with the vertical?","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$x=\\left(\\frac{q^2l^2}{2\\pi\\varepsilon_0m^2g}\\right)^{1/3}$","is_correct":false},{"key":"B","content":"$x=\\left(\\frac{q^2l}{2\\pi\\varepsilon_0mg}\\right)^{1/3}$","is_correct":true},{"key":"C","content":"$x=\\left(\\frac{q^2l}{2\\pi\\varepsilon_0mg}\\right)^{1/2}$","is_correct":false},{"key":"D","content":"$x=\\left(\\frac{q^2l^2}{2\\pi\\varepsilon_0m^2g^2}\\right)^{1/3}$","is_correct":false}],"correct_answer":"B","explanation":"$\\frac{x}{2l}=\\frac{q^2}{4\\pi\\varepsilon_0x^2mg}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":12,"question":"In the reported figure, there is a cyclic process ABCDA on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $A\\to B$ and $C\\to D$ are $T_1$ and $T_2$ ($T_1>T_2$) respectively. Choose the correct option out of the following for work done if processes BC and DA are adiabatic.","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q12.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$W_{BC}+W_{DA}>0$","is_correct":false},{"key":"B","content":"$W_{AB}\u003cW_{CD}$","is_correct":false},{"key":"C","content":"$W_{AB}=W_{DC}$","is_correct":false},{"key":"D","content":"$W_{AD}=W_{BC}$","is_correct":true}],"correct_answer":"D","explanation":"$W_{BC}=nC_v(T_1-T_2)=-W_{DA}=W_{AD}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":13,"question":"A body takes 4 min. to cool from 61°C to 59°C. If the temperature of the surroundings is 30°C, the time taken by the body to cool from 51°C to 49°C is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"3 min.","is_correct":false},{"key":"B","content":"4 min.","is_correct":false},{"key":"C","content":"6 min.","is_correct":true},{"key":"D","content":"8 min.","is_correct":false}],"correct_answer":"C","explanation":"Rate $\\propto$ (mean excess): 30 vs 20, so time $4\\times\\frac{30}{20}=6$ min.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":14,"question":"The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $2\\times10^{-9}$ J per molecule is:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$0.75\\times10^{11}$","is_correct":false},{"key":"B","content":"$1.5\\times10^{11}$","is_correct":true},{"key":"C","content":"$3\\times10^{11}$","is_correct":false},{"key":"D","content":"$6\\times10^{11}$","is_correct":false}],"correct_answer":"B","explanation":"$PV=\\frac23N\\bar E\\Rightarrow N=\\frac{3\\times2.026\\times10^2}{2\\times2\\times10^{-9}}\\approx1.5\\times10^{11}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":15,"question":"Assertion A: If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that $|\\vec{AB}|=|\\vec{BC}|=|\\vec{CD}|$, then $\\vec{AB}+\\vec{AC}+\\vec{AD}=4\\vec{AO}+\\vec{OB}+\\vec{OC}$. Reason R: Polygon law of vector addition yields $\\vec{AB}+\\vec{BC}+\\vec{CD}=\\vec{AD}=2\\vec{AO}$. In the light of the above statements, choose the most appropriate answer from the options given below:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q15.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Both A and R are correct and R is the correct explanation of A.","is_correct":false},{"key":"B","content":"Both A and R are correct but R is not the correct explanation of A.","is_correct":true},{"key":"C","content":"A is correct but R is not correct.","is_correct":false},{"key":"D","content":"A is not correct but R is correct.","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":16,"question":"The figure shows two solid discs with radius R and r respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (which is $\\perp$ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q16.png","type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$2R^2:r^2$","is_correct":false},{"key":"B","content":"$R^2:r^2$","is_correct":false},{"key":"C","content":"$2R^4:r^4$","is_correct":true},{"key":"D","content":"$2r^4:R^4$","is_correct":false}],"correct_answer":"C","explanation":"$\\frac{MR^2/2}{(Mr^2/R^2)r^2/4}=\\frac{2R^4}{r^4}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":17,"question":"Match List-I with List-II. List-I: (a) MI of the rod (length L, Mass M, about an axis $\\perp$ to the rod passing through the midpoint) (b) MI of the rod (length L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end) (c) MI of the rod (length 2L, Mass M, about an axis $\\perp$ to the rod passing through its midpoint) (d) MI of the rod (length 2L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end). List-II: (i) $8ML^2/3$ (ii) $ML^2/3$ (iii) $ML^2/12$ (iv) $2ML^2/3$. 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)-(iii), (b)-(iv), (c)-(i), (d)-(ii)","is_correct":false},{"key":"B","content":"(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)","is_correct":true},{"key":"C","content":"(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)","is_correct":false},{"key":"D","content":"(a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":18,"question":"Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. The masses of A, B and C are m, 2m and 2m respectively. A moves towards B with a speed of 9 m/s and makes an elastic collision with it. Thereafter B makes a completely inelastic collision with C. All motions occur along same straight line. The final speed of C is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"6 m/s","is_correct":false},{"key":"B","content":"3 m/s","is_correct":true},{"key":"C","content":"4 m/s","is_correct":false},{"key":"D","content":"9 m/s","is_correct":false}],"correct_answer":"B","explanation":"$v_B=\\frac{2m}{3m}\\cdot9=6$; then $6/2=3$ m/s.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":19,"question":"A ball is thrown up with a certain velocity so that it reaches a height 'h'. Find the ratio of the two different times of the ball reaching $\\frac h3$ in both the directions.","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac{\\sqrt3-\\sqrt2}{\\sqrt3+\\sqrt2}$","is_correct":true},{"key":"B","content":"$\\frac{\\sqrt3-1}{\\sqrt3+1}$","is_correct":false},{"key":"C","content":"$\\frac13$","is_correct":false},{"key":"D","content":"$\\frac{\\sqrt2-1}{\\sqrt2+1}$","is_correct":false}],"correct_answer":"A","explanation":"$t=\\frac{u\\mp\\sqrt{u^2-2gh/3}}{g}$ with $u^2=2gh$: ratio $\\frac{\\sqrt3-\\sqrt2}{\\sqrt3+\\sqrt2}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":20,"question":"Assertion A: If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm. Reason R: Least Count $=\\frac{\\text{Pitch}}{\\text{Total divisions on circular scale}}$. In the light of the above statements, choose the most appropriate 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 A and R are correct and R is the correct explanation of A.","is_correct":false},{"key":"B","content":"Both A and R are correct and R is NOT the correct explanation of A.","is_correct":false},{"key":"C","content":"A is correct but R is not correct.","is_correct":false},{"key":"D","content":"A is not correct but R is correct.","is_correct":true}],"correct_answer":"D","explanation":"Pitch 1 mm, LC = 0.02 mm = 0.002 cm, so A is wrong.