Untitled Document

Question Bank No: 1

1. Equation of the line passing through (1,1,1) and parallel to the plane 2x+3y+z+5=0 is

a)		$\frac{x - 1}{- 1} = \frac{y - 1}{1} = \frac{z - 1}{- 1}$
b)		$\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{1}$
c)		$\frac{x - 1}{3} = \frac{y - 1}{2} = \frac{z - 1}{1}$
d)		$\frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 1}{1}$

2. The sine of angle between the straight line $\frac{x - 3}{3} = \frac{y - 3}{4} = \frac{z - 4}{5} and the plane .$ 2x - 2y + z =5 is

a)		$\frac{2 \sqrt{3}}{5}$
b)		$\frac{\sqrt{2}}{10}$
c)		$\frac{4}{5 \sqrt{2}}$
d)		$\frac{10}{6 \sqrt{5}}$

3. The angle between two lines $\frac{x + 1}{2} = \frac{y + 4}{2} = \frac{z + 1}{2} is$

a)		$\cos^{- 1} (\frac{1}{9})$
b)		$\cos^{- 1} (\frac{2}{9})$
c)		$\cos^{- 1} (\frac{3}{9})$
d)		$\cos^{- 1} (\frac{4}{9})$

4. The distance of the point (2,3,4) from the plane 3x-6y+2z+11=0 is

a)		9
b)		10
c)		2
d)		1

5. The equation of the plane through(2,3,4) and parallel to the plane x+2y+4z=5 is

a)		x+2y+4z=10
b)		x+2y+4z=3
c)		x+y+2z=2
d)		x+2y+4z=24

6. The equation of a plane which cuts equal intercepts of unit length on the axes is

a)		x+y+z=0
b)		x+y+z=1
c)		x+y-z=1
d)		$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

7. The direction cosines of the normal to the plane x+2y-3z+4=0 are

a)		$\frac{1}{\sqrt{14}}, \frac{- 2}{\sqrt{14}} \frac{- 3}{\sqrt{14}}$
b)		$\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}} \frac{- 3}{\sqrt{14}}$
c)		$\frac{- 1}{\sqrt{14}}, \frac{2}{\sqrt{14}} \frac{3}{\sqrt{14}}$
d)		$\frac{1}{\sqrt{14}}, \frac{- 2}{\sqrt{14}} \frac{- 3}{\sqrt{14}}$

8. The perpendicular distance of the point (2,4,-1) from the line $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{- 9} is$

a)		3
b)		5
c)		7
d)		9

9. The co-ordinate of a point P are (3,12,4) w:r: to the orgin o, then the direction cosines of OP are

a)		3,12,4
b)		$\frac{1}{4}, \frac{1}{3}, \frac{1}{2}$
c)		$\frac{3}{\sqrt{13}}, \frac{1}{\sqrt{13}}, \frac{2}{\sqrt{13}}$
d)		$\frac{3}{13}, \frac{12}{13}, \frac{4}{13}$

10. Direction cosines of the line that makes equal angles with the three axes in a space are

a)		${}^{\pm}{\frac{1}{3}, {}^{\pm}{\frac{1}{3}}, {}^{\pm}{\frac{1}{3}}}$
b)		${}^{\pm}{\frac{6}{7}, {}^{\pm}{\frac{2}{7}}, {}^{\pm}{\frac{3}{7}}}$
c)		${}^{\pm}{\frac{1}{\sqrt{3}}, {}^{\pm}{\frac{1}{\sqrt{3}},} {}^{\pm}{\frac{1}{\sqrt{3}}}}$
d)		$\pm \sqrt{\frac{1}{7}}, \pm \sqrt{\frac{3}{14}}, \pm \sqrt{\frac{1}{14}}$

11. The angle between the line $\frac{x + 1}{3} = \frac{y - 1}{2} = \frac{z - 2}{4}$ and the plane 2x+y-3z+4=0

a)		$\cos^{- 1} (\frac{- 4}{\sqrt{406}})$
b)		$\sin^{- 1} (\frac{- 4}{\sqrt{406}})$
c)		$30^{0}$
d)		none of these

