of a point which lies on both of these lines, isA iˆ 2jˆ kˆ B 2iˆ jˆ kˆ C iˆ jˆ kˆ D non existent as the lines are skew Q.3 P, Q have position vectors a&b relative to the origin 'O' & X,
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Q.1 A (1, 1, 3), B (2, 1, 2) & C ( 5, 2, 6) are the position vectors of the vertices of a triangle ABC The
length of the bisector of its internal angle at A is :
Q.2 Let r a l and r b m be two lines in space where a 5iˆ jˆ kˆ, b iˆ 7jˆ kˆ,
kˆjˆiˆ4
l and m 2iˆ 5jˆ kˆ then the p.v of a point which lies on both of these lines, is(A) iˆ 2jˆ kˆ (B) 2iˆ jˆ kˆ (C) iˆ jˆ kˆ (D) non existent as the lines are skew
Q.3 P, Q have position vectors a&b relative to the origin 'O' & X, Y divide PQ internally and externally
respectively in the ratio 2 : 1 Vector XY=
Q.4 Let p is the p.v of the orthocentre &g is the p.v of the centroid of the triangle ABC where circumcentre
is the origin Ifp = Kg , then K =
Q.5 A vector a has components 2p & 1 with respect to a rectangular cartesian system The system is rotated
through a certain angle about the origin in the counterclockwise sense If with respect to the new system,ahas components p + 1 & 1 then ,
(C) p = 1 or p = 1/3 (D) p = 1 or p = 1
Q.6 The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) & b (0, 1, 1) is:
Q.7 Four points A(+1, –1, 1) ; B(1, 3, 1) ; C(4, 3, 1) and D(4, – 1, 1) taken in order are the vertices of
(A) a parallelogram which is neither a rectangle nor a rhombus
(A) are collinear (B) form an equilateral triangle
(C) form a scalene triangle (D) form a right angled triangle
Q.9 If the vectors a 3iˆ jˆ 2kˆ ,b i 3j 4k & c 4i 2j 6k constitute the sides of a ABC,
then the length of the median bisecting the vector c is
Vector & 3D
DPP-1
Trang 3ABC, the point P is, its
(A) centroid (B) orthocentre (C) incentre (D) circumcentre
Q.11 A vector of magnitude 10 along the normal to the curve 3x2 + 8xy + 2y2 – 3 = 0 at its point P(1, 0) can
be
(A) 6iˆ 8jˆ (B) 8iˆ 3jˆ (C) 6iˆ 8jˆ (D) 8iˆ 6jˆ
Q.12 Consider the points A, B and C with position vectors 2iˆ 3jˆ kˆ , iˆ 2jˆ kˆ and 7iˆ kˆ
respectively
Statement-1: The vector sum, AB BC CA = 0
because
Statement-2: A, B and C form the vertices of a triangle
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.(C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true
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Then which one of the following pair(s) are in the same plane
(A) only L1L2 (B) only L2L3 (C) only L3L1 (D) L1L2 and L2L3
Q.3 The acute angle between the medians drawn from the acute angles of an isosceles right angled triangle is:
(A) cos 1 2 3 (B) cos 1 3 4 (C) cos 1 4 5 (D) none
Q.4 If e1 &e2 are two unit vectors and is the angle between them , then cos 2 is
2 1
ee2
ee
Q.5 The vectors 3i 2j k ,i 3j 5k &2i j 4k form the sides of a triangle Then triangle is
(A) an acute angled triangle (B) an obtuse angled triangle
(C) an equilateral triangle (D) a right angled triangle
Q.6 If the vectors 3p q ; 5p 3q and 2p q ; 4p 2q are pairs of mutually perpendicular vectors then
Statement-2: A, B and C form the vertices of a triangle
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.(C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true
Q.8 The set of values of c for which the angle between the vectors cx i 6j 3k & x i 2j 2cx k is
acute for every x R is
Q.10 If the vector 6i 3j 6k is decomposed into vectors parallel and perpendicular to the vector i j k
then the vectors are :
(A) i j k & 7i 2j 5k (B) 2 i j k & 8i j 4k
(C) + 2 i j k & 4i 5j 8k (D) none
DPP-2
Trang 5Q.1 If a b c = 0 , a = 3 , b = 5 , c = 7 , then the angle between a &b is :
Q.2 The lengths of the diagonals of a parallelogram constructed on the vectors p 2a b &q a b ,
where a &b are unit vectors forming an angle of 60º are :
(A) 3 & 4 (B) 7 & 13 (C) 5 & 11 (D) none
Q.3 Let a b c, , be vectors of length 3, 4, 5 respectively Let a be perpendicular to b c,b toc a & c
Q.5 The set of values of x for which the angle between the vectors a xiˆ 3jˆ kˆ and b xiˆ xjˆ kˆ
acute and the angle between the vector b and the axis of ordinates is obtuse, is
(A) 1 < x < 2 (B) x > 2 (C) x < 1 (D) x < 0
Q.6 If a vector a of magnitude 50 is collinear with vector b 6i 8j 15k
2 and makes an acute angle withpositive z-axis then :
Q.7 A, B, C & D are four points in a plane with pv's a b c, , & d respectively such that
ac
·dbcb
·d
a = 0 Then for the triangle ABC, D is its
(A) incentre (B) circumcentre (C) orthocentre (D) centroid
Q.8 Let A & B be two non parallel unit vectors in a plane If ( A B) bisects the internal angle between
A & B, then is equal to
Q.9 Image of the point P with position vector 7i j 2k in the line whose vector equation is,
r =9iˆ 5jˆ kˆ iˆ 3jˆ kˆ) has the position vector
(A) ( 9, 5, 2) (B) (9, 5, 2) (C) (9, 5, 2) (D) none
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Q.11 A tangent is drawn to the curve y = 82
x at a point AA (x1 , y1) , where x1 = 2 The tangent cuts the x-axis
at point B Then the scalar product of the vectors AB & OB is
Q.12 L1 and L2 are two lines whose vector equations are
L1 : r cos 3 iˆ 2sin jˆ cos 3 kˆ
L2 : r aiˆ bjˆ ckˆ ,where and are scalars and is the acute angle between L1 and L2
If the angle ' ' is independent of then the value of ' ' is
(A)
Trang 7Q.1 Cosine of an angle between the vectors a b and a b if |a| = 2, |b| = 1 and a^b = 60° is
Q.2 An arc AC of a circle subtends a right angle at the centre O The point B divides the arc in the ratio 1 : 2
If OA a & OB b , then the vector OC in terms of a&b, is
(A) 3a 2b (B) – 3a b (C) 2a 3b (D) – 2a 3b
Q.3 For two particular vectors AandB it is known that A B = B A What must be true about the two
vectors?
(A) At least one of the two vectors must be the zero vector
(B) A B = B A is true for any two vectors
(C) One of the two vectors is a scalar multiple of the other vector
(D) The two vectors must be perpendicular to each other
Q.4 'P' is a point inside the triangle ABC , such that BC PA + CA PB + ABAB PC = 0 , then for the
triangle ABC the point P is its :
(A) incentre (B) circumcentre (C) centroid (D) orthocentre
Q.5 The vector equations of two lines L1 and L2 are respectivly
)kˆjˆiˆ3(kˆ9jˆ9iˆ
17
I L1 and L2 are skew lines
II (11, –11, –1) is the point of intersection of L1 and L2
III (–11, 11, 1) is the point of intersection of L1 and L2
IV cos–1
35
3 is the acute angle between L1 and L2then , which of the following is true?
