Let D denote the parallelogram in the xy -plane with vertices (0,0), (12, 2), (6, 8), (18, 10). Consider the change of variables x = 6u + 3v, y= u+ 4v. Find P+Q+R: l, zydA = f PQ SS R du dv. 0 0

Q: The equation of the plane that contains the point (1, 2, 3) and the line x = 4t, y = t, z = t is…

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Q: Find the equation of the plane through the point (2,4,3)(2,4,3) that is perpendicular to the line…

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Q: Find a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

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Q: Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5,  yz

A: The trace of the plane in yz plane is obtained by substituting x=0 in the given equation of the…

Q: Let P(x1, y1) and Q(x2, y2) be two points in the plane. Prove that the equation of the line through…

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Q: Calculate , z² dS if S is the prism whose base is the triangle in the limited xy plane by the x axis…

A: Here given figure is Since prism has 5 surfaces (1)-OAB (2) OADE (3) CDE (4) OBCD (5) ABCE

Q: Let L be a line which passes through the point (-4,0, 1) and is perpendicular to the plane x + 2y +…

A: Please refer the attached image for complete solution.

Q: Let L be a line which passes through the point (-5,0, 1) and is perpendicular to the plane x + 2y +…

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Q: Find the distance between the given parallel planes: (a) 2x - 3y + z = 4 and 4x - 6y + 2z = 3 (b)…

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Q: Find the intersection of the plane – 5x + 4y – 6z = – 267 and the line T(t) = (6, – 7,6)t + (1, – 7,…

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Q: Find a plane through P (2, 1, 1) and perpendicular to the line of intersec- tion of the planes:…

A: find a plane through P2,1,1 and perpendicular to the line of intersection of the planes  2x+y-z=3 ,…

Q: Find an equation of the plane through (-3, 8, – 7) that is perpendicular to the planes 6x – y +z = 4…

A: We need to find equation of plane.

Q: At what point does the line :LiX==€ 1=10+8€ 14. -2y t52 +16:0 10+3€ z=41+3€ intersect with the plane…

A: Question is based on 3d Geometry.

Q: Find the trace of the plane in the given coordinate plane.3x − 9y + 4z = 5,  xz

A: Let the plane P whose equation is:                                          3x-9y+4z=5…

Q: Let L be a line which passes through the point (-8, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Find the point at which the line intersects the given plane. x = 2 − 2t,    y = 3t,    z = 1 + t;…

A: Given that  x = 2 − 2t,    y = 3t,    z = 1 + t;    x + 2y − z = 4 first we will substitute values…

Q: Let L be a line which passes through the point (-5, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: 24. The plane that contains the line x = 1 + t, y = 2 – t, z = 4 – 3t and is parallel to the plane…

A: To find out equation of plane first we change the equation of line in symmetrical form then find out…

Q: If possible, determine the vector, parametric,symmetrical and reduced line r that: a) passes…

A: Given the lines S1:t·1,2,3, t∈ℝ and S2:y=x+2, z=x. To find a line L passing through the point P1,1,1…

Q: Let L be a line which passes through the point (-4, 0, 1) and is perpendicular to the plane z + 2y +…

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Q: Determine whether the plane x + y + z = 9 and the line x = t, y = -2t + 1, z = t + 2 are parallel,…

A: The given equation of line is in parametric form, x=ty=-2tz=t+2 Convert it into the cartesian form,…

Q: Find an equation of the plane through (– 1, 3, –4) that is perpendicular to the planes 8x – y + z =…

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Q: Find an equation of the plane through (-7, 2, –6) that is perpendicular to the planes 4x – y + z = 4…

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Q: Find an equation of the plane through (-8, 6, - 6) that is perpendicular to the planes 9x - y + z =…

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Q: The distance from the point P(1,2,3) to the plane x +y + z = 0 is

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Q: Find the condition for the points (a,0) ,(h,k),(0,b) when ab is not equal to 0 to be collinear

