Algebra PYQ — Page 17

If A and B are square matrices of order 3 such that $|A| = 3$ and $|B| = -1$, then $|3AB|$ is equal to

For what value of k, the following system of equations has infinitely many solutions? $x + 2y = 5, 3x + ky = 15$

If $A = \begin{bmatrix} x+z & 2 & -3 \\ x & 0 & 4 \\ 3 & x-y & 0 \end{bmatrix}$ is a skew-symmetric matrix, then which of the following are true? (A) $y > z > x$ (B) $x > y$ (C) $x + y + z > 0$ (D) $z > x$ Choose the correct answer from the options given below:

Let $\vec{a}, \vec{b}$ be two vectors such that $|\vec{a}| = 2, |\vec{b}| = 3, \vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}|$ is equal to

If $A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$, then $|adj A|$ is equal to

The corner points of the feasible region associated with the LPP: Maximise $Z = px + qy$, $p,q > 0$ subject to $2x + y \leq 10$, $x + 3y \leq 15$, $x, y \geq 0$ are (0, 0), (5, 0), (3, 4) and (0, 5). If optimum value occurs at both (3, 4) and (0, 5), then

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + \hat{j}$ be three vectors. If $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$, then the value of $\lambda$ is

The following system of equations: $x + y - z = 7$ $4x + \lambda y - \lambda z = 3$ $3x + 2y - 4z = 5$ does not possess a solution if the value of $\lambda$ is:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert $, then | (I) $\vec{a}$ and $\vec{b}$ are orthogonal | | (B) Projection vector of $\vec{a}$ on $\vec{b}$ | (II) $[0, 12]$ | | (C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert $, then | (III) $\vec{a} = \pm \vec{b}$ | | (D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$, then the range of $\vert \lambda \vec{a}\vert $ | (IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$ | Choose the correct answer from the options given below:

If A and B are independent events, then which of the following is/are true? (A) $\bar{A}$ and B are independent events (B) $P(A \cap B) = 0$ (C) $\bar{A}$ and $\bar{B}$ are independent events (D) $P(A \cap B) = P(A) + P(B)$ Choose the correct answer from the options given below:

For an LPP: Maximize $z = 3x + 9y$, $x \geq 0, y \geq 0$, the feasible region OAB is shown in the figure, then the other constraints are 

A fair coin is tossed a fixed number of times. If the probability of getting 11 heads is equal to the probability of getting 13 heads, then the probability of getting 2 heads is:

If A and B are invertible matrices of the same order, then $(AB)^{-1}$ is equal to

If A and B are symmetric matrices of the same order, then which of the following are true? (A) AB - BA is a skew symmetric matrix (B) AB is a symmetric matrix (C) AB is a scalar matrix (D) AB + BA is a symmetric matrix Choose the correct answer from the options given below:

The corner points of the bounded feasible region determined by a set of constraints (linear inequalities) are A(0, 5), B(3, 5), C(5, 0) and D(4, 1) and the objective function is z = $px$ + 2$qy$ where p,q > 0. The condition on $p$ and $q$ such that the maximum z occurs at B and D, is:

Two cards are drawn simultaneously at random from a well shuffled pack of 52 Cards. Let X be the random variable which denotes number of kings in the draw. Then the probability distribution of X is

Consider the linear programming problem(LPP): *Minimize* $Z = x + y$ $x + 2y \leq 4,$ $3x + y \geq 3,$ $4x + 3y \geq 6,$ $x, y \geq 0.$ Which of the following is correct for the above linear programming problem (LPP): (A) The LPP has a bounded feasible region. (B) The LPP has a unique optimal solution. (C) The optimal value of the LPP exists at the point (3/2, 0) (D) The corner points of the feasible region are (3/2, 0), (3/5, 6/5), (2/5, 6/5) and (4, 0) Choose the **correct** answer from the options given below:

Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = 10x$. Then (Where $\mathbb{R}$ is the set of real numbers)

A die is tossed 6 times and getting "1 or 5" is considered a success. The probability of getting at least one success in six tosses is:

The region represented by the system of inequalities $x, y \geq 0, y \leq 6, x + y \leq 3$

If $A$ is a $3 \times 3$ matrix such that $|adj A| = 9$ and $|kA^{-1}| = 9$, then the value of $k$ are:

A random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | P(X) | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a | Then the values of 'a' and P(0 < X < 5) respectively are

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

The corner points of the bounded feasible region associated with the LPP: Maximize $Z=px+qy$, $p,q>0$ are $(0, 0)$, $(3.5, 0)$, $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$. If the optimum value of Z occurs at both $\left(\frac{112}{59}, \frac{135}{59}\right)$ and $(0, 3)$, then