Practice Test


1) If a line is equally inclined to co-ordinate axes, then its each direction angle is of measure


2) Direction cosines of a line, whose direction ratios are -2, 1, 2 are


3) If P is (-1, 1, 2), then direction cosines of line OP are


4) Direction cosines of the line passing through A(2, 3, -1) and B(-3, -4, 2) are


5) If the join of A(4,1, 2), B (5, b, 0) is parallel to the join of C(2, 1, 1), D(3, 3, -1), then b =


6) If l, m, n are direction cosines of a line then vector li + mj+ nk is a


7) Angle between lines whose direction ratios are 1, 2, 1 and 2, -3, 4 is


8) If direction ratios of two lines are 2, -6, -3 and 4, 3, -1, then direction ratios of a line perpendicular to them are


9) Number of lines in space which are equally inclined to three co-ordinate axes are


10) Projection of the line segment joining the points (-1, 0, 3) and (2, 5, 1) on the line whose direction Ratios are 6, 2, 3 is


11) Direction Cosines of a normal to the XOY-plane are


12) A mirror and a source of light are situated at the origin O and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are 1, -1, 1, then direction ratios of the reflected ray are


13) l=m=n=1 are the direction cosines of


14) If P(x, y, z) is a point in space at a distance r from the origin O, then direction Cosines of the line OP


15) Co-ordinates of the foot of the perpendicular from the point (a, b, c) on the Z-axis are


16) In three dimensional space, the equation xy = 0 represents


17) In space, the equation 3x - 4y = 0 represents


18) Equation of the line passing through the point (0, 1, 2), and equally inclined to the co-ordinate axes, are


19) Equations of line joining (2, -3, 1) and (3, -4, -5) are


20) Equations of line joining (1, 2, 3) and (-3, 4, 3) are


21) Equation of the line passing through (a, b, c), and parallel to Z-axis are


22) Equation of the line passing through (a, b,c) and perpendicular to Z-axis are


23) Equations of the line passing through the origin and the point (a, b, c) are


24) Line joining the points (2, -3, 1) and (3, -4, -5) intersects the plane 2x + y + z = 7 in the point


25) Co-ordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y - 4z + 1 = 0 are


26) Measure of angle between the lines 3x + 2y + z - 3 = 0, x + y - 2z - 3 = 0 and 2x - y - z = 0, 7x + 10y - 8z = 15 is


27) Distance of the point (1, -2, 3) from the plane x - y + z = 5, measured parallel to the line whose direction cosines are proportional to 2, 3, -6 is


28) Equations of the line of intersection of the two planes y = b, z = c in the symmetrical form are


29) Equation of plane passing through A (0, -1, 3) and perpendicular to the join of A to B (1, 3, 5) is


30) Equation of plane passing through (2, 6, 1) and perpendicular to the join (-3, 1, 2) and (4, -3, 6) is


31) Equation of plane which bisects the join of A (3, 7, 12) and B (-5, -3, -2) at right angles is


32) Equation of plane passing through P (a, a, 0) and normal to OP is


33) Equation of plane passing through A (a, b, c) and normal to OA is


34) If perpendiculars from the point P (a, b ,c) drawn to Y Z - and Z X - planes meet them in the points L and M respectively, then equation of plane OLM is


35) Equation of plane passing through (0, 0, a) and perpendicualr to two planes x - y - z = 0 and x - 2y = 0 is


36) Equation of plane passing through (-1, -1, 2) and perpendicular to two planes x - 2y + z = 4 and x + 2y - 2z + 4 = 0 is


37) Equation of plane, passing through (-1, 3, 2) and perpendicualr to two planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8 is


38) Equation of plane passing through two points (-1, 2, 0), (1, 1, 2) and perpendicualr to the plane x + 2y + 2z = 4 is


39) Equation of plane passing through two points (-1, 1, 1), (1, -1, 1) and perpendicular to the plane x + 2y + 2z = 5 is


40) Equation of plane passing through (0, -2, 3) and containing the X - axis, is


41) Equation of plane passing through (4, 0, 3) and containing the Y - axis, is


42) Equation of plane passing through (2, -4, 0) and containing the Z - axis, is


43) Equation of plane passing through two points (0, 1, 3), (2, 4, 5), and parallel to X - axis, is


44) Equation of plane passing through two points (2, 3, 1), (4, -5, 3), and parallel to Y - axis, is


45) Equation of plane passing through two points (2, 2, 0), (4, 0, 0), and parallel to Z - axis, is


46) Equation of plane passing through two points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9 is


47) Equation of plane passing through three points (2, 2, -1), (3, 4, 2) and (7, 0, 6) is


