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What are Partial Fractions? Steps for Using Partial Fractions Types of Partial Fractions Notes Tip:Whenever you see a rational function, first try to write it in one of the forms above. This will make integration much easier! Example of Partial Fraction Decomposition Express as a sum of partial fractions. Step 1: Identify the Form: The

Trigonometry has a wide set of identities that simplify calculations and problem-solving. Here are the most important formulae, grouped by type, for your reference. 1. Trigonometric Functions of Sum and Difference of Two Angles 2. Product to Sum (or Product to Series) Formula 3. Sum to Product (or Series to Product) Formulas 4. Double Angle

Question 1. Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = -3, and at x = 5. Solution: To prove that a function is continuous at a point , we need to show: This requires that: 1. is defined,2. The left-hand limit (LHL) and right-hand limit

Question 1: Evaluate Minors and Cofactors of the elements of the following determinant: (i) Solution: Element a₁₁ = 2 Element a₁₂ = -4 Element a₂₁ = 0 Element a₂₂ = 3 Summary Table: (ii) Solution: Element a₁₁ = a Element a₁₂ = c Element a₂₁ = b Element a₂₂ = d Summary Table: Question 2:

Question 1: Find the area of the triangle with vertices at the points given: (i) Solution:Using the formula for the area of triangle: Substituting the points: Expanding: (ii) Solution:Substituting the points: Expanding: (iii) Solution:Substituting the points: Expanding: Question 2: Show that points are collinear. Solution:If points are collinear the area of a triangle formed by

Evaluate the determinants in Exercises 1 and 2 Question 1: Solution: Question 2 (i) Solution: (ii) Solution: Question 3: If , Then show that Solution: Now, , Hence Proved Question 4: If , then show that |3A| = 27 |A| Solution: determinant of triangular matrix is product of diagonal elements: Now, This is an upper

Question 1: If and are symmetric matrices, prove that is a skew‑symmetric matrix. Solution: Because and are symmetric, their transposes equal themselves: and . Consider the transpose of : By definition, a matrix is skew‑symmetric when . Hence is skew‑symmetric. Question 2: Show that is symmetric or skew‑symmetric according as is symmetric or skew‑symmetric. Solution: Step 1 Recall

Question 1: Matrices A and B will be inverse of each other only if (A) (B) (C) , (D) Solution: Two square matrices of the same order, say and , are inverses of each other if and only if each is the multiplicative inverse of the other, i.e., This relation guarantees that both left‑ and