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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 […]

Solve the following questions. The questions are designed to test your conceptual understanding of algebra at a Class 9 level. This sheet includes evaluation, factorization, simplification, and solving equations. 📘 Section A: Evaluate the Expression Q1. Evaluate: Q2. Evaluate: Q3. Evaluate: Q4. Evaluate: Q5. Evaluate: Q6. Evaluate: Q7. Evaluate: Q8. Evaluate: Q9. Evaluate: Q10. Evaluate:

Solve the following questions. Show all necessary steps. Focus on evaluating, simplifying, and solving algebraic expressions and equations. 📘 Section A: Evaluate the Expression Q1. Evaluate: when Q2. Evaluate: when Q3. Evaluate: when Q4. Evaluate: when Q5. Evaluate: when Q6. Evaluate: when Q7. Evaluate: when Q8. Evaluate: when Q9. Evaluate: when Q10. Evaluate: when 📗

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

Question 1: Find the transpose of each of the following matrices: (i) (ii) (iii) Solution 1: (i) Let . Then (ii) Let . Then (iii) Let . Then Question 2 If and verify that(i) (ii) Solution 2 (i) Compute : Then Now compute : Thus (i) is verified. (ii) Compute : Then Thus (ii) is

Question 1:  Let Find: (i)  (ii)  (iii)  (iv)  (v)  Solution(i)   (ii)  (iii)  (iv)   Row 1 × Col 1 :  Row 1 × Col 2 :  Row 2 × Col 1 :  Row 2 × Col 2 :  (v)  Row 1 × Col 1 :  Row 1 × Col 2 :  Row 2 × Col 1 :  Row 2 × Col 2 :  Question 2: Compute the following (i)  (ii)  (iii)  (iv)  Solutions: (i) (ii) Add entry‑wise: (iii) (iv) Use : Question 3. Compute (i) Solution Entry (1,1):  Entry (1,2):  Entry (2,1):  Entry (2,2):  (ii) The first matrix is 3 × 1 (column) and the second