1. The unit vector perpendicular to both the lines. Ans:, (2)
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- Kelley Garrett
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2 1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and i 7j 7k i 7j 5k ) 2) i 7j 5k 7i 7j k 3) 4) i 7j 5k Ans:, (2) 5 3 is
3 Solution: Consider i j k a b1.2.3 ab i 3 4 j 9 2 k 6 1 i 7 j 5k x 1 y 2 z 1 x 2 y 2 z 3 and ab ab i 7j 5k Unit vector ab 5 3
4 2. The straight lines x = 1 + s, y = -3 - s, z = 1 + s t and x, y = 1 + t, z = 2 t with parameters s 2 and t respectively are coplanar then = 1) 0 2) -1 3) 4) -2 Ans: -2, (4) 1 2
5 Solution: x = 1 + s, y = -3 - s, z = 1 + s and y = 1 + t, z = 2 t x 1 = s, y + 3 = - s, z 1 = s, 2x = t, y 1 = t, z - 2 = -t y 3 z 1 x z 2 s s t x 1 = s,,,, y 1 = t, 1 2 x 1 y 3 z 1 x y 1 z 2 and 1 1/ x t 2 t 1 x 1 y 3 z 1 and x 0 y 1 z 2 l1 1 m 1 1 n1 l2 m2 1 n2 1 2
6 The lines are coplanar if x x y y z z l m n l m n = -5 hence = -2
7 3. The distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is ) 2) 3) 4) Ans:, (3) 2
8 Solution: Consider 2x + y + 2z = 8 i.e. 2x + y + 2z 8 = 0 (multiply by 2) i.e. 4x + 2y + 4z 16 = 0 and 4x + 2y + 4z + 5 = 0 Distance between parallel lines
9 4. A tetrahedron has vertices at 0 = (0, 0, 0) A = (1, 2, 1) B = (2, 1, 3) and C = (-1, 1, 2) then the acute angle between the faces OAB and ABC is 1) 2) 3) ) cos cos 31 Ans: cos, (1)
10 Solution: Let a and b be the normals to the faces OAB and ABC respectively And a OA b OB i j k OAOB i j 3k , 1 AB OB OA AC OC OA i j k AB AC i 5 j 3k =(a 1, b 1, c 1 ) =(a 2, b 2, c 2 )
11 cos a 1a2 b b2 c1c 2 a b c a b c cos 35
12 5. The direction cosines of the line passing through the points (2, 3, 5) and (-1, 2, 4) are ),, 2) ,, ,, ) 4) 3 1 1,, ,, Ans:, (1)
13 Solution: Let P=(2,3,5)=(x 1,y 1,z 1 ) and Q=(-1,2,4)=(x 2,y 2,z 2 ) PQ x2 x1 y2 y1 z2 z D. C. S.,,,, PQ PQ PQ
14 6. The x co-ordinate of a point on the line joining the points Q (2, 2, 1) and R (5, 1, -2) is 4. Then its Z co-ordinate. 1) -1 2) 1 3) 3 4) -2 Ans: -1, (1)
15 Solution: Let P (X, Y, Z) divide QR in the ratio : = = 4 2 = 2 Hence Z co-ordinate of P , y, z,,
16 7. The distance of the point whose position vector (2i + j k) from the plane r (i 2 j + 4 k ) = 9 is 13 1) 2) 3) 4) Ans:, (1) 21
17 Solution: a 2i j k n i 2 j 4k and d 9 Required distance a. n d n
18 8. The ratio in which the line joining (2, 4, 5) (3, 5, -4) is divided by yz plane is 1) 2:3 externally 2) 3:2 internally 3) -2 : 3 internally 4) 4 : -3 externally Ans: 2:3 externally, (1)
19 Solution: Let the point P = (x, y, z) divides the line joining A=(2,4,5) and B = (3, 5, -9) in the ratio : 1=m:n B 1A P,, this point lies in YOZ plane x = P divides AB externally in the ratio 2:3
20 9. The angle between the planes and r i 4 j 2k 2 is r 3i j k 1 1) 2) cos ) 4) cos cos cos Ans:, (1) cos
