Nội dung text 3D GEOMETRY.pdf
CHAPTER 11 THREE DIMENSIONAL GEOMETRY Exercise 1: NCERT Based Topic-wise MCQs 11.1 & 11.2 INTRODUCTION, DIRECTION COSINES & DIRECTION RATIOS OF A LINE 1. If two straight lines whose direction cosines are given by the relations l + m − n = 0,3l 2 + m2 + cnl = 0 are parallel, then the positive value of c is: NCERT Page-465/N-380 (a) 6 (b) 4 (c) 3 (d) 2 2. Under what condition do ⟨ 1 √2 , 1 2 , k⟩ represent direction cosines of a line? NCERT Page-465/N-379 (a) k = 1 2 (b) k = − 1 2 (c) k = ± 1 2 (d) k can take any value 3. A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2 β = 3sin2 θ, then cos2 θ equals NCERT Page-466/N-379 (a) 2 5 (b) 1 5 (c) 3 5 (d) 2 3 4. If the direction cosines of a line are ( 1 c , 1 c , 1 c ) then (a) 0 < c < 1 (b) c > 2 (c) c > 0 (d) c = ±√3 5. The direction cosines l, m, n, of one of the two lines connected by the relations NCERT Page-465/N-380 l − 5m + 3n = 0,7l 2 + 5m2 − 3n 2 = 0 are (a) 1 √14 , 2 √14 , 3 √14 (b) −1 √14 , 2 √14 , 3 √14 (c) 1 √14 , −2 √14 , 3 √14 (d) 1 √14 , 2 √14 , −3 √14
6. The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is : NCERT/ Page-466/N-379 (a) π/3 (b) π/4 (c) π/2 (d) 0 7. Direction ratios of two lines are a, b, c and 1 bc , 1 ca , 1 ab , The lines are (a) Mutually perpendicular (c) Coincident (b) Parallel (d) None of these 8. Three lines with direction ratios ⟨1,1,2⟩,⟨√3 − 1, −√3 − 1,4⟩ and ⟨−√3 − 1, √3 − 1,4⟩ formNCERT/ Page-466/N-380 (a) a right angled triangle (c) an equilateral triangle (b) a scalene triangle (d) None of these 9. A = [ l1 m1 n1 l2 m2 n2 l3 m3 n3 ] and B = [ p1 q1 r1 p2 q2 r2 p3 q3 r3 ], where pi , qi , ri are the co-factors of the elements ll , mi , ni for i = 1,2,3. If (l1, m1, n1 ), (l2, m2, n2 ) and (l3, m3, n3 ) are the direction cosines of three mutually perpendicular lines then (p1, q1, r1 ), (p2, q2, r2 ) and (p3, q3, r3 ) are NCERT Page-466/N-378 (a) the direction cosines of three mutually perpendicular lines (b) the direction ratios of three mutually perpendicular lines which are not direction cosines (c) the direction cosines of three lines which need not be perpendicular (d) the direction ratios but not the direction cosines 10. A line makes angles of 45∘ and 60∘ with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is. NCERT Page-466/N-378 (a) 30∘ or 60∘ (b) 60∘ or 90∘ (c) 90∘ or 120∘ (d) 60∘ or 120∘ 11. The projections of the segment PQ on the co-ordinate axes are −9,12, −8 respectively. The direction cosines of the line PQ are NCERT Page-467/N-379 (a) < −9 √17 , 12 √17 , −8 √17 > (b) ⟨−9,12, −8⟩ (c) < − 9 289 , 12 289 , −8 289 > (d) < − 9 17 , 12 17 , −8 17 > 12. The projections of a vector on the three coordinate axis are 6, −3,2 respectively. The direction cosines of the vector are NCERT Page-465/N-379 (a) 6 5 , −3 5 , 2 5 (b) 6 7 , −3 7 , 2 7 (c) −6 7 , −3 7 , 2 7 (d) 6, −3,2
