MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q. 1
SPECIAL DPP ON VECTOR
A (1, 1, 3), B B(2, (2, 1, 2) & C (5, 2, 6) are the t he position vectors of o f the vertices of a triangle ABC. The length of the bisecto bisectorr of its internal angle at A is is : (A) 10 4
Q.2 Q.2
DPP. NO.- 1
(B) 3 10 4
(C) 10
(D) none
P, Q have ave pos posiition tion vecto ectors rs a & b relative to the origi o rigin n 'O' & X, Y divide PQ internally and externally
respectively respect ively in the ratio 2 : 1. Vecto Vectorr XY = (A) Q. 3
3 b a 2
(B)
4 a 3
b
(C)
5 b 6
a
(D)
4 b a 3
Let p is the p.v. p.v. of the orthocentre ort hocentre & g is the p.v. of the centroid cent roid of the triangle t riangleABC where circumcentr circumcentree
is the origin. If p = K g , then K = (A) 3 (B) 2
(C) 1/3
(D) 2/3
Q. 4
A vector a has components 2p & 1 with respect to t o a rectangular rect angular cartesian system system.. The system is is rotated rotat ed through a certain angle about the oorigin rigin in the counterclockwise sense. If with respect to the t he new system, system,a has components p + 1 & 1 then , (A) p = 0 (B) p = 1 or p = 1/3 (C) p = 1 or p = 1/3 (D) p = 1 or p = 1
Q.5
The num numbe berr of vectors vectors of uni unitt leng length th perpen perpendi dicul cular ar to vectors vectors a = (1, 1, 0) & b (0, 1, 1) is: (A) 1 (B) 2 (C) 3 (D)
Q.6
Four poi points nts A(+1, A(+1, –1, –1, 1) ; B( B(1, 1, 3, 1) ; C(4, C(4, 3, 1) and and D D(4, (4, – 1, 1) taken taken in in order order are the the verti vertices ces of (A) a parallelogram which is is neither a rectangle rect angle nor a rhombus (B) rhombus (C) an isosceles trapezium (D) a cycl c yclic ic quadrilateral.
Q. 7
Let
; , & be distinct real numbers. The points whose position vector's are i j k
and i j k i j k
(A) are collinear (C) form a scalene triangle Q. 8
constitute ˆ , b & c 4 i 2 j 6 k If the vectors a 3ˆi jˆ 2 k constitut e the sides sides of a ABC, i 3 j 4 k then the length of the t he median median bisecting the vector c is
(A) 2 Q.9
(B) form an equilateral triangle (D) form a right angled triangle
(B) 14
(C) 74
(D) 6
P be a poi point interi nterior or to the acute acute trian triangl glee ABC. ABC. If P A P B P C is a null vector then t hen w.r.t. w.r.t. tthe he triangle ABC, the point po int P is, is, its (A) centroid (B) orthocentre (C) incentre (D) circumcentre
Dpp's on Vector Vector & 3D
[1]
Q.10
A vector of magni magnitude tude 10 al along the norm normal al to the curv curvee 3x2 + 8xy + 2y2 – 3 = 0 at its point P(1, 0) can be (A) 6iˆ 8 jˆ (B) 8ˆi 3 jˆ (C) 6iˆ 8 jˆ (D) 8ˆi 6 jˆ
Q.11 Q.11
Let A(0, A(0, –1, 1), B(0, B(0, 0, 1), 1), C(1, C(1, 0, 1) are the the vertic vertices es of a ABC. If R and r denotes denot es the circumradius circumradius and inradius of ABC, then (A) tan
3 8
r has value equal to R
(B) cot
3 8
(C) tan
(D) cot
12
12
Q.12 If A(0, A(0, 1, 0) 0),, B (0, 00,, 0), C(1, C(1, 0, 1) 1) are are the verti vertices ces of a ABC. Column-I
Column-II
(A) Orthocentre of ABC.
(P)
2 2
(B) Circumcentre of ABC.
(Q)
3 2
(C) Area (ABC).
(R)
3 3
(D) Distance be betwe tween or ortthoce ocentre an and ce centroi troidd.
(S)
(E) Distan tance be betwe tween or orth thoc ocen entr tree an and ci circu rcumcentre ntre..
(T)
(F) Dis Distan tance bet betw ween een ci circum cumcen centre tre an and cen centroi troidd.
(U)
1 , 1 , 1 2 2 2
(G) Incentre of ABC.
(V)
1 , 1 , 1 3 3 3
(H) Centroid of ABC
3 6 (0 (0,, 0, 0)
1 2 1 , , (W) 1 2 3 1 2 3 1 2 3
Dpp's on Vector Vector & 3D
[2]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
SPECIAL DPP ON VECTOR
DPP. NO.- 2
Q.1
If the three points with position vectors (1, a, b) ; (a, 2, b) and (a, b, 3) are collinear in space, then the value of a + b is (A) 3 (B) 4 (C) 5 (D) none
Q.2
Consider the following 3 lines in space ˆ ( 2ˆi 4 jˆ k ˆ) L1 : r 3ˆi jˆ 2k ˆ ( 4ˆi 2 jˆ 4k ˆ) L2 : r ˆi jˆ 3k
ˆ t (2ˆi jˆ 2k ˆ) L3 : r 3ˆi 2 jˆ 2k 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) cos1 2 3 (B) cos1 3 4 (C) cos1 4 5 (D) none
Q.4
, i 3 j 5 k & 2 i j 4 k form the sides of a triangle. Then triangle is The vectors 3 i 2 j k (A) an acute angled triangle (B) an obtuse angled triangle (C) an equilateral triangle (D) a right angled triangle
Q.5
If the vectors 3p q ; 5 p 3q and 2p q ; 4 p 2q are pairs of mutually perpendicular vectors then
sin ( p q ) is (A) 55 4 Q.6
(B) 55 8
(C) 3 16
Consider the points A, B and C with position vectors respectively. Statement-1: The vector sum, A B B C C A = 0
(D)
247 16
ˆ , ˆi 2 jˆ 3k ˆ and 7ˆi k ˆ 2ˆi 3 jˆ 5k
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. Q.7
Q.8
The set of values of c for which the angle between the vectors cx i 6 j 3 k & x i 2 j 2 cx k is acute for every x R is (A) (0, 4/3) (B) [0, 4/3] (C) (11/9, 4/3) (D) [0, 4/3) ˆ . If nˆ is a unit vector such that u · nˆ 0 and v · nˆ 0 , then Let u ˆi jˆ , v ˆi jˆ and w ˆi 2 jˆ 3k | w · nˆ | is equal to (A) 1 (B) 2 (C) 3 (D) 0
