Title XI - maths - chapter 12 - 3D Geometry-LEVEL-VI(11.03.2015)-(296-320).pdf ✅

JEE ADVANCED - VOL - IV Narayana Junior Colleges 296 Narayana Junior Colleges THREE DIMENSIONAL GEOMETRY LEVEL-VI SINGLE ANSWER QUESTIONS 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 2. Consider the three points P, Q, R whose coordinates are respectively (2, 5, –4), (1, 4, –3), (4, 7, –6) then the ratio in which the point Q divides PR. (A)1 : 3 (B)1 : 2 (C)–1 : 3 (D)–1 : 2 3. Let r a       and r b m       be two lines in space where ˆ ˆ ˆ a 5i j 2k ˆ    , ˆ ˆ ˆ b i 7 j 8k      , ˆ ˆ ˆ     4i j k   and m 2i 5j 7k ˆ ˆ ˆ     then the p.v. of a point which lies on both of these lines is (A)ˆ ˆ ˆ i 2j k   (B) ˆ ˆ ˆ 2i j k   (C)ˆ ˆ ˆ i j 2k   (D) non existent s the lines are skew 4. L1 and L2 are two lines whose vector equations are 1       L : r cos 3 i 2 sin j cos 3 k   ˆ ˆ ˆ             2   L : r ai bj ck ˆ ˆ ˆ      , 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  (D) 2  5. If three lines 1L x y z :   2 : 2 3 y z L x   3 1 1 1 : x y z L a b c      form a triangle of area 6 sq.units, then for the point of intersection a, ,    of L2 and L3 ,  = (A) 2 (B) 4 (C)6 (D) 8 6. Image of the point P with position vector ˆ ˆ ˆ 7i j 2k   in the line whose vector equation is   ˆ ˆ ˆ ˆ ˆ ˆ r 9i 5j 5k i 3j 5k         has the position vector. (A)9,5, 2 (B)9,5, 2  (C)9, 5, 2    (D) none 7. The intercept made by the plane r.n q    on the x-axis is (A) q ˆ i.n (B) ˆ i.n q (C)    ˆ i.n q (D) q n  8. ABC is any triangle and O is any point in the plane of the triangle. AO, BO, CO meet the sides BC, CA, AB in D, E, F respectively. Find OD OE OF AD BE CF   . (A)1 (B)2 (C)–1 (D)–2 9. If from the point P(f,g,h) perpendicular PL, PM be drawn to YZ and ZX planes then the equation of the plane OLM is (A) x y z 0 f g h    (B) x y z 0 f g h    (C) x y z 0 f g h    (D) x y z 0 f g h     10. If the distance from 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 p q p q 1   2    equals (A)4950 (B) 5050 (C)5150 (D) none
JEE ADVANCED - VOL - IV Narayana Junior Colleges 297 Narayana Junior Colleges THREE DIMENSIONAL GEOMETRY 11. Consider the following 3 lines in space 1   L : r 3i j 2k 2i 4j k ˆ ˆ ˆ ˆ ˆ ˆ         2   L : r i j 3k 4i 2j 4k ˆ ˆ ˆ ˆ ˆ ˆ         3   L : r 3i 2j 2k t 2i j 2k ˆ ˆ ˆ ˆ ˆ ˆ        Then which one of the following pair(s) are in the same plane. (A)only L1 L2 (B) only L2 L3 (C)only L3 L1 (D) L1 L2 and L2 L3 12. Position vectors of the four angular points of a tetrahedron ABCD are A 3, 2,1    ; B 3,1,5  ; C 4,0,3   and D 1,0,0  . Acute angle between the plane faces ADC and ABC is (A) 1 5 tan 2        (B) 1 2 cos 5        (C) 1 5 cosec 2        (D) 1 3 cot 2        13. If a plane passing through the point (1,2,3) cuts+ve directions of co-ordinate axes in A, B&C, then the minumum volume of the tetrahedron formed by origin and A,B,C is cubic units a) 9 2 b) 9 c) 18 d) 27 14. A, B,C, D are 4 complanar points and A;B:C:D’ are their projections on any plane. If  is the angle between plane of ABCD and plane of projections then ' ' ' ' Volumeof tetrahedron AB C D Volumeof tetrahedron A BCD  (A) 1 (B) 2 (C) 2 cos  (D) cos  15. Let a point R lies on the plane x y z    3 0 and P be the point 1, 1, 1 . A point Q lies on PR such that 2 2 PQ PR k   0 then the equation of locus of Q is A)         2 2 2 2 4 1 1 1 1 1 x y z k x y z                       B)         2 2 2 2 4 1 1 1 1 1 x y z k x y z                       C)         2 2 2 2 4 1 1 1 1 1 x y z k x y z                       D)         2 2 2 2 1 1 1 1 1 1 1 4 x y z k x y z           16. Let OABC be tetrahedron, Let the mid points of edges OA & OB and OC be 1 1 1 A B C , , respectively while those of edges AB, BC and AC be R, P and Q respectively.If OA is (A) 2 2 QB RC 1 1  (B) 2 2 QA RC 1 1  (C) 2 2 QC RC 1 1  (D) None 17. Let 1 2 3    , , and 4 be the areas of the trianglular faces of tetrahedron and 1 2 3 4 h h h h , , ,& be the corresponding altitudes of the tetrahedron, then the minimum value of   1 . 4 i J i J h       = Given volume of the tetrahedron is 5cubic units. (A) 240 (B) 225 (C) 160 (D) 180 18. A line is drawn from the point P(1,1,1) and Perpendicular to a line with direction ratios (1,1,1) to interset the plane x 2y 3z 4    at Q. The locus of point Q is A) x y 5 z 2 1 2 1     
JEE ADVANCED - VOL - IV Narayana Junior Colleges 298 Narayana Junior Colleges THREE DIMENSIONAL GEOMETRY B) x y 5 z 2 2 1 1      C) x y z   D) x y z 2 3 5   19. Three positive real numbers x,y,z satisfy the equations 2 2 2 2 x 3xy y 25, y z 9      and 2 2 x xz z 16    . Then the value of xy 2yz 3xz   is A) 18 B) 24 C) 30 D) 36 20. Three straight lines mutually perpendicular to each other meet in a point P and one of them intersects the x- axis and another intersects the y- axis, while the third line passes through a fixed point (0,0,c) on the Z- axis. Then the locus of P is A) 2 2 2 x y z 2cx 0     B) 2 2 2 x y z 2cy 0     C) 2 2 2 x y z 2cz 0     D)   2 2 2 x y z 2c x y z 0       21. Perpendiculars are drawn from points on the line 2 1 2 1 3 x y z      to the plane x y z   3 . The feet of perpendiculars lie on the line A) 1 2 5 8 13 x y z      B) 1 2 2 3 5 x y z      C) 1 2 4 3 7 x y z      D) 1 2 2 7 5 x y z      22. Consider the lines 1 2 1 3 4 3 3 : , : 2 1 1 1 1 2 x y z x y z L L          and the planes 1 2 P x y z P x y z :7 2 3, :3 5 6 4.       Let ax by cz d    the equation of the plane passing through the point of intersection of lines L and L 1 2 and perpendicular to planes P and P 1 2 . Match List -I with List - II and select the correct answer using the code given below the lists : List I  List II  P Q R S  a  b c d  1 2 3 4 13 3 1 2 Codes : P Q R S (A) 3 2 4 1 (B) 1 3 4 2 (C) 3 2 1 4 (D) 2 4 1 3 23. The shortest distance from the point (1,2,3) to 2 2 2 x y z xy yz zx 0       is A) 1 2 B) 1 C) 2 D) 1 2 24. A rigid body rotates about an axis through the origin with an angular velocity 10 3 radians/s