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1 million = 10⁶

Any number can be expressed as a decimal number between 1.0 and 10.0 (including 1.0) multiplied by a power of 10. Such form of a number is called its standard form or scientific notation.

1 million = 10,00,000

= 10⁶

(a) 3² ______15

3² < 15

Where, 3² = 3 × 3 = 9

Therefore, 9 < 15

(b) 2³ ______ 3²

2³ < 3²

Where, 2³ = 2 × 2 × 2 = 8

3² = 3 × 3 = 9

Therefore, 8 < 9

(c) 7⁴ ______5⁴

7⁴ > 5⁴

Where, 7⁴ = 7 × 7 × 7 × 7 = 2401

5 × 5 × 5 × 5 = 625

Therefore, 2401 > 625

(d) 10,000 ______ 10⁵

10,000 < 100000

Where, 10 × 10 × 10 × 10 × 10= 100000

Therefore, 10,000 < 100000

(e) 6³ _____4⁴

6³ < 4⁴

Where, 6³ = 6 × 6 × 6 = 216

4⁴ = 4 × 4 × 4 × 4 = 256

Therefore, 216 < 256

Correct option is (D) 1/2

\(10^{-3x}=8\)

\(\Rightarrow\) \((10^{-x})^3=2^3\)

\(\Rightarrow\) \(10^{-x}=2\)

\(\Rightarrow\) \((10^{-x})^{-1}=2^{-1}\)

\(\Rightarrow\) \(10^x=2^{-1}=\frac{1}{2}\)

Correct option is  D) 1/2

False.

We know that, 1 million = 10,00,000

= 10⁶

False

Take RHS = 6 x 10⁵ + 6 x 10² 

= 6 x 100000 + 6 x 100 

= 600000 + 600 = 600600 ≠ LHS

Hence, RHS ≠ LHS

True

If x is a rational number and m and n are positive integers, then

a^m x aⁿ = a^m+n

Correct option is (B) \(\frac{1}{4\sqrt{2}}\)

\(x=\sqrt{2}=2^\frac12\)

Then \(x^{-5}=\frac1{x^5}=\frac1{(2^\frac12)^5}\) \(=\frac1{2^\frac52}=\frac1{2^{2+\frac12}}\) \(=\frac{1}{2^2.2^\frac12}=\frac{1}{4\sqrt{2}}\)

Correct option is   B) \(\frac{1}{4\sqrt2}\)

True

∴ 16³ =(4²)³ [∴ 16 = 4 x 4 = 4²]

= 4⁶

Now, in 4⁹ and 4⁶, 4⁹ > 4⁶ as powers 9 > 6

False.

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m × aⁿ = a^m+n

correct option is A) Rabi

correct option is B) Ooty

True

Take RHS = 8 x 10⁵ + 7 x 10⁴ + 6 x 10³ +5 x 10² + 4 x 10¹ + 3 x 10°

= 8 x 100000 + 7 x 10000 + 6 x 1000 + 5 x 100 + 4 x 10 + 3 x 1 [∴ a° = 1]

= 800000 + 70000 + 6000 + 500+ 40+3 = 876543 =

LHS

Hence, RHS = LHS

Correct option is (D) 1

\(\cfrac{2^{n+2}-2^{n+1}}{4\times2^{n-1}}=\cfrac{2^{n+1}(2-1)}{2^2\times2^{n-1}}=\cfrac{2^{n+1}}{2^{n+1}}=1\)

Correct option is  D) 1

C) Annual average temperature increases towards poles

Correct option is C) July, August

In present times, polarization of classes and class solidarity is associated with the segregation within a society which are immediate cause of emergence of income inequality, economic displacements, formation of various social groups from high income to low income, economic restructuring particularly in cities, economic inequality etc.

Given, diameter of red blood cell = 7 x 10^-6 m
and diameter of platelet = 2.33 x 10^-6 m

We know that, two numbers written in scientific notation can be compared. The number with the larger power of 10 is greater than the number with the smaller power of 10. If the powers of ten are the same, then the number with the larger factor is the larger number. Therefore, red blood cell has a greater diameter than a platelet.

Correct option is C) 10th June

D) Gujarat, Rajasthan, Punjab and Haryana

Correct option is (B) 6.22 x 10^-7

\(\frac{622}{100,00,00,000}=\frac{622}{10^9}=\frac{6.22\times10^2}{10^9}\) \(=6.22\times10^{2}\times10^{-9}\)

\(=6.22\times10^{2-9}\)

= \(6.22\times10^{-7}\)

B) 6.22 × 10^-7

1. Jaipur has an extreme climate whereas Chennai is with moderate climate. 

2. Though the highest average temperature for both regions are at near 40°C, the lowest average temperatures for Jaipur is at 6°C whereas that is for Chennai is 18°C. 

3. The monthly differences in highest and lowest temperatures also vary. 

4. For Jaipur they are around 16°C whereas for Chennai they are around 10°C. 

5. The atmosphere of Chennai will be pleasant with these temperatures whereas that is not the case with Jaipur.

correct option is B) Kodaikanal

D) Nearness to sea

Correct option is C) 1,2,4 only

1. The wettest months for Chennai are October and November. 

2. It is because it receives rainfall during the retreat of southwest monsoon or during the north-east monsoon. 

3. Whereas the wettest months for Jaipur are July and August. 

4. It receives rainfall during the southwest monsoon. 

5. In this way Chennai and Jaipur are different.

B) Industrial Revolution

Correct option is (C) 1

\((7^{-1}-18^{-1})^0=1\)  \((\because a=0\,for\,a\in\mathbb{R}-\{0\})\)

Correct option is C) 1

Correct option is (D) 1/1000

\(10^{-3}=\frac1{10^3}\)

= \(\frac{1}{1000}\)

Correct option is   D) \(\frac{1}{1000}\)

The expression for 8^-2 as a power with the base 2 is (2)^-6.

