Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Three Dimensional Geometry

Multiple Choice Questions

21.

If the line $\frac{x - 4}{1} = \frac{y - 2}{1} = \frac{z - k}{2}$ lies on the plane 2x – 4y + z = 7, then what is the value of k ?

22.

A point on the line $\frac{x - 1}{1} = \frac{y - 3}{2} = \frac{z + 2}{7}$ has coordinates

(3, 5, 4)
(2, 5, 5)
( - 1, - 1, 5)
(2, - 1, 0)

23.
A point on a line has coordinates $(p + 1, p - 3, \sqrt{2} p)$ where p is any real number. What are the direction cosines of the line ?

$\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$

$\frac{1}{\sqrt{2}}, \frac{1}{2}, - \frac{1}{2}$

Cannot be determined due to insufficient data

$\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}$

we have co-ordinates of point, $(p + 1, p - 3, \sqrt{2} p)$ these are the intercepts of respectively.

Then

x = p + 1

x - 1 = p ...(i)

y = p - 3

y = 3 = p ... (ii)

We can write these three equations as,

$z = \sqrt{2} p \Rightarrow \frac{z}{\sqrt{2}} = p$ ...(iii)

We can write these three equations a,

$\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 0}{\sqrt{3}} = p$

We can compare this with standard form

$\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c} = r$

Therefore, direction ratios are a, b, c then 1, 1, $\sqrt{2}$

Direction cosines are l, m, n

We know

$l = \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}}, m = \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}}, n = \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}} l = \frac{1}{\sqrt{1^{2} + 1^{2} + {(\sqrt{2})}^{2}}}, m = \frac{1}{\sqrt{1^{2} + 1^{2} + {(\sqrt{2})}^{2}}}, n = \frac{1}{\sqrt{1^{2} + 1^{2} + {(\sqrt{2})}^{2}}} l = \frac{1}{2}, m = \frac{1}{2}, n = \frac{1}{\sqrt{2}} Then direction cosines are \frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}$

24.

What is the equation of the plane which cuts an intercept 5 units on the z-axis and is parallel to xy-plane ?

x + y = 5
z = 5
z = 0
x + y + z = 5

25.

Into how many compartments do the coordinate planes divide the space ?

26.

If a line has direction ratios < a + b, b + c, c + a >, then what is the sum of the square of its direction cosines ?

(a + b + c)²
2(a + b + c)
3
1

27.

What is the perpendicular distance from the point (2, 3, 4) to the line $\frac{x - 0}{1} = \frac{y - 0}{0} = \frac{z - 0}{0}$ ?

6 units
5 units
3 units
2

28.

What is the length of the diameter of the sphere whose centre is at (1, - 2, 3) and which touches the plane 6x – 3y + 2z – 4 = 0 ?

1 unit
2 units
3 units
4 units

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

23. A point on a line has coordinates p + 1, p - 3, 2p where p is any real number. What are the direction cosines of the line ? 12, 12, 12 12, 12, 12 12, 12, - 12 Cannot be determined due to insufficient data