21 Solution: r. n 1 n 3i j k 1 1 r. n 2 n i 4 j 2k 2 2 cos
22 10. The distance between the planes and r. 6i 3 j 9k 13 0 is r. 2i j 3k 4 5 1) 2) 3) 4) Ans:, (3) 3 14
23 Solution: Consider 1 r. 2i j 3k 4 i. e. r. n P and r. 6i 3 j 9k r. 2i j 3 k i. e. r. n P 3 2 Required distance P 4 13/ 3 1 P / 3 25 n
24 11. The planes bx ay = n, cy bz = l, az cx = m intersect in a line if 1) al bm cn 1 2) al bm cn 0 3) 4) al bm cn 1 0 al bm cn 1 Ans: al bm cn 0, (2)
25 Solution: Consider bx ay n, cy bz l and az cx m bcx acy cn acy abz al abz bcx bm bcx acy acy abz abz bcx cn al bm al bm cn 0
26 12. If a plane meets the co-ordinate axes as A, B and C such that the centroid of the triangle is the point (l, r, r 2 ) then the equation of the plane is 1) x + ry + r 2 z = 3r 2 2) r 2 x + ry + z = 3r 2 3) x + ry + r 2 z = 3 4) r 2 x + ry + z = 3 Ans: r 2 x + ry + z = 3r 2, (2)
27 Solution: x y z 1 Let the plane is a b c. This meets the coordinate axes in A = (a, 0, 0), B = (0, b, 0) &C=(0,0,C) a b c a b c r r,, a b c 2 1 r and r a 3 b 3r c 3r x y z 1 multiply by 3r 2 3 3r 3r 2 2 r x ry z 3r Centroid of the triangle 2
28 13. Equation of the plane parallel to the planes x + 2y + 3z 5 = 0, x + 2y + 3z 7 = 0 and equi distant from them is 1) x + 2y + 3z 6 = 0 2) x + 2y + 3z 1 = 0 3) x + 2y + 3z 8 = 0 4) x + 2y + 3z 3 = 0 Ans: x + 2y + 3z 6 = 0, (1)
29 Solution: Any plane parallel to the given planes is x + 2y + 3z + k = 0 k 5 k 7 i. e. k 5 7 k k 5 7 k k 5 7 k k k 7 5 2x12 k 6 The required plane is x + 2y + 3z 6 = 0
30 14. If the plane 2x y + z = 0 is parallel to the line 2x 1 2 y z 1 then the value of a is 2 2 a 1) 4 2) -4 3) 2 4) -2 Ans: -2, (4)
31 Solution: The plane is parallel to the given line is the normal to the plane perpendicular to the given line 2 (2) + (-1) (2) + 1 (a) = a = 0, 2 + a = 0 a = -2
32 15. The foot of the perpendicular from the point P (1, 3, 4) to the plane 2x y+z+3 = 0 is 1) (3, 5, -2) 2) (-3, 5, 2) 3) (3, -5, 2) 4) (-1, 4, 3) Ans: (-1, 4, 3), (4)
33 Solution: The foot of the perpendicular from (x 1, y 1, z 1 ) in a plane ax + by + cz + d = 0 is given by x x1 y y1 z z ax 1 1 by1 cz1 d a b c a b c Required foot of the perpendicular is (-1, 4, 3) x 1 y 3 z x 1 x 3 z x 1 2 y 3 1 z 4 1 x 1 y 4 z 3
34 16. The image of the point (-1, 3, 4) in the plane x 2y = 0 is ),, 4 2) 3) (8, 4, 4) 4) 17 19,, ,, Ans:,, 4, (1) 5 5
35 Solution: The image (x, y, z) of the point (x 1, y 1, z 1 ) in a plane ax + by + cz + d = 0 is x x1 y y1 z z 2 ax 1 1 by1 cz1 d a b c a b c Image of (-1, 3, 4) in the plane x xy = 0 is x y z x 1 y 3 z
36 x 1 14 y 3 14 z 4 14 and x 1 y z x 1 y z 4 Required image x, y, z,, 4
37 17. A straight line makes an angle of 60 0 with each of y and z axes, inclines with x axis at angle 1) ) ) ) 60 0 Ans: 45 0 ; (1)
38 Solution: = 60 0 = 60 0 and cos 2 + cos 2 + cos 2 = 1 cos cos cos 2 = cos cos cos
