Algebra PYQ — Page 14

Which of the following statements are correct? (A) The mean and variance of the Poisson distribution are equal. (B) The mean and variance of a Binomial distribution are equal. (C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$. (D) If the variance of a Poisson distribution is 2, then P(X = 2) = $2e^{-2}$ Choose the **correct** answer from the options given below:

The value of k for which the system of equations $x + y + z = 1$ $x - ky + z = 1$ $x - y + z = 1$ has more than one solutions is

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

The objective function of an LPP is $z = ax + \beta y, (a, \beta > 0)$ in that has to be maximized/minimized subject to constraints $x + y \leq 2$, $x \geq 0$, $y \geq 0$. Then max (z) $-$ min (z) is equal to

Two persons A and B throw a die alternately till one of them gets a 'three' and wins the game. The probability of A's winning if A starts first is

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---| | P(X) | 0 | k | 2k | 2k | 3k | k² | 2k² | 7k² + k | The value of $P(4 < x < 7)$ is equal to

If $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix}$, then which of the following statements are TRUE? (A) AB is defined (B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2 (C) BA is defined (D) AB and BA both are defined and AB = BA Choose the correct answer from the options given below:

If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

For the linear programming problem(LPP), Maximize $Z = 4x + y$ $x + y \leq 5$ $3x + y \leq 9$ $x,y \geq 0$. Which of the following are NOT true? (A) The given LPP has unbounded feasible region. (B) The corner points of the feasible region are (0,0), (0, 5), (3, 2) and (3, 0). (C) The optimal value of the objective function is 12. (D) The given LPP has a unique optimal solution. Choose the correct answer from the options given below:

If $A = [a_{ij}]_{2×2}$ where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ and $I$ is the identity matrix of order 2, then $(A^2 - 3A + 4I)$ is (A) Symmetric Matrix (B) Skew-symmetric Matrix (C) Non-singular Matrix (D) Square Matrix Choose the correct answer from the options given below:

The matrix $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is a (A) Null matrix (B) Unit matrix (C) Symmetric matrix (D) Skew-symmetric matrix Choose the correct answer from the options given below:

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Angle between $\vec{i} - \vec{j}$ and $\vec{j} + \vec{k}$ | (I) 0 | | (B) Angle between $2\vec{j} - \vec{k}$ and $\vec{j} + 2\vec{k}$ | (II) $\frac{2\pi}{3}$ | | (C) Angle between $\vec{i} + 2\vec{j}$ and $5\vec{i} + 10\vec{j}$ | (III) $\frac{\pi}{6}$ | | (D) Angle between $\sqrt{3}\vec{i} + \vec{j}$ and $\vec{i}$ | (IV) $\frac{\pi}{2}$ | Choose the correct answer from the options given below:

The feasible region associated with the inequality $2x + 3y > 4$ is

If matrices $A = [1 \quad 2 \quad 3]$ and $B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $BA$ is equal to:

Which of the following statements are correct? (A) If E and F are independent events then $P(E \cap F) = P(E) \cdot P(F)$ (B) If E and F are mutually exclusive events, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$ (C) The conditional probability of an event E, given the occurrence of the event F is given by $\frac{P(E \cap F)}{P(F)}, P(F) \neq 0$ (D) If E and F be the events associated with the sample space S of an experiment, then $P(\overline{E}|F) = 2 - P(E|F)$ Choose the correct answer from the options given below:

If A be a square matrix of order 3 such that $|A| = 2$, then $|adj(2A)|$ is equal to

If A and B are symmetric matrices, then AB - BA is

If X is a random variable and $a$, $b$ are real numbers, then which of the following statements are true? (A) $Var(aX+b) = a^2 Var(X)$ (B) $E(aX+b)= a E(X) + b$ (C) $E(aX+b)= a E(X) - E(b)$ (D) $Var(aX+b)= a Var(X) + b$ Choose the correct answer from the options given below:

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | then, which of the following is correct?

If $A = \begin{bmatrix} 3 & 2a \\ 1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 \\ b & 5 \end{bmatrix}$ both are singular matrices, then $a + b$ is equal to

A unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is

If the area of a triangle whose vertices are $(-2, 4)$, $(2, -6)$ and $(k, 4)$, $(k > 0)$ is 35 squnits, then the value of k is

The diagonal elements of a skew symmetric matrix are all