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Q: Q1. A) Determine if the plane given by -x + 2z = 10 and the line given by %3D (x = 5,y = 2 - t,z =…

A: Consider the given values. -x+2z=10 And, (x=5, y=2-t, z=10+4t)

Q: Find the distance between the given parallel planes. 4x – 5y + z = 16, 8x – 10y + 2z = 3

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Q: Find the plane containing the intersecting lines x = -1+t, y = 2+ t, z =1–t; -0 <t < o∞ | L2: x = 1…

A: Find the planes containing the intersecting lines: L1:x=-1+t,y=2+t,z=1-t for all…

Q: Let L be a line which passes through the point (-8, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Let L be a line which passes through the point (-4, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Find the Polnt otintersectton of the lines: l,: x = 2+t, y =3+2t,z z 4 +3t X = S+ 2, y= 2S+4, 2=…

A: Here we will find the point of intersection of given lines,

Q: Let L be a line which passes through the point (-7,0, 1) and is perpendicular to the plane x + 2y +…

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Q: A. Show that the planes (x+ y– z = 1 ) and (2x – 3y+ 4z = 5 ) are neither parallel nor…

A: let the normal vectors for the vectors be a1, a2 thus a1=&lt;1,1,-1&gt; and a2=&lt;2,-3,4&gt;. Let…

Q: Let L be a line which passes through the point (-8,0, 1) and is perpendicular to the plane z + 2y +…

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Q: Find the distance between the given parallel planes. 2x - 3y + z = 8, 4x - 6y + 2z = 1

A: The solution are next step

Q: Find the point of intersection between the line r(t) = + t and the plane 4x + 5y + 3z = 5. o a. (…

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Q: Suppose we have: u = and v = . Based on their relative positions in the xy-plane, in which…

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Q: Show that the line in which the planes x + 2y - 2z = 5 and 5x - 2y - z = 0 intersect is parallel to…

A: Find the vectors normal to the planes x + 2y - 2z = 5 and 5x - 2y - z = 0.

Q: Find the distance between the following parallel planes: (a) 3x – 4y +z=1 and 6x – 8y + 2z =3 (b) –…

A: Solve the above questions.

Q: 02. A. Show that the plane 3x + y - z = 9 and the line {x = 2,y = -1+ 3t ,z = 1+ 3t} are parallel,…

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Q: giren points wll, 2), x (4,-3), y (-6, +) and z (-4, +) cartesjan plane on Since tnow that A Cb), B…

A: To find:- How opposite and adjacent sides related to given slopes  And What specific geometric…

Q: find the (dis tan ant) between the planes ml and m2 when ml =: 3x - 2y+z = 9 and m2 =: 6x - 4y+ 2z =…

A: Consider the equation of planes m1: A1x+B1y+C1z+D1=0  m2: A2x+B2y+C2z+D2=0Two planes are parallel if…

Q: Let L be a line which passes through the point (-4, 0, 1) and is perpendicular to the plane x + 2y +…

A: The line L is perpendicular to the plane, x+2y+4z = 0 Hence the direction ratios of the line L is…

Q: Find a parametrization for the line segment beginning at P, and ending at P2. P (2,-3,2) and P2(0, –…

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Q: Let L be a line which passes through the point (-3, 0, 1) and is perpendicular to the plane x + 2y +…

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Q: Determine an equation of the form Ax t t Cz + D=O By for the plane that contains the line (x,y,2)=…

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Q: an) Determine the vecter equation.of the line af intersection of the twu planes = -l 3x+ Zy +4Z X ty…

A: given that two planes 3x+2y+4z=-1 and x+y+z=1. we need to find the vector equation for the line of…

Q: At what point does the line r(t) = (6- 3t)i + (-2 + 2t)j + (-1- 3t)k intersect the yz-plane %3D O a.…

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Q: Does the line r(t) = intersect the plane 5x - 2y - z = 21 and if so, where?

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.