Author name: Ankur Negi

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