Algebra PYQ — Page 5

If $\det \begin{pmatrix} 2x & 5 \\ 8 & x \end{pmatrix} = \det \begin{pmatrix} 6 & -2 \\ 1 & 1 \end{pmatrix}$, then the value of $x$ is

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is

If X is a normal variate with mean 16 and standard deviation 4, then the value of standard normal variate Z corresponding to X = 17 is:

In 5 trials of binomial distribution, the probability of 3 successes is 4 times the probability of 2 successes. The probability of success in each trial is:

The standard deviation of the number of tails in three tosses of a coin is:

If the corner points of the bounded feasible region for a Linear Programming Problem (LPP) are A(0,2), B(3, 0), C(2, 3) and D(3, 1), then the maximum value of the objective function $Z = 4x + 2y$ occurs at

Match List-I with List-II Let $f: A \rightarrow B$ be a function given by $f(x) = x^2$ | List-I | List-II | |---|---| | **Domain and Co-domain** | **Kind** | | (A) $A = \mathbb{R}$ and $B = \mathbb{R}$ | (I) $f$ is both one-one and onto | | (B) $A = \mathbb{R}$ and $B = [0, \infty]$ | (II) $f$ is one-one but not onto | | (C) $A = B = [0, \infty]$ | (III) $f$ is not one-one but onto | | (D) $A = [0, \infty]$ and $B = \mathbb{R}$ | (IV) $f$ is neither one-one nor onto | Choose the correct answer from the options given below:

Let A = $[a_{ij}]_{n \times n}$ be a matrix. Then Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $A^T = A$ | (I) A is a singular matrix | | (B) $A^T = -A$ | (II) A is a non-singular matrix | | (C) $\vert A\vert = 0$ | (III) A is a skew symmetric matrix | | (D) $\vert A\vert \neq 0$ | (IV) A is a symmetric matrix | Choose the correct answer from the options given below:

If A and B are invertible matrices then which of the following statement is NOT correct?

A letter is known to have come either from KOLKATA or TATANAGAR. On the envelope just two consecutive letters TA are visible. The probability that letter has come from TATANAGAR is

If A and B are independent events, then which of the following statements are TRUE? (A) $P(A \cap B) = P(A).P(B)$ (B) $P(A \cap B) = P(A) - P(B)$ (C) $P(A \cup B) = P(A) + P(B) - P(A).P(B)$ (D) $P(A \cap B) = P(A). P(B|A)$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Mathematical Statement | Value | | (A) $\hat{i} \cdot (\hat{j} \times \hat{k})$ | (I) $-\hat{k}$ | | (B) $\hat{j} \cdot (\hat{i} \times \hat{k})$ | (II) 1 | | (C) $\hat{i} \times (\hat{j} \times \hat{k})$ | (III) -1 | | (D) $\hat{j} \times \hat{i}$ | (IV) $\vec{0}$ | Choose the correct answer from the options given below:

If $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|2\vec{a}+\vec{b}| = 2\sqrt{3}$ then $|\vec{a}-\vec{b}|$ is:

The corner points of the bounded feasible region determined by the system of linear inequalities are $(0, 0)$, $(2, 4)$, $(0, 5)$ and $(4, 0)$. If the maximum value of $z = ax + by$, where $a, b > 0$ occurs at both $(2, 4)$ and $(4, 0)$, then

If the random variable X has the following probability distribution: | X | 0 | 1 | 2 | Otherwise | |---|---|---|---|---| | P(X) | K | 3K | 5K | 0 | then K is equal to

The inverse of the matrix $A = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{8} \end{bmatrix}$ is

Let $M_{ij}$ and $A_{ij}$ denote respectively minors and co-factors of the element in the $i^{th}$ row and $j^{th}$ column of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 2 & 3 \\ 4 & -1 & 0 \end{bmatrix}$. Then: (A) $M_{32} = 6$ (B) $M_{23} = 9$ (C) $A_{32} = -6$ (D) $A_{23} = -9$ Choose the correct answer from the options given below:

The corner points of the bounded feasible region determined by the system of linear constraints are $(0, 0), (5, 0), (6, 5), (6, 8), (4, 10), (0,8)$. Let $z = 3x - 4y$ be the objective function. Then the minimum value of the objective function z occurs at

There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1, then the probability that all the lines are busy?

The system of equations $x + ky = 0$ $3x + 5y = 0$ has infinitely many solutions, then k is equal to

The mean of the number of heads in a simultaneous toss of three coins is

A die is thrown 6 times. If getting an odd number is a success, then the probability of at least 5 successes is

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

The solution set of the inequation $|2x - 3| ≤ 4$ is