Read the following passage and answer the questions given below. There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. (i) What is the probability that the shell fired from exactly one of them hit the plane? (ii) If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?

Read the following passage and answer the questions given below. …

Ritika starts walking from his house to shopping mall. Instead of going to the mall directly, she first goes to a ATM, from there to her daughter's school and then reaches the mall. ln the diagram, A, B, C, and D represent the coordinates of House, ATM, School and Mall respectively.

Based on the above information, answer the following questions.

Distance between House (A) and ATM (B) is:

$3\text{ units}$
$3\sqrt{2}\text{ units}$
$\sqrt{2}\text{ units}$
$4\sqrt{2}\text{ units}$

Distance between ATM (B) and School (C) is:

$\sqrt{2}\text{ units}$
$2\sqrt{2}\text{ units}$
$3\sqrt{2}\text{ units}$
$4\sqrt{2}\text{ units}$

Distance between School (C) and Shopping mall (D) is:

$3\sqrt{2}\text{ units}$
$5\sqrt{2}\text{ units}$
$7\sqrt{2}\text{ units}$
$10\sqrt{2}\text{ units}$

What is the total distance travelled by Ritika:

$4\sqrt{2}\text{ units}$
$6\sqrt{2}\text{ units}$
$8\sqrt{2}\text{ units}$
$9\sqrt{2}\text{ units}$

What is the extra distance travelled by Ritika in reaching the shopping mall?

$3\sqrt{2}\text{ units}$
$5\sqrt{2}\text{ units}$
$6\sqrt{2}\text{ units}$
$7\sqrt{2}\text{ units}$

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Family photography is all about capturing groups of people that have family ties. These range from the small group, such as parents and their children. New-born photography also falls under this umbrella. Mr Ramesh, His wife Mrs Saroj, their daughter Sonu and son Ashish line up at random for a family photograph, as shown in figure.

(i) Find the probability that daughter is at one end, given that father and mother are in the middle.

(ii) Find the probability that mother is at right end, given that son and daughter are together.

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Between students of class $XII$ of two schools $A$ and $B$ basketball match is organised. For which, a team from each school is chosen, say $T_1$ be the team of school $A$ and $T_2$ be the team of school $B.$ These teams have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probability of $T_1$ winning, rawmg an osrng a game against $T_2$ are $\frac{1}{2},\ \frac{3}{10}$ and $\frac{1}{5}$ respecnvely. Each team gets $2$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by team $A$ and

$B$ respectively, after two games. Based on the above information, answer the following questions.

$P(T_2$ winning a match against $T_1)$ is equal to:

$\frac{1}{5}$
$\frac{1}{6}$
$\frac{1}{3}$
None of these

$P(T_2$ drawing a match against $T_1)$ is equal to:

$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{6}$
$\frac{3}{10}$

$P(X > Y)$ is equal to:

$\frac{1}{4}$
$\frac{5}{12}$
$\frac{1}{20}$
$\frac{11}{20}$

$P(X = Y)$ is equal to:

$\frac{11}{100}$
$\frac{1}{3}$
$\frac{29}{100}$
$\frac{1}{2}$

$P(X + Y = 8)$ is equal to:

$0$
$\frac{5}{12}$
$\frac{13}{36}$
$\frac{7}{12}$

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A trust fund has $₹ 35000$ that must be invested in two different types of bonds, say $\mathrm{X}$ and $\mathrm{Y}$. The first bond pays $10 \%$ interest p.a. which will be given to an old age home and second one pays $8 \%$ interest p.a. which will be given to WWA (Women Welfare Association). Let A be a $1 \times 2$ matrix and B be a $2 \times 1$ matrix, representing the investment and interest rate on each bond respectively.

(i) Represent the given information in matrix algebra.

(ii) If ₹ 15000 is invested in bond $\mathrm{X}$, then find total amount of interest received on both bonds?

(iii) If the trust fund obtains an annual total interest of ₹ 3200 , then find the investment in two bonds.

OR

If the amount of interest given to old age home is ₹500, then find the amount of investment in bond Y.

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A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and Bare respectively sitting
on the plane represented by the equation$ \vec{\text{r}}.(\hat{\text{i}}+\hat{\text{j}}+\hat{2\text{k}})=5$ and $ \vec{\text{r}}.(\hat{2\text{i}}-\hat{\text{j}}+\hat{\text{k}})=6$ to cheer up the team of their respective schools.

Based on the above information, answer the following questions.

The cartesian equation of the plane on which students of school A are seated is:

2x - y + z = 8
2x + y + z = 8
x + y + 2z = 5
x + y + z = 5

The magnitude of the normal to the plane on which students of school Bare seated, is:

$\sqrt{5}$
$\sqrt{6}$
$\sqrt{3}$
$\sqrt{2}$

The intercept form of the equation of the plane on which students of school Bare seated is:

$\frac{\text{x}}{6}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{(-6)}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{3}=1$

Which of the following is a student of school B?

