Untitled Document

Question Bank No: 1

1. If a line makes an angle of $\frac{π}{4}$ with the positive direction of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

a)		$\frac{π}{3}$
b)		$\frac{π}{4}$
c)		$\frac{π}{2}$
d)		$\frac{π}{6}$

2. Let L be the line of intersection of the planes 2x + 3y + z =1 and x+ 3y +2z=2. If L makes an angle $α$ with the positive x-axis, then cos $α =$

a)		$\frac{1}{2}$
b)		1
c)		$\frac{1}{\sqrt{2}}$
d)		$\frac{1}{\sqrt{3}}$

3. The two lines x= ay + b, z= cy + d and x= $a^{'} y + b^{'}$ , z= $c^{'} y + d^{'}$ are perpendicular, if

a)		$\frac{a}{a^{'}} + \frac{c}{c^{'}} = - 1$
b)		$\frac{a}{a^{'}} + \frac{c}{c^{'}} = 1$
c)		$a a^{'} + c c^{'} = - 1$
d)		$a a^{'} + c c^{'} = 1$

4. If the angle $θ$ between the line $\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$ and the plane $2 x - y + \sqrt{λ} z + 4 = 0$ is such that sin $θ = \frac{1}{3},$ the value of $λ is$

a)		$- \frac{3}{5}$
b)		$\frac{5}{3}$
c)		$- \frac{4}{3}$
d)		$\frac{3}{4}$

5. The angle between the lines 2x = 3y=-z and 6x = -y = - 4z is

a)		$0^{\circ}$
b)		$30^{\circ}$
c)		$60^{\circ}$
d)		$9 0^{\circ}$

6. A line is perpendicular to x + 2y + 2z = 0 and passes through (0, 1, 0). The perpendicular distance of this line from the origin is

a)		2
b)		$\sqrt{3}$
c)		$\frac{\sqrt{5}}{2}$
d)		$\frac{\sqrt{5}}{3}$

7. A plane passes through (1, -2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z=4. The distance of the plane from the point (1, 2, 2) is

a)		0
b)		1
c)		$\sqrt{2}$
d)		$2 \sqrt{2}$

8. The equation of the plane containing the line 2x-y+z-3=0, 3x+y+z=5 and at a distance of $\frac{1}{\sqrt{6}}$ from the point (2, 1, -1) is

a)		x + y+ z - 3 =0
b)		2x - y - z -3 = 0
c)		2x + y + z - 3 =0
d)		62x + 29 y + 19 z - 105=0

9. A variable plane at a distance of 1 unit from the origin cuts the coordinate axis at A, B, C. If the centroid D(x, y, z) of triangle ABC satisfies the relation $\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = k$ , then k=

a)		3
b)		1
c)		$\frac{1}{3}$
d)		9

10. If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then k=

a)		$\frac{3}{2}$
b)		$\frac{9}{2}$
c)		$- \frac{2}{9}$
d)		$- \frac{3}{2}$

11. If the line $\frac{x - 4}{1} = \frac{y - 2}{1} = \frac{z - k}{2}$ lies on the plane 2x - 4y +z=7, then k=

a)		0
b)		-7
c)		7
d)		-1

12. A plane which passes through the point (3, 2, 0) and the line $\frac{x - 4}{1} = \frac{y - 7}{5} = \frac{z - 4}{4}$ is

a)		x - y + z=1
b)		x + y +z=5
c)		x+ 2y- z=1
d)		2x - y+z=5

13. The direction ratios of normal to the plane through (1, 0, 0), (0, 1, 0) ahich makes an angle $\frac{π}{4}$ with the plane x+ y= 3 are

a)		1, $\sqrt{2, 1}$
b)		1,1, $\sqrt{2}$
c)		1,1,2
d)		$\sqrt{2},$ 1,1

14. The lines x=ay+b, z=cy+d and x= $a^{'} y + b^{'},$ z= $c^{'} y + d^{'}$ are perpendicular if and only if

a)		$a a^{'} + b b^{'} + c c^{'} + 1 = 0$
b)		$a a^{'} + b b^{'} + c c^{'} = 0$
c)		$(a + a^{'}) (b + b^{'}) (c + c^{'}) = 0$
d)		$a a^{'} + c c^{'} + 1 = 0$

15. The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{1}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar if k=

a)		0 or -1
b)		1 or -1
c)		0 or -3
d)		3 or -3

16. A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1), B (2, 1, 3), C (- 1, 1, 2) .$ The angle between the faces OAB and ABC is

a)		$\cos^{1} (\frac{19}{35})$
b)		$\cos^{1} (\frac{17}{31})$
c)		$30^{\circ}$
d)		$90^{\circ}$

17. If the lines x=1+s, y= $- 3 - λ s, z = 1 + λ s$ and x= $\frac{t}{2}$ , y=1+t, z=2 $-$ t are coplanar, then $λ =$

a)		$-$ 2
b)		$-$ 1
c)		$- \frac{1}{2}$
d)		0

18. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x= y+ a=z and s+a= 2y = 2z. The coordinates of each of the point of intersection are

a)		(3a, 3a, 3a), (a, a, a)
b)		(3a, 2a, 3a), (a, a, a)
c)		(3a, 2a, 3a), (a, a, 2a)
d)		(2a, 3a, 2a), (2a, a, a)

