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

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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?

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The Indian Coast Guard (ICG) while patrolling, saw a suspicious boat with four men. They were nowhere looking like fishermen. The soldiers were closely observing the movement of the boat for an opportunity to seize the boat. They observe that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of coast guard helicopter and boat are (2, 3, 5) and (1, 4, 2) respectively.

Based on the above information, answer the following questions.

If the line joining the positions of the helicopter and boat is perpendicular to the plane in which boat moves, then equation of plane is:

x - y + 3z = 2
x + y + 3z = 2
x - y + 3z = 3
x + y + 3z = 3

If the soldier decides to shoot the boat at given instant of time, where the distance measured in metres then what is the distance that bullet has to travel?

$\sqrt{5}\text{m}$
$\sqrt{8}\text{m}$
$\sqrt{10}\text{m}$
$\sqrt{11}\text{m}$

If the speed of bullet is 30m/ sec, then how much time will the bullet take to hit the boat after the shot is fired?

30 seconds
1 second
$\frac{1}{2}\text{second}$
$\frac{\sqrt{11}}{30}\text{seconds}$

At the given instant of time, the equation of line passing through the positions of helicopter and boat is:

$\frac{\text{x}}{1}=\frac{\text{y}}{-1}=\frac{\text{z}}{3}$
$\frac{\text{x}-1}{1}=\frac{\text{y}-4}{-1}=\frac{\text{z}-2}{3}$
$\frac{\text{x}}{1}=\frac{\text{y}}{1}=\frac{\text{z}}{-3}$
$\frac{\text{x}-1}{1}=\frac{\text{y}-4}{1}=\frac{\text{z}-2}{-3}$

At a different instant of time, the boat moves to a different position along the planar surface. What should be the coordinates of the location of the boat for the bullet to hit the boat if soldier shoots the bullet along the line whose equation is $\frac{\text{x}-1}{1}=\frac{\text{y}-1}{-2}=\frac{\text{z}-2}{3}?$

$\Big(\frac{1}{2},\frac{1}{2},\frac{1}{2}\Big)$
$\Big(\frac{3}{4},\frac{3}{2},\frac{5}{4}\Big)$
$\Big(\frac{1}{3},\frac{1}{4},\frac{1}{5}\Big)$
None of these

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Three slogans on chart papers are to be placed on a school bulletin board at the points A, Band C displaying A (Hub of Learning), B (Creating a better world for tomorrow) and C (Education comes first). The coordinates of these points are (1, 4, 2), (3, -3, -2) and (-2, 2, 6) respectively.

Based on the above information, answer the following questions.

Let $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ be the position vectors of points A, B and C respectively, then $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$ is equal to:

$2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}$
$2\hat{\text{i}}-3\hat{\text{j}}-6\hat{\text{k}}$
$2\hat{\text{i}}+8\hat{\text{j}}+3\hat{\text{k}}$
$2(7\hat{\text{i}}+8\hat{\text{j}}+3\hat{\text{k}})$

Which of the following is not true?

$\overline{\text{AB}}+\overline{\text{BC}}+\overline{\text{CA}}=\vec{0}$
$\overline{\text{AB}}+\overline{\text{BC}}-\overline{\text{AC}}=\vec{0}$
$\overline{\text{AB}}+\overline{\text{BC}}-\overline{\text{CA}}=\vec{0}$
$\overline{\text{AB}}-\overline{\text{CB}}+\overline{\text{CA}}=\vec{0}$

Area of $\triangle\text{ABC}$ is:

19 sq. units
$\sqrt{1937}\text{sq}.\text{units}$
$\frac{1}{2}\sqrt{1937}\text{sq}.\text{units}$
$\sqrt{1837}\text{sq}.\text{units}$

Suppose, if the given slogans are to be placed on a straight line, then the value of $|\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}|$ will be equal to:

-1
-2
2
0

If $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}},$ then unit vector in the direction of vector $\vec{\text{a}}$ is:

$\frac{2}{7}\hat{\text{i}}-\frac{3}{7}\hat{\text{j}}-\frac{6}{7}\hat{\text{k}}$
$\frac{2}{7}\hat{\text{i}}+\frac{3}{7}\hat{\text{j}}+\frac{6}{7}\hat{\text{k}}$
$\frac{3}{7}\hat{\text{i}}+\frac{2}{7}\hat{\text{j}}+\frac{6}{7}\hat{\text{k}}$
None of these

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A child cut a pizza with a knife. Pizza is circular in shape which is represented by $x^2+y^2=4$ and sharp edge of knife represents a straight line given by $\text{x}=\sqrt{3\text{y}}$ Based on the above information, answer the following questions.

