JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank

done Self Evaluation Test - Introduction to Three Dimensional Geometry Total Questions - 30

• question_answer1)
  If the sum of the squares of the distance of the point (x, y, z) from the points (a, 0, 0) and (-a, 0, 0) is \[2{{c}^{2}}\], then which one of the following is correct?
  A)
  \[{{x}^{2}}+{{a}^{2}}=2{{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done clear
  B)
  \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done clear
  C)
  \[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] done clear
  D)
  \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}+{{y}^{2}}+{{z}^{2}}\] done clear
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• question_answer2)
  The ordered pair \[(\lambda ,\,\,\mu )\] such that the points \[(\lambda ,\mu ,-6),\] (3, 2, -4) and (9, 8, -10) become collinear is
  A)
  (3, 4) done clear
  B)
  (5, 4) done clear
  C)
  (4, 5) done clear
  D)
  (4, 3) done clear
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• question_answer3)
  The co-ordinates of the point in which the line joining the points (3, 5, -7) and (-2, 1, 8) is intersected by the plane yz are given by
  A)
  \[\left( 0,\,\,\frac{13}{5},\,\,2 \right)\] done clear
  B)
  \[\left( 0,-\frac{13}{5},-2 \right)\] done clear
  C)
  \[\left( 0,-\frac{13}{5},\frac{2}{5} \right)\] done clear
  D)
  \[\left( 0,\,\,\frac{13}{5},\,\,\frac{2}{5} \right)\] done clear
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• question_answer4)
  If P (3, 2, - 4), Q (5, 4, - 6) and R (9, 8, -10) are collinear, then R divides PQ in the ratio
  A)
  3 :2 internally done clear
  B)
  3:2 externally done clear
  C)
  2:1 internally done clear
  D)
  2:1 externally done clear
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• question_answer5)
  Points ( -2, 4, 7), (3, -6, -8) and (1, -2, -2) are
  A)
  Collinear done clear
  B)
  Vertices of an equilateral triangle done clear
  C)
  Vertices of an isosceles triangle done clear
  D)
  None of these done clear
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• question_answer6)
  Which of the following statement is true?
  A)
  The point A(0, -1), B(2, 1), C(0, 3) and D(-2, 1) are vertices of a rhombus. done clear
  B)
  The points A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) are vertices of a square. done clear
  C)
  The points A(-2, -1), B(1, 0), C(4, 3) and D(1, 2) are vertices of a parallelogram. done clear
  D)
  None of these done clear
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• question_answer7)
  Find the equation of set points P such that \[P{{A}^{2}}+P{{B}^{2}}=2{{K}^{2}},\] where A and B are the points (3, 4, 5) and (-1, 3, -7), respectively:
  A)
  \[{{K}^{2}}-109\] done clear
  B)
  \[2{{K}^{2}}-109\] done clear
  C)
  \[3{{K}^{2}}-109\] done clear
  D)
  \[4{{K}^{2}}-10\] done clear
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• question_answer8)
  ABC is a triangle and AD is the median. If the coordinates of A are (4, 7, -8) and the coordinates of centroid of the triangle ABC are (1, 1, 1), what are the coordinates of D?
  A)
  \[\left( -\frac{1}{2},\,\,2,\,\,11 \right)\] done clear
  B)
  \[\left( -\frac{1}{2},-2,\frac{11}{2} \right)\] done clear
  C)
  \[(-1,2,11)\] done clear
  D)
  \[(-5,-11,19)\] done clear
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• question_answer9)
  The equation of locus of a point whose distance from the y-axis is equal to its distance from the point (2, 1, -1) is
  A)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\] done clear
  B)
  \[{{x}^{2}}-4x+2z+6=0\] done clear
  C)
  \[{{y}^{2}}-2y-4x+2z+6=0\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}=0\] done clear
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• question_answer10)
  The ratio in which the join of points (1, -2, 3) and (4, 2, -1) is divided by XOY plane is
  A)
  1:3 done clear
  B)
  3:1 done clear
  C)
  -1:3 done clear
  D)
  None of these done clear
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• question_answer11)
  Points (1, 1, 1), (-2, 4, 1), (-1, 5, 5) and (2, 2, 5) are the vertices of a
  A)
  Rectangle done clear
  B)
  Square done clear
  C)
  Parallelogram done clear
  D)
  Trapezium done clear
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• question_answer12)
  If x co-ordinates of a point P of line joining the points Q (2, 2, 1) and R (5, 2, - 2) is 4, then the z-coordinates of P is
  A)
  -2 done clear
  B)
  -1    done clear
  C)
  1 done clear
  D)
  2 done clear
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• question_answer13)
  A parallelopiped is formed by planes drawn through the points (2, 4, 5) and (5, 9, 7) parallel to the coordinate planes. The length of the diagonal of the parallelepiped is        
  A)
  8 done clear
  B)
  4     done clear
  C)
  7 done clear
  D)
  11 done clear
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• question_answer14)
  The co-ordinates of the points A and B are (2, 3, 4) and (-2, 5, -4) respectively. If a point P moves so that \[P{{A}^{2}}-P{{B}^{2}}=k\] where k is a constant, then the locus of P is
  A)
  \[-8x+4y-16z+16=k\] done clear
  B)
  \[-8x-4y-16z-16=k\] done clear
  C)
  \[-8x+4y-16z-16=k\] done clear
  D)
  None of these done clear
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• question_answer15)
  What is the perpendicular distance of the point P(6,7, 8) from xy-plane?
  A)
  8 done clear
  B)
  7 done clear
  C)
  6 done clear
  D)
  None of these done clear
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• question_answer16)
  P(a, b, c); Q(a+2, b+2, c - 2) and R (a + 6, b + 6, c - 6) are collinear.

