Free Vector Algebra 02 Practice Test - 12th Grade - Commerce
Question 1
The three points whose position vectors are ^i+2^j+3^k,3^i+4^j+7^k and−3^i−2^j−5^k
SOLUTION
Solution : C
If A, B, C are the given points respectively, then
−−→OA=^i+2^j+3^k,−−→OB=3^i+4^j+7^k,−−→OC=−3^i−2^j−5^k,−−→AB=−−→OB−−−→OA=2^i+2^j+4^k,−−→AC=−−→OC−−−→OA=−4^i−4^j−8^k=−2−−→AB
∴−−→AB,−−→AC are collinear ⇒A,B,C are collinear.
Question 2
If a,b represent −−→AB,−−→BC respectively of a regular hexagon ABCDEF then −−→CD,−−→DE,−−→EF,−−→FA are
SOLUTION
Solution : A
ABCDEF is a regular hexagon
⇒⃗AD=2⃗BC,⃗ED=⃗AB,⃗FE=⃗BC,⃗FA=⃗DC
Given ⃗AB=a,⃗BC=b
Now ⃗AB+⃗BC+⃗CD=⃗AD⇒a+b+⃗CD=2⃗BC
⇒⃗CD=2b−(a+b)=b−a⃗DE=⃗BA=−⃗AB=−a,⃗EF=⃗CB=−⃗BC=−b⃗FA=⃗DC=−⃗CD=−(b−a)=a−b
Question 3
If A=(1,3,-5) and B=(3,5,-3), then the vector equation of the plane passing through the midpoint of AB and perpendicular to AB is
SOLUTION
Solution : A
−−→AB=−−→OB−−−→OA=(3^i+5^j−3^k)−(^i+3^j−5^k)=2^i+2^j+2^k
Midpoint of AB is (2, 4, -4)
Vector equation of the plane is [r−(2^i+4^j−4^k)].(2^i+2^j+2^k)=0
⇒r.(^i+^j+^k)=2+4−4⇒r.(^i+^j+^k)=2
Question 4
A unit vector perpendicular to the plane determined by the points P(1,-1,2), Q(2,0,-1) and R(0,2,1) is
SOLUTION
Solution : A
−−→OP=^i−^j+2^k,−−→OQ=2^i−^k,−−→OR=2^j+^k⇒−−→PQ=−−→OQ−−−→OP=^i+^j−3^k,−−→PR=−−→OR−−−→OP=−^i+3^j−^k
−−→PQ×−−→PR=∣∣ ∣ ∣∣^i^j^k11−3−13−1∣∣ ∣ ∣∣=8^i+4^j+4^k;|−−→PQ×−−→PR|=√64+16+16=√96=4√6
Required unit vectors = ±8^i+4^j+4^k4√6=±2^i+^j+^k√6
Question 5
If x.a=0,x×b=c×b then x =
SOLUTION
Solution : A
x×b=c×b⇒(x−c)×b=0⇒x−c is parallel to b
⇒x−c=λb for some scalar λ⇒x=c+λb
x.a=0⇒(c+λb).a=0⇒c.a+λb.a=0⇒λ=−c.ab.a⇒x=c−c.ab.ab
Question 6
If the vector a is perpendicular to b and c, |a|=2, |b|=3, |c|=4 and the angle between b and c is 2π3 then |[a b c ]| =
SOLUTION
Solution : C
|[abc]|=|a.(b×c)|=|a||b×c||cos(a,b×c)|=|a||b×c|=|a||b|c|sin(b,c)
=2.3.4sin(2π3)=24.√32=12√3.
Question 7
The volume of the tetrahedron with vertices at (1,2,3), (4,3,2), (5,2,7), (6,4,8) is
SOLUTION
Solution : D
[−−→AB−−→AC−−→AD]=∣∣ ∣∣31−1404525∣∣ ∣∣=3(0−8)−1(20−20)−1(8−0)=−24−0−8=−32
Volume of the tetrahedron =16(32)=163 cubic unit.
Question 8
If a,b,c are linearly independent, then [2a+b, 2b+c, 2c+a][a, b, c]=
SOLUTION
Solution : A
[2a+b,2b+c,2c+a][a,b,c]=∣∣ ∣∣210021102∣∣ ∣∣=2(4−0)−1(0−1)=8+1=9
Question 9
Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then −−→OA+−−→OB+−−→OC+−−→OD equals
SOLUTION
Solution : D
Since, the diagonals of a parallelogram bisect each other. Therefore, P is the middle point of AC and BD both.
∴−−→OA+−−→OC=2−−→OP and −−→OB+−−→OD=2−−→OP⇒−−→OA+−−→OB+−−→OC+−−→OD=4−−→OP
Question 10
If ⃗a=^i+2^j+2^k and ⃗b=3^i+6^j+2^k, then the vector in the direction of ⃗a and having magnitude as |⃗b|, is
SOLUTION
Solution : C
The required vectors
=|⃗b|^a=|⃗b||⃗a|⃗a=73(^i+2^j+2^k)