Tiêu đề XII - maths - chapter 9 - VECTOR ALGEBRA PART-3- LEVEL-6 (11.03.2015)(160-185).pdf ✅

Narayana Junior Colleges JEE ADVANCED - VOL - I VECTOR ALGEBRA 160 Narayana Junior Colleges LEVEL-VI SINGLE ANSWER QUESTIONS 1. The unit vector ^ ^ ^ c if i j k     bisects the angle between vectors c  and ^ ^ 3 4 i j  is (A) ^ ^ ^ 1 11 10 2 15 i j k          (B) ^ ^ ^ 1 11 10 2 15          i j k (C) ^ ^ ^ 1 11 10 2 15 i j k         (D) ^ ^ ^ 1 11 10 2 15 i j k         2. A man travelling east at 8km/h finds that the wind seems to blow directly from the north. On doubling the speed, he finds that it appears to come from the north-east. The velocity of the wind is (A) 0    45 , 8 2 v (B) 0     45 , 8 2 v (C) 0    90 , 4 2 v (D) 0    60 , 8 2 v 3. ABCD is a parallelogram . If L and M be the middle points of BC and CD, respectively express AL  and AM  in terms of AB  and AD  . If AL AM K AC   .    then the value of K is (A) 1/2 (B) 3/2 (C) 2 (D) 1 4. If the vectors a  and  b are linearly independent satisfying  3 tan 1 3 sec 2 0        a b     , then the most general values of  are (A) , 6 n n Z     (B) 11 2 , 6 n n Z     (C) 2 , 6 n n Z     (D) 11 2 , 6 n n Z     5. If b  is vector whose initial point divides the join of ^ 5i and ^ 5 j in the ratio k:1 and whose terminal point is the origin and b  37  , then k lies in the interval (A) 1 6, 6        (B)   1 , 6 , 6            (C) 0,6 (D) 1 6, 5         6. Let 2 2 x y   3 3 be the equation of an ellipse in the xy-plane. A and B are two points whose position vectors are  3 i and ^ ^   3 2 i k then the position vector of a point P on the ellipse such that   APB  / 4 is (A) ^  j (B) ^ ^ i j         (C) ^ i (D) k  7. If vectors ^ ^ ^ a i j k    3 2 ,  ^ ^ ^ b i j k     3 4  and ^ ^ ^ c i j k    4 2 6  constitute the sides of ABC , then the length of the median bisecting the vector c 
Narayana Junior Colleges VECTOR ALGEBRA JEE ADVANCED - VOL - I Narayana Junior Colleges 161 (A) 2 (B) 14 2 (C) 74 (D) 6 8. Let AC be an arc of a circle, subtending a right angle at the centre O. The point B divides the arc AC in the ratio 1:2 . If OA a and OB b       , then calculate OC  in terms of a and b   (A) 2 3 b a    (B) 2 3 b a    (C)   2 3 b a   (D) 3 2 b a    9. ABC is a triangle and O any point in the same plane. AO, BO and CO meet the sides BC, CA and AB respectively at points D, E and F then OD OE OF AD BE CF   (A) 1 (B) 1/2 (C) 1/3 (D) 2 10. In triangle ABC, 0   A 30 , H is the orthocenter and D is the mid-point of BC. Segment HD is produced to T such that HD=DT. The length AT is equal to A) 2BC B) 3BC C) 4 3 BC D) 1 3 BC 11. The value of  so that the points P, Q, R and S on the sides OA, OB, OC and AB of a regular tetrahedron OABC are coplanar when 1 1 1 , , 3 2 3 OP OQ OR OA OB OC    and OS AB   then A) 1 2   B)   1 C)   0 D) for no value of  12. In triangle ABC, AD and AD' are the bisectors of the angle A meeting BC in D and D' respectively. A' is the mid point of DD '; B ' and C' are the points on CA and AB similarly obtained then A B C ', ', ' forms A) Equilateral triangle B) Isosceles triangle C) Scalene triangle D) Straight line 13. Any plane cuts the sides AB, BC, CD, DA of a quadrilateral in P, Q, R, S respectively, and if 1, AP PB   2 , BQ QC   3 , CR RD   4 DS SA   . Then     1 2 3 4  A) 1 B) 2 C) 3 D) 4 14. Points X and Y are taken on the sides QR and RS respectively of a parallelogram PQRS. So that QX XR  4 and RY YS 4 the line XY cuts the line PR at Z then PZ ZR  A) 1 4 B) 21 5 C) 21 4 D) 16 3 15. Let OABCD is a pentagon in which the side OA and CB are parallel and the sides OD and AB are parallel and OA CB : 2 :1  and OD AB : 1:3  . If the diagonal OC and AD meet at X then OX XC  A) 1 3 B) 1 5 C) 2 3 D) 2 5 16. Let a i j b j k     , . If ‘m’ be slope of tangent to the curve y f x    at x  1 where 3 4 2 3 4 x f x x           4 3 x R          then the vector of magnitude “m” along the bisector of angle between a  and b  is A)   3 3 2 2 i j k   B)   2 2 3 i j k   C)   2 2 3 3 i j k   D) 2 i j k   17. If the resultant of two forces is equal in magnitude to one of the components and perpendicular to it in direction , find the other component using the vector method
Narayana Junior Colleges JEE ADVANCED - VOL - I VECTOR ALGEBRA 162 Narayana Junior Colleges (A) 0 135 (B) 0 120 (C) 0 90 (D) 0 45 18. If the vector tan , 1,2 sin 2 b            and 3 tan , tan , sin 2 c             are orthogonal and a vector a  1,3,sin 2  makes an obtuse angle with the z-axis then the value of  is A)   1   4 1 tan 2 n     B)   1   4 2 tan 2 n     C)   1   4 1 tan 2 n     D)   1   4 2 tan 2 n     19. Image of the point P with position vector ˆ ˆ ˆ 7 2 i j k   in the line whose vector equation is   ˆ ˆ ˆ ˆ ˆ ˆ r i j k i j k       9 5 5 3 5   has the position vector A) ˆ ˆ ˆ    9 5 2 i j k B) ˆ ˆ ˆ 9 5 2 i j k   C) ˆ ˆ 9 5 2 i j k   D) ˆ ˆ ˆ 9 5 2 i j k   20. Foot of the perpendicular from the point P a   to the line r b tc     A)   2 a b c. b c c         B) 2 a c. b c c               C)   2 b c c. a c c                 D) None 21. The reflection of the point P a   with respect to the line r b tc      A)   2 . 2 2 a b c b a c c           B)   2 a b c. b a c c           C)   2 a b c. b a a c                   D) None 22. The lar  distance from A1,4, 2  from the segment BC where B   2,1, 2 , C0, 5,1   A) 3 26 7 B) 26 7 C) 3 7 D) 26 23. If a b c , , are unit vectors such that a b c     2 3 3 2 2 , angle between a and b is  , angle between a and c is  and angle between b and c varies in 2 , 2 3         then the greatest value of 4cos 6cos    A) 2 2 5  B)   2 2 5 C) 2 2 5  D) 42 24. Let P, Q, R and S be the points on the plane with position vectors –2 ˆ ˆ ˆ ˆ ˆ i j, 4i,3i 3j   and ˆ ˆ   3i 2j respectively. The quadrilateral PQRS must be a [IIT JEE 2010] (A) parallelogram, which is neither a rhombus nor a rectangle (B) square (C) rectangle, but not a square (D) rhombus, but not a square 25. Two adjacent sides of a parallelogram ABCD are given by AB 2i 10j 11k ˆ ˆ ˆ     and AD i 2j 2k ˆ ˆ ˆ      . The side AD is rotated by an acute angle a in the plane of the
Narayana Junior Colleges VECTOR ALGEBRA JEE ADVANCED - VOL - I Narayana Junior Colleges 163 parallelogram so that AD becomes 1 AD . If 1 AD makes a right angle with the side AB, then the cosine of the angle a is given by [IIT JEE 2010] (A) 8 9 (B) 17 9 (C) 1 9 (D) 4 5 9 26. The vector A  satisfying the vector equation A B a A B b     ,       and A a. 1    , where a  and b  are given vectors is A)     2 2 b a a a 1 a       B)   2 a b a a      C)     2 2 2 2 a a b b 1 1 a b      D) None of these 27. The point of intersection of the lines r a b a       and r b a b        A) a b    B) a b    C) 2 3 a b    D) 3 2 a b    28. In a regular tetrahedron of side a the distance of any vertex from opposite face is A) 2 3 a B) 1 3 a C) 2 3 a D) 1 2 a 29. the shortest distance between two opposite edges regular tetrahedron of side k is A) 2 k B) 3 k C) 2 k D) 3 k 30. The shortest distance between the lines r i j k i j k       3 15 9 2 7 5           and r i j k i j k         9 2 3            is A) 34 B) 3 C) 4 3 D) 2 3 31. If a plane is parallel to two vectors ˆ ˆ ˆ i j k   and 2ˆi and another plane is parallel to two other vectors ˆ ˆ i j  and ˆ ˆ i k  , then the acute angle between 2ˆ ˆ i j  and the line of intersection of the planes will be A) 1 3 cos 30        B) 1 2 cos 30        C) 1 1 cos 10        D) 1 19 cos 30        32. If the vector x  satisfying x a x b c d     .        be given by     2 . a d c x a a a c a               , then   A) 2 a c. a   B) 2 a b. b   C) 2 c d. c   D) 2 a x. a   33. The lar  distance from the point A a   to the plane r n q .    is A) q a n. n     B) a n. q n     C) q a n. n     D) None 34. The reflection of the point a  in the plane r n q .    A) 2 . 2 q a n a n n                 B) 2 q a n. a n n                 C) 2 a n. a n n               D) None 35. The reflection of the line r a tb      in the plane r n q .    A) 2 q a n. r a n b n                     B) 2 2 . . 2 2 q a n b n r a n t b n n n                                 C) 2 . b n r a b n n           D) None