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### Vector Dpp (1-10) 12th ) WA.pdf

1. Q.1 A(1, 1, 3), B(2, 1, 2) & C(5, 2, 6) aretheposition vectors of the vertices ofa triangleABC.The length ofthebisector ofitsinternal angleatAis : (A) 10 4 (B) 3 10 4 (C) 10 (D) none Q.2 Let l      a r and m b r       be two lines in space where k̂ 2 j ˆ î 5 a     , k̂ 8 j ˆ 7 î b      , k̂ j ˆ î 4     l  and k̂ 7 ĵ 5 î 2 m     then the p.v. of a point which lies on both of these lines, is (A) k̂ j ˆ 2 î   (B) k̂ j ˆ î 2   (C) k̂ 2 ĵ î   (D)non existent as thelines areskew Q.3 P, Q have position vectors   a b & relative to the origin 'O'& X,YdividePQ  internallyand externally respectivelyin the ratio 2:1. Vector XY  = (A)   3 2   b a  (B)   4 3   a b  (C)   5 6   b a  (D)   4 3   b a  Q.4 Let  p isthep.v.oftheorthocentre&  g isthep.v.ofthecentroidofthetriangleABCwherecircumcentre is the origin. If  p = K  g , then K = (A) 3 (B) 2 (C) 1/3 (D) 2/3 Q.5 Avector  a hascomponents 2p& 1with respect to arectangularcartesian system. Thesystem is rotated throughacertainangleabouttheorigininthecounterclockwisesense.Ifwithrespecttothenewsystem,  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.6 The number of vectors of unit length perpendicular to vectors  a = (1, 1, 0) &  b (0, 1, 1) is: (A) 1 (B) 2 (C) 3 (D)  Q.7 Four pointsA(+1, –1, 1) ; B(1, 3, 1) ; C(4, 3, 1) and D(4, – 1, 1) takes in order are the vertices of (A)a parallelogramwhichis neitherarectanglenorarhombus (B)rhombus (C)anisosceles trapezium (D)acyclicquadrilateral. Q.8 The vector equations of the two lines L1 and L2 are given by L1 : ) k̂ 3 j ˆ 2 î ( k̂ 13 ĵ 9 î 2 r         ; L2 : ) k̂ 3 j ˆ 2 î ( k̂ p j ˆ 7 î 3 r           then the lines L1 and L2 are (A) skew lines for all R p (B) intersectingfor all R p and the point of intersection is (–1, 3, 4) (C) intersectinglines for p =– 2 (D)intersectingforall real R p Q.9 Let ,  &  be distinct real numbers. The points whose position vector's are       i j k   ;       i j k   and       i j k   : (A)arecollinear (B)formanequilateraltriangle (C)form ascalenetriangle (D)formaright angledtriangle Q.10 The lines ) k̂ ĵ 2 î 3 ( k̂ 2 r       and ) k̂ 2 j ˆ 4 î 6 ( k̂ 3 j ˆ 2 î 3 r         are (A)non-intersecting (B) non-coplanar (C) intersects exactlyat one point (D) intersect at more than one point CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 1
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 Considerthefollowing3linesinspace L1 : ) k̂ ĵ 4 î 2 ( k̂ 2 j ˆ î 3 r         ; L2 : ) k̂ 4 j ˆ 2 î 4 ( k̂ 3 j ˆ î r         L3 : ) k̂ 2 j ˆ î 2 ( t k̂ 2 j ˆ 2 î 3 r        Thenwhichoneofthe followingpair(s)areinthe sameplane. (A) onlyL1L2 (B) onlyL2L3 (C) onlyL3L1 (D) L1L2 and L2L3 Q.3 Theacuteanglebetweenthemediansdrawnfromtheacuteanglesofanisoscelesrightangledtriangleis: (A) cos-1 2 3 (B) cos-1 3 4 (C) cos-1 4 5 (D) none Q.4 If   e e 1 2 & are two unit vectors and  is theangle between them, then cos  2  is : (A) 1 2 1 2   e e  (B) 1 2 1 2   e e  (C)   e e 1 2 2 . (D)     e x e e e 1 2 1 2 2 Q.5 The vectors3 2    i j k   ,    i j k   3 5 &2 4    i j k   form the sides of a triangle. Then triangle is : (A)anacuteangledtriangle (B)anobtuseangled triangle (C)anequilateraltriangle (D)arightangledtriangle Q.6 If the vectors 3p q  ;5 3 p q  and 2p q  ; 4 2 p q  are pairs of mutually