Algebra PYQ — Page 2

Match List-I with List-II | List-I | List-II | | :--- | :--- | | **Type of matrix** | **Conditions** | | (A) Square matrix A | (I) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = k & , i = j \end{cases}$, where $k \neq 0$ is constant. | | (B) Scalar Matrix A | (II) $A = [a_{ij}]_{m \times m}$ | | (C) Diagonal matrix A | (III) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = 1 & , i = j \end{cases}$ | | (D) Identity matrix A | (IV) $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0$, $i \neq j$ | Choose the correct answer from the options given below:

The maximum value of the objective function $z = 2x + 3y$ of an L.P.P. subjected to the constraints $x - y \le 1$, $x + y \le 3$, $x, y \ge 0$ is

A can solve 90% problems and B can solve 70% problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to

For some constant 'k', if the system of linear equations $2x - y + 3z = 1$ $x - 2y + z = 3$ $kx + y - z = 0$ has a unique solution, then

The relation R on the set of real numbers defined by $R = \{(a, b): a \leq b^2\}$ is (A) Reflexive (B) Not symmetric (C) Neither reflexive nor transitive (D) Transitive Choose the correct answer from the options given below:

If A and B are two events such that P(A) ≠ 0 and P(B | A) = 1 then

Let X denotes the number of heads in a simultaneous toss of three coins, then $P(0 < X < 3)$ is

Assume that R is a relation on the set Z of integers and it is given by $(x, y) \in R \Leftrightarrow |x - y| \leq 1$. Then, R is

The feasible region represented by the constraints: $x + 2y \geq 100$, $2x - y \leq 0$, $2x + y \leq 200$, $x \geq 0$, $y \geq 0$ of an LPP is: 

If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

The corner points of the feasible region of a LPP with the constraints $x + 2y \leq 40$, $3x + y \geq 30$, $4x + 3y \geq 60$, $x, y \geq 0$ are

If $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}$ then

If $\begin{bmatrix} 1 & 0 & 0 \\ 0 & y+1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2x & \\ -2 & \\ z-3 & \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 1 \end{bmatrix}$ then $x + y + z$ is

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is

If a computer code is correctly programmed, it gives 90% acceptable results. But if it is not correctly programmed, it gives only 40% acceptable results. From previous experience, it is observed that only 80% of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results, then the approximate probability that the code is correctly programmed is

Let E and F are events associated with an experiment. If $P(E) = 0.4$, $P(F) = 0.8$ and $P(F|E) = 0.6$, then $P(E|F)$ is

Let $f: \mathbb{R}$ -> $\mathbb{R}$ be a function defined as $f(x) = x^4$. Which one of the following is true?

If $c_{ij}$ denotes the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3 \end{bmatrix}$ then the value of $c_{21} \cdot c_{33}$ is

The function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = |x|$ ($\mathbb{R}$ is the set of real numbers) is

A relation $f: N \rightarrow N$ be defined by $f(x) = x^2$, $x \in N$ (Set of Natural numbers). Then $f(x)$ is

Which one of the following represents the correct feasible region determined by the following constraints $x - y \geq 5$, $5x - 5y \leq 16$

The random variable X can take values 0, 1, 2. If $P(X=0)=P(X=1)=\alpha$, and $E(X^2)=E(X)$, then which of the following are correct? (A) $E(X) = 2-3\alpha$ (B) $E(X^2) = 4+7\alpha$ (C) $\alpha = \frac{1}{2}$ (D) $\alpha = \frac{1}{5}$ Choose the **correct** answer from the options given below:

Which of the region shown in the given figures represents the feasible region bounded by the following constraints? $4x + y \geq 80$, $2x + y \geq 60$, $x + y \leq 80$, $x \geq 0$, $y \geq 0$ 

Match List-I with List-II Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$. | List-I | List-II | | --- | --- | | (A) $\vec{a} \cdot \vec{b}$ | (I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2} \vec{b}$ | | (B) $\vec{a} \times \vec{b}$ | (II) $\vec{a} \cdot \vec{b} = 0$ | | (C) Projection vector of $\vec{a}$ on $\vec{b}$ ($\ne{0}$) | (III) $\vert \vec{a}\vert \vert \vec{b}\vert \sin \theta \, \hat{n}$ where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ | | (D) $\vec{a}$ and $\vec{b}$ are orthogonal vectors | (IV) $\vert \vec{a}\vert \vert \vec{b}\vert \cos \theta$ |