Maths Quiz on Vector Algebra for IIT JEE & Engineering Exam

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Maths Quiz on Vector Algebra
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JEE Maths Quiz on Vector Algebra

JEE Maths Quiz on Vector Algebra : In this article you will get to Online test for JEE Main, JEE Advanced, UPSEE, WBJEE and other engineering entrance examinations that will help the students in their preparation. These tests are free of cost and will useful in performance and inculcating knowledge. In this post we are providing you quiz on Vector Algebra.

Quiz on Vector Algebra

 

Q1. The magnitudes of mutually perpendicular forces a, b and c are 2, 10 and 11, respectively. Then the magnitude of its resultant is ______.
a) 10 
b) 12
c) 15
d) 17

View Answer

c) 15

Solution:

R = √[22 + 102 + 112].

= √[4 + 100 + 121]

= 15

Q2. A unit vector a makes an angle π / 4 with z-axis. If a + i + j is a unit vector, then a is equal to _________.
a) [−i / 2] − [j / 2] – [k / √2]
b) [−i / 2] + [j / 2] + [k / √2]
c) [i / 2] − [j / 2] + [k / √2]
d) [−i / 2] − [j / 2] + [k / √2]

View Answer

d) [−i / 2] − [j / 2] + [k / √2]

Solution:

Let a = li + mj + nk, where l+ m2 + n2 = 1. a makes an angle π / 4 with z−axis.

Hence, n = 1 / √2, l+ m2 = 1 / 2 …..(i)

Therefore, a = li + mj + k / √2

a + i + j = (l + 1) i + (m + 1) j + k / √2

Its magnitude is 1, hence (l + 1)2 + (m + 1)2 = 1 / 2 …..(ii)

From (i) and (ii),

2lm = 1 / 2

⇒ l = m = −1 / 2

Hence, a = [−i / 2] − [j / 2] + [k / √2].

Q3. Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a . b + b . c + c . a is ________.
a) 25
b) -25
c) -15
d) 15

View Answer

b) -25

Solution:

Since a + b + c = 0

On squaring both sides, we get

|a|2 + |b|2 + |c|2 + 2 (a . b + b . c + c . a) = 0

⇒ 2 (a . b + b . c + c . a) = − (9 + 16 + 25)

⇒ a . b + b . c + c . a = −25

Q4. a. [(b + c) × (a + b + c)] is equal to ______.
a) 0
b) 1
c) 2
d) 4

View Answer

a) 0

Solution:

a. [(b + c) × (a + b + c)]

= a . (b × a + b × b + b × c) + a . (c × a + c × b + c × c)

= [aba] + [abb] + [a b c] + [aca] + [a c b] + [a c c]

= 0 + 0 + [abc] + 0 − [abc] + 0

= 0

Q5. Let a, b, c be distinct non-negative numbers. If the vectors ai + aj + ck, i + k and ci + cj + bk lie in a plane, then c is
a) The arithmetic mean of a and b
b) The geometric mean of a and b
c) The harmonic mean of a and b
d) Equal to zero

View Answer

b) The geometric mean of a and b

Q6. Let a = 2i + j − 2k and b = i + j. If c is a vector such that a . c = |c|, |c − a| = 2√2 and the angle between (a × b) and c is 30o, then |(a × b) × c| = _________.
a) 3/2
b) 2/3
c) 3/4
d) 4/5

View Answer

a) 3/2

Q7. If b and c are any two non-collinear unit vectors and a is any vector, then (a . b) b + (a . c) c + [a . (b × c) / |b × c|] (b × c) = ___________.
a) b
b) a
c) b2
d) a

View Answer

d) a

Solution:

Let i be a unit vector in the direction of b, j in the direction of c.

Note that b = i and c = j

We have b × c = |b| |c| sinαk = sinαk, where k is a unit vector perpendicular to b and c. ⇒ |b × c| = sinα

⇒ k = [b × c] / [|b × c|]

Any vector a can be written as a linear combination of i, j and k.

Let a = a1i + a2j + a3k

Now a . b = a . i = a1, a . c = a . j = a2 and {[a] . [b × c] / [|b × c|]} = a . k = a3

Thus, (a . b) b + (a . c) c + {[a] . [(b × c) / |b × c|] * [(b × c)|]}

= a1b + a2c + a3 [b × c] / [|b × c|]

= a1i + a2j + a3k

= a

Q8. Let b = 4i + 3j and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively are given by _________.
a) [2 / 5] i + [7 / 5] j
b) [−2 / 5] i – [11 / 5] j
c) [−2 / 5] i + [11 / 5] j
d) [2 / 5] i + [11 / 5] j

View Answer

c) [−2 / 5] i + [11 / 5] j

Solution:

Let r = λb + μc and c = ± (xi + yj).

Since c and b are perpendicular, we have 4x + 3y = 0

⇒ c = ±x (i − 43j), {Because, y = [−4 / 3]x}

Now, projection of r on b = [r . b] / [|b|] = 1

⇒ [(λb + μc) . b] / [|b|]

= [λb . B] / [|b|] = 1

⇒ λ = 1 / 5

Again, projection of r on c = [r . c] / [|c|] = 2

This gives μx = [6 / 5]

⇒ r = [1 / 5] (4i + 3j) + [6 / 5] (i − [4 / 3]j)

= 2i−j or

r = [1 / 5] (4i + 3j) − [6 / 5] (i − [4 / 3]j)

= [−2 / 5] i + [11 / 5] j

Q9. If a, b and c are unit vectors, then |a − b|2 + |b − c|2 + |c − a|2 does not exceed ______.
a) 4
b) 6
c) 8
d) 9

View Answer

d) 9

Solution:

|a − b|2 + |b − c|2 + |c − a|= 2 (a+ b2 + c2) − 2 (a * b + b * c + c * a)

= 2 * 3 − 2 (a * b + b * c + c * a)

= 6 − {(a + b + c)− a2− b− c2}

= 9 − |a + b + c| 2 ≤ 9

Q10. A vector has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p + 1 and 1 with respect to the new system, then find p.
a) 1
b) -1/3
c) option (a) or (b)
d)None of these

View Answer

 

Solution:

If x, y are the original components; X, Y the new components and α is the angle of rotation, then x = X cosα − Y sinα and y = X sinα + Y cosα

Therefore, 2p = (p + 1) cosα − sinα and 1 = (p + 1) sinα + cosα

Squaring and adding, we get 4p+ 1 = (p + 1)+ 1

⇒ p + 1 = ± 2p

⇒ p = 1 or −1 / 3

 



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