO$_4^{2-}$ B. SiCl$_4$ and TiCl$_4$ C. NH$_3$ and NO$_3^-$ D. BCl$_3$ and BrCl$_3$","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"A and B only","is_correct":true},{"key":"B","content":"A and C only","is_correct":false},{"key":"C","content":"B and C only","is_correct":false},{"key":"D","content":"C and D only","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":32,"question":"In Freundlich adsorption isotherm, slope of AB line is:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q32.png","type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"n with (n, 0.1 to 0.5)","is_correct":false},{"key":"B","content":"$\\log n$ with (n > 1)","is_correct":false},{"key":"C","content":"$\\log\\frac1n$ with (n \u003c 1)","is_correct":false},{"key":"D","content":"$\\frac1n$ with $\\left(\\frac1n=0\\text{ to }1\\right)$","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":33,"question":"Consider the elements Mg, Al, S, P and Si, the correct increasing order of their first ionization enthalpy is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Al \u003c Mg \u003c Si \u003c S \u003c P","is_correct":true},{"key":"B","content":"Mg \u003c Al \u003c Si \u003c P \u003c S","is_correct":false},{"key":"C","content":"Mg \u003c Al \u003c Si \u003c S \u003c P","is_correct":false},{"key":"D","content":"Al \u003c Mg \u003c S \u003c Si \u003c P","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":34,"question":"Which of the following ore is concentrated using group 1 cyanide salt?","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Calamine","is_correct":false},{"key":"B","content":"Malachite","is_correct":false},{"key":"C","content":"Siderite","is_correct":false},{"key":"D","content":"Sphalerite","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":35,"question":"(A) $\\mathrm{HOCl+H_2O_2\\rightarrow H_3O^++Cl^-+O_2}$ (B) $\\mathrm{I_2+H_2O_2+2OH^-\\rightarrow2I^-+2H_2O+O_2}$. Choose the correct option.","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"H$_2$O$_2$ acts as oxidising agent in equations (A) and (B).","is_correct":false},{"key":"B","content":"H$_2$O$_2$ acts as reducing agent in equations (A) and (B).","is_correct":true},{"key":"C","content":"H$_2$O$_2$ act as oxidizing and reducing agent respectively in equations (A) and (B).","is_correct":false},{"key":"D","content":"H$_2$O$_2$ acts as reducing and oxidising agent respectively in equations (A) and (B).","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":36,"question":"Al$_2$O$_3$ was leached with alkali to get X. The solution of X on passing of gas Y, forms Z. X, Y and Z respectively are:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"X = Na[Al(OH)$_4$], Y = SO$_2$, Z = Al$_2$O$_3$","is_correct":false},{"key":"B","content":"X = Al(OH)$_3$, Y = SO$_2$, Z = Al$_2$O$_3$.xH$_2$O","is_correct":false},{"key":"C","content":"X = Al(OH)$_3$, Y = CO$_2$, Z = Al$_2$O$_3$","is_correct":false},{"key":"D","content":"X = Na[Al(OH)$_4$], Y = CO$_2$, Z = Al$_2$O$_3$.xH$_2$O","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":37,"question":"The electrode potential of M$^{2+}$/M of 3d-series elements shows positive value for:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Fe","is_correct":false},{"key":"B","content":"Co","is_correct":false},{"key":"C","content":"Zn","is_correct":false},{"key":"D","content":"Cu","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":38,"question":"The major components in \"Gun Metal\" are:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Cu, Sn and Zn","is_correct":true},{"key":"B","content":"Cu, Zn and Ni","is_correct":false},{"key":"C","content":"Cu, Ni and Fe","is_correct":false},{"key":"D","content":"Al, Cu, Mg and Mn","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":39,"question":"The gas released during anaerobic degradation of vegetation may lead to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Acid rain","is_correct":false},{"key":"B","content":"Global warming and cancer","is_correct":true},{"key":"C","content":"Corrosion of metals","is_correct":false},{"key":"D","content":"Ozone hole","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":40,"question":"Which of the following compound gives pink colour on reaction with phthalic anhydride in conc. H$_2$SO$_4$ followed by treatment with NaOH?","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q40.png","type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"4-propylphenol","is_correct":false},{"key":"B","content":"2-propylphenol","is_correct":true},{"key":"C","content":"4-methyl-6-propylbenzene-1,3-diol type (two OH meta to each other, CH3 and propyl on ring)","is_correct":false},{"key":"D","content":"4-methyl-5-propylbenzene-1,2-diol type (two OH ortho to each other)","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":41,"question":"What is the major product formed by HI on reaction with $\\mathrm{(CH_3)_3C-CH=CH_2}$?","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\mathrm{(CH_3)_3C-CH_2-CH_2I}$","is_correct":false},{"key":"B","content":"$\\mathrm{(CH_3)_2CI-CH(CH_3)_2}$","is_correct":true},{"key":"C","content":"$\\mathrm{(CH_3)_3C-CHI-CH_3}$","is_correct":false},{"key":"D","content":"$\\mathrm{(CH_3)_2CH-CHI-CH_2CH_3}$","is_correct":false}],"correct_answer":"B","explanation":"Secondary cation undergoes a methyl shift to a tertiary cation: 2-iodo-2,3-dimethylbutane.