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Three Dimensional Geometry Maths - Three Dimensional Geometry

Gujarat Board (GSEB) - STD 12 Science - Maths (GSEB - English Medium). 818 questions in this chapter.

Question types

Three Dimensional Geometry question types

818 questions across 7 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.

818
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7
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5
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Sample Questions

Three Dimensional Geometry questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1M.C.Q (1 Marks)1 Mark
The direction cosines of the normal to the plane 2x - 3y - 6z - 3 = 0 are:
  1. $\frac{2}{7},\frac{-3}{7},\frac{-6}{7}$
  2. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
  3. $\frac{-2}{7},\frac{-3}{7},\frac{-6}{7}$
  4. None of these
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Q 2M.C.Q (1 Marks)1 Mark
The points A(1, 1, 0), B(0, 1, 1), C(1, 0, 1) and $\text{D}\big(\frac{2}{3},\frac{2}{3},\frac{2}{3}\big)$
  1. Coplanar
  2. Non-coplanar
  3. Vertices of a parallelogram
  4. None of these
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Q 3M.C.Q (1 Marks)1 Mark
If $\cos\alpha,\cos\beta,\cos\gamma$ are the direction cosines of a vector $\vec{\text{a}}$ then $\cos2\alpha+\cos2\beta+\cos2\gamma$ is equal to:
  1. 2
  2. 3
  3. -1
  4. 0
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Q 5M.C.Q (1 Marks)1 Mark
A point P lies on the line segment joining the points (-1, 3, 2) and (5, 0, 6). If x-coordinate of P is 2, then its z-coordinate is:
  1. $-1$
  2. $4$
  3. $\frac{3}{2}$
  4. $8$
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Q 6Assertion (A) & Reason (B) MCQ1 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If the cartesian equation of a line is $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$ then its vector form is $\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}).$
Reason: The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}$ is $\frac{\text{x}+3}{-2}=\frac{\text{y}-4}{4}=\frac{\text{z}+8}{-5}.$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
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Q 7Assertion (A) & Reason (B) MCQ1 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Points A(4, 0, 4), B(1, 2, 3), C(-2, 4, 2) are collinear.
Reason: Three points A, B, C are collinear if AB + BC = AC and AB, BC < AC.
  1. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion.
  2. Both Assertion & Reason are individually true but Reason is not the, correct (proper) explanation of Assertion.
  3. Assertion is true but Reason is false.
  4. Assertion is false but Reason is true.
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Q 8Assertion (A) & Reason (B) MCQ1 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The points (1, 2, 3), (-2, 3, 4) and (7, 0, 1) are collinear
Reason: If a line makes angles $\frac{\pi}{2}, \frac{3\pi}{4}$ and $\frac{\pi}{4}$ with X, Y, and Z - axes respectively, then its direction cosines are $0,\frac{-1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
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Q 101 Marks1 Mark
$\text{Find}\ \lambda\ \text{if}\ (2\hat{\text{i}}+6\hat{\text{j}}+14\hat{\text{k}})\times(\hat{\text{i}}-\lambda\hat{\text{i}}+7\hat{\text{k}})=\overrightarrow{0}\dot{}$
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Q 111 Marks1 Mark
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
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Q 142 Marks2 Marks
Write the condition for the lines $\vec{\text{r}}=\vec{\text{a}}_1+\lambda\vec{\text{b}}_1$ and $\vec{\text{r}}=\vec{\text{a}}_2+\mu\vec{\text{b}}_2$ to be intersecting.
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Q 172 Marks2 Marks
Find the vector equation of the plane which is at a distance of 5 units from the orgin and its normal vector is $2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}.$
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Q 182 Marks2 Marks
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1), (4, 3, -1).
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Q 193 Marks3 Marks
Find the angle between the following pairs of lines:
  1. $\vec{\text{r}}=2\hat{\text{i}}-5\hat{\text{j}}+\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\Big)\ \text{and}$
$\vec{\text{r}}=7\hat{\text{i}}-6\hat{\text{k}}+\mu\Big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\Big)$
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Q 203 Marks3 Marks
Find the length of the perpendicular drow from the point (5, 4, -1) to the line $\vec{\text{r}}=\hat{\text{i}}+\lambda\big(2\hat{\text{i}}+9\hat{\text{j}}+5\hat{\text{k}}\big).$
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Q 213 Marks3 Marks
Find the equation of the line in vector and in cartesian form that passes through the point with position vector $2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$ and is in the direction $\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}.$
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Q 223 Marks3 Marks
Show that the line through points (1, -1, 2) and (3, 4, -2) is perpendicular to the line throught the points (0, 3, 2) and (3, 5, 6).
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Q 233 Marks3 Marks
Find the angle between the following pair of lines:
  1. $\frac{\text{x}}{2}=\frac{\text{y}}{2}=\frac{\text{z}}{1}\ \text{and}\ \frac{\text{x}-5}{4}=\frac{\text{y}-2}{1}=\frac{\text{z}-3}{8}$
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Q 244 Marks4 Marks
If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.
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Q 254 Marks4 Marks
Find the equation of the plane through the points (2, 2, -1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.
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Q 264 Marks4 Marks
Find the angle between the lines whose direction ratios are proportional to a, b, c and b - c, c - a, a - b.
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Q 274 Marks4 Marks
Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes $\text{x + 2y +3z = 5 and 3x + 3y + z} = 0$
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Q 284 Marks4 Marks
Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from the plane 2x - y + z + 1 = 0. Also, find the image of the point in the plane.
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Q 29Case study (4 Marks)4 Marks
A mobile tower stands at the top of a hill. Consider the surface on which tower stand as a plane having points A(0, 1, 2), B(3, 4, -1), and C(2, 4, 2) on it. The mobile tower is tied with 3 cables from the point A, Band C such that it stand vertically on the ground. The peak of the tower is at the point ( 6, 5, 9), as shown in the figure.

