Algebra PYQ — Page 19

The minimum value of $Z = 2x + y$ subjected to $x + y \geq 10, 2x + 3y \leq 26, x, y \geq 0$ is

About linear programming problem (LPP), which of the following statements are correct? (A) In a LPP, the linear inequalities or restrictions on the variables are called linear constraints. (B) If the feasible region for an LPP is unbounded, then the maximum or minimum value of the objective function $Z = ax + by$ never exists. (C) The feasible region for an LPP is always convex. (D) The common region determined by all the linear constraints of an LPP is called the feasible region. Choose the correct answer from the options given below:

A player participates in 3 matches against three teams T₁, T₂ and T₃.The probability of winning a match against teams T₁, T₂ and T₃ are 0.2, 0.3 and 0.9 respectively. If 'wins' can be regarded as independent events, then the probability that he (A) wins all the 3 matches is 0.054 (B) wins no match is 0.054 (C) wins exactly two matches is 0.348 (D) wins exactly one match is 0.542 Choose the correct answer from the options given below:

An objective function $z = ax + by$ is maximum at points (15,15) and (0, 20). If $a, b \geq 0$ and $ab = 27$, then the maximum value of the objective function is

Which of the given values of $x$ and $y$ make the following pair of matrices equal ? $\begin{bmatrix}2x-1 & 4\\y-1 & 3+2x\end{bmatrix}$ and $\begin{bmatrix}0 & y-2\\5 & 4\end{bmatrix}$

A linear programming problem is as follows: Minimize $z = 2x + 3y$ Subject to the constraints $x \ge 3, x \le 9, y \ge 0, x - y \ge 0, x + y \le 14$. The feasible region has 5 corner points including

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix}$, then values of x and y, are:

In a linear programming problem(LPP), the maximum value of the objective function $Z = 2x + 5y$ subjected to the constraints: $2x + 3y ≤ 6$ $2x + y ≤ 4$ $x, y ≥ 0$ is

If $x, y$ and $z$ are real number such that $x + y + z = 0$, then value of $\begin{vmatrix}3x & -x+y & -x+z\\x-y & 3y & z-y\\x-z & y-z & 3z\end{vmatrix}$ is

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

The probability distribution of a random variable x is given below. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | k/3 | k/2 | k/4 | k/7 | Then the value of k is

If X is a random variable with probability distribution as given below: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | k | 2k | k | 3k | Then, the variance of the distribution is

The minimum value of $\begin{vmatrix}2 & 2 & 2 \\ 2 & 2+x & 2 \\ 2 & 2 & 2+x\end{vmatrix}$, $x \in R$ is

Let A and B be two events. Then which of the following statements are TRUE? (A) $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \neq 0$ (B) $P(B') = 1 + P(B)$ (C) $P(A \cup B) = P(A) + P(B) + P(A \cap B)$ (D) $P(A \cap B) = P(A).P(B)$ If A and B are independent events Choose the correct answer from the options given below:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 100x + 1$, where $\mathbb{R}$ is a set of real numbers, then

If P and Q are non-singular square matrices of the same order, then $(PQ^{-1})^{-1}$ equals

Let $A = [a_{ij}]_{3 \times 3}$ such that $|A| = -5$. Then the value of $\det(5A)$ is equal to

The integral value of k for which the system of linear equations $kx + y + 2z = 0$, $ky = x - 3z$ and $2x + y + kz = 0$ has a non-zero solution is

If matrix $A = \begin{bmatrix} x & 2 & 3 \\ a & y & -5 \\ b & c & 0 \end{bmatrix}$ is a skew-symmetric matrix, then (A) $x + y + c = 5$ (B) $c = 5$ (C) $a + b + c = 0$ (D) $a + b - c = 10$ Choose the correct answer from the options given below:

Three bad eggs are mixed with 7 good ones. If two eggs are drawn one by one without replacement, then the probability distribution of the number (X) of bad eggs drawn is: | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/4 | 1/2 | 1/4 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 15/61 | 20/61 | 26/61 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 7/15 | 7/15 | 1/15 | | X | 0 | 1 | 2 | |---|---|---|---| | P(X) | 1/8 | 1/4 | 5/8 |

If A is an invertible matrix of order 2, then $\det(( {adj } A)^{-1})$ is equal to

If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $A² = KA - 2I$, then the value of K is

A problem in Mathematics is given to two students X and Y whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. The probability that only X solves the problem, is:

If $y$ is normal distribution random variable with mean $\mu = 10$ and standard deviation $\sigma = 2$. $z$ is standard normal variable and $F(Z)$ is cumulative distribution function, then which of the following are true? [Given that $F(1.5) = 0.9332$, $F(3) = 0.9986$, $F(2.25) = 0.9878$ and $F(1) = 0.8413$] (A) $P(X < 13) = 0.9332$ (B) $P(X > 16) = 0.9986$ (C) $P(12 < X < 14.5) = 0.1465$ (D) $P(X > 8) = 0.8413$ Choose the correct answer from the options given below: