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JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) Class 12 - JEE Advanced Level Test: Three Dimensional 3D Geometry- 3

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 questions and answers have been prepared according to the JEE exam syllabus.The JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 below.
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JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

Detailed Solution for JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 1

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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

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JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 3

In the adjacent figure ‘P’ is any arbitrary interior point of the triangle ABC such that the lines AA1,BB1,CC1 are concurrent at P. Value of   is always equal to

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 5

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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 9

Minimum value of x2 + y2 + z2 when ax+by+cz = p is

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 10

The direction cosines of a line equally inclined to three mutually perpendicular lines having ℓ1, m1, n1, ; ℓ2, m2, n2 ; ℓ3, m3, n3 are

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 12

If the line joining the origin and the point (–2, 1, 2) makes angle θ1 ,θand θ3 with the positive direction of the coordinate axes, then the value of cos 2θ1 + cos 2θ2 + cos 2θ3 is

JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 14

Equation of the plane passing through A(x1, y1, z1) d containing the line 

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 15

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

Detailed Solution for JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 15

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 16

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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 17

The ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18) is

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 18

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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 19

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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 20

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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 21

The direction cosines of the lines bisecting the anglebetween the lines whose direction cosines are ℓ1, m1, n1 and ℓ2, m2, n2 and the angle between these lines is 0, are 

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 23

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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - 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

*Multiple options can be correct
JEE Advanced Level Test: Three Dimensional 3D Geometry- 3 - Question 25

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

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