Algebra PYQ — Page 23

If $A = \begin{bmatrix} a & a & a \\ o & a & a \\ o & o & a \end{bmatrix}$, then $|adj A|$ is equal to

If the system of equations $x + 2y + 3z = 10$ $-x + y + \lambda z = 20$ $2x + 3y + \lambda z = 0$ does not possess a unique solution, then $\lambda$ is equal to

If A and B are matrices of same order, then $(AB^T - BA^T)$ is always

If E and F are independent events associated with an experiment, then which one of the following statements is correct?

If $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = px + qy$ where $p, q > 0$. Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is

The probability distribution of a ramdom variable $X$ is: $P(X=x)=\begin{cases}kx^2,& x=1,2,3\\2kx,&x=4,5,6\\0,&\text{otherwise}\end{cases}$ Where $K$ is a constant Match List-I with List-II | List-I | List-II | |---|---| | (A) $k$ | (I) 7/22 | | (B) $P(X \geq 4)$ | (II) 1/44 | | (C) $P(X < 4)$ | (III) 95/22 | | (D) $E[X]$ | (IV) 15/22 | Choose the correct answer from the options given below:

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

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:

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 $\begin{bmatrix}0 & y-2\\5 & 4\end{bmatrix}$

The point which provides the optimal solution of the linear programming problem maximize $z = 21x + 35y$ $3x + 2y \leq 30$ $4x + 5y \leq 60$ $x \geq 0, y \geq 0$ has the coordinates

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

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

The value of $\begin{vmatrix}265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181\end{vmatrix}$ is

Which of the following statements are true? (A) The vector joining the points P(2, 3, 0) and Q(-1,-2,-4) directed from P to Q is $\vec{PQ} = -3\hat{i} - 5\hat{j} - 4\hat{k}$ (B) Projection of a vector $\vec{a}$ on other vector $\vec{b}$ is $\frac{\vec{a}.\vec{b}}{|\vec{a}|}$ (C) If $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ then $\vec{a} + \vec{b} = -\hat{i} + 2\hat{j} + 6\hat{k}$ (D) If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then $\cos \theta = \frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$ Choose the correct answer from the options given below:

The probability of a man hitting a target is 1/2. How many times must he fire so that the probability of hitting the target at least once is more than 90%?

If a random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | K | K/2 | K/4 | K/8 | then, Match List-I with List-II | List-I | List-II | |---|---| | (A) The value of K is | (I) 2/15 | | (B) P(0 < X < 2) is | (II) 1/15 | | (C) P(1 < X < 3) is | (III) 8/15 | | (D) P(X > 2) is | (IV) 4/15 | Choose the correct answer from the options given below: