Algebra PYQ — Page 46

If $0 < x < \pi$ and the matrix $\begin{bmatrix} 4\sin x & -1 \\ -3 & \sin x \end{bmatrix}$ is singular, then the values of $x$ are :

If $\vec{a}$ and $\vec{b}$ are two perpendicular vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $\theta$ is the angle between $\vec{a}$ and $(\vec{a} - \vec{b})$, then $\cos\theta$ is equal to:

In a triangle, $\triangle ABC$, the sides AB and AC are represented by vectors $\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} - \hat{k}$ respectively. The length of median drawn from vertex A to BC is:

If $x-y=2$ and $y-z=3$ then value of $\begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix} =$

Which is the most suitable definition for random variable among the options given below:

If P is matrix of order $m \times n$ and Q is a matrix such that PQ' and Q'P are both computable, then the order of matrix Q is

Consider the linear programming problem : Minimize $z = 50x + 70y$ Subject to $2x+y \geq 8$, $x+2y \geq 10$, $x \geq 0$, $y \geq 0$ The minimum value of objective function is :

Two numbers are selected at random (without replacement) from the first three positive integers. Let $X$ denotes the larger of the two integers, then the probability distribution of $X$ is

Bag A contains 2 red and 3 white balls, Bag B contains 3 red and 2 white balls. If a ball is drawn at random and is found to be red, then the probability that it was drawn from bag B, is :

The matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is: A. a square matrix B. a scalar matrix C. a diagonal matrix D. an identity matrix Which of the above statements are true? Choose the correct answer from the options given below:

The random variable $X$ has a probability distribution $P(X=x) = \begin{cases} 5k, & x=0 \\ 2k, & x=1 \\ 3k, & x=2 \\ 0, & \text{otherwise.} \end{cases}$ Then, the value of $E(X)$ is :

The Relation $R = \{(x, y) : x \leq y^2\}$ defined on the set $\mathbf{R}$ of Real numbers is : (A) reflexive but not symmetric (B) neither reflexive nor symmetric (C) neither reflexive nor transitive (D) reflexive but not transitive (E) not reflexive but symmetric Choose the correct answer from the options given below :

If $2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}$, then the value of $x-y$ is :

If A and B are square matrices of same order n, then identify correct statements from the statements given below: A. $|adj\ A| = |A|^{n-1}$ B. $|A \cdot B| = |B| \cdot |A|$ C. $adj\ A' = (adj\ A)'$ D. $adj\ AB = (adj\ A) \cdot (adj\ B)$ E. $|A^n| = |A|^n$ Choose the correct answer from the options given below:

The probability that a question is guessed by a student and found to be correct is.

If $A$ is square matrix of order 3 and $A \cdot (Adj.(A)) = 10I$, then the value of $\frac{1}{25}|Adj.(A)|$ is

Assume $P$, $Q$, $R$ and $S$ are matrices of order $2 \times m$, $k \times n$, $m \times 2$ and $2 \times 3$ respectively. The restrictions on $k$, $m$ and $n$, so that $PQ + RS$ is defined are

If the mean and variance of a binomially distributed random variable X are 4 and 2 respectively, then $P(X = 2)$ is equal to

The system of equations $3x + 4y = 5$, $6x + 7y = -8$ is written in matrix form as

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A) (A) $(AB)^{-1} = B^{-1} A^{-1}$ (B) $(A+B)' = A' \times B'$ (C) $(AB)' = A'B'$ (D) $|kA| = k^3 |A|$ (E) $|A'| = |A|$ Choose the correct answer from the options given below :

If $A$ is a skew-symmetric matrix and $n$ is an odd positive integer, then $A^n$ is :

The equations in terms of $x$ and $y$ are:

The value of $\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{j} \times \hat{i})$ is

There are three identical boxes I, II and III, each containing two balls. In box I, both balls are red, In box II, both balls are blue and box III contains one blue ball and one red ball. A boy randomly chooses a box and takes out a ball at random from it. If the ball is red, then the probability that the other ball in the box is also red colour is: