Algebra PYQ — Page 27

It is known that 3% of plastic bags manufactured in a factory are defective. Using the Poisson distribution on a sample of 100 bags, the probability of at most one defective bag is:

If $A = \begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}$, $B = \begin{bmatrix}a & 1 \\ b & -1\end{bmatrix}$ and $(A + B)^2 = A^2 + B^2$ then

Let $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2 \end{bmatrix}$. $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$ (A) $M_{23} = 6$ (B) $A_{22} = -8$ (C) $A_{13} = 7$ (D) $M_{32} = -5$ Choose the correct answer from the options given below:

If $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct? (A) $|\vec{p}| = |\vec{q}| = 1$ (B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors (C) $\vec{p}$ and $\vec{q}$ are collinear vectors (D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$ Choose the correct answer from the options given below:

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$. If |adj A| = |A|², then which of the following statements are correct? (A) A is a skew symmetric matrix. (B) A is a non-singular matrix. (C) A is a square matrix of order 4. (D) A is a symmetric matrix. Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}$ then $|A. adj A|$ is

If $A = \begin{bmatrix} 4 & 3 \\ 2 & -1 \\ 1 & 0 \end{bmatrix}$ and $B^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & -1 \end{bmatrix}$, then $A - B$ is equal to

If $A = \begin{bmatrix} 0 & l & -3 \\ -2 & 0 & 1 \\ m & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix, then

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ then

If $A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 1 & 2 \\ -1 & 0 & 1 \end{bmatrix}$, then which of the following statement(s) is/are correct? (A) A is singular matrix (B) |3A| = 135 (C) |adj A| = 125 (D) $|A^{-1}| = \frac{1}{5}$ Choose the correct answer from the options given below:

The mean of the number of heads in the two tosses of a coin is

The probability that a leap year selected at random will have 53 Mondays is

If $P$, $Q$ and $R$ are three singular matrices given by $P = \begin{bmatrix} 2 & 3a \\ 4 & 3 \end{bmatrix}$, $Q = \begin{bmatrix} b & 5 \\ 2a & 6 \end{bmatrix}$ and $R = \begin{bmatrix} a^2 + b^2 - c & 1 - c \\ c + 1 & c \end{bmatrix}$, then the value of $(2a + 6b + 17c)$ is

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & -5 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & -3 \\ 4 & -5 \\ 2 & -1 \end{bmatrix}$ and $BA = [b_{ij}]$, then $(b_{23} - b_{31})$ is equal to

If the corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 2), (3, 0), (6, 0), (6, 8) and (0,5), then which of the following are correct for the objective function $Z = 4x + 6y$? (A) The minimum value of the objective function occurs at (0, 2) and (3, 0) only. (B) The minimum value of the objective function occurs at the mid-point of the line segment joining the points (0, 2) and (3, 0) only. (C) The minimum value of the objective function occurs at every point of the line segment joining the points (0, 2) and (3, 0). (D) The difference between the maximum value and minimum value of the objective function is 60. Choose the correct answer from the options given below:

If $\vec{a} = 2\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = -\hat{i} + \hat{k}$ are three vectors, then the area (in sq. units) of the parallelogram whose diagonals are $(\vec{b} + \vec{c})$ and $(\vec{a} + \vec{c})$ is

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Match List-I with List-II | List-I | List-II | |---|---| | **X** | **Probability, P(X)** | | (A) 4 | (I) $\frac{1}{6}$ | | (B) 5 | (II) $\frac{5}{36}$ | | (C) 6 | (III) $\frac{1}{12}$ | | (D) 7 | (IV) $\frac{1}{9}$ | Choose the correct answer from the options given below:

If the system of equations $x - 3y + 5z = 3$ $x - 2y + 4z = 4$ $2x - 7y + \lambda z = 5$ has infinite number of solutions, then the value of $\lambda$ is:

For the matrix $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$, which of the following statements are correct? (A) The order of the matrix is 3 × 3 (B) |A| = 21 (C) $|adj\ A| = 225$ (D) A is skew symmetric matrix Choose the correct answer from the options given below:

If the matrix $A = \begin{bmatrix} x & 2 & y \\ -2 & 0 & 3 \\ -1 & z & 0 \end{bmatrix}$ is skew-symmetric, then the value of $2x - 3y + 5z$ is equal to

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$. If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to

The probability that A hits a target is $\frac{1}{5}$ and the probability that B hits it is $\frac{2}{3}$. The probability that the target will be hit if both A and B shoot at it independently is:

A die is rolled in such a way that an even number is twice likely to occur as an odd number. If the die is rolled twice, then the mean of the number of perfect squares in two tosses is:

The inverse of the matrix $A = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{8} \end{bmatrix}$ is