Mohit sitting at (1, 2, 1)
Ravi sitting at (0, 1, 2)
Khushi sitting at (3, 1, 1)
Shewta sitting at (2, -1, 2)

The distance of the plane, on which students of school Bare seated, from the origin is:

6 units
$\frac{1}{\sqrt{6}}\text{ units}$
$\frac{5}{\sqrt{6}}\text{ units}$
$\sqrt{6}\text{ units}$

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Let $\text{A}=\begin{bmatrix}1&0\\2&1\end{bmatrix},$ and $U_1, U_2$ are e first and second columns respectively of a $2 \times 2$ matrix $U.$ Also, let the column matrices $U_1$ and $U_2$ satisfying $\text{AU}_1=\begin{bmatrix}1\\0\end{bmatrix}$ and $\text{AU}_2=\begin{bmatrix}2\\3\end{bmatrix}.$ Based on the above information, answer the following questions.

The matrix $U_1 + U_2$ is equal to:

$\begin{bmatrix}1\\-1\end{bmatrix}$
$\begin{bmatrix}2\\-2\end{bmatrix}$
$\begin{bmatrix}3\\-3\end{bmatrix}$
$\begin{bmatrix}4\\-4\end{bmatrix}$

The value of $|U|$ is:

$2$
$-2$
$3$
$-3$

If $\text{X}=\begin{bmatrix}3&2\end{bmatrix}\text{U}\begin{bmatrix}3\\2\end{bmatrix},$ then the value of $|X| =$

$3$
$-3$
$-5$
$5$

The minor of element at the position $a_{22}$ in $U$ is:

$1$
$2$
$-2$
$-1$

If $\text{U}=[\text{a}_\text{ij}]_{2\times2},$ then the value of $a_{11}A_{11 }+ a_{12}A_{12},$ where $A_{ij}$ denotes the cofactor of $a_{ij},$ is:

$1$
$2$
$-3$
$3$

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Read the following text carefully and answer the questions that follow:
Team $\text{P, Q, R}$ went for playing a tug of war game. .
Teams $\text{P, Q, R}$ have attached a rope to a metal ring and is trying to pull the ring into their own areas $($team areas when in the given figure below$)$.
Team $P$ pulls with force$F _1=4 \hat{i}+0 \hat{j} KN$
Team $Q$ pull with force $F _2=-2 \hat{i}+4 \hat{j} KN$
Team $R$ pulls with force $F _3=-3 \hat{i}-3 \hat{j} KN$

$i.$ What is the magnitude of the teams combined force? $(1)$
$ii.$ Find the magnitude of Team $B. (1)$
$iii.$ Which team will win the game? $(2)$
$OR$
Find the probability that she gets grade $A$ in at least one subject. $(2)$

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The Government declare that farmers can get ₹ 300 per quintal for their onions on 1st July and after that, the price will be dropped by ₹ 3 per quintal per extra day. Govind's father has 80 quintals of onions in the field on 1st July and he estimates that the crop is increasing at the rate of 1 quintal per day.

(i) If $x$ is the number of days after $1^{\text {st }}$ July, then express price and quantity of onion and the revenue as a function of $x$.

(ii) Find the number of days after 1st July, when Govind's father attains maximum revenue.

(iii) On which day should Govind's father harvest the onions to maximize his revenue?

OR

Find the maximum revenue collected by Govind's father.

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Consider the following diagram, where the forces in the cable are given.

Based on the above information, answer the following questions.

The cartesian equation of line along EA is:

$\frac{\text{x}}{-4}=\frac{\text{y}}{3}=\frac{\text{z}}{12}$
$\frac{\text{x}}{-4}=\frac{\text{y}}{3}=\frac{\text{z}-24}{12}$
$\frac{\text{x}}{-3}=\frac{\text{y}}{3}=\frac{\text{z}-12}{12}$
$\frac{\text{x}}{3}=\frac{\text{y}}{4}=\frac{\text{z}-24}{12}$

The vector $\overline{\text{ED}}$ is:

$8\hat{\text{i}}-6\hat{\text{j}}+24\hat{\text{k}}$
$-8\hat{\text{i}}-6\hat{\text{j}}+24\hat{\text{k}}$
$-8\hat{\text{i}}-6\hat{\text{j}}-24\hat{\text{k}}$
$8\hat{\text{i}}+6\hat{\text{j}}+24\hat{\text{k}}$

The length of the cable EB is:

24 units
26 units
27 units
25 units

The length of cable EC is equal to the length of:

EA
EB
ED
All of these

The sum of all vectors along the cables is:

$96\hat{\text{i}}$
$96\hat{\text{j}}$
$-96\hat{\text{k}}$
$96\hat{\text{k}}$

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Three friends A, Band Care playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw.

Based on the above information, answer the following questions.

P(A | B) =

$\frac{1}{6}$
$\frac{1}{3}$
$\frac{1}{2}$
$\frac{2}{3}$

P(B | C) =

$\frac{2}{3}$
$\frac{1}{12}$
$\frac{1}{9}$
$0$

$\text{P}(\text{A}\cap\text{B}|\text{C})=$

$\frac{1}{6}$
$\frac{1}{2}$
$\frac{1}{12}$
$\frac{1}{3}$

P(A | C)

$\frac{1}{4}$
$1$
$\frac{2}{3}$
None of these.

$\text{P}(\text{A}\cup\text{B}|\text{C})=$

0
$\frac{1}{2}$
$\frac{2}{3}$
1

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