39 18. The points (5, 2, 4) (6, -1, 2) and (8, -7, k) are collinear if K is equal to 1) -2 2) 2 3) 3 4) -1 Ans: -2, (1)
40 Solution: Let A = (5, 2, 4) B = (6, -1, 2) and C = (8, -7, k) and A, B, C are collinear Direction ratio s of AB = Direction ratio s of BC (6-5, -1-2, 2-4) = (8-6, -7+1, k-2) (1, -3, -2) = (2, -6, k-2) 2 6 k 2 k k 2 4 k 4 2 k 2
41 19. If is the angle given by cos where,, are the angles made by a line with the positive directions of the axes of reference then is equal to 1) /3 2) /6 3) /2 4) / cos cos cos sin sin sin Ans: /3, (1)
42 Solution: cos cos cos cos sin sin sin 1 cos cos
43 20. The shortest distance between the lines x 1 y 2 z and x 2 y 4 z is ) 2) 3) 4) Ans:, (2) 6
44 Solution: Consider x 1 y 2 z and x 2 y 4 z (x 1,y 1,z 1 ) =(1,2,3) and (x 2,y 2,z 2 )=(2,4,5) (a 1,b 1,c 1 )=(2,3,4) and (a 2,b 2,c 2 )=(3,4,5) x2 x1 y2 y1 z2 z a1 b1 c a2 b2 c shortest dis tan ce b1 c2 b2c1 c1a 2 a1c 2 ( a1b 2 a2b1) 6 1
45 21. The plane XOZ divides the join of (1, -1, 5) and (2, 3, 5) in the ratio :1 then is 1) -3 2) -1/3 3) 3 4) 1/3 Ans: 1/3, (4)
46 Solution: Since XOZ plane i.e. y = 0 divide the join of P=(1, -1, 5) and Q=(2, 3, 5) in the ratio :l ,, x, 0, / 3 1
47 22.The projections of segment PQ on the coordinate planes are -9, 12, -8 respectively. Then direction cosines of PQ are 1) ),, , 12, ),, 4) ,, Ans:,,, (4)
48 Solution: Length PQ Directions cosines are ,,
49 23. A line makes angles,,, with the four diagonals of a cube, then cos 2 + cos 2 + cos 2 + cos 2 1) 0 2) 2 3) 4/3 4) 5/3 has the value Ans: 4/3, (3)
50 Solution: cos 2 + cos 2 + cos 2 + cos
51 24. The angle between diagonals of a cube are 1) /3 2) /4 3) cos 4) cos 3 Ans: cos, (3) 1 1 3
52 Solution: Let the sides of the cube of length 1 unit. Direction ratios of the diagonals of the cube are (a 1,b 1, c 1 )=(-1,1,1) and (a 2,b 2,c 2 )=(-1,-1,1)
53 25. The equation of the plane which bisects the line joining (2, 3, 4) and (6, 7, 8) is 1) x + y + z 15 = 0 2) x y + z 15 = 0 3) x y z 15 = 0 4) x + y + z + 15 = 0 Ans: x + y + z 15 = 0, (1)
54 Solution: Midpoint of (2, 3, 4) and (6, 7, 8) is ,, 4, 5, This lies on x + y + z = x + y + z 15 = 0 is equation of the plane
55 26. The lines are x 1 y 2 z 3 x y 2 z 3 and ) Parallel 2) Intersecting 3) Skew 4) at right angles Ans: at right angles, (4)
56 Solution: Consider x 1 y 2 z 3 x y 2 z 3 and The direction ratios of the line are (a 1,b 1, c 1 )=(1,2,3) and (a 2,b 2,c 2 )=(2,2,-2) a 1 a 2 + b 1 b 2 + c 1 c 2 = 1 (2) + 2 (2) + 3 (-2) lines are at right angles = = 0
57 27. The lines are x 1 y 1 z 3 x 2 y 3 z 4 and ) Parallel 2) Coinside 3) Skew 4) Perpendicular Ans: Perpendicular, (4)
58 Solution: Consider x 1 y 1 z 3 x 2 y 3 z 4 and The direction ratios of the line are (a 1,b 1,c 1 )=(2,3,0) and (a 2,b 2,c 2 )=(0,0,1) Therefore a 1 a 2 +b 1 b 2 +c 1 c 2 =2(0)+3(0)+0(1)=0 Hence the two lines are perpendicular to each other.