48) Equation of plane passing through three points (1, 1, 1), (1, -1, 1) and (-7, -3, -5) is


49) Equation of plane passing through (5, 2, -1), (2, 2, 3) and the origin is


50) Equation of plane passing through (1, 2, 3) and perpendicular to the vector 3i - 4j + k is


51) Equation of plane passing through (2, 0, 5), and parallel to the vectors i - j + k and 3i + 2j - k, is


52) Point P divides the join of A (4, -2, -1) and B (1, 4, 2) internally in the ratio 2 : 1. Equation of plane passing through P and perpendicular to line A B is


53) Equation of plane passing through (1, 0, 1), (3, 1, 2) and parallel to the vector i - j + 2k is


54) Equation of plane passing through (2, 1, 0), and perpendicular to the unit vector k along the Z-axis, is


55) A non-zero vector parallel to the plane 3x + y - z = 2 and perpendicular to the vector i + 2k is


56) A non-zero vector perpendicular to the two planes x + 2y - z + 1 = 0 and 2x - y + z + 9 = 0 is


57) Measure of angle between the planes 5x - 2y + 3z - 7 = 0 and 15x - 6y + 9z + 5 = 0 is


58) Measure of angle between the planes x + y + 2z = 3 and 2x - y + 2z = 5 is


59) Equation of plane through (2, -1, 3) and making equal intercepts on the co-ordinate axes is


60) Equation of plane passing through (2, 1, 3), making equal intercepts on X - axes and Y - axes, and having Z - intercept 4, is


61) Equation of plane passing through (1, -3, 5), whose y - and z - intercepts are both double the x- intercept, is


62) Equation of plane passing through (-4, 0, 4) and making intercepts 4 and 3 on X - axis and Y - axis respectively, is


63) A plane bisects the join of (1, 2, 3) and (3, 4, 5) at right angles. Its interecetps on the co-ordinates axes are


64) Length of the normal to the plane 2x - 2y - z + 3 = 0 is


65) Measure of angle between the lines 2x = 3y = -z and 6x = -y = -4y is


66) Equation of plane passing through (2, 3, 4) and parallel to the plane x + 2y + 4z = 5 is


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


68) The XOY - plane divides the join of the points (a, b, c) and (-a, -b, -c) in the ratio


69) Equation of plane passing through (2, -3, 1) and perpendicular to the join of (3, 4, -1) and (2, -1, 5) is


70) Equation of plane passing thorugh (2, 3, 1) and (4, -5, 3) and parallel to the X - axis is


71) Equation of plane perpedicular to the YZ - plane and passing through (1, -2, 4) and (3, -4, 5) is


72) Equation of plane containing the points A (1, 0, 1) and B (3, 1, 2), and parallel to the line joining the origin to the point C (1, -1, 2) is


73) Equation of plane passing through (1, 1, 1) and the line of intersection of planes x + 2y - z + 1 = 0 and 3x - y - 4z + 3 = 0, is


74) Equation of plane passing through the line of intersection of planes 2x = y and y = 3z, and perpendicular to plane 4x + 5y - 3z = 8, is


75) Equation of plane passing through (-1, 3, 2) and perpendicular to the two planes x + 2y + 2z = 5, and 3x + 3y + 2z = 8, is


76) Equation of plane passing through (1, 1, 0), (-2, 2, -1) and (1, 2, 1) is


77) Measure of angle between the lines x = 1, y = 2 and y = -1, z = 0 is


78) Four points (0, 4, 3), (-1, -5, -3), (-2, -2, 1) and (1, 1, -1) lie in the plane


79) Equation of plane passing through (-2, -2, 2), and containing the line joining the points (1, 1, 1) and (1, -1, 2), is


80) If l, m, n are the direction Cosines of a normal to a plane passing through the point (1, 2, 3), then the equation of the plane is


81) The line x = 1, y = 2 is


82) Locus of the equation xy + yz = 0 is


83) Measure of the acute angle between the plane 5x - 4y + 7z = 12 and the Y - axis is


84) The three planes x + y = 0, y + z = 0 and z + x = 0


85) The direction Ratios of a normal to the plane passing through the points (0, 0, 1), (0, 1, 2) and (1, 2, 3) are


86) If the plane x + 2y + kz = 0 and 2x + y - 2z = 0 are mutually perpendicular, then : k =


87) The direction Cosines of a normal to the plane x + 2y - 3z + 4 = 0 are


88) Projection of the line segment joining A (-1, 0, 3) and B (2, 5, 1) on the line whose d.R.s. are 6, 2, 3 is