Dpp's on Vector & 3D
[3]
Q.9
Q.10
is decomposed into vectors parallel and perpendicular to the vector If the vector 6 i 3 j 6 k i j k then the vectors are :
(A) i j k & 7 i 2 j 5 k
(B) 2 i j k & 8 i j 4 k
(C) + 2 i j k & 4 i 5 j 8 k
(D) none
l
ˆ and m 2ˆi 5 jˆ 7 k ˆ then the p.v. of a point which lies on both of these lines, is 4ˆi jˆ k
ˆ (A) ˆi 2 jˆ k
Q.11 (i)
ˆ , b ˆi 7 jˆ 8k ˆ, Let r a l and r b m be two lines in space where a 5ˆi jˆ 2k
ˆ (B) 2ˆi jˆ k
Let A(1, 2, 3), B(0, 0, 1), C(–1, 1, 1) are the vertices of a ABC. The equation of internal angle bisector through A to side BC is ˆ µ(3ˆi 2 jˆ 3k ˆ) ˆ µ(3ˆi 4 jˆ 3k ˆ) (A) r ˆi 2 jˆ 3k (B) r ˆi 2 jˆ 3k ˆ µ(3ˆi 3 jˆ 2k ˆ) (C) r ˆi 2 jˆ 3k
(ii)
ˆ t ( 7ˆi 10 jˆ 2k ˆ) (D) r k
The equation of median through C to side AB is ˆ p(3ˆi 2k ˆ) ˆ p(3ˆi 2k ˆ) (A) r ˆi jˆ k (B) r ˆi jˆ k ˆ p( 3ˆi 2k ˆ) (C) r ˆi jˆ k
(iv)
ˆ µ(3ˆi 3 jˆ 4k ˆ) (D) r ˆi 2 jˆ 3k
The equation of altitude through B to side AC is ˆ t ( 7ˆi 10 jˆ 2k ˆ) ˆ t ( 7ˆi 10 jˆ 11k ˆ) (A) r k (B) r k ˆ t ( 7ˆi 10 jˆ 2k ˆ) (C) r k
(iii)
ˆ (D) non existent as the lines are skew (C) ˆi jˆ 2k
ˆ p(3ˆi 2 jˆ) (D) r ˆi jˆ k
The area (ABC) is equal to (A)
9 2
(B)
17 2
(C)
17 2
Dpp's on Vector & 3D
(D)
7 2
[4]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
DPP. NO.- 3
If a b c = 0, a = 3 , b = 5 , c = 7, then the angle between a & b is :
(A) Q.2
SPECIAL DPP ON VECTOR
6
(B) 2 3
(C) 5 3
(D)
The lengths of the diagonals of a parallelogram constructed on the vectors p
3
2 a b & q a 2 b ,
where a & b are unit vectors forming an angle of 60º are : (A) 3 & 4 Q.3
(B) 7 & 13
(C) 5 & 11
(D) none
(C) 10 5
(D) 5 2
Let a , b , c be vectors of length 3, 4, 5 respectively. Let a be perpendicular to b c , b to c a & c
to a b . Then a b c is : (A) 2 5
(B) 2 2
Given a parallelogram ABCD. If | AB | = a , | AD | = b & | AC | = c , then DB . AB has the value 3 a 2 b 2 c2 (A) 2
Q.5
Q.4
a 2 3 b 2 c2 (B) 2
a 2 b 2 3 c2 (C) 2
(D) none
ˆ and b 2x ˆi x jˆ k ˆ The set of values of x for which the angle between the vectors a x ˆi 3 jˆ k
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
and makes an acute angle with If a vector a of magnitude 50 is collinear with vector b 6 i 8 j 15 k
2
positive z-axis then :
(A) a 4 b Q.7
(B) a 4 b
(C) b 4 a
(D) none
A, B, C & D are four points in a plane with pv's a , b , c & d respectively such that
a d · b c b d · ca = 0. Then for the triangle ABC, D is its (A) incentre Q.8
(C) orthocentre
(D) centroid
a and b are unit vectors inclined to each other at an angle , (0, ) and a b < 1. Then
, 2 (A) 3 3 Q.9
(B) circumcentre
2 , (B) 3
(C) 0,
3
, 3 (D) 4 4
in the line whose vector equation is, Image of the point P with position vector 7 i j 2 k ˆ (ˆi 3 jˆ 5k ˆ ) has the position vector r = 9ˆi 5 jˆ 5k (A) ( 9, 5, 2) (B) (9, 5, 2) (C) (9, 5, 2) (D) none
Dpp's on Vector & 3D
[5]
Q.10
Q.11
Let a , b , c are three unit vectors such that a b c is also a unit vector. If pairwise angles between a , b , c are 1, 2 and 3 rexpectively then cos 1 + cos 2 + cos 3 equals (A) 3 (B) 3 (C) 1 (D) 1 8 at a point A (x1 , y1) , where x1 = 2. The tangent cuts the x-axis x2 at point B. Then the scalar product of the vectors AB & OB is
A tangent is drawn to the curve y = (B) 3
(A) 3 Q.12
(D) 6
(C) 6
L1 and L2 are two lines whose vector equations are
L1 : r cos 3 ˆi
ˆ 2 sin jˆ cos 3 k
ˆ , L2 : r aˆi b jˆ ck 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)
6
(B)
4
(C)
3
Dpp's on Vector & 3D
(D)
2
[6]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
SPECIAL DPP ON VECTOR
(C) 3
7
(B) 9
21
(D) none
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) 3 a 2 b (B) – 3 a 2 b (C) 2 a 3 b Q.3
DPP. NO.- 4
Cosine of an angle between the vectors a b and a b if | a | = 2, | b | = 1 and a ^ b = 60° is (A) 3 7
Q.2
(D) – 2 a 3 b
For two particular vectors A and B 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
triangle ABC the point P is its : (A) incentre (B) circumcentre Q.5
'P' is a point inside the triangle ABC , such that BC PA + CA PB + AB PC = 0 , then for the (C) centroid
(D) orthocentre
L1 and L2 are skew lines (11, –11, –1) is the point of intersection of L1 and L2 (–11, 11, 1) is the point of intersection of L1 and L2
II III
cos –1 3 35 is the acute angle between L1 and L2 then , which of the following is true? (A) II and IV (B) I and IV (C) IV only IV
(D) III and IV
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 with a & a = b = c = 2 . Then the value of a . b + b . c + c . a : (B) is 3
(A) is 3 Q.7
The vector equations of two lines L1 and L2 are respectivly ˆ (3ˆi jˆ 5k ˆ ) and r 15ˆi 8 jˆ k ˆ ( 4ˆi 3 jˆ ) r 17ˆi 9 jˆ 9k I
Q.6
(C) is 0
(D) cannot be evaluated
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 (A) 2n ±
3 where n is an integer.
