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ax + by = 1---(1)

bx + ay = \(\frac{(a+b)^2}{a^2+b^2-1}\)---(2)

∴ a²x + aby = a

-  b²x\(\pm\)aby = \(-\frac{(a+b)^2b}{a^2+b^2-1}\) 

(a² - b²)x = a \(-\frac{(a+b)^2b}{a^2+b^2-1}\)

 = \(\frac{a^3+ab^2-a-a^2b-b^3-2ab^2}{a^2+b^2-1}\) 

 = \(\frac{a^3-ab^2-a^2b-b^3-a}{a^2+b^2-1}\) 

∴ x = \(\frac{a(a^2-b^2)-b(a^2+b^2)-a}{(a^2-b^2)(a^2+b^2-1)}\) 

∴ y = \(\frac{1-ax}b\) 

= \(\frac1b(1-\frac{a^4-a^2b^2-a^3b-ab^3-a^2}{(a^2-b^2)(a^2+b^2-1)})\) 

 = \(\frac1b(\frac{a^4+a^2b^2-a^2-a^2b^2-b^4+b^2-a^4+a^2b^2+a^3b+ab^3+a^2}{(a^2-b^2)(a^2+b^2-1)})\)

 = \(\frac{-b^4+ab^3+a^2b^2+a^3b+b^2}{(a^2-b^2)b(a^2+b^2-1)}\) 

Hence, x = \(\frac{a^3-ab^2-a^2b-b^3-a}{(a^2-b^2)(a^2+b^2-1)}\) 

and y = \(\frac{-b^4+ab^3+a^2b^2+a^3b+b^2}{(a^2-b^2)b(a^2+b^2-1)}\)

Shelf space is wasted when a store: Overstocks one item while under stocking another, fails to refill bins or shelves when the basic stock is reduced/ Stocks the same items in several places/ Uses the wrong shelf profile/ Does not relate shelf space to customer demand fluctuations.

[3     -4      3

-2.      3     -2

8.    -12     9] is the inverse of the given matrix

\(A = \begin{bmatrix}3&0&-1\\2&3&0\\0&4&1\end{bmatrix}\)

A = IA

⇒ \( \begin{bmatrix}3&0&-1\\2&3&0\\0&4&1\end{bmatrix}\) = \( \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}A\)

R₂ →3R₂ - 2R₁

\( \begin{bmatrix}3&0&-1\\0&9&2\\0&4&1\end{bmatrix}\)= \( \begin{bmatrix}1&0&0\\-2&3&0\\0&0&1\end{bmatrix}A\)

R₁ → \(\frac{R_1}3\) 

\( \begin{bmatrix}1&0&-1/3\\0&9&2\\0&4&1\end{bmatrix}\) = \( \begin{bmatrix}1/3&0&0\\-2&3&0\\0&0&1\end{bmatrix}A\) 

R₃ → 9R₃ - 4R₂

\( \begin{bmatrix}1&0&-1/3\\0&9&2\\0&0&1\end{bmatrix}\) = \( \begin{bmatrix}1/3&0&0\\-2&3&0\\8&-12&9\end{bmatrix}A\)

R₁ → R₁ + \(\frac13R_3\)

R₂ → R₂ - 2R₃

\( \begin{bmatrix}1&0&0\\0&9&0\\0&0&1\end{bmatrix}\) = \( \begin{bmatrix}3&-4&3\\-2&3&-2\\8&-12&9\end{bmatrix}A\)

∴ A^-1 = \( \begin{bmatrix}3&-4&3\\-2&3&-2\\8&-12&9\end{bmatrix}\)

(i) (2x^-2 y³)³ = 2³(x^-2)³ (y³)³

 = 8x^-6y⁹

 = 8 \(\frac{y^9}{x^6}\) 

(ii) (625)^-1/4 = (5⁴)^-1/4 = 5^-1 = 1/5.

I have not found answer

The value of Pi (π) is the ratio of the circumference of a circle to its diameter and is approximately equal to 3.14159. 

In a circle, if you divide the circumference (is the total distance around the circle) by the diameter, you will get exactly the same number. Whether the circle is big or small, the value of pi remains the same. The symbol of Pi is denoted by π and pronounced as “pie”. It is the 16th letter of the Greek alphabet and used to represent a mathematical constant.

 (i) \((\frac{-3}{17})+(\frac{-12}5)=(\frac{-12}5)+(\frac{-3}{17})\) 

(By cumulative law of addition)

(ii) -9 + (-\(\frac{21}8)=(-\frac{21}8)\) + (-9)

(By cumulative law of addition)

 (iii) \((\frac{-8}{13}+\frac37)+(\frac{-13}4)=(\frac{-8}{13})+(\frac37+(-\frac{13}4))\) 

(By associative law of addition)

(iv) -12 + \((\frac7{12}+(\frac{-9}{11}))=(-12+\frac7{12})+(\frac{-9}{11})\) 

(By associative law of addition)

Carcinogens are factors capable of producing neoplasias. Any mutagen, a substance that can induce DNA mutation, is a potential carcinogen. Examples of carcinogens are radiation, nitrous acid, many substances inhaled through tobacco smoking and the human papilloma virus (HPV).

