Algebra PYQ — Page 25

If $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$, then which of the following statements is/are correct? (A) $\vec{a}$ and $\vec{b}$ are collinear (B) $\vec{a}$ and $\vec{b}$ are perpendicular (C) Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ (D) $|\vec{a} + \vec{b}| = 2\sqrt{5}$ Choose the **correct** answer from the options given below:

The random variable X has the following probability distribution | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X) | a | a | b | b | such that E(x²) = 2E(x), then the value of b is:

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

If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is /are possible? (A) A= B (B) P (A) = P(B) (C) A ⊂ B but A ≠ B (D) A∩ B = ɸ Choose the correct answer from the options given below:

If the system of equations $kx + y + z = 0$, $x + ky - z = 0$, $x - y + z = 0$ has a non-zero solution, then the possible values of $k$ are:

Match List-I with List-II Let A & B are two events such that P(A)=0.8, P(B)=0.5, P(B|A)=0.4 | List-I | List-II | | :--- | :--- | | (A) $P(A \cap B)$ | (I) 0.2 | | (B) $P(A \mid B)$ | (II) 0.32 | | (C) $P(A \cup B)$ | (III) 0.64 | | (D) $P(A')$ | (IV) 0.98 | Choose the correct answer from the options given below:

If $\vec{a}$ is any vector, then $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to

The value of $\begin{vmatrix} 1 & bc & bc(b+c) \\ 1 & ca & ca(c+a) \\ 1 & ab & ab(a+b) \end{vmatrix}$ is

If $M = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $N = [7 \quad 1 \quad -4]$, then $(MN)^T$ will be equal to:

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2+1}$ is (where $\mathbb{R}$ is a set of real number)

The corner points of the bounded feasible region determined by the system of linear constraints are (15,0), (40,0), (4,18) and (6, 12). If objective function is Z = 30x + 20y, then the sum of the maximum and the minimum values of Z is

A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$, then $f$ is:

If the roots of the equation x² - 5x + k = 0 are in the ratio 2:3, then the value of k is:

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ 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 value of $(A-2I)^2 - (2A + I)^2 + 11A$ is:

The graph given below represents which of the following function? 

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

Which of the following are the properties of Normal Distribution function f(x) and Normal probability curve: (A) The probability of success remains the same in each trial and the number of trials is small in number. (B) The curve is bell-shaped and is symmetrical about the mean. (C) If set of n trials are repeated N times, then frequency f(r) of r successes is given by f(r) = N.p(r) = N$e^{-m\frac{m^r}{r!}}$, r=0,1,2,... (D) As x increases numerically, f(x) decreases rapidly and the maximum value of f(x) occurs at x=μ(mean) Choose the correct answer from the options given below:

In a linear programming problem, the constraints on decision variables $x$ and $y$ are $y-2x \leq 0$, $y \geq 0$, $0 \leq x \leq 5$. The feasible region of the above problem:

If A and B are symmetric matrices of order 3 x 3 then the matrix $2AB - BA$ is:

Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success, then the probability of at most 3 successes is

If $\begin{vmatrix} 1 & \cos \theta & 0 \\ \sin \theta & 1 & \cos \theta \\ |\cos \theta & 1 & -\sin \theta| \end{vmatrix} = A\sin \theta + B\cos \theta + C\sin \theta\cos \theta$ then:

If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$ then the value of $C_{23} + C_{31} - C_{22}$ is

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$, and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$, then