An Overview of Ncert Books Class 12 Maths Chapter 11 Free Download
FAQs on Ncert Books Class 12 Maths Chapter 11 Free Download
1. What are the most important topics in Class 12 Maths Chapter 11, Three Dimensional Geometry, for the board exam?
For the CBSE board exams, the most important and frequently tested topics from Three Dimensional Geometry are:
- Shortest distance between two skew lines: This is a very common high-scoring question.
- Equation of a plane: Questions on finding the equation of a plane in various forms (e.g., passing through three points, intersection of two planes, normal form).
- Equation of a line: Both vector and Cartesian forms are crucial.
- Angle between two lines, two planes, or a line and a plane.
- Distance of a point from a plane.
Focusing on these areas will cover the majority of expected questions.
2. How many marks does the Three-Dimensional Geometry unit carry in the CBSE Class 12 Maths exam for 2025-26?
As per the CBSE 2025-26 syllabus, the unit 'Vectors and Three-Dimensional Geometry' combined holds a weightage of 14 marks. Students can expect a significant portion of these marks, including a possible long-answer question, to come from Three-Dimensional Geometry, making it a very important chapter for scoring well.
3. Which types of questions are frequently asked from 3D Geometry for 3 or 5 marks?
Based on board trends, the expected question types are:
- 3-Mark Questions: These often involve finding the angle between two lines or two planes, finding the vector and Cartesian equations of a line passing through given points, or determining if lines are coplanar.
- 5-Mark Questions: These are typically more complex. The most expected 5-mark question is finding the shortest distance between two skew lines. Other possibilities include finding the coordinates of the foot of the perpendicular from a point to a plane or finding the image of a point in a plane.
4. What are the essential formulas from Chapter 11 needed to solve important exam questions?
To excel in this chapter, you must master the following formulas:
- Equation of a Line: Vector form r = a + λb and Cartesian form (x-x₁)/a = (y-y₁)/b = (z-z₁)/c.
- Equation of a Plane: Vector form r . n̂ = d and Cartesian form ax + by + cz = d.
- Angle between two lines: cos θ = |(a₁a₂ + b₁b₂ + c₁c₂) / (√(a₁²+b₁²+c₁²) * √(a₂²+b₂²+c₂²))|.
- Shortest Distance between Skew Lines: d = |(b₁ × b₂) . (a₂ - a₁) / |b₁ × b₂||.
- Distance of a Point from a Plane: d = |(Ax₁ + By₁ + Cz₁ - D) / √(A² + B² + C²)|.
5. What is the key difference between direction cosines and direction ratios, and why is this important for solving problems?
The key difference lies in their uniqueness. For any given line in space, the direction cosines (l, m, n) are unique and represent the cosines of the angles the line makes with the x, y, and z axes. They always satisfy the relation l² + m² + n² = 1. In contrast, direction ratios (a, b, c) are not unique; they are any three numbers proportional to the direction cosines (e.g., if (l,m,n) are DCs, then (kl, km, kn) are DRs for any k≠0). This distinction is vital because when a problem requires a unit vector along a line, you must use direction cosines, as they are derived from a vector of magnitude one.
6. How do you find the shortest distance between two skew lines, and why is this concept a favourite for board examiners?
The shortest distance between two skew lines (lines that are not parallel and do not intersect) is the length of the perpendicular line segment between them. It is calculated using the formula d = |(b₁ × b₂) . (a₂ - a₁) / |b₁ × b₂||, where a₁ and a₂ are position vectors of points on the lines, and b₁ and b₂ are the direction vectors. This question is considered important for board exams because it effectively tests a student's understanding of multiple vector concepts in a single problem: position vectors, direction vectors, the cross product (to find a vector perpendicular to both lines), and the dot product (to find the projection). It is a comprehensive application of vector algebra in 3D space.
7. What is a common mistake students make when finding the equation of a plane passing through the intersection of two other planes?
A common and time-consuming mistake is trying to find the equation of the line of intersection first and then finding the equation of the plane. The correct and most efficient method is to use the 'family of planes' concept. If the equations of the two intersecting planes are P₁ = 0 and P₂ = 0, then the equation of any plane passing through their intersection is given by P₁ + λP₂ = 0, where λ is a constant. You can find the value of λ by using the other given condition in the problem (e.g., the plane passes through a specific point). Forgetting this direct approach leads to lengthy calculations and a higher chance of errors.
8. When should I use the vector form versus the Cartesian form of equations for lines and planes to solve problems more easily?
Choosing the right form can significantly simplify a problem. Here is a general guideline:
- Use the vector form when the problem is conceptual or involves geometrical interpretations like finding the shortest distance, projections, or angles using dot/cross products. It keeps the calculations cleaner and more intuitive.
- Use the Cartesian form when the problem involves specific points, finding intersection points of lines/planes, or dealing with intercepts on the axes. It is more algebraic and direct for calculations involving coordinates.
For example, for the shortest distance between skew lines, the vector formula is almost always faster. For finding the point where a line intersects a plane, the Cartesian form is often more straightforward.