perpendicular vectors then sin p q        is (A) 55 4 (B) 55 8 (C) 3 16 (D) 247 16 Q.7 Ifthevectors   i x j yk   2 3 &   i x j yk   3 2 areorthogonaltoeachother,thenthelocusofthepoint(x,y)is (A) a circle (B)anellipse (C) a parabola (D)astraightline Q.8 The set of values of c for which the angle between the vectors cx i j k      6 3 & x i j cxk      2 2 is acute for everyx  R is (A) 0 4 3 ,       (B) 0 4 3 ,       (C) 11 9 4 3 ,       (D) 0 4 3 ,       Q.9 Let j ˆ î u    , j ˆ î v    and k̂ 3 j ˆ 2 î w     . If n̂ is a unit vector such that 0 n̂ · u   and 0 n̂ · v   , then | n̂ · w |  is equal to (A) 1 (B) 2 (C) 3 (D) 0 Q.10 Ifthevector 6 3 6    i j k   isdecomposedintovectorsparallel andperpendiculartothevector    i j k   then the vectors are : (A)      i j k   & 7 2 5    i j k   (B)  2     i j k   & 8 4    i j k   (C) + 2     i j k   & 4 5 8    i j k   (D) none CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 2
3. Q.1 If    a b c   = 0 ,  a = 3 ,  b = 5 ,  c = 7 , then the angle between   a b & is : (A)  6 (B) 2 3  (C) 5 3  (D)  3 Q.2 The lengths of the diagonals of a parallelogram constructedon the vectors    p a b   2 &    q a b   2 , 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 to   c a  &  c to   a b  . Then    a b c   is : (A) 2 5 (B) 2 2 (C) 10 5 (D) 5 2 Q.4 Given a parallelogramABCD. If  AB = a , AD  = b & AC  = c , then DB AB   . has the value: (A) 3 2 2 2 2 a b c   (B) a b c 2 2 2 3 2   (C) a b c 2 2 2 3 2   (D) none Q.5 The set ofvalues of x for which the angle between the vectors k̂ j ˆ 3 î x a     and k̂ j ˆ x î x 2 b     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 Ifavector  a ofmagnitude50is collinearwithvector  b i j k    6 8 15 2    andmakes anacuteanglewith positivez-axisthen: (A)   a b  4 (B)   a b   4 (C)   b a  4 (D) none Q.7 A, B, C & D are four points in a plane with pv's    a b c , , &  d respectively such that         a c · d b c b · d a              = 0.Then forthe triangleABC, Dis its : (A)incentre (B)circumcentre (C) orthocentre (D)centroid Q.8 Let  A &  B betwononparallelunitvectorsin aplane. If      A B  bisects theinternalanglebetween  A &  B , then  is equal to : (A) 1/2 (B) 1 (C) 2 (D) 4 Q.9 Image of the point P with position vector 7 2    i j k   in the line whose vector equation is,  r =   9 5 5 3 5       i j k i j k       has the position vector : (A) ( 9, 5, 2) (B) (9, 5,  2) (C) (9,  5,  2) (D) none Q.10 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 rexpectivelythen cos 1 + cos 2 + cos 3 equals (A) 3 (B)  3 (C) 1 (D)  1 Q.11 AnarcAC ofacirclesubtends a right angleat thecentreO.Thepoint Bdividesthearcin theratio 1 : 2. If a OA    & b OB    , then the vector  OC in terms of b & a   , is (A) b 2 a 3    (B) – b 2 a 3    (C) b 3 a 2    (D) – b 3 a 2    CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 3
4. Q.1 Cosine of an angle between the vectors b a    and b a    if | a |  = 2, | b |  = 1 and b ^ a   = 60° is (A) 7 3 (B) 21 9 (C) 7 3 (D) none Q.2 Atangent is drawn to the curve y= 8 2 x at a pointA A(x1 , y1) , where x1 = 2. The tangent cuts the x-axis at point B. Then the scalar product of the vectorsAB  &OB  is : (A) 3 (B)  3 (C) 6 (D)  6 Q.3 The two vectors (x2  1)  i + (x + 2)  j + x2  k & 2 3    i x j k   are orthogonal : (A) for no real value of x (B) for x =  1 (C) for x = 1 2 (D) for x =  1 2 & x = 1 