Cerebrovascular accident (CVA), also known as stroke, is the generic name given to infarction (tissue and cellular death by hypoxia) of areas of the brain due to vascular obstruction or hemorrhages. CVAs are divided into ischemic and hemorrhagic. In the ischemic CVA blocking of arteries that carry blood to the brain occurs; its cause is generally atherosclerosis (atheroma formation) of these vessels. In the hemorrhagic CVA there is rupture of one or more blood vessels of the brain with blood leakage, increasing intracranial pressure and thus interruption of blood flow in some areas of the brain. The severity of the stroke depends on the function performed by the affected area of the brain, for example, motor function, visual function, vegetative function, etc., and on the size of the involved area.

The main risk factors for cerebrovascular accidents are hypertension, hypercholesterolemia, tobacco smoking and old age.

In neoplastic tumors a phenomenon called angiogenesis occurs. Angiogenesis is the formation of new blood vessels. Neoplastic cells induce the formation of new blood vessels to irrigate and drain the neoplastic tissue. Angiogenesis is important because the tumor growth depends on it. A lot of research on cancer has tried to discover  natural and synthetic substances to inhibit angiogenesis.

Neoplasia is any abnormal and uncontrolled proliferation of cells of an organism. Neoplasias can be benign or malign. Benign neoplasias are those in which the cell proliferation is limited to a given site of the body and so neoplastic cells do not spread to other close regions or at distance through the circulation. Malign neoplasias are those in which the neoplastic cells disseminate at distance to other sites and organs of the body, a process called metastasis, where they continue to proliferate. Malign neoplasias injure tissues and if not eradicated they are fatal. Benign neoplasia can also be deadly when it forms a tumor that grows and compresses vital organs.

Neoplasias appear due to DNA mutations in genes that regulate the cellular proliferation thus making the cell lose its capacity to control and limit its division by mitosis. The cell then divides continuously and uncontrollably and this defect is transmitted to its daughter cells.

Excluding skin cancer, that are the more easily detected and so cases are registered in larger number, the main types of cancer in men are prostate cancer, lung cancer, stomach cancer; in women, breast cancer, colon and rectal cancer and lung cancer are of great incidence. Other common cancers are ovarian cancer, pancreatic cancer, liver cancer, esophageal cancer, brain cancer and the leukemias and lymphomas (blood cancers).

Epithelial cancers, of the skin as well as of the internal organs, are more common because epithelial tiss

The proportional incidence of the many types of cancers varies according to the considered population.

Cancers are malign neoplasias, i.e., abnormal and uncontrolled proliferation of cells that can disseminate to other sites of the body. Cancer dissemination at distance usually occurs through blood or lymphatic vessels.

Not every tumor is neoplastic and not every neoplasia creates tumor. Tumor is the generic name given to the abnormal increase in mass or volume of any area of the body (for example, the enlarged tonsils during throat infection are a kind of tumor, any inflammation creating a swelled area characterize a tumor, etc.). Neoplasias can form tumors, some of them very large, by aggregation of neoplastic cells in the region where the neoplasia began or in distant implantations. Cancer is a synonym for malign neoplasia.

Myocardial infarction is the condition in which an area of this tissue or the entire heart muscle dies by hypoxia due to lack of blood irrigation. Myocardial infarction is a severe disease since on the dependence of its extension the heart can fail, i.e., it can no longer pump blood to the lungs or to the body or it can even stop beating (causing death). 

The main cause of myocardial infarction is coronary obstruction, blocking of the arteries that carry arterial blood to the heart muscle. Other events like hemodynamic shock (circulation stoppage due to large hemorrhages, for example) can also cause myocardial infarction.

The main risk factor for skin cancer is solar exposition of the skin without protection against ultraviolet radiation (a potential carcinogen).

The most lethal skin cancer is melanoma.

The main risk factor for lung cancer is tobacco smoking. The large number of cases of this type of cancer is due to the increased number of smokers worldwide.

Alzheimer’s disease is a degenerative disease of the central nervous system in which the patient has progressive dementia and alteration of mental functions.

The disease generally appears after 40 years of age and it is more frequent in the elderly. Image studies of the brain show broad loss of brain tissue. (The Alzheimer’s disease should not be confused with other mental deteriorations common in the elderly.)

If the cancer is in its initial stage treatment is often done by surgical removal of the neoplastic tissue. Cancers already disseminated are often treated with radiation (radiotherapy) and anti-mitotic drugs (chemotherapy).