(B) n ±
3
(C) 2n ±
Dpp's on Vector & 3D
2 3
(D) n ±
2 3
[7]
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 (A) 2 Q.9
(B) 7
(C) 14
(D) 14
If a and b are non zero, non collinear, and the linear combination
(2x y)a 4 b 5a ( x 2 y) b holds for real x and y then x + y has the value equal to (A) – 3 (B) 1 (C) 17 (D) 3
| |
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 between C E & CA is (where CA = 12)
Q.11
(A)
3 7 8
(B)
3 8 17
(C)
3 7 8
(D)
3 8 17
If p 3 a 5 b ; q 2a b ; r a 4 b ; s a b are four vectors such that sin p q = 1 and sin r s = 1 then cos a b is :
(A) Q.12
19 5 43
(B) 0
(C) 1
(D)
19 5 43
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 A N = K A C and A M =
AB . If B N and C M are orthogonal 3
then the value of K is equal to (A)
1 5
(B)
1 4
(C)
1 3
Dpp's on Vector & 3D
(D)
1 2
[8]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
SPECIAL DPP ON VECTOR
If u
q . s (C) q p p . s
(D) q p for all scalars
a b ; v a b and | a | | b | = 2 then | u v | is equal to
2 16 (a b . )2
(A)
Q.5
q . p (B) q p p . s
(A) p . s
Q.4
(B) atleast one x in (– 1, 0) (D) no value of x.
If p & s are not perpendicular to each other and r x p qx p & r . s = 0, then r =
Q.3
V1 3ax 2ˆi 2( x 1) jˆ and V2 b( x 1)ˆi x 2 jˆ , where ab < 0. The vector V1 and V2
Let
are linearly dependent for (A) atleast one x in (0, 1) (C) atleast one x in (1, 2) Q.2
DPP. NO.- 5
16 (a b.) 2
(B) 2
4 (a b.) 2
(C) 2
2 . ) 2 4 (a b
(D)
If u and v are two vectors such that | u | 3 ; | v | 2 and | u v | 6 then the correct statement is (A) u ^ v (0, 90°) (B) u ^ v (90°, 180°) (C) u ^ v = 90° (D) ( u v) u 6 v
If A = (1, 1, 1) , C = (0, 1, 1) are given vectors, then a vector B satisfying the equation A x B = C and
A . B = 3 is :
5 2 2 (B) , ,
(A) (5, 2, 2) Q.6
3 3 3
2 2 , 5 3 3 3
2 5 , 2 3 3 3
(D) ,
(C) ,
Given a parallelogram OACB. The lengths of the vectors OA , OB & AB are a, b & c respectively. The
scalar product of the vectors OC & OB is : a 2 3 b 2 c2 (A) 2
Q.7
3 a 2 b 2 c2 (B) 2
Vectors a & b make an angle = (A) 225
(B) 250
3 a 2 b 2 c2 (C) 2
2 . If a = 1 , b = 2 then a 3
(C) 275
2
3 b x 3 a b = (D) 300
2 AD which is , 3
Q.8
a 2 3 b 2 c2 (D) 2
In a quadrilateral ABCD, AC is the bisector of the A B BA C D is 15 AC = 3 AB = 5 A D then cos
| | | | | |
(A)
14 7 2
(B)
21 7 3
(C)
2 7
Dpp's on Vector & 3D
(D)
2 7 14
[9]
Q.9
If the two adjacent sides of two rectangles are represented by the vectors p 5a 3b ; q a 2b
and r 4a b ; s a b respectively, then the angle between the vectors x
y
1 p r s and 3
1 r s 5
19 (A) is –cos –1 5 43 (C) is –
cos –1
19 (B) is cos –1 5 43
19 5 43
(D) cannot be evaluated
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
(C) a straight line parallel to OA Q.11
(B) a circle with centre O radius equal to OA (D) none of these
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 ( p q )( p q 1) equals 2 (A) 4950 (B) 5050
(C) 5150
Dpp's on Vector & 3D
(D) none
[10]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
SPECIAL DPP ON VECTOR
DPP. NO.- 6
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
; & are so placed that the end point of one The vectors a = i 2 j 3 k b = 2 i j k c = 3 i j 4 k 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 triangle
Q.3
Given the vectors ˆ u 2ˆi jˆ k
ˆ v ˆi jˆ 2k ˆ w ˆi k If the volume of the parallelopiped having – c u , v and c w as concurrent edges, is 8 then 'c' can be equal to (A) ± 2 (B) 4 (C) 8 (D) can not be determined
Q.4
Q.5
, c i 2j ; (a b ) = /2, a c ˆ , b i j k Given a xˆi y jˆ 2k
4 then
(A) [a b c] 2 = | a |
(D) [a b c] = a
(B) [a b c] = | a |
(C) [a b c] = 0
2
The set of values of m for which the vectors i j mk , i j ( m 1) k & i j m k are non-coplanar : (A) R
(B) R {1}
(C) R {2}
(D)
Q.6
& c i c j b k lie in a Let a, b, c be distinct non-negative numbers. If the vectorsa i a j ck , i k plane, then c is : (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
; c c i c j c k be Let a a 1 i a 2 j a 3 k ; b b1 i b 2 j b 3 k three non-zero vectors such 1 2 3
a1 b1
c1
2
that c is a unit vector perpendicular to both a & b . If the angle between a & b is then a 2 b 2 c 2 = 6
(A) 0 1 4
a 3 b3
c3
(B) 1
(C) (a12 + a22 + a32) (b12 + b22 + b32)
(D)
3 (a 2 + a22 + a32) (b12 + b22 + b32) (c12 + c22 + c32) 4 1
Dpp's on Vector & 3D
[11]
Q.8
For three vectors u , v , w which of the following expressions is not equal to any of the remaining three? (A) u . ( v x w ) (B) ( v x w ) . u (C) v . ( u x w ) (D) ( u x v ) . w
Q.9
The vector c is perpendicular to the vectors a = (2,
3, 1) , b = (1, 2, 3) and satisfies the
Then the vector c = condition c . i 2 j 7 k = 10.
