Integration by Substitution Method When the variable in a defined integral is changed, the substitution in terms of new variable should be effected at three places. i. In the integration ii. In the differential, say dx iii. In the limits Evaluate: . Solution: , = , = , = , = , Let -cosx + Read more about Integration by Substitution Method[…]
Mean – Problems 1. Mean of 9 observations is 100 and mean of 6 observations is 80, then the mean of 15 observations is Solutions: 9 observations is 100 6 observations is 80 n₁ = 9 and n₂ = 6 x̄₁ = 100 and x̄₂ = 80 We know that , , . 2. Mean Read more about Mean – Problems[…]
Variance The variance of a variate is the square of standard deviation and it is denoted by σ² or Var(x). Var(x) = σ² = (standard deviation) Coefficient of dispersion = σ/x Step I: Firstly, write the given data, if it is given Step II: If either change of origin or change of scale is given, Read more about Variance[…]
Properties of Standard Deviation It is independent of origin It is dependent of change of scale It is not less than mean deviation from mean Let n₁ and n₂ be the sizes of two series. Their means and standard deviations are x̄₁, x̄₂ and σ₁, σ₂ respectively. Let x̄ denote the combined mean of two Read more about Properties of Standard Deviation[…]
Pair of straight lines – Theorem Statement: The product of the perpendicular from (α, β) to the pair of lines ax² + 2hxy + by² = 0 is . Proof: Let ax² + 2hxy + by² = 0 … (1) Represent lines l₁ x + m₁ y = 0 and l₂ x + m₂ y = Read more about Pair of straight lines – Theorem[…]
Inverse Trigonometric Functions – Problems If a function is one – one and onto from A to B, then function g which associates each elements y ϵ B to one and only one element x ϵ A such that y = f(x), then g is called the inverse function of f, denoted by x = Read more about Inverse Trigonometric Functions – Problems[…]
Positional Measures and Mean These are the values of the variable which divide the total frequency into a number of equal parts example median divides the total frequency into two equal parts. Some of these are as follows (i) Quartiles: Those values of the variable which divides total distribution into four equal parts are known as Read more about Positional Measures and Mean[…]
Geometrical Mean for Ungrouped and Grouped Data Geometric Mean (G.M): The nth root of the product of the values is called Geometric Mean. Geometric Mean for Ungrouped Data: If x₁, x₂, …, xn be n observations, then geometric mean is given by , , , . Geometric Mean for Grouped Data: If x₁, x₂, …, Read more about Geometrical Mean for Ungrouped and Grouped Data[…]
Matrices Problems i) Inverse Matrix: If , then A⁻¹ is equal to Solution: , |A| = 1 and adj (A) = A ∴ . ii) Unity Matrix: Element of a matrix A of order 10 x 10 are defined as aij = ωi + j (Where, ω is cube root of unity), then tr (A) Read more about Matrices Problems[…]
Rolle’s Theorem Let f be real valued function defined on the closed interval [a, b] such that It is continuous on the closed interval [a, b]. It is differential on the open interval (a, b) and F (a) = f (b) Then there exists c ϵ (a, b) such that f’(c) = 0 Geometrical Interpretation Read more about Rolle’s Theorem[…]