Algebra PYQ — Page 33

The angle between two lines whose direction ratios are propotional to $1,1,-2$ and $(\sqrt{3}-1),(-\sqrt{3}-1),-4$ is :

There are two bags. Bag-1 contains $4$ white and $6$ black balls and Bag-2 contains $5$ white and $5$ black balls. A die is rolled, if it shows a number divisible by 3, a ball is drawn from Bag-1, else a ball is drawn from Bag-2. If the ball drawn is not black in colour, the probability that it was not drawn from Bag-2 is :

Let $X$ denote the number of hours you play during a randomly selected day. The probability that $X$ can take values $x$ has the following form, where $c$ is some constant. $\mathrm{P}(\mathrm{X}=\mathrm{x})=\left\{\begin{array}{lll} 0.1, & \text { if } \mathrm{x}=0 \\ \mathrm{cx}, & \text { if } \mathrm{x}=1 \text { or } \mathrm{x}=2 \\ \mathrm{c}(5-\mathrm{x}), & \text { if } \mathrm{x}=3 \text { or } \mathrm{x}=4 \\ 0, & \text { otherwise } \end{array}\right.$ Match List-I with List-II : | List-I | List-II | | --- | --- | | (A) $ c $ | (I) 0.75 | | (B) $ P(X \leq 2) $ | (II) 0.3 | | (C) $ P(X = 2) $ | (III) 0.55 | | (D) $ P(X \geq 2) $ | (IV) 0.15 |

If $A$ is a square matrix and $I$ is an identity matrix such that $A^{2}=A$, then $A(I-2 A)^{3}+2 A^{3}$ is equal to :

If $A, B$ and $C$ are three singular matrices given by $A=\left[\begin{array}{ll}1 & 4 \\ 3 & 2 a\end{array}\right], B=\left[\begin{array}{ll}3 b & 5 \\ a & 2\end{array}\right]$ and $C=\left[\begin{array}{cc}a+b+c & c+1 \\ a+c & c\end{array}\right]$, then the value of $a b c$ is :

A person wants to invest an amount of ₹ $75,000$. He has two options A and B yielding $8 \%$ and $9 \%$ return respectively on the invested amount. He plans to invest at least ₹ $15,000$ in Plan A and at least ₹ $25,000$ in Plan B. Also he wants that his investment in Plan A is less than or equal to his investment in Plan B. Which of the following options describes the given LPP to maximize the return (where $x$ and $y$ are investments in Plan A and Plan B respectively) ?

There are $6$ cards numbered $1$ to $6$, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then $\mathrm{P}(\mathrm{X}>3)$ is :

An objective function $Z=a x+b y$ is maximum at points $(8,2)$ and $(4,6)$. If $a \geq 0$ and $b \geq 0$ and $a b=25$, then the maximum value of the function is equal to :

If $A$ and $B$ are symmetric matrices of the same order, then $A B-B A$ is a :

If $P=\left[\begin{array}{r}-1 \\ 2 \\ 1\end{array}\right]$ and $Q=\left[\begin{array}{lll}2 & -4 & 1\end{array}\right]$ are two matrices, then $(P Q)^{\prime}$ will be :

Which of the following cannot be the direction ratios of the straight line $\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}$ ?

The unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$, is :

The solution set of the inequality $|3 x| \geq|6-3 x|$ is :

The probability of a shooter hitting a target is $\frac34$. How many minimum number of times must he fire so that the probability of hitting the target at least once is more than $90 \%$ ?

The corner points of the feasible region determined by $x+y \leq 8, 2 x+y \geq 8, x \geq 0, y \geq 0$ are $A(0,8), B(4,0)$ and $C(8,0)$. If the objective function $Z=a x+$ by has its maximum value on the line segment $A B$, then the relation between $a$ and $b$ is :

Match the options of **List-I** to **List-II** : | List-I | List-II | | --- | --- | | (A) $ k $ | (I) $ \frac{7}{10} $ | | (B) $ P(X < 3) $ | (II) $ \frac{53}{100} $ | | (C) $ P(X > 2) $ | (III) $ \frac{1}{10} $ | | (D) $ P(2 < X <7 ) $ | (IV) $ \frac{3}{10} $ | Choose the **correct** answer from the options given below :

If $A = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and B is a square matrix of order 3, then |AB| is equal to:

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Then variance of the number of kings is

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Then, E (X) is :

The number of ways to arrange the letters of the word "COMMITTEE" is:

If A is a square matrix of order 3, B = kA and |B| = $x$|A| then,

Which of the following graphs represent a function ?

The number of all onto functions from the set {1, 2, .......n} to itself is

The value of the determinant $\begin{vmatrix} a\cos\theta & b\sin\theta & 0 \\ -b\sin\theta & a\cos\theta & 0 \\ 0 & 0 & c \end{vmatrix}$ is :