89) If the d.R.s. of two lines are 1, -2, -2 and 2, -2, 1, then the measures of the acute angle between them is


90) If the line joining the points P (-2, 1, -8) and Q (a, b, c) is in the direction of the vector 6i + 2j + 3k, then the respective values of a, b, c are


91) The d.C.s. of the line 3x - 1 = 2y - 3 = 4z - 1 are


92) Equation of the plane passing through the points (1, 0, 1), (-1, 1, 1) and (2, 3, 1) is


93) Equation of the plane passing through the origin, and parallel to the vectors i + 3j - k and j + k, is


94) The plane 3x + y - 2z = 4 meets the Y-axis in the point


95) Equation of the plane passing through the points A = (2, 1, -1), B (3, 1, 2), and parallel to X - axis, is


96) If the points (3, 9, x), (-4, 4, 4), (4, 5, 1) and (0, -1, -1) are coplanar, then : x =


97) XY-plane divides the segment joining the points A (2, 4, 5) and B (-4, 3, -2) in the ratio


98) Equation of the plane which bisects the join of the points A (3, -1, 4) and B (-3, 5, 2) at right angles is


99) If the foot of the perpendicular from the origin to a plane is (2, -3, 4), then the equation of the plane is


100) If a plane makes intercepts 2, 2, 1 on X-, Y- and Z- axes respectively, then its perpendicular distance from the origin is


101) A plane is parallel xy-plane, so it is perpendicular to


102) The locus of a point for which y = 0, z = 0 is


103) The point (- 2, - 3, - 4) lies in the


104) L is the foot of the perpendicular drawn from a point P( 3, 4, 5 ) on the xy-plane. The coordinates of point L are


105) The distance of point P ( 3, 4, 5 ) from the yz-plane is


106) What is the length of foot of perpendicular drawn from the point P ( 3, 4, 5 ) on y-axis ?


107) If x-coordinate of a point P of line joining the points Q ( 2, 2, 1 ) and R ( 5, 2, - 2 ) is 4, then the z - coordinate of P is


108) Distance of the point (1, 2, 3) from the coordinate axes are


109) If the sum of the squares of the distance of a point from the three coordinate axes be 36, then its distance from the origin is


110) The coordinates of a point which is equidistant from the point ( 0, 0, 0 ), ( a, 0, 0 ), ( 0, b, 0 ), ( 0, 0, c ) are given by


111) If a parallelopiped is formed by planes drawn through the points ( 5, 8, 10 ) and ( 3, 6, 8) parallel to the coordinate planes, then the length of diagonal of the parallelopiped is


112) Find the coordinates of the point which divides the line segment joining the points ( - 2, 3, 5) and ( 1, - 4, 6) in the ratio 2 : 3 externally.


113) Find the ratio in which the YZ - plane divides the line segment formed by joining the points ( - 2, 4, 7) and ( 3, - 5, 8).


114) Find the length of the medians of the triangle with vertices A ( 0, 0, 6 ), B ( 0, 4, 0 ) and C ( 6, 0, 0 ).


115) Find the coordinates of the points which trisect the line segment joining the points P ( 4, 2, - 6 ) and Q ( 10, - 16, 6 ).


116) The points A ( 5, - 1, 1 ), B ( 7, - 4, 7 ), C (1, - 6, 10 ) and D ( - 1, - 3, 4 ) are vertices of a


117) The points ( 5, - 4, 2 ), ( 4, - 3, 1 ), ( 7, - 6, 4 ) and ( 8, - 7, 5 ) are the vertices of


118) The mid-points of the sides of a triangle are ( 5, 7, 11 ), ( 0, 8, 5 ) and ( 2, 3, - 1). Then, the vertices are


119) The area of the triangle, whose vertices are at the points ( 2, 1 , 1 ), ( 3, 1, 2 ) and ( - 4, 0, 1 ) is


120) If vertices of a triangle are A ( 1, - 1, 2 ), B ( 2, 0, - 1 ) and C ( 0, 2, 1 ), then the area of a triangle is


121) The triangle formed by the points ( 0, 7, 10 ), ( - 1, 6, 6 ), ( - 4, 9, 6) is


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


123) The point A ( 1, - 1, 3 ), B ( 2, - 4, 5 ) and C ( 5, - 13, 11 ) are


124) If orthocentre and circumcentre of a triangle are respectively ( 1, 1, 1 ) and ( 3, 2, 2 ), then the coordinates of its centroid is


125) The incentre of triangle with vertices A ( 1, 1, 2 ), B ( 2, 1, 3 ) and C ( 1, - 1, 3 ) is