19. The distance between parallel planes 2x + y + 2z =8 and 4x + 2y + 4z + 5=0 is

a)		$\frac{3}{2}$
b)		$\frac{5}{2}$
c)		$\frac{7}{2}$
d)		$\frac{9}{2}$

20. A tetrahedron of volume V = 5 has three of its vertices at the points. A (2, 1, -1), B (3, 0, 1) and C (2, -1, 1). The fourth vertex D lies on the y axis. Then D is the point

a)		(0, 8, 0)
b)		(0, -7, 0)
c)		(0, 8, 0) or (0, -7, 0)
d)		none of these

21. The equation of the XOY plane is

a)		x = 0
b)		y = 0
c)		z = 0
d)		z = c, c ≠ 0

22. The direction cosines of x-axis are

a)		<1, 0, 0>
b)		<0, 0, 1>
c)		<0, 1, 0>
d)		<1, 1, 1>

23. If l, m, n are the d,c.'s of a line, then

a)		$1^{2}$ + $m^{2}$ + $n^{2}$ = 0
b)		$1^{2}$ + $m^{2}$ + $n^{2}$ = 1
c)		l + m + n = 1
d)		None of these

24. A straight line which makes and angle of $60^{o}$ with each of y and z axes, inclines with x-axis at an angle

a)		$45^{o}$
b)		$30^{o}$
c)		$75^{0}$
d)		$60^{o}$

25. If a, b, y are the angles which a half ray makes with the positive directions of the axes, then $\sin^{2}$ a + $\sin^{2}$ b + $\sin^{2}$ y is equal to

a)		1
b)		2
c)		0
d)		-1

26. The locus of $x^{2}$ + $y^{2}$ + $z^{2}$ = 0 is

a)		(0, 0, 0)
b)		a sphere
c)		a circle
d)		none of these

27. The projection of line joining (3, 4, 5) and (4, 6, 3) on the line joining (-1, 2, 4) and (1, 0, 5) is

a)		4/3
b)		2/3
c)		8/3
d)		1/3

28. For every point (x, y, z) on xy-plane

a)		x = 0
b)		y = 0
c)		z = 0
d)		x = 0, z = 0

29. The line x + 3 = y – 2 = z +1 and the plance 4x + 5y + 3z – 5 = 0 intersect at 3 -2 1

a)		(3, 1, -2)
b)		(3, -2, 1)
c)		(2, -1, 3)
d)		(-1, -2, -3)

30. The ratio in which the line joining points, (2, 4, 5) (3, 5, -4) is divided by yz plane, is:

a)		2 : 3
b)		3 : 2
c)		– 2 : 3
d)		4 : 3

31. The locus of the equation

a)		an empty set
b)		a degenerate set
c)		a sphere
d)		a pair of planes

32. Two lines which do not lie in the same plane are called

a)		parallel
b)		intersecting
c)		co-incident
d)		skew

33. skew lines are:

a)		perpendicular lines
b)		parallel lines
c)		coplanar lines
d)		non-coplanar lines

34. The lines l1 and l2 intersect. The shortest distance between them is

a)		positive
b)		negative
c)		zero
d)		infinity

35. The points (5, 0, 2), (2, -6, 0), (4, -9, 6) and (7, -3, 8) are vertices of a

a)		square
b)		rhombus
c)		rectangle
d)		parallelogram

36. The angle between the lines x = 1, y = 2 and y = -1 and z = 0 is

a)		$90^{o}$
b)		$30^{0}$
c)		$60^{o}$
d)		$0^{o}$

37. Volume of a tetrahedron is K (area of one face) (length perpendicular from the opposite vertex upon it) where K is

a)		$\frac{1}{2}$
b)		$\frac{1}{3}$
c)		$\frac{1}{4}$
d)		$\frac{1}{6}$

38. A point (x, y, z) moves parallel to xy plane. Which of the three variables x, y, z remain fixed

a)		z
b)		x
c)		y and z
d)		x and y

39. A plane meets the coordinate axes at A, B, C such that the centre of the triangle is (3, 3, 3). The equation of the plane is

a)		x + y + z = 3
b)		x + y + z = 9
c)		3x + 3y + 3z = 1
d)		9x + 9y + 9z = 1

40. The intercepts of the plane 2x – 3y + 4z = 12 on the co=ordinate axes are given by

a)		2, -3, 4
b)		6, -4, -3
c)		6, -4, 3
d)		3, -2, 1.5

41. The equation $x^{2}$ + $y^{2}$ + $z^{2}$ + 2ux + 2vy + 2wz + d = 0 represents a sphere iff $u^{2}$ + $v^{2}$ + $w^{2}$ - d is

a)		zero or negative
b)		negative
c)		zero
d)		positive