367293
Assertion : According to the principle of homogeneity of dimensions, only that formula is correct in which the dimensions of L.H.S. equal to dimensions of R.H.S. Reason : The time period of a pendulum is given by the formula, \(T=2 \pi \sqrt{g / l}\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Let us write the dimension of quantities on two sides of the given relation \(L . H . S=T=[T]\), \(R.H.S. = 2\pi \sqrt {g/l} = \sqrt {\frac{{\left[ {L{T^{ - 2}}} \right]}}{{[L]}}} = \left[ {{T^{ - 1}}} \right]\) ( \(\because 2 \pi\) has no dimensions). As dimensions of L.H.S. is not equal to dimensions of R.H.S Therefore according to principle of homogeneity of dimensions the relation \(T=2 \pi \sqrt{g / l}\) is not valid. So correct option is (3)
PHXI02:UNITS AND MEASUREMENTS
367294
Assertion : \(L / R\) and \(C R\) both have same unit. Reason : \(L / R\) and \(C R\) both have dimension of time.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Unit of quantity \((L / R)\) is henry ohm. As henry \(=\) ohm \(\times\) sec hence unit of \(L / R\) is sec i.e. \([L / R]=[T]\). Similarly, unit of product \(C R\) is farad \(\times\) ohm or, \(\dfrac{\text { coulomb }}{\text { volt }} \times \dfrac{\text { volt }}{\mathrm{amp}}=\dfrac{\sec \times \mathrm{amp}}{\mathrm{amp}}=\mathrm{sec}\) i.e. \([C R]=[T]\) therefore \([L / R]\) and \([C R]\) both have the same dimensions. So correct option is (1).
PHXI02:UNITS AND MEASUREMENTS
367295
The equation of stationary wave is \(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\). Which of the following is not correct?
1 The dimensions of \(n\) is \(\left[L T^{-1}\right]\)
2 The dimensions of \(x\) is \([L]\)
3 The dimensions of \(n t\) is \([L]\)
4 The dimensions of \(\dfrac{n}{\lambda}\) is \([T]\)
Explanation:
\(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\) As angle is dimensionless quantity.Therefore, Dimensions of \(n=\) Dimensions of \(\left(\dfrac{\lambda}{t}\right)\) \([n]=\dfrac{[L]}{[T]}=\left[L T^{-1}\right]\) Dimensions of \(x=\) Dimensions of \(\lambda\) \([x]=[L]\) Dimensions of \(n t=\) Dimensions of \(\lambda\) \([n t]=[L]\) Dimensions of \(\dfrac{n}{\lambda}=\left[T^{-1}\right]\) Option (4) is wrong.
JEE - 2024
PHXI02:UNITS AND MEASUREMENTS
367296
The velocity \(v\) (in \(cm{s^{ - 1}}\) ) of a particle is given in terms of time \(t\) (in second) by the equation \(v=a t+\dfrac{b}{t+c}\). The dimensions of \(a, b\) and \(c\) are
Given, \(v=a t+\dfrac{b}{t+c}\) or \([a t]=[v]=\left[L T^{-1}\right]\) \(\therefore[a]=\dfrac{\left[L T^{-1}\right]}{[T]}=\left[L T^{-2}\right]\) Dimension of \(c=[t]=[T]\) (we can add quantities of same dimensions only). \(\left[\dfrac{b}{t+c}\right]=[v]=\left[L T^{-1}\right]\) or \([b]=\left[L T^{-1}\right][T]=[L]\) So, correct option is (3) \(a=\left[L T^{-2}\right] b=[L] c=[T]\)
PHXI02:UNITS AND MEASUREMENTS
367297
The position of the particle moving along \(y\) axis is given as \(y=A t^{2}-B t^{3}\), where \(y\) is measured in metre and \(t\) in second. Then, the dimensions of \(B\) are
1 \(\left[L T^{-2}\right]\)
2 \(\left[L T^{-1}\right]\)
3 \(\left[L T^{-3}\right]\)
4 \(\left[M L T^{-2}\right]\)
Explanation:
As, \(y=B\left[T^{3}\right]\) So, \([L] = B\left[ {{T^3}} \right]\) \( \Rightarrow B = \left[ {L{T^{ - 3}}} \right]\)
367293