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":21,"question":"Suppose two planets (spherical in shape) of radii R and 2R, but mass M and 9M respectively have a centre to centre separation 8R as shown in the figure. A satellite of mass 'm' is projected from the surface of the planet of mass 'M' directly towards the centre of the second planet. The minimum speed 'v' required for the satellite to reach the surface of the second planet is $\\sqrt{\\frac a7\\frac{GM}{R}}$ then the value of 'a' is ________. [Given: The two planets are fixed in their position]","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q21.png","type":"numerical","difficulty":"hard","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"4","explanation":"Null point where $\\frac{GM}{x^2}=\\frac{9GM}{(8R-x)^2}$: $x=2R$. Energy needed to reach it from the surface of M: $\\frac12v^2=\\left(-\\frac{GM}{2R}-\\frac{9GM}{6R}\\right)-\\left(-\\frac{GM}{R}-\\frac{9GM}{7R}\\right)=\\frac{2GM}{7R}$, so $v^2=\\frac{4}{7}\\frac{GM}{R}$ and $a=4$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":22,"question":"In a uniform magnetic field, the magnetic needle has a magnetic moment $9.85\\times10^{-2}$ A/m$^2$ and moment of inertia $5\\times10^{-6}$ kg m$^2$. If it performs 10 complete oscillations in 5 seconds then the magnitude of the magnetic field is ________ mT. [Take $\\pi^2$ as 9.85]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"8","explanation":"$B=\\frac{4\\pi^2I}{mT^2}=\\frac{4(9.85)(5\\times10^{-6})}{9.85\\times10^{-2}\\times0.25}=8$ mT.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":23,"question":"A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $10^{-6}$ m$^2$ stretched by an amount 0.04 m. The velocity of the projected stone is ________ m/s. (Young's modulus of rubber $=0.5\\times10^9$ N/m$^2$)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"20","explanation":"$U=\\frac{YA\\Delta l^2}{2l}=4$ J $=\\frac12(0.02)v^2\\Rightarrow v=20$ m/s.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":24,"question":"A transistor is connected in common emitter circuit configuration, the collector supply voltage is 10 V and the voltage drop across a resistor of 1000 $\\Omega$ in the collector circuit is 0.6 V. If the current gain factor ($\\beta$) is 24, then the base current is ________ $\\mu$A. (Round off to the Nearest Integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"25","explanation":"$I_C=0.6$ mA, $I_B=25\\ \\mu$A.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":25,"question":"A prism of refractive index $n_1$ and another prism of refractive index $n_2$ are stuck together (as shown in the figure). $n_1$ and $n_2$ depend on $\\lambda$, the wavelength of light, according to the relation $n_1=1.2+\\frac{10.8\\times10^{-14}}{\\lambda^2}$ and $n_2=1.45+\\frac{1.8\\times10^{-14}}{\\lambda^2}$. The wavelength for which rays incident at any angle on the interface BC pass through without bending at that interface will be ________ nm.","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q25.png","type":"numerical","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"600","explanation":"$n_1=n_2\\Rightarrow\\frac{9\\times10^{-14}}{\\lambda^2}=0.25\\Rightarrow\\lambda=600$ nm.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":26,"question":"A particle of mass $9.1\\times10^{-31}$ kg travels in a medium with a speed of $10^6$ m/s and a photon of a radiation of linear momentum $10^{-27}$ kg m/s travels in vacuum. The wavelength of photon is ________ times the wavelength of the particle.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"910","explanation":"Ratio $=\\frac{mv}{p}=\\frac{9.1\\times10^{-25}}{10^{-27}}=910$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":27,"question":"A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance 200 $\\mu$F is first charged and later connected with resistor 'R'. If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of 'R' should be ________ $\\Omega$.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"150","explanation":"$RC=\\tau\\Rightarrow R=\\frac{30\\times10^{-3}}{200\\times10^{-6}}=150\\ \\Omega$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":28,"question":"In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius 0.5 Å. If the speed of electron is $2.2\\times10^6$ m/s, then the current associated with the electron will be ________ $\\times10^{-2}$ mA. [Take $\\pi$ as $\\frac{22}{7}$]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"112","explanation":"$I=\\frac{ev}{2\\pi r}=1.12$ mA $=112\\times10^{-2}$ mA.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":29,"question":"Consider an electrical circuit containing a two way switch 'S'. Initially S is open and then $T_1$ is connected to $T_2$. As the current in $R=6\\ \\Omega$ attains a maximum value of steady state level, $T_1$ is disconnected from $T_2$ and immediately connected to $T_3$. Potential drop across $r=3\\ \\Omega$ resistor immediately after $T_1$ is connected to $T_3$ is ________ V. (Round off to the Nearest Integer)","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q29.png","type":"numerical","difficulty":"medium","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"3","explanation":"Steady current $6/6=1$ A through L continues through $r$: $3$ V.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":30,"question":"The amplitude of upper and lower side bands of A.M. wave where a carrier signal with frequency 11.21 MHz, peak voltage 15 V is amplitude modulated by a 7.7 kHz sine wave of 5 V amplitude are $\\frac a{10}$ V and $\\frac b{10}$ V respectively. Then the value of $\\frac ab$ is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Physics","marks":4,"negative_marks":0,"options":[],"correct_answer":"1","explanation":"Both sidebands have amplitude A_m/2: ratio 1.