Q.4 'P' is a point inside the triangle ABC , such that BC PA        + CA PB        + AB AB PC        = 0 , then for the triangleABC thepoint Pisits : (A)incentre (B)circumcentre (C) centroid (D)orthocentre Q.5 The vector equations of two lines L1 and L2 are respectivly ) k̂ 5 j ˆ î 3 ( k̂ 9 j ˆ 9 î 17 r         and ) j ˆ 3 î 4 ( k̂ j ˆ 8 î 15 r        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 L2 then,whichofthefollowingis true? (A) II and IV (B) I and IV (C) IV only (D) III and IV Q.6 Giventhreevectors  a ,  b &  c eachtwoofwhicharenoncollinear.Furtherif     a b  iscollinearwith  c ,     b c  is collinear with  a &   a  =   b  =   c  = 2 . Then the value of  a .  b +  b .  c +  c .  a : (A) is 3 (B) is 3 (C) is 0 (D) cannot be evaluated CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 4
5. Q.7 The area of the triangle whose vertices areA (1, 1, 2) ; B(2, 1, 1) ; C(3, 1, 2) is : (A) 13 (B) 2 13 (C) 13 (D) none Q.8 If the vector product of a constant vector OA  with a variable vectorOB  in a fixed plane OAB be a constant vector,then locus of Bis : (A)a straightlineperpendicularto OA  (B) a circle with centre O radius equal to OA  (C)a straight lineparallel to OA  (D) none of these Q.9 If OA a    ; OB b    ;OC a b    2 3   ; OD a b      2 , thelength ofOA  is three times the length ofOB  andOA  is prependicular to DB  then BD AC OD OC                   . is (A) 7   a b  2 (B) 42   a b  2 (C) 0 (D) all correct Q.10 IntheisoscelestriangleABC | | AB  =| | BC  =8,apointEdividesABinternallyintheratio 1:3,thenthe cosine of the angle between CE  & CA  is (where | | CA  = 12) (A)  3 7 8 (B) 3 8 17 (C) 3 7 8 (D)  3 8 17 Q.11 If    p a b   3 5 ;    q a b   2 ;    r a b   4 ;    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 (B) 0 (C) 19 5 43 (D) 1
6. Q.1 If   e e 1 2 & are two unit vectors and  is the angle between them , then sin  2  is : (A) 1 2 1 2   e e  (B) 1 2 1 2   e e  (C)   e e 1 2 2 . (D)     e x e e e 1 2 1 2 2 Q.2 If   p s & are not perpendicular to each other and     r xp qxp  &   r s . = 0, then  r = (A)   p s . (B)       q q s p s p        . . (C)       q q p p s p        . . (D)   q p   for all scalars  Q.3 If b a u      ; b a v      and | b | | a |    = 2 then | v u |    is equal to (A)   2 ) b . a ( 16 2    (B)   2 ) b . a ( 16 2    (C)   2 ) b . a ( 4 2    (D)   2 ) b . a ( 4 2    Q.4 If A1, A2, A3,...... , An are the vertices of a regular plane polygon with n sides & O is its centre then OA x OA i i i n             1 1 1 = (A) (1  n) OA x OA 2 1   (B) (n 1) OA x OA 2 1   (C) n OA x OA 2 1   (D) none Q.5 If  A = (1,1, 1) ,  C = (0, 1, 1) aregiven vectors, then a vector  B satisfyingthe equation  A x  B =  C and  A .  B = 3 is : (A) (5, 2, 2) (B) 5 3 2 3 2 3 , ,       (C) 2 3 5 3 2 3 , ,       (D) 2 3 2 3 5 3 , ,       Q.6 If  a i j k       &  b i j k       2 , then the vector  c such that   a c . = 2 &   a c  =  b is (A) 1 3      i j k   2 (B) 1 3         i j k 2 5 (C) 1 3      i j k   2 5 (D) 1 3         i j k 2 5 Q.7 ForanyfourpointsP,Q,R,S PQ RS QR PS RP QS            isequalto4timestheareaofthetriangle (A) PQR (B) QRS (C) PRS (D) PQS Q.8 Vectors   a b & make an angle  = 2 3  . If  a = 1 ,  b = 2 then           a b x a b   3 3 2 = (A) 225 (B) 250 (C) 275 (D) 300 CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 5