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":42,"question":"Which of the following reagent is used for the following reaction? $\\mathrm{CH_3CH_2CH_3\\xrightarrow{?}CH_3CH_2CHO}$","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Copper at high temperature and pressure","is_correct":false},{"key":"B","content":"Molybdenum oxide","is_correct":true},{"key":"C","content":"Manganese acetate","is_correct":false},{"key":"D","content":"Potassium permanganate","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":43,"question":"The product formed in the first step of the reaction of $\\mathrm{CH_3-CH_2-CHBr-CH_2-CHBr-CH_3}$ with excess Mg/Et$_2$O (Et = C$_2$H$_5$) is:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q43.png","type":"mcq_single","difficulty":"hard","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"a dimer: two C6 chains joined at C-3/C-5 (Wurtz-type coupling product)","is_correct":false},{"key":"B","content":"1-ethyl-2-methylcyclopropane","is_correct":false},{"key":"C","content":"CH3-CH2-CH(-)-CH2-CH(-)-CH3 bridged dimer (as drawn)","is_correct":false},{"key":"D","content":"$\\mathrm{CH_3CH_2-CH(MgBr)-CH_2-CH(MgBr)-CH_3}$","is_correct":true}],"correct_answer":"D","explanation":"With excess Mg each C–Br bond forms a Grignard centre: the first-step product is the bis-Grignard CH3CH2CH(MgBr)CH2CH(MgBr)CH3 (official key).","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":44,"question":"What is the final product (major) 'A' in the given reaction? 1-(2-methylcyclohexyl)propan-1-ol drawn as ring–CH(OH)–CH$_3$... (a 2-methylcyclohexyl ring bearing –CH(OH)CH$_3$) $\\xrightarrow{\\text{HCl}}$ 'A'","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-feb-24-shift1/q44.png","type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"chloride at the side-chain carbon (ring-CH(Cl)CH3, no rearrangement)","is_correct":false},{"key":"B","content":"alkene ring-CH=CH2","is_correct":false},{"key":"C","content":"exocyclic alkene ring=CH-CH3","is_correct":false},{"key":"D","content":"rearranged tertiary chloride: Cl and CH2CH3 on the same ring carbon","is_correct":true}],"correct_answer":"D","explanation":"The secondary carbocation undergoes a 1,2-hydride shift to the tertiary ring carbon; chloride then attacks there.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":45,"question":"Identify products A and B. 1-methylcyclopentene $\\xrightarrow{\\text{dil. KMnO}_4\\text{, 273 K}}$ A $\\xrightarrow{\\text{CrO}_3}$ B","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"A: 1-methylcyclopentane-1,2-diol; B: 2-methylcyclopent-2-en-1-one","is_correct":false},{"key":"B","content":"A: 1-methylcyclopentane-1,2-diol; B: 2-hydroxy-2-methylcyclopentan-1-one","is_correct":true},{"key":"C","content":"A: 2-methylcyclopentan-1-ol; B: 2-methylcyclopentan-1-one","is_correct":false},{"key":"D","content":"A: OHC-CH2CH2CH2-CO-CH3; B: HOOC-CH2CH2CH2-CO-CH3","is_correct":false}],"correct_answer":"B","explanation":"Syn-dihydroxylation; CrO3 oxidises only the secondary OH.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":46,"question":"In the following reaction the reason why meta-nitro product also formed is: aniline $\\xrightarrow{\\text{Conc. HNO}_3\\text{, Conc. H}_2\\text{SO}_4\\text{, 288 K}}$ p-nitroaniline [A] 51% + m-nitroaniline [B] 47% + o-nitroaniline [C] 2%","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"–NH$_2$ group is highly meta-directive","is_correct":false},{"key":"B","content":"–NO$_2$ substitution always takes place at meta-position","is_correct":false},{"key":"C","content":"Formation of anilinium ion","is_correct":true},{"key":"D","content":"low temperature","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":47,"question":"'A' and 'B' in the following reactions are: aniline $\\xrightarrow{\\text{NaNO}_2/\\text{HCl, KCN}}$ (A) $\\xrightarrow{\\text{SnCl}_2/\\text{HCl/H}_3\\text{O}^+}$ (B)","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"(A) benzenediazonium chloride, (B) chlorobenzene","is_correct":false},{"key":"B","content":"(A) benzenediazonium chloride, (B) benzaldehyde","is_correct":false},{"key":"C","content":"(A) benzonitrile, (B) chlorobenzene","is_correct":false},{"key":"D","content":"(A) benzonitrile, (B) benzaldehyde","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":48,"question":"Match List I with List II. List I (Monomer Unit): (a) Caprolactum (b) 2-Chloro-1,3-butadiene (c) Isoprene (d) Acrylonitrile. List II (Polymer): (i) Natural rubber (ii) Buna-N (iii) Nylon 6 (iv) Neoprene. Choose the correct answer from the options given below:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"(a)$\\to$(i), (b)$\\to$(ii), (c)$\\to$(iii), (d)$\\to$(iv)","is_correct":false},{"key":"B","content":"(a)$\\to$(iv), (b)$\\to$(iii), (c)$\\to$(ii), (d)$\\to$(i)","is_correct":false},{"key":"C","content":"(a)$\\to$(ii), (b)$\\to$(i), (c)$\\to$(iv), (d)$\\to$(iii)","is_correct":false},{"key":"D","content":"(a)$\\to$(iii), (b)$\\to$(iv), (c)$\\to$(i), (d)$\\to$(ii)","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":49,"question":"Out of the following, which type of interaction is responsible for the stabilisation of $\\alpha$-helix structure of proteins?","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"vander Waals forces","is_correct":false},{"key":"B","content":"Covalent bonding","is_correct":false},{"key":"C","content":"Ionic bonding","is_correct":false},{"key":"D","content":"Hydrogen bonding","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":50,"question":"Given below are two statements: Statement I: Colourless cupric metaborate is reduced to cuprous metaborate in a luminous flame. Statement II: Cuprous metaborate is obtained by heating boric anhydride and copper sulphate in a non-luminous flame. In the light of the above statements, choose the most appropriate answer from the options given below.","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Both Statement I and Statement II are true","is_correct":false},{"key":"B","content":"Both Statement I and Statement II are false","is_correct":true},{"key":"C","content":"Statement I is true but Statement II is false","is_correct":false},{"key":"D","content":"Statement I is false but Statement II is true","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":51,"question":"4.5 g of compound A (MW = 90) was used to make 250 mL of its aqueous solution. The molarity of the solution in M is $x\\times10^{-1}$. The value of x is ________. (Rounded off to the nearest integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":"0.05 mol / 0.25 L = 0.2 M.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":52,"question":"The coordination number of an atom in a body-centered cubic structure is ________. [Assume that the lattice is made up of atoms.]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"8","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":53,"question":"A proton and a Li$^{3+}$ nucleus are accelerated by the same potential. If $\\lambda_{Li}$ and $\\lambda_p$ denote the de Broglie wavelengths of Li$^{3+}$ and proton respectively, then the value of $\\frac{\\lambda_{Li}}{\\lambda_p}$ is $x\\times10^{-1}$. The value of x is ________. (Rounded off to the nearest integer) [Mass of Li$^{3+}$ = 8.3 mass of proton]","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":"$\\sqrt{\\frac{1}{8.3\\times3}}\\approx0.2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":54,"question":"For the reaction $A_{(g)}\\to B_{(g)}$, the value of the equilibrium constant at 300 K and 1 atm is equal to 100.0. The value of $\\Delta_rG$ for the reaction at 300 K and 1 atm in J mol$^{-1}$ is $-xR$, where x is ________. (Rounded off to the nearest integer) [R = 8.31 J mol$^{-1}$K$^{-1}$ and ln 10 = 2.3]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"1380","explanation":"$\\Delta G^\\circ=-RT\\ln100=-300\\times4.6R=-1380R$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":55,"question":"When 9.45 g of ClCH$_2$COOH is added to 500 mL of water, its freezing point drops by 0.5°C. The dissociation constant of ClCH$_2$COOH is $x\\times10^{-3}$. The value of x is ________. (Rounded off to the nearest integer) [$K_{f(H_2O)}$ = 1.86 K kg mol$^{-1}$]","image_url":null,"type":"numerical","difficulty":"hard","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"36","explanation":"$m=0.2$; expected $\\Delta T=0.372$, so $i\\approx1.344$, $\\alpha\\approx0.344$; $K_a=\\frac{0.2\\alpha^2}{1-\\alpha}\\approx0.036=36\\times10^{-3}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":56,"question":"At 1990 K and 1 atm pressure, there are equal number of Cl$_2$ molecules and Cl atoms in the reaction mixture. The value of $K_p$ for the reaction $\\mathrm{Cl_{2(g)}\\rightleftharpoons2Cl_{(g)}}$ under the above conditions is $x\\times10^{-1}$. The value of x is ________. (Rounded off to the nearest integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"5","explanation":"$K_p=\\frac{0.5^2}{0.5}=0.5$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":57,"question":"The reaction of sulphur in alkaline medium is given below: $\\mathrm{S_{8(s)}+a\\,OH^-_{(aq)}\\rightarrow b\\,S^{2-}_{(aq)}+c\\,S_2O_3^{2-}{}_{(aq)}+d\\,H_2O_{(l)}}$. The values of 'a' is ________. (Integer answer)","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"12","explanation":"$\\mathrm{S_8+12OH^-\\to4S^{2-}+2S_2O_3^{2-}+6H_2O}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":58,"question":"Gaseous cyclobutene isomerizes to butadiene in a first order process which has a 'k' value of $3.3\\times10^{-4}$ s$^{-1}$ at 153°C. The time in minutes it takes for the isomerization to proceed 40% to completion at this temperature is ________. (Rounded off to the nearest integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"26","explanation":"$t=\\frac{\\ln(1/0.6)}{3.3\\times10^{-4}}\\approx1548$ s $\\approx26$ min.