Coronary disease, or coronary insufficiency, is a disease in which there is total or partial obstruction of one or more of the arteries that irrigate the heart musculature, i.e., obstruction of the coronary arteries. The disease is formed by slow and gradual formation of atheroma plaques inside the coronaries. The fatty plaques grow and block the flow of blood, a process known as atherosclerosis (do not confuse with arteriosclerosis which is the hardening of the arteries generally cause by chronic high blood pressure).

The main risk factors for coronary disease are tobacco smoking, diabetes mellitus, hypertension, hypercholesterolemia (high level of bad cholesterol and low level of good cholesterol), stress, alcoholism and sedentary lifestyle.

Coronary disease may present in two manners, as angina pectoris or as myocardial infarction. If the arterial obstruction is not complete and extensive the patient often feels chest pain (angina pectoris), mainly when performing physical exercise or in any situation when the heart needs more oxygen. If the obstruction of one or more coronaries is complete or blood cannot irrigate some regions of the heart muscle (myocardium) the infarction occurs and the muscle cells of the affected area die.

Hypertension is a disease in which the arterial blood pressure, during systole or during diastole, is abnormally high.

Hypertension, or high blood pressure, is a condition that must be diagnosed and treated since it produces irreversible injuries in arteries and, later, it causes other severe diseases in organs like the heart, brain, kidneys, retina, etc.

The main degenerative diseases of the nervous system are Alzheimer’s disease and Parkinson’s disease.

Degenerative diseases of the nervous system are caused by progressive tissue degradation or loss of neurons in some regions of the nervous system.

The main degenerative diseases of the heart are heart failure, arrhythmias, valvular heart diseases, coronary insufficiency and myocardial infarction.

The main human degenerative diseases are divided into three groups: cardiovascular diseases, neoplastic diseases and degenerative diseases of the nervous system. The main cardiovascular diseases are hypertension, the cardiopathies, including coronary disease and myocardial infarction, and the cerebrovascular accidents (CVAs, or strokes). Neoplasias are benign tumors and cancers. The main degenerative diseases of the nervous system are Alzheimer's disease and Parkinson's disease.

Parkinson’s disease is a degenerative disease of the nervous system in which the main manifestations are progressive motor disturbances, like tremors of feet, hands and mandibles (jaws) and walking and balance impairments. Parkinson’s disease is due to the degeneration of dopaminergic motor neurons, i.e., motor neurons that use dopamine as a neurotransmitter, located in a specific region of the brain, the mesencephalon. Such degeneration creates deficiency of dopamine in the nervous system. (Parkinson’s disease should not be confused with other causes of tremors, like the use of some medicines.) 

Risk factors for a disease are everything that contributes to increase the risk of the disease to appear. For example, for most cardiovascular diseases, tobacco smoking and diabetes mellitus are both important risk factors; for some cancers having a genetic component a positive family history is a risk factor, etc.

The two mains species of taenias that cause disease in humans are Taenia solium, or the pork tapeworm, and Taenia saginata, or the beef tapeworm.

Besides these species there are still the fish tapeworm, or Diphyllobothrium latum, and a tapeworm, Echinococcus granulosus, which lives in dogs and cause in humans the severe disease known as hydatid disease, or echinococcosis.

Tapeworms have hooks and sucking structures on their heads (scolex) that fixate the parasite in the gut wall; these structures often do not injure the host tissue. The parasite obtains food and makes gas exchange through absorption and diffusion across its skin; since it is a platyhelminth it does not have a digestive system or a circulatory system.

Taenias are monoecious (hermaphrodite), the same individual has female and male reproductive organs and undergoes self fecundation.

Schistosomiasis has acute and chronic phases. Days after the infection the cercarial dermatitis appears at the site where the worm penetrated the skin. In one or two months the acute phase begins and fever, cough, muscle pain and nausea occur but soon disappear; in some people this phase may not manifest. In the chronic phase of the disease the worm infestation can affect the bowels, causing diarrhea, bloody feces, tiredness and hepatosplenomegaly (enlargement of the liver and spleen). The excessive pressure upon the hepatic portal vein produces varix (collateral circulation) in the esophagus and the patient may have hemorrhages and bloody vomiting that may even cause death.

Degenerative diseases are non infectious prevalent diseases whose incidences increase with aging.

Taenias, also know as tapeworms, are platyhelminth animals (flatworms). The main diseases caused by taenias are taeniasis and cysticercosis.

Taeniasis patients may develop the most severe form of the worm infection, cysticercosis, because their feces contain eggs and pregnant proglottids of the taenia and there is risk of self infection due to bad hygienic habits, like not washing the hands after defecation. If these individuals ingest the eggs of the parasite they can develop cysticercosis too.

The most important human worm infections are schistosomiasis, taeniasis, cysticercosis, ascariasis and ancylostomiasis (hookworm infection).