  Consider the following statements:

  1. R divides PQ internally in the ratio 3:2

  2. R divides PQ externally in the ratio 3:2

  3. Q divides PR internally in the ratio 1:2

  Which of the statements given above is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  1 and 3 done clear
  D)
  2 and 3 done clear
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• question_answer17)
  What is the locus of a point which is equidistant from the points (1, 2, 3) and (3, 2, - 1)?
  A)
  \[x+z=0\] done clear
  B)
  \[x-3z=0\] done clear
  C)
  \[x-z=0\] done clear
  D)
  \[x-2z=0\] done clear
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• question_answer18)
  If x co-ordinates of a point P of line joining the points Q (2, 2, 1) and R (5, 2, - 2) is 4, then the z-coordinates of P is
  A)
  -2 done clear
  B)
  -1    done clear
  C)
  1 done clear
  D)
  2 done clear
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• question_answer19)
  L is the foot of the perpendicular drawn from a point P(6, 7, 8) on the xy-plane. What are the coordinates of point L?
  A)
  (6, 0, 0) done clear
  B)
  (6, 7, 0) done clear
  C)
  (6, 0, 8) done clear
  D)
  None of these done clear
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• question_answer20)
  If the origin is shifted (1, 2 -3) without changing the directions of the axis, then find the new coordinates of the point (0, 4, 5) with respect to new frame.
  A)
  (-1, 2, 8) done clear
  B)
  (4, 5, 1) done clear
  C)
  (3, -2, 4) done clear
  D)
  (6, 0, 8) done clear
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• question_answer21)
  The ratio in which the line joining (2, 4, 5), (3, 5,-  4) is divided by the yz plane, is
  A)
  2 : 3 done clear
  B)
  3 : 2 done clear
  C)
  -2 : 3 done clear
  D)
  4 : - 3 done clear
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• question_answer22)
  The points (5, 2, 4), (6, -1, 2) and (8, -7, k) are collinear if k is equal to
  A)
  -2 done clear
  B)
  2    done clear
  C)
  3 done clear
  D)
  -1 done clear
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• question_answer23)
  What are coordinates of the point equidistant from the points (a, 0, 0), (0, a, 0), (0, 0, a) and (0, 0, 0)?
  A)
  \[\left( \frac{a}{3},\frac{a}{3},\frac{a}{3} \right)\] done clear
  B)
  \[\left( \frac{a}{2},\frac{a}{2},\frac{a}{2} \right)\] done clear
  C)
  \[(a,a,a)\] done clear
  D)
  \[(2a,2a,2a)\] done clear
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• question_answer24)
  The xy-plane divides the line joining the points (-1, 3, 4) (2, -5, 6)
  A)
  Internally in the ratio 2:3 done clear
  B)
  Externally in the ratio 2:3 done clear
  C)
  Internally in the ratio 3:2 done clear
  D)
  Externally in the ratio 3:2 done clear
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• question_answer25)
  Let A(4, 7, 8), B(2, 3, 4), C(2, 5, 7) be the vertices of a triangle ABC. The length of internal bisector of \[\angle A\] is
  A)
  \[\frac{\sqrt{34}}{2}\] done clear
  B)
  \[\frac{3}{2}\sqrt{34}\] done clear
  C)
  \[\frac{2}{3}\sqrt{34}\] done clear
  D)
  \[\frac{\sqrt{34}}{3}\] done clear
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• question_answer26)
  A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are the vertices of a triangle ABC. If the bisector of \[\angle ABC\] meets BC at D, then coordinates of D are
  A)
  \[\left( \frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] done clear
  B)
  \[\left( -\frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] done clear
  C)
  \[\left( \frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] done clear
  D)
  None of these done clear
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• question_answer27)
  The coordinates of point in xy-plane which is equidistant from three points A (2, 0, 3), B (0, 3, 2) and C (0, 0, 1) are
  A)
  (3, 2, 0) done clear
  B)
  (3, 4, 0) done clear
  C)
  (0, 0, 3) done clear
  D)
  (2, 3, 0) done clear
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• question_answer28)
  Ratio in which the zx-plane divides the join of (1, 2 3) and (4, 2, 1).
  A)
  1:1 internally done clear
  B)
  1:1 externally done clear
  C)
  2:1 internally done clear
  D)
  2: 1 externally done clear
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• question_answer29)
  The points (4, 7, 8), (2, 3, 4), (-1, -2, 1) and (1, 2, 5) are the vertices of a
  A)
  Parallelogram done clear
  B)
  Rhombus done clear
  C)
  Rectangle done clear
  D)
  Square done clear
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• question_answer30)
  In \[\Delta ABC\] the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, \[\frac{A{{B}^{2}}+B{{C}^{2}}+C{{A}^{2}}}{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}\] is equal to
  A)
  2 done clear
  B)
  4     done clear
  C)
  8 done clear
  D)
  16 done clear
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