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(b) The sum of these numbers is 663.821. The number with least decimal places is 227.2 is correct to only one decimal places and the closest number to it 664.

This is a multiple choice answer as classified in NCERT Exemplar

(b) The zero after points are not significant. The number can be written as 0.069, Hence number of significant figure are four.

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t₁=39.6s, t₂=39.9s and t₃=39.5s

Least count of measuring instrument =0.1s

Precision in the measurement =least count of the measuring instrument =0.1s

So mean value of 20 observation

Mean value of t= $\frac{t_1+t_2+t_3}{3}=\frac{39.6+39.9+39.5}{3}=39.7s$

Absolute error in

 t₁= t-t₁ = 39.7-39.6 =0.1s

In t₂= t-t₂ = 39.7-39.9=-0.2s

In t₃= t-t₁ = 39.7-39.5= 0.2s

Mean error = $\frac{0.1+0.2+0.2}{3}=0.17s\approx 0.2$

Accuracy in the measurement = $?0.2s$

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For dimensional consistency LHS = RHS

[L]= [A]=L

As wt-kx should be dimensionless [wt]= [kx]=1

w [T]=k [L]=1

So w= [T]^-1 and k= [L]^-1

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The formula for solid angle is

 solid angle = area/ (distance)²

= $\frac{1{cm}^2}{5{cm}^2}=\frac{1}{25}=4\times {10}^{-2}steridian$

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As we know, $\theta =\frac{l}{r}radian$

$\theta =\frac{\pi }{6}=\frac{l}{31}cm$

L=31 $\times \frac{\pi }{6}cm=\frac{31\times 3.14}{6}cm=16.22cm$

Rounding off to three significant figures it would be 16.2cm

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This is a short answer type question as classified in NCERT Exemplar

(a) Plane angle = l/r radian its si unit is radian but no dimension formula.

(b) Strain= $\frac{?L}{L}=\frac{changeinlength}{length}$ so no dimension

And g =9.8m/s²

(c) Gravitational constant G= 6.67 $\times {10}^{-11}N{m}^2/{kg}^2$

(d) Reynold’s number is a number having no dimension.

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F= [MLT^-2]=100N…….1

Length L= 10m………….2

Time t= 100s……….3

Substituting the value of L and T  in eqn 1,2,3

M $\times 10\times {100}^{-2}=100$

$\frac{M\times 10}{100\times 100}=100$

M=100 $\times 1000kg$

M= 10⁵kg

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R_me= distance of moon from earth

R_se=distance of sun from earth

Let angle made by sun and moon is .

$\theta =\frac{A_{sun}}{R_{se}^2}=\frac{A_{moon}}{R_{me}^2}$

$\theta =\frac{\pi R_s^2}{R_{se}^2}=\frac{\pi R_m^2}{R_{me}^2}$

( $\frac{R_s}{R_{se}}$ )2 = ( $\frac{R_m}{R_{me}}$ )2

( $\frac{R_s}{R_m}$ )= ( $\frac{R_{se}}{R_{me}}$ )

Hence radius of sun and moon represents their sizes respectively.

This is a short answer type question as classified in NCERT Exemplar

It is given that 50 VSD $\approx 49\mathrm{M}\mathrm{S}\mathrm{D}$

1MSD = 50/49VSD

1VSD= 49/50MSD

Minimum inaccuracy = 1MSD-1VSD= 1MSD-49/50MSD=1/50 MSD

1MSD= 0.5mm

Minimum inaccuracy = 1/50 $\times$ 0.5mm=0.01mm

This is a short answer type question as classified in NCERT Exemplar

Distance of galaxy = 10²⁵m

Speed of light = 3 $\times$ 10⁸ m/s

T = distance /speed = $\frac{{10}^{25}}{3\times {10}^8}\approx \frac{1}{3}\times {10}^{17}$

$=\frac{10}{3}\times {10}^{16}s$

$3.33\times {10}^{16}$ s

This is a short answer type question as classified in NCERT Exemplar

(a) A wall clock can measure time correctly upto one second .

(b) A stop watch can measure time correctly upto a fraction of a second .

(c) A digital watch upto the same precision

(d) An atomic clock measure more accurately.

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(a)

 l=R_E; r= 60R_E

$\theta =\frac{R_E}{60R_E}=\frac{1}{60}rad=\frac{1}{60}\times \frac{180}{\pi }degree=\frac{3}{\pi }\approx 1^o$

Hence angle subtend is $2^o$

(b) But in 2^nd case it is given that moon is (½)° and earth is 2^o so it become 4 times.

$\frac{diameterofearth}{diameterofmoon}=\frac{2/\pi }{1/2\pi }=4$

(c) To measure parallax sun is about 400 times the earth moon distance hence $\frac{r_{sun}}{r_{moon}}=400$

To finding ratio of

$\frac{D_{sun}}{r_{sun}}=\mathrm{}\frac{D_{sun}}{r_{moon}}$

$\frac{D_{sun}}{D_{moon}}=400$

$\frac{D_{earth}}{D_{moon}}=4$

so $\frac{D_{sun}}{D_{earth}}=100$

so $\frac{D_{sun}}{D_{earth}}=100$

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Because they cannot be derived from one another as they are independent quantities.

Also we can say that all other quantities in mechanics can be expressed in terms of length mass and time.

This is a short answer type question as classified in NCERT Exemplar

Since f ( $\theta )$ is a sum of different powers of $\theta$  an we know that it is a dimensionless quantity. By principle of homogeneity as RHS is dimensionless. Hence LHS should also be dimensionless.