Assertion : According to the principle of homogeneity of dimensions, only that formula is correct in which the dimensions of L.H.S. equal to dimensions of R.H.S. Reason : The time period of a pendulum is given by the formula, \(T=2 \pi \sqrt{g / l}\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Let us write the dimension of quantities on two sides of the given relation \(L . H . S=T=[T]\), \(R.H.S. = 2\pi \sqrt {g/l} = \sqrt {\frac{{\left[ {L{T^{ - 2}}} \right]}}{{[L]}}} = \left[ {{T^{ - 1}}} \right]\) ( \(\because 2 \pi\) has no dimensions). As dimensions of L.H.S. is not equal to dimensions of R.H.S Therefore according to principle of homogeneity of dimensions the relation \(T=2 \pi \sqrt{g / l}\) is not valid. So correct option is (3)
PHXI02:UNITS AND MEASUREMENTS
367294
Assertion : \(L / R\) and \(C R\) both have same unit. Reason : \(L / R\) and \(C R\) both have dimension of time.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Unit of quantity \((L / R)\) is henry ohm. As henry \(=\) ohm \(\times\) sec hence unit of \(L / R\) is sec i.e. \([L / R]=[T]\). Similarly, unit of product \(C R\) is farad \(\times\) ohm or, \(\dfrac{\text { coulomb }}{\text { volt }} \times \dfrac{\text { volt }}{\mathrm{amp}}=\dfrac{\sec \times \mathrm{amp}}{\mathrm{amp}}=\mathrm{sec}\) i.e. \([C R]=[T]\) therefore \([L / R]\) and \([C R]\) both have the same dimensions. So correct option is (1).
PHXI02:UNITS AND MEASUREMENTS
367295
The equation of stationary wave is \(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\). Which of the following is not correct?
1 The dimensions of \(n\) is \(\left[L T^{-1}\right]\)
2 The dimensions of \(x\) is \([L]\)
3 The dimensions of \(n t\) is \([L]\)
4 The dimensions of \(\dfrac{n}{\lambda}\) is \([T]\)
Explanation:
\(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\) As angle is dimensionless quantity.Therefore, Dimensions of \(n=\) Dimensions of \(\left(\dfrac{\lambda}{t}\right)\) \([n]=\dfrac{[L]}{[T]}=\left[L T^{-1}\right]\) Dimensions of \(x=\) Dimensions of \(\lambda\) \([x]=[L]\) Dimensions of \(n t=\) Dimensions of \(\lambda\) \([n t]=[L]\) Dimensions of \(\dfrac{n}{\lambda}=\left[T^{-1}\right]\) Option (4) is wrong.
JEE - 2024
PHXI02:UNITS AND MEASUREMENTS
367296
The velocity \(v\) (in \(cm{s^{ - 1}}\) ) of a particle is given in terms of time \(t\) (in second) by the equation \(v=a t+\dfrac{b}{t+c}\). The dimensions of \(a, b\) and \(c\) are
Given, \(v=a t+\dfrac{b}{t+c}\) or \([a t]=[v]=\left[L T^{-1}\right]\) \(\therefore[a]=\dfrac{\left[L T^{-1}\right]}{[T]}=\left[L T^{-2}\right]\) Dimension of \(c=[t]=[T]\) (we can add quantities of same dimensions only). \(\left[\dfrac{b}{t+c}\right]=[v]=\left[L T^{-1}\right]\) or \([b]=\left[L T^{-1}\right][T]=[L]\) So, correct option is (3) \(a=\left[L T^{-2}\right] b=[L] c=[T]\)
PHXI02:UNITS AND MEASUREMENTS
367297
The position of the particle moving along \(y\) axis is given as \(y=A t^{2}-B t^{3}\), where \(y\) is measured in metre and \(t\) in second. Then, the dimensions of \(B\) are
1 \(\left[L T^{-2}\right]\)
2 \(\left[L T^{-1}\right]\)
3 \(\left[L T^{-3}\right]\)
4 \(\left[M L T^{-2}\right]\)
Explanation:
As, \(y=B\left[T^{3}\right]\) So, \([L] = B\left[ {{T^3}} \right]\) \( \Rightarrow B = \left[ {L{T^{ - 3}}} \right]\)
367293
