Choose the correct option :

- The planes intersect in a line if-

none of these.

Solution.

The equation of any plane passing through the line of intersection of two planes can be written as

Comparing and we get,

From we get,

So, option (c) is correct.

2. The point in which the line meets the plane is-

Solution.

Let the straight line meets the plane at the point

So, the required point is

So, option (a) is correct.

3. The coordinates of the point where the line joining the points and meets the xy plane are-

(a) (b) (c) (d) none of these

Solution.

Let

The equation of the straight line passing through and is given by

Let the straight line (1) intersects the plane xy at the point

So, from (1) and (2) we get,

So, the point is

Hence, option (b) is correct.

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4. The line joining the points and meets the plane is-

(a) (b) (c) (d)

Solution.

The equation of the straight line passing through the points and is given by

So, the point lies on the plane.

So, the required point is

So, option (b) is correct.

5. A plane meets the coordinate axes in A, B,C such that the centroid of the triangle ABC is the point Then the equation of the plane where is –

(a) (b) (c) (d)

Solution.

If is the centroid of the triangle ABC, then the equation of the plane is

So, option (d) is correct.

6. State which of the following statement is true ?

(a) The plane through the points whose coordinates are and respectively, passes through the point for all values of

(b) The equation of the plane which is parallel to the plane and passing through the point is

(c) The equation of the line of intersection of planes and is

(d) The equation of the straight line and the plane are perpendicular to each other.

Solution.

Option (a) is correct.

Explanation.

The equation of the plane through the points and is

Clearly, the point satisfies the equation of the plane (1).

7. The equation of the plane passing through the point having as the direction ratios of the normal to the plane is-

(a) (b) (c) (d) none of these

Solution.

The equation of the plane can be written as

where the values of are respectively and

So, by we get,

So, option (c) is correct.

8. The value of for which the straight line is parallel to the plane is – (a) (b) (c) (d)

Solution.

The direction ratios of are and the direction ratios of are

The vector equation of the plane perpendicular to and is given by

Now, is perpendicular to

So, option (a) is correct.

9. The intercept made by the plane on the x-axis is – (a) (b) (c) (d)

Solution.

Let

So, the required intercept is given by

So, option (a) is correct.

10. A unit vector parallel to the intersection of planes and is-

(a) (b) (c) (d)

Solution. Let the planes and be denoted by and respectively.

Hence, the required unit vector is given by

So, option (c) is correct.

11. The line will not meet the plane if-

(a) (b)

(c) (d)

Solution.

If then because the line will not meet the plane

So, option (c) is correct.

12. The ratio in which the plane divides the line joining the points and is-

(a) (b) (c) (d)

Solution.

Let the given plane divides the line joining the points and in the ratio

By our supposition,

So, from (1) we get,

Hence, option (d) is correct.

13. The lines and are coplanar if-

(a) (b) (c) (d)

Solution.

The given lines are coplanar if

So, option (b) is correct.

14. The plane , which passes through the point and the line is

(a) (b) (c) (d)

Solution.

The equation of the plane passing through the point is

Since the point lies on the plane (1), so

Since the direction ratios of and are perpendicular to each other , we get

Hence, from (1), (2) and (3) we get,

Hence, option (a) is correct.