## Three Dimensional Geometry Quiz

Important topics for Maths has been designed in such a way that it offers very practical and application-based learning to further make it easier for students to understand every concept or topic by correlating it with the day-to-day experiences.

Q1. A tetrahedron has vertices O(0,0,0),A(1,2,1),B(2,1,3) and C(-1,1,2), then angle between faces OAB and ABC will be:
•  cos^(-1)⁡(17/31)
•  30°
•  90°
•  cos^(-1)⁡(19/35)
Solution

Q2.Let the equations of a line and a plane be (x+3)/2=(y-4)/3=(z+5)/2 and 4x-2y-z=1, respectively, the
•  The line is parallel to the plane
•  The line is perpendicular to the plane
•  The line lies in the plane
•  None of these
Solution
4(2)-2(3)-1(2)=0 Also, point (-3,4,-5) does not lie on the plane Therefore, the line is parallel to the plane

Q3.  The length of the perpendicular from the origin to the plane passing through the point a and containing the line r =b +Î»c ⃗is
•   [a ⃗b c ]/|a ×b +b ×c +c ×a |
•  [a b c ]/|a ×b +b ×c |
•  [a b c ]/|b ×c +c ×a |
•  [a b c ]/|c ×a +a ×b |
Solution
Q5.The intercept made by the plane r ∙n =q on the x-axis is
•   q/(i ̂∙n )
•  (i ̂∙n )/q
•   (i ̂∙n )/q
•  q/(|n|)
Solution
x intercept is say x_1 ⇒ Plane passes through it ∴x_1 i ̂∙n ⃗=q ⇒x_1=q/(i ̂∙n ⃗ )

Q6. What is the nature of the intersection of the set of planes x+ay+(b+c)z+d=0,x+by+(c+a)z+d=0 and x+cy+(a+b)z+d=0?
•  They meet at a point
•  b) They form a triangular prism
• They pass through a line
•  They are at equal distance from the origin
Solution

Q7.Which of the following are equations for the plane passing through the points P(1,1,-1),Q(3,0,2) and R(-2,1,0)?
•  (2i ̂-3j ̂+3k ̂ )∙((x+2) i ̂+(y-1) j ̂+zk ̂ )=0
•  x=3-t,y=-11t,z=2-3t
•  (x+2)+11(y-1)=3z
•  (2i ̂-j ̂+3k ̂ )×(-3i ̂+k ̂ )∙((x+2) i ̂+(y-1) j ̂+zk ̂ )=0
Solution
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Q8.The line through i ̂+3j ̂+2k ̂ and ⊥ to the line r =(i ̂+2j ̂-k ̂ )+Î»(2i ̂+j ̂+k ̂) and r =(2i ̂+6j ̂+k ̂ )+Î¼(i ̂+2j ̂+3k ̂) is
•  r =(i ̂+2j ̂-k ̂ )+Î»(-i ̂+5j ̂-3k ̂)
•  Tr =i ̂+3j ̂+2k ̂+Î»(i ̂-5j ̂+3k ̂)
•  r =i ̂+3j ̂+2k ̂+Î»(i ̂+5j ̂+3k ̂)
•  r =r =i ̂+3j ̂+2k ̂+Î»(-i ̂-5j ̂-3k ̂)
Solution
The required line passes through the point i ̂+3j ̂+2k ̂ and is perpendicular to the lines r ⃗=(i ̂+2j ̂-k ̂ )+Î»(2i ̂+j ̂+k ̂) and r ⃗=(2i ̂+6j ̂+k ̂ )+Î¼(i ̂+2j ̂+3k ̂); therefore it is parallel to the vector b ⃗=(2i ̂+6j ̂+k ̂ )×Î¼(i ̂+2j ̂+3k ̂ )=(i ̂-5j ̂+3k ̂) Hence, the equation of the required line is r ⃗=(i ̂+3j ̂+2k ̂ )+Î»(i ̂-5j ̂+3k ̂)

Q9.Equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and ⊥ to the plane 2x+6y+6z-1=0 is
•  3x+4y+5z=9
•  3x+4y-5z=9
•  3x+4y-5z=9
•  None of these
Q10. The intersection of the spheres x^2+y^2+z^2+7x-2y-z=13 and x^2+y^2+z^2-3x+3y+4z=8 is the same as the intersection of one of the spheres and the plane
•  x-y-z=1
•  x-2y-z=1
•  x-y-2z=1
• 2x-y-z=1
Solution
The given sphere are x^2+y^2+z^2+7x-2y-z-13=0 (i) and x^2+y^2+z^2-3x+3y+4z-8=0 (ii) Subtracting (ii) from (i), we get 10x-5y-5z-5=0 ⇒ 2x-y-z=1

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