12. A point (x,y,z) moves parallel to x-axis which of the three variables x,y,z remain fixed

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

13. Lines $\bar{r} = \vec{a_{1}}$ + t $\bar{b_{1}}$ and $\bar{r} = \vec{a_{2}}$ + s $\bar{b_{2}}$ are parallel if

a)		$\bar{b_{1}}$ is parallel to $\bar{a_{2}}$ $- \bar{a_{1}}$
b)		$\bar{b_{2}}$ is parallel to $\bar{a_{2}}$ $- \bar{a_{1}}$
c)		$\bar{b_{1}}$ = $λ \bar{b_{2}} for some λ$
d)		none of these

14. The plane x-2y+z-6=0 and the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are related as

a)		parallel to the plane
b)		at right angles to the plane
c)		lies in the plane
d)		meet the plane obliquelly

15. The value of K so that $\frac{x - 3}{- 3} = \frac{y - 2}{2 k} = \frac{z - 3}{2} and \frac{x - 1}{3 k} = \frac{y - 1}{1} = \frac{z - 6}{- 5}$ may be perpendicular given by

a)		-10
b)		$\frac{10}{7}$
c)		$\frac{- 10}{7}$
d)		$\frac{- 7}{10}$

16. The distance between the parallel planes az+by+cz+ $d_{1} = 0$ and az+by+cz+ $d_{2} = 0 is$

a)		$\frac{\| d_{2} - d_{1} \|}{\sqrt{a^{2} + b^{2}} + c^{2}}$
b)		$\frac{\| d_{2} + d_{1} \|}{\sqrt{a^{2} + b^{2}} + c^{2}}$
c)		$\frac{d_{1}}{\sqrt{a^{2} + b^{2}} + c^{2}}$
d)		none of these

17. The line $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ is parallel to the plane

a)		2x+y-2z=0
b)		3x+4y+5z=7
c)		x+y+z=z
d)		2x+3y+4z=0

18. The lines $\frac{x}{1} = \frac{y}{2} = \frac{z}{3} and \frac{x - 1}{- 2} = \frac{y - 2}{- 4} = \frac{z - 3}{- 6} are$

a)		parallel
b)		intersecting
c)		skew
d)		coincidents

19. The points(5,-4,2), (4,-3,1) are vertices of a

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

20. The lines $\frac{x - 1}{2} = \frac{y - 2}{4} = \frac{z - 3}{7}$ and $\frac{x - 1}{4} = \frac{y - 2}{5} = \frac{z - 3}{7} are$

a)		Parallel
b)		intersecting
c)		skew lines
d)		perpendicular

21. The equations of the line through the point(2,3,-5) and equally inclined to the axis are

a)		x-2=y-3=z=5
b)		$\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{- 5}$
c)		$\frac{x}{2} = \frac{y}{3} = \frac{z}{- 5}$
d)		none of these

22. The lines $l_{1}$ and $l_{2}$ intersect the shortest distance between them is

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

23. If <a, b, c> and < $a^{1}, b^{1}, c^{1} >$ are the direction ratios of two perpendicular lines then

a)		$\frac{a}{a^{1}} = \frac{b}{b^{1}} = \frac{c}{c^{1}}$
b)		${aa}^{1} + b b^{1} + {cc}^{1} = 0$
c)		${aa}^{1} + b b^{1} + {cc}^{1} = 1$
d)		a+b+c= $a^{1} + b^{1} + c^{1}$

24. The angle between the diagonals of a cube are

a)		$\frac{π}{3}$
b)		$\frac{π}{4}$
c)		$\cos^{- 1} (\frac{1}{3})$
d)		$\cos^{- 1} (\frac{1}{\sqrt{3}})$

25. In three dimensional space,the equation 3y+4z=0 represents

a)		a plane containing x-axis
b)		a plane containing y-axis
c)		a plane containing z-axis
d)		a line with direction numbers(0,3,4)