7. Q.9 In a quadrilateralABCD, AC  is the bisector of the AB AD          which is 2 3  , 15| | AC  = 3| | AB  = 5| | AD  then cos BA CD          is : (A)  14 7 2 (B)  21 7 3 (C) 2 7 (D) 2 7 14 Q.10 If the two adjacent sides of two rectangles are represented bythe vectors    p a b   5 3 ;    q a b    2 and    r a b    4 ;    s a b    respectively, then the angle between the vectors       x p r s    1 3 and   s r 5 1 y      (A) is –cos–1 19 5 43       (B) is cos–1 19 5 43       (C) is – cos–1 19 5 43       (D) cannot be evaluated Q.11 Let w , v , u    be such that 3 w , 2 v , 1 u       . 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 w v u      equals (A) 2 (B) 7 (C) 14 (D) 14
8. Q.1 For non-zero vectors    a b c , , ,    a x b c . =    a b c holds if and onlyif ; (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 j k   2 3 ;  b =2   i j k   &  c =3 4    i j k   are so placed that the end point of one vector is the startingpoint of the next vector.Then thevectors are : (A) not coplanar (B) coplanar but cannot forma triangle (C) coplanar but can form a triangle (D) coplanar&can form aright angled triangle Q.3 The value of                 a b c a b a b c      2 , , is equal to the box product : (A)      a b c (B) 2      a b c (C) 3      a b c (D) 4      a b c Q.4 Given k̂ 2 j ˆ y î x a    , b i j k       , c i j     2 ;( ) a b  = /2,a c   4 then (A) [ ] a b c 2 = | | a (B) [ ] a b c = | | a (C)[ ] a b c = 0 (D)[ ] a b c = | | a 2 Q.5 The set of values of m for which the vectors    i j mk   ,   ( )  i j m k   1 &     i j mk   are non-coplanar: (A) R (B) R {1} (C) R  {2} (D)  Q.6 Let a,b,c be distinct non-negativenumbers.Ifthe vectorsa i a j ck      ,   i k  &ci c j bk      liein a plane, thenc is : (A) theA.M. of a & b (B) the G. M. of a & b (C) the H. M. of a & b (D) equal to zero. Q.7 Let  a a i a j a k    1 2 3    ;  b b i b j b k    1 2 3    ;  c c i c j c k    1 2 3    be three non-zero vectors such that  c is aunitvectorperpendiculartoboth   a b & .Iftheanglebetween   a b & is  6 then 2 3 3 3 2 2 2 1 1 1 c b a c b a c b a = (A) 0 (B) 1 (C) 1 4 (a1 2 + a2 2 + a3 2) (b1 2 + b2 2 + b3 2) (D) 3 4 (a1 2 + a2 2 + a3 2) (b1 2 + b2 2 + b3 2) (c1 2 + c2 2 + c3 2) CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 6
9. Q.8 The shortest distancebetweenthe skew lines,  r1 =         3 6 4 3 2      i j i j k  and  r2 =         2 7 4      i k i j k  is : (A) 9 (B) 6 (C) 3 (D) none Q.9 The vector  c is perpendicular to the vectors  a = (2,  3, 1) ,  b = (1,  2, 3) and satisfies the condition    c i j k .      2 7 = 10. Then the vector  c = (A) (7, 5, 1) (B) ( 7,  5, 1) (C) (1, 1, 1) (D) none Q.10 Let  a i j     ,  b j k     &    c a b     . If the vectors ,    i j k   2 , 3 2    i j k   &  c are coplanar then   is : (A) 1 (B) 2 (C) 3 (D)  3 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 tangential speed 20 m/sec. is : (A) x2 + y2 + z2  x y  yz  z x  1 = 0 (B) x2 + y2 + z2  2 x y  2 yz  2 z x  1 = 0 (C) x2 + y2 + z2  x y  yz  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 j k   2 2 . The speed of the particle at the instant it passes through the point with p.v. . 2 3 5    i j k   is : (A)  2 (B) 2  (C)  2 (D) none