59 28. The lines and are x 1 y 2 z x 1 y 2 z ) Parallel 2) Intersecting 3) Skew lines 4) Perpendicular Ans: Intersecting, (2)
60 Solution: Since both the lines are passing through (1, 2, 3) The lines are intersecting
61 x 2 y 3 z The line is 1) Parallel to x axis 2) parallel to y-axis 3) Parallel to z-axis 4) lies in a plane parallel to xy plane Ans: lies in a plane parallel to xy plane Ans; (4)
62 Solution: Since the direction cosines of the normal to the xy plane are 0, 0, 1 and direction ratios of the given lines are 1, 2, 0 and 1 (0) + 2(0) + 0 (1) = 0 Normal of xy plane is perpendicular to the line. Hence the given line lies in a plane parallel to xy plane
63 30. If the centroid of a tetrahedron OABC where 0 = (0, 0, 0) A = (a, 2, 3) B = (1, b, 2) and C = (2, 1, c) is (1, 2, -1) then the distance of P (a, b, c) from origin is 1) 2) 3) 4) Ans: 107, (2)
64 Solution: G = (1, 2, -1) = Centroid of the tetrahedron x x x x a , b, c a 3 b 3 c a 3 4 b 3 8 c 5 4 a 1 b 5 c 9 OP=Distance of P from origin P ( a, b, c) 1, 5,
65 x 2 y 3 z The line is parallel to the plane 1) 2x + y 2z = 0 2) 3x + 4y + 5z = 0 3) x + y + z = 2 4) 2x + 3y + 4z = 0 Ans: 2x + y 2z = 0, (1)
66 Solution: Since 3 (2) + 4 (1) + 5 (-2) = = 0 The line is parallel to 2x + y 2z = 0
67 32.If (2, 3, -1) is the foot of the perpendicular from (4, 2, 1) to a plane, then the equation of the plane is 1) 2x y 2z 3 = 0 2) 2x + y 2z 9 = 0 3) 2x y + 2z 5 = 0 4) 2x y + 2z + 1 = 0 Ans: 2x y + 2z + 1 = 0, (4)
68 Solution: The line joining the given points is normal to the plane Direction ratios of the normal are (2, -1, 2) and the point (2, 3, -1) lies in the plane Required plane is 2x y +2z = 2 (2) + 3 (-1) + (-1) (2) = = -1 2x y + 2z + 1 = 0
69 x 1 y 3 z The line meets the plane 3x + 4y + 5z = 5 at the point 1)(5, 15, -14) 2) (3, 4, 5) 3) (1, 3, -2) 4) (3, 10, -12) Ans: (5, 15, -14) ; (1)
70 Solution: Any point on the lines (r-1, 3r-3, -2r-2) 3(r-1) + 4 (3r-3) + 5(-2r-2) = 5 5r = 30 r = 6 P = (5, 15, -14)
71 34. P is the point on the line segment joining the point (3, 2, -1) and (6, 2, -2). If x co-ordinate of P is 5. Then its y co-ordinate is 1) 2 2) 1 3) -1 4) -2 Ans: 2, (1)
72 Solution: Let P divide the line segment A=(3,2,-1) & B=(6, 2, -2) In the ratio : 1 (x, y, z) Gien B1p ,, Given = = 5 3 = 2 and y
73 35. If,, are the angles that a line makes with the positive direction of x, y, z axis respectively, then the direction cosines of a line are 1) sin, sin, sin 2) cos, cos, cos 3) tan, tan, tan 4) cos 2, cos 2, cos 2 Ans: cos, cos, cos, (2)
74 36. The distance of a point P (a, b, c) from x axis is 1) a 2 2 c 2) a 2 2 b 2 2 3) b c 4) b c 2 2 b c 2 2 Ans:, (3)
75 Solution: The distance of P (a, b, c) from Q (a, 0, 0) is PQ a a b 0 c 0 0 b c b c
76 37. The equation of x axis in space are 1) x = 0, y = 0 2) x = 0, z = 0 3) x = 0 4) y = 0, z = 0 Ans: y = 0, z = 0, (4)
77 Solution: Equations of x axis in space are y = 0 and z = 0
78 38. A line makes equal angles with co-ordinate axis, direction cosines of the line are 1) 1, 1, 1 2) 1 1 1,, ),, 4) 1 1 1,, Ans:,,, (2) 3 3 3
79 Solution: Let the line makes angle with x, y and z axis The Direction cosines are cos, cos, cos and cos 2 + cos 2 + cos 2 = 1 3 cos 2 = cos Direction cos ines,,
