Algebra PYQ — Page 42

The angle between the vectors $\hat{i} - \hat{j}$ and $\hat{j} - \hat{k}$ is:

Read the following statements carefully: A. Determinant is a square matrix B. If A be any given square matrix of order n, then $A(\text{adj}A) = (\text{adj}A)A = |A|I$. C. If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. D. If A is a nonsingular matrix, then its inverse does not exist Which of the above statements are true? Choose the correct answer from the options given below:

$\begin{vmatrix} a+b & 1 & 0 \\ a^2-b^2 & a-b & 1 \\ a^3+b^3 & a^2+b^2+ab & a^2-b^2 \end{vmatrix} =$

Objective function of LPP is:

Match List I with List II | List I | List II | |---|---| | A. Range of $\lvert x\rvert$ | I. $(-5, \infty)$ | | B. Range of $9x^2 + 6x - 5$ for all $x \geq 0$ | II. $[0, \infty)$ | | C. Domain of $\dfrac{1}{\sqrt{x+5}}$ | III. $\{(1,1), (2,2), (3,3)\}$ | | D. Smallest equivalence relation on Set $\{1,2,3\}$ | IV. $[-5, \infty)$ | Choose the correct answer from the options given below:

Match List I with List II: Given that A and B are invertible matrices of size $3 \times 3$ | List I | List II | |---|---| | A. $\lvert AB \rvert$ | I. $\frac{1}{\lvert A \rvert}$ | | B. $\lvert \operatorname{Adj} A \rvert$ | II. $\lvert A \rvert \lvert B \rvert$ | | C. $\lvert A^{-1} \rvert$ | III. $B^{-1} \cdot A^{-1}$ | | D. $(AB)^{-1}$ | IV. $\lvert A \rvert^2$ | Choose the correct answer from the options given below:

Match List - I with List - II. | List - I | List - II | |---|---| | (A) Two events E and F will be independent if $P(E'F')$ is equal to | (I) $1 - P(E/F)$ | | (B) If $P(F) \neq 0$, then $P(E'/F)$ is equal to | (II) $P(E) = P(F)$ | | (C) If E and F are independent events, then | (III) $P(E \cap F') = P(E) \cdot P(F')$ | | (D) If $P(E \cap F) \neq 0$ and $P(E/F) = P(F/E)$, then | (IV) $[1-P(E)][1-P(F)]$ | Choose the correct answer from the options given below :

Match List - I with List - II. | | List - I | | List - II | |---|---|---|---| | (A) | $A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$, AB will be | (I) | $\begin{bmatrix} 75 & 117 & 72 \\ 35 & 39 & 24 \end{bmatrix}$ | | (B) | $P = \begin{bmatrix} 8 & 45 & 30 \\ 17 & 19 & 5 \end{bmatrix}$, $Q = \begin{bmatrix} 67 & 72 & 42 \\ 8 & 30 & 19 \end{bmatrix}$, P+Q will be | (II) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 39 & 24 \end{bmatrix}$ | | (C) | $\begin{bmatrix} 85 & 42 & 69 \\ 73 & 42 & 50 \end{bmatrix} - \begin{bmatrix} 10 & -75 & -3 \\ 38 & 3 & 26 \end{bmatrix}$ | (III) | $\begin{bmatrix} 75 & 117 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | | (D) | $2 \begin{bmatrix} 34 & 36 & 30 \\ 12 & 18 & 20 \end{bmatrix} + \begin{bmatrix} 7 & 55 & 12 \\ 1 & 13 & -16 \end{bmatrix}$ | (IV) | $\begin{bmatrix} 75 & 127 & 72 \\ 25 & 49 & 24 \end{bmatrix}$ | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $\begin{bmatrix}0 & -5 & 9 \\ 5 & 0 & -3 \\ -9 & 3 & 0\end{bmatrix}$ | (I) Scalar matrix | | (B) $\begin{bmatrix}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{bmatrix}$ | (II) Diagonal matrix | | (C) $\begin{bmatrix}1 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 7\end{bmatrix}$ | (III) Symmetric matrix | | (D) $\begin{bmatrix}3 & -2 & 1 \\ -2 & -5 & 6 \\ 1 & 6 & 0\end{bmatrix}$ | (IV) Skew-symmetric matrix | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) $A$ is a square matrix of order $3$ and $\lvert 2A \rvert = k\lvert A \rvert$, then $k$ is | (I) $0$ | | (B) Value of $\begin{vmatrix}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{vmatrix}$ is | (II) $3$ | | (C) Matrix $\begin{bmatrix}5-x & x+1 \\ 2 & 4\end{bmatrix}$ is singular, then $x =$ | (III) $8$ | | (D) If $A = (a_{ij}) = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}$, then the minor of the element $a_{23}$ is | (IV) $7$ | Choose the correct answer from the options given below :

