Practice Test

Q1) The points (2,3,4), (-1,-2,1) and (5,8,7)are Show Answer

Q2) The points P( 4, 5, 10) , Q(2, 3, 4) , and R(1, 2, -1) are three vertices of a parallelogram PQRS. The coordinates of S are Show Answer

Q3) A line passes through the points (6, -7, -1) and (2, -3, 1). The direction cosines of the line so directed that the angles made by it with the positive direction of x- axis is acute , is Show Answer

Q4) Lines OA and OB are drawn from O with direction cosines proportional to (1, -2, -1) and (3, -2, 3) respectively. The direction ratios of the normal to the pane AOB are Show Answer

Q5) If a line lies in the octant OXYZ and it makes equal angles with the axes , then Show Answer

Q6) The points (5,4,2), (6,-1,2 ) and (8,-7,k) are collinear, if k is equal to Show Answer

Q7) if O is the origin and OP = 3 with direction ratios -1,2,-2, then coordinates of P are Show Answer

Q8) The direction cosines l, m and n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angles between them is Show Answer

Q9) The angle between a line whose direction ratios are in the ratio 2 : 2 : 1 and a line joining ( 3, 1, 4 )to ( 7, 2, 12 ) is Show Answer

Q10) The coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 4z - 12 = 0 is Show Answer

Q11) The equation of the plane that passes through the points ( 1,1,0) (1,2,1) and ( -2,2,-1) is Show Answer

Q12) The equation of the plane through the intersection of the planes 3x- y +2z -4 =0 and x + y + z - 2 = 0 and the point (2, 2, 1) is Show Answer

Q13) The equation of the plane through the line of intersection of the planes x + y + z = 1 and
2x + 3y + 4z = 5 which is perpendicular to the plane
x - y + z = 0 is, Show Answer

Q14) Two planes 7x + 5y + 6z +30 = 0 and 3x - y - 10z +4 = 0 are Show Answer

Q15) The distance of the point ( 3, -2, 1) from the plane 2x - y + 2z + 3 = 0 is, Show Answer

Q16) The equation of the plane which is perpendicular to the plane
5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z - 4 = 0 and 2x + y - z + 5 = 0 is Show Answer

Q17) The coordinates of the point where the line through ( 5, 1, 6 ) and ( 3, 4, 1 ) cross the YZ-plane, is Show Answer

Q18) If O be the origin and the coordinates of P be ( 1, 2, -3), then the equation of the plane passing through P and perpendicular to OP is, Show Answer

Q19) Distance between the two planes 2x + 3y +4z = 4 and 4x +6y +8z =12 is Show Answer

Q20) The planes 2x - y + 4z = 5 and
5x - 25y + 10z = 6 are Show Answer

Q21) The points ( 0, -1, -1 ), (-4, 4, 4 ), ( 4, 5, 1 ) and ( 3, 9, 4 ) are Show Answer

Q22) The reflection of the point ( 2, -1, 3 ) in the plane 3x - 2y - z = 9 is Show Answer

Q23) Let A ( 1, 1, 1 ), B ( 2, 3, 5 ) and C ( -1, 0, 2 ) be three points, then equation of a plane parallel to the plane ABC which is at a distance 3 is Show Answer

Q24) Two systems of rectangular axes have the same origin. If plane cut the intercepts a', b', c' on coordinates axes for Ist system and intercepts a', b', c' on second system, then pick the correct alternatives Show Answer

Q25) The line x + 2y - z - 3 = 0,
x + 3y - z - 4 = 0 is parallel to Show Answer

Q26) There is a point P(a, a, a ) on the line passing through the origin and equally inclined with axes. The equation of plane perpendicular to OP and passing through P cuts the intercepts on axes. the sum of whose reciprocals is Show Answer

Q27) The equation of the plane through the points (1, 2, 3 ), ( -1, 4, 2 ) and ( 3, 1, 1 ) is Show Answer

Q28) If for a plane, the intercepts on the coordinate axes are 8, 4, and 4, then the length of the perpendicular from the origin to the plane is Show Answer

Q29) If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is ( 1, 2, 4 ), then the equation of the plane is Show Answer

Q30) If the distance of the point ( 1, 1, 1 ) from the origin is half its distance from the plane x + y + z + k = 0, then k is equal to Show Answer

