Algebra PYQ — Page 13

Given a matrix A of order 3x3. If |A|=3 then the value of |A(adj A)| is:

For the L.P.P. Maximize z = 10x + 6y subjected to 3x + y ≤ 12, 2x + 5y ≤ 34, x, y ≥ 0. Then the feasible region represented by system of inequalities is

Probability distribution of random variable X is | X | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | P(X) | 2/11 | 1/11 | 4/11 | 3/11 | 1/11 | Then the value of E(X) is

Which of the following statement('s) is/are TRUE? (A) Skew symmetric matrix of even order is always symmetric (B) Skew symmetric matrix of odd order is non-singular (C) Skew symmetric matrix of odd order is singular (D) Skew symmetric matrix is always square matrix Choose the correct answer from the options given below:

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

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.2 | 0.1 | 0.3 | 0.4 | The variance of X will be

If $A = [a_{ij}]_{3 \times 2}$ where $a_{ij} = i + j$, then (A) A is a square matrix (B) $a_{21} + a_{32} = 8$ (C) Number of elements in A is 6 (D) Transpose of $A = \begin{bmatrix}2 & 3 \\ 3 & 4 \\ 4 & 5\end{bmatrix}$ Choose the correct answer from the options given below:

If A is a singular matrix, then A{adj A} is equal to

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then which of the following statements are true? (A) A and B are independent events (B) $P(A | B) = \frac{3}{4}$ (C) $P(A' | B') = \frac{5}{8}$ (D) $P(A' | B) = \frac{1}{4}$ Choose the correct answer from the options given below:

The probability distribution of a random variable $x$ is, $P(x) = \frac{k}{2^x}, x = 0, 1, 2, 3$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) $\frac{2}{15}$ | | (B) $P(x = 1)$ | (II) $\frac{1}{5}$ | | (C) $P(1 < x < 3)$ | (III) $\frac{8}{15}$ | | (D) $P(x \geq 2)$ | (IV) $\frac{4}{15}$ | Choose the correct answer from the options given below:

Linear inequalities corresponding to the shaded feasible region OABCO in the given figure are 

For the linear programing problem, $Minimize(Z) = 60x + 30y$ subject to: $2x - y \geq -5; 3x + y \geq 3; 2x - 3y \leq 12; x, y \geq 0$ the optimal value of $z$ is

Solution of the inequality $\frac{2x+3}{4x-5} \geq 0$ is

If $a_{ij}=i+3j$, then the matrix of order 2 with elements as $a_{ij}$ is

Relation R on the set $A = \{1, 2, ..., 15\}$ defined as $R = \{(x, y): y - 4x = 0\}$ is

In a college, 30% students fail in physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is

If $A$ is an invertible matrix of order 3, then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\vert \text{adj} A\vert $ | (I) $8\vert A\vert $ | | (B) $\vert A(\text{adj} A)\vert $ | (II) $\vert A\vert ^2$ | | (C) $\vert 2A\vert $ | (III) $\frac{1}{\vert A\vert }$ | | (D) $\vert A^{-1}\vert $ | (IV) $\vert A\vert ^3$ | Choose the correct answer from the options given below:

If A is an invertible matrix, then which of the following statement(s) is/are TRUE? (A) $|A^{-1}| = |A|$ (B) $(A^{-1})^{-1} = A$ (C) $A^{-1} = \frac{adj A}{|A|}$ (D) $(A^T)^{-1} = (A^{-1})^T$ Choose the correct answer from the options given below:

Which of the following statements are true? (A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$. (B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ (C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$ (D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$ Choose the correct answer from the options given below:

If the binomial distribution $X\sim B(n, p)$ of mean 3 and variance $\frac{3}{2}$, $(p + q) = 1$, then which of the following is/are TRUE? (A) $q = \frac{1}{2}$, $n = 6$ (B) $P(X \leq 5) = \frac{63}{64}$, $p = \frac{1}{2}$ (C) $q = \frac{1}{3}$, $p = \frac{2}{3}$ (D) $P(X = 4) = \frac{15}{64}$, $n = 6$ Choose the correct answer from the options given below:

If $A = [a_{ij}]$ is skew symmetric matrix of order 'n', then

A random variable X has the following probability distribution: | X | 2 | 3 | 4 | 5 | |---|---|---|---|---| | P(X) | 5/k | 7/k | 9/k | 11/k | Then the value of $\frac{k}{4}$ is:

Let A be a matrix such that $A = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix}$. Then which of the following are TRUE? (A) A is non-singular matrix (B) $A^T = A$ (C) A is not invertible matrix (D) A is not skew-symmetric matrix Choose the *correct* answer from the options given below:

Which of the following statements are correct? (A) Inverse of a matrix, if it exists, is unique (B) $(kA)' = -kA'$ (where k is any real number) (C) For an invertible matrix $A$, $(A^{-1})^{-1} = A$ (D) For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$ Choose the **correct** answer from the options given below: