Algebra PYQ — Page 6

If $\vec{a}$, $\vec{b}$ and $\sqrt{3}\vec{a} - \vec{b}$ are three unit vectors, then the angle between $\vec{a}$ and $\vec{b}$ is:

Let N be set of natural numbers, then the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f(x) = \begin{cases} \frac{n+1}{2} & \text{, if } n \text{ is odd} \\ \frac{n}{2} & \text{, if } n \text{ is even} \end{cases}$, is

The normal distribution curve is symmetrical about [$\mu$ = mean, $\sigma$= standard deviation]

If a, b, c are positive real numbers, then the least value of $(a+b+c)(ab+bc+ca)$ is:

Which of the following are correct? (A) For a square matrix A, if A³ = I, then A⁻¹ = A². (B) The determinant of only a square matrix can be defined. (C) If A is a square matrix of order 3, then the number of minors of the matrix A is 3. (D) If A and B are two non-singular matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹. Choose the correct answer from the options given below:

Which of the following statements is (are) true? (A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix (B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix (C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix (D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix (E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix Choose the correct answer from the options given below:

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

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

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

Let A be any skew- symmetric matrix (where $A^T$ is Transpose of matrix A). Then which of the following statements are correct? (A) $A^2$ is a symmetric matrix (B) $A^2$ is a skew- symmetric matrix (C) $A^T A = -A^2$ (D) $A^T A - AA^T = O$ Choose the correct answer from the options given below:

Which of the following statement is/are correct? (A) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$ (B) $A = [a_{ij}]_{m \times m}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$ (C) A square matrix $A = [a_{ij}]$ is called a skew symmetric matrix, if $a_{ij} = -a_{ji}$ for all $i, j$ (D) The multiplication of diagonal matrices of same order is commutative Choose the correct answer from the options given below:

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:

Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution. | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(x) | $\frac{25}{72}k$ | $\frac{15}{72}k$ | $\frac{3}{72}k$ | $\frac{1}{360}k$ | If value of k is equal to $\frac{m}{n} \cdot \gcd(m,n) = 1$, then $m + n$ is equal to

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

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the matrix $(2I+A)^3 - 19A - 3I$ is equal to

If the optimal value of the objective function $z = px + y$ of an L.P.P occurs at two corner points (2, 11) and (4, 5) of its bounded feasible region, then its optimal value is

If $X$ is a normal variate with mean 12 and standard deviation 4, then $P[X \ge 20]$ is: [Given that: $P[0 \le Z \le 2] = 0.4772]$

The inequality $\frac{5x - 2}{3} - \frac{7x - 3}{5} > \frac{x}{4}$ holds when

Let $A = [a_{ij}]$ be a square matrix of order 3 with each entry either 0 or 1, then the number of all such possible matrices is:

For the linear programming problem (LPP), $Maximize Z = 7x + 9y$, subject to constraints, $x - y \le -1, -x + y \le 0, x, y \ge 0.$ Which of the following is correct?

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. The expected value of the amount(in Rs.) he wins or loses is:

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

Match List-I with List-II if $P(A) = \frac{3}{7}$, $P(B) = \frac{4}{7}$ and $P(A \cup B) = \frac{5}{7}$ | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) $\dfrac{2}{3}$ | | (B) $P(A \mid B)$ | (II) $\dfrac{5}{7}$ | | (C) $P(B \mid A)$ | (III) $\dfrac{1}{2}$ | | (D) $P(A' \cup B')$ | (IV) $\dfrac{2}{7}$ | Choose the correct answer from the options given below:

The area of a triangle whose vertices are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ is given by the absolute value of