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  1. Question bank on Vector, 3D and Complex Number & Misc. Subjective Problem Select the correct alternative : (Only one is correct) Q.1 If  a = 11 1 ,  b = 23 ,   a b  = 30 , then   a b  is : (A) 10 (B) 20 (C) 30 (D) 40 Q.2 ThepositionvectorofapointPmovinginspaceisgivenby k̂ ) t sin 5 ( ĵ ) t cos 4 ( î ) t cos 3 ( R OP      . The time 't' when the point P crosses the plane 4x – 3y + 2z = 5 is (A) 2  sec (B) 6  sec (C) 3  sec (D) 4  sec Q.3 Indicate the correct order sequence in respect of the following : I. The lines 3 4 x   = 1 6 y   = 1 6 y   and 1 1 x   = 2 2 y   = 2 3 z  are orthogonal. II. The planes 3x – 2y – 4z = 3 and the plane x – y – z = 3 are orthogonal. III. The function f (x) = ln(e–2 + ex) is monotonic increasing  x R. IV. If g is the inverse of the function, f (x) = ln(e–2 + ex) then g(x) = ln(ex – e–2). (A) FTFF (B)TFTT (C) FFTT (D)FTTT Q.4* If 2 1 z z is purelyimaginarythen 2 1 2 1 z z z z   is equal to : (A) 1 (B) 2 (C) 3 (D) 0 Q.5 In a regulartetrahedron, the centres ofthe four faces are thevertices of a smallertetrahedron. The ratio of the volume ofthe smaller tetrahedron to that of the larger is n m ,where m and n arerelativelyprime positive integers. The value of (m + n)is (A) 3 (B) 4 (C) 27 (D) 28 Q.6 If    a b c , , arenon-coplanarunitvectorssuchthat      a x bxc b c ( ) ( )   1 2 thentheanglebetween b & a   is (A) 3 /4 (B) /4 (C) /2 (D)  Q.7 The sineof angle formed bythe lateral faceADC and plane of thebaseABC of thetetrahedronABCD where A  (3, –2, 1); B  (3, 1, 5); C  (4, 0, 3) and D  (1, 0, 0) is (A) 2 29 (B) 5 29 (C) 3 3 29 (D) 2 29 Q.8* Let z be a complex number, then the region represented by the inequality z + 2< z + 4is given by : (A) Re (z) >  3 (B) Im(z) < 3 (C) Re (z) <  3 & Im (z) >  3 (D) Re (z) <  4 & Im (z) >  4 Q.9 The volume of the parallelpiped whose edges are represented by the vectors  a i j k    2 3 4    ,  b i j k    3 2    ,  c i j k       2 is : (A) 7 (B) 5 (C) 4 (D) none
  2. Q.10 Let w , v , u    bethevectors suchthat 0 w v u       ,if 5 w & 4 v , 3 u       then the value of u . w w . v v . u         is : (A) 47 (B) 25 (C) 0 (D) 25 Q.11 Let  a = j ˆ î  ,  b= k̂ j ˆ  ,  c= î k̂  . If  d is a unit vector such that      a d b c d . [ , , ]   0 then  d (A) ) k̂ 2 j ˆ î ( 6 1    (B) ) k̂ j ˆ î ( 3 1    (C) ) k̂ j ˆ î ( 3 1    (D) ± k̂ Q.12* If z be a complex number for which z z   1 2 , then the greatest value of z is : (A) 2 1  (B) 3 1  (C) 2 2 1  (D) none Q.13 If the non  zero vectors   a b & are perpendicular to each other, then the solution of the equation ,    r a b   is : (A)         r xa a a a b    1 . (B)         r xb b b a b    1 . (C)    r xa b   (D)    r xb a   where x is anyscalar. Q.14 Thelines k 4 z 1 3 y 1 2 x       and 1 5 z 2 4 y k 1 x      arecoplanar if (A) k = 1 or – 1 (B) k = 0 or – 3 (C) k = 3 or – 3 (D) k = 0 or – 1 Q.15 Whichoneof thefollowingstatementis INCORRECT ? (A) If   n a . = 0 ,   n b . = 0 &   n c . = 0 for some non zero vector  n , then     a b c = 0 (B) there exist a vector having direction angles  = 30º &  = 45º (C) locus of point for which x = 3 & y = 4 is a line parallel to the z - axis whose distance from the z-axis is 5 (D)InaregulartetrahedronOABCwhere'O'istheorigin,thevector OA OB OC      isperpendicular to the planeABC. Q.16* Given that the equation, z2 +(p+ iq)z+ r+ is = 0 has a real root where p, q, r, s  R.Then which one is correct (A) pqr = r2 + p2s (B) prs = q2 + r2p (C) qrs = p2 + s2q (D) pqs = s2 + q2r Q.17 The distance ofthe point (3, 4, 5) from x-axis is (A) 3 (B) 5 (C) 34 (D) 41 Q.18 Givennonzerovectors C and B , A    ,thenwhich oneofthefollowingis false? (A)Avector orthogonal to C and B A     is ± C ) B A (      (B)Avector orthogonal to B A    and B A    is ± B A    (C)Volume of the parallelopiped determined by C and B , A    is | C · B A |     (D)Vector projection of B onto A   is | B | B · A   
  3. Q.19 Given three vectors    a b c , , such that they are nonzero, noncoplanar vectors, then which of the followingarenoncoplanar. (A)       a b b c c a    , , (B)       a b b c c a    , , (C)       a b b c c a    , , (D)       a b b c c a    , , Q.20* The sum i + 2i2 + 3i3 + ........ + 2002i2002, where i = 1  is equal to (A) – 999 + 1002i (B) – 1002 + 999i (C) – 1001 + 1000i (D) – 1002 + 1001i Q.21 Locus of the point P, for which OP  represents a vector with direction cosine cos  = 1 2 ( 'O' is the origin) is : (A)Acircle parallel to yz plane withcentre on the xaxis (B) a coneconcentric with positive xaxis havingvertex at the originand the slant height equal to the magnitudeofthevector (C)a rayemanatingfrom the origin and making an angleof60ºwith xaxis (D) a disc parallel to yz plane withcentre on xaxis & radius equal to| | OP  sin 60º Q.22 Alinewithdirectionratios(2,1,2)intersectsthelines ) k̂ j ˆ î ( j ˆ r        and ) k̂ ĵ î 2 ( î r        at A and B, then l (AB) is equal to (A) 3 (B) 3 (C) 2 2 (D) 2 Q.23 The vertices of a triangle areA(1, 1, 2), B(4, 3, 1) & C(2, 3, 5).The vector representing the internal bisector of theangleAis : (A)    i j k   2 (B) 2 2    i j k   (C) 2 2    i j k   (D) 2 2    i j k   Q.24* Lowest degree ofa polynomial with rational coefficients if one of its rootis, i  2 is (A) 2 (B) 4 (C) 6 (D) 8 Q.25 Aplane vector has components 3 & 4 w.r.t. the rectangular cartesian system. This system is rotated through an angle  4 in anticlockwisesense.Then w.r.t. thenew system the vectorhascomponents : (A) 4, 3 (B) 7 2 1 2 , (C) 1 2 7 2 , (D) none Q.26 Let  a = xi + 12j  k ;  b = 2i + 2xj + k &  c = i + k . If the ordered set      b c a is left handed , then: (A) x  (2, ) (B) x  (,  3) (C) x  ( 3, 2) (D) x  { 3, 2} Q.27 If k̂ j ˆ î cos    , k̂ j ˆ cos î    & k̂ cos j ˆ î    (      2 n ) are coplanar then the value of            2 ec cos 2 ec cos 2 ec cos 2 2 2 equal to (A) 1 (B)  (C) 3 (D) none of these Q.28* The straight line (1 + 2i)z + (2i – 1) z = 10i on the complex plane, has intercept onthe imaginaryaxis equal to (A) 5 (B) 2 5 (C) – 2 5 (D) – 5
  4. Q.29 The perpendicular distance of an angular point of a cube of edge 'a' from the diagonal which does not pass thatangularpoint, is (A) a 3 (B) a 2 (C) a 2 3 (D) a 3 2 Q.30 Which one of the following does not hold for the vector V  =      a x b x a ? (A) perpendicular to  a (B) perpendicular to  b (C) coplanar with  a &  b (D) perpendicular to  a x  b . Q.31* Let z1,z2 &z3 bethecomplex numbersrepresentingthevertices ofatriangleABC respectively.If P is a point representing the complex number z0 satisfying; a(z1 z0) + b(z2 z0) + c(z3 z0) = 0, then w.r.t. the triangleABC, the point Pis its : (A) centroid (B) orthocentre (C) circumcentre (D) incentre Q.32 Given the position vectors of the vertices of a triangleABC, ) a ( A  ; ) b ( B ; ) c ( C .Avector r  is parallel to the altitude drawn from the vertexA, making an obtuse angle with the positiveY-axis. If | r |  = 34 2 ; k̂ 3 j ˆ î 2 a     ; k̂ 4 j ˆ 2 î b     ; k̂ 2 j ˆ î 3 c     , then r  is (A) k̂ 6 j ˆ 8 î 6    (B) k̂ 6 j ˆ 8 î 6   (C) k̂ 6 j ˆ 8 î 6    (D) k̂ 6 j ˆ 8 î 6   Q.33* The complex number, z = 5 + i has anargument which is nearlyequal to : (A) /32 (B) /16 (C) /12 (D) /8 Q.34 If k̂ j ˆ î a     , k̂ j ˆ î b     , k̂ j ˆ 2 î c     , then the value of c . c b . c a . c c . b b . b a . b c . a b . a a . a                   equal to (A) 2 (B) 4 (C) 16 (D) 64 Q.35* If the equation x2 + ax + b = 0 where a, b  R has a non real root whose cube is 343 then (7a + b) has the value (A) 98 (B) – 49 (C) – 98 (D) 49 Direction for Q.36 to Q.40. Let k̂ 3 j ˆ 2 î A     and k̂ 5 j ˆ 4 î 3 B     Q.36 The value of the scalar 2 2 ) B · A ( | B A |       is equal to (A) 8 (B) 10 7 (C) 7 10 (D) 64 Q.37 Equation of a line passing through the point with position vector j ˆ 3 î 2  and orthogonal to the plane containingthevectors A  and B  ,is (A) k̂ j ˆ ) 3 2 ( î ) 2 ( r          (B) k̂ j ˆ ) 3 2 ( î ) 2 ( r          (C) k̂ j ˆ ) 3 2 ( î r         (D) none
  5. Q.38 Equation of a plane containing the point with position vector ) k̂ j ˆ î (   and parallel to the vectors A  and B  , is (A) x + 2y + z = 0 (B) x – 2y – z – 2 = 0 (C) x – 2y + z – 4 = 0 (D) 2x + y + z – 1 = 0 Q.39 Volume of the tetrahedron whose 3 coterminous edges are the vectors A  , B  and k̂ 4 j ˆ î 2 C     is (A) 1 (B) 4/3 (C) 8/3 (D) 8 Q.40 Vector component of A  perpendicular to thevector B  is given by (A) 2 B ) B A ( B       (B) 2 B ) B A ( A       (C) 2 A ) B A ( B       (D) 2 A ) B A ( A       Select the correct alternatives : (More than one are correct) Q.41 If a, b, c are different real numbers and a i b j ck      ; bi c j a k      & ci a j bk      are position vectors of threenon-collinear pointsA, B &C then: (A) centroid oftriangleABC is   a b c i j k     3    (B)   i j k   is equallyinclinedto the threevectors (C)perpendicular from the originto theplane of triangleABC meet at centroid (D)triangleABCis an equilateraltriangle. Q.42 The vectors    a b c , , are of thesame length & pairwise form equal angles. If  a i j     &  b j k     , the pv's of  c can be : (A) (1, 0, 1) (B)         4 3 1 3 4 3 , , (C) 1 3 4 3 1 3 , ,        (D)         1 3 4 3 1 3 , , Q.43* Which of the following locii of z on the complex plane represents a pair of straight lines? (A) Re z2 = 0 (B) Im z2 = 0 (C) z + z = 0 (D) z1= zi Q.44 If    a b c , , &  d are linearlyindependent set of vectors &K a K b K c K d 1 2 3 4        = 0 then : (A) K1 + K2 + K3 + K4 = 0 (B) K1 + K3 = K2 + K4 = 0 (C) K1 + K4 = K2 + K3 = 0 (D) none of these Q.45 If    a b c , , &  d are the pv's of the pointsA, B, C & D respectivelyin three dimensional space & satisfy the relation3 2 2     a b c d    =0, then : (A)A, B, C & D are coplanar (B) the line joiningthe points B& D divides the line joining the pointA& C in the ratio 2 : 1. (C) theline joiningthe pointsA& C divides the line joiningthepoints B & D in the ratio 1: 1 (D) the four vectors    a b c , , &  d arelinearlydependent. Q.46* If z3  i z2  2 i z  2 = 0 then z can be equal to : (A) 1  i (B) i (C) 1 + i (D)  1  i
  6. Q.47 If  a &  b are two non collinear unit vectors &  a ,  b , x  a  y  b form a triangle, then : (A) x =  1 ; y = 1 &   a +  b  = 2 cos   a b            2 (B) x =  1 ; y = 1 & cos   a b        +   a +  b  cos      a a b ,          =  1 (C)   a +  b  =  2 cot   a b            2 cos   a b            2 & x =  1 , y = 1 (D) none Q.48 The lines with vector equations are ;  r1 =         3 6 4 3 2    ~  i j i j k  and  r2 =         2 7 4    ~  i j i j k  are such that : (A) theyare coplanar (B) theydo not intersect (C) theyare skew (D) the angle between them is tan1 (3/7) Q.49* Given a, b,x, yR thenwhich of the followingstatement(s) hold good? (A) (a + ib) (x + iy)–1 = a – ib  x2 + y2 = 1 (B) (1–ix) (1 + ix)–1 = a – ib  a2 + b2 = 1 (C) (a + ib) (a – ib)–1 = x – iy  |x + iy| = 1 (D) (y – ix) (a + ib)–1 = y + ix |a – ib| = 1 Q.50 The acute angle that the vector 2 2    i j k   makes with the plane contained by the two vectors 2 3    i j k   and    i j k   2 is given by: (A) cos–1 1 3       (B) sin–1 1 3       (C) tan–1   2 (D) cot–1   2 Q.51 ThevolumeofarighttriangularprismABCA1B1C1 isequalto3.Ifthepositionvectorsoftheverticesof the baseABC areA(1, 0, 1) ; B(2,0, 0) and C(0, 1, 0) the position vectors of the vertexA1 can be: (A) (2, 2, 2) (B) (0, 2, 0) (C) (0,  2, 2) (D) (0,  2, 0) Q.52* If xr = CiS  2r       for 1  r  n r, n  N then : (A) Limit n   Re xr r n         1 =  1 (B) Limit n   Re xr r n         1 = 0 (C) Limit n   Im xr r n         1 = 1 (D) Limit n   Im xr r n         1 = 0
  7. Q.53 If a line has a vector equation,    r i j i j     2 6 3      then which of the following statements holds good ? (A)the lineis parallel to 2 6   i j  (B) the line passes through thepoint 3 3   i j  (C) the line passes through the point   i j  9 (D)thelineis parallel to xyplane Q.54 If    a b c , , are non-zero, non-collinear vectors such that a vector  p = a b cos     2      a b c and a vector  q = a c cos           a c b then  p +  q is (A) parallel to  a (B) perpendicular to  a (C) coplanarwith   b c & (D) coplanar with a  and c  Q.55* The greatest valueof the modulus ofof the complex number 'z' satisfyingtheequality z z  1 = 1 is (A)   1 5 2 (B) 3 5 2  (C) 3 5 2  (D) 5 1 2  SUBJECTIVE: Q.1 Let j ˆ î 3 a    and ĵ 2 3 î 2 1 b    and b ) 3 q ( a x 2       , b q a p y       . If y x    , then express p as a function of q, say p = f (q), (p  0 & q  0) and find the intervals of monotonicity of f (q). Q.2 Using onlythe limit theorems 1 x x n Lim 1 x   l = 1 and x 1 e Lim x 0 x   = 1. Evaluate 1 x x n x x Lim x 1 x     l . Q.3 The three vectors  a i j k    4 2    ;  b i j k    2   and  c i k   2  are all drawn from the point with p.v. j ˆ î . Findthe equationof theplanecontainingtheir endpoint in scalar dot product form. Q.4 (cos cos ) 2 1 2 1 2 2 n n x x dx     z   where n N  Q.5 Let points P, Q & R have position vectors, 1 r  = 3i  2j  k; 2 r  = i + 3j + 4k & 3 r  = 2i+j2k respectively, relative toan origin O. Find the distance of Pfrom the plane OQR. Q.6 Evaluate:     3 1 dx ) 3 x )( 2 x )( 1 x ( Q.7 Given that vectors    A B C , , form a triangle such that    A B C   , find a,b,c,d such that the area of the triangle is5 6 where  A = ai + bj + ck;  B= di + 3j + 4k &  C =3i + j  2k.
  8. Q.8                       1 n 0 k n 1 k n k n dx x n 1 k n k x n Lim Q.9 Find the distance of the point P(i + j + k) from the plane L which passes through the three points A(2i + j + k), B(i +2j + k), C(i+ j + 2k).Also find thepvofthefoot of the perpendicularfrom Pon the plane L. Q.10 Evaluate: (a)   dx x cos x sin x cos x sin 3 4 4 , x         2 , 0 ; (b)   dx x cos x sin x cos x sin 3 4 4 Q.11 Findtheequationofthestraightlinewhichpassesthroughthepointwithpositionvector a  ,meetstheline c t b r      and is parallel to the plane 1 n . r    . Q.12 Integrate:   x sin x cos dx 3 3 . Q.13 Findtheequationofthelinepassingthroughthepoint(1, 4,3)whichis perpendiculartobothofthelines 2 1 x  = 1 3 y  = 4 2 z  and 3 2 x  = 2 4 y  = 2 1 z   Also find all points on this linethe square of whose distance from (1, 4,3) is 357. Q.14 1 n n 2 2 n 2 n 1 n n Lim               Q.15 If z-axis be vertical, find the equation of the line of greatest slope through the point (2, –1, 0) on the plane 2x + 3y – 4z = 1. Q.16 Let I =    2 0 dx x sin b x cos a x cos and J =    2 0 dx x sin b x cos a x sin , where a > 0 and b > 0. Compute the values of Iand J.
  9. ANSWER KEY Select the correct alternative : (Only one is correct) Q.1 B Q.2 B Q.3 C Q.4 A Q.5 D Q.6 A Q.7 B Q.8 A Q.9 A Q.10 B Q.11 A Q.12 A Q.13 A Q.14 B Q.15 B Q.16 D Q.17 D Q.18 D Q.19 A Q.20 D Q.21 B Q.22 A Q.23 D Q.24 B Q.25 B Q.26 C Q.27 B Q.28 A Q.29 D Q.30 B Q.31 D Q.32 A Q.33 B Q.34 C Q.35 A Q.36 C Q.37 A Q.38 C Q.39 B Q.40 A Select the correct alternatives : (More than one are correct) Q.41 A,B,C,D Q.42 A,D Q.43 A,B Q.44 A,B,C Q.45 A,C,D Q.46 B,C,D Q.47 A,B Q.48 B,C,D Q . 49 A,B,C,D Q.50 B,D Q.51 A,D Q . 52 A,D Q.53 B,C,D Q.54 B,C Q.55 B,D SUBJECTIVE: Q.1 p = ,decreasing in q  (–1, 1), q  0 Q.2 – 2 Q.3 = 3 Q.4 Q.5 3 units Q.6 1/2 Q.7 (8, 4, 2, 11) or (8, 4, 2, 5) Q.8 Q.9 Q.10 (a) C – where t = cot 2 x; (b) + C, where t = tan 2 x Q.11 Q.12 2 [tan –1 (sin x + cos x) + ln + C Q.13 , ; (–9, 20, 4) ; (11, –12, 2) Q.14 e –1 Q.15 Q.16 ,
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