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Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced


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20 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced

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Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 1

 and  be three non-zero vectors such that   is a unit vector perpendicular to both the vectors   and  If the angle between then

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 1






Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 2

The number of vectors of unit length perpendicular to vectors 

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 2

We know that if  then

as   represent two vectors in opp. directions. 

∴ We have two possible values of

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 3

Let   be three vectors. A vector in the plane of whose projection on  

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 3

We have


Given magnitude of projection of 

∴ The required vector is either,

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 4

The vector 

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 4


*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 5

If    and  are linearly dependent vectors and 

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 5

Given that,   and   are linearly dependent,

NOTE THIS STEP : 

for some scalars l and m not all zeros.

Þ l + 4m = 1 ...(1)
l + 3m = α ...(2)
l + 4m = β ...(3)

From (1) and (3) we have, β = 1

Also given that 

Substituting the value of  β we get α2 = 1

⇒ a = ±1

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 6

For three vectors u, v, w which of the following expression is not equal to any of the remaining three?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 6

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 7

Which of the following expressions are meaningful?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 7

As dot product of two vectors gives a scalar quantity.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 8

Let a and b be two non-collinear unit vectors. If u = a – (a . b) b and  v = a × b, then | v | is

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 8

We have






*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 9

Let  be vector parallel to line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors  and that  P2 is parallel to  then the angle between vector   and a given vector

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 9

Normal to plane P1 is



Now, angle between   is given by

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 10

The  vector (s) which is/are coplanar with vectors  and  and perpendicular to the vector  is/are

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 10

∵ Required vector is coplanar with 

∴a and d are the correct options

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 11

If the straight lines  are coplanar, then the plane (s) containing these two lines is (are)

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 11

For given lines to be coplanar, we should have

For k = 2, obviously the plane y + 1 = z is common in both lines.
For k = – 2, the plane is given by

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 12

A line l passing through the origin is perpendicular to the lines

Then, the coordinate(s) of the point(s) on l2 at a distance of  from the point of intersection of l and l1 is (are)

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 12

The given lines are


Let direction ratios of l be a, b, c then as 
∴ a + 2b + 2c = 0
2a + 2b + c = 0

Any point on ℓ1 is (t + 3, 2t – 1, 2t + 4) and any point on ℓ is (2λ, –3λ, 2λ)


*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 13

Two lines  are coplanar. Then a can take value(s)

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 13

As L1, L2 are coplanar, therefore

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 14

Let  be three vectors each of magnitude   and the angle between each pair of them is   is a non-zero vector perpendicular to    is a non-zero vector perpendicular to 

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 14


Angle between each pair is 







(d) is not correct.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 15

From a point P(λ, λ, λ), perpendicular PQ and PR are drawn respectively on the lines y = x, z = 1 and y = – x, z = – 1. If P is such that ∠QPR is a right angle, then the possible value(s) of ∠ is/(are)

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 15

Lines are x = y, z = 1

Let Q (a, a, 1) and R (– β,β, – 1) Direction ratios of PQ are λ – α, λ – α, λ – 1 and direction ratios of PR are λ + β, λ – β, λ + 1
∴ PQ is perpendicular to line (1)

∴ – (λ + β) + λ – β = 0  ⇒ b = 0
∴ dr’s of PQ are 0, 0, λ – 1 and dr’s of PR are λ, λ, λ + 1
∵ ∠QPR = 90° ⇒ (λ – l) (λ + 1) = 0 ⇒ λ = 1 or – 1
But for λ = 1, we get point Q itself
∴ we take λ = – 1

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 16

In R3, consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be the plane, different from P1 and P2, which passes through the intersection of P1 and P2. If the distance of the point (0, 1, 0) from P3 is 1 and the distance of a point (α, β, γ) from P3 is 2, then which of the following relations is (are) true?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 16

P3 : x + λy + z – 1 = 0


*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 17

In R3, let L be  a straight line passing through the origin. Suppose that all the  points on L are at a constant distance from the two planes P1 : x + 2y – z + 1 = 0 and P2 : 2x – y + z – 1 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following points lie (s) on M?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 17





For locus of M,


On checking the given point, we find    and  satisfy the above eqn.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 18

Let ΔPQR  be a triangle.  then which of the following is (are) true?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 18









*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 19

Consider a pyramid OPQRS located in the first octant (x >  0, y > 0, z > 0) with O as origin, and OP and OR along the x–axis and the y–axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point, T of diagonal OQ such that TS = 3. Then

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 19

The coordinates of vertices of pyramid OPQRS will be
O(0, 0, 0), P (3, 0, 0), Q (3, 3, 0), R (0, 3, 0), 

dr's of OQ = 1, 1, 0
dr's of OS = 1, 1, 2
∴ acute angle between OQ and OS

⇒ 2x – 2y = 0 or x – y = 0
length of perpendicular from P (3, 0, 0) to plane x – y = 0

If ON is perpendicular to RS, then N 

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 20

Let be a unit vector in  R3 and  Given that there exists a vector   in Which of the following statement(s) is (are) correct?

Detailed Solution for Test: MCQs (One or More Correct Option): Vector Algebra and Three Dimensional Geometry | JEE Advanced - Question 20



where α is the angle between
From (i) and (ii) cos α = 1 ⇒ a = 0°

 is perpendicular to the plane containing 
 is perpendicular to 

Clearly there can be infinite many choices for 

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