Important Questions Class 11 Mathematics Chapter 12
Important Questions for CBSE Class 11 Mathematics Chapter 12 – Introduction to Three-Dimensional 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 three-dimensional coordinate systems. It explains the Z-coordinate 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 Chapter-12 Important Questions
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Study Important Questions for Class 11 Mathematics Chapter 12 – Introduction to Three-Dimensional 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 y-z plane
Ans: (2,3,4)
Q4. Find the image of (5,2,−7) in the x-y plane.
Ans: (5,2,7)
Q5. A point lies on X-axis. What are the coordinates of the point?
Ans: (a,0,0)
Q6. Write the name of the plane in which the x-axis and y-axis 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 3|AP|=2|AQ|.
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 (y-2)2 (2-3)2
|AP|= x2+ y2+ z2 +4x−4y−6z+17
|AQ|= (x+1)2 (y-4)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.