10. Q.1 Let  a i j k       2 3 ;  b i j k     2   &  c j k   10  &         a x b x c ua vb w c    then : (A) u = 17, v =  9, w = 0 (B) u =  9, v = 17, w = 0 (C) u = 9, v =  17, w = 0 (D) u = 9, v = 17, w = 1 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 AM  &BD  is : (A)  cos1 1 5 13       (B)  cos1 1 13 5       (C) cos1 1 5 13       (D) cos1 1 13 5       Q.3 The set of values of 'm' for which the vectors  a mi m j m k       ( ) ( )  1 8 ,  b m i m j m k       ( ) ( ) ( )  3 4 5 &  c m i m j m k       ( ) ( ) ( )  6 7 8 are non-coplanar is : (A) R (B) R {1} (C) R  {1, 2} (D)  Q.4 IfA(– 4, 0, 3) ; B (14, 2, –5) then which one of the following points lie on the bisector of the angle betweenOA andOB ('O' is the origin of reference) (A) (2, 1, –1) (B) (2, 11, 5) (C) (10, 2, –2) (D) (1, 1, 2) Q.5 If the vectorsa i j k      ,    i b j k   &    i j ck   (a  b  c  1) are coplanar then the value of 1 1 1 1 1 1      a b c = (A) 1 (B) 1 (C) 0 (D) none Q.6 The volume of the tetrahedron formed bythe coterminus edges c , b , a    is 3. Then the volume of the parallelepiped formed bythecoterminus edges a c , c b , b a          is (A) 6 (B) 18 (C) 36 (D) 9 Q.7 Given unit vectors    m n p , & suchthat angle between   m n & = angle between  p and    m x n   6 then      n p m = (A) 3 4 (B) 3 4 (C) 1 4 (D) none CLASS : XII (ABCD) SPECIAL DPP ON VECTOR DPP. NO.- 7
11. Q.8  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 (D) 5 12  Q.9 The position vectors of the points A, B, C arerespectively(1, 1, 1); (1, 1, 2) ; (0, 2, 1).Aunit vector parallel to the plane determined byABC & perpendicular to the vector (1, 0, 1), is (A) ± ) k̂ ĵ 5 î ( 3 3 1   (B) ± ) k̂ ĵ 5 î ( 3 3 1   (C) ± ) k̂ ĵ 5 î ( 3 3 1   (D) ± ) k̂ ĵ 5 î ( 3 3 1   Q.10 Theanglesof atriangle, two ofwhosesidesarerepresentedbythevectors 3    a b   and     b a b a   .  ; where  b is a non-zero vector and  a is a unit vector in the direction of  a , are : (A) tan1 1 3       ; tan1 1 2       ; tan1 3 2 1 2 3           (B) tan1   3 ; tan1 1 3       ; cot 1 (0) (C) tan 1   3 ; tan1(2) ; tan 1 3 2 2 3 1           (D) tan1   3 ; tan1   2 ; tan1 2 3 3 2 1           Q.11 Avectorofmagnitude5 5 coplanarwithvectors j ˆ 2 î & k̂ 2 j ˆ andtheperpendicularvector k̂ 2 j ˆ î 2   is (A) ± 5   k̂ 8 j ˆ 6 î 5   (B) ± 5   k̂ 8 j ˆ 6 î 5   (C) ± 5 5   k̂ 8 j ˆ 6 î 5   (D) ±   k̂ 8 j ˆ 6 î 5   Q.12 The altitude of a parallelopiped whose three coterminous edges are the vectors, k̂ ĵ î A     ; k̂ j ˆ 4 î 2 B     & k̂ 3 j ˆ î C     with A  andB  as the sides of the base of the parallelopiped, is (A) 19 2 (B) 19 4 (C) 19 38 2 (D) none
12. Q.1 If the plane x + y+ z + d = 0 bisect the line segment joining the points (1, 2, 3) and (3, 2, 1), then d is equal to : (A) 6 (B) – 6 (C) 4 (D) – 4 Q.2 The value of 'a' for which the lines 1 2 x  = 3 13 z 2 9 y    and 3 2 z 2 7 y 1 a x        intersect , is (A) – 5 (B) – 2 (C) 5 (D) – 3 Q.3 The intercept madebythe plane q n . r    on the x-axis is (A) n . î q  (B) q n . î  (C)  q n . î  (D) | n | q  Q.4 The distance between the parallel planes given by the equations,   k̂ j ˆ 2 î 2 . r    + 3 = 0 and    r i j k .    