Other important worm infections caused by platyhelminthes are: fascioliasis, caused by Fasciola hepatica, which attacks the liver and the bile ducts; hydatid disease, also known as echinococcosis, caused by a taenia-like parasite and transmitted by feces of dogs in contact with sheep, with a clinical picture similar to cysticercosis. Other important infections caused by roundworms are: strongyloidiasis, agent Strongyloides stercoralis, a common opportunistic disease in AIDS; filariasis, also known as elephantiasis, transmitted by mosquitoes of the Culex genus, caused by Wuchereria bancrofti and other thread-like worms and manifested by obstruction and fibrosis of lymphatic vessels that cause swelling (lymphedema) of limbs; enterobiasis (pinworm infection), caused by Enterobius vermicularis, a worm that parasites the colon and the human perianal region; cutaneous larva migrans, an ancylostomiasis of the skin caused by Ancylostoma braziliensis.

The main known human diseases of such type are the Creutzfeldt-Jacob disease (CJD), the kuru and the Gerstmann-Sträussle-Scheinken disease (GSS). The hypothesis that many other diseases of unknown etiological agents are actually caused by self-replicating infectious proteins is strong.

Worm infections are human diseases caused by platyhelminthes (flatworms) or nematodes (roundworms).

Schistosomiasis is a worm infection caused by schistosomes, a species of flatworms (platyhelminthes). The disease is prevalent in Latin America and in the Far East. The main species of schistosome found in Latin America is Schistosoma mansoni.

A prion is an infectious (transmissible) protein able to replicate by transforming other proteins into a copy of the prion. The mechanism of copying is not yet understood by science.

The hypothesis come out from research about a nervous system disease known as Creutzfeldt-Jacob disease, epidemiologically associated to a bovine disease called bovine spongiform encephalitis (the mad cow disease). Research discovered that the infectious agent that causes those diseases, suprisingly, was a protein capable of copying itself and of being transmitted by ingestion (the reason why meat from contaminated animals cannot be consumed), inoculation and even heredity.

Ascaris, or Ascaris lumbricoides, is an animal of the nematode phylum, i.e., a roundworm. Ascaris causes ascariasis, a common worm infection of the intestine.

The main measures to prevent schistosomiasis are: information for infected individuals to look for treatment and to not disseminate the disease; chemical and biological combat against the vector snail; basic sanitary conditions to avoid contamination of lakes, rivers and other water streams by infected feces; to avoid contact with fresh water suspected of contamination.

In the pulmonary phase the ascaris infestation causes cough, hemoptysis, dyspnea, fever, fatigue and may cause a special kind of pneumonia called eosinophilic pneumonia. In the intestinal phase the symptoms are due to spoliation of nutrients of the host and thus hunger and weight loss may appear. Masses of ascaris inside the bowels can cause severe intestinal obstruction. 

The main prophylactic measures against taeniasis and cysticercosis are: not to ingest raw or badly cooked swine or bovine meat; sanitary education of the people; appropriate destination of feces; adequate treatment of infected people.

Schistosomes are dioecious, i.e., the species has separated sexes, male and female individuals.

The main prophylactic measures against hookworm disease are: to avoid walking barefoot on soils suspected of contamination; basic sanitary conditions and appropriate destination of feces; treatment of infected people.

Solution:

As we know that the work done by a body is dot product of force and displacement.

Thus, W = F.S

F = 5i+6j-4k

S = 6i+5k

W = (5i+6j-4k).(6i+0j+5k)

By dot product we have,

W=30+0-20

W= 10 j

As we have work done by body and displacement both are in vector form ,

Work done = force x displacementAs given in question, W=(5i+6j-4k)(6i+0i+5k)by multiplying, we get 30 + 0 -20                      =10unitsthis is the work done by force.

Adult ascaris that live within the human intestine can release up to 200 thousand eggs a day. The eggs are eliminated with human feces and mature in the environment under some heat and moisture conditions. Humans may ingest mature eggs through food contaminated by human feces or through bad hygienic habits. The eggs again inside the human intestine release larvae that cross the enteric mucosa and gain the circulation reaching the lungs. In the lungs the larva mature and go to the airway and to the pharynx when they are then swallowed. Within the gut the larvae develop into adult worms.

The ascaris live within the human gut and feed from the food ingested by the infected person.

Male and female adult schistosomes live within blood vessels of the human intestines. The females release eggs that trespass the vessel walls and enter the intestinal lumen being eliminated with the human feces. When in water the egg releases a larva called miracidium that then swims to reach the snail that is the intermediate host. Within the snail miracidia transform into cercariae, another kind of schistosome larva. The cercariae leave the snail and swim until penetrating through the human skin. In humans the cercaria gains the circulation infecting mainly the blood vessels of the intestines, liver or lungs and develops into the adult form of the parasite.