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":31,"question":"The parameters of the unit cell of a substance are $a=2.5$, $b=3.0$, $c=4.0$, $\\alpha=90°$, $\\beta=120°$, $\\gamma=90°$. The crystal system of the substance is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Triclinic","is_correct":false},{"key":"B","content":"Hexagonal","is_correct":false},{"key":"C","content":"Orthorhombic","is_correct":false},{"key":"D","content":"Monoclinic","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":32,"question":"Given below are two statements: Statement I: Rutherford's gold foil experiment cannot explain the line spectrum of hydrogen atom. Statement II: Bohr's model of hydrogen atom contradicts Heisenberg's uncertainty principle. In the light of the above statements, choose the most appropriate 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":"Both statement I and statement II are true.","is_correct":true},{"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":false},{"key":"D","content":"Statement I is false but statement II is true.","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":33,"question":"For a reaction of order n, the unit of the rate constant is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"mol$^{1-n}$ L$^{1-n}$ s$^{-1}$","is_correct":false},{"key":"B","content":"mol$^{1-n}$ L$^{1-n}$ s","is_correct":false},{"key":"C","content":"mol$^{1-n}$ L$^{2n}$ s$^{-1}$","is_correct":false},{"key":"D","content":"mol$^{1-n}$ L$^{n-1}$ s$^{-1}$","is_correct":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":34,"question":"Match List-I with List-II: List-I: (a) NaOH (b) Be(OH)$_2$ (c) Ca(OH)$_2$ (d) B(OH)$_3$ (e) Al(OH)$_3$. List-II: (i) Acidic (ii) Basic (iii) Amphoteric. Choose the most appropriate 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)-(ii), (b)-(i), (c)-(ii), (d)-(iii), (e)-(iii)","is_correct":false},{"key":"B","content":"(a)-(ii), (b)-(ii), (c)-(iii), (d)-(i), (e)-(iii)","is_correct":false},{"key":"C","content":"(a)-(ii), (b)-(iii), (c)-(ii), (d)-(i), (e)-(iii)","is_correct":true},{"key":"D","content":"(a)-(ii), (b)-(ii), (c)-(iii), (d)-(ii), (e)-(iii)","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":35,"question":"The statement that is INCORRECT about Ellingham diagram is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"provides idea about reduction of metal oxide.","is_correct":false},{"key":"B","content":"provides idea about the reaction rate.","is_correct":true},{"key":"C","content":"provides idea about free energy change.","is_correct":false},{"key":"D","content":"provides idea about changes in the phases during the reaction.","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":36,"question":"The product obtained from the electrolytic oxidation of acidified sulphate solutions, is:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"HO$_3$SOSO$_3$H","is_correct":false},{"key":"B","content":"HO$_2$SOSO$_2$H","is_correct":false},{"key":"C","content":"HO$_3$SOOSO$_3$H","is_correct":true},{"key":"D","content":"HSO$_4^-$","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":37,"question":"Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Lithium halides are some what covalent in nature. Reason R: Lithium possess high polarisation capability. In the light of the above statements, choose the most appropriate 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":"Both A and R are true and R is the correct explanation of A","is_correct":true},{"key":"B","content":"Both A and R are true but R is NOT the correct explanation of A","is_correct":false},{"key":"C","content":"A is true but R is false","is_correct":false},{"key":"D","content":"A is false but R is true","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":38,"question":"The oxidation states of 'P' in H$_4$P$_2$O$_7$, H$_4$P$_2$O$_5$ and H$_4$P$_2$O$_6$, respectively, are:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"6, 4 and 5","is_correct":false},{"key":"B","content":"5, 4 and 3","is_correct":false},{"key":"C","content":"5, 3 and 4","is_correct":true},{"key":"D","content":"7, 5 and 6","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":39,"question":"The type of hybridisation and magnetic property of the complex $[\\mathrm{MnCl_6}]^{3-}$, respectively, are:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"d$^2$sp$^3$ and paramagnetic","is_correct":false},{"key":"B","content":"sp$^3$d$^2$ and diamagnetic","is_correct":false},{"key":"C","content":"sp$^3$d$^2$ and paramagnetic","is_correct":true},{"key":"D","content":"d$^2$sp$^3$ and diamagnetic","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":40,"question":"The number of geometrical isomers found in the metal complexes $[\\mathrm{PtCl_2(NH_3)_2}]$, $[\\mathrm{Ni(CO)_4}]$, $[\\mathrm{Ru(H_2O)_3Cl_3}]$ and $[\\mathrm{CoCl_2(NH_3)_4}]^+$ respectively, are:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1, 1, 1, 1","is_correct":false},{"key":"B","content":"2, 0, 2, 2","is_correct":true},{"key":"C","content":"2, 1, 2, 1","is_correct":false},{"key":"D","content":"2, 1, 2, 2","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":41,"question":"Which one of the following statements is NOT correct?","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"The dissolved oxygen concentration below 6 ppm inhibits fish growth","is_correct":false},{"key":"B","content":"Eutrophication indicates that water body is polluted","is_correct":false},{"key":"C","content":"Eutrophication leads to increase in the oxygen level in water","is_correct":true},{"key":"D","content":"Eutrophication leads to anaerobic conditions","is_correct":false}],"correct_answer":"C","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":42,"question":"Which one among the following chemical tests is used to distinguish monosaccharide from disaccharide?","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Seliwanoff's test","is_correct":false},{"key":"B","content":"Barfoed test","is_correct":true},{"key":"C","content":"Tollen's test","is_correct":false},{"key":"D","content":"Iodine test","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":43,"question":"Staggered and eclipsed conformers of ethane are:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Rotamers","is_correct":true},{"key":"B","content":"Mirror images","is_correct":false},{"key":"C","content":"Enantiomers","is_correct":false},{"key":"D","content":"Polymers","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":44,"question":"The correct order of stability of given carbocations is: A: benzyl cation $\\mathrm{C_6H_5\\overset{+}{C}H_2}$; B: vinyl cation $\\mathrm{CH_2=\\overset{+}{C}H}$; C: ethyl cation $\\mathrm{CH_3-\\overset{+}{C}H_2}$; D: ethynyl cation $\\mathrm{HC\\equiv\\overset{+}{C}}$","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"D > B > C > A","is_correct":false},{"key":"B","content":"A > C > B > D","is_correct":true},{"key":"C","content":"C > A > D > B","is_correct":false},{"key":"D","content":"D > B > A > C","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":45,"question":"Presence of which reagent will affect the reversibility of the following reaction, and change it to a irreversible reaction: $\\mathrm{CH_4+I_2\\xrightleftharpoons[\\text{Reversible}]{h\\nu}CH_3-I+HI}$","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Concentrated HIO$_3$","is_correct":true},{"key":"B","content":"HOCl","is_correct":false},{"key":"C","content":"Liquid NH$_3$","is_correct":false},{"key":"D","content":"dilute HNO$_2$","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":46,"question":"Which one of the following compounds will give orange precipitate when treated with 2,4-dinitrophenyl hydrazine?","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q46.png","type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"acetophenone (C6H5COCH3)","is_correct":true},{"key":"B","content":"ethyl benzoate (C6H5COOCH2CH3)","is_correct":false},{"key":"C","content":"ethyl salicylate (2-HO-C6H4-COOCH2CH3)","is_correct":false},{"key":"D","content":"salicylic acid (2-HO-C6H4-COOH)","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":47,"question":"Cyclohexylmethanol $\\xrightarrow{\\text{H}_3\\text{PO}_4,\\ 120^\\circ\\text{C}}$ A (Major Product) $\\xrightarrow{\\text{(BH}_3)_2\\text{, H}_2\\text{O}_2/\\overline{\\text{O}}\\text{H, H}_2\\text{O}}$ P (Major Product). Consider the above reaction and identify the Product P:","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q47.png","type":"mcq_single","difficulty":"hard","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1-methylcyclohexan-1-ol","is_correct":false},{"key":"B","content":"cyclohexylmethanol","is_correct":false},{"key":"C","content":"2-methylcyclohexan-1-ol","is_correct":true},{"key":"D","content":"2-(hydroxymethyl)cyclohexan-1-ol","is_correct":false}],"correct_answer":"C","explanation":"Ring-expansion-free dehydration via hydride shift gives 1-methylcyclohexene; anti-Markovnikov hydroboration-oxidation gives 2-methylcyclohexanol.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":48,"question":"Given below are two statements: Statement I: Aniline is less basic than acetamide. Statement II: In aniline, the lone pair of electrons on nitrogen atom is delocalised over benzene ring due to resonance and hence less available to a proton. Choose the most appropriate option:","image_url":null,"type":"mcq_single","difficulty":"easy","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":false},{"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":true}],"correct_answer":"D","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":49,"question":"Match List-I with List-II: List-I (Drug): (a) Furacin (b) Arsphenamine (c) Dimetone (d) Valium. List-II (Class of Drug): (i) Antibiotic (ii) Tranquilizers (iii) Antiseptic (iv) Synthetic antihistamines. Choose the most appropriate match:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)","is_correct":true},{"key":"B","content":"(a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)","is_correct":false},{"key":"C","content":"(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)","is_correct":false},{"key":"D","content":"(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)","is_correct":false}],"correct_answer":"A","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":50,"question":"The compound 'A' (5-methylpyrimidine-2,4(1H,3H)-dione, i.e. thymine) is a complementary base of ________ in DNA strands.","image_url":"https://gyanvapi.kineticsoftwaresolution.com/storage/questions/jee-main/jee-main-2021-jul-27-shift1/q50.png","type":"mcq_single","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":1,"options":[{"key":"A","content":"Guanine","is_correct":false},{"key":"B","content":"Adenine","is_correct":true},{"key":"C","content":"Cytosine","is_correct":false},{"key":"D","content":"Uracil","is_correct":false}],"correct_answer":"B","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":51,"question":"The density of NaOH solution is 1.2 g cm$^{-3}$. The molality of this solution is ________ m. (Round off to the Nearest Integer) [Use: Atomic masses: Na: 23.0 u O: 16.0 u H: 1.0 u; Density of H$_2$O: 1.0 g cm$^{-3}$]","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"5","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":52,"question":"The difference between bond orders of CO and NO$^\\oplus$ is $\\frac x2$ where $x$ = ________. (Round off to the Nearest Integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"0","explanation":"Both are isoelectronic with N2 (bond order 3): x = 0.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":53,"question":"For water at 100°C and 1 bar, $\\Delta_{vap}H-\\Delta_{vap}U$ = ________ $\\times10^2$ J mol$^{-1}$. (Round off to the Nearest Integer) [Use: R = 8.31 J mol$^{-1}$ K$^{-1}$] [Assume volume of H$_2$O(l) is much smaller than volume of H$_2$O(g). Assume H$_2$O(g) can be treated as an ideal gas]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"31","explanation":"$RT=8.31\\times373\\approx3100$ J $=31\\times10^2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":54,"question":"1.46 g of a biopolymer dissolved in a 100 mL water at 300 K exerted an osmotic pressure of $2.42\\times10^{-3}$ bar. The molar mass of the biopolymer is ________ $\\times10^4$ g mol$^{-1}$. (Round off to the Nearest Integer) [Use: R = 0.083 L bar mol$^{-1}$ K$^{-1}$]","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"15","explanation":"$M=\\frac{1.46\\times0.083\\times300}{0.1\\times2.42\\times10^{-3}}\\approx1.5\\times10^5$ g/mol.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":55,"question":"$\\mathrm{PCl_5\\rightleftharpoons PCl_3+Cl_2}$, $K_c=1.844$. 3.0 moles of PCl$_5$ is introduced in a 1 L closed reaction vessel at 380 K. The number of moles of PCl$_5$ at equilibrium is ________ $\\times10^{-3}$. (Round off to the Nearest Integer)","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"1396","explanation":"$\\frac{x^2}{3-x}=1.844\\Rightarrow x\\approx1.604$; PCl$_5\\approx1.396$ mol.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":56,"question":"The conductivity of a weak acid HA of concentration 0.001 mol L$^{-1}$ is $2.0\\times10^{-5}$ S cm$^{-1}$. If $\\Lambda_m^\\circ$(HA) = 190 S cm$^2$ mol$^{-1}$, the ionization constant ($K_a$) of HA is equal to ________ $\\times10^{-6}$. (Round off to the Nearest Integer)","image_url":null,"type":"numerical","difficulty":"medium","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"12","explanation":"$\\Lambda_m=20$, $\\alpha=\\frac{20}{190}$, $K_a=\\frac{C\\alpha^2}{1-\\alpha}\\approx12\\times10^{-6}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":57,"question":"CO$_2$ gas adsorbs on charcoal following Freundlich adsorption isotherm. For a given amount of charcoal, the mass of CO$_2$ adsorbed becomes 64 times when the pressure of CO$_2$ is doubled. The value of n in the Freundlich isotherm equation is ________ $\\times10^{-2}$. (Round off to the Nearest Integer)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"17","explanation":"$2^{1/n}=64\\Rightarrow n=\\frac16\\approx17\\times10^{-2}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":58,"question":"The number of geometrical isomers possible in triamminetrinitrocobalt (III) is X and in trioxalatochromate (III) is Y. Then the value of X+Y is ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":"fac/mer gives X = 2; [Cr(ox)3]3- has no geometrical isomers (Y = 0).","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":59,"question":"In gaseous triethyl amine the \"–C–N–C–\" bond angle is ________ degree.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"108","explanation":null,"source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":60,"question":"An organic compound is subjected to chlorination to get compound A using 5.0 g of chlorine. When 0.5 g of compound A is reacted with AgNO$_3$ [Carius Method], the percentage of chlorine in compound A is ________ when it forms 0.3849 g of AgCl. (Round off to the Nearest Integer) (Atomic masses of Ag and Cl are 107.87 and 35.5 respectively)","image_url":null,"type":"numerical","difficulty":"easy","subject":"Chemistry","marks":4,"negative_marks":0,"options":[],"correct_answer":"19","explanation":"$\\frac{35.5}{143.37}\\times\\frac{0.3849}{0.5}\\times100\\approx19\\%$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":61,"question":"Let P and Q be two distinct points on a circle which has center at $C(2,3)$ and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set $\\{P,Q\\}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\{(2+2\\sqrt2,3+\\sqrt5),(2-2\\sqrt2,3-\\sqrt5)\\}$","is_correct":false},{"key":"B","content":"$\\{(2+2\\sqrt2,3-\\sqrt5),(2-2\\sqrt2,3+\\sqrt5)\\}$","is_correct":false},{"key":"C","content":"$\\{(-1,5),(5,1)\\}$","is_correct":true},{"key":"D","content":"$\\{(4,0),(0,6)\\}$","is_correct":false}],"correct_answer":"C","explanation":"$P,Q=C\\pm(-3,2)$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":62,"question":"Let $\\vec a=\\hat i+\\hat j+2\\hat k$ and $\\vec b=-\\hat i+2\\hat j+3\\hat k$. Then the vector product $(\\vec a+\\vec b)\\times((\\vec a\\times((\\vec a-\\vec b)\\times\\vec b))\\times\\vec b)$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$5(30\\hat i-5\\hat j+7\\hat k)$","is_correct":false},{"key":"B","content":"$7(30\\hat i-5\\hat j+7\\hat k)$","is_correct":false},{"key":"C","content":"$5(34\\hat i-5\\hat j+3\\hat k)$","is_correct":false},{"key":"D","content":"$7(34\\hat i-5\\hat j+3\\hat k)$","is_correct":true}],"correct_answer":"D","explanation":"$(\\vec a-\\vec b)\\times\\vec b=\\vec a\\times\\vec b$; $\\vec a\\times(\\vec a\\times\\vec b)=7\\vec a-6\\vec b=(13,-5,-4)$; $\\times\\vec b=(-7,-35,21)$; $(0,3,5)\\times(-7,-35,21)=(238,-35,21)$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":63,"question":"If the coefficients of $x^7$ in $\\left(x^2+\\frac{1}{bx}\\right)^{11}$ and $x^{-7}$ in $\\left(x-\\frac{1}{bx^2}\\right)^{11}$, $b\\ne0$, are equal, then the value of b is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$-1$","is_correct":false},{"key":"B","content":"2","is_correct":false},{"key":"C","content":"$-2$","is_correct":false},{"key":"D","content":"1","is_correct":true}],"correct_answer":"D","explanation":"${}^{11}C_5b^{-5}={}^{11}C_6b^{-6}\\Rightarrow b=1$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":64,"question":"If the area of the bounded region $R=\\left\\{(x,y):\\max\\{0,\\log_ex\\}\\le y\\le2^x,\\ \\frac12\\le x\\le2\\right\\}$ is, $\\alpha(\\log_e2)^{-1}+\\beta(\\log_e2)+\\gamma$, then the value of $(\\alpha+\\beta-2\\gamma)^2$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1","is_correct":false},{"key":"B","content":"2","is_correct":true},{"key":"C","content":"4","is_correct":false},{"key":"D","content":"8","is_correct":false}],"correct_answer":"B","explanation":"$\\frac{4-\\sqrt2}{\\ln2}-(2\\ln2-1)$: $(4-\\sqrt2-2-2)^2=2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":65,"question":"Let $A=\\begin{bmatrix}1&2\\\\-1&4\\end{bmatrix}$. If $A^{-1}=\\alpha I+\\beta A$, $\\alpha,\\beta\\in\\mathbb{R}$, I is a $2\\times2$ identity matrix, then $4(\\alpha-\\beta)$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"2","is_correct":false},{"key":"B","content":"4","is_correct":true},{"key":"C","content":"5","is_correct":false},{"key":"D","content":"$\\frac83$","is_correct":false}],"correct_answer":"B","explanation":"$A^2-5A+6I=0\\Rightarrow A^{-1}=\\frac56I-\\frac16A$; $4(\\alpha-\\beta)=4$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":66,"question":"Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2x-6y+6=0$. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"2","is_correct":false},{"key":"B","content":"4","is_correct":true},{"key":"C","content":"$(3\\sqrt2+2)$","is_correct":false},{"key":"D","content":"$3(\\sqrt2-1)$","is_correct":false}],"correct_answer":"B","explanation":"PACB is a square: $A(-1,3)$, $B(1,1)$; $D=(1,5)$; area 4.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":67,"question":"Let $\\mathbb{C}$ be the set of all complex numbers. Let $S_1=\\{z\\in\\mathbb{C}\\,|\\,|z-3-2i|^2=8\\}$, $S_2=\\{z\\in\\mathbb{C}\\,|\\,\\mathrm{Re}(z)\\ge5\\}$ and $S_3=\\{z\\in\\mathbb{C}\\,|\\,|z-\\bar z|\\ge8\\}$. Then the number of elements in $S_1\\cap S_2\\cap S_3$ is equal to:","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":"1","is_correct":true},{"key":"C","content":"2","is_correct":false},{"key":"D","content":"Infinite","is_correct":false}],"correct_answer":"B","explanation":"Only $(5,4)$ satisfies all three.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":68,"question":"Let the plane passing through the point $(-1,0,-2)$ and perpendicular to each of the planes $2x+y-z=2$ and $x-y-z=3$ be $ax+by+cz+8=0$. Then the value of $a+b+c$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"5","is_correct":false},{"key":"B","content":"3","is_correct":false},{"key":"C","content":"4","is_correct":true},{"key":"D","content":"8","is_correct":false}],"correct_answer":"C","explanation":"Normal $(2,1,-1)\\times(1,-1,-1)=(-2,1,-3)$: plane $2x-y+3z+8=0$, sum 4.