8^-2 

= (2 × 2 × 2)^-2 

= (2³)^-2

To add the numbers given in standard form, we first convert them into numbers with equal exponents.

e.g. 2.46 x 10⁶   + 24.6 x 105 

= 2.46 x 10⁵ + 2.46 x 10⁶ 

= 4.92 x 10⁶

Given, 3 x 10^-7   = 3.0 x 10^-7   

Now, placing decimal seven place towards left of original position, we get 0.0000003.

Hence, the value of 3 x 10^-7 is equal to 0.0000003.

For standard form, 0.000000008 = 0.8 x 10^-8= 8 x  10^-9 =8.0 x 10^-9

Hence, the standard form for 0.000000008 is 8.0 x 10^-9

For standard form, 32500000000 = 3250 x 10² x 10² x 10³

= 3250 x 10⁷ 

= 3.250 x 10¹⁰ or 3.25 x 10¹⁰

Hence, the standard form for 32500000000 is 3.25 x 10¹⁰.

If 36 = 6 × 6 = 6², then 1/36 expressed as a power with the base 6 is 6^-2.

Explanation: 

36 = 6 × 6 = 6²

1/36 = 1/6² = 6^-2

Given, 3⁵ ÷ 3^–6 

= (3)^5-(-6)

= (3)⁵⁺⁶

= (3)¹¹

For usual form, 2.3 x 10^-10 

= 0.23 x 10^-11

= 0.00000000023

Hence, the usual form for 2.3 x 10^-10 is 0.00000000023.

The origins of Indian cricket, that is, cricket played by Indians are to found in Bombay and the first Indian community to start playing the game was the small community of Zoroastrians, the Parsis. Parsi clubs were funded and sponsored by Parsi businessmen like the Tatas and the Wadias.

By the 1890 s, Hindus and Muslims were busy gathering funds and support for a Hindu Gymkhana and an Islam Gymkhana. 

 In the late nineteenth century, many Indian institutions and movements were organised around the idea of religious community because the colonial state encouraged these divisions and was quick to recognize communal institutions.

The Clinton government tended to focus on ‘Soft issues’ like democracy promotion, climate change and world trade rather than on the hard politics of military power.

Women are subjected to many public restrictions.

1. In our political system the head of the state exercises only nominal powers. The President of India is like the Queen of Britain whose functions are to a large extent ceremonial. ThePresident supervises the overall functioning of all the political institutions in the country so that they operate in harmony to achieve the objectives of the state.

2. The President represents the entire nation but can never claim the kind of direct popular mandate that the Prime Minister can. The same is true of his powers. All government activities do take place in the name of the President. All laws and major decisions of the government are issued in his name, all international treaties and agreements are made in his name but the President exercises these powers only on the advice of the Council of Ministers.

Amnesty International

The correct answer is 25/24

“We can identify three constraints on American Power” which were actually not in operation in the years following 9/11. Hence the US could establish its hegemony. Recently all these constraints are slowly beginning to operate in the following ways: 

1. The US bears institutional architecture in the American State itself. It refers division of powers between the three branches of government where American military’s executive branch can place significant brakes upon the unrestrained and immoderate exercise. 

2. The second constraint on American hegemony emerges from open nature of American society. American society and suffering from a deep skepticism towards purposes and methods of government in America despite an imposition of particular perspectives on domestic opinion in the US. This is a huge constraint on US military action overseas. 

3. The third constraint on US hegemony is the possession of NATO to moderate the exercise of the US hegemony today. The US has an enormous interest in keeping the alliance of democracies to follow the market economies alive and it may be possible to its allies in NATO to moderate the exercise of the US hegemony through their own liberal policies to fulfill their own ends.

Goods that can be consumed by people without reducing the amount of available goods for others are known as the global public goods.

Examples: Fresh air, roads, sea-lanes of communications (SLoCs).

The global public goods can be consumed by one person without reducing the amount of the goods available for someone else: 

1. Sea lanes of communications (SLoCs) is the naval power of the hegemon to underwrite the law of the sea and to ensure freedom of navigational international waters. These sea routes are commonly used by merchant ships.

2. Internet is the direct outcome of a US military research project that began in 1950. Even today, internet relies on a global network of satellites.

The US domination in military, economic, cultural aspects over other nations to show her supremacy is known as US hegemony. 

Its constraints are as follows- 

1. The institutional architecture of American State itself i.e. they follow the system of division of powers between three organs of government. 

2. The open nature of American Society and Political Culture i.e. the American mass media may promote a particular issue on domestic public opinion but never opposed the purposes and methods of government in American Political Culture.

By the beginning of the 21st century, America became a huge Superpower. But American dominance is controlled within America itself. 

There are three controls that keep America in check.

• The first control comes from the Constitution. It is based on the division of Powers are divided among the Executive, Legislature and Judiciary. This provision is capable of preventing America embarking on a military adventure.
• The second control comes from the openness seen in the American system. Public opinion is very strong in America. The mass media keeps a close watch on the government and its activities. 
• The third control comes from NATO (North Atlantic Treaty Organization.

The period of the beginning of US hegemony by 1991.

A formal document containing an order of the court to the government.

The heart and soul of our Constitution.