De Moivre’s Theorem It is customary to write cisθ for cosθ + i sinθ. Thus, we may state the De Moivre’s theorem as (cisθ)ⁿ = cis(nθ) if n ϵ z. (cosθ + i sinθ)⁻ⁿ = cos(-nθ) + i sin(-nθ) = cos(nθ) – isin(nθ) provided n is an integer. (cosθ + isinθ) (cosθ – isinθ) = Read more about De Moivre’s Theorem[…]
Theorems on Derivatives 1. . 2. where k is any constant. 3. . In general , 4. . Example: Find for y = x sinx logx Solution: Given that y = x sinx logx We can use the formula , y’ = d/dx (x sinx logx) = sinx logx d/dx(x) + x logx d/dx(sinx) + Read more about Theorems on Derivatives[…]
Cross Product of Vector Let , be two vectors. The cross product or vector product or skew product of vectors , is denoted by and is defined as follows. i) If or or , are parallel then . ii) If , , , are not parallel then where is a unit vector perpendicular to Read more about Cross Product of Vector[…]
Vectors – Problems Key points: let , be two vectors dot product (or) scalar product (or) direct product (or) inner product denoted by which is defined as where . (i) The product , is zero when (or) (or) θ = 90°. (ii) The angle between the vectors is . Problems: 1. Find the angle between Read more about Vectors – Problems[…]
Equation of Tangents and Normal Let P (x₁, y₁) be any point on the curve y = f(x) If a tangent at P makes an angle θ with the positive direction of the x – axis, then dy /dx = tanθ. Equation of Tangent: Equation of a tangent at point P (x₁, y₁) is . Read more about Equation of Tangents and Normal[…]
Binomial Theorem – Problems Key Points: 1. . 2. . 3. Number of terms in (x + a) ⁿ is (n + 1), where n is a positive integer Examples 1: Expand by using binomial theorem Solution: . . Examples 2: Find the 10th term in . Solution: . We know that . . . Read more about Binomial Theorem – Problems[…]
Trace, Transpose of a Matrix – Problems 1. If the trace of the Matrix . Solution: Given . Trace of matrices is defined as Tr(D) = = (x – 1) + (x² – 2) + (x – 3) + (x² – 6) = 0 x – 1 + x² – 2 + x- 3 + Read more about Trace, Transpose of a Matrix – Problems[…]
Evaluation of Trigonometric Limits (where θ is in radians). Proof: 1. Consider a circle of radius such that ∠AOB = θ, where θ is measured in radius and its value is very small. Suppose the tangent at A meets. OB produced at P. from Fig. we have Area of ΔABC < area of sector Read more about Evaluation of Trigonometric Limits[…]
Limits – II Evaluation of Algebraic limits using some standard limits: We know that binomial expansion for any rational power. . Where |x| < 1 1. Theorem: If n ϵ Q, then . Proof: We have . . . . . (when x → 0, (1 + x) ⁿ = 1 + n x) = Read more about Limits – II[…]
Limits – I Limits of The Form . 1. Form 0⁰, ∞⁰. Let then . . . . Example 1: Evaluate . Solution: Given that . . . (since x increases faster than loge x when x → ∞) = e⁰ = 0 Example 2: Evaluate . Solution: Given that . Let . . . Read more about Limits – I[…]