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Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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DIRECTIONS (Q. 7-9) : Each question has matching lists. The codes for the lists have choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

Q. 7. Match List I with List II and select the correct answer using the code given below the lists :

List I                                                                                                                             List II

P. Volume of parallelepiped determined by vectors                                               1. 100
 Then the volume of the parallelepiped determined by vectors
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEMatrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. Volume of parallelepiped determined by vectors  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                     2. 30
 Then the volume of the parallelepiped determined by vectors
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

R. Area of a triangle with adjacent sides determined by vectors                           3. 24
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEMatrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then the area of the triangle with adjacent sides
 determined by vectors  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

S. Area of a parallelogram with adjacent sides determined by vectors                 4. 60
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then the area of the parallelogram with adjacent sides
 determined by vectors Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. (c)

Solution. Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 8. Consider the lines Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand the planes P1 : 7x + y + 2z = 3, P2 = 3x + 5y – 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point of intersection of lines L1 and L2, and perpendicular to planes P1 and P2.

Match List I with List II and select the correct answer using the code given below the lists :

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. (a)

Solution. Any point on L1 is (2λ + 1, – λ, λ – 3) and that on L2 is (μ + 4, μ – 3, 2μ –3)

For point of intersection of L1 and L2 2λ + 1 = μ + 4, – λ = μ – 3, λ – 3 = 2μ – 3 ⇒ λ = 2, μ = 1

∴ Intersection point of L1 and L2 is (5, –2, – 1)

∵ ax + by + cz = d is perpendicular to p1 & p2

∴ 7a + b + 2c = 0 and 3a + 5b – 6c = 0

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Equation of plane is x – 3y – 2z = d

As it passes through (5, –2, –1)

∴ 5 + 6 + 2 = d = 13

∴ a = 1, b = –3, c = –2, d = 13

or (P)  → (3) (Q)  → (2)  (R)  → (4) (S)  → (1)


Q. 9. Match List I with List II and select the correct answer using the code given below the lists :

List - I                                                                                                               List - II

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                               1. 1

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. Let A1, A2, ..., An (n > 2) be the vertices of a regular                                  2. 2
 polygon of n sides with its  centre at the origin. LetMatrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

be the position vector of the point Ak, k = 1, 2, ....., n.

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 then the minimum value of n is

R. If the normal from the point P(h, 1) on the ellipse                                        3. 8
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is perpendicular  to the line x + y = 8, then the
 value of h is

S. Number of positive solutions satisfying the equation                                      4. 9
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. (a)

Solution. 

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Q(3) ... A1, A2, A3, ... An are the vertices of a regular  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are their position vectors.

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence given equation reduces to

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

It is perpendicular to x + y = 8

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Only one +ve solution is there

Hence (a) is the correct option.


DIRECTIONS (Q. 10-11) :  Each question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column-II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column-II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example :

If the correct matches are A-p, s and t; B-q and r; C-p and q; and D-s then the correct darkening of bubbles will look like the given.

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 10. Match the following :

Column I                                                                                                                   Column II

(A) In R2, if the magnitude of the projection vector of the vector                         (p) 1
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  then possible value
 of |α| is/are

(B) Let a and b be real numbers such that the function                                           (q) 2

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Then possible value of a is (are)

(C) Let  ω ≠ 1 be a complex cube root of unity.                                                        (r) 3

If Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 then possible value (s) of n is (are) 

(D) Let the harmonic mean of two positive real numbers a and b be 4.                 (s) 4
 If q is a positive real number such that a, 5, q, b is an arithmetic
 progression, then the value(s) of | q – a | is (are)

Ans. (A) → q; (B) → p, q; (C) → p, q, s, t; (D) → q, t

Solution. 

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴–3a – 2 = b + a2
⇒ a2 – 3a + 2 = 0 (using (i))
⇒ a = 1, 2

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEMatrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 4n + 3 should be an integer other than multiple of 3.

∴ n = 1, 2, 4, 5

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Putting values of a and b in eqn(i) 

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 11. Match the following :

Column I                                                                                                                Column II

(A) In a triangle DXYZ, let a, b, and c be the lengths                                             (p) 1
 of the sides opposite to the angles X, Y and Z,
 respectively. If 2(a2 – b2) = c2 and  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 , then possible values of n for which  cos(nπλ) = 0 is (are) 

(B) In a triangle DXYZ, let a, b and c be the lengths of the                                      (q) 2
 sides opposite to the angles X, Y, and Z respectively. If
 1 + cos 2X – 2cos 2Y = 2 sin X sin Y, then possible value
 (s) of a/b is (are)

(C) In R2, let  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE be the position                                      (r) 3
 vectors of X , Y and Z with respect to the origin O,
 respectively. If the distance of Z from the bisector of the
 acute angle of Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then possible
 value(s) of |β| is (are)

(D) Suppose that F(a) denotes the area of the region bounded                              (s) 5
 by x = 0, x = 2, y2 = 4x and y =| αx- 1 | + | αx- 2 | +αx ,
 where a∈ {0, 1} . Then the value(s) of  Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE , when
 a = 0 and a = 1, is (are)

                                                                                                                                       (t) 6

Ans. (A)→p, r, s;  (B) → p; (C) → p, q; (D)→s, t

Solution. (A) 2(a2 – b2) = c
⇒ 2(sin2x – sin2y) = sin2z
⇒ 2sin(x + y) sin(x – y) = sin2z
⇒ 2sin(x – y) = sin z     (∵sin(x + y) = sin z)

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(B) 1 + cos2X – 2cos2Y = 2sinXsinY
⇒ 2cos2X – 2cos2Y = 2sinXsinY
⇒ 1 – sin2X – 1 + 2sin2Y = sinXsinY

⇒ sin2X + sinXsinY – 2sin2Y = 0
⇒ (sinX – sinY) (sinX + 2sinY) = 0

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

By symmetry, acute angle bisector of ÐXOY is y = x.

∴ Distance of Z from bisector

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ |β| = 1, 2 (D)

For α = 0, y = 3

For α = 1, y = |x – 1| + |x – 2| + x

Case I 

F(aα) is the area bounded by x = 0, x = 2, y2 = 4x and y = 3

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Case II

F(α) is the area bounded by x = 0, x = 2, y2 = 4x and y = |x – 1| + |x – 2| + x

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Matrix-Match Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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