(A) II and IV (B) I and IV (C) IV only (D) III and IV
Q.6 Given three vectors a ,b &c each two of which are non collinear Further if a b is collinear with
c, b c is collinear witha & a = b = c = 2 Then the value of a.b +b.c + c.a :
Q.7 For some non zero vector V, if the sum of V and the vector obtained from V by rotating it by an angle
2 equals to the vector obtained from V by rotating it by then the value of , is
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Q.8 Let u,v,w be such that u 1, v 2, w 3 If the projection of v along u is equal to that of w
along u and vectors v , w are perpendicular to each other then u v w equals
Q.9 If a and b are non zero, non collinear, and the linear combination
b)y2x(a5b4a)y
x
2
( holds for real x and y then x + y has the value equal to
Q.10 In the isosceles triangle ABC | |A B =| |BC = 8 , a point E divides AB internally in the ratio 1 : 3, then the
cosine of the angle betweenC E & CA is (where | |CA = 12)
Q.11 If p 3a 5b ; q 2a b ; r a 4b ; s a b are four vectors such that
sin p q = 1 and sin r s = 1 then cos a b is :
(A) 19
5 43
Q.12 Given an equilateral triangle ABC with side length equal to 'a' Let M and N be two points respectively
on the side AB and AC much that AN = KAC and AM =
3
BA
If BN and CM are orthogonalthen the value of K is equal to
(A)
5
1
(B) 4
1
(C) 3
1
(D) 21
Trang 9Q.1 If e1 &e2 are two unit vectors and is the angle between them , then sin 2 is :
)b.a(16
23
3
53
23
3
23
53, ,
Q.6 Given a parallelogram OACB The lengths of the vectors OA,OB &AB are a, b & c respectively TheThe
scalar product of the vectorsOC&OB is :
Q.7 Vectors a &b make an angle = 2
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Q.9 If the two adjacent sides of two rectangles are represented by the vectors p 5a 3 ; qb a b
and r 4a b; s a b respectively, then the angle between the vectorsx 1 p r s
Q.10 If the vector product of a constant vector OA with a variable vector OB in a fixed plane OAB be a
constant vector, then locus of B is :
(A) a straight line perpendicular to OA (B) a circle with centre O radius equal to OA
(C) a straight line parallel to OA (D) none of these
Q.11 If the distance from the point P(1, 1, 1) to the line passing through the points Q(0, 6, 8) and
R(–1, 4, 7) is expressed in the form p q where p and q are coprime, then the value of
2
)1qp)(
Trang 11Q.1 For non-zero vectors a b c, , , a x b c =a b c holds if and only if ;
(A) a b = 0, b c = 0 (B) c a = 0, a b = 0
(C) a c = 0, b c = 0 (D) a b =b c =c a = 0
Q.2 The vectors a =i 2j 3k ; b =2 i j k & c =3i j 4k are so placed that the end point of one
vector is the starting point of the next vector Then the vectors are
(A) not coplanar (B) coplanar but cannot form a triangle
(C) coplanar but can form a triangle (D) coplanar & can form a right angled triangleQ.3 Given the vectors
kˆjˆiˆ2u
kˆ2jˆiˆv
kˆiˆw
If the volume of the parallelopiped having – uc , v and wc as concurrent edges, is 8 then 'c' can beequal to
Q.4 Given a xiˆ yjˆ 2kˆ, b i j k , c i 2 ; (j a b = /2, a c) 4 then
(A) the A.M of a & b (B) the G M of a & b
(C) the H M of a & b (D) equal to zero
Q.7 Leta a i1 a j2 a k3 ; b b i1 b j2 b k3 ; c c i1 c j2 c k3 be three non-zero vectors such
thatc is a unit vector perpendicular to both a&b If the angle betweena &b is
6 then
2
3 3 3
2 2 2
1 1 1
cba
cba
cba
(A) u (v x w ) (B) (v x w ) u (C) v ( u x w ) (D) ( u x v) w
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Q.9 The vectorc is perpendicular to the vectors a = (2, 3, 1) , b = (1, 2, 3) and satisfies the
conditionc i 2j 7k = 10 Then the vectorc =
(A) (7, 5, 1) (B) ( 7, 5, 1) (C) (1, 1, 1) (D) none
Q.10 L et a i j , b j k & c a b If the vectors , i 2j k , 3i 2j k & c are coplanar
then is :
Q.11 A rigid body rotates about an axis through the origin with an angular velocity 10 3 radians/sec
If points in the direction of i j k then the equation to the locus of the points having tangentialspeed 20 m/sec is :
(A) x2 + y2 + z2 x y y z z x 1 = 0
(B) x2 + y2 + z2 2 x y 2 y z 2 z x 1 = 0
(C) x2 + y2 + z2 x y y z z x 2 = 0
(D) x2 + y2 + z2 2 x y 2 y z 2 z x 2 = 0
Q.12 A rigid body rotates with constant angular velocity about the line whose vector equation is,
r= i 2j 2k The speed of the particle at the instant it passes through the point with p.v
kˆjˆ
3
1
kˆjˆ2
3
1
kˆjˆ2
3
1
kˆjˆ2iˆ3
One or more than one is/are correct:
Q.15 If a b c, , be three non zero vectors satisfying the condition a b c & b c a then which of the
following always hold(s) good?