The point(s) of intersection of the edge of knife (line) and pizza shown in the figure is (are).

$(1, \sqrt{3}),(-1,-\sqrt{3})$
$(\sqrt{3},1),(-\sqrt{3,}-1)$
$(\sqrt{2,}0),(0,\sqrt{3})$
$(-\sqrt{3,}),(1,-\sqrt{3})$

Which of the following shaded portion represent the smaller area bounded by pizza and edge of knife in first quadrant?

Value of area of the region bounded by circular pizza and edge of knife in first quadrant is.

$\frac{\pi}{2}\text{ sq.units}$
$\frac{\pi}{3}\text{ sq.units}$
$\frac{\pi}{5}\text{ sq.units}$
$\pi\text{ sq.units}$

Area of each slice of pizza when child cut the pizza into 4 equal pieces is.

$\pi\text{ sq.units}$
$\frac{\pi}{2}\text{ sq.units}$
$3\pi\text{ sq.units}$
$2\pi\text{ sq.units}$

Area of whole pizza is.

$3\pi\text{ sq.units}$
$2\pi\text{ sq.units}$
$5\pi\text{ sq.units}$
$4\pi\text{ sq.units}$

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Read the following text carefully and answer the questions that follow:
The relation between the height of the plant$ (y$ in $\ cm)$ with respect to exposure to sunlight is governed by the following equation $y=4 x-\frac{1}{2} x^2$ where x is the number of days exposed to sunlight.

$i.$ Find the rate of growth of the plant with respect to sunlight. $(1)$
$ii.$ What is the number of days it will take for the plant to grow to the maximum height? $(1)$
$iii.$ Verify that height of the plant is maximum after four days by second derivative test and find the maximum height of plant. $(2)$
OR
What will be the height of the plant after $2$ days? $(2)$

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Shreya got a rectangular parallelepiped shaped box and spherical ball inside it as return gift. Sides of the box are x, 2x, and $\frac{\text{x}}{3},$ while radius of the ball is r.

Based on the above information, answer the following questions.

If S represents the sum of volume of parallelepiped and sphere, then Scan be written as.

$\frac{4\text{x}^3}{3}+\frac{2}{2}\pi\text{r}^2$
$\frac{2\text{x}^2}{3}+\frac{4}{3}\pi\text{r}^2$
$\frac{2\text{x}^3}{3}+\frac{4}{3}\pi\text{r}^3$
$\frac{2}{3}\text{x}+\frac{4}{3}\pi\text{r}$

If sum of the surface areas of box and ball are given to be constant $k^2$ then x is equal to.

$\sqrt{\frac{\text{k}^2-4\pi\text{r}^2}{6}}$
$\sqrt{\frac{\text{k}^2-4\pi\text{r}}{6}}$
$\sqrt{\frac{\text{k}^2-4\pi}{6}}$
$\text{None of these}$

The radius of the ball, when Sis minimum, is.

$\sqrt{\frac{\text{k}^2}{54+\pi}}$
$\sqrt{\frac{\text{k}^2}{54+4}}$
$\sqrt{\frac{\text{k}^2}{64+3\pi}}$
$\sqrt{\frac{\text{k}^2}{4\pi+3}}$

Relation between length of the box and radius of the ball can be represented as.

$\text{x} = \frac{2}{\text{r}}$
$\text{x}=\frac{\text{r}}{2}$
$\text{x}=\frac{2}{\text{r}}$
$\text{x}=3\text{r}$

Minimum value of S is.

$\frac{\text{k}^2}{2(3\pi+54)^\frac{2}{3}}$
$\frac{\text{k}}{2(3\pi+54)^\frac{3}{2}}$
$\frac{\text{k}^3}{2(4\pi+54)^\frac{1}{2}}$
$\text{None of these}$

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An Apache helicopter of the enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $(3,7)$ want to shoot down the helicopter when it is nearest to him.

(i) If $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ be the position of a helicopter on curve $\mathrm{y}=\mathrm{x}^2+7$, then find distance $\mathrm{D}$ from $\mathrm{P}$ to soldier place at $(3,7)$.

(ii) Find the critical point such that distance is minimum.

(iii) Verify by second derivative test that distance is minimum at $(1,8)$.

Find the minimum distance between soldier and helicopter?

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Consider the curve $x^2+y^2=16$ and line $y = x$ in the first quadrant. Based on the above information, answer the following questions.

Point of intersection of both the given curves is.

$(0, 4)$
$(0, 2\sqrt{2})$
$(2\sqrt{ 2},2\sqrt{2})$
$(2,\sqrt{2},4)$

Which of the following shaded portion represent the area bounded by given two curves?

