ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths Solutions Chapter-2 Section-B. Step by step Solutions of ML Aggarwal ISC Understanding APC Mathematics Class-12 Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths Solutions

Class:	12th
Subject:	Mathematics
Chapter :	Ch-2 Three Dimensional Geometry Section-B

Board	ISC Board
Writer	ML Aggarwal ISC Understanding ( Vol-II)
Publications	APC Arya Publications ( 2020-21 )

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-: Select Topics :-

Exe-2.1, 

Exe-2.2, 

Exe-2.3, 

Exe-2.4, 

Exe-2.5,

Exe-2.6,

Exe-2.7,

Exe-2.8,

Chapter Test

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ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths Solutions

Direction Cosines of a Line:

If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos α, m = cos β and n = cos γ. Also, sum of squares of direction cosines of a line is always 1,
i.e. l² + m² + n² = 1 or cos² α + cos² β + cos² γ = 1
Note: Direction cosines of a directed line are unique.

Straight line:

A straight line is a curve, such that all the points on the line segment joining any two points of it lies on it.

Equation of Line Passing through Two Given Points:

Vector form: 𝑟⃗ =𝑎⃗ +𝜆(𝑏⃗ −𝑎⃗ ), λ ∈ R, where a and b are the position vectors of the points through which the line is passing.

Condition of Perpendicularity:

Two lines are said to be perpendicular, when in vector form 𝑏1⃗⋅𝑏2⃗=0; in cartesian form a₁a₂ + b₁b₂ + c₁c₂ = 0
or l₁l₂ + m₁m₂ + n₁n₂ = 0 [direction cosine form]

Plane:

A plane is a surface such that a line segment joining any two points of it lies wholly on it. A straight line which is perpendicular to every line lying on a plane is called a normal to the plane.

Equations of a Plane in Normal form

Vector form: The equation of plane in normal form is given by 𝑟⃗ ⋅𝑛⃗ =𝑑, where 𝑛⃗  is a vector which is normal to the plane.
Cartesian form: The equation of the plane is given by ax + by + cz = d, where a, b and c are the direction ratios of plane and d is the distance of the plane from origin.
Another equation of the plane is lx + my + nz = p, where l, m, and n are direction cosines of the perpendicular from origin and p is a distance of a plane from origin.
Note: If d is the distance from the origin and l, m and n are the direction cosines of the normal to the plane through the origin, then the foot of the perpendicular is (ld, md, nd).

Cartesian form:

If the equation of planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, then equation of any plane passing through the intersection of planes is a₁x + b₁y + c₁z – d₁ + λ (a₂x + b₂y + c₂z – d₂) = 0
where, λ is a constant and calculated from given condition.

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Exercise – 2.1,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1279 to 1280)

Question 1: If a line makes angles 90 , 60 and 30 with the positive direction of x, y and z- axes respectively, find its direction cosines.

Question 2: (i) Can the number …………… of a line ?

Question 3: …………………..

………………………..

Question 23: The figure shows a rectangular box of size a, b and c units ………………… sign cannot be +ve or -ve.

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Exercise – 2.2,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1290 to 1292)

Question 1: (i) Find the vector equation of the line …………… the vector 3i + 2j – 2k.

Question 2: Find the cartesian equation of the line which passes through the point …………….. of the vector i + 2j – k .

Question 3: ……………….

…………………….

Question 20: A line with direction number < 2, 7, -5 > is ……………. and the length intercepted on it.

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Exercise – 2.3,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1299 to 1301)

Question 1: Find the angle between each of the following pairs of lines :

…………………

Question 2: Find the angle between each of the following pairs of lines :

………………………

Question 3: (i) show that the lines……………….

Question 4: ………………….

……………………….

Question 13: Find the equation of a line passing through the point ……………….. .

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Exercise – 2.4,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1310)

Question 1: Find the shortest distance between each of the following pairs of lines :

……………………….

Question 2: Find the shortest distance between the lines ……………………….. .

Question 3: ……………………..

……………………..

Question 10: Find the vector equation of the lines joining the following pair of vertices and then find the shortest distance between the lines :

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Exercise – 2.5,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1326 to 1328)

Question 1: (i) Find the direction cosine of the normal to the plane 2x + 3y – z – 7 = 0.

Question 2: Find the cartesian equation of the following planes ;

………………………..

Question 3: ………………….

…………………….

Question 25: Prove that the normals to planes ………… are perpendicular to each other .

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Exercise – 2.6,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1348 to 1350)

Question 1: (i) Write the intercept cut off by the plane 2x + y – z = 5 on x- axis.

Question 2: Find the intercepts made by the plane ………… on the coordinate axes.

Question 3: …………………

…………………….

Question 26: Find the equation of the plane passing through the line of ………….. ratio of a normal to this plane.

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Exercise – 2.7,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1363 to 1365)

Question 1: Find the angle between the plane :

……………………

Question 2: Show that the following pairs of planes are …….. to each other ;

Question 3: ………………….

…………………

Question 26: Find the equation of the plane which contain the line of intersection of the plane …………….. parallel to the plane obtained above.

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Exercise – 2.8,

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1374 to 1375)

Question 1: In each of the following problems, find the perpendicular distance :

…………………..

Question 2: (i) Find the distance of the point whose position vector is ………… = 9.

Question 3: …………………..

…………………….

Question 15: State when the line ……… Also find the distance between the line and the plane.

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Chapter Test

ML Aggarwal Three Dimensional Geometry ISC Class-12 Understanding APC Maths

(page-1376 to 1377)

Question 1: Find the ration in which the join of the points ………… point of division.

Question 2: The cartesian equations of a line are …………….. and parallel to the given lines.

Question 3: …………………….

Question 31: Find the distance between the planes ……… = 0.

-: End of Three Dimensional Geometry ML Aggarwal ISC Class-12 Solutions :-

Return to :- ML Aggarwal ISC Class-12 APC Understanding Maths Solutions

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