NCERT Solutions for Class 12 Science Maths Chapter 4 – Determinants
Explore the comprehensive NCERT Solutions for Class 12 Science Maths Chapter 4 on Determinants, featuring easy-to-follow step-by-step explanations. These solutions have gained immense popularity among Class 12 Science students, serving as a valuable resource for efficiently completing homework assignments and preparing for exams. All the questions and answers from Chapter 4 of the NCERT Book for Class 12 Science Maths are readily available here, providing students with free access to essential study materials. Page No 108: Question 1: Evaluate the determinants in Exercises 1 and 2. ANSWER: = 2(−1) − 4(−5) = − 2 + 20 = 18 Page No 108: Question 2: Evaluate the determinants in Exercises 1 and 2. (i) (ii) ANSWER: (i) = (cos θ)(cos θ) − (−sin θ)(sin θ) = cos2θ+ sin2θ = 1 (ii) = (x2 − x + 1)(x + 1) − (x − 1)(x + 1) = x3 − x2 + x + x2 − x + 1 − (x2 − 1) = x3 + 1 − x2 + 1 = x3 − x2 + 2 Page No 108: Question 3: If, then show that ANSWER: The given matrix is. Page No 108: Question 4: If, then show that ANSWER: The given matrix is. It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C1) for easier calculation. From equations (i) and (ii), we have: Hence, the given result is proved. Page No 108: Question 5: Evaluate the determinants (i) (iii) (ii) (iv) ANSWER: (i) Let. It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation. (ii) Let. By expanding along the first row, we have: (iii) Let By expanding along the first row, we have: (iv) Let By expanding along the first column, we have: Page No 109: Question 6: If, find. ANSWER: Let By expanding along the first row, we have: Page No 109: Question 7: Find values of x, if ANSWER: (i) (ii) Page No 109: Question 8: If, then x is equal to (A) 6 (B) ±6 (C) −6 (D) 0 ANSWER: Answer: B Hence, the correct answer is B. Page No 119: Question 1: Using the property of determinants and without expanding, prove that: ANSWER: Page No 119: Question 2: Using the property of determinants and without expanding, prove that: ANSWER: Here, the two rows R1 and R3 are identical. Δ = 0. Page No 119: Question 3: Using the property of determinants and without expanding, prove that: ANSWER: Page No 119: Question 4: Using the property of determinants and without expanding, prove that: ANSWER: By applying C3 → C3 + C2, we have: Here, two columns C1 and C3 are proportional. Δ = 0. Page No 119: Question 5: Using the property of determinants and without expanding, prove that: ANSWER: Applying R2 → R2 − R3, we have: Applying R1 ↔R3 and R2 ↔R3, we have: Applying R1 → R1 − R3, we have: Applying R1 ↔R2 and R2 ↔R3, we have: From (1), (2), and (3), we have: Hence, the given result is proved. Page No 120: Question 6: By using properties of determinants, show that: ANSWER: We have, Here, the two rows R1 and R3 are identical. ∴Δ = 0. Page No 120: Question 7: By using properties of determinants, show that: ANSWER: Applying R2 → R2 + R1 and R3 → R3 + R1, we have: Page No 120: Question 8: By using properties of determinants, show that: (i) (ii) ANSWER: (i) Applying R1 → R1 − R3 and R2 → R2 − R3, we have: Applying R1 → R1 + R2, we have: Expanding along C1, we have: Hence, the given result is proved. (ii) Let. Applying C1 → C1 − C3 and C2 → C2 − C3, we have: Applying C1 → C1 + C2, we have: Expanding along C1, we have: Hence, the given result is proved. Page No 120: Question 9: By using properties of determinants, show that: ANSWER: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Applying R3 → R3 + R2, we have: Expanding along R3, we have: Hence, the given result is proved. Page No 120: Question 10: By using properties of determinants, show that: (i) (ii) ANSWER: (i) Applying R1 → R1 + R2 + R3, we have: Applying C2 → C2 − C1, C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved. (ii) Applying R1 → R1 + R2 + R3, we have: Applying C2 → C2 − C1 and C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved. Page No 120: Question 11: By using properties of determinants, show that: (i) (ii) ANSWER: (i) Applying R1 → R1 + R2 + R3, we have: Applying C2 → C2 − C1, C3 → C3 − C1, we have: Expanding along C3, we have: Hence, the given result is proved. (ii) Applying C1 → C1 + C2 + C3, we have: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Expanding along R3, we have: Hence, the given result is proved. Page No 121: Question 12: By using properties of determinants, show that: ANSWER: Applying R1 → R1 + R2 + R3, we have: Applying C2 → C2 − C1 and C3 → C3 − C1, we have: Expanding along R1, we have: Hence, the given result is proved. Page No 121: Question 13: By using properties of determinants, show that: ANSWER: Applying R1 → R1 + bR3 and R2 → R2 − aR3, we have: Expanding along R1, we have: Page No 121: Question 14: By using properties of determinants, show that: ANSWER: Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Applying C1 → aC1, C2 → bC2, and C3 → cC3, we have: Expanding along R3, we have: Hence, the given result is proved. Page No 121: Question 15: Choose the correct answer. Let A be a square matrix of order 3 × 3, then is equal to A. B. C. D. ANSWER: Answer: C A is a square matrix of order 3 × 3. Hence, the correct answer is C. Page No 121: Question 16: Which of the following is correct? A. Determinant is a square matrix. B. Determinant is a number associated to a matrix. C. Determinant is a number associated to a square matrix. D. None of these ANSWER: Answer: C We know that to every square matrix, of order n. We can associate a number called the determinant of square matrix A, where element of A. Thus, the determinant is a number associated to a square matrix. Hence, the correct answer is C. Page No …
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