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":59,"question":"Number of amphoteric compounds among the following is ________. (A) BeO (B) BaO (C) Be(OH)$_2$ (D) Sr(OH)$_2$","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":60,"question":"The stepwise formation of $[\\mathrm{Cu(NH_3)_4}]^{2+}$ is given below: $\\mathrm{Cu^{2+}+NH_3\\xrightleftharpoons{K_1}[Cu(NH_3)]^{2+}}$; $\\mathrm{[Cu(NH_3)]^{2+}+NH_3\\xrightleftharpoons{K_2}[Cu(NH_3)_2]^{2+}}$; $\\mathrm{[Cu(NH_3)_2]^{2+}+NH_3\\xrightleftharpoons{K_3}[Cu(NH_3)_3]^{2+}}$; $\\mathrm{[Cu(NH_3)_3]^{2+}+NH_3\\xrightleftharpoons{K_4}[Cu(NH_3)_4]^{2+}}$. The value of stability constants $K_1$, $K_2$, $K_3$ and $K_4$ are $10^4$, $1.58\\times10^3$, $5\\times10^2$ and $10^2$ respectively. The overall equilibrium constants for dissociation of $[\\mathrm{Cu(NH_3)_4}]^{2+}$ is $x\\times10^{-12}$. The value of x is ________. (Rounded off to the nearest integer)","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"1","explanation":"$\\beta_4=7.9\\times10^{11}$; dissociation $1/\\beta_4\\approx1.27\\times10^{-12}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":61,"question":"Let $f:\\mathbb{R}\\to\\mathbb{R}$ be defined as $f(x)=2x-1$ and $g:\\mathbb{R}-\\{1\\}\\to\\mathbb{R}$ be defined as $g(x)=\\frac{x-\\frac12}{x-1}$. Then the composition function $f(g(x))$ is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"one-one but not onto","is_correct":true},{"key":"B","content":"onto but not one-one","is_correct":false},{"key":"C","content":"neither one-one nor onto","is_correct":false},{"key":"D","content":"both one-one and onto","is_correct":false}],"correct_answer":"A","explanation":"$f(g(x))=\\frac{x}{x-1}$: injective, range excludes 1.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":62,"question":"Let p and q be two positive numbers such that $p+q=2$ and $p^4+q^4=272$. Then p and q are roots of the equation:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$x^2-2x+136=0$","is_correct":false},{"key":"B","content":"$x^2-2x+16=0$","is_correct":true},{"key":"C","content":"$x^2-2x+8=0$","is_correct":false},{"key":"D","content":"$x^2-2x+2=0$","is_correct":false}],"correct_answer":"B","explanation":"With $s=pq$: $(4-2s)^2-2s^2=272\\Rightarrow s=16$ (or $-8$).","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":63,"question":"The system of linear equations $3x-2y-kz=10$, $2x-4y-2z=6$, $x+2y-z=5m$ is inconsistent if:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$k\\ne3, m\\ne\\frac45$","is_correct":false},{"key":"B","content":"$k=3, m=\\frac45$","is_correct":false},{"key":"C","content":"$k=3, m\\ne\\frac45$","is_correct":true},{"key":"D","content":"$k\\ne3, m\\in\\mathbb{R}$","is_correct":false}],"correct_answer":"C","explanation":"$\\Delta=24-8k=0$ at $k=3$; then consistent only for $m=\\frac45$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":64,"question":"The value of $-{}^{15}C_1+2\\cdot{}^{15}C_2-3\\cdot{}^{15}C_3+\\dots-15\\cdot{}^{15}C_{15}+{}^{14}C_1+{}^{14}C_3+{}^{14}C_5+\\dots+{}^{14}C_{11}$ is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$2^{16}-1$","is_correct":false},{"key":"B","content":"$2^{13}-14$","is_correct":true},{"key":"C","content":"$2^{13}-13$","is_correct":false},{"key":"D","content":"$2^{14}$","is_correct":false}],"correct_answer":"B","explanation":"First sum $=\\sum(-1)^rr\\binom{15}{r}=0$; second $=2^{13}-\\binom{14}{13}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":65,"question":"If $e^{(\\cos^2x+\\cos^4x+\\cos^6x+\\dots\\infty)\\log_e2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\\frac{2\\sin x}{\\sin x+\\sqrt3\\cos x}\\left(0\u003cx\u003c\\frac\\pi2\\right)$ is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac12$","is_correct":true},{"key":"B","content":"$\\sqrt3$","is_correct":false},{"key":"C","content":"$\\frac32$","is_correct":false},{"key":"D","content":"$2\\sqrt3$","is_correct":false}],"correct_answer":"A","explanation":"$2^{\\cot^2x}=8\\Rightarrow x=\\frac\\pi6$: value $\\frac12$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":66,"question":"$\\lim_{x\\to0}\\frac{\\int_0^{x^2}(\\sin\\sqrt t)\\,dt}{x^3}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac23$","is_correct":true},{"key":"B","content":"$\\frac32$","is_correct":false},{"key":"C","content":"$\\frac1{15}$","is_correct":false},{"key":"D","content":"0","is_correct":false}],"correct_answer":"A","explanation":"L\\'Hopital: $\\frac{2x\\sin x}{3x^2}\\to\\frac23$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":67,"question":"The function $f(x)=\\frac{4x^3-3x^2}{6}-2\\sin x+(2x-1)\\cos x$:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"increases in $\\left[\\frac12,\\infty\\right)$","is_correct":true},{"key":"B","content":"decreases in $\\left[\\frac12,\\infty\\right)$","is_correct":false},{"key":"C","content":"increases in $\\left(-\\infty,\\frac12\\right]$","is_correct":false},{"key":"D","content":"decreases in $\\left(-\\infty,\\frac12\\right]$","is_correct":false}],"correct_answer":"A","explanation":"$f'=(2x-1)(x-\\sin x)\\ge0$ for $x\\ge\\frac12$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":68,"question":"A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1050","is_correct":false},{"key":"B","content":"1625","is_correct":true},{"key":"C","content":"560","is_correct":false},{"key":"D","content":"575","is_correct":false}],"correct_answer":"B","explanation":"$\\binom62\\binom84+\\binom63\\binom86+\\binom64\\binom88=1050+560+15$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":69,"question":"If $f:\\mathbb{R}\\to\\mathbb{R}$ is a function defined by $f(x)=[x-1]\\cos\\left(\\frac{2x-1}{2}\\right)\\pi$, where [·] denotes the greatest integer function, then f is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"discontinuous only at $x=1$","is_correct":false},{"key":"B","content":"discontinuous at all integral values of x except at $x=1$","is_correct":false},{"key":"C","content":"continuous only at $x=1$","is_correct":false},{"key":"D","content":"continuous for every real x","is_correct":true}],"correct_answer":"D","explanation":"The cosine factor vanishes at every integer, cancelling the jumps.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":70,"question":"If $\\int\\frac{\\cos x-\\sin x}{\\sqrt{8-\\sin2x}}dx=a\\sin^{-1}\\left(\\frac{\\sin x+\\cos x}{b}\\right)+c$, where c is a constant of integration, then the ordered pair (a, b) is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"(3, 1)","is_correct":false},{"key":"B","content":"(1, 3)","is_correct":true},{"key":"C","content":"(−1, 3)","is_correct":false},{"key":"D","content":"(1, −3)","is_correct":false}],"correct_answer":"B","explanation":"$u=\\sin x+\\cos x$, $8-\\sin2x=9-u^2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":71,"question":"The area (in sq. units) of the part of the circle $x^2+y^2=36$, which is outside the parabola $y^2=9x$, is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$24\\pi+3\\sqrt3$","is_correct":false},{"key":"B","content":"$24\\pi-3\\sqrt3$","is_correct":true},{"key":"C","content":"$12\\pi+3\\sqrt3$","is_correct":false},{"key":"D","content":"$12\\pi-3\\sqrt3$","is_correct":false}],"correct_answer":"B","explanation":"Inside both: $12\\pi+3\\sqrt3$; outside $=36\\pi-(12\\pi+3\\sqrt3)$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":72,"question":"The population $P=P(t)$ at time 't' of a certain species follows the differential equation $\\frac{dP}{dt}=0.5P-450$. If $P(0)=850$, then the time at which population becomes zero is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\log_e9$","is_correct":false},{"key":"B","content":"$\\frac12\\log_e18$","is_correct":false},{"key":"C","content":"$\\log_e18$","is_correct":false},{"key":"D","content":"$2\\log_e18$","is_correct":true}],"correct_answer":"D","explanation":"$P-900=-50e^{t/2}$; zero at $t=2\\ln18$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":73,"question":"A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\\frac14$. Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"A only","is_correct":false},{"key":"B","content":"B only","is_correct":true},{"key":"C","content":"C only","is_correct":false},{"key":"D","content":"All the three","is_correct":false}],"correct_answer":"B","explanation":"$\\frac1a+\\frac1b=\\frac12$: the line passes through (2, 2).","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":74,"question":"The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$x=a$","is_correct":false},{"key":"B","content":"$x=-\\frac a2$","is_correct":false},{"key":"C","content":"$x=0$","is_correct":true},{"key":"D","content":"$x=\\frac a2$","is_correct":false}],"correct_answer":"C","explanation":"Locus $y^2=2a\\left(x-\\frac a2\\right)$: directrix $x=0$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":75,"question":"If the tangent to the curve $y=x^3$ at the point $P(t,t^3)$ meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"0","is_correct":false},{"key":"B","content":"$2t^3$","is_correct":false},{"key":"C","content":"$-t^3$","is_correct":false},{"key":"D","content":"$-2t^3$","is_correct":true}],"correct_answer":"D","explanation":"$Q=(-2t,-8t^3)$; ordinate $\\frac{2t^3-8t^3}{3}=-2t^3$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":76,"question":"The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$, is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$6x-5y+2z+10=0$","is_correct":false},{"key":"B","content":"$11x+y+17z+38=0$","is_correct":true},{"key":"C","content":"$6x-5y-2z-2=0$","is_correct":false},{"key":"D","content":"$3x-10y-2z+11=0$","is_correct":false}],"correct_answer":"B","explanation":"Normal $(3,1,-2)\\times(2,-5,-1)\\propto(11,1,17)$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":77,"question":"The distance of the point (1, 1, 9) from the point of intersection of the line $\\frac{x-3}{1}=\\frac{y-4}{2}=\\frac{z-5}{2}$ and the plane $x+y+z=17$ is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$2\\sqrt{19}$","is_correct":false},{"key":"B","content":"$19\\sqrt2$","is_correct":false},{"key":"C","content":"$\\sqrt{38}$","is_correct":true},{"key":"D","content":"38","is_correct":false}],"correct_answer":"C","explanation":"Intersection (4, 6, 7); distance $\\sqrt{38}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":78,"question":"An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac1{32}$","is_correct":false},{"key":"B","content":"$\\frac3{16}$","is_correct":false},{"key":"C","content":"$\\frac5{16}$","is_correct":false},{"key":"D","content":"$\\frac12$","is_correct":true}],"correct_answer":"D","explanation":"$\\binom n2=\\binom n3\\Rightarrow n=5$; probability $\\frac12$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":79,"question":"Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"25","is_correct":false},{"key":"B","content":"30","is_correct":false},{"key":"C","content":"$20\\sqrt3$","is_correct":false},{"key":"D","content":"$25\\sqrt3$","is_correct":true}],"correct_answer":"D","explanation":"$h\\cdot3h=75^2\\Rightarrow h=25\\sqrt3$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":80,"question":"The statement among the following that is a tautology is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$A\\wedge(A\\vee B)$","is_correct":false},{"key":"B","content":"$A\\vee(A\\wedge B)$","is_correct":false},{"key":"C","content":"$[A\\wedge(A\\to B)]\\to B$","is_correct":true},{"key":"D","content":"$B\\to[A\\wedge(A\\to B)]$","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":81,"question":"If the least and the largest real values of $\\alpha$, for which the equation $z+\\alpha|z-1|+2i=0$ ($z\\in\\mathbb{C}$ and $i=\\sqrt{-1}$) has a solution, are p and q respectively; then $4(p^2+q^2)$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"10","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":82,"question":"Let $B_i$ ($i=1,2,3$) be three independent events in a sample space. The probability that only $B_1$ occur is $\\alpha$, only $B_2$ occurs is $\\beta$ and only $B_3$ occurs is $\\gamma$. Let p be the probability that none of the events $B_i$ occurs and these 4 probabilities satisfy the equations $(\\alpha-2\\beta)p=\\alpha\\beta$ and $(\\beta-3\\gamma)p=2\\beta\\gamma$ (All the probabilities are assumed to lie in the interval (0, 1)). Then $\\frac{P(B_1)}{P(B_3)}$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"6","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":83,"question":"Let $P=\\begin{bmatrix}3&-1&-2\\\\2&0&\\alpha\\\\3&-5&0\\end{bmatrix}$, where $\\alpha\\in\\mathbb{R}$. Suppose $Q=[q_{ij}]$ is a matrix satisfying $PQ=kI_3$ for some non-zero $k\\in\\mathbb{R}$. If $q_{23}=-\\frac k8$ and $|Q|=\\frac{k^2}{2}$, then $\\alpha^2+k^2$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"17","explanation":"$|P|=12\\alpha+20=2k$ and $q_{23}=\\frac{-k(3\\alpha+4)}{|P|}=-\\frac k8$ give $\\alpha=-1$, $k=4$: 17.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":84,"question":"Let M be any $3\\times3$ matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of $M^TM$ is seven, is ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"540","explanation":"Sum of squares of entries $=7$: seven 1s ($\\binom97=36$) or one 2 and three 1s ($9\\binom83=504$): 540.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":85,"question":"Let $A=\\{n\\in\\mathbb{N}: n$ is a 3-digit number$\\}$, $B=\\{9k+2: k\\in\\mathbb{N}\\}$ and $C=\\{9k+l: k\\in\\mathbb{N}\\}$ for some $l$ $(0\u003cl\u003c9)$. If the sum of all the elements of the set $A\\cap(B\\cup C)$ is $274\\times400$, then $l$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"5","explanation":"3-digit numbers $\\equiv2\\pmod9$ sum to 54650; the rest 54950 matches $l=5$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":86,"question":"The minimum value of $\\alpha$ for which the equation $\\frac{4}{\\sin x}+\\frac{1}{1-\\sin x}=\\alpha$ has at least one solution in $\\left(0,\\frac\\pi2\\right)$ is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"9","explanation":"Min of $\\frac4s+\\frac1{1-s}$ on (0,1) is $(2+1)^2=9$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":87,"question":"If $\\int_{-a}^{a}(|x|+|x-2|)dx=22$, $(a>2)$ and [x] denotes the greatest integer $\\le x$, then $\\int_{a}^{-a}(x+[x])dx$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"3","explanation":"$2a^2+4=22\\Rightarrow a=3$; $-\\int_{-3}^{3}[x]dx=3$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":88,"question":"If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"3","explanation":"$r^2=2^2+5=9$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":89,"question":"Let three vectors $\\vec a$, $\\vec b$ and $\\vec c$ be such that $\\vec c$ is coplanar with $\\vec a$ and $\\vec b$, $\\vec a\\cdot\\vec c=7$ and $\\vec b$ is perpendicular to $\\vec c$, where $\\vec a=-\\hat i+\\hat j+\\hat k$ and $\\vec b=2\\hat i+\\hat k$, then the value of $2|\\vec a+\\vec b+\\vec c|^2$ is ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"75","explanation":"$\\vec c=\\frac52\\vec a+\\frac12\\vec b$; $|\\vec a+\\vec b+\\vec c|^2=\\frac{75}{2}$, so $2|\\cdot|^2=75$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":90,"question":"$\\lim_{n\\to\\infty}\\tan\\left\\{\\sum_{r=1}^{n}\\tan^{-1}\\left(\\frac{1}{1+r+r^2}\\right)\\right\\}$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"1","explanation":"Telescoping to $\\frac\\pi2-\\frac\\pi4$: value 1.","source":"JEE Main 2021 official paper","shown_locale":"en"}]}} . The metal sheet is heated uniformly, by a small temperature $\\Delta T$, so that its new temperature is $T+\\Delta T$. Calculate the increase in the volume of the metal box.","text":"Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is '$\\alpha

