Algebra PYQ — Page 6

For any events A and B of a sample space S, which of the following statements are TRUE? (A) $P(S | B) = 1$ (B) $P(A \cap B) = P(A) + P(B) + P(A \cup B)$ (C) $P(\bar{A} | B) = 1 - P(A | B)$ (D) $P(A | B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0$ Choose the correct answer from the options given below:

Consider the LPP: Maximize $z = x + y$ subject to the constraints $x + 2y \leq 70$, $2x + y \leq 95$, $x,y \geq 0$. The optimal feasible solution is

Let $A = [a_{ij}]_{3×2}$ and $B = [b_{ij}]_{3×4}$ be two matrices. Then the order of the matrix $(A^T . B)^T$ is:

If $A = \begin{bmatrix} 2 & 1 & -1 \\ 0 & 1 & 2 \\ 2 & -1 & \lambda \end{bmatrix}$ is a singular matrix, then the value of $\lambda$ is

Assume A, B and C are matrices of order $m \times n$, $n \times 3$ and $3 \times q$ respectively. The restrictions on $_{m,n}$ and $_q$ so that $AB + BC$ is defined are

If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true? (A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$ (B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$ (C) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$, $i = 1,2,3$ (D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$, $i = 1,2,3$ Choose the correct answer from the options given below:

A box contains 10 balls, each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, then the probability that none is marked with the digit 0 is:

The number of all possible matrices of order $2 \times 3$ with entries -1 or 1 is

A lot of 50 watches is known to have 10 defective watches. If 8 watches are selected one by one with a replacement at random, then the probability that there will be at least one defective watch is:

If A and B are two invertible matrices, then which of the following statements are correct? (A) $|A^{-1}| = |A|^{-1}$ (B) $adjA = |A|A^{-1}$ (C) $(AB)^{-1} = A^{-1}B^{-1}$ (D) $(A + B)^{-1} = A^{-1} + B^{-1}$ Choose the **correct** answer from the options given below:

An urn I contains 3 white and 4 blue balls, while urn II contains 5 white and 6 blue balls. One ball is drawn at random from one of the urns and it is found to be white. The probability that it was drawn from urn II is

The minimum value of the objective function $z = x + 2y$ of an L.P.P. subject to constraints $2x + y \geq 3, \frac {x} {2} + 2y \geq 6, x \geq 0, y \geq 0$ is:

Match **List-I** with **List-II** Consider two vectors $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -3\hat{i} - 6\hat{j} + 3\hat{k}$, then | List-I | List-II | |---|---| | (A) Angle between $\vec{a}$ and $\vec{b}$ is | (I) $\cos^{-1}\left(\frac{1}{\sqrt{6}}\right)$ | | (B) Angle between $\vec{a}$ and $x$-axis is | (II) $\cos^{-1}\left(\frac{2}{\sqrt{6}}\right)$ | | (C) Angle between $\vec{b}$ and $x$-axis is | (III) $\pi$ | | (D) Angle between $\vec{a}$ and $y$-axis is | (IV) $\cos^{-1}\left(-\frac{1}{\sqrt{6}}\right)$ | Choose the **correct** answer from the options given below:

Let $A = [a_{ij}]$ be a square matrix of order 3 with $|A| = 2$ and let $C = [c_{ij}]$ where $c_{ij} =$ cofactor of $a_{ij}$ in A. Then $|C|$ is equal to:

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

One person speaks truth in 60% of the cases and another person in 80% of the cases. They are likely to agree in stating the same fact in

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

If $z = 3x + 4y$ be the objective function of a of a linear programming problem (LPP) and (3, 1), (2, 4), (0, 4), (5, 0) be corner points of the bounded feasible region. Then the maximum value of objective function 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

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:

Which of the following inequalities are NOT correct? (A) If $a > 1, b > 1,$ then $\log_b a + \log_a b \leq 2$ (B) For any real number $x, (9^x + 9^{1-x}) \geq 9$ (C) If $a,b,c$ are non-zero real numbers of the same sign, then $\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \leq 3$ (D) If $a,b,c$ are three distinct real numbers, then $(a + b)(b + c)(c + a) \geq 8abc$ 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?

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: