If in a population of size ”N” the birth rate is represented as “b” and the death rate as “d”, the increase or decrease in “N” during a unit time period ”t” will be: The equation given above can also be represented as: What does ”r” represent. Write any one significance of calculating ”r” for any population?

r here represents the intrinsic rate of increase of population and it is equal to the difference in the natality (birth rate) and mortality (death rate). The 'r' is called the 'intrinsic rate of natural increase. It' is a very important parameter chosen for assessing impacts of any biotic or abiotic factor on population growth. If the value of 'r' is increasing it signifies the growth of the population and if it is going down then the population growth has slowed down.

The probability of finding a green signal on a busy crossing X is 30 % . What is the probability of finding a green signal on X on two consecutive days out of three?

Let A be the event of finding a green signal on a busy crossing X The probability of finding a green signal on a busy crossing X is 30 % . P A = 30 100 = 3 10 The probability of not finding a green signal on a busy crossing X is P A ¯ = 1 - 3 10 = 7 10 . The probability of finding a green signal on X on two consecutive days out of three = P A P A P A ¯ + P A ¯ P A P A = 3 10 × 3 10 × 7 10 + 7 10 × 3 10 × 3 10 = 63 500 .

Find g o f and f o g when f : R → R and g : R → R is defined by f ( x ) = x 2 + 2 x + 3 and g ( x ) = 3 x − 4 .

Given f ( x ) = x 2 + 2 x + 3 and g ( x ) = 3 x - 4 Therefore, f o g x = f g x = f 3 x - 4 = 3 x - 4 2 + 2 3 x - 4 + 3 = 9 x 2 - 24 x + 16 + 6 x - 8 + 3 = 9 x 2 - 18 x + 11 And g ∘ f x = g f x = g x 2 + 2 x + 3 = 3 x 2 + 2 x + 3 - 4 = 3 x 2 + 6 x + 9 - 4 = 3 x 2 + 6 x + 5

Let f : R → R and g : R → R be defined by f ( x ) = x + 1 and g ( x ) = x − 1 . Show that f o g = g o f = I R .

Given, f : R → R and g : R → R ⇒ f o g : R → R and g o f : R → R (Also, we know that I R : R → R ) So, the domains of all f o g , g o f and I R are the same. ( f o g ) ( x ) = f ( g ( x ) ) = f ( x − 1 ) = x − 1 + 1 = x = I R ( x ) . . . ( 1 ) ( g o f ) ( x ) = g ( f ( x ) ) = g ( x + 1 ) = x + 1 − 1 = x = I R ( x ) . . . ( 2 ) From ( 1 ) and ( 2 ) , ( f o g ) ( x ) = ( g o f ) ( x ) = I R ( x ) , ∀ x ∈ R Hence , f o g = g o f = I R

Let R + be the set of all non-negative real numbers. If f: R + → R + and g : R + → R + are defined as f ( x ) = x 2 and g ( x ) = + x , find f o g and g o f . Are they equal functions ?

Given, f : R + → R + , f x = x 2 g : R + → R + , g x = x g ∘ f x = g f x = g x 2 = x 2 = x f o g x = f g x = f x = x 2 = x g ∘ f x = f o g x = x ∀ x ∈ R + Hence, f ∘ g and g ∘ f are identity function.

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Important Questions on Electrostatic Potential and Capacitance

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If 8 individuals in a population of 80 butterflies die in a week, calculate the death rate of population of butterflies during that period.

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The principle of competitive exclusion was stated by

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State two observations made by the German naturalist, Alexander von Humboldt during his extensive explorations in South American Jungles.

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Explain the difference between commensalism and mutualism types of interactions, with the help of a suitable example of each.

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Four persons can hit a target correctly with probabilities

\frac{1}{2}, \frac{1}{3}, \frac{1}{4}

and

\frac{1}{8}

respectively. If all hit at the target independently, then the probability that the target would be hit, is

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Differentiate between an ‘Expanding age pyramid’ and a ‘Stable age pyramid’. Substantiate your answer with diagrams.

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Match the following and select the correct option:

(a)	Earthworm	(p)	Pioneer species
(b)	Succession	(q)	Detrivore
(c)	Ecosystem service	(r)	Natality
(d)	Population growth	(s)	Pollination

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If in a population of size ”N” the birth rate is represented as “b” and the death rate as “d”, the increase or decrease in “N” during a unit time period ”t” will be:

Question Image

The equation given above can also be represented as:

Question Image

What does ”r” represent. Write any one significance of calculating ”r” for any population?

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A die marked

1, 2, 3

in red

4, 5, 6

in green tossed. Let

A

be the event "number is even" and

B

be the event "number is marked red". Find whether the events

A

and

B

are independent or not.

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The probability of finding a green signal on a busy crossing

X

30 %

. What is the probability of finding a green signal on

X

on two consecutive days out of three?

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Physics>Electricity and Magnetism>Electrostatic Potential and Capacitance>Electrostatics of Conductors

Explain Commensalism and mutualism with examples

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f (x) = x + 1

, find

\frac{d}{d x} (fof) (x)

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An association of individuals of different species living in the same habitat and having functional interactions is:

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A problem in mathematics is given to three students

A, B, C

and their respective probability of solving the problem is

\frac{1}{2}, \frac{1}{3}

and

\frac{1}{4}

, the probability that the problem will be solved is

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For

x ϵ (0, \frac{3}{2}),

let

f (x) = \sqrt{x}, g (x) = t a n x

and

h (x) = \frac{1 - x^{2}}{1 + x^{2}}

. If

ϕ (x) = (h o f) o g) (x),

then

ϕ (\frac{π}{3})

is equal to:

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Find

g o f

and $f o g$ when

f : R \to R

and

g : R \to R

is defined by

f (x) = x^{2} + 2 x + 3

and

g (x) = 3 x - 4

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Find

f o g

and

g o f

if :

f (x) = x + 1, g (x) = 2 x + 3

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Let

f : R \to R

and

g : R \to R

be defined by

f (x) = x + 1

and

g (x) = x - 1

. Show that

f o g = g o f = I_{R}

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Let

R^{+}

be the set of all non-negative real numbers. If f:

R^{+} \to R^{+}

and

g : R^{+} \to R^{+}

are defined as

f (x) = x^{2}

and

g (x) = + \sqrt{x}

, find

f o g

and

g o f

. Are they equal functions ?

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If the prime sign

(^{'})

represents differentiation w.r.t.

x

and

f^{'} (x) = \sin x + \sin 4 x \cos x

, then

f^{'} (2 x^{2} + \frac{π}{2})

x = \sqrt{\frac{π}{2}}

is equal to

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Important Questions on Electrostatic Potential and Capacitance

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