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This mock test of Test: Three Dimensional 3D Geometry (Competition Level) - 2 for JEE helps you for every JEE entrance exam.
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QUESTION: 1

Equation of plane which passes through the point of intersection of l ines and and at greatest distance from the point (0, 0, 0) is

Solution:

x = 3λ + 1, y = λ + 2 and 2λ + 3 x = m + 3, y = 2m + 1 and 3m + 2 Lines intersect.

Therefore 3λ + 1 = m + 3 and λ + 2 = 2λ + 1

3λ - u - 2 = 0

λ - 2u + 1 = 0

Apply ‘2312’ we get

λ/(-1-4) + u(-2-3) = 1(-6+1)

λ = 1 and u = 1

Therefore, point of intersection is (4, 3, 5).

Now plane passing through (4, 3, 5) and at maximum distance from the origin must have directions of the normal as 4 − 0, 3 − 0 and 5 − 0.

Therefore, equation of required plane is (x − 4)4 + (y − 3)3 + (z − 5)5 = 0 or 4x + 3y + 5z = 16 + 9 + 25 ⇒ 4x + 3y + 5z = 50

QUESTION: 2

The base of the pyramid AOBC is an equilateral triangle OBA with each side equal to 4√2, 'O'is the origin of reference, AC is perpendicular to the plane of Δ OBC and = 2. Then the cosine of the angle between the skew straight lines one passing through A and the mid point of OB and the other passing through O and the mid point of BC is

Solution:

QUESTION: 3

In the adjacent figure ‘P’ is any arbitrary interior point of the triangle ABC such that the lines AA_{1},BB_{1},CC_{1} are concurrent at P. Value of is always equal to

Solution:

QUESTION: 4

A tetrahedron has vertices at O(0, 0, 0), A(1, 2, 1), B(2, 1, 3) and C(–1, 1, 2). Then the angle between the face OAB and ABC will be

Solution:

QUESTION: 5

The two lines x = ay + b, z = cy + d and = a' y + b', z = c' y + d' will be perpendicular, iff

Solution:

QUESTION: 6

ABC is a triangle where A = (2, 3, 5), B = (–1, 2, 2) and C(λ, 5, μ). If the median through A is equally inclined to the axes then

Solution:

QUESTION: 7

A mirror and a source of light are situated at the origin O and at a point on OX, respectively. A ray of light from the source strikes the mirror and is reflected. If the D.r.’s of the normal to the plane are 1, –1, 1, then D.C.’s of the reflected ray are

Solution:

QUESTION: 8

The equation of motion of a point in space is x = 2t, y = –4t, z = 4t where t measured in hours and the co-ordinates of moving point in kilometers. The distance of the point from the starting point O(0, 0, 0) in 10 hours is

Solution:

QUESTION: 9

Minimum value of x^{2} + y^{2} + z^{2} when ax+by+cz = p is

Solution:

QUESTION: 10

The direction cosines of a line equally inclined to three mutually perpendicular lines having ℓ_{1}, m_{1}, n_{1}, ; ℓ_{2}, m_{2}, n_{2} ; ℓ_{3}, m_{3}, n_{3} are

Solution:

QUESTION: 11

The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are

Solution:

QUESTION: 12

If the line joining the origin and the point (–2, 1, 2) makes angle θ_{1 ,}θ_{2 }and θ_{3} with the positive direction of the coordinate axes, then the value of cos 2θ_{1} + cos 2θ_{2} + cos 2θ_{3} is

Solution:

QUESTION: 13

The square of the perpendicular distance of point P(p, q, r) from a line through A(a, b, c) an whose direction cosine are (ℓ) , m, n is

Solution:

*Multiple options can be correct

QUESTION: 14

Equation of the plane passing through A(x_{1}, y_{1}, z_{1}) d containing the line

Solution:

*Multiple options can be correct

QUESTION: 15

The equation of the line x + y + z – 1 = 0, 4x + y – 2z + 2 = 0 written in the symmetrical form is

Solution:

*Multiple options can be correct

QUESTION: 16

The acute angle that the vector makes with the plane contained by the two vectors and is given by

Solution:

*Multiple options can be correct

QUESTION: 17

The ratio in which the sphere x^{2} + y^{2} + z^{2} = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18) is

Solution:

*Multiple options can be correct

QUESTION: 18

The equation of the planes through the origin which are parallel to the line and distance 5/3 from it are

Solution:

*Multiple options can be correct

QUESTION: 19

If the edges of a rectangular parallelopiped are 3, 2, 1 then the angle between a pair of diagonals is given by

Solution:

*Multiple options can be correct

QUESTION: 20

Consider the lines x/2 = y/3 = z/5 and x/1 = y/2 = z/3 equation of the line which

Solution:

*Multiple options can be correct

QUESTION: 21

The direction cosines of the lines bisecting the anglebetween the lines whose direction cosines are ℓ_{1}, m_{1}, n_{1} and ℓ_{2}, m_{2}, n_{2} and the angle between these lines is 0, are

Solution:

*Multiple options can be correct

QUESTION: 22

The equation og line AB is . Through a point P(1, 2, 5), line PN is drawn perpendicular to AB and line PQ is drawn parallel to the plane 3x + 4y + 5z = 0 to meet AB is Q. Then

Solution:

*Multiple options can be correct

QUESTION: 23

The planes 2x – 3y – 7z = 0, 3x – 14y – 13z = 0 and 8x – 31y – 33z = 0

Solution:

*Multiple options can be correct

QUESTION: 24

If the length of perpendicular drawn from origin on a plane is 7 units and its direction ratios are –3, 2, 6, then that plane is

Solution:

*Multiple options can be correct

QUESTION: 25

Let a perpendicular PQ be drawn from P(5, 7, 3) to the line when Q is the foot.

Then

Solution:

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