Assertion : According to the principle of homogeneity of dimensions, only that formula is correct in which the dimensions of L.H.S. equal to dimensions of R.H.S. Reason : The time period of a pendulum is given by the formula, \(T=2 \pi \sqrt{g / l}\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Let us write the dimension of quantities on two sides of the given relation \(L . H . S=T=[T]\), \(R.H.S. = 2\pi \sqrt {g/l} = \sqrt {\frac{{\left[ {L{T^{ - 2}}} \right]}}{{[L]}}} = \left[ {{T^{ - 1}}} \right]\) ( \(\because 2 \pi\) has no dimensions). As dimensions of L.H.S. is not equal to dimensions of R.H.S Therefore according to principle of homogeneity of dimensions the relation \(T=2 \pi \sqrt{g / l}\) is not valid. So correct option is (3)
PHXI02:UNITS AND MEASUREMENTS
367294
Assertion : \(L / R\) and \(C R\) both have same unit. Reason : \(L / R\) and \(C R\) both have dimension of time.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Unit of quantity \((L / R)\) is henry ohm. As henry \(=\) ohm \(\times\) sec hence unit of \(L / R\) is sec i.e. \([L / R]=[T]\). Similarly, unit of product \(C R\) is farad \(\times\) ohm or, \(\dfrac{\text { coulomb }}{\text { volt }} \times \dfrac{\text { volt }}{\mathrm{amp}}=\dfrac{\sec \times \mathrm{amp}}{\mathrm{amp}}=\mathrm{sec}\) i.e. \([C R]=[T]\) therefore \([L / R]\) and \([C R]\) both have the same dimensions. So correct option is (1).
PHXI02:UNITS AND MEASUREMENTS
367295
The equation of stationary wave is \(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\). Which of the following is not correct?
1 The dimensions of \(n\) is \(\left[L T^{-1}\right]\)
2 The dimensions of \(x\) is \([L]\)
3 The dimensions of \(n t\) is \([L]\)
4 The dimensions of \(\dfrac{n}{\lambda}\) is \([T]\)
Explanation:
\(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\) As angle is dimensionless quantity.Therefore, Dimensions of \(n=\) Dimensions of \(\left(\dfrac{\lambda}{t}\right)\) \([n]=\dfrac{[L]}{[T]}=\left[L T^{-1}\right]\) Dimensions of \(x=\) Dimensions of \(\lambda\) \([x]=[L]\) Dimensions of \(n t=\) Dimensions of \(\lambda\) \([n t]=[L]\) Dimensions of \(\dfrac{n}{\lambda}=\left[T^{-1}\right]\) Option (4) is wrong.
JEE - 2024
PHXI02:UNITS AND MEASUREMENTS
367296
The velocity \(v\) (in \(cm{s^{ - 1}}\) ) of a particle is given in terms of time \(t\) (in second) by the equation \(v=a t+\dfrac{b}{t+c}\). The dimensions of \(a, b\) and \(c\) are
Given, \(v=a t+\dfrac{b}{t+c}\) or \([a t]=[v]=\left[L T^{-1}\right]\) \(\therefore[a]=\dfrac{\left[L T^{-1}\right]}{[T]}=\left[L T^{-2}\right]\) Dimension of \(c=[t]=[T]\) (we can add quantities of same dimensions only). \(\left[\dfrac{b}{t+c}\right]=[v]=\left[L T^{-1}\right]\) or \([b]=\left[L T^{-1}\right][T]=[L]\) So, correct option is (3) \(a=\left[L T^{-2}\right] b=[L] c=[T]\)
PHXI02:UNITS AND MEASUREMENTS
367297
The position of the particle moving along \(y\) axis is given as \(y=A t^{2}-B t^{3}\), where \(y\) is measured in metre and \(t\) in second. Then, the dimensions of \(B\) are
1 \(\left[L T^{-2}\right]\)
2 \(\left[L T^{-1}\right]\)
3 \(\left[L T^{-3}\right]\)
4 \(\left[M L T^{-2}\right]\)
Explanation:
As, \(y=B\left[T^{3}\right]\) So, \([L] = B\left[ {{T^3}} \right]\) \( \Rightarrow B = \left[ {L{T^{ - 3}}} \right]\)
367293
