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Matrices and Determinants are essential topics in JEE Mathematics that test conceptual understanding and application in solving systems of equations, transformations, and inverse operations. This practice set is designed to reinforce these foundational concepts. Topic: Matrices and Determinants 1. If A = [[2, 3], [4, 5]], then |A| is: a) 2 b) -2 c) -1 d) 10 2. If A is a 2×2 matrix such that A² = I, then A is: a) Invertible b) Non-invertible c) Null matrix d) Diagonal 3. For a 3×3 matrix A, the value of |kA| is: a) k³|A| b) k²|A| c) k|A| d) |A| 4. If A is a singular matrix, then |A| is: a) 1 b) -1 c) 0 d) Infinite 5. If A and B are square matrices of the same order and AB = BA, then: a) A and B are equal b) A is symmetric c) A and B commute d) B is a zero matrix 6. If A is an orthogonal matrix, then A⁻¹ is: a) A b) Aᵀ c) -A d) None of these 7. The adjoint of a diagonal matrix is: a) Same diagonal matrix b) Zero matrix c) Inverse matrix d) Transpose 8. The determinant of a skew-symmetric matrix of odd order is: a) 0 b) 1 c) -1 d) Not defined 9. Which matrix has no inverse? a) Identity matrix b) Orthogonal matrix c) Singular matrix d) Diagonal matrix 10. Which of the following is true for all invertible matrices A and B? a) (AB)⁻¹ = A⁻¹B⁻¹ b) (AB)⁻¹ = B⁻¹A⁻¹ c) (A + B)⁻¹ = A⁻¹ + B⁻¹ d) A⁻¹ + B⁻¹ = (AB)⁻¹ Submit Answers Matrices and Determinants PYQs Theory Explanation by Rahul Sir 1. If A = [[2, 3], [4, 5]], then |A| is: Correct Answer: b) -2 Theory: For a 2 × 2 matrix A = [[a, b], [c, d]], its determinant is calculated using |A| = ad − bc. Here, A = [[2, 3], [4, 5]], so |A| = (2)(5) − (3)(4) = 10 − 12 = −2. The determinant is a scalar value associated with a square matrix and is extremely important for studying invertibility, solving linear equations, and calculating areas or volumes. If the determinant is zero, the matrix is singular and has no inverse. A non-zero determinant indicates that the matrix is non-singular and therefore invertible. 2. If A is a 2×2 matrix such that A² = I, then A is: Correct Answer: a) Invertible Theory: Given A² = I, we can write AA = I. By definition, a matrix A is invertible if there exists a matrix B such that AB = BA = I. Here, A itself acts as its inverse because AA = I. Therefore, A⁻¹ = A. This also shows that A must be non-singular, so its determinant cannot be zero. Such a matrix is called an involutory matrix because its square is the identity matrix. Examples include I and certain reflection matrices. Whenever a square matrix satisfies A² = I, its inverse exists immediately without needing the usual adjoint or determinant formula. 3. For a 3×3 matrix A, the value of |kA| is: Correct Answer: a) k³|A| Theory: For an n × n matrix A, multiplying every element of A by a scalar k multiplies its determinant by kⁿ. Therefore, |kA| = kⁿ|A|. Since A is a 3 × 3 matrix, n = 3, giving |kA| = k³|A|. This rule follows from the fact that a determinant is multilinear in the rows or columns. Each of the three rows receives a factor k, so the total factor becomes k × k × k = k³. Students should remember the general formula |kA| = kⁿ|A| for an n-order square matrix. 4. If A is a singular matrix, then |A| is: Correct Answer: c) 0 Theory: A square matrix is called singular precisely when its determinant is zero. Thus, if A is singular, |A| = 0. The determinant provides an important test for invertibility: a square matrix A has an inverse if and only if |A| ≠ 0. When |A| = 0, the rows or columns of the matrix are linearly dependent, meaning one row or column can be expressed as a combination of others. Consequently, the matrix does not have a unique inverse. This concept is fundamental in solving simultaneous linear equations, finding rank, and applying determinant-based methods. 5. If A and B are square matrices of the same order and AB = BA, then: Correct Answer: c) A and B commute Theory: Matrix multiplication is generally not commutative, meaning AB is usually different from BA. However, when two square matrices satisfy AB = BA, they are said to commute with each other. Therefore, the given condition directly means that A and B commute. This does not imply that A = B, that either matrix is symmetric, or that B is a zero matrix. The commutative property is a special relationship between particular matrices. For example, a matrix always commutes with the identity matrix and with its own powers. Recognizing the phrase AB = BA is an important examination shortcut. 6. If A is an orthogonal matrix, then A⁻¹ is: Correct Answer: b) Aᵀ Theory: A square matrix A is orthogonal if AᵀA = AAᵀ = I. Comparing this with the definition of an inverse, the matrix that multiplies A to produce I is its inverse. Hence, for an orthogonal matrix, A⁻¹ = Aᵀ. This is one of the most important properties of orthogonal matrices. It also means that the rows and columns of A form orthonormal sets. Orthogonal matrices preserve lengths and angles under transformations, which is why they are important in geometry, coordinate transformations, and linear algebra. Their determinant is always either +1 or −1. 7. The adjoint of a diagonal matrix is: Correct Answer: a) Same diagonal matrix (only in the general structural sense: it remains diagonal) Theory: The adjoint, or adjugate, of a square matrix is the transpose of its cofactor matrix. For a diagonal matrix, all off-diagonal