80 3, and 39. If a line makes angles with x, y and z axis respectively then its direction cosines are 1 1 1) 0,, 2) ,, ) 1,, 4) ,, Ans: 0,,, (1)
81 Solution: 3, and Directions cos ines cos, cos, cos 0,,
82 40. If a line makes angles,, with the direction of the co-ordinate axes then the value of sin 2 + sin 2 + sin 2 is 1) 2 2) -2 3) 1 4) -1 Ans: 2, (1)
83 Solution: We know that cos 2 + cos 2 + cos 2 = 1 sin 2 + sin 2 + sin 2 = 1- cos cos cos 2 = 3 (cos 2 + cos 2 + cos 2 ) = 3-1 = 2
84 41. If a line makes an angle of 4 with each of y and z axis then the angle which it makes with x axis is 2 2 1) 2) 3) 4) Ans:, (1)
85 Solution: 4 4 and cos 2 + cos 2 + cos 2 = 1 cos 2 +(1/2)+(1/2)= 1 cos 2 =1-1=0 cos = 0 = cos 90 0 = 90 0
86 42. Distance of a point (,, ) from y axis is 1) 2) 3) 4) 2 2 Ans: 2 2, (4)
87 Solution: Let P = (,, ) and Q lies on y axis = (0,, 0) PQ
88 43. If the direction cosines of a line are k, k, k then 1)K > 0 2) 0 < K < 1 3) K = 1 4) 1 1 K ork K ork 3 3 Ans:, (4)
89 Solution: l k, m k, n k l m n 1 3k 1 k k 3 3
90 44. The distance of the plane from the origin is 1) 1 2) 7 3) 1/7 4) r. i j k Ans: 1, (1)
91 Solution: r. i j k is of the form r. nˆ d When d is the distance of the plane from origin d = 1
92 45. The sine of the angle between the line x 2 y 3 z and the plane 2x 2y + z = 5 is 1) 2) 3) 4) Ans:, (2)
93 Solution: If is the angle between the given line and the plane Then b 3i 4 j 5k n 2i 2 j k bn sin bn
94 46. The reflection of the point (,, ) in the xy plane is 1) (,, 0) 2) (0, 0, ) 3) (-, -, ) 4) (,, -) Ans: (,, -), (4)
95 Solution: The reflection of (,, ) in xy plane is (,,-)
96 47. The area of the quadrilateral ABCD where A = (0, 4, 1) B = (2, 3, -1) C = (4, 5, 0) and D = (2, 6, 2) is equal to 1) 9 sq units 2) 18 sq. units 3) 27 sq. units 4) 81 sq. units Ans: 9 sq units, (1)
97 Solution: AD = D A = 2i + 2j + k AB B A 2i j 2k i j k AD AB i 4 1 j 4 2 k i 6 j 6k AD AB sq units
98 48. The locus represented by xy + yz = 0 is 1) a pair of perpendicular lines 2) a pair of parallel lines 3) a pair of parallel planes 4) a pair of perpendicular planes Ans: a pair of perpendicular planes Ans; (4)
99 Solution: The given equation is xy + yz = 0 y(x+z)=0 y=0 and (x+z)=0 which are a pair of perpendicular planes.
100 49. The plane 2x 3y + 6z 11 = 0 makes an angle sin -1 () with x axis the value of is equal to 1) 3 2 2) 3) 4) Ans: (3), 2/7
101 Solution: 2 i. 2i 3 j 6k sin sin sin
102 50. The vector equation of the line is 1) (5i 4j + 6k) + (3i + 7j + 2k) 2) (5i + 4j + 6k) + (3i + 7j + 2k) 3) (5i 4j - 6k) + (3i + 7j + 2k) 4) (5i 4j + 6k) + (3i - 7j - 2k) x 5 y 4 z Ans; (1), (5i 4j + 6k) + (3i + 7j + 2k)
103 Solution: a 5i 4 j 6k b 3i 7 j 2k Vector equation is r a b = (5i 4j + 6k) + (3i + 7j +2k)
104 51. The angle between the line and the plane r i j k is r i j k i j k 1) 2) sin sin 3) 4) sin sin Ans: (2), sin
105 Solution: Consider the line and the plane r i j k i j k r. 3i 4 j k 5 0 b 2i j k n 3i 4 j k Let be the angle between the line and the plane b. n sin bn sin
106 52. The angle between the planes and r. i j 4 is 1 5 1) cos 2) 58 cos 2 7 r. 2i 3 j k ) cos 4) cos Ans: cos, (2)