(B) ( 7, 5, 1)
(A) (7, 5, 1) Q.10
(D) none
& c a b . If the vectors , i 2 j k , 3 i 2 j k & c are coplanar Let a i j , b j k
then
is
(A) 1 Q.11
(C) (1, 1, 1)
(B) 2
(D) 3
(C) 3
A rigid body rotates about an axis through the origin with an angular velocity 10 3 radians/sec.
then the equation to the locus of the points having tangential If points in the direction of i j k speed 20 m/sec. is (A) x2 + y2 + z2 x y y z z x 1 = 0 (B) x2 + y2 + z2 2 x y 2 yz 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 yz 2 z x 2 = 0
Q.12 A rigid body rotates with constant angular velocity about the line whose vector equation is,
r = i
2 j 2 k . The speed of the particle at the instant it passes through the point with p.v..
ˆ is : 2ˆi 3 jˆ 5k
(A) 2
(B) 2
(C)
ˆ; Q.13 Given 3 vectors V1 aˆi b jˆ ck
2
(D) none
ˆ; V2 bˆi c jˆ ak
V3
ˆ cˆi a jˆ bk
In which one of the following conditions V1, V2 and V3 are linearly independent? (A) a + b + c = 0 and a2 + b2 + c2 ab + bc + ca (B) a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca (C) a + b + c 0 and a2 + b2 + c2 = ab + bc + ca (D) a + b + c 0 and a2 + b2 + c2 ab + bc + ca Q.14
(A)
1 ˆ 3ˆi 2 jˆ 5k 3
(B)
1 3
ˆ ˆi 2 jˆ 5k
(C)
1 ˆi 2 jˆ 5k ˆ 3
One or more than one is/are correct:
Q.15
, then the vector such that a . c = 2 & a c = If a i j k & b i 2 j k c b is
(D)
1 ˆ 3ˆi 2 jˆ k 3
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
2
(C) a b c = c
(B) a b c = b
(D) b = c
Dpp's on Vector & 3D
[12]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
SPECIAL DPP ON VECTOR
DPP. NO.- 7
ˆ ; The altitude of a parallelopiped whose three coterminous edges are the vectors, A ˆi jˆ k
ˆ & C ˆi jˆ 3k ˆ with A and B as the sides of the base of the parallelopiped, is B 2ˆi 4 jˆ k
(A) 2 Q.2
(B) 4
19
Consider ABC with A c
(a ) ; B
(C) 2 38 19
( b) & C
(D) none
( c) . If b . (a
c) = b . b a . c ; b a = 3;
b = 4 then the angle between the medians AM & BD is
1 5 13 1 (C) cos1 5 13
1 13 5 1 (D) cos1 13 5
(A) cos1
Q.3
19
(B) cos1
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 tetrahedronABCD 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 (A) 6 (B) 18 (C) 36 Q.6
(D) 9
Given unit vectors m , n & p such that angle between m & n = angle between p and m n 6
then n p m = (A) 3 4 Q.7
(B) 3/4
(C) 1/4
(D) none
a , b and c 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 : (A) /6 (B) /4 (C) /3
Dpp's on Vector & 3D
(D) 5 12
[13]
Q.8
Q.9
k, b
4i 3j 4 k and c i j k are linearly dependent vectors & c (A) = 1, = 1 (B) = 1, = ±1 (C) = 1, = ±1 (D) = ±1, = 1
If a
i
j
3 , then
ˆ and the perpendicular vector 2ˆi jˆ 2k ˆ is A vector of magnitude 5 5 coplanar with vectors ˆi 2 jˆ & jˆ 2k ˆ (A) ± 5 5ˆi 6 jˆ 8k
(B) ±
ˆ 5 5ˆi 6 jˆ8k
ˆ (C) ± 5 5 5ˆi 6 jˆ 8k
ˆ (D) ± 5ˆi 6 jˆ8k
Paragraph for questions nos. 10 to 12 ˆ , q ˆ and r ˆi jˆ 3k ˆ and let s be a unit vector, then Consider three vectors p ˆi jˆ k 2ˆi 4 jˆ k
Q.10 p, q and r are (A) linearly dependent (B) can form the sides of a possible triangle (C) such that the vectors (q r ) 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 = u p vq w r , then (u + v + w) equals to (A) 8 (B) 2 (C) – 2
(D) 4
Q.12 the magnitude of the vector ( p · s )(q r ) + (q · s )( r p) + ( r · s )( p q ) is (A) 4 (B) 8 (C) 18 (D) 2 One or more than one is/are correct:
Q.13
( A B) A 0
(i)
(iii) A · B 6 Which one of the following holds good?
(ii)
B· B 4
(iv)
B· C 6
(C) A · A 8
(D) A · C 9
Let a , b, c are non zero vectors such that they are not orthogonal pairwise and such that V1 a ( b c )
and V2
(C) b and c are orthogonal
(a b) c . If V1 V2 then which of the following hold(s) good?
(A) a and b are orthogonal
Q.15
(B) A · (B C) 0
(A) A B 0 Q.14
Given the following information about the non zero vectors A, B and C
(B) a and c are collinear
(D) b (a c ) when is a scalar..