if   points in the direction of ˆ ˆ ˆ i j k   then the equation to the locus of the points having tangential speed 20 m/sec. is (A) 2 2 2 x y z xy yz zx 1 0        (B) 2 2 2 x y z 2xy 2yz 2zx 1 0        (C) 2 2 2 x y z xy yz zx 2 0        (D) 2 2 2 x y z 2xy 2yz 2zx 2 0        25. A point Q at a distance 3 from the point P(1, 1, 1) lying on the line joining the points A(0, –1, 3) and P has the coordinates (A) (2, 3, –1) (B)(4, 7, –5) (C)(0, –1, 3) (D)(–2, –5, 7) 26. Let PM be the perpendicular from the point P(1, 2, 3) to XY plane. If OP makes an angle q with the positive direction of the z-axis and OM makes an angle f with the positive
JEE ADVANCED - VOL - IV Narayana Junior Colleges 299 Narayana Junior Colleges THREE DIMENSIONAL GEOMETRY direction of x-axis, where O is the origin then (q and f are acute angles) (A) 5 tan 3   (B) 2 sin sin 14    (C)tan 2   (D) 1 cos cos 14    27. If the direction ratios of a line are 1 ,1 , 2     , and it makes an angle 60° with the y-axis then  is (A) 1 3  (B) 2 5  (C) 1 3  (D) 2 5  28. The line x y z , x y z         2 3 0 3 4 0 is parallel to (A) XY plane (B)YZ plane (C) ZX plane (D)Z-axis 29. A variable plane makes with the coordinate planes, a tetrahedron of constant volume 3 64k . Then the locus of the centroid of tetrahedron is the surface (A) 3 xyz k  6 (B) 2 xy yz zx k    6 (C) 2 2 2 2 x y z k    8 (D) 2 2 2 2 x y z k8        30. The angle beetween the line x + 2y + 3z = 0 = 3x + 2y + z and the y-axis is (A) 1 2 sec–1 3 (B)2sec–1 3 (C) 1 2 cos 6        (D) 1 2sec (4)  31. I f p1 , p2 , p3 denote the perpendicular distances of the plane 2x – 3y + 4z + 2 = 0 from the parallel planes, 2x 3y 4z 6 0, 4x 6y 8z 3 0         and 2x 3y 4z 6 0     respectively, then (A) 1 2 3 p 8p p 0    (B) 3 2 p 16p  (C) 2 1 8p p  (D) 1 2 3 p 2p 3p 29.    32. The line whose vector equations are   ˆ ˆ ˆ ˆ ˆ ˆ r 2i 3j 7k 2i pj 5k         and   ˆ ˆ ˆ ˆ ˆ ˆ r i 2 j 3k 3i pj pk         are perpendicular for all values of   and if p is equal to (A) -1 (B)2 (C) 5 (D)6 33. Consider the lines x 2 y 1, z 2 3 2      and x 1 2y 3 z 5 1 3 2      is (A) Angle between two lines 90° (B)the second line passes through 3 1, , 5 2         (C)Angle between two lines 45° (D)Angle between two lines is 30° 34. The equation of the bisector planes of an angle between the planes 2x-3y+6z+8=0 and x-2y+2z+5=0 (A)x+5y+4z+11=0 (B)x-5y-4z+11=0 (C)13x – 23y+32z+59=0 (D) x 5y 4z 11 0     35. Let A  be vector parallel to line of intersection of planes P1 and P2 . Plane P1 is parallel to the vectors ˆ ˆ ˆ ˆ 2 j 3k and 4 j 3k   and that P2 is parallel to ˆ ˆ ˆ ˆ j k and 3i 3 j   , then the angle between vector A  and a given vector ˆ ˆ ˆ 2i j 2k   is (A) 2  (B) 4  (C) 6  (D) 3 4  36. Consider the lines x = y = z and the line 2x + y + z –1 = 0 = 3x + y + 2z – 2 is (A)The shortest distance between the two lines is 1 2 (B)The shortest distance between the two lines is 2 (C) plane containing 2nd line parallel to 1st line is y z   1 0