107 Solution: Consider r 2i 3 j k 1 and r i j 4 n1 2i 3j k n2 i j n1 n cos n1 n cos cos
108 53. The equation of the plane passing through the points (2, 1, 0) (5, 0, 1) and (4, 1, 1) is 1) x + y 2z 3 = 0 2) x + y + 2z + 3 = 0 3) x y -2z + 3 = 0 4) x y + 2z 3 = 0 Ans: (1), x + y 2z 3 = 0
109 Solution: Equation of the plane is x 2 y 1 z x 2 y 1 z (x 2) (-1 0) (y-1) (3-2) + z (0+2) = 0 -x + 2 y z = 0 -x y + 2z + 3 = 0 x + y 2z 3 = 0
110 54. The two lines x = ay + b, z = cy + d & x = a y + b and z = c y + d are perpendicular to each other if a a c a c c a c 1) 2) ) aa cc 1 4) aa cc 1 Ans: aa cc 1, (3)
111 Solution: x = ay + b, z = cy + d and x = a y + b, z = c y + d x-b=ay, z-d=cy and x-b =a y, z-d =c y The lines are x b y 0 z d a 1 c The lines are perpendicular if and x b y 0 z d a 1 c aa 1.1 c. c 0 aa cc 1
112 55. The value of k such that x 4 y 2 z k lies on the plane 2x 4y + z = 7 is 1) 7 2) -7 3) 4 4) no real value Ans: (1), 7
113 Solution: Equation the line lies on the plane, The point (4, 2, k) lies on the plane 2x 4y + z = 7 2 (4) 4 (2) + k = k = 7 k = 7
114 56. If the lines and 3 2 intersect then k = x 1 y 1 z x 3 y k z ) 2) 3) 1 4) Ans:, (3)
115 Solution: The lines intersect means the lines are coplanar x x y y z z 31 k l m n 0 i. e l m n k (3-4) (k +1) (2-4) + (-1) (2-3) = 0 2 (-1) + 2 (k + 1) -1 (-1) = k = 0 2k = -1 k 1 2
116 57. The lines and are coplanar if x 2 y 3 z 4 x 1 y 4 z k k 2 1 1) k = 0 or k = -3 2) k = 1 or k = -1 3) k = 0 or k = -4 4) k = 3 or k = -3 Ans;(1), k = 0 or k = -3
117 Solution: The lines are coplanar or intersect k 0 k (1 + 2k) +1 (1 + k 2 ) + (-1) (2 k) = k k k = 0 k 2 + 3k = 0 k (k + 3) = 0 k = 0 or k = k 0 k 2 1
118 58. The equation of the plane through (0, 7, -7) and (-1, 3, -2) and perpendicular to -3x + 2y + z = 7 is 1) x + y + z = 6 2) 2x y + z =4 3) x + y + z = 0 4) x + 2y + 2z = 0 Ans: (3) x + y + z = 0
119 Solution: Required equation is x x y y z z x 0 y 7 z x x y y z z 0 i. e l m n x y 7 z x (4 + 10) (y 7) (1-15) + (z + 7) (2 + 12) = 0 14x + 14y z + 98 = 0 (dividing by 14 we get) x + y + z = 0
120 59. The angle between the planes 2x + 4z = 5 and 2x y = 7 is 1) cos 2) cos ) ) cos Ans: cos, (1) 5 1 2
121 Solution: cos cos 5 1 2
122 60. The equation of the plane passing through the point (1, -2, 5) and perpendicular to the line joining the origin and (3, 1, -1) is 1) 3x + 4y + z = 0 2) 3x + y z + 4 = 0 3) 3x + 4y z + 7 = 0 4) 3x 4y + z 7 = 0 Ans: (2), 3x + y z + 4 = 0
123 Solution: The equation of the plane is a (x x ) + b (y y ) + c (z z ) = 0 Where a, b, c, are direction ratios of the normal (3 0) (x 1) + (1 0) (y + 2) + (-1-0) (z 5) = 0 3 (x-1) + 1 (y+2) -1 (z-5) = 0 3x 3 + y + 2 z + 5 = 0 3x + y z + 4 = 0
124
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0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 2) 8 3) 3 4) 6 2 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation
CLASS IX : CHAPTER - 1 NUMBER SYSTEM
MCQ WORKSHEET-I CLASS IX : CHAPTER - 1 NUMBER SYSTEM 1. Rational number 3 40 is equal to: (a) 0.75 (b) 0.1 (c) 0.01 (d) 0.075. A rational number between 3 and 4 is: (a) 3 (b) 4 3 (c) 7 (d) 7 4 3. A rational