Algebra PYQ — Page 9

If the minimum value of the objective function $Z = ax + by$ of an LPP occurs at two points $(3, 5)$ and $(5, 3)$, then

Let the corner points of the bounded feasible region of the linear programming problem (LPP) $Z = ax+by$ be: (0, 0), (2, 0), (20/19, 45/19) and (0, 3). If the optimal value of Z occurs at both points (2, 0) and (20/19, 45/19), then the relation between a and b is:

If $\vec{a} = 2\hat{i} + m\hat{j} - n\hat{k}$ and $\vec{b} = l\hat{i} - 3\hat{j} + 4\hat{k}$ such that $\vec{a} = 2\vec{b}$ then the value of $14{l} + m + n$ is:

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$, $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

Consider a L.P.P, Maximize $Z = 3x + 5y$ subject to constraints $2x + 6y ≤ 6$, $x - y ≥ 0$, $x ≥ 0$, $y ≥ 0$. Then which of the following are true? (A) The feasible region of L.P.P is bounded region (B) The corner points of the feasible region are (0, 0), (2, 2), (0, 1) (C) Maximum value of Z is 9 (D) Point (1, 3) lies in the feasible region Choose the correct answer from the options given below:

In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown. The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3. If X denotes the amount which the man wins or loses, then which of the following are correct? (A) All the possible values of X are 8, 5, 2 and -1. (B) The probability distribution of X is: | X | 8 | 5 | 2 | -1 | -12 | |---|---|---|---|---|---| | P(X) | 1/2 | 1/4 | 1/8 | 1/16 | 1/16 | (C) The mean value of X is 75/16. (D) The variance of X is 6615/256. Choose the correct answer from the options given below:

Let A= {1, 2, 3}, then the possible equivalence relations on A are: (A) {(1, 1), (2, 2) , (3, 3)} (B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} (C) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} (D) {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} Choose the correct answer from the options given below:

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

With respect to the following shaded feasible region (ABCDEFA), the maximum value of the objective function z = 3x + 4y – 2 is at point(s): 

If $\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$, then the value of $a + 2b - 3c + 4d$ is equal to

If $A$ and $B$ square matrices of order 3 such that $|A| = -1$, $|B| = 5$, then the value of $|2AB|$ is:

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let z = px + qy where p, q > 0. Then the condition on p and q so that the maximum value of z occurs at (15, 15) and (0, 20) is

The position vector of a point which divides the line joining the points with position vectors $(\vec{a} - 2\vec{b})$ and $(2\vec{a} + \vec{b})$ externally in the ratio 2:1, is

If $\frac{3x - 5}{6} + 8 \geq 4 + \frac{2x}{3}$, then

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of the same order as A, then $(I+A)^2-3A$ is equal to

If a matrix has 8 elements then the possible order(s) it may have (A) $8 \times 1$ (B) $5 \times 3$ (C) $6 \times 2$ (D) $2 \times 4$ Choose the correct answer from the options given below:

Let X be random variable which assumes $x_1$, $x_2$, $x_3$, $x_4$ such that 2P(X=$x_1$)= 3P(X=$x_2$)=P(X=$x_3$)=5P(X=$x_4$) , then the probability distribution of X is

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

A manufacturing unit makes two models, 'classic' and 'supreme' of the scooter. Each piece of the classic model requires 9 labour hours for assembling and 1 labour hour for finishing. Each piece of supreme model requires 12 labour hours for assembling and 3 labour hour for finishing. For assembling and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of ₹ 10000 on each piece of the classic model and ₹ 15000 on each piece of the supreme model. Which of the following options describes the given Linear Programming Problem (LPP) to maximize the profit Z (Max Z) (where x and y are the number of pieces of the classic model and the supreme model respectively)?

A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. The probability that the problem is solved is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) If $A$ is a non-singular matrix of order $n$, then $\vert A(\text{adj} A)\vert $ is equal to | (I) $\vert A\vert ^{n-1}$ | | (B) If $A$ is a non-singular matrix of order $n$, then $\vert \text{adj}(\text{adj} A)\vert $ is equal to | (II) $\vert A\vert ^{n-2} A$ | | (C) If $A$ is a non-singular matrix of order $n$, then $\text{adj}(\text{adj} A)$ is equal to | (III) $\vert A\vert ^n$ | | (D) If $A$ is a non-singular matrix of order $n$, then $\vert (\text{adj} A)\vert $ is equal to | (IV) $\vert A\vert ^{(n-1)^2}$ | Choose the correct answer from the options given below:

Which of the following are correct? (A) If a ≡ b(mod n), then -a ≡ -b (mod n) (B) If a + b = c, then a(mod n) + b(mod n) ≡ (a + b + c)(mod n). (C) If a ≡ b (mod n), then ka ≡ kb(mod n), ∀ k ∈ I. (D) If a ≡ b (mod n), then (a + k) ≡ (b + k)(mod n), ∀ k ∈ I. Choose the correct answer from the options given below:

Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black then the probability that the ball is drawn from box III, is:

The function $f: [0, \infty) \rightarrow \mathbb{R}$ defined by, $f(x) = 2x^2 + 3$, is