Integer Answer Type Questions: Vector Algebra and Three Dimensional Geometry | JEE Advanced | 35 Years Chapter wise Previous Year Solved Papers for JEE PDF Download

Q. 1. If   are vectors in space given by   and   then find the value of            (2010)

Ans. 5

Solution.

Q. 2. If the distance between the plane Ax – 2y + z = d and the plane containing the lines   and  , then find |d|.             (2010)

Ans. 6

Solution. The equation of plane containing the lines

∴ Distance between x – 2 y +z=0 and Ax –2 y + z = d

= Perpendicular distance between parallel planes  (∴ A= 1)

Q. 3. Let   be three given vectors.    is a vector such that   then the value of              (2011)

Ans. 9

Solution.

Q. 4. If are unit vectors satisfying             (2012)

Ans. 3

Solution.   are units vectors such that 

Q. 5. Consider the set of eight vectors  Three non- coplanar vectors can be chosen from V in 2^p ways. Then p is           (JEE Adv. 2013)

Ans. 5

Solution. Given 8 vectors are

(1, 1, 1), (–1, –1, –1); (–1, 1, 1), (1, –1, –1); (1, –1, 1),
(–1, 1, –1); (1, 1, –1), (–1, –1, 1)
These are 4 diagonals of a cube and their opposites.
For 3 non coplanar vectors first we select 3 groups of diagonals and its opposite in ⁴C₃ ways. Then one vector from each group can be selected in 2 × 2 × 2 ways.

∴ Total ways = ⁴C₃ × 2 × 2× 2 = 32 = 25

∴ p = 5

Q. 6. A pack contains n cards numbered from 1 to n . Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k – 20 =          (JEE Adv. 2013)

Ans. 5

Solution. Let k, k + 1 be removed from pack.

∴ (1 + 2 + 3 + ... + n) – (k + k + 1) = 1224

Q. 7. Let  be three non-coplanar unit vectors suchthat the angle between every pair of them is π/3. If   where p, q and r are scalars, then the value of    (JEE Adv. 2014)

Ans. 4

Solution. 

Taking its dot product with   we get

From (1) and (3), p = r Using (2) q = – p

Q. 8. Suppose that are three non-coplanar vectors in   Let the components of a vector  be 4, 3 and 5, respectively. If the components of this vector  are x, y and z, respectively, then the value of 2x + y + z is         (JEE Adv. 2015)

Ans. 9

Solution. 

⇒  – x + y – z = 4
x – y – z = 3
x + y + z = 5

Solving above equation 

∴ 2x + y + z = 9