Limits, Continuity and Differentiability – Standard Results on Limits To evaluate trigonometric limits, reduce the terms of the function in terms of sinθ and cosθ. Remove the positive and negative signs in between two terms i.e., express the function in product form. Arrange terms and take help of the following standard results. Trigonometric Limits: i) Read more about Limits, Continuity and Differentiability – Standard Results on Limits[…]

Limits, Continuity and Differentiability – Method of Rationalization Rationalization method is used when, we have radical signs in an expression (like ½, ⅓ etc) and there exists a negative sign between two terms of an algebraic expression. After rationalization, the terms are factorized which on cancellation gives the required result. Example: is equal to Solution: Read more about Limits, Continuity and Differentiability – Method of Rationalization[…]

Differential Equations Reducible to Homogeneous Form A differential equation of the form , where can be reduced to homogeneous form by taking new variable x and y such that x = X + h and y = Y + k, where h and k are constants to be so chosen as to make the given Read more about Differential Equations Reducible to Homogeneous Form[…]

Homogeneous Differential Equations A differential equation of the form , where f₁ (x, y) and f₂ (x, y) are homogeneous functions of x and y of the same degree, is called a homogeneous equation. For solving such equations, put y = υx and , These substitutions transform the given equation into an equation of the Read more about Homogeneous Differential Equations[…]

Differential Equations Reducible to Variable Separable Method Differential equation of the first order cannot be solved directly by variable separable method. But by some substitution, we can reduce it to a differential equation with separable variable. Let the differential equation is of the form , Can be reduced to variable separable form by the substitution Read more about Differential Equations Reducible to Variable Separable Method[…]

Properties of Triangles – EX-RADII Definition: The internal bisector of angle A and external bisectors of angles B, C of ΔABC are concurrent. The point of concurrence is called ex-centre of ΔABC opposite to the vertex A. It is denoted by I₁. The point I₁ is equidistant to the sides of the triangle. The circle Read more about Properties of Triangles – EX-RADII[…]

Inverse Trigonometry – Theorem Statement: Tan⁻¹ x + Tan⁻¹ y = Tan⁻¹ for x > 0, y > 0, xy < 1 = π + Tan⁻¹ for x > 0, y > 0, xy > 1. Proof: Case (i): Suppose x > 0, y > 0, xy < 1 x > 0 → 0 < Tan⁻¹ Read more about Inverse Trigonometry – Theorem[…]

Square Root of A Complex Number Statement: √a + ib = ± (x + iy) where . Proof: √a + ib = ± (x + iy) → a + ib = (x + iy)² = x² – y² + 2ixy a = x² – y² … (1) b = 2xy … (2) ∴ (x² + Read more about Square Root of A Complex Number[…]

Functions – Theorems Statement: If f: A → B, g: B → C are two one one onto functions then gof: A → C is also one one onto. i) If f: A → B, g: B → C are two one one functions then gof: A → C is also one one. Proof: Let x₁, Read more about Functions – Theorems[…]

Point of Contact – Circles 1. The condition that the straight-line lx + my + n = 0 may touch the circle x² + y² = a² is n² = a² (l² + m²) and the point of contact is . Proof: Given line is lx + my + n = 0 … (1) Given Read more about Point of Contact – Circles[…]