Part 19 – ProbabilityThis section contains 10 multiple-choice questions from previous IIT JEE exams focused on Probability. These questions are designed to test your understanding of fundamental probability concepts including independent and dependent events, Bayes’ theorem, and conditional probability. Part 19 – IIT JEE Maths Practice Paper: Probability Instructions: Each correct answer gives 4 marks. No negative marking. Choose the best answer for each question and click “Submit” to view your score and explanations. Two dice are thrown simultaneously. What is the probability that the sum is divisible by 4? a) 1/4 b) 1/3 c) 5/18 d) 1/2 A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is either a red card or a king? a) 7/13 b) 4/13 c) 15/26 d) 8/13 An unbiased coin is tossed 5 times. What is the probability of getting at least 3 heads? a) 1/2 b) 26/32 c) 5/16 d) 11/32 A bag contains 3 red and 5 black balls. Two balls are drawn without replacement. What is the probability both are black? a) 5/14 b) 5/7 c) 5/8 d) 10/21 If A and B are two independent events such that P(A) = 1/3 and P(B) = 1/4, then P(A ∪ B) is: a) 1/2 b) 7/12 c) 1/3 d) 3/4 What is the probability of getting a sum of 7 or 11 when two dice are rolled? a) 2/9 b) 1/6 c) 1/4 d) 5/36 A and B throw a die alternatively. The one who gets a 6 first wins. If A starts, then what is the probability that A wins? a) 5/11 b) 6/11 c) 1/2 d) 7/13 If P(E) = 0.3 and P(F) = 0.4 and E and F are mutually exclusive, find P(E ∪ F). a) 0.12 b) 0.7 c) 1 d) 0.1 If two events A and B are such that P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3, then P(A|B) is: a) 0.3 b) 0.6 c) 0.9 d) 0.5 In how many ways can 3 boys and 2 girls be selected from 5 boys and 4 girls? a) 40 b) 60 c) 100 d) 10 Submit

Boost your IIT JEE preparation with Part 18 of our Maths Practice Paper series. This set includes carefully selected Binomial Theorem questions based on previous years’ IIT JEE papers. Practice and test your understanding with instant evaluation — answers, score, and explanations included. IIT JEE Maths Practice Paper – Part 18: Binomial Theorem 1. The middle term in the expansion of (1 + x)18 is: 9th term 10th term 8th term 11th term 2. The coefficient of x3 in the expansion of (2 + x)5 is: 40 80 20 60 3. In the expansion of (1 – 3x)4, the term independent of x is: 81 -81 1 -1 4. The general term in the expansion of (a + b)n is: nCr an-r br nCr ar bn-r nCr an br None of the above 5. In the expansion of (1 + x)10, the sum of coefficients is: 10 0 1024 512 Submit

Boost your problem-solving skills in Permutations and Combinations with 10 carefully selected multiple-choice questions from previous IIT JEE exams. This set helps you master arrangements, selections, and advanced counting principles essential for scoring well in combinatorics. IIT JEE Maths Practice Paper – Part 17: Permutations and Combinations In how many ways can the letters of the word “APPLE” be arranged? a) 120 b) 60 c) 240 d) 100 How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition? a) 120 b) 625 c) 360 d) 256 Number of ways to choose 3 balls from 5 red and 4 green balls: a) 56 b) 84 c) 126 d) 36 How many ways can 3 boys and 2 girls be seated in a row such that the girls do not sit together? a) 72 b) 36 c) 144 d) 60 Number of circular permutations of 6 distinct objects: a) 720 b) 120 c) 60 d) 5040 A committee of 3 is to be formed from 4 men and 3 women. Number of ways it can be done: a) 35 b) 20 c) 30 d) 25 From the word “BANANA”, how many unique permutations can be formed? a) 60 b) 120 c) 360 d) 720 The number of permutations of 5 different books taken 3 at a time: a) 10 b) 60 c) 20 d) 15 If 5 people sit around a round table, how many seating arrangements are possible? a) 120 b) 24 c) 60 d) 30 The number of ways to divide 10 students into 2 groups of 5 each: a) 126 b) 252 c) 462 d) 180 Submit