Algebra PYQ — Page 18

For the system $\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$ which of the following statements are correct? (A) The system has no solution. (B) The system is consistent. (C) It has infinitely many solutions. (D) It has a unique solution. Choose the correct answer from the options given below:

The probabilities of occurrance of two events A and B are 0.45 and 0.20 respectively. The probability of their simultaneous occurrence is 0.06. The probability that neither A nor B occurs is

The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

Let P and Q be any two invertible matrices of the same order. Then Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Matrix** | **Equivalent matrix** | | (A) $(P Q)^{-1}$ | (I) $Q^{-1}P$ | | (B) $(P^{-1}Q)^{-1}$ | (II) $Q P^{-1}$ | | (C) $(P Q^{-1})^{-1}$ | (III) $Q^{-1}P^{-1}$ | | (D) $(P^{-1}Q^{-1})^{-1}$ | (IV) Q P | Choose the **correct** answer from the options given below:

A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have atleast one boy is:

For the system of linear equations $x + y + z = 5000$ $6x + 7y + 8z = 35800$ $6x + 7y - 8z = 7000$ the values of x, y and z are:

A bag contain 8 blue and 12 green balls. Two balls are drawn in succession without replacement. The probability that first is blue and second is green is

If a matrix P is both symmetric and skew-symmetric, then

The area of triangle with vertices P, Q, R is given by (where $\vec{AB}$ = position vector of point B – position vector of point A)

A random variable X has the following probability distribution: | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | 0.1 | 0.2 | 0.3 | 0.4 | The variance of the X will be:

Suppose X has Poisson distribution such that $3 P(X=1) = 2 P(X=2)$ then $P(X>0)$ is:

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

If $|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1$, then angle between $\vec{a}$ and $\vec{r}$ is

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

Let $R = \{(L_1, L_2): L_1 \perp L_2\ $ where $L_1, L_2 \in L$ (set of straight line in a plane)}, then

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

If $\begin{bmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{bmatrix}$ is singular matrix, where $x \in \mathbb{N}$ (where N set of natural number), then x is equal to

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 length of line segment joining the points with position vectors $2\hat{i} - 2\hat{j} + 3\hat{k}$ and $5\hat{i} + 2\hat{j} + 3\hat{k}$ 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:

Mean and variance of a binomial distribution are 6 and 2 respectively. The probability of 2 successes will be

Which of the region shown in the given figures represents the feasible region bounded by the following constraints? $4x + y \geq 80$, $2x + y \geq 60$, $x + y \leq 80$, $x \geq 0$, $y \geq 0$