STD 11 - Units,dimensions… PYQs: JEE Mains Physics

151. Which of the following physical quantities have the same dimensions? 2022 Hard

152. A physical quantity \(y\) is represented by the formula \(y=m^{2}\, r^{-4}\, g^{x}\,l^{-\frac{3}{2}}\). If the percentage error found in \(y, m, r, l\) and \(g\) are \(18,1,0.5,4\) and \(p\) respectively, then find the value of \(x\) and \(p\). 2021 Medium

153. A student determined Young's Modulus of elasticity using the formula \(Y=\frac{M g L^{3}}{4 b d^{3} \delta} .\) The value of \(g\) is taken to be \(9.8 \,{m} / {s}^{2}\), without any significant error, his observation are as following.

Physical Quantity	Least count of the Equipment used for measurement	Observed value
Mass \(({M})\)	\(1\; {g}\)	\(2\; {kg}\)
Length of bar \((L)\)	\(1\; {mm}\)	\(1 \;{m}\)
Breadth of bar \((b)\)	\(0.1\; {mm}\)	\(4\; {cm}\)
Thickness of bar \((d)\)	\(0.01\; {mm}\)	\(0.4 \;{cm}\)
Depression \((\delta)\)	\(0.01\; {mm}\)	\(5 \;{mm}\)

Then the fractional error in the measurement of \({Y}\) is 2021 Hard

154. Asseretion \(A:\) 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: 2021 Medium

155. If \(C\) and \(V\) represent capacity and voltage respectively then what are the dimensions of \(\lambda,\) where \(\frac{ C }{ V }=\lambda ?\) 2021 Hard

156. If \({E}\) and \({H}\) represents the intensity of electric field and magnetising field respectively, then the unit of \(E/H\) will be : 2021 Hard

157. If e is the electronic charge, \(c\) is the speed of light in free space and \(h\) is Planck's constant, the quantity \(\frac{1}{4 \pi \varepsilon_{0}} \frac{| e |^{2}}{h c}\) has dimensions of ....... 2021 Hard

158. If \({E}, {L}, {m}\) and \({G}\) denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of \({P}\) in the formula \({P}={EL}^{2} {m}^{-5} {G}^{-2}\) are 2021 Medium

159. If force \(({F})\), length \(({L})\) and time \(({T})\) are taken as the fundamental quantities. Then what will be the dimension of density 2021 Hard

160. If time \((t)\), velocity \((u)\), and angular momentum \((I)\) are taken as the fundamental units. Then the dimension of mass \(({m})\) in terms of \({t}, {u}\) and \({I}\) is 2021 Medium

161. If velocity \([V],\) time \([T]\) and force \([F]\) are chosen as the base quantities, the dimensions of the mass will be 2021 Medium

162. In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are \(50\) divisions on the circular scale, and the main scale moves by \(0.5 \,{mm}\) on a complete rotation. For a particular observation the reading on the main scale is \(5\, {mm}\) and the \(20^{\text {th }}\) division of the circular scale coincides with reference line. Calculate the true reading. (in \({mm}\)) 2021 Hard

163. In a typical combustion engine the work done by a gas molecule is given \(W =\alpha^{2} \beta e ^{\frac{-\beta x ^{2}}{ KT }}\), where \(x\) is the displacement, \(k\) is the Boltzmann constant and \(T\) is the temperature. If \(\alpha\) and \(\beta\) are constants, dimensions of \(\alpha\) will be 2021 Hard

164. In order to determine the Young's Modulus of a wire of radius \(0.2\, cm\) (measured using a scale of least count \(=0.001\, cm )\) and length \(1 \,m\) (measured using a scale of least count \(=1\, mm\) ), a weight of mass \(1\, kg\) (measured using a scale of least count \(=1 \,g\) ) was hanged to get the elongation of \(0.5\, cm\) (measured using a scale of least count \(0.001\, cm\) ). What will be the fractional error in the value of Young's Modulus determined by this experiment? (in \(\%\)) 2021 Hard

