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Important Questions Class 11 Mathematics Chapter 12
Important Questions for CBSE Class 11 Mathematics Chapter 12 – Introduction to ThreeDimensional Geometry
Chapter 12 of Class 11 Mathematics serves as an introduction to 3D geometry for students. Geometry is a mathematical branch that studies points, lines, and solid shapes in threedimensional coordinate systems. It explains the Zcoordinate as well. This chapter should be studied thoroughly so that students can have a clear understanding of the material and avoid confusion.
Extramarks has provided Important Questions for Class 11 Mathematics Chapter 12 to help students perform well in exams. Students can easily access this set of questions from the Extramarks website.
CBSE Class 11 Mathematics Chapter12 Important Questions
(add important questions)
Study Important Questions for Class 11 Mathematics Chapter 12 – Introduction to ThreeDimensional Geometry
Students can refer to the sample Chapter 12 Class 11 Mathematics Important Questions discussed here. They can also click the link below to access the complete set of Mathematics Class 11 Chapter 12 Important Questions.
1 Mark Answers and Questions
Q1. Name the octant in which the following lie: (5,2,3)
Ans: Octant I
Q2. Name the octant in which the following lie: (−5,4,3)
Ans: Octant II
Q3. Find the image of (−2,3,4) in the yz plane
Ans: (2,3,4)
Q4. Find the image of (5,2,−7) in the xy plane.
Ans: (5,2,7)
Q5. A point lies on Xaxis. What are the coordinates of the point?
Ans: (a,0,0)
Q6. Write the name of the plane in which the xaxis and yaxis are taken together.
Ans: X Y Plane
Q7. The point (4,−3,−6) lie in which octants?
Ans: VIII
Q8. The point (2,0,8) lies in which plane?
Ans: XZ Plane
Q9. A point is in the XZ plane. What is the value of y coordinates?
Ans: Zero
Q10. What are the coordinates of the XY plane?
Ans: (x,y,0)
Q11. The point (−4,2,5) lies in which octant.
Ans: Octant II
Q12. The distance from the origin to point (a,b,c) is:
Ans: Distance from origin= a2+b2+ c2
4 Marks Answers and Questions
Q1. Determine the points in the x y plane which is equidistant from these point A (2,0,3), B(0,3,2), and C(0,0,1)
Ans: Since the z coordinate in the XY plane is zero. So, let P(x, y, 0) be a point in the xy plane, such that PA=PB=PC. Now, PA=PB
PA2 = PB2
⇒(x−2)2+(y−0)2+(0−3)2=(x−0)2+(y−3)2+(0−2)2
2x−3y=0…..(i)
PB=PC
⇒PB2=PC2
⇒(x−0)2+(y−3)2+(0−2)2=(x−0)2+(y−0)2+(0−1)2T
⇒−6y+12=0⇒y=2……..(ii)
Put y=2 in (i) we get x=3
Hence the points required are (3,2,0).
Q2. If P and Qbe the points (3.4,5) and (−1,3,7) respectively. Find the equation of the set points A such that AP2+AQ2=K2 where K is a constant.
Ans: Let coordinates of point P be(x,y,z)
AP2=(x−3)2+(y−4)2+(z−5)2
=x2−6x+9+y2−8y+16+z2−10z+25
=x2+y2+z2−6x−8y−10z+50
AQ2=(x+1)2+(y−3)2+(z−7)2
=x2+2x+1+y2−6y+9+z2−14+49
=x2+y2+z2+2x−6y−14z+59
AP2+AQ2=K2
2(x2+y2+z2)−4x−14y−24z+109=K2
x2+y2+z2−2x−7y−12z= k2 – 1092
6 Marks Answers and Questions
Q1. If P and Q are the points (−2,2,3) and (−1,4,−3) respectively, then find the locus of A such that 3AP=2AQ.
Ans: The given points P(−2,2,3) and Q(−1,4,−3)
Suppose the coordinates of point A(x,y,z)
AP= (x+2)2 (y2)2 (23)2
AP= x2+ y2+ z2 +4x−4y−6z+17
AQ= (x+1)2 (y4)2 (z+3)2
AQ= x2+ y2+ z2 +2x−8y−6z+26
9AP2=4AQ2
9(x2+y2+z2+4x−4y−6z+17)=4(x2+y2+z2+2x−8y+6z+26)
(5×2+5y2+5z2+28x−4y−30z+49=0)
Q2. Show that the plane px+qy+rz+s=0 divides the line joining the points (x1,y1,z1) and (x2,y2,z2) in the ratio px1 + qy1+ rz1+spx2+ qy2+ rz2+s
Ans: Let the plane px+qy+rz+s=0 divide the line joining the points (x1,y1,z1) and(x2,y2,z2) in the ratio λ=1.
∴x=2×2 + x1λ+1 = y = λy2+y1λ+1 = z= λz2+z1λ + 1
∵ Plane px+qy+rz+s=0 Passing through (x0y,z)
p (λx2+x1)λ+1 + q (λy2+y1)λ+1 + r (λz2+z1)2+1 + s = 0
p(λx2+x1)+q(λy2+y1)+r(λz2+z1)+s(λ+1)=0
λ(px2+qy2+rz2+s)+(px1+qy1+rz1+s)=0
λ=− (px1+qy1+rz1+s)(px2+qy2+rz2+s)
Hence proved.
Q.1 If A(2, 2, 3) and B(13,3,13) are two points, the equation of set of points P such that 3PA = 2PB is
Marks:1
Ans
Let the coordinates of point P be (x, y, z).
Now, PA = {(x + 2)^{2}+ (y – 2)^{2} + (z – 3)^{2}}.
PA^{2} = (x + 2)^{2 }+ (y – 2)^{2 }+ (z – 3)^{2}
PB = {(x – 13)^{2}+ (y + 3)^{2} + (z – 13)^{2}}
PB^{2} = (x – 13)^{2 }+ (y + 3)^{2 }+ (z – 13)^{2}
Therefore, from the given condition,
3PA = 2PB 9PA^{2} = 4PB^{2}
9[(x + 2)^{2} + (y – 2)^{2} +(z – 3)^{2}] = 4[(x – 13)^{2} + (y + 3)^{2} +(z – 13)^{2}]
On solving this equation we get,
5(x^{2 }+ y^{2 }+ z^{2}) +140x – 60y – 50z – 1235 = 0.
Q.2 By using distance formula, prove that the three points ( – 4,6,10), (2,4,6) and (14, 0, – 2) are collinear.
Marks:4
Ans
Let points A(4,6,10), B(2,4,6) and C(14,0,2) are given. AB=(2+4)2+(46)2+(610)2=36+4+16=56=214 BC=(142)2+(04)2(26)2 =144+16+64=224=414 AC=(14+4)2+(06)2+(210)2=324+36+144=504=614 AB+BC=214+414=614= AC Thus, AtB and C are collinear.
Q.3 Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, 8) is divided by the XZplane.
Marks:2
Ans
LetXZplanedividesthesegmentjoiningA(4,8,10)andB(6,10,8) atP(x,y,z)in the ratiok:1.Then the coordinates of Pare1(4)+k(6)k+1,1(8)+k(10)k+1,1(10)k(8)k+14+6kk+1,8+10kk+1,108kk+1Since,
PliesonXZplaneSo,itsYcoordinateiszero,i.e.,8+10kk+1=08+10k=010k=8k=810k=45Therefore, XZplane divides AB externallyin the ratio4:5.
Q.4 Name the octants in which the following points lie: (7,4,3) and (5,3,2).
Points (7,4,3) and (5,3,2) lie in V and VII Octants respectively.
Marks:1
Ans
Points (7,4,3) and (5,3,2) lie in V and VII Octants respectively.
Q.5 Find the point in XYplane which is equidistant from three points A(2,0,3), B(0,3,2) and C(0,0,1).
Marks:4
Ans
The zcoordinate = 0 on xyplane.
Let P (x, y, 0) be a point on xyplane such that PA = PB = PC
Now, PA = PB
PA^{2} = PB^{2} (x 2)^{2} + (y0)^{2} + (03)^{2} = (x0)^{2} + (y 3)^{2} + (02)^{2}
4x6y = 0 or 2x 3y = 0 …(1)
PB = PC
PB^{2} = PC^{2}
(x 0)^{2} + (y3)^{2} + (0 2)^{2} = (x 0)^{2} + (y0)^{2} + (01)^{2} 6y + 12 = 0 or y = 2 …(2)
By (1), we get x = 3
Hence, the required point is (3, 2, 0).
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FAQs (Frequently Asked Questions)
1. How many exercises are there in Chapter 12 of Class 11 Mathematics?
There are three exercises (12.1, 12.2, and 12.3) and miscellaneous exercises in this chapter. Practicing these numerical problems will help the students become perfect with the concept and also help them to think logically. Class 11 Mathematics Chapter 12 Important Questions and answers provided by Extramarks will come in handy if students have any confusion, and they can rely upon these solutions with full confidence.
2. What are the x and yaxis in the plane known as? Also what is the form of coordinates of points XY in a plane?
The Xaxis and the Yaxis together are known as the XY plane, and the form in which the coordinates of points on the XY plane is (x, y, 0). Students should know this and implement it while doing the problems or answering the questions, which might be in the form of fillintheblanks or true or false. Understanding the basic concepts in the chapter is very important. Hence, students must practise stepwise solutions to understand the main topics and concepts of the chapter.
3. What is the concept of 3D Geometry covered in Chapter 12 for Class 11 Maths.
Introduction to Three Dimensional Geometry is covered in Chapter 12 Class 11 Mathematics covers topics such as finding the point coordinates in a given space, calculating the difference between two points using the section formula and the distance formula, and a brief overview of the cartesian coordinate system. Extramarks provides students with answers, study material, and stepbystep solutions to all of the exercises in this chapter. These responses have been verified by experienced experts.
4. Where can I find Important Questions for Chapter 12 of Class 11 Maths?
The Important Questions of Class 11 Maths Chapter 12 are available on the Extramarks website. These questions will allow students to put what they learned in Chapter 12 into practise. Students will also gain an understanding of the types of questions that may be asked on the exam, which will aid in their preparation. Students should pay close attention to this chapter because it is an introduction with many unfamiliar terms. Focus and attention are required to understand and grasp the concepts.