Q. & A. 711 to 1707 with solved Papers css 1971 to date

Ibn al-Lais: He found geometrical method o? solving cubical equations.
Al-Khujandi: He proved the so-called fermats problem for cubic powers.
Al-Karkhi: Al-Karkhi of the 11th century is considered to be one of greatest Arab mathematicians who wrote an arithmetic book al-Kafi fi Hisab and another one in algebra called al-Fakhri. In these books he developed approximate method of finding square root, theory of indices, theory of surds, theory of mathematical’ inductions and theory of quadratic intermediate equations etc. Umar Khayyam: Khayyam was the most glamorous figure of the llth century and a poet of renown. According to Mortiz Cantor: ”Has better claim to immortality as one of the greatest mathematicians of all times. He made great advance in the theory of equation. He found and proved binomial theorem for positive integral indices”.
In the field of geometry the Arabs started with the translations of Euclid and the conic section of Apollonios and thus preserved the work of these Greek Masters for the modern world. Having completed that by the 9th century, they soon began making new discoveries in the field. Thus Hussain, Ahsan and Muhammad,

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the three sons of Musa Ibn Shakir discovered the method of trisecting the angle by means of the geometry of motions. Ibn ul-Haitham: His book on geometrical optics, deals with problems, which would be difficult to solve even now. It is this book that was translated by Roger Bacon and published in his ”Opus majus”. Later, the Arabs developed the geometry of conic section to a great extent, but the crowning discovery is that of Abu Jafar Muhammad Ibn Muhammad Ibn al- Hasan commonly known as Nasir al-Din al-Tusi, the greatest savant of the 13th century. He separated trigonometry from astronomy, but his greatest contribution is the recognition and explanation of weakness in Euclid’s theory of the parallels and may thus be credited as founder of non-Euclidian geometry. Trigonometry is the branch of mathematics, which the Arabs cultivated most diligently because of its application to astronomy. The first steps in this science date from al- Bartani, who was the first to use the expression of ”sine” and ’cosine’ and prepared tables for ’sin’ ’con’ ’tan’ and ’cot’; etc. These tables were translated in the 12th century into Latin and in Spanish in the 13th century. His basic formula and proof of spherical trigonometry are well-known and earned him the title of the best observer of the sky and the best Arab astronomer. His book on the motion of stars was translated by Plato of Tivoli, in the 12th century. It is from this translation, that the word of ’sine’ spread in all European languages. With these trigonometric tables, they calculated the height of mountains, distance of in-accessible points and breadth of rivers. Modern mathematicians are said to have rediscovered it some 500 years later, Their knowledge of applied mathematics evident from all those wonderful examples of Arab.
Thus we find Muslim scientists deeply interested in the functioning and the making of clocks. Now clocks are machines which, by recording the progress of time from one moment to another, indicate the living nature of the universe and thus of the eveliving God. Al-Biruni and other Muslim astronomers, as we remember, envisaged the universe as a dynamic ’becoming’, and thus as a proof of God’s eternal manifestation in the phenomenal world. Everything that denoted the close relationship between the world and its maker was of supreme concern to Muslim thinkers and scientists. Movement produced solely by man, such as we find in mechanical contraptions, was of less interest to them than movement caused by forces that are more obvious manifestations of divine
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creativeness. Thus we find Muslim scientists devoting much labour to the making of clocks moved by water or anercury or, even, by burning candles. The force of water, or mercury, or fire, and the movement produced by any of these, seemed a direct illustration of the living nature of the universe and of the deity from which it derived its life.
One of Islam’s most famous clocks was that at Damascus made by Muhammad ibn AM which was run by water. Thanks to the account of it written by its maker’s son Rizwan, in 1203, that clock remained famous until the sixteenth century. Many books often embellished by fascinating illustrations, were also written bv Muslims on other clocks and on all manner of hydrostatic automata In their lighter moments, and to satisfy the fancies of their patrons Muslims scientists would turn their attention to mechanical toys These toys were extremely popular with the princes and the gentlemen and ladies at the various Arab courts. This fashion incidentally, was not confined to Muslim courts.- It was equally popular at the princely courts of Renaissance Italy, and at the various European courts of that time.
Abul Ali Hasan ibn al Haytham (Alhazen) of Basrah (965) was the first scholar to attempt to refute the then generally accepted optical doctrines of Euclid and Ptolemy. According to those doctrines, the eye received images of various objects by sending out visual rays to those objects. In his book On Optics, Alhazen proved that the process is the very reverse. In so doing, he laid the foundations of modern optics. His own classical formula was: ’It is not a ray that leaves the eye and meets the object that gives rise to vision. Rather the form of the perceived object passes into the eye and is transmuted by its transparent body’, i.e the lens. Alhazen also did essential spadework for the much later discovery in Italy of magnifying lenses. Most of the medieval writers on optics, including Roger Bacon, used Alhazen’s findings as their starting point, especially his Optica the-saurus, a book that left its mark even on Leonardo da Vinci and Johann Kepler. Alhazen’s writings are rooted in very sound mathematical knowledge, a knowledge that enabled him to propound well-nigh revolutionary doctrines on such subjects as the halo and the rainbow, eclipses and shadows, and on spherical and parabolic mirrors.
Alhazen’s work was carried on by a number of Muslim scientists, one of the last of real importance being the Persian Kamal

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al-Din, of the fourteenth century, best known for his explanation of primary and secondary rainbows and his experiments with the camera obscura. It is perhaps worth mentioning that the three most famous of the questions whose explanations Frederick II of Hohenstauffen sought from Muslim scientists dealt with optics. They were: what is the cause of the illusion of floating specks before the eyes of those suffering from incipient cataract? Why do oars or lances, partly covered with water, appear to be bent? And why does Canopus appear bigger when near the horizon, whereas the absence of moisture in the southern deserts precludes moisture as an explanation? Botanical Sciences
One of the features that invariably roused the curiosity of Muslim travellers in foreign countries was plant life. They studied, collected and described plants, and Europe can thank them for the source of most distinguished contributions to early botanical science. With their strongly developed practical sense they were particularly interested in plants that might serve some utilitarian purpose, whether in agriculture or in medicine. As Professor Sarton points out: By far the most important herbalistic tradition in almost every respect was the Arabic or Muslim one.... these excellent tendencies, without equivalent in Christendom, were continued during the first half of the thirteenth century by an admirable group of four botanist. One of these Ibn al-Baitar, compiled the most elaborate Arabic work on the subject (botany), in fact the most important for the whole period extending from Dioscordies down to the sixteenth century, it was a true encyclopedia on the subject, incorporating the whole of Greek and Arabic experience. Botany was only part of Ibn alBaitar’s life-interest. He also described more than 1,400 medicinal drugs and plants throughout the Mediterranean area. As has already been indicated, the Muslims took the keen interest in pharmacology, and produced a very large number of books on that subject. Many of the receipts contained in the writings of the fourteenth-century Jewish author Kohen al-Attar and of the sixteenth-century Dawud alAntaki were adopted by European druggists; and so were some of the original Arab or Persian names such as syrup from the Arab sharab. rob for a particular mixture of honey and fruit juice, and julep from the Persian gulab, or rose-water, for a medicinal aromatic drink.
Since farming provided the chief source of income for the caliphs and other Muslim princes, and since most of their subjects