Algebra PYQ — Page 8

A relation R on the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x,y): 3x - y = 0\}$ is

Match List-I with List-II | List-I | List-II | | :--- | :--- | | (Matrix) | (Inverse of the Matrix) | | (A) $\begin{pmatrix} 1 & 7 \\ 4 & -2 \end{pmatrix}$ | (I) $\begin{pmatrix} 2/15 & 1/10 \\ -1/15 & 1/5 \end{pmatrix}$ | | (B) $\begin{pmatrix} 6 & -3 \\ 2 & 4 \end{pmatrix}$ | (II) $\begin{pmatrix} 1/5 & -2/15 \\ -1/10 & 7/30 \end{pmatrix}$ | | (C) $\begin{pmatrix} 5 & 2 \\ -5 & 4 \end{pmatrix}$ | (III) $\begin{pmatrix} 1/15 & 7/30 \\ 2/15 & -1/30 \end{pmatrix}$ | | (D) $\begin{pmatrix} 7 & 4 \\ 3 & 6 \end{pmatrix}$ | (IV) $\begin{pmatrix} 2/15 & -1/15 \\ 1/6 & 1/6 \end{pmatrix}$ | Choose the correct answer from the options given below:

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ is

If a person A speaks the truth in 80% cases and the person B speaks the truth in 75% cases, then the probability that they contradict each other in a statement is

An objective function $z = ax + by$ is maximum at points (15,15) and (0, 20). If $a, b \geq 0$ and $ab = 27$, then the maximum value of the objective function is

Which of the following is incorrect about the Linear Programming Problem (LPP)?

If the points P, Q, R with position vectors $5\hat{i} + \lambda\hat{j}$, $20\hat{i} - \hat{j}$ and $15\hat{i} - 6\hat{j}$ respectively are collinear, then the value of $\lambda$ is

The difference of two different skew-symmetric matrices is:

The feasible region of a linear programming problem is bounded. The corresponding objective function is Z= 3x-4y. The objective function attains

Match List-I with List-II | List-I | List-II | |---|---| | **(Matrix)** | **(Determinant)** | | (A) $\begin{bmatrix} 1 & 7 \\ -3 & 5 \end{bmatrix}$ | (I) 24 | | (B) $\begin{bmatrix} -2 & 5 \\ -3 & -3 \end{bmatrix}$ | (II) 32 | | (C) $\begin{bmatrix} -12 & 8 \\ -16 & 8 \end{bmatrix}$ | (III) 21 | | (D) $\begin{bmatrix} 15 & 9 \\ -21 & -11 \end{bmatrix}$ | (IV) 26 | Choose the correct answer from the options given below:

In a linear programming problem(LPP), the maximum value of the objective function $Z = 2x + 5y$ subjected to the constraints: $2x + 3y ≤ 6$ $2x + y ≤ 4$ $x, y ≥ 0$ is

Let P and Q be any two invertible matrices of the same order. Then Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Matrix** | **Equivalent matrix** | | (A) $(P Q)^{-1}$ | (I) $Q^{-1}P$ | | (B) $(P^{-1}Q)^{-1}$ | (II) $Q P^{-1}$ | | (C) $(P Q^{-1})^{-1}$ | (III) $Q^{-1}P^{-1}$ | | (D) $(P^{-1}Q^{-1})^{-1}$ | (IV) Q P | Choose the **correct** answer from the options given below:

Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II | List-I | List-II | |---|---| | (A) If Var (X) = $a$, then Var (2X + 3) is | (I) 11$a$ | | (B) If E (X) = $a$, then E (2X) is | (II) 6$a$ | | (C) If Var (X) = $a$, then Var(3X - $a$) + Var ($\sqrt{2}x + \beta$) is | (III) 4$a$ | | (D) If E (X) = $\frac{5a}{12}$, then E (12X + $a$) is | (IV) 2$a$ | Choose the correct answer from the options given below:

Consider the following L.P.P Max. z = 5x + 2y; subject to -2x - 3y ≤ -6, x - 2y ≤ 2, 3x + 2y ≤ 12, -3x + 2y ≤ 3 and x, y ≥ 0 then

If the system of equation $x - y + z = 4$ $x - 2y - 2z = 9$ $2x + y + \lambda z = 1$ has a unique solution, then

If A and B are two independent events, then which of the following is/are true? (A) $P(A \cap B) = 0$ (B) $P(A \cup B) = 1 - P(A')P(B')$ (C) $P(A \cup B) = P(A)P(B)$ (D) $P(A \cap B) = P(A)P(B)$ Choose the correct answer from the options given below:

Let A, B, C, D and E be matrices of order $2 \times n, 3 \times k, 2\times p, n \times 3$ and $p \times k$ respectively. Choose the correct statement(s) from the following? (A) EB + DB will be defined if $k = 3, p = n$. (B) EB + DB will be defined if $k = 2, p = 3$. (C) If n = p = 2, then the order of the matrix $5A^2 - 3C$ is $2 \times 2$. (D) If n = p, then the order of the matrix $5A^2 - 3C$ is $p \times k$. Choose the correct answer from the options given below:

Under which of the following conditions the Poisson distribution is the limiting case of, binomial distribution: (A) The number of trials is indefinitely large. (B) The probability of success for each trial is indefinitely small. (C) The product of the number of trials and the probability of success for each trial is finite. (D) The probability of success for each trial is indefinitely large. Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}$, $X = \begin{bmatrix} \alpha \\ -2 \end{bmatrix}$, $B = \begin{bmatrix} 7 \\ 32 \end{bmatrix}$ and $AX = B$, then the value of the $\alpha$ is

If A is a non-singular matrix of order 3 such that $|adj(A)| = 121$, then $|AA^T|$ is equal to:

If $A$ is a square matrix of order 3 and $|A| = 5$, then the value of $|-AA^T|$ is

If the system of equations $2x + 3y = 10$, $x + ky = 4$ has a unique solution, then

It is given that 3% of items manufactured by an industry are defective. The probability that a packet of 250 items contains one defective item is: [Given: $e^{-7.5} \approx 0.000553$]

The integral value of k for which the system of linear equations $kx + y + 2z = 0$, $ky = x - 3z$ and $2x + y + kz = 0$ has a non-zero solution is