. The metal sheet is heated uniformly, by a small temperature $\\Delta T$, so that its new temperature is $T+\\Delta T$. Calculate the increase in the volume of the metal box.","acceptedAnswer":{"@type":"Answer","text":"D. $3a^3\\alpha\\Delta T$"},"suggestedAnswer":[{"@type":"Answer","text":"A. $4\\pi a^3\\alpha\\Delta T$"},{"@type":"Answer","text":"B. $4a^3\\alpha\\Delta T$"},{"@type":"Answer","text":"C. $\\frac43\\pi a^3\\alpha\\Delta T$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Y, K and $\\eta$ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.","text":"Y, K and $\\eta$ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.","acceptedAnswer":{"@type":"Answer","text":"C. $K=\\frac{Y\\eta}{9\\eta-3Y}$ N/m$^2$","comment":"From $Y=\\frac{9K\\eta}{3K+\\eta}$: $K(9\\eta-3Y)=Y\\eta$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $Y=\\frac{9K\\eta}{2\\eta+3K}$ N/m$^2$"},{"@type":"Answer","text":"B. $Y=\\frac{9K\\eta}{3K-\\eta}$ N/m$^2$"},{"@type":"Answer","text":"D. $\\eta=\\frac{3YK}{9K+Y}$ N/m$^2$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph?","text":"If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph?","acceptedAnswer":{"@type":"Answer","text":"A. a constant negative, then jumps to a constant positive value (single step)","comment":"Constant negative slope then constant positive slope."},"suggestedAnswer":[{"@type":"Answer","text":"B. a sinusoid-like curve"},{"@type":"Answer","text":"C. a square wave alternating repeatedly between positive and negative values"},{"@type":"Answer","text":"D. a triangular (rising then falling) curve"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"1 mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes. $A\\to B$: Isothermal expansion at temperature T so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$. $B\\to C$: Isobaric compression at pressure $P_2$ to initial volume $V_1$. $C\\to A$: Isochoric change leading to change of pressure from $P_2$ to $P_1$. Total workdone in the complete cycle ABCA is:","text":"1 mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes. $A\\to B$: Isothermal expansion at temperature T so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$. $B\\to C$: Isobaric compression at pressure $P_2$ to initial volume $V_1$. $C\\to A$: Isochoric change leading to change of pressure from $P_2$ to $P_1$. Total workdone in the complete cycle ABCA is:","acceptedAnswer":{"@type":"Answer","text":"D. $nRT\\left(\\ln2-\\frac12\\right)$","comment":"$nRT\\ln2-P_2V_1=nRT\\ln2-\\frac{nRT}{2}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. 0"},{"@type":"Answer","text":"B. $nRT\\ln2$"},{"@type":"Answer","text":"C. $nRT\\left(\\ln2+\\frac12\\right)$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Match List I with List II. List I: (a) Isothermal (b) Isochoric (c) Adiabatic (d) Isobaric. List II: (i) Pressure constant (ii) Temperature constant (iii) Volume constant (iv) Heat content is constant. Choose the correct answer from the options given below:","text":"Match List I with List II. List I: (a) Isothermal (b) Isochoric (c) Adiabatic (d) Isobaric. List II: (i) Pressure constant (ii) Temperature constant (iii) Volume constant (iv) Heat content is constant. Choose the correct answer from the options given below:","acceptedAnswer":{"@type":"Answer","text":"D. (a)$\\to$(ii), (b)$\\to$(iii), (c)$\\to$(iv), (d)$\\to$(i)"},"suggestedAnswer":[{"@type":"Answer","text":"A. (a)$\\to$(i), (b)$\\to$(iii), (c)$\\to$(ii), (d)$\\to$(iv)"},{"@type":"Answer","text":"B. (a)$\\to$(iii), (b)$\\to$(ii), (c)$\\to$(i), (d)$\\to$(iv)"},{"@type":"Answer","text":"C. (a)$\\to$(ii), (b)$\\to$(iv), (c)$\\to$(iii), (d)$\\to$(i)"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be:","text":"In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be:","acceptedAnswer":{"@type":"Answer","text":"B. $A\\sqrt{\\frac{M}{M+m}}$","comment":"Momentum conserved at the mean position: $A\\'=\\frac{Mv}{(M+m)\\omega\\'}=A\\sqrt{\\frac{M}{M+m}}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $A\\sqrt{\\frac{M+m}{M}}$"},{"@type":"Answer","text":"C. $A\\sqrt{\\frac{M-m}{M}}$"},{"@type":"Answer","text":"D. $A\\sqrt{\\frac{M}{M-m}}$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"A cell $E_1$ of emf 6 V and internal resistance 2 $\\Omega$ is connected with another cell $E_2$ of emf 4 V and internal resistance 8 $\\Omega$ (as shown in the figure). The potential difference across points X and Y is:","text":"A cell $E_1$ of emf 6 V and internal resistance 2 $\\Omega$ is connected with another cell $E_2$ of emf 4 V and internal resistance 8 $\\Omega$ (as shown in the figure). The potential difference across points X and Y is:","acceptedAnswer":{"@type":"Answer","text":"C. 5.6 V","comment":"Net emf 2 V, $I=0.2$ A; $V_{XY}=4+0.2\\times8=5.6$ V (E2 is being charged)."