If A, B, C and 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
(C) ( A B) (C D) 0
(B) ( A C) · ( B D) 0
(D) ( A C) ( B D) 0
Dpp's on Vector & 3D
[14]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
Q.2
DPP. NO.- 8
ˆ , q ˆ and r ˆi jˆ 3k ˆ . If p , q Consider three vectors p ˆi jˆ k and r denotes the 2ˆi 4 jˆ k position vector of three non-collinear points then the equation of the plane containing these points is (A) 2x – 3y + 1 = 0 (B) x – 3y + 2z = 0 (C) 3x – y + z – 3 = 0 (D) 3x – y – 2 = 0
The intercept made by the plane r . n q on the x-axis is ˆi . n (B) q
q (A) ˆ i .n Q.3
SPECIAL DPP ON 3-D
q (D) | n |
(C) ˆi . n q
If the distance between the planes 8x + 12y – 14z = 2 and 4x + 6y – 7z = 2 N ( N 1) 1 where N is natural then the value of is 2 N (B) 5050 (C) 5150 (D) 5151
can be expressed in the form (A) 4950 Q.4
A plane passes through the point P(4, 0, 0) and Q(0, 0, 4) and is parallel to the y-axis. The distance of the plane from the origin is (A) 2
Q.5
(C)
(D) 2 2
2
If from the point P (f, g, h) perpendiculars PL, PM be drawn to yz and zx planes then the equation to the plane OLM is (A)
Q.6
(B) 4
x y z f g h
0
(B)
x y z f g h
0
(C)
x y z f g h
0
(D)
x f
y g
z h
0
If the plane 2x – 3y + 6z – 11 = 0 makes an angle sin –1(k) with x-axis, then k is equal to (A) 3 2
(B) 2/7
(C)
2 3
(D) 1
Q.7
The plane XOZ divides the join of (1, –1, 5) and (2, 3, 4) in the ratio : 1 , then is (A) – 3 (B) – 1/3 (C) 3 (D) 1/3
Q.8
A variable plane forms a tetrahedron of constant volume 64 K 3 with the coordinate planes and the origin, then locus of the centroid of the tetrahedron is (A) x3 + y3 + z3 = 6K 3 (B) xyz = 6k 3 (C) x2 + y2 + z2 = 4K 2 (D) x –2 + y –2 + z –2 = 4K –2
Q.9
Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the area of triangles ABC, ACD and ADB be 3, 4 and 5 sq. units respectively. Then the area of the triangle BCD, is (A) 5 2
(B) 5
(C) 5
2
Dpp's on Vector & 3D
(D) 5/2 [15]
Q.10
Equation of the line which passes through the point with p. v. (2, 1, 0) and perpendicular to the plane ˆ is containing the vectors ˆi jˆ and jˆ k (A) r = (2, 1, 0) + t (1, –1, 1) (C) r = (2, 1, 0) + t (1, 1, –1) where t is a parameter
Q.11
(B) r = (2, 1, 0) + t (–1, 1, 1) (D) r = (2, 1, 0) + t (1, 1, 1)
Which of the following planes are parallel but not identical? P1 : 4x – 2y + 6z = 3 P2 : 4x – 2y – 2z = 6 P3 : –6x + 3y – 9z = 5 P4 : 2x – y – z = 3 (A) P2 & P3 (B) P2 & P4 (C) P1 & P3
(D) P1 & P4
Q.12 A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to the coordinate planes then which of the following is not the length of an edge of this rectangular parallelopiped (A) 2 (B) 4 (C) 6 (D) 8 ˆ ) ( ˆi 2 jˆ 3k ˆ ) in the scalar dot product form is Q.13 Vector equation of the plane r ˆi jˆ (ˆi jˆ k ˆ) 7 (A) r .(5ˆi 2 jˆ 3k
ˆ) 7 (B) r .(5ˆi 2 jˆ 3k
ˆ) 7 (C) r .(5ˆi 2 jˆ 3k
ˆ) 7 (D) r .(5ˆi 2 jˆ 3k
Q.14 The vector equations of the two lines L1 and L2 are given by ˆ (ˆi 2 jˆ 3k ˆ ) ; L2 : r 3ˆi 7 jˆ pk ˆ (ˆi 2 jˆ 3k ˆ) L1 : r 2ˆi 9 jˆ 13k then the lines L1 and L2 are (A) skew lines for all p R (B) intersecting for all p R and the point of intersection is (–1, 3, 4) (C) intersecting lines for p = – 2 (D) intersecting for all real p R Q.15 Consider the plane (x, y, z) = (0, 1, 1) + (1, – 1, 1) + (2, – 1, 0). The distance of this plane from the origin is (A) 1/3
(B) 3 2
(C)
32
Dpp's on Vector & 3D
(D) 2
3
[16]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
The value of 'a' for which the lines (A) – 5
Q.2
Q.3
Q.4
Q.5
SPECIAL DPP ON 3-D
(B) – 2
DPP. NO.- 9
x a y7 z2 x 2 y 9 z 13 = and intersect, is 2 1 3 2 3 1 (C) 5 (D) – 3
Given A (1, –1, 0) ; B(3, 1, 2) ; C(2, –2, 4) and D(–1, 1, –1) which of the following points neither lie on AB nor on CD? (A) (2, 2, 4) (B) (2, –2, 4) (C) (2, 0,1) (D) (0, –2, –1) x 1 y 2 z 3 , which one of the following is incorrect? 1 2 3 x y z (A) it lies in the plane x – 2y + z = 0 (B) it is same as line 1 2 3 (C) it passes through (2, 3, 5) (D) it is parallel to the plane x – 2y + z – 6 = 0
For the line
Given planes P1 : cy + bz = x P2 : az + cx = y P3 : bx + ay = z P1, P2 and P3 pass through one line, if (A) a2 + b2 + c2 = ab + bc + ca (C) a2 + b2 + c2 = 1
(B) a2 + b2 + c2 + 2abc = 1 (D) a2 + b2 + c2 + 2ab + 2bc + 2ca + 2abc = 1
x x1 y y1 z z1 is 0 1 2 (A) parallel to x-axis (C) perpendicular to YOZ plane
(B) perpendicular to x-axis (D) parallel to y-axis
The line
Q.6
x 2 y3 z4 x 1 y 4 z 5 and are coplanar if k 1 1 k 2 1 (A) k = 0 or – 1 (B) k = 1 or – 1 (C) k = 0 or – 3 (D) k = 3 or – 3
Q.7
The line
The lines
(A) ± 1 Q.8
x2 3
y 1 z 1 intersects the curve xy = c2, in xy plane if c is equal to 2 1
(B) ± 1/3
(C) ±
5
(D) none
The line which contains all points (x, y, z) which are of the form (x, y, z) = (2, –2, 5) + (1, –3, 2) intersects the plane 2x – 3y + 4z = 163 at P and intersects the YZ plane at Q. If the distance PQ is a b where a, b N and a > 3 then (a + b) equals (A) 23 (B) 95 (C) 27 (D) none
