Position of Roots of a Polynomial Equation Some results on roots of an equation: The following are results on the roots of a polynomial equation with rational coefficients: An equation of degree n has n roots, real or imaginary. Imaginary roots always occur in pairs i.e., if 2 – 3i is a root of an equation, then Read more about Position of Roots of a Polynomial Equation[…]
Fundamental Principles of Counting Fundamental Principles of Multiplication: If there are two jobs such that one of them can be completed in m ways and when t has been completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be m x Read more about Fundamental Principles of Counting[…]
Selection of One or More Objects Selection from Different Objects: The number of ways of selecting one or more items from a group of n distinct item is 2ⁿ – 1. Example: A man has 6 friends, in how many ways may he invite one or more of them to dinner? Solution: The man has Read more about Selection of One or More Objects[…]
Permutations Each of the different arrangements which can be made by taking some or all of a number of distinct objects is called a permutation. If n is a natural number and r is a positive integer satisfying 1 ≤ r ≤ n then the natural number is denoted by the symbol ⁿPr or P Read more about Permutations[…]
Trigonometric Ratios of Multiple and Substitute Angles 1. Sin2A = 2sinA cosA 2. Cos2A = cos²A – sin²A 3. Cos2A = 2cos²A – 1 (or) 1 + cos2A = cos²A 4. Cos2A = 1 – 2sin²A (or) 1 – cos2A = 2cos²A 5. 6. 7. The above relation are true for all values of angle Read more about Trigonometric Ratios of Multiple and Substitute Angles[…]
Angle of Elevation and Depression of A Point Suppose a straight-line AX is drawn in the horizontal direction. Then the angle XAP where P is a point above AX is called the angle of elevation of P as seen from A. similarly the angle XAQ where Q is below AX, is called the angle of Read more about Angle of Elevation and Depression of A Point[…]
Circular Permutations The number of circular permutation of n distinct objects is (n – 1)! The number of circular permutation of n things in which clockwise and anti-clockwise arrangements give rise to different permutations is (n – 1)! If anti clockwise and clockwise order of arrangements are not distinct. Then the number of circular permutations Read more about Circular Permutations[…]
Half Angle Formula In any triangle A, B, C Example: If P₁ , P₂ and P₃ are the altitudes of a triangle ABC from the vertical A, B and C respectively and Δ denotes its area then prove that Solution: We have Now, L H S = , , , , , And , , Read more about Half Angle Formula[…]
Continuous Functions and Discontinuous Functions Continuous Functions: A function f(x) is said to be continuous, if it is continuous at each point of its domain. A function f(x) is said to be everywhere continuous if it is continuous on the entire real line (-∞, ∞) i.e., on R.Some Fundamental Results on Continuous Functions: Here, we Read more about Continuous Functions and Discontinuous Functions[…]
Evaluation of Trigonometric, Exponential and Logarithm Limits In order to evaluate trigonometric limits the following results are very useful: does not exist does not exist does not exist does not exist. Expansions useful in Evaluation of Limits: Evaluate: Solution: we have, Evaluation of Exponential and Logarithm Limits: Share Tweet View Email Print Follow