Match List - I with List - II. | | List - I (Two given vector) | | List - II (Projection of $\vec{a}$ on $\vec{b}$) | |---|---|---|---| | (A) | $\vec{a} = \hat{i} - \hat{j}$, $\vec{b} = \hat{i} + \hat{j}$ | (I) | $\frac{2}{\sqrt{5}}$ | | (B) | $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = 2\hat{i} - \hat{k}$ | (II) | 0 | | (C) | $\vec{a} = \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \hat{k}$ | (III) | $\sqrt{2}$ | | (D) | $\vec{a} = 2\hat{i} + 3\hat{j}$, $\vec{b} = \hat{i} - \hat{k}$ | (IV) | $\frac{1}{\sqrt{2}}$ | Choose the correct answer from the options given below :

Match List - I with List - II. | List - I | List - II | |---|---| | (A) The value of $\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})$ | (I) 16 | | (B) If $\lvert\vec{a}\rvert=10$, $\lvert\vec{b}\rvert=2$ and $\vec{a}\cdot\vec{b}=12$, then the value of $\lvert\vec{a}\times\vec{b}\rvert$ is | (II) $\frac{\pi}{4}$ | | (C) If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then the value of $\theta$, for which $\vec{a}\cdot\vec{b} = \lvert\vec{a}\times\vec{b}\rvert$ is | (III) 14 | | (D) If $\vec{a}$ and $\vec{b}$ are perpendicular and $\vec{a} = 2\hat{i}+4\hat{j}+\lambda\hat{k}$ and $\vec{b} = 3\hat{i}-5\hat{j}+\hat{k}$, then the value of $\lambda$ is | (IV) 1 | Choose the correct answer from the options given below :

Let X be the random variable with probability distribution given by the following table. | X | 0 | 1 | 2 | 3 | |---|---|---|---|---| | P(X = x) | $\frac{1}{8}$ | k | $\frac{3}{8}$ | $\frac{1}{8}$ | The value of $P(X \leq 1)$ is:

Let X be a discrete random variable and probability distribution X is | X | $-1$ | 0 | 1 | |---|---|---|---| | P(X) | $\frac{1}{2}$ | $\frac{1}{5}$ | $\frac{3}{10}$ | Then E(X) is equal to :

Let $R : \mathbb{R} \to \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers and R be a relation. An element $(x, y) \in R$ if $x + y - \sqrt{2}$ belongs to the set of irrational numbers. Then the relation R is :

Let the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ and $AB = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$ then order of B is :

Let L be the set of all lines in a plane and R be the relation in L defined as $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2, \text{ where } l_1, l_2 \in L\}$. Choose the correct answer :

Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$. The range of $f$ is :

Let $\vec{OA} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{OB} = \hat{i} + \hat{j} - \hat{k}$. Then A. The magnitude of vector $\vec{OA}$ is 6 B. The magnitude of vector $\vec{OB}$ is $\sqrt{3}$ C. The vector $\vec{AB}$ is $(-\hat{i} + 2\hat{j} - 2\hat{k})$ D. $\vec{OA} \cdot \vec{OB} = 0$ E. $\vec{OA} \parallel \vec{OB}$ Choose the correct answer from the options given below:

Let $\vec{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k}$. Let $\vec{c}$ be a vector such that $\vec{c} \times \vec{a} = \vec{b} \times \vec{c}$ and $\vec{c}\cdot(2\vec{a}-3\vec{b}) = 238\sqrt{2}$ then $|\vec{c}|^2$ is equal to :

Let $A$ and $B$ be two non-singular, square matrices of same order, and A. $(AB)^{-1} = B^{-1} \cdot A^{-1}$ B. $(A+B)^{-1} = B^{-1} + A^{-1}$ C. $adj. A = |A| \cdot A^{-1}$ D. $det(A^{-1}) = [det A]^{-1}$ Choose the correct answer from the options given below

Let $A = (a_{ij})$ and $B = (b_{ij})$ are square matrices of same order. (A) The number of possible matrices of order $2 \times 2$ with entries $-1, 0, 1$ is 81. (B) $A + A'$ is skew symmetric matrix (C) $A \cdot A^{-1} = 0$, $|A| \neq 0$ (D) A is skew symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$ (E) $(AB)' = A'B'$ Choose the correct answer from the options given below :

Let A be a non singular square matrix of $2 \times 2$. Then, $|adj\ A|$ is equal to

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A) (A) $(AB)^{-1} = B^{-1} A^{-1}$ (B) $(A+B)' = A' \times B'$ (C) $(AB)' = A'B'$ (D) $|kA| = k^3 |A|$ (E) $|A'| = |A|$ Choose the correct answer from the options given below :