Based on the above information, answer the following questions.

  1. The equation of plane passing through the points A, Band C is:
  1. 3x - 4y + z = 0
  2. 3x - 2y + z = 0
  3. 3x - 2y + z = 0
  4. 4x - 3y + 3z = 0
  1. The height of the tower from the ground is:
  1. 6 units
  2. 5 units

  3. $\frac{17}{\sqrt{14}}\text{units}$

  4. $\frac{5}{\sqrt{14}}\text{units}$

  1. The equation of line of perpendicular drawn from the peak of tower to the ground is:

  1. $\frac{\text{x}-6}{3}=\frac{\text{y}-4}{-2}=\frac{\text{z}-9}{1}$

  2. $\frac{\text{x}-6}{3}=\frac{\text{y}-5}{-2}=\frac{\text{z}-9}{1}$

  3. $\frac{\text{x}-6}{3}=\frac{\text{y}-4}{2}=\frac{\text{z}-9}{1}$

  4. $\frac{\text{x}-6}{3}=\frac{\text{y}-5}{2}=\frac{\text{z}-9}{1}$

  1. The coordinates of foot of perpendicular drawn from the peak of tower to the ground are:

  1. $\Big(\frac{33}{14},\frac{104}{14},\frac{109}{14}\Big)$

  2. $\Big(\frac{33}{14},\frac{109}{14},\frac{104}{14}\Big)$

  3. $\Big(\frac{33}{14},\frac{105}{14},\frac{109}{14}\Big)$

  4. None of these
  1. The area of $\triangle\text{ABC}$ is:

  1. $\frac{1}{2}\sqrt{14}\text{sq}.\text{units}$

  2. $\frac{3}{2}\sqrt{14}\text{sq}.\text{units}$

  3. $\sqrt{14}\text{sq}.\text{units}$

  4. $2\sqrt{14}\text{sq}.\text{units}$

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Q 30Case study (4 Marks)4 Marks
Consider the following diagram, where the forces in the cable are given.

Based on the above information, answer the following questions.

  1. The cartesian equation of line along EA is:

  1. $\frac{\text{x}}{-4}=\frac{\text{y}}{3}=\frac{\text{z}}{12}$

  2. $\frac{\text{x}}{-4}=\frac{\text{y}}{3}=\frac{\text{z}-24}{12}$

  3. $\frac{\text{x}}{-3}=\frac{\text{y}}{3}=\frac{\text{z}-12}{12}$

  4. $\frac{\text{x}}{3}=\frac{\text{y}}{4}=\frac{\text{z}-24}{12}$

  1. The vector $\overline{\text{ED}}$ is:

  1. $8\hat{\text{i}}-6\hat{\text{j}}+24\hat{\text{k}}$

  2. $-8\hat{\text{i}}-6\hat{\text{j}}+24\hat{\text{k}}$

  3. $-8\hat{\text{i}}-6\hat{\text{j}}-24\hat{\text{k}}$

  4. $8\hat{\text{i}}+6\hat{\text{j}}+24\hat{\text{k}}$

  1. The length of the cable EB is:
  1. 24 units
  2. 26 units
  3. 27 units
  4. 25 units
  1. The length of cable EC is equal to the length of:
  1. EA
  2. EB
  3. ED
  4. All of these
  1. The sum of all vectors along the cables is:

  1. $96\hat{\text{i}}$

  2. $96\hat{\text{j}}$

  3. $-96\hat{\text{k}}$

  4. $96\hat{\text{k}}$

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Q 31Case study (4 Marks)4 Marks
A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and Bare respectively sitting 
on the plane represented by the equation$ \vec{\text{r}}.(\hat{\text{i}}+\hat{\text{j}}+\hat{2\text{k}})=5$ and $ \vec{\text{r}}.(\hat{2\text{i}}-\hat{\text{j}}+\hat{\text{k}})=6$ to cheer up the team of their respective schools. 


Based on the above information, answer the following questions. 

  1. The cartesian equation of the plane on which students of school A are seated is:
  1. 2x - y + z = 8
  2. 2x + y + z = 8
  3. x + y + 2z = 5
  4. x + y + z = 5
  1. The magnitude of the normal to the plane on which students of school Bare seated, is:

  1. $\sqrt{5}$

  2. $\sqrt{6}$

  3. $\sqrt{3}$

  4. $\sqrt{2}$

  1. The intercept form of the equation of the plane on which students of school Bare seated is:

  1. $\frac{\text{x}}{6}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$

  2. $\frac{\text{x}}{3}+\frac{\text{y}}{(-6)}+\frac{\text{z}}{6}=1$

  3. $\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$

  4. $\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{3}=1$

  1. Which of the following is a student of school B?
  1. Mohit sitting at (1, 2, 1)
  2. Ravi sitting at (0, 1, 2)
  3. Khushi sitting at (3, 1, 1)
  4. Shewta sitting at (2, -1, 2)
  1. The distance of the plane, on which students of school Bare seated, from the origin is:
  1. 6 units

  2. $\frac{1}{\sqrt{6}}\text{ units}$

  3. $\frac{5}{\sqrt{6}}\text{ units}$

  4. $\sqrt{6}\text{ units}$

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Q 32Case study (4 Marks)4 Marks
Suppose the floor of a hotel is made up of mirror polished Kota stone. Also, there is a large crystal chandelier attached at the ceiling of the hotel. Consider the floor of the hotel as a plane having equation x - 2y + 2z = 3 and crystal chandelier at the point (3, -2, 1).

Based on the above information, answer the following questions.

  1. The d.r'.s of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is:
  1. < 1, 2, 2 >
  2. < 1, -2, 2 >
  3. < 2, 1, 2 >
  4. < 2, -1, 2 >
  1. The length of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is:
  1. $\frac{2}{3}\text{units}$
  2. 3 units
  3. 2 units
  4. None of these
  1. The equation of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is:

  1. $\frac{\text{x}-3}{1}=\frac{\text{y}-2}{-2}=\frac{\text{z}-1}{2}$

  2. $\frac{\text{x}-3}{1}=\frac{\text{y}+2}{-2}=\frac{\text{z}-1}{2}$

  3. $\frac{\text{x}+3}{1}=\frac{\text{y}+2}{-2}=\frac{\text{z}-1}{2}$

  4. None of these
  1. The equation of plane parallel to the plane x - 2y + 2z = 3, which is at a unit distance from the point (3, -2, 1) is:
  1. x - 2y + 2z = 0
  2. x - 2y + 2z = 6
  3. x - 2y + 2z = 12
  4. Both (b) and (c)
  1. The image of the point (3, -2, 1) in the given plane is:

  1. $\Big(\frac{5}{3},\frac{2}{3},\frac{-5}{3}\Big)$

  2. $\Big(\frac{-5}{3},\frac{-2}{3},\frac{5}{3}\Big)$

  3. $\Big(\frac{-5}{3},\frac{2}{3},\frac{5}{3}\Big)$

  4. None of these
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Q 33Case study (4 Marks)4 Marks
If a1, b1, c1, and a2, b2, c2 are direction ratios of two lines say L1 and L2 respectively. Then L1 || L2 iff $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$ and $\text{L}_1\perp\text{L}_2$ iff a1a2 + b1b2 + c1c2 = 0.

Based on the above information, answer the following questions.

  1. If l1, m1,  n1, and l2, m2, n2 are the direction cosines of L1 and L2 respectively, then L1 will be perpendicular to L2, iff:
  1. l1l2 + m1m2 + n1n2 = 0
  2. l1m2 + m1l2 + n1n2 = 0

  3. $\frac{\text{l}_1}{\text{l}_2}=\frac{\text{m}_1}{\text{m}_2}=\frac{\text{n}_1}{\text{n}_2}$

  4. None of these
  1. If l1, m1, n1 and l2, m2, n2 are direction cosines of L1 and L2 respectively, then L1 will be parallel to L2, iff:
  1. l1l2 + m1m2 + n1n2 = 0
  2. l1m2 + m1l2 + n1n2 = 0

  3. $\frac{\text{l}_1}{\text{l}_2}=\frac{\text{m}_1}{\text{m}_2}=\frac{\text{n}_1}{\text{n}_2}$

  4. m1n2 + m2n2 + l1l2 = 0
  1. The coordinates of the foot of the perpendicular drawn from the point A(1, 2, 1) to the line joining B(1, 4, 6) and C(5, 4, 4), are:
  1. (1, 2, 1)
  2. (2, 4, 5)
  3. (3, 4, 5)
  4. (3, 4, 5)
  1. The direction ratios of the line which is perpendicular to the lines with direction ratios proportional to (1, -2, -2) and (0, 2, 1) are:
  1. < 1, 2, 1 >
  2. < 2,-1, 2 >
  3. < -1,2, 2 >
  4. None of these
  1. The lines $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{-2}=\frac{\text{z}-2}{0}$ and $\frac{\text{x}-1}{1}=\frac{\frac{\text{y}+3}{2}}{\frac{3}{2}}=\frac{\text{z}+5}{2}$ are:
  1. Parallel.
  2. Perpendicular.
  3. Skew lines.
  4. Non-intersecting.
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