11.3 EQUATION OF A LINE IN SPACE 13. If (2,3,9), (5,2,1), (1, λ, 8) and (λ, 2,3) are coplanar, then the product of all possible values of λ is : (a) 21 2 NCERT Page-487/N-382 (b) 59 8 (c) 57 8 (d) 95 8 14. Let the lines x−1 λ = y−2 1 = z−3 2 and x+26 −2 = y+18 3 = z−28 λ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P? NCERT Page-488/N-382 (a) (0, −2, −2) (b) (−5,0, −1) (c) (3, −1,0) (d) (0,4,5). 15. The points A(1,2,3), B(−1, −2, −3) and C(2,3,2) are three vertices of a parallelogram ABCD. The equation of CD is NCERT Page-470/N-382 (a) x 1 = y 2 = z 2 (b) x+2 1 = y+3 2 = z−2 2 (c) x 2 = y 3 = z 2 (d) x−2 1 = y−3 2 = z−2 2 16. The vector equation of the symmetrical form of equation of straight line x−5 3 = y+4 7 = z−6 2 is NCERT Page-470/N-382 (a) r⃗ = (3iˆ + 7ˆj + 2kˆ ) + μ(5iˆ + 4j − 6kˆ ) (b) r⃗ = (5iˆ + 4ˆj − 6kˆ ) + μ(3iˆ + 7j + 2kˆ ) (c) r⃗ = (5iˆ − 4ˆj − 6kˆ ) + μ(3iˆ − 7j − 2kˆ ) (d) r⃗ = (5iˆ − 4jˆ + 6kˆ ) + μ(3iˆ + 7j + 2kˆ ) 17. If vector equation of the line x−2 2 = 2y−5 −3 = z + 1, is r⃗ = (2iˆ + 5 2 jˆ − kˆ ) + λ (2iˆ − 3 2 jˆ + pkˆ ) then p is equal to NCERT Page-470/N-382 (a) 0 (b) 1 (c) 2 (d) 3 18. The lines whose vector equations are r = 2iˆ − 3jˆ + 7kˆ + λ(2iˆ + pjˆ + 5kˆ ) and r = iˆ − 2jˆ + 3kˆ + μ(3iˆ + pjˆ + pkˆ ) are perpendicular for all values of λ and μ if p = NCERT Page-470/N-382 (a) 1 (b) -1 (c) -6 (d) 6 19. If the length of the perpendicular drawn from the point P(a, 4,2), a > 0 on the line x+1 2 = y−3 3 = z−1 −1 is 2√6 units and Q(α1, α2, α3 ) is the image of the point P in this line, then a + ∑i=1 3 αi is equal to: NCERT Page-475/ N-382 (a) 7 (b) 8
(c) 12 (d) 14 20. The length of the perpendicular drawn from the point (3, −1,11) to the line x 2 = y−2 3 = z−3 4 is : (a) √29 NCERT Page-475/N-382 (b) √33 (c) √53 (d) √66 21. The foot of the perpendicular from (2,4, −1) to the line x + 5 = 1 4 (y + 3) = − 1 9 (z − 6) NCERT Page-475/N-382 (a) (−4,1, −3) (c) (−4, −1,3) (b) (4, −1, −3) (d) (−4, −1, −3) 11.4 ANGLE BETWEEN TWO LINES 22. If the two lines l1: x−2 3 = y+1 −2 , z = 2 and l2: x−1 1 = 2y+3 α = z+5 2 perpendicular, then an angle between the lines l2 and l3: 1−x 3 = 2y−1 −4 = z 4 is : NCERT Page-476/N-384 (a) cos−1 ( 29 4 ) (b) sec−1 ( 29 4 ) (c) cos−1 ( 2 29) (d) cos−1 ( 2 √29) 23. Let a⃗ = iˆ + jˆ + 2kˆ , b⃗⃗ = 2iˆ − 3jˆ + kˆ and c⃗ = iˆ − jˆ + kˆ be the three given vectors. Let v⃗, be a vector in the plane of a⃗ and b⃗⃗ whose projection on c⃗ is 2 √3 . If v⃗ ⋅ jˆ = 7, then v⃗ ⋅ (iˆ + kˆ ) is equal to : NCERT Page-491/N-385 (a) 6 (b) 7 (c) 8 (d) 9 24. The lines x = ay + b, z = cy + d and x = a ′y + b ′ , z = c ′y + d ′ are perpendicular if NCERT Page-471 (a) aa′ + bb′ + cc′ + 1 = 0 (b) aa′ + bb′ + 1 = 0 (c) bb′ + cc′ + 1 = 0 (d) aa′ + cc′ + 1 = 0 25. If the equations of two lines l1 and l2 are given by r = a⃗1 + λb⃗⃗ 1 and r = a⃗2 + λb⃗⃗ 2, where λ, μ are parameter then angle θ between them is given by. NCERT Page-472/N-384 (a) cos θ = a⃗⃗1⋅a⃗⃗2 |a⃗⃗1||a⃗⃗2| (b) cos θ = b⃗⃗ 2⋅b⃗⃗ 1 |b⃗⃗ 1||b⃗⃗ 2| (c) cos θ = a⃗⃗1⋅b⃗⃗ 2 |a⃗⃗1||b⃗⃗ 2| (d) cos θ = a⃗⃗2⋅b⃗⃗ 1 |a⃗⃗2|| ⃗b⃗ 1| 26. The equation of two lines through the origin, which intersect the line x−3 2 = y−3 1 = z 1 at angles of π 3 each, Are NCERT Page-471/N-384