4 4 2   + 5 = 0 is (A) 2 1 (B) 3 1 (C) 4 1 (D) 6 1 Q.5 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 Q.6 GivenA(1, –1, 0) ; B(3, 1, 2) ; C(2, –2, 4) and D(–1, 1, –1) which of the following points neitherlie on AB nor on CD? (A) (2, 2, 4) (B) (2, –2, 4) (C) (2, 0,1) (D) (0, –2, –1) Q.7 Iffrom thepoint P (f,g,h)perpendiculars PL, PMbedrawn toyz andzx planes then theequation to the plane OLM is (A) 0 h z g y f x    (B) 0 h z g y f x    (C) 0 h z g y f x    (D) 0 h z g y f x     Q.8 If the plane 2x – 3y+ 6z – 11 = 0 makes an angle sin–1(k) with x-axis, then k is equal to (A) 2 3 (B) 7 2 (C) 3 2 (D) 1 Q.9 The plane XOZ divides the join of (1, –1, 5) and (2, 3, 4) in the ratio  : 1 , then  is (A) – 3 (B) 3 1  (C) 3 (D) 3 1 Q.10 The equation of the right bisector planeof the segment joining (2, 3, 4) and (6,7, 8) is (A) x + y + z + 15 = 0 (B) x + y + z – 15 = 0 (C) x – y + z – 15 = 0 (D) None of these Q.11 Forthe line 3 3 z 2 2 y 1 1 x      , which one ofthefollowingis incorrect? (A) it lies in the plane x – 2y+ z = 0 (B) it is same as line 3 z 2 y 1 x   (C) it passes through (2, 3, 5) (D) it is parallel to the plane x – 2y+ z – 6 = 0 CLASS : XII (ABCD) SPECIAL DPP ON 3-D DPP. NO.- 8, 9
13. Q.12 Avariableplaneformsatetrahedronofconstantvolume64K3 withthecoordinateplanesandtheorigin, then locus of the centroid of the tetrahedron is (A) x3 + y3 + z3 = 6K2 (B) xyz = 6k3 (C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2 Q.13 If the planes x = cy + bz ; y = az + cx ; z = bx + ay pass through the same line then a2 + b2 + c2 + 2abc has the value equal to (A) 1 (B) – 1 (C) 0 (D) 3 Q.14 Theline 2 z z 1 y y 0 x x 1 1 1      is (A)paralleltox-axis (B) perpendicularto x-axis (C) perpendiculartoYOZplane (D) parallelto y-axis Q.15 The lines k 4 z 1 3 y 1 2 x       and 1 5 z 2 4 y k 1 x      are coplanar if (A) k = 0 or – 1 (B) k = 1 or – 1 (C) k = 0 or – 3 (D) k = 3 or – 3 Q.16 The line 1 1 z 2 1 y 3 2 x       intersects the curve xy= c2, inxyplane if cis equal to (A) ± 1 (B) ± 3 1 (C) ± 5 (D) none Q.17 Equationofalinewhichpassesthroughapointwithp.v. c  ,paralleltotheplane n . r   =1andperpendicular totheline b t a r      is (A) n ) a c ( c r           (B) ) n a ( c r         (C) ) n b ( c r         (D) a ) n . b ( c r         Q.18 The distance of the plane passing through the point P(1, 1, 1) and perpendicular to the line 4 1 z 0 1 y 3 1 x      fromtheoriginis (A) 4 3 (B) 3 4 (C) 5 7 (D) 1 Q.19 If   k̂ 4 j ˆ î 2 j ˆ î r        and   k̂ j ˆ 2 î r     = 3 are the equations of a line and a plane respectively thenwhichofthefollowingistrue? (A) line is perpendicular totheplane (B)lineliesintheplane (C) line is parallel to the planebut does not liein the plane (D)linecutstheplane obliquely Q.20 LetABCD be a tetrahedron such that the edgesAB,AC andAD are mutuallyperpendicular. Let the area of trianglesABC,ACD andADB be 3, 4and 5 sq. units respectively.Then the area of the triangle BCD, is (A) 2 5 (B) 5 (C) 2 5 (D) 2 5 Q.21 Let L1 be the line ) k̂ 2 î ( k̂ ĵ î 2 r1        and let L2 be the line ) k̂ ĵ î ( ĵ î 3 r2        . Let  betheplanewhichcontains theline L1 andisparallel to L2. Thedistanceoftheplane from the originis (A) 7 2 (B) 7 1 (C) 6 (D) none
14. Q.1 If the distancebetween a point P and the point (1, 1, 1) on the line 3 1 x  = 4 1 y  = 12 1 z  is 13, then the coordinates of P are (A) (3, 4, 12) (B)       13 12 , 13 4 , 13 3 (C) (4, 5, 13) (D) (40, 53, 157) Q.2 GivenaparallelogramOACB.The lengths ofthevectors OA  ,OB  &AB  area, b &crespectively.The The scalar product of the vectorsOC OB   & is : (A) a b c 2 2 2 3 2   (B) 3 2 2 2 2 a b c   (C) 3 2 2 2 2 a b c   (D) a b c 2 2 2 3 2   Q.3 Twovectors areequal onlyif theyhaveequal component in (A)agivendirection (B)twogivendirections (C)threegivendirections (D)inanyarbitrarydirection Q.4 A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to the coordinateplanesthenwhichofthefollowingisnotthelengthofanedgeofthisrectangularparallelopiped (A) 2 (B) 4 (C) 6 (D) 8 Q.5 If c , b , a    are threenon-coplanar & r , q , p    are reciprocal vectors to c & b , a    respectively, then     r n q m p c n b m a              is equal to : (where l, m, n are scalars) (A) l2 + m2 + n2 (B) l m + m n + n l (C) 0 (D) none of these Q.6 In a trapezium the vector BC AD     . We will then find that  p AC BD     is collinear with AD  . If  p AD    thenwhich oneis true. (A)  =  + 2 (B)  +  = 1 (C)  =  + 1 (D)  =  + 1 Q.7 If the vectors k̂ 2 j ˆ î 3 a     ,  b i j k        3 4 &  c i j k    4 2 6    constitute the sides of a ABC, thenthe lengthofthe medianbisectingthevector  c is : (A) 2 (B) 14 (C) 74 (D) 6 CLASS : XII (ABCD) MISCELLANEOUS DPP. NO.- 10
15. Q.8 If    a b c , , are linearly independent vectors, then which one of the following set of vectors is linearly dependent ? (A)       a b b c c a    , , (B)       a b b c c a    , , (C)       a xb bxc cxa , , (D) none Q.9 Thetripleproduct               d a a x b x cxd  . simplifiesto: (A)          b d a c d . (B)          b c a b d . (C)         b a a b d . (D) none Q.10 The vector  a = 4  i + 3  k ; b  =14  i + 2  j  5  k arelaid offfrom onepoint. Thevector  d which being laidoff from the same point dividingthe anglebetweenthevectors  a &  b in equalhalves &havingthe magnitude 6 is (A)  i +  j + 2  k (B)    i j k   2 (C)    i j k   2 (D)    i j k   2 2 Q.11 A unit vector parallel to the intersection of the planes,    r i j k .      = 5 and    r i j k .    2 3   = 4is (A) 2 5 3 38    i j k   (B) 2 5 3 38    i j k   (C)    2 5 3 38    i j k (D)    2 5 3 38    i j k Q.12 AplaneP1 has theequation2x – y+z =4 and theplaneP2 has theequation x +ny+2z =11. If theangle between P1 and P2 is 3  then the value(s) of 'n' is (are) (A) 2 7 (B) 17, –1 (C) –17, 1 (D) – 2 7 Q.13 Ifthevectors b and a   arelinearlyindependent satisfying     0 b 2 sec 3 a 1 tan 3         then the most general values of  are (A) n – 6  (B) 2n ± 6 11 (C) n ± 6  (D) 2n + 6 11 where nis an integer.
16. Q.14 Vector equation of the plane ) k̂ 3 j ˆ 2 î ( ) k̂ ĵ î ( j ˆ î r            in the scalar dot product form is (A) 7 ) k̂ 3 j ˆ 2 î 5 .( r     (B) 7 ) k̂ 3 j ˆ 2 î 5 .( r     (C) 7 ) k̂ 3 j ˆ 2 î 5 .( r     (D) 7 ) k̂ 3 j ˆ 2 î 5 .( r     Q.15 The three vectors   ,   ,   i j j k k i    taken two at a time form three planes. The three unit vectors drawn perpendicularto thesethree planes form a parallelopiped ofvolume : (A) 1 3 (B) 4 (C) 3 3 4 (D) 4 3 3 Q.16 If   x y & are two non collinear vectors and a, b, c represent the sides of a  ABC satisfying (a  b)  x + (b  c)  y + (c  a)    x y  = 0 then  ABC is : (A)anacuteangletriangle (B)anobtuseangletriangle (C)aright angletriangle (D)ascalenetriangle Q.17 Given     a i j b j k c pi k       2 2 2  ;  ;  arelinearlyindependentvectors, then (A) p = 8 (B) p = 4 (C) p  8 (D) p  4 Q.18 Let    a b c , , be three noncoplanar vectors &    p q r , , are vectors defined bythe relations ] c b a [ b a r ; ] c b a [ a c q ; ] c b a [ c b p                         Then the value of the expression( ). ( ). ( ).          a b p b c q c a r      is equal to : (A) 0 (B) 1 (C) 2 (D) 3 Q.19 Given three non – zero, non – coplanar vectors    a b and c , ,     r pa qb c 1    and     r a pb qc 2    if the vectors   r r 1 2 2  and2 1 2   r r  are collinearthen (p, q) is (A) (0, 0) (B) (1, –1) (C) (–1, 1) (D) (1, 1) Q.20 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 nb       =0 implies : (A) l m + m n + nl = 0 (B) l + m + n = 0 (C) l 2 + m 2 + n 2 = 0 (D) l 3 + m 3 + n 3 = 0
