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Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry

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6 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry

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Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 1

Consider the lines Q. The unit vector perpendicular to both L1 and L2 is

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 1

Vector in the direction of
Vector in the direction of
∴ Vector perpendicular to both L1 and L2

Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 2

Consider the lines Q. The shortest distance between L1 and L2 is

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 2

The shortest distance between L1 and L2 is

Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 3

Consider the lines Q. The distance of the point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose normal is perpendicular to both the lines L1 and L2 is

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 3

The plane passing through (–1, –2, –1) and having normal along
– 1(x + 1) – 7(y + 2) + 5(z + 1) = 0
or x + 7y - 5z +10 = 0
∴ Distance of point (1, 1, 1) from the above plane is

Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 4

Consider the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5.

STATEMENT-1 : The parametric equations of the line of intersection of the given planes are x = 3 + 14t, y = 1 + 2t, z = 15t.  because

STATEMENT-2 : The vector   is parallel to the line of intersection of given planes.

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 4

The line of intersection of given plane is 3x -6y - 2z -15 = 0 = 2x + y - 2z-5
For z = 0 , we obtain x = 3 and y =-1
∴Line passes through (3, –1, 0).
Let a, b, c be the d’rs of line of intersection, then 3a -6b -2c = 0 and 2a + b - 2c = 0

Solving the above equations using cross product method, we get a :b :c = 14 : 2 : 15

∴ Eqn. of line is

whose parametric form is x = 3 + 14t , y = 1 + 2t, z= 15t

∴ Statement-I is false (value of y is not matching).
Since dr’s of line intersection of given planes are 14, 2,15

∴ is parallel to this line.
∴ Statement 2 is true.

Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 5

Let the vectors   epresent the sides of a regular hexagon.

STATEMENT-1 :

STATEMENT-2 :

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 5

Statement-1 is true.

∴ Statement-2 is false.

Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 6

Consider three planes

P1 : x – y + z = 1

P2 : x + y – z = 1

P3 : x – 3y + 3z = 2

Let L1, L2, L3 be the lines of intersection of the planes P2 and P3, P3 and P1, P1 and P2, respectively.
STATEMENT - 1Z :  At least two of the lines L1, L2 and L3 are non-parallel and

STATEMENT - 2 :  The three planes doe not have a common point.

Detailed Solution for Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - Question 6

The given planes are
P1 : x – y + z = 1 ...(1)
P2 : x + y – z = –1 ...(2)
P3 : x – 3y + 3z = 2 ...(3)

Line L1 is intersection of planes P2 and P3.
∴ L1 is parallel to the vector

Line L2 is intersection of P3 and P1

∴ L2 is parallel to the vector

Line L3 is intersection of P1 and P2

∴ L3 is parallel to the vector

Clearly lines L1, L2 and L3 are parallel to each other.

∴ Statement–1 is False

Also family of planes passing through the intersection of P1 and P2 is P1 + λP2 = 0 .If plane P3 is represented by P1 + λP2 = 0 for some value of l then the three planes pass through the same point.
Here P1 + λP2 = 0

This will be identical to P3 if

...(1)

and taking

∴There is no value of λ which satisfies eq (1).

∴ The three planes do not have a common point.
⇒ Statement 2 is true.

∴ (d) is the correct option.

Maths 35 Years JEE Main & Advanced Past year Papers

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Maths 35 Years JEE Main & Advanced Past year Papers

132 docs|70 tests