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Given Form Kepler;s III law, 

T∝ r ^{\(\frac{3}{2}\)}

T is also function of g and R

⇒ T∝r^{\(\frac{3}{2}\)}R^yg^x= kr^{\(\frac{3}{2}\)}R^yg^x

[k = dimension less constant of proportionality]

Substituting dimensionless in each term-

∴ [L⁰M⁰T^-1] = k[L^3/2M⁰T⁰][L¹M⁰T^-2]^x [L]^y 

= k[L^x+y+3/2T^-2x]

For L, 0 = \(\frac{3}{2}\)+x+y

For T, 1 = 0 – 2x 

⇒ x = \(-\frac{1}{2}\)

Therefore, 0 = \(\frac{3}{2}\) \(-\frac{1}{2}\)+ y

⇒ y = -1

Thus,

T = kr^3/2g^−1/2 R^-1

^{=\(\frac{k}{R}\sqrt\frac{r^3}{g}\)}

Measurement means comparison of a physical quantity with its unit to find out how many times the unit is contained in the given physical quantity. Measurement = Numerical value × unit

Correct option is: (D) parallax method

The term measurement means the comparison of a quantity with its unit to find out how many times the unit is contained in the given physical quantity.

(a) Mass: The mass of a body is the quantity of matter contained in it. Its SI unit is kg.

(b) Weight: The weight of a body is the force with which a body is attracted towards the center of the earth. W = mg. 

1. The state in which the momentum of a system is constant in the absence of an external unbalanced force is called mechanical equilibrium. 

2. A particle is said to be in mechanical equilibrium, if no net force is acting upon it. 

3. In case of a system of bodies to be in mechanical equilibrium, the net force acting on any part of the system should be zero i.e., the velocity or linear momentum of all parts of the system must be constant or zero. There should be no acceleration in any part of the system.

4. Mathematically, for a system in mechanical equilibrium, ∑\(\vec{F} = 0.\)

5. In case of rotating fan, if linear velocity is zero, then the linear momentum is zero. That means there is no net force acting on the fan. Hence, the fan is in mechanical equilibrium.

Commensalism is species interaction in which one species (the commensal) is benefitted whereas the host is neither benefitted nor affected.

Example: Sucker fish on Shark.

1. Amensalism 

2. Mutualism

1. Random error (accidental):

The errors which are caused due to minute change in experimental conditions like temperature, pressure change in gas or fluctuation in voltage, while the experiment is being performed are called random errors.

2. They can be positive or negative.

3. Random error cannot be eliminated completely but can be minimized by taking multiple observations and calculating their mean.

We have,

θ = 1^o30’ = (60’ + 30’) = 90’

= (90 × 60)’’

= 5400”

= (5400)(4.85 × 10^-6 ) rad [ ∵ 1’’ = 4.85 × 10^-6 rad]

and b = 1.3 × 10⁷ m.

using, \(D= \frac{b}{\theta}\) , we get

\(D = \frac{1.3\times10^7}{(5400(4.85\times10^{-6}))}\)

\(= \frac{1.3\times10^{11}}{54\times4.85}\)

= 4.96 × 10⁸ m

(a) Precision is given by the least count of the instrument.

For 20 oscillations, precision = 0.1 s

For 1 oscillation, precision = 0.005 s.

Average time for 20 oscillations

\(t=\frac{39.6+39.9+39.5}{3}s=39.7sec\)

Absolute errors, 

∆t₁ = t − t₁

= 39.7 – 39.6

= 0.1 s.

∆t₂ = t − t₂

= 39.7 – 39.9 = −0.2s

∆t₃ = t − t₃

= 39.7 – 39.5 

= 0.2s

Mean absolute error,

\(=\frac{|Δt_1|+|Δt_2|+|Δt_3|}{3}\)

\(=\frac{0.1+0.2+0.3}{3}\)=\(\frac{0.5}{3}\)

= 0.17

≈ ±0.2s.

(a) Precision is given by the least count of the instrument.

For 20 oscillations, precision = 0.1 s

For 1 oscillation, precision = 0.005 s.

(b) Average time t = (39.6 + 39.9 + 39.5/3)s = 39.6s

period = 39.6/20 = 1.98 s

Max. observed error = (1.995 - 1.980)s = 0.015s

a = a_mean ± ∆ a_mean signifies that the actual value of a lies between (a_mean – ∆ a_mean) and (amean + ∆ a_mean).

