Free Vector Algebra 01 Practice Test - 12th Grade - Commerce
Question 1
The vector a^i+b^j+c^k is a bisector of the angle between the vectors ^i+^j and ^j+^k if
SOLUTION
Solution : B
a^i+b^j+c^k=Angle bisector of ^i+^j and ^j+^k=m [^i+^j+^j+^k√2]⇒a=m√2,b=√2.m,c=m√2
Question 2
If |a+b| = |a-b| then (a,b) =
SOLUTION
Solution : D
|a+b|=|a−b|⇒|a+b|2=|a−b|2⇒(a+b)2=(a−b)2⇒a2+b2+2a.b=a2+b2−2a.b⇒4a.b=0⇒a.b=0⇒(a,b)=90∘
Question 3
If the angle θ between the vectors a=2x2^i+4x^j+^k and b=7^i−2^j+x^k is such that 90∘ < θ < 180∘
then x lies in the interval:
SOLUTION
Solution : A
90∘<θ<180∘⇒a.b<0⇒(2x2^i+4x^j+^k).(7^i−2^j+x^k)<0⇒14x2−8x+x<0⇒14x2−7x<0⇒7x(2x−1)<0⇒0<x<12
Question 4
A unit vector perpendicular to the plane of a=2^i−6^j−3^k,b=4^i+3^j−^k is
SOLUTION
Solution : C
a×b=∣∣ ∣ ∣∣^i^j^k2−6−343−1∣∣ ∣ ∣∣=^i(6+9)−^j(−2+12)+^k(6+24)=15^i−10^j+30^k
|a×b|=√225+100+900=35
Unit vector normal to the plane = 15^i−10^j+30^k35=3^i−2^j+6^k7
Question 5
If a is any vector then (a×^i)2+(a×^j)2+(a×^k)2 =
SOLUTION
Solution : B
(a×^i)2+(a×^j)2+(a×^k)2=2a2
Question 6
The area of the parallelogram whose diagonals are ^i−3^j+2^k,−^i+2^j is
SOLUTION
Solution : B
Vector area =12(a×b)=12∣∣ ∣ ∣∣^i^j^k1−32120∣∣ ∣ ∣∣=12[(0−4)−^j(0+2)+^k(2−3)]=12(−4^i−2^j−^k)
Area =12√16+4+1=12√21sq. units
Question 7
The volume of the parallellopiped whose coterminal edges are 2^i−3^j+4^k,^i+2^j−2^k,3^i−^j+^k is
SOLUTION
Solution : C
Volume =| ∣∣ ∣∣2−3412−23−11∣∣ ∣∣|=|2(2−2)+3(1+6)+4(−1−6)|=|0+21−28|=7 cubic units
Question 8
If a, b, c are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then
SOLUTION
Solution : A
The position vector of the centroid of the triangle is →a+→b+→c3
Since, the triangle is an equilateral, therefore the orthocenter coincides with the centroid and hence →a+→b+→c3=→0⇒→a+→b+→c=0
Question 9
If ⃗a=^i+^j+^k,⃗b=2^i+^j+^k and ⃗c=^i+x^j+y^k, are linearly dependent and |⃗c|=√3 then (x,y) is
SOLUTION
Solution : A
Given that the three vectors are linearly dependent so
⃗c=l⃗a+m⃗b
⇒l+2m=1
l−m=x
⇒x=3y−2
I +m =y
Also, x2+y2+1=3
10y2−12y+2=0
⇒y=1,15
x=1,−75
Question 10
If ⃗a ′=^i+^j,⃗b ′=^i+^j+2^k and ⃗c ′=2^i+^j−^k. Then altitude of the parallelopiped formed by the vectors ⃗a,⃗b,⃗c having base formed by ⃗b and ⃗c is (⃗a,⃗b,⃗c and ⃗a′,⃗b′,⃗c′ are reciprocal system of vectors)
SOLUTION
Solution : D
Volume of the parallelepiped formed by ⃗a ′,⃗b ′,⃗c ′ is 4
∴ Volume of the parallelepiped formed by ⃗a,⃗b,⃗c is 14
⃗b×⃗c=14⃗a ′∴∣∣⃗b×⃗c∣∣=√24=12√2
∴ length of altitude = 14×2√2=1√2.