Assertion : According to the principle of homogeneity of dimensions, only that formula is correct in which the dimensions of L.H.S. equal to dimensions of R.H.S. Reason : The time period of a pendulum is given by the formula, \(T=2 \pi \sqrt{g / l}\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Let us write the dimension of quantities on two sides of the given relation \(L . H . S=T=[T]\), \(R.H.S. = 2\pi \sqrt {g/l} = \sqrt {\frac{{\left[ {L{T^{ - 2}}} \right]}}{{[L]}}} = \left[ {{T^{ - 1}}} \right]\) ( \(\because 2 \pi\) has no dimensions). As dimensions of L.H.S. is not equal to dimensions of R.H.S Therefore according to principle of homogeneity of dimensions the relation \(T=2 \pi \sqrt{g / l}\) is not valid. So correct option is (3)
PHXI02:UNITS AND MEASUREMENTS
367294
Assertion : \(L / R\) and \(C R\) both have same unit. Reason : \(L / R\) and \(C R\) both have dimension of time.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Unit of quantity \((L / R)\) is henry ohm. As henry \(=\) ohm \(\times\) sec hence unit of \(L / R\) is sec i.e. \([L / R]=[T]\). Similarly, unit of product \(C R\) is farad \(\times\) ohm or, \(\dfrac{\text { coulomb }}{\text { volt }} \times \dfrac{\text { volt }}{\mathrm{amp}}=\dfrac{\sec \times \mathrm{amp}}{\mathrm{amp}}=\mathrm{sec}\) i.e. \([C R]=[T]\) therefore \([L / R]\) and \([C R]\) both have the same dimensions. So correct option is (1).
PHXI02:UNITS AND MEASUREMENTS
367295
The equation of stationary wave is \(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\). Which of the following is not correct?
1 The dimensions of \(n\) is \(\left[L T^{-1}\right]\)
2 The dimensions of \(x\) is \([L]\)
3 The dimensions of \(n t\) is \([L]\)
4 The dimensions of \(\dfrac{n}{\lambda}\) is \([T]\)
Explanation:
\(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\) As angle is dimensionless quantity.Therefore, Dimensions of \(n=\) Dimensions of \(\left(\dfrac{\lambda}{t}\right)\) \([n]=\dfrac{[L]}{[T]}=\left[L T^{-1}\right]\) Dimensions of \(x=\) Dimensions of \(\lambda\) \([x]=[L]\) Dimensions of \(n t=\) Dimensions of \(\lambda\) \([n t]=[L]\) Dimensions of \(\dfrac{n}{\lambda}=\left[T^{-1}\right]\) Option (4) is wrong.
JEE - 2024
PHXI02:UNITS AND MEASUREMENTS
367296
The velocity \(v\) (in \(cm{s^{ - 1}}\) ) of a particle is given in terms of time \(t\) (in second) by the equation \(v=a t+\dfrac{b}{t+c}\). The dimensions of \(a, b\) and \(c\) are
Given, \(v=a t+\dfrac{b}{t+c}\) or \([a t]=[v]=\left[L T^{-1}\right]\) \(\therefore[a]=\dfrac{\left[L T^{-1}\right]}{[T]}=\left[L T^{-2}\right]\) Dimension of \(c=[t]=[T]\) (we can add quantities of same dimensions only). \(\left[\dfrac{b}{t+c}\right]=[v]=\left[L T^{-1}\right]\) or \([b]=\left[L T^{-1}\right][T]=[L]\) So, correct option is (3) \(a=\left[L T^{-2}\right] b=[L] c=[T]\)
PHXI02:UNITS AND MEASUREMENTS
367297
The position of the particle moving along \(y\) axis is given as \(y=A t^{2}-B t^{3}\), where \(y\) is measured in metre and \(t\) in second. Then, the dimensions of \(B\) are
1 \(\left[L T^{-2}\right]\)
2 \(\left[L T^{-1}\right]\)
3 \(\left[L T^{-3}\right]\)
4 \(\left[M L T^{-2}\right]\)
Explanation:
As, \(y=B\left[T^{3}\right]\) So, \([L] = B\left[ {{T^3}} \right]\) \( \Rightarrow B = \left[ {L{T^{ - 3}}} \right]\)
367293
Assertion : According to the principle of homogeneity of dimensions, only that formula is correct in which the dimensions of L.H.S. equal to dimensions of R.H.S. Reason : The time period of a pendulum is given by the formula, \(T=2 \pi \sqrt{g / l}\).