},"suggestedAnswer":[{"@type":"Answer","text":"A. 2.0 V"},{"@type":"Answer","text":"B. 3.6 V"},{"@type":"Answer","text":"D. 10.0 V"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"A current through a wire depends on time as $i=\\alpha_0t+\\beta t^2$ where $\\alpha_0=20$ A/s and $\\beta=8$ As$^{-2}$. Find the charge crossed through a section of the wire in 15 s.","text":"A current through a wire depends on time as $i=\\alpha_0t+\\beta t^2$ where $\\alpha_0=20$ A/s and $\\beta=8$ As$^{-2}$. Find the charge crossed through a section of the wire in 15 s.","acceptedAnswer":{"@type":"Answer","text":"C. 11250 C","comment":"$q=10(225)+\\frac83(3375)=11250$ C."},"suggestedAnswer":[{"@type":"Answer","text":"A. 260 C"},{"@type":"Answer","text":"B. 2100 C"},{"@type":"Answer","text":"D. 2250 C"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be:","text":"Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be:","acceptedAnswer":{"@type":"Answer","text":"D. 1 : 4"},"suggestedAnswer":[{"@type":"Answer","text":"A. 1 : 2"},{"@type":"Answer","text":"B. 2 : 1"},{"@type":"Answer","text":"C. 4 : 1"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is –Q. The electric field at the centre of cube is:","text":"A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is –Q. The electric field at the centre of cube is:","acceptedAnswer":{"@type":"Answer","text":"C. $\\frac{-2Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$","comment":"Equivalent to $-2Q$ at the origin: $E=\\frac{2Q}{4\\pi\\varepsilon_0(3a^2/4)}$ pointing to the origin."},"suggestedAnswer":[{"@type":"Answer","text":"A. $\\frac{-Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$"},{"@type":"Answer","text":"B. $\\frac{Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$"},{"@type":"Answer","text":"D. $\\frac{2Q}{3\\sqrt3\\pi\\varepsilon_0a^2}(\\hat x+\\hat y+\\hat z)$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"When an emitter current is changed by 4 mA, the collector current changes by 3.5 mA. The value of $\\beta$ will be:","text":"When an emitter current is changed by 4 mA, the collector current changes by 3.5 mA. The value of $\\beta$ will be:","acceptedAnswer":{"@type":"Answer","text":"A. 7","comment":"$I_B$ change 0.5 mA: $\\beta=7$."},"suggestedAnswer":[{"@type":"Answer","text":"B. 0.875"},{"@type":"Answer","text":"C. 0.5"},{"@type":"Answer","text":"D. 3.5"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E. The transitions A, B and C respectively represent:","text":"In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E. The transitions A, B and C respectively represent:","acceptedAnswer":{"@type":"Answer","text":"D. The series limit of Lyman series, third member of Balmer series and second member of Paschen series."},"suggestedAnswer":[{"@type":"Answer","text":"A. The first member of the Lyman series, third member of Balmer series and second member of Paschen series."},{"@type":"Answer","text":"B. The ionization potential of hydrogen, second member of Balmer series and third member of Paschen series."},{"@type":"Answer","text":"C. The series limit of Lyman series, second member of Balmer series and second member of Paschen series."}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Given below are two statements: Statement I: Two photons having equal linear momenta have equal wavelengths. Statement II: If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above statements, choose the correct answer from the options given below.","text":"Given below are two statements: Statement I: Two photons having equal linear momenta have equal wavelengths. Statement II: If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above statements, choose the correct answer from the options given below.","acceptedAnswer":{"@type":"Answer","text":"C. Statement I is true but Statement II is false"},"suggestedAnswer":[{"@type":"Answer","text":"A. Both Statement I and Statement II are true"},{"@type":"Answer","text":"B. Both Statement I and Statement II are false"},{"@type":"Answer","text":"D. Statement I is false but Statement II is true"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"The focal length f is related to the radius of curvature r of the spherical convex mirror by:","text":"The focal length f is related to the radius of curvature r of the spherical convex mirror by:","acceptedAnswer":{"@type":"Answer","text":"D. $f=+\\frac12r$"},"suggestedAnswer":[{"@type":"Answer","text":"A. $f=r$"},{"@type":"Answer","text":"B. $f=-r$"},{"@type":"Answer","text":"C. $f=-\\frac12r$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.","text":"In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.","acceptedAnswer":{"@type":"Answer","text":"A. 4 : 1","comment":"$\\left(\\frac{3+1}{3-1}\\right)^2=4$."},"suggestedAnswer":[{"@type":"Answer","text":"B. 2 : 1"},{"@type":"Answer","text":"C. 1 : 4"},{"@type":"Answer","text":"D. 3 : 1"}]}]}