Q.9
ˆ (ˆi 2k ˆ ) and let L be the line r Let L1 be the line r 1 2ˆi jˆ k 2 2
ˆ). 3ˆi jˆ (ˆi jˆ k Let be the plane which contains the line L1 and is parallel to L2. The distance of the plane from the
origin is (A) 1/7
(B)
27
(C)
6
Dpp's on Vector & 3D
(D) none
[17]
Q.10
Q.11
Q.12
The value of m for which straight line 3x – 2y + z + 3 = 0 = 4x – 3y + 4z + 1 is parallel to the plane 2x – y + mz – 2 = 0 is (A) –2 (B) 8 (C) – 18 (D) 11 ˆ where t A straight line is given by r (1 t ) ˆi 3t jˆ (1 t ) k x + y + cz = d then the value of (c + d) is (A) – 1 (B) 1 (C) 7
R. If this line lies in the plane (D) 9
The distance of the point (–1, –5, – 10) from the point of intersection of the line
x 2 y 1 z 2 = = 2 4 12
and the plane x – y + z = 5 is (A) 2 11 Q.13
(B) 126
(C) 13
(D) 14
P( p) and Q(q ) are the position vectors of two fixed points and R ( r ) is the position vector of a variable
point. If R moves such that ( r p) ( r q ) 0 then the locus of R is (A) a plane containing the origin 'O' and parallel to two non collinear vectors O P and O Q (B) the surface of a sphere described on PQ as its diameter. (C) a line passing through the points P and Q (D) a set of lines parallel to the line PQ. MATCH THE COLUMN:
Q.14 Consider the following four pairs of lines in column-I and match them with one or more entries in column-II. Column-I
(A)
(B)
(C) (D)
Column-II
L1 : x = 1 + t, y = t, z = 2 – 5t L2 : r ( 2,1,3) + (2, 2, – 10)
x 1 y 3 z 2 = = 1 2 2 z2 x2 y6 L2 : = = 3 1 1 L1 : x = – 6t, y = 1 + 9t, z = – 3t L2 : x = 1 + 2s, y = 4 – 3s, z = s x y 1 z2 L1 : = = 1 3 2 x 3 y2 z 1 L2 : = = 4 3 2 L1 :
(P)
non coplanar lines
(Q)
lines lie in a unique plane
(R)
infinite planes containing both the lines
(S)
lines are not intersecting at a unique point
Q.15 P(0, 3, – 2); Q(3, 7, – 1) and R(1, – 3, – 1) are 3 given points. Let L1 be the line passing through P and ˆ. Q and L2 be the line through R and parallel to the vector V ˆi k Column-I
Column-II
(A) (B) (C)
perpendicular distance of P from L2 shortest distance between L1 and L2 area of the triangle PQR
(P) (Q) (R)
(D)
distance from (0, 0, 0) to the plane PQR
(S)
Dpp's on Vector & 3D
7 3 2 6 19 147
[18]
MATHEMATICS Daily Practice Problems Target IIT JEE 2010 CLASS : XIII (VXYZ)
Q.1
MISCELLANEOUS
DPP. NO.- 10
If a , b, c are three non-coplanar & p, q , r are reciprocal vectors to a , b & c respectively, then
a m b nc p mq nr
is equal to : (where l, m, n are scalars) (B) l m + m n + n l (C) 0 (D) none of these
(A) l2 + m2 + n2 Q.2
(A) 0 Q.3
If A, B & C are three non-coplanar vectors, then ( A B C) ·[(A B) (A C)] equals
(B) [ A B C ]
(D) [ A B C ]
(C) 2 [ A B C ]
A plane P1 has the equation 2x – y + z = 4 and the plane P2 has the equation x + ny + 2z = 11. If the angle
between P1 and P2 is then the value(s) of 'n' is (are) 3 (A) 7/2 (B) 17, –1 (C) –17, 1 Q.4
The three vectors i j , j k , k i taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume : (A) 1/3
Q.5
(D) – 7/2
(B) 4
(C) 3 3 4
(D) 4 3 3
ABC satisfying
If x & y are two non collinear vectors and a, b, c represent the sides of a
(a b) x + (b c) y + (c a) x y = 0 then ABC is (A) an acute angle triangle (B) an obtuse angle triangle (C) a right angle triangle (D) a scalene triangle Q.6
Given three non – zero, non – coplanar vectors a , b, c and r1
the vectors r1 2 r 2 and 2 r1 r 2 are collinear then (p, q) is (A) (0, 0) (B) (1, –1) (C) (–1, 1) Q.7
pa qb c and r2 a pb qc if (D) (1, 1)
If the vectors a , b , c are non-coplanar and l, m, n are distinct scalars, then
a mb n c b mc na c ma n b = 0 implies : (A) l m + mn + n l = 0 (C) l 2 + m 2 + n 2 = 0 Q.8
(B) l + m + n = 0 (D) l 3 + m 3 + n 3 = 0
Let r 1 , r 2 , r 3 ........ r n be the position vectors of points P1, P2, P3,.....Pn relative to the origin O. If the
vector equation a1 r 1 a 2 r 2 .......... a n r n other origin provided (A) a1 + a2 + ..... + an = n (C) a1+ a2 +...+ an= 0 Q.9
(B) a1 + a2 + ..... + an = 1 (D) none
The orthogonal projection A' of the point A with position vector (1, 2, 3) on the plane 3x – y + 4z = 0 is (A) (–1, 3, –1)
Q.10
0 holds, then a similar equation will also hold w.r.t. to any
1 5 (B) , ,1 2 2
(C)
1 , 5 ,1 2 2
If aˆ and bˆ are unit vectors then the vector defined as V (A) aˆ bˆ
(B) bˆ aˆ
(D) (6, –7, –5)
(aˆ bˆ) (aˆ bˆ) is collinear to the vector
(C) 2aˆ bˆ Dpp's on Vector & 3D
(D) aˆ 2 bˆ [19]
Q.11
If aˆ and bˆ are orthogonal unit vectors then for any non zero vector r , the vector ( r aˆ ) equals (A) [ r aˆ bˆ] ( aˆ bˆ)
(B) [ r aˆ bˆ] aˆ ( r · aˆ )(aˆ bˆ)
(C) [ r aˆ bˆ] bˆ ( r · bˆ)( bˆ aˆ )
(D) [ r aˆ bˆ] bˆ ( r · aˆ )(aˆ bˆ)
Paragraph for Question Nos. 12 to 13
Q.12 Q.13
