Class 12 Three Dimensional Geometry Mock Test (CBSE)

Topics in Three Dimensional Geometry

• Direction cosines and direction ratios of a line
• Equation of a line in space
• Angle between two lines
• Shortest distance between two lines

Three Dimensional Geometry practice questions with answers

1. Q1Multiple choice·1 mark
   A line has direction ratios $2, -2, 1$. Its direction cosines are:

   1. (a)$\left(\frac{2}{\sqrt{3}}, -\frac{2}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$
   2. (b)$\left(\frac{2}{9}, -\frac{2}{9}, \frac{1}{9}\right)$
   3. (c)$\left(\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\right)$
   4. (d)$\left(\frac{2}{5}, -\frac{2}{5}, \frac{1}{5}\right)$
   Show answer and step marks

   1. Correct option: (c) $\left(\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\right)$, since $\sqrt{2^2 + (-2)^2 + 1^2} = 3$.+1

   Answer: (c)
2. Q2Assertion and reason·1 mark
   Assertion (A): If the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{1} = \frac{y-1}{2} = \frac{z}{k}$ are perpendicular, then the value of $k$ is $-2$. Reason (R): Two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are perpendicular if and only if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer and step marks

   1. Correct option: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).+1

   Answer: (a)
3. Q3Multiple choice·1 mark
   If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of the coordinate axes, then the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ is:

   1. (a)$2$
   2. (b)$3$
   3. (c)$0$
   4. (d)$1$
   Show answer and step marks

   1. Correct option: (a) $2$. Using the relation $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$, we get $(1 - \sin^2 \alpha) + (1 - \sin^2 \beta) + (1 - \sin^2 \gamma) = 1$, which simplifies to $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2$.+1

   Answer: (a)
4. Q4Multiple choice·1 mark
   The lines $\vec{r} = \hat{i} + \lambda(\hat{j} + \hat{k})$ and $\vec{r} = \hat{j} + \mu(\hat{i} + \hat{k})$ have shortest distance equal to zero. This is because the lines:

   1. (a)intersect each other
   2. (b)are parallel to each other
   3. (c)are perpendicular to each other
   4. (d)lie on opposite sides of the origin
   Show answer and step marks

   1. Correct option: (a) intersect each other, since $(\hat{j}-\hat{i})\cdot((\hat{j}+\hat{k})\times(\hat{i}+\hat{k})) = 0$ and the lines meet at $(1, 1, 1)$, so the shortest distance is $0$.+1

   Answer: (a)
5. Q5Short answer·3 marks
   A gardener lays a straight water pipe in a kitchen garden. Taking a corner of the garden as origin, with coordinates in metres (the z-axis is vertical), the pipe starts at the tap A(2, 1, 1) and ends at a sprinkler at B(5, 5, 3). The pipe lies along the line AB. (a) Find the direction ratios of the line AB. [1] (b) Write the vector equation of the line AB. [1] (c) Determine whether the point C(8, 9, 5) lies on the line AB. [1]
   Show answer and step marks

   1. (a) Direction ratios of AB are $5-2,\ 5-1,\ 3-1$, that is $3, 4, 2$.+1
   2. (b) The line passes through A with position vector $2\hat{i}+\hat{j}+\hat{k}$ and is parallel to $3\hat{i}+4\hat{j}+2\hat{k}$, so $\vec{r}=(2\hat{i}+\hat{j}+\hat{k})+\lambda(3\hat{i}+4\hat{j}+2\hat{k})$.+1
   3. (c) $\overrightarrow{AC}=(8-2)\hat{i}+(9-1)\hat{j}+(5-1)\hat{k}=6\hat{i}+8\hat{j}+4\hat{k}=2(3\hat{i}+4\hat{j}+2\hat{k})$, which is parallel to the direction of AB with $\lambda=2$, so C lies on AB.+1

   Answer: (a) $3, 4, 2$; (b) $\vec{r}=(2\hat{i}+\hat{j}+\hat{k})+\lambda(3\hat{i}+4\hat{j}+2\hat{k})$; (c) Yes, C lies on the line AB (at $\lambda=2$).

More Class 12 Maths mock tests