17. Q.21 Let n 3 2 1 r ........ r , r , r     be the position vectors of points P1, P2, P3,.....Pn relative to the origin O. If the vectorequation 0 r a .......... r a r a n n 2 2 1 1        holds, thenasimilarequationwillalsoholdw.r.t.toany otheroriginprovided (A) a1 + a2 + ..... + an = n (B) a1 + a2 + ..... + an = 1 (C) a1+ a2 +...+ an= 0 (D) a1 = a2 = a3 = ...... = an = 0 Q.22 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) (B)        1 , 2 5 , 2 1 (C)         1 , 2 5 , 2 1 (D) (6, –7, –5) Direction for Q.23 & Q.24 Consider a plane x + y – z = 1 and the pointA(1, 2, –3) AlineLhastheequation x = 1 + 3r y = 2 – r z = 3 + 4r Q.23 The co-ordinate of a point B ofline L, such thatAB is parallel tothe plane, is (A) 10, –1, 15 (B) –5, 4, –5 (C) 4, 1, 7 (D) –8, 5, –9 Q.24 EquationoftheplanecontainingthelineLandthepointAhas theequation (A) x – 3y + 5 = 0 (B) x + 3y – 7 = 0 (C) 3x – y – 1 = 0 (D) 3x + y – 5 = 0 Direction for Q.25 to Q.28 Consider a triangular pyramidABCD the position vectors of whose angular points areA(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.25 The length of the vector G A is (A) 17 (B) 3 51 (C) 9 51 (D) 4 59 Q.26 AreaofthetriangleABC insq. units is (A) 24 (B) 6 8 (C) 6 4 (D) none Q.27 The length of the perpendicular from thevertex D on theoppositeface is (A) 6 14 (B) 6 2 (C) 6 3 (D) none Q.28 Equationof theplaneABC is (A) x + y + 2z = 5 (B) x – y – 2z = 1 (C) 2x + y – 2z = 4 (D) x + y – 2z = 1
18. Direction for Q.29 to Q.31 (3 questions) Given four pointsA(2, 1, 0) ; B(1, 0, 1) ; C(3, 0, 1) and D(0, 0, 2). The point D lies on a line Lorthogonal to the plane determined bythe pointA, B and C. Q.29 Equationof theplaneABC is (A) x + y + z – 3 = 0 (B) y + z – 1 = 0 (C) x + z – 1 = 0 (D) 2y + z – 1 = 0 Q.30 EquationofthelineLis (A) ) k̂ î ( k̂ 2 r      (B) ) k̂ ĵ 2 ( k̂ 2 r      (C) ) k̂ j ˆ ( k̂ 2 r      (D) none Q.31 Perpendiculardistanceof Dfrom theplaneABC, is (A) 2 (B) 2 1 (C) 2 (D) 2 1
19. DPP-1 Q.1 B Q.2 A Q.3 D Q.4 A Q.5 B Q.6 B Q.7 D Q.8 C Q.9 B Q.10 D DPP-2 Q.1 B Q.2 D Q.3 C Q.4 A Q.5 D Q.6 B Q.7 A Q.8 D Q.9 C Q.10 A DPP-3 Q.1 D Q.2 B Q.3 D Q.4 A Q.5 D Q.6 B Q.7 C Q.8 B Q.9 B Q.10 D Q.11 B DPP-4 Q.1 A Q.2 A Q.3 D Q.4 A Q.5 A Q.6 B Q.7 A Q.8 C Q.9 A, C Q.10 C Q.11 C DPP-5 Q.1 B Q.2 B Q.3 B Q.4 A Q.5 B Q.6 B Q.7 B Q.8 D Q.9 C Q.10 B Q.11 C DPP-6 Q.1 D Q.2 B Q.3 C Q.4 D Q.5 A Q.6 B Q.7 C Q.8 A Q.9 A Q.10 D Q.11 C Q.12 A DPP-7 Q.1 B Q.2 A Q.3 A Q.4 D Q.5 A Q.6 C Q.7 A Q.8 A Q.9 A Q.10 B Q.11 D Q.12 C DPP-8-9 Q.1 B Q.2 D Q.3 A Q.4 D Q.5 A Q.6 A Q.7 A Q.8 B Q.9 D Q.10 B Q.11 C Q.12 B Q.13 A Q.14 B Q.15 C Q.16 C Q.17 C Q.18 C Q.19 B Q.20 A Q.21 A DPP-10 Q.1 C Q.2 D Q.3 D Q.4 B Q.5 A Q.6 D Q.7 D Q.8 B Q.9 A Q.10 A Q.11 A, C Q.12 C Q.13 D Q.14 C Q.15 D Q.16 A Q.17 C Q.18 D Q.19 D Q.20 B Q.21 C Q.22 B Q.23 D Q.24 B Q.25 B Q.26 C Q.27 A Q.28 D Q.29 B Q.30 C Q.31 D ANSWER KEY
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