Consider a plane x + y – z = 1 and the point A(1, 2, –3) A line L has the equation x = 1 + 3r y = 2 – r z = 3 + 4r The co-ordinate of a point B of line L, such that AB is parallel to the plane, is (A) 10, –1, 15 (B) –5, 4, –5 (C) 4, 1, 7 (D) –8, 5, –9 Equation of the plane containing the line L and the point A has the equation (A) x – 3y + 5 = 0 (B) x + 3y – 7 = 0 (C) 3x – y – 1 = 0 (D) 3x + y – 5 = 0 Paragraph for Question Nos. 14 to 17
Consider a triangular pyramid ABCD the position vectors of whose angular points are A(3, 0, 1) ; B(–1, 4, 1); C(5, 2, 3) and D(0, –5, 4). Let G be the point of intersection of the medians of the triangle BCD. Q.14
The length of the vector A G is (A) 17
(B) 51 3
(C)
51 9
(D) 59 4
Q.15 Area of the triangle ABC in sq. units is (A) 24 Q.16
(B) 8 6
(C) 4 6
(D) none
The length of the perpendicular from the vertex D on the opposite face is (A) 14
(B) 2
6
(C) 3
6
Q.17 Equation of the plane ABC is (A) x + y + 2z = 5 (B) x – y – 2z = 1
6
(C) 2x + y – 2z = 4
(D) none (D) x + y – 2z = 1
Paragraph for Question Nos. 18 to 20
Consider the three vectors p, q and r such that ˆ ; p r = q c p and p · r = 2 q ˆi jˆ k
ˆ; p ˆi jˆ k
Q.18 The value of p q r is (A) – Q.19
5 2c | r |
(B) –
(C) 0
(D) greater then zero
If x is a vector such that p q r x = p q r , then x is ˆ) (A) c (ˆi 2 jˆ k
(B) a unit vector
1 ˆ) (ˆi 2 jˆ k 2 If y is a vector satisfying (1 + c) y = p (q r ) then the vectors x , y, r (A) are collinear (B) are coplanar (C) represent the coterminus edges of a tetrahedron whose volume is 'c' cubic units. (D) represent the coterminus edges of a parallelepiped whose volume is 'c' cubic units
(C) indeterminate, as p q r Q.20
8 3
(D) –
Dpp's on Vector & 3D
[20]
[REASONING
TYPE]
x 4 y 5 z 1 x 2 y 1 z and 2 4 1 3 2 3 Statement-1: The lines intersect. Statement-2: They are not parallel. (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.21 Given lines
Q.22
Consider three vectors a , b and c
ˆ a ) · b k ˆ a b (ˆi a ) · b ˆi ( jˆ a ) · b jˆ ( k
Statement-1:
ˆ · c) k ˆ c (ˆi · c ) ˆi ( jˆ · c) jˆ ( k (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. Statement-2:
Q.23
Consider the lines
L1 : r a b and L2 : r b a , where a and b are non zero and non collinear vectors. Statement-1: L1 and L2 are coplanar and the plane containing these lines passes through origin.
(a b) · ( b a ) 0 and the plane containing L1 and L2 is [ r a b] = 0 which passes through origin. (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. Statement-2:
Q.24
Statement-1:
ˆ be vertical. The line of greatest slope on a plane with Let the vector a ˆi jˆ k
ˆ is along the vector ˆi 4 jˆ 2k ˆ. normal b 2ˆi jˆ k Statement-2:
If a is vertical, then the line of greatest slope on a plane with normal b is along the
vector (a b) b . (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. [MULTIPLE OBJECTIVE TYPE] Select the correct alternative(s): (More than one are correct)
Q.25
If A(a ) ; B( b ) ; C (c ) and D(d ) are four points such that ˆ ; b 2ˆi 8 jˆ ; c ˆi 3 jˆ 5k ˆ ; d 4ˆi jˆ 7k ˆ a 2ˆi 4 jˆ 3k d is the shortest distance between the lines AB and CD, then which of the following is True?
(A) d = 0, hence AB and CD intersect
(C) AB and CD are skew lines and d =
(B) d = 23 13
(D) d =
[ AB CD BD] | AB CD | [ AB CD AC] | AB CD |
Dpp's on Vector & 3D
[21]
Q.26
Consider four points A(a ) ; B( b ) ; C( c ) and D(d ) , such that GA GB GC GD 0 for a point G( g ), if (A) G is the centroid of the tetrahedron ABCD (B) G lies on the line joining each of A, B, C, D to the centroid of the triangle formed by the other three (C) p.v. of G is collinear with the p.v. of the centroids of the triangle formed by any three of the four given points. (D) ABCD is parallelogram with G as the point of intersection of the diagonals AC and BD.
Q.27 Given the equations of the line 3x – y + z + 1 = 0, 5x + y + 3z = 0. Then which of the following is correct ? (A) symmetical form of the equations of line is x 2
1 5 z 8 = 8 1 1
y
1 5 y 8 = 8 = z (B) symmetrical form of the equations of line is 2 1 1 (C) equation of the plane through (2, 1, 4) and prependicular to the given lines is 2x – y + z – 7 = 0 (D) equation of the plane through (2, 1, 4) and prependicular to the given lines is x + y – 2z + 5 = 0 x
Q.28
Given three vectors ˆ; U 2ˆi 3 jˆ 6k
ˆ; V 6ˆi 2 jˆ 3k
ˆ W 3ˆi 6 jˆ 2k
Which of the following hold good for the vectors U, V and W ?
(A) U, V and W are linearly depedent
(B) ( U V) W 0
(C) U, V and W form a triplet of mutually perpendicular vectors
(D) U ( V W ) 0 Q.29
Consider the family of planes x + y + z = c where c is a parameter intersecting the coordinate axes at P, Q, R and , , are the angles made by each member of this family with positive x, y and z axis. Which of the following interpretations hold good for this family. (A) each member of this family is equally inclined with the coordinate axes. (B) sin2 + sin2 + sin2 = 1 (C) cos2 + cos2 + cos2 = 2 (D) for c = 3 area of the triangle PQR is 3 3 sq. units.
