Class 12th

Determinants 4

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Determinants * Let A be the set of square matrices and B is a set of real or complex numbers. If a function, ƒ: A → B is defined by ƒ(x) = b, such that X ∈ A and b ∈ B, then ƒ(x) is called the determinant of X. * The determinant of A ... Read more

Matrices 3

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Fundamental of Matrices * A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. Column matrix: A matrix is said to be column matrix if it has only one column. Row matrix: A matrix is said to be a row ... Read more

Inverse Trigonometric Functions 2

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Basic Concept * The principal value branch of sin^{-1} is [ - \frac{π}{2} , \frac{π}{2}  ]. * The principal value branch of cos^{-1} is [ 0 , π ]. * The principal value branch of cosec^{-1}is [  -\frac{π}{2} , \frac{π}{2}  ]  - { 0 }. * The principal value branch of tan^{-1} is ( - ... Read more

Relations and Functions 1

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Types of Relations * A relation R from a non-empty set A to another non-empty set B is the subset of A× B. * The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. * If n(A) = p, n(B) ... Read more

Chapter 16. Environmental Issues

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Air Pollution And Its Control * An undesirable change in the physical, chemical or biological characteristics of air, water, land or soil that may have detrimental effect on living organisms is called pollution. * Substances that bring about such undesirable changes are called pollutants. * Common air pollutants are automobiles exhausts, chemicals and toxic gases ... Read more

Chapter 15. Biodiversity And Conservation

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Need for Conservation of Biodiversity *  Various human activities such as over exploitation, fragmentation of habitats and the introduction of exotic species cause loss of species diversity. * Currently, more than 15,500 species worldwide are on the verge of extinction. * The reasons to conserve biodiversity are grouped into three categories: narrowly utilitarian, broadly utilitarian ... Read more

Chapter 14. Ecosystem

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Structure and Functions of an Ecosystem * An ecosystem is a stable and functional unit where all living organisms interact among themselves and the surrounding physical environment, leading to a flow of energy and nutrients. * Typically, an ecosystem has two major components-abiotic and biotic. * The structure of an ecosystem involves: • The composition ... Read more