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Let us write the dimension of quantities on two sides of the given relation \(L . H . S=T=[T]\), \(R.H.S. = 2\pi \sqrt {g/l} = \sqrt {\frac{{\left[ {L{T^{ - 2}}} \right]}}{{[L]}}} = \left[ {{T^{ - 1}}} \right]\) ( \(\because 2 \pi\) has no dimensions). As dimensions of L.H.S. is not equal to dimensions of R.H.S Therefore according to principle of homogeneity of dimensions the relation \(T=2 \pi \sqrt{g / l}\) is not valid. So correct option is (3)
PHXI02:UNITS AND MEASUREMENTS
367294
Assertion : \(L / R\) and \(C R\) both have same unit. Reason : \(L / R\) and \(C R\) both have dimension of time.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Unit of quantity \((L / R)\) is henry ohm. As henry \(=\) ohm \(\times\) sec hence unit of \(L / R\) is sec i.e. \([L / R]=[T]\). Similarly, unit of product \(C R\) is farad \(\times\) ohm or, \(\dfrac{\text { coulomb }}{\text { volt }} \times \dfrac{\text { volt }}{\mathrm{amp}}=\dfrac{\sec \times \mathrm{amp}}{\mathrm{amp}}=\mathrm{sec}\) i.e. \([C R]=[T]\) therefore \([L / R]\) and \([C R]\) both have the same dimensions. So correct option is (1).
PHXI02:UNITS AND MEASUREMENTS
367295
The equation of stationary wave is \(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\). Which of the following is not correct?
1 The dimensions of \(n\) is \(\left[L T^{-1}\right]\)
2 The dimensions of \(x\) is \([L]\)
3 The dimensions of \(n t\) is \([L]\)
4 The dimensions of \(\dfrac{n}{\lambda}\) is \([T]\)
Explanation:
\(y=2 a \sin \left(\dfrac{2 \pi n t}{\lambda}\right) \cos \left(\dfrac{2 \pi x}{\lambda}\right)\) As angle is dimensionless quantity.Therefore, Dimensions of \(n=\) Dimensions of \(\left(\dfrac{\lambda}{t}\right)\) \([n]=\dfrac{[L]}{[T]}=\left[L T^{-1}\right]\) Dimensions of \(x=\) Dimensions of \(\lambda\) \([x]=[L]\) Dimensions of \(n t=\) Dimensions of \(\lambda\) \([n t]=[L]\) Dimensions of \(\dfrac{n}{\lambda}=\left[T^{-1}\right]\) Option (4) is wrong.
JEE - 2024
PHXI02:UNITS AND MEASUREMENTS
367296
The velocity \(v\) (in \(cm{s^{ - 1}}\) ) of a particle is given in terms of time \(t\) (in second) by the equation \(v=a t+\dfrac{b}{t+c}\). The dimensions of \(a, b\) and \(c\) are
Given, \(v=a t+\dfrac{b}{t+c}\) or \([a t]=[v]=\left[L T^{-1}\right]\) \(\therefore[a]=\dfrac{\left[L T^{-1}\right]}{[T]}=\left[L T^{-2}\right]\) Dimension of \(c=[t]=[T]\) (we can add quantities of same dimensions only). \(\left[\dfrac{b}{t+c}\right]=[v]=\left[L T^{-1}\right]\) or \([b]=\left[L T^{-1}\right][T]=[L]\) So, correct option is (3) \(a=\left[L T^{-2}\right] b=[L] c=[T]\)
PHXI02:UNITS AND MEASUREMENTS
367297
The position of the particle moving along \(y\) axis is given as \(y=A t^{2}-B t^{3}\), where \(y\) is measured in metre and \(t\) in second. Then, the dimensions of \(B\) are
1 \(\left[L T^{-2}\right]\)
2 \(\left[L T^{-1}\right]\)
3 \(\left[L T^{-3}\right]\)
4 \(\left[M L T^{-2}\right]\)
Explanation:
As, \(y=B\left[T^{3}\right]\) So, \([L] = B\left[ {{T^3}} \right]\) \( \Rightarrow B = \left[ {L{T^{ - 3}}} \right]\)