1. Groups of species with comparable role and niche dimensions within a community are termed ‘guilds’. 

2. Species that occupy the same niche in different geographical regions, are termed ‘ecological equivalents’

(d) Commensalism

Methods to minimise effect of systematic errors:

1. By using correct instrument. 

2. Following proper experimental procedure. 

3. Removing personal error.

1. Systematic errors are errors that are not determined by chance but are introduced by an inaccuracy (involving either the observation or measurement process) inherent to the system.

2. Classification of errors:

Errors are classified into following two groups:

3. Systematic errors:

• Instrumental error (constant error),
• Error due to imperfection in experimental technique,
• Personal error (human error).

4. Random error (accidental error)

Yes because the least accurate measurement, i.e., least count of a screw gauge.

= \(\frac{Pitch}{number\, of\, division\, on\, circular\, scale}\)

∴ Least count is inversely proportional to number of division on the circular scale.

So, with the increase in number of division on the circular scale, the least count will improve. In simple words, accuracy of screw gauge will increase.

1. Centre of mass of an object allows us to apply Newton’s laws of motion to finite objects (objects of measurable size) by considering these objects as point objects. 

2. For objects in non-uniform gravitational field or whose size is comparable to that of the Earth (size at least few thousand km), the position of centre of mass will differ than that of centre of gravity.

Light influences growth, pigmentation, migration and reproduction. The intensity and frequency of light influences metabolic activity, induce gene mutations (UV, X-rays). Light is essential for vision. This is proved by the poorly developed or absence of eyes in cave dwelling organisms. Diapause is also influenced by light in animals. Gonads of birds become more active with increasing light in summer. Light influences the locomotion and movement of lower animals.

(a) 1

(b) 3

(c) 4

(d) 4

(e) 4

(f) 4

(a) Extinction

(b) Commensalism

Some populations, as in a population of small mammals, increase slowly at first then more rapidly and gradually slow down as environmental resistance increases whereby equilibrium is reached and maintained. Their growth is represented by S shaped growth curve.

Gangetic Dolphin.

Derived quantities may get errors due to individual errors of fundamental quantities, such type of errors are called as combined errors.

Animals are known to modify their response to environmental changes (stress) in reasonably short time spans. This is known as Acclimatization. This is observed when people who have moved from the plains to higher altitudes show enhanced RBC count within a few days of settling in their new habitat. This helps them cope with lower atmospheric oxygen and higher oxygen demand.

Given area of the house in photograph = 1.75 cm².

Area of the house on screen = 1.55 m² = 1.55 x 10⁴ cm².

Magnification in area of photograph

= (1.55 x 10⁴/1.75) 

= 8.857 x 10³ cm²

Linear magnification 

= √(areal magnification)

= √(8.587 x 10³)

= 94.1 cm².

(D) \(\frac{1}{n^2}\)

1 V.S.D. = \(\frac{(n-1)}{n}\) M.S.D.

LC. = 1 M.S.D. – 1 V.S.D.

= 1 M.S.D. – \(\frac{(n-1)}{n}\) M.S.D.

= \(\frac{1}{n}\) M.S.D.

= \(\frac{1}{n}\times\frac{1}{n}\) cm

∴ L.C. = \(\frac{1}{n^2}\) cm

Yes, the least count of the screw gauge is given by

= Pitch/No.of dimensions on circular scale

i.e., least count of screw gauge is inversely proportional to the number of divisions on circular scale. So, with the increase in number of the divisions on the circular scale, the least count will improve. Thus, the accuracy of screw gauge will increase. 

Area of the house = 1.75 cm²

Let’s assume that the house is square in shape.

Length of a side of the house = √175 cm

Area of the house on the screen 1.55 m²

Length of a side of the house on the screen

= √1.55 m = √1.55 × 100 cm

Linear magnification = length on the screen/length on the side

= ((√1.55 x 100 cm)/√1.75)

= 94.1

Exponential growth shows J-shaped growth patterns.

(c) Catadromous

(b) Precipitation decreases

Van’t Hoff’s rule states that with the increase of every 10°C, the rate of metabolic activity is doubled or the reaction rate is halved with the decrease of 10°C.

In some aquatic environments, an inverse relationship between water temperature and fish meristic characters is observed – lower the temperature, more the vertebrae.

1. Extremely cold climate 

2. Low biotic diversity 

3. Simple vegetation structure 

4. Limitation of drainage 

5. Short season of growth and reproduction 

6. Energy and nutrients in the form of dead organic material 

7. Large population oscillations

Regions with great amount of biodiversity are called ‘Biodiversity Hotspots’.

Correct answer is (c) 34

The importance of biodiversity can be in three forms, viz. economic value, medicinal value and environmental value:

1. Economic Value: Biodiversity has immense economic value. We get food, fuel, fodder, timber, industrial raw materials, etc. directly from biodiversity. We are able to get different types of food, like rice, wheat, pulses, vegetables, fruits, etc. because of biodiversity.

2. Medicinal Value: Herbs are being used as cure for many diseases since ancient times. As per one estimate, about 40% of the medicines are obtained from plants. Many incurable diseases have appeared on earth from times to time. Cure for such disease has been searched in the biodiversity. For example, malaria was considered as incurable but bark of cinchona gave its cure.

3. Environmental Value: Biodiversity has immense environmental value, as it helps in conservation of food chain, nutrient cycle, etc.

Random errors can be decreased by having repeated observations. The mean value of these observation will be close to the actual or true value of the measured quantity.

There are three levels of biodiversity, viz. species diversity, genetic diversity and ecosystem diversity:

(1) Species Diversity: The total number of species of plants and animals in a given region is called species diversity of that region. Species diversity is considered as the most common meaning of biodiversity. This is taken as a unit of measurement for balance of any ecosystem. Population of microorganisms is impossible to count because they can be in millions or even billions in numbers. Only 10 gm of soil may contain a billion bacteria and more than 50,000 fungi. Hence, number of microorganisms is not counted while defining the biodiversity at a place.

(2) Genetic Diversity: The variation among members of a single species because of genes is called genetic diversity. This type of diversity is seen in different population groups of a species or even among different members of a population. For example, different varieties of rice show genetic diversity. Similarly, differences in height, skin colour, facial features, hair colour, etc. among human beings show genetic diversity. A higher level of genetic diversity among members of a species ensures that the species is at lower risk of becoming extinct; because genetic diversity increases the ability to adapt to the environment. Genetic diversity is responsible for giving birth to a new variety of a species.

(3) Ecosystem Diversity: The system of interaction among different creatures and between living beings and abiotic components in a region is called ecosystem. There are many types of ecosystem on earth, e.g. grasslands, mountains, desert, marshes, ocean, river valley, tropical forest, etc. Each ecosystem has its unique geographical and environmental features. Such features are responsible for diversity among creatures in a particular region. This type of diversity is called ecosystem diversity. In other words, the differences in geographical and environmental conditions in different ecosystems is are called ecosystem diversity.

The measured (nominal) volume of the block is,

Therefore,

V = l x w x h

= (1.37 × 4.11 × 2.56) cm³ .

= 14.41 cm³

However, each of these measurement has an uncertainty of + 0.01 cm, the least count of the Vernier caliper. We can say that the values of length, width and height should be written as

l = (1.37 cm ± 0.01 cm)

w = (4.11 cm ± 0.01 cm)

h = ( 2.56 cm ± 0.01 cm)

We thus find that the lower limit, of the volume of the block, is given

V_min= 1.36 cm × 4.10 cm × 2.55 cm

= 14.22 cm²

This is 0.19 cm³ lower than the (nominal) measured value.

The upper limit can also be calculated:

V_min = 1.38 cm × 4.12 cm × 2.57 cm

= 14.61 cm³

This is 0.20 cm3 higher than the measured value. As a practical rule, we choose the higher of these two deviations (from the measured value) as the uncertainty, in our result. We, therefore, should report the volume of the block as (14.41 cm³ ± 0.20 cm³ ).

When more measurements are taken, the likelihood of random errors is decreased. Hence a set of 100 measurements is more reliable than a set of 5 measurements.

Increasing the number of observations, increases the reliability as the mean error is also reduced. The best possible value is.

a_mean = (a₁ + a₂ +a₃ +........ +a_n)/n

This will give a better value, if n is large.

Niche of an organism can be defined as the total position and function of an individual in its environment.

Correct Answer is: (b) Niche

(a) Species → Population → Community → biome

Allen’s rule: Warm blooded animals living in cold climates have shorter limbs, ears and other appendages than the same species in warm climates.

Beegmann’s rule: Birds and mammals have larger body size in cold regions than warmer regions.