Dpp's on Vector & 3D
[22]
[MATCH THE COLUMN]
Q.30
Column-I
(A)
(B)
Column-II
(P)
Centre of the parallelop iped whose 3 coterminou s edges OA , OB and OC have position vectors a , b and c respectively where O is the origin, is
OABC is a tetrahedron where O is the origin. Positions (Q) vectors of its angular points A, B and C are a , b and c respectively. Segments joining each vertex with the centroid of the opposite face are concurrent at a point P whose p.v.' s are
ab c
a b c 3
(C)
(D)
Let ABC be a triangle the positionvectors its angular points of are a , b and c respectively. If | a b | | b c | | c a | then the p.v. of the orthocentr e of the triangle is
(R)
(S)
Let a , b , c be 3 mutually perpendicu lar vectors of the same magnitude. If an unknown vector x satisfies the equation a ( x b ) a b ( x c ) b c ( x a) c 0. Then x is given by
Q.31
(A)
(B)
(C)
a b c 2
Column-I
4
a b c
Column-II
Let O be an interior point of ABC such that O A 2 O B 3 O C 0 , then the ratio of the area of ABC to the area of AOC, is with O is the origin Let ABC be a triangle whose centroid is G, orthocentre is H and circumcentre is the origin 'O'. If D is any point in the plane of the triangle such that no three of O, A, B, C and D are
(P)
0
(Q)
1
(R)
2
collinear satisfying the relation A D B D C H 3 H G H D then the value of the scalar '' is If a , b, c and d are non zero vectors such that no three of them are in the same plane and no two are orthogonal then the value of the scalar
(S)
3
( b c) · ( a d ) ( c a ) · ( b d ) is (a b) · (d c ) [SUBJECTIVE TYPE]
Q.32
Given a tetrahedron D-ABC with AB = 12 , CD = 6. If the shortest distance between the skew lines AB and CD is 8 and the angle between them is
Q.33
ˆ, Given A 2ˆi 3 jˆ 6k
6
, then find the volume of tetrahedron.
ˆ and C ˆi 2 jˆ k ˆ. B ˆi jˆ 2k
Compute the value of A A (A B) · C .
Dpp's on Vector & 3D
[23]
ANSWER
3 4 3 3 3 . Q 8 4 ) B ( ; S ) A ( 1 3 . Q ) B ( ; S ) A ( Q ) C ( ; R S ) D ( ; Q ) C ( ; R C , B , A 9 2 . Q D , C , B 8 2 . Q D , B 6 2 . Q D , C , B 5 2 . Q D 4 2 . Q A 3 2 . Q A 0 2 . Q D 9 1 . Q B 8 1 . Q D 7 1 . Q A 6 1 . Q C 3 1 . Q D 2 1 . Q C 1 1 . Q B 0 1 . Q B 9 . Q C 6 . Q A 5 . Q D 4 . Q C 3 . Q D 2 . Q A p D 0 1 p
2 3 . Q 0 3 . Q 7 2 . Q 2 2 . Q 5 1 . Q 8 . Q 1 . Q
) B ( ; R ) A ( 5 1 . Q S P ( , Q ) B ( , R ) A ( S ) D ( ; P ) C ( ; Q , ) D ( , S , Q ) C C 3 1 . Q C 2 1 . Q D 1 1 . Q A 0 1 . Q B 9 . Q A 7 . Q C 6 . Q B 5 . Q B 4 . Q C 3 . Q A 2 . Q D 9 p p D C 4 1 . Q C 3 1 . Q B 2 1 . Q C 1 1 . Q A 0 1 . Q A 9 . Q B 7 . Q B 6 . Q A 5 . Q D 4 . Q D 3 . Q A 2 . Q D D 8 p p C , B 5 1 . Q D , B D , B , A 3 1 . Q A 2 1 . Q B 1 1 . Q C 0 1 . Q D 9 . Q D 7 . Q A 6 . Q C 5 . Q A 4 . Q D 3 . Q A 2 . Q C 7 p p D C , A 4 1 . Q D 3 1 . Q A 2 1 . Q C 1 1 . Q D 0 1 . Q A 9 . Q C 7 . Q B 6 . Q A 5 . Q D 4 . Q A 3 . Q B 2 . Q D p D 6 p
4 1 . Q 8 . Q 1 . Q
D , B , A D 1 2 . Q C B 4 1 . Q B B 7 . Q D
C
C D
A
B C
KEY
5 1 . Q 8 . Q 1 . Q 4 1 . Q 8 . Q 1 . Q 5 1 . Q 8 . Q 1 . Q
D 6 . Q
A 1 1 . Q C 0 1 . Q B 5 . Q C 4 . Q B 3 . Q 5 p p D
B 9 . Q C 2 . Q
C 8 . Q A 1 . Q
B 6 . Q
A 2 1 . Q D 1 1 . Q C 0 1 . Q A 5 . Q A 4 . Q C 3 . Q p p D 4 -
B 9 . Q B 2 . Q
C 8 . Q A 1 . Q
C 7 . Q
B 6 . Q
A 2 1 . Q A 1 1 . Q D 0 1 . Q D 5 . Q A 4 . Q D 3 . Q 3 p p D
B 9 . Q B 2 . Q
B 8 . Q D 1 . Q
D 7 . Q
i i ( , D ) i ( 1 1 . Q A 0 1 . Q B ) v i ( , B ) i i i ( , C ) C 6 . Q B 5 . Q D 4 . Q C 3 . Q p p D 2 -
A 9 . Q D 2 . Q
C 8 . Q B 1 . Q
D 7 . Q
A 7 . Q
B 7 . Q
D 6 . Q
( 2 ) C ( ; U ) B ( ; T ) A 1 . Q ) E ( ; R ) D ( ; P ( ; W ) G ( ; S ) F ( ; Q V ) H B 1 1 . Q A 0 1 . Q A 9 . Q D 8 . Q B 5 . Q B 4 . Q A 3 . Q D 2 . Q B 1 . Q 1 p p D
Dpp's on Vector & 3D
[24]
