let 'a' be any positive integer and b=2
By euclid's division algorithm
a=bq+r       0≤r<b
a=2q+r       0≤r<2
(i.e) r =0,1
r=0 , a=2q+0=> a=2q
r=1, a=2q+1
if a is the form of 2q then 'a' is an even integer and positive odd integer is of the form 2q+1

Grouping objects

Grouping of objects is similar to putting objects into a container. You can move the objects as a group and apply global changes to the objects within the group. A group can always be undone and the objects that make up the group can always be manipulated separately.

The process of unification of Germany:

1. The middle-class Germans had tried to fight for a united nation governed by an elected parliament in 1848 but were repressed by the monarchy and military.

2. Otto von Bismarck, chief minister of Prussian led the movement for unification of Germany. He was supported by the bureaucracy and the Prussian army.

3. After three wars with Austria, Denmark, and France, Prussia won and unified Germany. The Prussian king, William I, was proclaimed the German emperor in January 1871.

The 1848 revolution of the liberals refers to the various national movements pioneered by educated middle classes alongside the revolts of the poor, unemployed and starving peasants and workers in Europe. While in countries like France, food shortages and widespread unemployment during 1848 led to popular uprisings, in other parts of Europe (such as Germany, Italy, Poland, and the Austro-Hungarian Empire), men and women of the liberal middle classes came together to voice their demands for the creation of nation-states based on parliamentary principles. The political, social and economic ideas supported by the liberals were:

•  Politically, they demanded constitutionalism with national unification, nation-state with a written constitution and parliamentary administration.
•  Socially, they wanted to rid society of its class-based partialities and birthrights. Serfdom and bonded labor had to be abolished.
•  Economically they demanded freedom of markets and right to property. The abolition of state-imposed restrictions on the movements of goods and capital.

The female figures as an allegory of the nation:Artists found a way out to represent a country in the form of a person. Then nations were portrayed as female figures.The female figure was chosen to personify the nation did not stand for any particular woman in real life. It gave the abstract idea of the nation a concrete form. Thus the female figure became an allegory of the nation.During the French Revolution artists used the formal allegory to portray idea such as Liberty Justice and the Republic.

The nationalism groups became more increasingly intolerant of each other & were ever ready to go to war by the last quarter of 19th century.The major European powers,in turn,manipulated the nationalist aspirations of the subject peoples in Europe to further their own imperialist aims.There was intense rivalry among the European powers over trade & colonies as well as naval & military might during this period.The rivalries were very evident in the way the Balkan problem unfolded.Each power-Russia,Germany,England,Austro-Hungary-was keen on countering the hold of other powers over the Balkans,& extending its own control over the area.

Giuseppe Mazzini was an Italian revolutionary in 19th century Europe. His contributions were:

• Mazzini was an Italian revolutionary who later became a member of the secret society of the Carbonari.
• At a young age of 24 in 1831, he was exiled for attempting a revolution in Liguria.
• To further spread the revolutionary ideas, he founded two more underground societies- ‘Young Italy’ in Marseilles and ‘Young Europe’ in Berne, whose members were like-minded young men from Poland, France, Italy and the German states.
• Mazzini believed that God had created nations to be the natural units of mankind. Thus, Italy cannot remain divided into various kingdoms. It has to be a single unified republic.
• Following his example, secret societies were set up in Germany, France, Switzerland and Poland. Mazzini’s opposition to monarchy and his vision of democratic republics scared the conservatives. Metternich described him as ‘the most dangerous enemy of our social order’.

Italy, before its unification, was divided into seven states. While northern states were under the Hapsburg dynasty of Austria, central states were under the control of the Pope. Southern states were ruled by the Bourbon dynasty of Spain. Giuseppe Garibaldi has been one of the most celebrated of all Italian freedom fighters. Sailor by profession, he joined the secret society, ‘Young Italy’. He supported King Victor Emmanuel II in his efforts to unify the Italian states. He organised an army to achieve the unification of Italy. People joined his army in large numbers and came to be known as ‘Red Shirts’.

By the end of 1860, he along with his army had conquered the states of Sicily and Naples which further paved the way for the unification of Italy

The history of nationalism in Britain unlike the rest of Europe because:

• In Britain the formation of the nation-state was not the result of a sudden upheaval or revolution.
• The primary identities of the people who inhabited the British Isles were ethnic ones - such as English, Welsh, Scot or Irish.
• The Act of Union (1707) between England and Scotland resulted in the formation of the 'United Kingdom of Great Britain' meant that England was able to impose its influence on Scotland. Scotland's distinctive culture and political institutions were systematically suppressed.
• The Scottish Highlanders were forbidden to speak their Gaelic language or wear their national dress and large numbers were forcibly driven out of their homeland.
• The English helped the Protestants of Ireland to establish their dominance over a largely Catholic country. Catholic revolts against British dominance were suppressed. Ireland was forcibly incorporated into the United Kingdom in 1801.
• The symbols of the new Britain - the British flag, the national anthem, the English language were actively promoted and the older nations survived only as subordinate partners in this union.

Sum of 3.25g and 32.507g

   3.025g

32.507g

____________

35.532

So, Sum of 3.25g and 32.507g is 35.532g

The Revolutions of 1830 were a revolutionary wave in Europe which took place in 1830. It included two "romantic nationalist" revolutions, the Belgian Revolution in the United Kingdom of the Netherlands and the July Revolution in France along with revolutions in Congress Poland and Switzerland. It was followed eighteen years later, by another and possibly even stronger wave of revolutions known as the Revolutions of 1848.

Language played an important role in Poland in developing nationalist sentiments. After Russian occupation, the Polish language was forced out of schools and the Rusian language was imposed everywhere. In 1831, an armed rebellion against Russian rule took place which was ultimately crushed. Following this, many members of the clergy in poland began to use language as a weapon of national resistance.

Polish was used for church gathering and all religious instruction. As a result, a large number of priest and bishops were put in jail or sent to Siberia by the Russian authorities as punishment for their refusal to preach in Russian. The use of Polish came to be seen as a symbol of the struggle against Russian Dominance.

Total number of outcomes when three dice are thrown = 6 x 6 x 6 = 216 

For sum of numbers to be 17,  possible ways are, (6, 6, 5), (6, 5,6), (5, 6,6), 

.'. Number of favorable outcomes = 3

  .'.  Required probability = 3/216 = 1/72 

Length of cuboid=8+8-8=24cm
b.=8cm
he.=8cm
area of cuboid=2(lb+BH+hl)
=2(24×8+8×8+8×24)
=2(192+64+192)
=2×448
=896cm²

 Italy  had a long history of political fragmentation are
(a) Italians were scattered over several dynastic states  as well  as  the multi-national Habsburg empire.
(b) During the middle of the 19th century, Italy was divided into seven states   of   which   only   one, Sardinia-Piedmont was ruled by an Italian Princely House.
(c) The   North    under   Austrian Habsburgs, the center was ruled by the pope and the southern regions were under the domination of the Bourbon kings of Spain.
(d) Even the Italian language had not any standard common form. It had many regional and local variations.  

The act of union of 1707 had a major impact on the Scottish people :
1.rhe Catholic clans that inhabited Scotland faced terrible repression when they tried to assert their freedom.
2.the Scottish Highlanders were forbidden to speak their Gaelic language.
3.they were not allowed to wear their national dresses.
4.a large number of people were driven out of their homeland.

• Political Fragmentation: Like Germany, Italy was also politically fragmented. During the middle of the 19th century, Italy was divided into seven states, of which only one, Sardinia- Piedmont was ruled by an Italian Prince.
• Role ofMazzini: Giuseppe Mazzini made efforts to unite Italian Republic. He had formed a secret society called ‘Young Italy’ for achieving his goal.
• Role of Count Cavour: He was the chief minister who led the movement to unify Italy. He formed a tactful diplomatic alliance with France and defeated the Austrian forces.
• Role of Giuseppe Garibaldi: Garibaldi also formed armed volunteers. In 1860, they marched into South Italy and the Kingdoms of Two Sicilies and succeeded in driving out the Spanish rulers.
• In 1861, Victor Emmanuel II was proclaimed King of United Italy.

(d) All of the above

1. Healthy Sports Environment A healthy sports environment plays a vital role in motivating the sportsperson. Healthy sport environment consists of proper humidity and temperature, smooth and clean sports fields, good quality of sports equipment and other facilities. .
2.  Positive Attitude For proper motivation, the coaches should try to encourage positive attitude among sportspersons. Players must think positively.
3.  Cash Prizes, Certificates and Trophies These are good incentives to sportspersons. Governments offer cash prizes to sportspersons who win.

When a ray of light is incident normally on a plane mirror it means that the angle of incidence is 0^ₒ. From the law of reflection, we find that angle of reflection is also 0. Thus, for normal incidence on a plane mirror, angle of incidence and angle of reflection are 0^ₒ​.

CLASS XII BOOK CHAPTER DELETED

Indian Society

1. The Market as a Social Institution

Social Change and Development in India

1. The Story of Indian Democracy

2. Globalisation and Social Change

3. Mass Media and Communications

Le Chatelier's principle states that when a system in chemical equilibrium is disturbed by a change of temperature, pressure, or a concentration, the system shifts in equilibrium composition in a way that tends to counteract this change of variable.