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ISC Class XII Notes 2026 : Mathematics

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ISC Class 12 Maths, Specimen Paper -2026, Unit 2 Algebra ( Matrices & Determinants] [10 M] Question no 1 [1 Marks each] Choose the correct option in subpart (i),(iv) & (vi) and answer the questions as instructed in subpart (xii) (i) If A and B are square matrices of order 3, A is non-singular matrix and AB = O, then the matrix B is: (a) unit matrix (b) Scalar matrix (c) non-singular matrix (d) null matrix 2 (iv) Consider = | 2 2 2 2 | Assertion: = 4 | | Reason: If all elements of one row or one column of a determinant are multiplied by a scalar k, then the value of the determinant is multiplied by k. Which of the following is correct? (a) Both Assertion and Reason are true, and Reason is the correct explanation for Assertion. (b) Both Assertion and Reason are true, but Reason is not the correct explanation for Assertion. (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. 0 0 (vi) IfA = (0 0) , 0 0 0 0 0 0 (a) ( 0 (b) (0 0) 0) 0 0 0 0 0 0 0 0 (c) ( 0 0 ) (d) ( 0 0 ) 0 0 0 0 (xii) The value of the determinant of a matrix A of order 3, is 3. If C is the matrix of cofactors of the matrix A ,then what is the value of determinant of 2? [sp2025] Question no 11 [6 Marks] [SP 2025] To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised Rs 410. Student of school B sold 20 paper bags. 10 scrap books and 20 pastel sheets and raised Rs 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised Rs 440. Answer the following question: (i) Translate the problem into a system of equations. (ii) Solve the system of equation by using matrix method. (iii) Hence, find the cost of one paper bag, one scrap book and one pastel sheet. ISC Class 12 Mathematics Board Exam 2025 Unit 2 Algebra [Matrices & Determinant] [10 Marks] Question 1 (i) (vi) (xi) (xiii) 0 ] , then A16 is: 0 0 (a) Unit matrix (c) Diagonal matrix If A = [ [1] (b) Null matrix (d) Skew matrix The existence of the unique solution of the system of equations x + = and 5 + = 2 depend on: (a) only (b) = 1 (b) both k and (d) k only Statement 1: If A is an invertible matrix , then ( 2 ) 1 = ( 1 )2 : Statement 2: If A is an invertible matrix, then | 1 | = | | 1 (a) Statement 1 is true, and Statement 2 is false. (b) Statement 2 is true, and Statement 1 is false. (c) Both the statement are true. (d) Both the statement are false. 0 1 2 For what value of x, is A = [ 1 0 3 ] a skew symmetric matrix? 3 0 Question 5 [1] [1] [2] Three shopkeepers Gaurav, Rizwan and jocob use carry bags made of polythene, handmade paper and newspaper. The number of polythene bags, handmade bags and newspaper bags used by Gaurav, Rizwan and jocob are ( 20, 30, 40) ,(30, 40, 20) and (40, 20, 30) respectively. One polythene bag costs 1, one handmade bag is for Rs 5 and one newspaper bag costs Rs 2. Gaurav, Rizwan and jocob spend Rs A, Rs B and Rs C respectively on these carry bags. Using the concepts of matrices and determinants, answer the following questions: i. ii. Represent the above information in Matrix form. Find the values of Rs A, Rs B and Rs C. Question 7 1 | If + + = 0 then show that 3 [4] 1 3 1 | = 0 , using properties of determinant. 3 ISC Class 12 Maths, Specimen Paper -2025, Unit 2 Algebra ( Matrices & Determinants] [10 M] Question no 1 [1 Marks each] choose the correct option answer the questions as instructed (i) a) c) (vi) A matrix which is both symmetric and skew symmetric matrix is a /an: Triangular matrix b) Identity matrix Diagonal matrix d) Null matrix 6 2 2 If | | = k , then what is the value of |3 | ? 3 a) (x) 6 b) 2k c) 3k d) 6k In a third order matrix denotes the elements of the row and the column. 0, = 1 , > { A = = 1 , < Assertion : Matrix A is not invertible. Reason: Determinant A = 0 Which of the following is correct? a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion . b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion . c) Assertion is true and Reason is false. d) Assertion is false and Reason is True. (xii). The value of the determine of a matrix A of order 3 is 3 . If C is the matrix of cofactors of the matrix A , then what is the value of determinant of 2 ? Question no 11[6 Marks] To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised Rs 410. Student of school B sold 20 paper bags. 10 scrap books and 20 pastel sheets and raised Rs 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised Rs 440. Answer the following question: (i) Translate the problem into a system of equations. (ii) Solve the system of equation by using matrix method. (iii) Hence, find the cost of one paper bag, one scrap book and one pastel sheet. ISC Class 12 Maths, Board Exam Paper -2024 Unit 2 Algebra ( Matrices & Determinants] [10 M] Question no 1 [1 marks each] Choose the correct option answer the questions as instructed (vii) Assertion: Let the matrices A = ( 3 2) 4 2) and B = ( be such that 100 = 100 5 4 5 3 Reason: AB = BA implies = for all positive integers n. (a) (b) (c) (d) Both Assertion and reason are true and reason is the correct explanation for assertion : Both Assertion and reason are true but reason is not the correct explanation for assertion : Assertion is true and Reason is false . Assertion is false and Reason is true. (x) If A and B are symmetric matrices of the same order , then AB BA is : a) b) c) d) Skew symmetric matrix Symmetric matrix Diagonal matrix Identity matrix Question no 3 [2 Marks] Find the value of the determinant given below, without expanding it any stage. | 1 ( + ) 1 ( + )| 1 ( + ) Question no 11 [6 Marks] A furniture factory uses three types of wood namely, teakwood, rosewood and satinwood for manufacturing three types of furniture that are, table, chair and cot. The wood requirements (in tones) for each type of furniture are given below: Teakwood Rosewood Satinwood Table 2 1 3 Chair 3 1 2 Cot 4 2 1 It is found 29 tonnes of teakwood , 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture. Using the above information , answer the following question: (i) (ii) (iii) Express the data given in the table above in the form of a set of simultaneous equations. Solve the set simultaneous equations formed in subpart (i) by matrix method. Hence , find the number of table (s) , chair (s) and cot (s) produced. ISC Class 12 Maths, Board Exam Paper -2023 Unit 2 Algebra ( Matrices & Determinants] [10 M] Question no 1[1 Marks] Choose the correct option answer the questions as instructed (ii) If A is a square matrix of order 3, then |2 | is equal to: a) 2| | c) 4| | b) 8 | | d) 6| | 0 ] (xii)For what value of k the matrix [ is a skew symmetric matrix? 6 0 Question 3 [2 Marks] Evaluate the following determinant without expanding : 5 | + 5 + 5 | + Question 11[6 Marks] Use matrix method to solve the following system of equation: 2 4 6 3 10 + + - 6 =4 5 + =1 9 20 + - =2

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