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JEE Advanced Number Systems — practice questions

Explore 15 free Number Systems questions in JEE Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Number Systems questions and exercises with answers

Showing questions 1–15 of 15.

If $A$ is a matrix of order $3 \times 4$ and $B$ is of order $4 \times 5$, then the order of the product $AB$ is:Read the answer and worked solution →For any square matrix $A$, the matrix $A + A'$ is:Read the answer and worked solution →The diagonal entries of any skew-symmetric matrix are:Read the answer and worked solution →If $A$ and $B$ are matrices for which $AB$ is defined, then $(AB)'$ equals:Read the answer and worked solution →A square matrix $A$ is **invertible** if and only if:Read the answer and worked solution →For a non-singular $n \times n$ matrix $A$, the relation between $\det(\operatorname{adj} A)$ and $\det A$ is:Read the answer and worked solution →A square matrix $A$ is called **idempotent** if:Read the answer and worked solution →The trace of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & 5 & 3 \\ 4 & -2 & 7 \end{bmatrix}$ is:Read the answer and worked solution →A square matrix $A$ satisfies $A^T A = I$. Such a matrix is called:Read the answer and worked solution →If $A$ is a square matrix with $|A| = 3$ and order $n = 3$, then $|\operatorname{adj}(A)|$ equals:Read the answer and worked solution →If $(d,m,n)$ are direction cosines of a line, which relation must hold?Read the answer and worked solution →Which non-parametric vector condition represents the line through points with position vectors $\bar a$ and $\bar b$?Read the answer and worked solution →If a line passes through points with position vectors $\bar a$ and $\bar b$, then its vector equation isRead the answer and worked solution →Which non-parametric vector equation represents the line passing through the point with position vector $\bar a$ and parallel to $\bar b$?Read the answer and worked solution →The vector equation of the line passing through the point with position vector $\bar a$ and parallel to the vector $\bar b$ isRead the answer and worked solution →

Worked practice questions

If $A$ is a matrix of order $3 \times 4$ and $B$ is of order $4 \times 5$, then the order of the product $AB$ is:

  1. $3 \times 4$
  2. $4 \times 5$
  3. $3 \times 5$
  4. $5 \times 3$
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Correct answer: C. $3 \times 5$

The product $A_{m \times n} \cdot B_{n \times p}$ is defined when the number of columns of $A$ equals the number of rows of $B$ ($n = 4$ here). The result has order $m \times p = 3 \times 5$.

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For any square matrix $A$, the matrix $A + A'$ is:

  1. Always symmetric
  2. Always skew-symmetric
  3. Always singular
  4. Always the zero matrix
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Correct answer: A. Always symmetric

**Taking the transpose:** $(A + A')' = A' + (A')' = A' + A = A + A'$. Since the matrix equals its own transpose it is symmetric. By contrast, $A - A'$ is always skew-symmetric. Every square matrix can be uniquely written as a sum $\frac{1}{2}(A + A') + \frac{1}{2}(A - A')$ of a symmetric and a skew-symmetric matrix.

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The diagonal entries of any skew-symmetric matrix are:

  1. Always $1$
  2. Always $0$
  3. Equal to each other but not necessarily $0$
  4. Always negative
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Correct answer: B. Always $0$

A skew-symmetric matrix satisfies $a_{ij} = -a_{ji}$ for all $i, j$. Setting $i = j$ gives $a_{ii} = -a_{ii}$, hence $2a_{ii} = 0$, so $a_{ii} = 0$. Every diagonal entry of a skew-symmetric matrix is therefore zero.

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