(A) a b c, , are orthogonal in pairs (B) a b c = b
Trang 13Q.1 The altitude of a parallelopiped whose three coterminous edges are the vectors,A iˆ jˆ kˆ ;
kˆjˆ4iˆ
2
B & C iˆ jˆ kˆ with A and B as the sides of the base of the parallelopiped, is
Q.2 Consider ABC with A (a); B ( )b & C ( )c If b (a c) = b b a c ; b a = 3;
c b = 4 then the angle between the medians A M & B D is
Q.3 If A (– 4, 0, 3) ; B (14, 2, –5) then which one of the following points lie on the bisector of the angle
between OA and OB ('O' is the origin of reference)
(A) (2, 1, –1) (B) (2, 11, 5) (C) (10, 2, –2) (D) (1, 1, 2)
Q.4 Position vectors of the four angular points of a tetrahedron ABCD are A(3, – 2, 1); B(3, 1, 5); C(4, 0, 3)
and D(1, 0, 0) Acute angle between the plane faces ADC and ABC is
(A) tan–1 5 2 (B) cos–1 2 5 (C) cosec–1 5 2 (D) cot–1 3 2
Q.5 The volume of the tetrahedron formed by the coterminus edgesa, b, c is 3 Then the volume of the
parallelepiped formed by the coterminus edgesa b, b c, c a is
Q.6 Given unit vectorsm n, &p such that angle betweenm&n = angle betweenp and m n 6
then n p m =
Q.7 a ,b andc be three vectors having magnitudes 1, 1 and 2 respectively If
a x (a x c) +b = 0, then the acute angle between a &c is :
Q.8 If a i j k b, 4i 3j 4k and c i j k are linearly dependent vectors & c 3, then
(A) = 1, = 1 (B) = 1, = ±1 (C) = 1, = ±1 (D) = ±1, = 1
Q.9 A vector of magnitude 5 5 coplanar with vectors iˆ 2jˆ & jˆ kˆ and the perpendicular vector 2iˆ jˆ kˆ is
(A) ± 5 5iˆ 6jˆ kˆ (B) ± 5 5iˆ 6jˆ kˆ
(C) ± 5 5 5iˆ 6jˆ kˆ (D) ± 5iˆ 6jˆ kˆ
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Consider three vectors p iˆ jˆ kˆ, q 2iˆ 4jˆ kˆ and r iˆ jˆ kˆ and let s be a unit vector, thenQ.10 p q and r are,
(A) linearly dependent
(B) can form the sides of a possible triangle
(C) such that the vectors (q )is orthogonal to p
(D) such that each one of these can be expressed as a linear combination of the other two
Q.11 if (p q) × r = p q wr, then (u + v + w) equals to
Q.12 the magnitude of the vector (p·s)(q ) + (q·s)(r p) + ( ·s)(p q) is
One or more than one is/are correct:
Q.13 Given the following information about the non zero vectors A,Band C
Which one of the following holds good?
(A) A B 0 (B) A (B C) 0 (C) A·A 8 (D) A·C 9
Q.14 Let a,b,c are non zero vectors and V1 a (b c) and V2 (a b) c If V1 V2 then which of
the following hold(s) good?
(A) a and b are orthogonal (B) a and c are collinear
(C) band c are orthogonal (D) b (a c) when is a scalar
Q.15 If A,B,Cand D are four non zero vectors in the same plane no two of which are collinear then which
of the following hold(s) good?
(A) (A B) (C D) 0 (B) (A C) (B D) 0
(C) (A B) (C D) 0 (D) (A C) (B D) 0