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":69,"question":"Let $\\alpha,\\beta$ be two roots of the equation $x^2+(20)^{\\frac14}x+(5)^{\\frac12}=0$. Then $\\alpha^8+\\beta^8$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"100","is_correct":false},{"key":"B","content":"10","is_correct":false},{"key":"C","content":"50","is_correct":true},{"key":"D","content":"160","is_correct":false}],"correct_answer":"C","explanation":"$\\alpha^2+\\beta^2=0$, $\\alpha^4+\\beta^4=-10$, $\\alpha^8+\\beta^8=100-50=50$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":70,"question":"Let $f:\\left(-\\frac\\pi4,\\frac\\pi4\\right)\\to\\mathbb{R}$ be defined as $f(x)=\\begin{cases}(1+|\\sin x|)^{\\frac{3a}{|\\sin x|}}, & -\\frac\\pi4\u003cx\u003c0\\\\ b, & x=0\\\\ e^{\\cot4x/\\cot2x}, & 0\u003cx\u003c\\frac\\pi4\\end{cases}$. If f is continuous at $x=0$, then the value of $6a+b^2$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"1+e","is_correct":true},{"key":"B","content":"1−e","is_correct":false},{"key":"C","content":"e","is_correct":false},{"key":"D","content":"e−1","is_correct":false}],"correct_answer":"A","explanation":"$e^{3a}=b=e^{1/2}$: $6a+b^2=1+e$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":71,"question":"Let $A=\\{(x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\,|\\,2x^2+2y^2-2x-2y=1\\}$, $B=\\{(x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\,|\\,4x^2+4y^2-16y+7=0\\}$ and $C=\\{(x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\,|\\,x^2+y^2-4x-2y+5\\le r^2\\}$. Then the minimum value of $|r|$ such that $A\\cup B\\subseteq C$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac{3+\\sqrt{10}}{2}$","is_correct":false},{"key":"B","content":"$1+\\sqrt5$","is_correct":false},{"key":"C","content":"$\\frac{2+\\sqrt{10}}{2}$","is_correct":false},{"key":"D","content":"$\\frac{3+2\\sqrt5}{2}$","is_correct":true}],"correct_answer":"D","explanation":"Need $r\\ge\\max(\\frac{\\sqrt{10}}2+1,\\sqrt5+\\frac32)=\\frac{3+2\\sqrt5}{2}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":72,"question":"If the mean and variance of the following data: 6, 10, 7, 13, a, 12, b, 12 are 9 and $\\frac{37}{4}$ respectively, then $(a-b)^2$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"16","is_correct":true},{"key":"B","content":"12","is_correct":false},{"key":"C","content":"24","is_correct":false},{"key":"D","content":"32","is_correct":false}],"correct_answer":"A","explanation":"$a+b=12$, $a^2+b^2=80$: $(a-b)^2=16$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":73,"question":"Let $y=y(x)$ be solution of the differential equation $\\log_e\\left(\\frac{dy}{dx}\\right)=3x+4y$, with $y(0)=0$. If $y\\left(-\\frac23\\log_e2\\right)=\\alpha\\log_e2$, then the value of $\\alpha$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$-\\frac14$","is_correct":true},{"key":"B","content":"$\\frac14$","is_correct":false},{"key":"C","content":"2","is_correct":false},{"key":"D","content":"$-\\frac12$","is_correct":false}],"correct_answer":"A","explanation":"$-\\frac{e^{-4y}}{4}=\\frac{e^{3x}}{3}-\\frac7{12}$; at the point $e^{-4y}=2$, $y=-\\frac14\\ln2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":74,"question":"If $\\sin\\theta+\\cos\\theta=\\frac12$, then $16(\\sin(2\\theta)+\\cos(4\\theta)+\\sin(6\\theta))$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"23","is_correct":false},{"key":"B","content":"−23","is_correct":true},{"key":"C","content":"27","is_correct":false},{"key":"D","content":"−27","is_correct":false}],"correct_answer":"B","explanation":"$\\sin2\\theta=-\\frac34$, $\\cos4\\theta=-\\frac18$, $\\sin6\\theta=-\\frac9{16}$: $16\\times(-\\frac{23}{16})=-23$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":75,"question":"The probability that a randomly selected 2-digit number belongs to the set $\\{n\\in\\mathbb{N}: (2^n-2)$ is a multiple of 3$\\}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac12$","is_correct":true},{"key":"B","content":"$\\frac13$","is_correct":false},{"key":"C","content":"$\\frac23$","is_correct":false},{"key":"D","content":"$\\frac16$","is_correct":false}],"correct_answer":"A","explanation":"Holds exactly for odd n: 45/90.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":76,"question":"A ray of light through $(2,1)$ is reflected at a point P on the y-axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of an ellipse with eccentricity $\\frac13$ and the distance of the nearer focus from this directrix is $\\frac{8}{\\sqrt{53}}$, then the equation of the other directrix can be:","image_url":null,"type":"mcq_single","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$2x-7y-39=0$ or $2x-7y-7=0$","is_correct":false},{"key":"B","content":"$11x+7y+8=0$ or $11x+7y-15=0$","is_correct":false},{"key":"C","content":"$2x-7y+29=0$ or $2x-7y-7=0$","is_correct":true},{"key":"D","content":"$11x-7y-8=0$ or $11x+7y+15=0$","is_correct":false}],"correct_answer":"C","explanation":"Reflected ray: $2x-7y+11=0$; $\\frac{a}{e}-ae=\\frac{8}{\\sqrt{53}}\\Rightarrow a=\\frac{3}{\\sqrt{53}}$; directrices $\\frac{2a}{e}=\\frac{18}{\\sqrt{53}}$ apart: $c=29$ or $-7$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":77,"question":"The compound statement $(P\\vee Q)\\wedge(\\sim P)\\Rightarrow Q$ is equivalent to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\sim(P\\Rightarrow Q)$","is_correct":false},{"key":"B","content":"$P\\wedge\\sim Q$","is_correct":false},{"key":"C","content":"$\\sim(P\\Rightarrow Q)\\Leftrightarrow P\\wedge\\sim Q$","is_correct":true},{"key":"D","content":"$P\\vee Q$","is_correct":false}],"correct_answer":"C","explanation":"The statement is a tautology, as is option C.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":78,"question":"Let $f:\\mathbb{R}\\to\\mathbb{R}$ be a function such that $f(2)=4$ and $f'(2)=1$. Then, the value of $\\lim_{x\\to2}\\frac{x^2f(2)-4f(x)}{x-2}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"4","is_correct":false},{"key":"B","content":"8","is_correct":false},{"key":"C","content":"12","is_correct":true},{"key":"D","content":"16","is_correct":false}],"correct_answer":"C","explanation":"L\\'Hopital: $2\\cdot2\\cdot4-4\\cdot1=12$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":79,"question":"The value of $\\lim_{n\\to\\infty}\\frac1n\\sum_{j=1}^{n}\\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$5+\\log_e\\left(\\frac32\\right)$","is_correct":false},{"key":"B","content":"$1+2\\log_e\\left(\\frac32\\right)$","is_correct":true},{"key":"C","content":"$2-\\log_e\\left(\\frac23\\right)$","is_correct":false},{"key":"D","content":"$3+2\\log_e\\left(\\frac23\\right)$","is_correct":false}],"correct_answer":"B","explanation":"$\\int_0^1\\frac{2x+8}{2x+4}dx=1+2\\ln\\frac32$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":80,"question":"The value of the definite integral $\\int_{-\\pi/4}^{\\pi/4}\\frac{dx}{(1+e^{x\\cos x})(\\sin^4x+\\cos^4x)}$ is equal to:","image_url":null,"type":"mcq_single","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":1,"options":[{"key":"A","content":"$\\frac{\\pi}{2\\sqrt2}$","is_correct":true},{"key":"B","content":"$-\\frac\\pi4$","is_correct":false},{"key":"C","content":"$-\\frac\\pi2$","is_correct":false},{"key":"D","content":"$\\frac{\\pi}{\\sqrt2}$","is_correct":false}],"correct_answer":"A","explanation":"$=\\int_0^{\\pi/4}\\frac{dx}{\\sin^4x+\\cos^4x}=\\int_0^1\\frac{1+t^2}{1+t^4}dt=\\frac{\\pi}{2\\sqrt2}$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":81,"question":"Let a plane P pass through the point $(3,7,-7)$ and contain the line, $\\frac{x-2}{-3}=\\frac{y-3}{2}=\\frac{z+2}{1}$. If distance of the plane P from the origin is d, then $d^2$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"3","explanation":"Plane $x+y+z=3$: $d^2=3$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":82,"question":"Let $f(x)=\\begin{vmatrix}\\sin^2x&-2+\\cos^2x&\\cos2x\\\\2+\\sin^2x&\\cos^2x&\\cos2x\\\\\\sin^2x&\\cos^2x&1+\\cos2x\\end{vmatrix}$, $x\\in[0,\\pi]$. Then the maximum value of $f(x)$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"6","explanation":"$f=4+2\\cos2x$, max 6.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":83,"question":"Let $F:[3,5]\\to\\mathbb{R}$ be a twice differentiable function on $(3,5)$ such that $F(x)=e^{-x}\\int_3^x(3t^2+2t+4F'(t))dt$. If $F'(4)=\\frac{\\alpha e^\\beta-224}{(e^\\beta-4)^2}$, then $\\alpha+\\beta$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"16","explanation":"$F=\\frac{x^3+x^2-36}{e^x-4}$; $F\\'(4)=\\frac{12e^4-224}{(e^4-4)^2}$: 16.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":84,"question":"Let $\\vec a=\\hat i+\\hat j+\\hat k$, $\\vec b$ and $\\vec c=\\hat j-\\hat k$ be three vectors such that $\\vec a\\times\\vec b=\\vec c$ and $\\vec a\\cdot\\vec b=1$. If the length of projection vector of the vector $\\vec b$ on the vector $\\vec a\\times\\vec c$ is $l$, then the value of $3l^2$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":"$\\vec b=\\frac{\\vec a-\\vec a\\times\\vec c}{3}=\\hat i$; projection on $(-2,1,1)$: $l^2=\\frac46$, $3l^2=2$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":85,"question":"Let the domain of the function $f(x)=\\log_4(\\log_5(\\log_3(18x-x^2-77)))$ be $(a,b)$. Then the value of the integral $\\int_a^b\\frac{\\sin^3x}{\\sin^3x+\\sin^3(a+b-x)}dx$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"1","explanation":"Domain $(8,10)$; integral $=\\frac{b-a}{2}=1$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":86,"question":"If $\\log_32$, $\\log_3(2^x-5)$, $\\log_3\\left(2^x-\\frac72\\right)$ are in an arithmetic progression, then the value of x is equal to ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"3","explanation":"$(t-5)^2=2t-7\\Rightarrow t=8$: $x=3$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":87,"question":"For real numbers $\\alpha$ and $\\beta$, consider the following system of linear equations: $x+y-z=2$, $x+2y+\\alpha z=1$, $2x-y+z=\\beta$. If the system has infinite solutions, then $\\alpha+\\beta$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"easy","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"5","explanation":"$\\alpha=-2$, $\\beta=7$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":88,"question":"Let $S=\\{1,2,3,4,5,6,7\\}$. Then the number of possible functions $f:S\\to S$ such that $f(m\\cdot n)=f(m)\\cdot f(n)$ for every $m,n\\in S$ and $m\\cdot n\\in S$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"hard","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"490","explanation":"$f(1)=1$; $f(2)\\in\\{1,2\\}$ with $f(4)=f(2)^2$, $f(6)=f(2)f(3)$: $(7+3)\\times7\\times7=490$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":89,"question":"If $y=y(x)$, $y\\in\\left[0,\\frac\\pi2\\right)$ is the solution of the differential equation $\\sec y\\frac{dy}{dx}-\\sin(x+y)-\\sin(x-y)=0$, with $y(0)=0$, then $5y'\\left(\\frac\\pi2\\right)$ is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"2","explanation":"$\\tan y=2-2\\cos x$; at $\\frac\\pi2$: $\\tan y=2$, $y\\'=2\\cos^2y=\\frac25$.","source":"JEE Main 2021 official paper","shown_locale":"en"},{"number":90,"question":"Let $f:[0,3]\\to\\mathbb{R}$ be defined by $f(x)=\\min\\{x-[x],1+[x]-x\\}$ where $[x]$ is the greatest integer less than or equal to x. Let P denote the set containing all $x\\in[0,3]$ where f is discontinuous, and Q denote the set containing all $x\\in(0,3)$ where f is not differentiable. Then the sum of number of elements in P and Q is equal to ________.","image_url":null,"type":"numerical","difficulty":"medium","subject":"Mathematics","marks":4,"negative_marks":0,"options":[],"correct_answer":"5","explanation":"Triangle wave: continuous; corners at 0.5,1,1.5,2,2.5: total 5.","source":"JEE Main 2021 official paper","shown_locale":"en"}]}} with the vertical?","text":"Two identical tennis balls each having mass 'm' and charge 'q' are suspended from a fixed point by threads of length 'l'. What is the equilibrium separation when each thread makes a small angle '$\\theta

with the vertical?","acceptedAnswer":{"@type":"Answer","text":"B. $x=\\left(\\frac{q^2l}{2\\pi\\varepsilon_0mg}\\right)^{1/3}$","comment":"$\\frac{x}{2l}=\\frac{q^2}{4\\pi\\varepsilon_0x^2mg}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $x=\\left(\\frac{q^2l^2}{2\\pi\\varepsilon_0m^2g}\\right)^{1/3}$"},{"@type":"Answer","text":"C. $x=\\left(\\frac{q^2l}{2\\pi\\varepsilon_0mg}\\right)^{1/2}$"},{"@type":"Answer","text":"D. $x=\\left(\\frac{q^2l^2}{2\\pi\\varepsilon_0m^2g^2}\\right)^{1/3}$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"In the reported figure, there is a cyclic process ABCDA on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $A\\to B$ and $C\\to D$ are $T_1$ and $T_2$ ($T_1>T_2$) respectively. Choose the correct option out of the following for work done if processes BC and DA are adiabatic.","text":"In the reported figure, there is a cyclic process ABCDA on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $A\\to B$ and $C\\to D$ are $T_1$ and $T_2$ ($T_1>T_2$) respectively. Choose the correct option out of the following for work done if processes BC and DA are adiabatic.","acceptedAnswer":{"@type":"Answer","text":"D. $W_{AD}=W_{BC}$","comment":"$W_{BC}=nC_v(T_1-T_2)=-W_{DA}=W_{AD}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $W_{BC}+W_{DA}>0$"},{"@type":"Answer","text":"B. $W_{AB}\u003cW_{CD}$"},{"@type":"Answer","text":"C. $W_{AB}=W_{DC}$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"A body takes 4 min. to cool from 61°C to 59°C. If the temperature of the surroundings is 30°C, the time taken by the body to cool from 51°C to 49°C is:","text":"A body takes 4 min. to cool from 61°C to 59°C. If the temperature of the surroundings is 30°C, the time taken by the body to cool from 51°C to 49°C is:","acceptedAnswer":{"@type":"Answer","text":"C. 6 min.","comment":"Rate $\\propto$ (mean excess): 30 vs 20, so time $4\\times\\frac{30}{20}=6$ min."},"suggestedAnswer":[{"@type":"Answer","text":"A. 3 min."},{"@type":"Answer","text":"B. 4 min."},{"@type":"Answer","text":"D. 8 min."}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $2\\times10^{-9}$ J per molecule is:","text":"The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $2\\times10^{-9}$ J per molecule is:","acceptedAnswer":{"@type":"Answer","text":"B. $1.5\\times10^{11}$","comment":"$PV=\\frac23N\\bar E\\Rightarrow N=\\frac{3\\times2.026\\times10^2}{2\\times2\\times10^{-9}}\\approx1.5\\times10^{11}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $0.75\\times10^{11}$"},{"@type":"Answer","text":"C. $3\\times10^{11}$"},{"@type":"Answer","text":"D. $6\\times10^{11}$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Assertion A: If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that $|\\vec{AB}|=|\\vec{BC}|=|\\vec{CD}|$, then $\\vec{AB}+\\vec{AC}+\\vec{AD}=4\\vec{AO}+\\vec{OB}+\\vec{OC}$. Reason R: Polygon law of vector addition yields $\\vec{AB}+\\vec{BC}+\\vec{CD}=\\vec{AD}=2\\vec{AO}$. In the light of the above statements, choose the most appropriate answer from the options given below:","text":"Assertion A: If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that $|\\vec{AB}|=|\\vec{BC}|=|\\vec{CD}|$, then $\\vec{AB}+\\vec{AC}+\\vec{AD}=4\\vec{AO}+\\vec{OB}+\\vec{OC}$. Reason R: Polygon law of vector addition yields $\\vec{AB}+\\vec{BC}+\\vec{CD}=\\vec{AD}=2\\vec{AO}$. In the light of the above statements, choose the most appropriate answer from the options given below:","acceptedAnswer":{"@type":"Answer","text":"B. Both A and R are correct but R is not the correct explanation of A."},"suggestedAnswer":[{"@type":"Answer","text":"A. Both A and R are correct and R is the correct explanation of A."},{"@type":"Answer","text":"C. A is correct but R is not correct."},{"@type":"Answer","text":"D. A is not correct but R is correct."}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"The figure shows two solid discs with radius R and r respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (which is $\\perp$ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)","text":"The figure shows two solid discs with radius R and r respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (which is $\\perp$ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)","acceptedAnswer":{"@type":"Answer","text":"C. $2R^4:r^4$","comment":"$\\frac{MR^2/2}{(Mr^2/R^2)r^2/4}=\\frac{2R^4}{r^4}$."},"suggestedAnswer":[{"@type":"Answer","text":"A. $2R^2:r^2$"},{"@type":"Answer","text":"B. $R^2:r^2$"},{"@type":"Answer","text":"D. $2r^4:R^4$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Match List-I with List-II. List-I: (a) MI of the rod (length L, Mass M, about an axis $\\perp$ to the rod passing through the midpoint) (b) MI of the rod (length L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end) (c) MI of the rod (length 2L, Mass M, about an axis $\\perp$ to the rod passing through its midpoint) (d) MI of the rod (length 2L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end). List-II: (i) $8ML^2/3$ (ii) $ML^2/3$ (iii) $ML^2/12$ (iv) $2ML^2/3$. Choose the correct answer from the options given below:","text":"Match List-I with List-II. List-I: (a) MI of the rod (length L, Mass M, about an axis $\\perp$ to the rod passing through the midpoint) (b) MI of the rod (length L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end) (c) MI of the rod (length 2L, Mass M, about an axis $\\perp$ to the rod passing through its midpoint) (d) MI of the rod (length 2L, Mass 2M, about an axis $\\perp$ to the rod passing through one of its end). List-II: (i) $8ML^2/3$ (ii) $ML^2/3$ (iii) $ML^2/12$ (iv) $2ML^2/3$. Choose the correct answer from the options given below:","acceptedAnswer":{"@type":"Answer","text":"B. (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)"},"suggestedAnswer":[{"@type":"Answer","text":"A. (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)"},{"@type":"Answer","text":"C. (a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)"},{"@type":"Answer","text":"D. (a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. The masses of A, B and C are m, 2m and 2m respectively. A moves towards B with a speed of 9 m/s and makes an elastic collision with it. Thereafter B makes a completely inelastic collision with C. All motions occur along same straight line. The final speed of C is:","text":"Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. The masses of A, B and C are m, 2m and 2m respectively. A moves towards B with a speed of 9 m/s and makes an elastic collision with it. Thereafter B makes a completely inelastic collision with C. All motions occur along same straight line. The final speed of C is:","acceptedAnswer":{"@type":"Answer","text":"B. 3 m/s","comment":"$v_B=\\frac{2m}{3m}\\cdot9=6$; then $6/2=3$ m/s."},"suggestedAnswer":[{"@type":"Answer","text":"A. 6 m/s"},{"@type":"Answer","text":"C. 4 m/s"},{"@type":"Answer","text":"D. 9 m/s"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"A ball is thrown up with a certain velocity so that it reaches a height 'h'. Find the ratio of the two different times of the ball reaching $\\frac h3$ in both the directions.","text":"A ball is thrown up with a certain velocity so that it reaches a height 'h'. Find the ratio of the two different times of the ball reaching $\\frac h3$ in both the directions.","acceptedAnswer":{"@type":"Answer","text":"A. $\\frac{\\sqrt3-\\sqrt2}{\\sqrt3+\\sqrt2}$","comment":"$t=\\frac{u\\mp\\sqrt{u^2-2gh/3}}{g}$ with $u^2=2gh$: ratio $\\frac{\\sqrt3-\\sqrt2}{\\sqrt3+\\sqrt2}$."},"suggestedAnswer":[{"@type":"Answer","text":"B. $\\frac{\\sqrt3-1}{\\sqrt3+1}$"},{"@type":"Answer","text":"C. $\\frac13$"},{"@type":"Answer","text":"D. $\\frac{\\sqrt2-1}{\\sqrt2+1}$"}]},{"@type":"Question","eduQuestionType":"Multiple choice","name":"Assertion A: If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm. Reason R: Least Count $=\\frac{\\text{Pitch}}{\\text{Total divisions on circular scale}}$. In the light of the above statements, choose the most appropriate answer from the options given below:","text":"Assertion A: If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm. Reason R: Least Count $=\\frac{\\text{Pitch}}{\\text{Total divisions on circular scale}}$. In the light of the above statements, choose the most appropriate answer from the options given below:","acceptedAnswer":{"@type":"Answer","text":"D. A is not correct but R is correct.","comment":"Pitch 1 mm, LC = 0.02 mm = 0.002 cm, so A is wrong."},"suggestedAnswer":[{"@type":"Answer","text":"A. Both A and R are correct and R is the correct explanation of A."},{"@type":"Answer","text":"B. Both A and R are correct and R is NOT the correct explanation of A."},{"@type":"Answer","text":"C. A is correct but R is not correct."}]}]}