Algebra PYQ — Page 32

If $A = \begin{bmatrix} 1 & 5 \\ 7 & 12 \end{bmatrix}, B = \begin{bmatrix} 9 & 1 \\ 7 & 8 \end{bmatrix}$ and C are three matrices such that $3A + 5B + 2C = 0$, then the matrix C is equal to

For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, if $X$ and $Y$ are square matrices of order $2$ such that $X Y=X$ and $Y X=Y$, then $\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)$ equals to :

If $A$ is a square matrix of order $4$ and $|A|=4$, then $|2 A|$ will be :

If $[\mathrm{A}]_{3 \times 2}[\mathrm{~B}]_{\mathrm{x} \times \mathrm{y}}=[\mathrm{C}]_{3 \times 1}$, then :

If $\mathrm{A}=\left[\begin{array}{ll}2 & 4 \\ 4 & 3\end{array}\right], \mathrm{X}=\left[\begin{array}{l}\mathrm{n} \\ 1\end{array}\right], \mathrm{B}=\left[\begin{array}{c}8 \\ 11\end{array}\right]$ and $A X=B$, then the value of $n$ will be :

A coin is tossed K times. If the probability of getting $3$ heads is equal to the probability of getting $7$ heads, then the probability of getting $8$ tails is :

Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be :

For a square matrix $A_{n \times n}$ (A) $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}$ (B) $|\mathrm{A}|=|\operatorname{adj} \mathrm{A}|^{\mathrm{n}-1}$ (C) $\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}|$ (D) $\left|\mathrm{A}^{-1}\right|=\frac{1}{|\mathrm{~A}|}$

Let R be the relation over the set A of all straight lines in a plane such that $l_{1} \mathrm{R} l_{2} \Leftrightarrow l_{1}$ is parallel to $l_{2}$. Then R is :

The corner points of the feasible region for an L.P.P. are $(0,10),(5,5),(5,15)$ and $(0,30)$. If the objective function is $Z=\alpha x+\beta y, \alpha, \beta>0$, the condition on $\alpha$ and $\beta$ so that maximum of $Z$ occurs at corner points $(5,5)$ and $(0,20)$ is :

Choose the **correct** answer from the options given below :

If f(x) = x² + 2x + 1, then f(3) equals

If the random variable $ X $ has the following distribution: | $X$ | 0 | 1 | 2 | otherwise | | --- | --- | --- | --- | --- | | $P(X)$ | $ k $ | $ 2k $ | $ 3k $ | $ 0 $ | Match List-I with List-II: | List-I | List-II | | --- | --- | | (A) $ k $ | (I) $ \frac{5}{6} $ | | (B) $ P(X < 2) $ | (II) $ \frac{4}{3} $ | | (C) $ E(X) $ | (III) $ \frac{1}{2} $ | | (D) $ P(1 \leq X \leq 2) $ | (IV) $ \frac{1}{6} $ |

A die is rolled thrice. What is the probability of getting a number greater than $4$ in the first and the second throw of dice and a number less than $4$ in the third throw ?

If the matrix $\left[\begin{array}{rrr}0 & -1 & 3 x \\ 1 & y & -5 \\ -6 & 5 & 0\end{array}\right]$ is skew-symmetric, then the value of $5 x-y$ is:

Three defective bulbs are mixed with $8$ good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly $1$ defective, more than $2$ defective, no defective and more than $1$ defective respectively are:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, where $\vec{a}$ and $\vec{b}$ are unit vectors and $|\vec{c}|=2$, then the angle between the vectors $\vec{b}$ and $\vec{c}$ is :

If the function $f: \mathbb{N} \rightarrow \mathbb{N}$ is defined as $f(n)=\left\{\begin{array}{ll}n-1, & \text { if } n \text { is even } \\ n+1, & \text { if } n \text { is odd }\end{array}\right.$, then (A) f is injective (B) f is into (C) f is surjective (D) f is invertible

The matrix $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ is a : (A) scalar matrix (B) diagonal matrix (C) skew-symmetric matix (D) symmetric matrix Choose the correct answer from the options given below :

 The feasible region represented by the constraints $4 x+y \geq 80, x+5 y \geq 115,3 x+2 y \leq 150, x, y \geq 0$ of an LPP is

$\Delta=\left|\begin{array}{ccc}1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1\end{array}\right|$ (A) $\Delta=2\left(1-\cos ^{2} x\right)$ (B) $\Delta=2\left(2-\sin ^{2} x\right)$ (C) Minimum value of $\Delta$ is $2$ (D) Maximum value of $\Delta$ is $4$ Choose the correct answer from the options given below :

Which one of the following represents the correct feasible region determined by the following constraints of an LPP? $x+y \geq 10,2 x+2 y \leq 25, x \geq 0, y \geq 0$

The probability of not getting $53$ Tuesdays in a leap year is :

If $(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b})=27$ and $|\vec{a}|=2|\vec{b}|$, then $|\vec{b}|$ is :