165. Match List\(-I\) with List\(-II\)

List\(-I\)	List\(-II\)
\((a)\) \(h\) (Planck's constant)	\((i)\) \(\left[ M L T ^{-1}\right]\)
\((b)\) \(E\) (kinetic energy)	\((ii)\) \(\left[ M L ^{2} T ^{-1}\right]\)
\((c)\) \(V\) (electric potential)	\((iii)\) \(\left[ M L ^{2} T ^{-2}\right]\)
\((d)\) \(P\) (linear momentum)	\(( iv )\left[ M L ^{2} I ^{-1} T ^{-3}\right]\)

Choose the correct answer from the options given below 2021 Medium

166. Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Magnetic Induction	\((i)\) \({ML}^{2} {T}^{-2} {A}^{-1}\)
\((b)\) Magnetic Flux	\((ii)\) \({M}^{0} {L}^{-1} {A}\)
\((c)\) Magnetic Permeability	\((iii)\) \({MT}^{-2} {A}^{-1}\)
\((d)\) Magnetization	\((iv)\) \({MLT}^{-2} {A}^{-2}\)

Choose the most appropriate answer from the options given below: 2021 Medium

167. Match List\(-I\) with List\(-II\).

List\(-I\)	List\(-II\)
\((a)\) \({R}_{{H}}\) (Rydberg constant)	\((i)\) \({kg} {m}^{-1} {s}^{-1}\)
\((b)\) \(h\) (Planck's constant)	\((ii)\) \({kg} {m}^{2} {s}^{-1}\)
\((c)\) \(\mu_{{B}}\) (Magnetic field energy density)	\((iii)\) \({m}^{-1}\)
\((d)\) \(\eta\) (coefficient of viscocity)	\((iv)\) \({kg} {m}^{-1} {s}^{-2}\)

Choose the most appropriate answer from the options given below: 2021 Hard

168. Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Torque	\((i)\) \({MLT}^{-1}\)
\((b)\) Impulse	\((ii)\) \({MT}^{-2}\)
\((c)\) Tension	\((iii)\) \({ML}^{2} {T}^{-2}\)
\((d)\) Surface Tension	\((iv)\) \({MI} {T}^{-2}\)

Choose the most appropriate answer from the option given below : 2021 Medium

169. Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Capacitance, \(C\)	\((i)\) \({M}^{1} {L}^{1} {T}^{-3} {A}^{-1}\)
\((b)\) Permittivity of free space, \(\varepsilon_{0}\)	\((ii)\) \({M}^{-1} {L}^{-3} {T}^{4} {A}^{2}\)
\((c)\) Permeability of free space, \(\mu_{0}\)	\((iii)\) \({M}^{-1} L^{-2} T^{4} A^{2}\)
\((d)\) Electric field, \(E\)	\((iv)\) \({M}^{1} {L}^{1} {T}^{-2} {A}^{-2}\)

Choose the correct answer from the options given below 2021 Medium

170. One main scale division of a vernier callipers is \(a\) \(cm\) and \(n ^{\text {th }}\) division of the vernier scale coincide with \(( n -1)^{\text {th }}\) division of the main scale. The least count of the callipers in \(mm\) is 2021 Medium

171. Student \(A\) and Student \(B\) used two screw gauges of equal pitch and \(100\) equal circular divisions to measure the radius of a given wire. The actual value of the radius of the wire is \(0.322\, {cm}\). The absolute value of the difference between the final circular scale readings observed by the students \(A\) and \(B\) is .... . [Figure shows position of reference \('O'\) when jaws of screw gauge are closed] Given pitch \(=0.1 \,{cm}\).

2021 Hard

172. The diameter of a spherical bob is measured using a vernier callipers. \(9\) divisions of the main scale, in the vernier callipers, are equal to \(10\) divisions of vernier scale. One main scale division is \(1\, {mm}\). The main scale reading is \(10\, {mm}\) and \(8^{\text {th }}\) division of vernier scale was found to coincide exactly with one of the main scale division. If the given vernier callipers has positive zero error of \(0.04\, {cm}\), then the radius of the bob is \(...... \,\times 10^{-2} \,{cm}\) 2021 Hard

173. The entropy of any system is given by \({S}=\alpha^{2} \beta \ln \left[\frac{\mu {kR}}{J \beta^{2}}+3\right]\) Where \(\alpha\) and \(\beta\) are the constants. \(\mu, J, K\) and \(R\) are no. of moles, mechanical equivalent of heat, Boltzmann constant and gas constant repectively. [Take \({S}=\frac{{dQ}}{{T}}\) ] Choose the incorrect option from the following: 2021 Hard

174. The force is given in terms of time \(t\) and displacement \(x\) by the equation \({F}={A} \cos {Bx}+{C} \sin {Dt}\) The dimensional formula of \(\frac{{AD}}{{B}}\) is - 2021 Medium

175. The period of oscillation of a simple pendulum is \(T =2 \pi \sqrt{\frac{ L }{ g }} .\) Measured value of \( L \) is \(1.0\, m\) from meter scale having a minimum division of \(1 \,mm\) and time of one complete oscillation is \(1.95\, s\) measured from stopwatch of \(0.01 \,s\) resolution. The percentage error in the determination of \(g\) will be ..... \(\%.\) 2021 Medium

176. The pitch of the screw gauge is \(1\, mm\) and there are \(100\) divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies \(8\) divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while \(72^{\text {nd }}\) division on circular scale coincides with the reference line. The radius of the wire is.........\(mm\) 2021 Hard

177. The radius of a sphere is measured to be \((7.50 \pm 0.85) \,cm .\) Suppose the percentage error in its volume is \(x\). The value of \(x\), to the nearest integer is .....\(\%\) 2021 Medium

178. The resistance \(R =\frac{ V }{ I },\) where \(V =(50 \pm 2) \;V\) and \(I=(20 \pm 0.2)\;A.\) The percentage error in \(R\) is \(x\%\). The value of \(x\) to the nearest integer is ......... 2021 Medium

179. The time period of a simple pendulum is given by \(T =2 \pi \sqrt{\frac{\ell}{ g }}\). The measured value of the length of pendulum is \(10\, cm\) known to a \(1\, mm\) accuracy. The time for \(200\) oscillations of the pendulum is found to be \(100\) second using a clock of \(1s\) resolution. The percentage accuracy in the determination of \('g'\) using this pendulum is \('x'\). The value of \('x'\) to the nearest integer is ...........\(\%\) 2021 Hard

180. The vernier scale used for measurement has a positive zero error of \(0.2\, mm\). If while taking a measurement it was noted that \('0'\) on the vernier scale lies between \(8.5\, cm\) and \(8.6\, cm\) vernier coincidence is \(6,\) then the correct value of measurement is ............. \(cm\). (least count \(=0.01\, cm )\) 2021 Medium

181. The workdone by a gas molecule in an isolated system is given by, \(W =\alpha \beta^{2} e ^{-\frac{ x ^{2}}{\alpha kT }},\) where \(x\) is the displacement, \(k\) is the Boltzmann constant and \(T\) is the temperature, \(\alpha\) and \(\beta\) are constants. Then the dimension of \(\beta\) will be 2021 Hard

182. Three students \(S_{1}, S_{2}\) and \(S_{3}\) perform an experiment for determining the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

Student No.	Length of pendulum \((cm)\)	No. of osci \((n)\)	Total time for osci	Time period \((s)\)
\(1.\)	\(64.0\)	\(8\)	\(128.0\)	\(16.0\)
\(2.\)	\(64.0\)	\(4\)	\(64.0\)	\(16.0\)
\(3.\)	\(20.0\)	\(4\)	\(36.0\)	\(9.0\)