26. The distance of the plane 2x-3y+6z+14=o from the orgin is

a)		14
b)		2
c)		-2
d)		11

27. A line makes angles $\propto, β, ϒ, δ with the four diagonal of a cube then$ $\cos^{2} \propto {+ \cos}^{2} β + \cos^{2} ϒ + \cos^{2} δ has the value .$

a)		0
b)		2
c)		$\frac{4}{3}$
d)		$\frac{5}{3}$

28. The direction cosines of the line joining the points(4,3,-5) and (-2,1-8)are

a)		<2, 4, -13>
b)		<6, 2, 3>
c)		< $\frac{6}{7}, \frac{2}{7}, \frac{3}{7} >$
d)		none of these

29. The shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5} is$

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

30. The line $\frac{x + 3}{3} = \frac{y - 2}{- 2} = \frac{z + 1}{1}$ and the plane 4x+ 5y+3z-5=0 intersect at

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

31. The distance of the point (x,y,z) from y axis is

a)		3
b)		5
c)		$\sqrt{34}$
d)		$\sqrt{41}$

32. If $\propto, β, ϒ$ are the angels which a half ray makes with +ve directions of the axes then sin ${}^{2}{\propto + \sin^{2} β + \sin^{2} ϒ}$ is equal to

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

33. If the direction cosines of a line are $(\frac{1}{c}, \frac{1}{c}, \frac{1}{c})$ then

a)		c > 0
b)		o < c < 1
c)		c = $\pm \sqrt{3}$
d)		c > 2

34. The points (5,2,4), (6,-1,2) and (8,-7,k) are collinear if K is equal to

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

35. A st:line which makes an angle of $60^{0}$ with each of y and z axis, inclines with x-axis at an angle.

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

36. Which of the following triiplets gives the direction cosines of a line

a)		<1,1,1>
b)		<1,-1,1>
c)		<1,-1,-1>
d)		< $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ >

37. The direction cisines of x-axis are

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

38. The projection of a line segment on the co-ordinate axes are 12,4,3 respectively the length of the line segment is

a)		19
b)		16
c)		13
d)		15

39. The angle between the planes 2x-y+z=6 and x+y+2z=7 is

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

40. The ratio in which the line joining (2,4,5), (3,5,-9) is divided by the yz plane is

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

41. The straight lines x + y = 0, 3x + y - 4 = 0 and x + 3y - 4 = 0 from a triangle which is:

a)		isosceles
b)		equilateral
c)		right angled
d)		none of these

42. The triangle whose vertices are (0, 0), (3.1, 0) and (0, 4.5) is

a)		equilateral
b)		rt. Angled
c)		isosceles
d)		acute angled

43. Slope of any line parallel to y-axis is

a)		1
b)		-1
c)		0
d)		not defined

44. Two points (a, 0) and (0, b) are joined by a straight line. Another point on this line is

a)		(3a, -2b)
b)		( $a^{2}$ , ab)
c)		(-3a, 2b)
d)		(a, b)

45. The point which divides the join of (1,2) and (3,4) externally in the ratio 1:1

a)		lies in the $1^{st}$ quadrant
b)		lies in the $2^{nd}$ quadrant
c)		lies in the $3^{rd}$ quadrant
d)		cannot be found

46. The number of points on x-axis which are at a distance c units (c $<$ 3) from (2, 3) is

a)		1
b)		2
c)		0
d)		3

47. The line passing through (1, 1) and parallel to the line 2x - 3y + 5 = 0 is

a)		3x + 2y = 5
b)		2x - 3y + 1 = 0
c)		3x -2y = 1
d)		2x + 3y = 5

48. The medians AD and BE of a triangle with vertices A(0, b), B(0, 0) and C (a, 0) are perpendicular to each other if

a)		b = $\sqrt{2 a}$
b)		a = $\sqrt{- 2 b}$
c)		b = $\sqrt{- 2 a}$ v-2a
d)		a = $\pm \sqrt{2 b}$

49. The perpendicular distance of the origin from the line 3x + 4y + 1 = 0 is

a)		-1
b)		1
c)		-1/5
d)		1/5

50. The equation y - y1 = m (x - x1) for different values of m and (x1, y1) fixed, represents

a)		A family of parallel lines
b)		A single line
c)		A family of lines which not concurrent
d)		A family of concurrent lines