None of these

The value of the integral $\int\limits_{0}^{2\sqrt{2}}\text{x}\text{dx}$ is.

The value of the integral $\int\limits_{2\sqrt{2}}^{0}\sqrt{16-\text{x}^2}\text{ dx}$ is.

$2(\pi-2)$
$2(\pi-8)$
$4(\pi-2)$
$4(\pi+2)$

Area bounded by the two given curves is.

$3\pi\text{ sq.units}$
$\frac{\pi}{2}\text{ sq.units}$
$\pi\text{ sq.units}$
$2\pi\text{ sq.units}$

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An architecture design a auditorium for a school for its cultural activities. The floor of the auditorium is rectangular in shape and has a fixed perimeter $P.$

Based on the above information, answer the following questions.

If $x$ and $y$ represents the length and breadth of the rectangular region, then relation between the variable is.

$x + y = P$
$x^2 + y^2 = P^2$
$2(x + y) = P$
$x + 2y = P$

The area (A) of the rectangular region, as a function of $x,$ can be expressed as.

$\text{A}=\text{px}+\frac{\text{x}}{2}$
$\text{A}=\frac{\text{px}+\text{x}^2}{2}$
$\text{A}=\frac{\text{px}-\text{2x}^2}{2}$
$\text{A}=\frac{\text{x}^2}{2}+\text{px}^2$

School's manager is interested in maximising the area of floor $'A'$ for this to be happen, the value of $x$ should be.

$\text{P}$
$\frac{\text{P}}{2}$
$\frac{\text{P}}{3}$
$\frac{\text{P}}{4}$

The value of $y,$ for which the area of floor is maximum, is.

$\frac{\text{P}}{2}$
$\frac{\text{P}}{3}$
$\frac{\text{P}}{4}$
$\frac{\text{P}}{16}$

Maximum area of floor is.

$\frac{\text{P}^2}{16}$
$\frac{\text{P}^2}{64}$
$\frac{\text{P}^2}{4}$
$\frac{\text{P}^2}{28}$

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A factory has three machines $A, B$ and $C$ to manufacture bolts. Machine A manufacture $30\%$, machine $B$ manufacture $20\%$ and machine C manufacture $50\%$ of the bolts respectively. Out of their respective outputs $5\%, 2\%$ and $4\%$ are defective. A bolt is drawn at random from total production and it is found to be defective.

Based on the above information, answer the following questions.

Probability that defective bolt drawn is manufactured by machine $A$, is:

$\frac{4}{13}$
$\frac{5}{13}$
$\frac{6}{13}$
$\frac{9}{13}$

Probability that defective bolt drawn is manufactured by machine $B,$ is:

$0.3$
$0.1$
$0.2$
$0.4$

Probability that defective bolt drawn is manufactured by machine $C,$ is:

$\frac{16}{39}$
$\frac{17}{39}$
$\frac{20}{39}$
$\frac{15}{39}$

Probability that defective bolt is not manufactured by machine $B,$ is:

$\frac{35}{39}$
$\frac{61}{39}$
$\frac{41}{39}$
None of these.

Probability that defective bolt is not manufactured by machine $C,$ is:

$0.03$
$0.09$
$0.5$
$0.9$

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A card is lost from a pack of $52$ cards. From the remaining cards two cards are drawn at random.

Based on the above information, answer the following questions.

The probability of drawing two diamonds, given that a card of diamond is missing, is:

$\frac{21}{425}$
$\frac{22}{425}$
$\frac{23}{425}$
$\frac{1}{425}$

The probability of drawing two diamonds, given that a card of heart is missing, is:

$\frac{26}{425}$
$\frac{22}{425}$
$\frac{19}{425}$
$\frac{23}{425}$

Let $A$ be the event of drawing two diamonds from remaining $51$ cards and $E_1, E_2, E_3$ and $E_4$ be the events that lost card is of diamond, club, spade and heart respectively, then the approximate value of $\displaystyle\sum_{\text{i}=1}^{4}\text{P(A|E}_\text{i})$ is:

$0.17$
$0.24$
$0.25$
$0.18$

AU of a sudden, missing card is found and, then two cards are drawn simultaneously without replacement. Probability that both drawn cards are king is:

$\frac{1}{52}$
$\frac{1}{221}$
$\frac{1}{121}$
$\frac{2}{221}$

If two cards are drawn from a well shuffled pack of $52$ cards, one by one with replacement, then probability of getting not a king in $1^{st}$ and $2^{nd}$ draw is:

$\frac{144}{169}$
$\frac{12}{169}$
$\frac{64}{169}$
None of these

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Probability — Maths STD 12 Science — Question

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