Q31) A plane makes intercepts 3 and 4 respectively on z-axis and x-axis. if plane is parallel to y-axis, then its equation is Show Answer

Q32) The equation of the plane through the intersection of the planes x + y + z = 1 and 2x + 3y - z + 4 = 0 and parallel to x-axis is Show Answer

Q33) The equation of the plane which bisects the line joining ( 2, 3, 4 ) and ( 6, 7, 8 ), is Show Answer

Q34) The points A (-1, 3, 0 ), B ( 2, 2, 1 ) and C ( 1, 1, 3 ) determine a plane. The distance from the plane to the point D ( 5, 7, 8 ) is Show Answer

Q35) If the foot of the perpendicular from the origin to a plane is ( a, b, c ), then equation of the plane is Show Answer

Q36) If the coordinates of the points A, B, C and D be ( 1, 2, 3 ), ( 4, 5, 7 ), ( -4, 3, -6 ) and ( 2, 9, 2 ) respectively, then the angle between the lines AB and CD is Show Answer

Q37) The coordinates of the foot of perpendicular drawn from the point A ( 1, 8, 4 ) to the line joining the points B ( 0, -1, 3 ) and C ( 2, -3, -1 ) is Show Answer

Q38) The lines x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular, if Show Answer

Q39) A line with positive direction cosines passes through the point P( 2, -1, 2 ) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals Show Answer

Q40) The coordinates of the point where the line through ( 3, -4, -5 ) and ( 2, -3, 1 ) crosses the plane passing through three points (2, 2, 1 ), (3, 0, 1 ) and ( 4, -1, 0 ) is, Show Answer

Q41) The equation of the line passing through the point ( 3, 0, 1 ) and parallel to the planes x + 2y = 0 and 3y - z = 0, is Show Answer

Q42) The ratio in which 2x + 3y + 5y = 1 divides the line joining the points (1, 0, -3) and (1, -5, 7) is Show Answer

Q43) If A ( 0, 0, 0 ), B ( a, 0, 0 ), C ( 0, b, 0 ), and D ( 0, 0, c ), are the vertices of a tetrahedron, then the volume of tetrahedron is Show Answer

Q44) The x-coordinate of a point on the line joining the points Q ( 2, 2, 1 ) and R ( 5, 1, -2 ) is 4, then its z-coordinate is Show Answer

Q45) The angle between the lines whose direction cosines are given by 2l - m + 2n = 0, lm + mn + nl = 0, is Show Answer

Q46) The angle between any two diagonals of a cube is Show Answer

Q47) A line segment has length 63 and direction ratios are 3, -2, 6. If the line makes an obtuse angle with x-axis, the components of the line vector are Show Answer

Q48) The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is Show Answer

Q49) The area of the triangle whose vertices are ( 1, 0, 1 ), ( 2, -1, 3 ) and ( -1, 2, -1 ), is Show Answer

Q50) A plane is such that the foot of perpendicular drawn from the origin to it is ( 2, -1, 1 ). The distance of ( 1, 2, 3 ) from the plane is
Show Answer

Q51) A and B are two given points. Let C divides AB internally and D divides AB externally in the same ratio. Then, AC, AB and AD are in Show Answer

Q52) The angle between a diagonal of a cube and an edge of the cube intersecting the diagonal is Show Answer

Q53) The equation of the plane through (3, 1, -3) and (1, -2, 2) and parallel to the line with DR's 1, 1, -2 is Show Answer

Q54) A plane passes through the point ( 1, -2, 3 ) and is parallel to the plane 2x - 2y + z = 0. The distance of the point ( -1, 2, 0 ) from the plane is Show Answer

Q55) The projection of the line segment joining ( 2, 5, 6 ) and ( 3, 2, 7 ) on the line with direction ratios 2, 1, -2 is Show Answer

Q56) If the points ( 1, 2, 3 ) and ( 2, -1, 0 ) lie on the opposite sides of the plane 2x + 3y - 2z = k, then Show Answer

Q57) The plane passing through the point ( -2, -2, 2 ) and containing the line joining the points ( 1, -1, 2 ) and ( 1, 1, 1 ) makes intercepts on the coordinate axes and the sum of whose length is Show Answer

Q58) OABC is regular tetrahedron of unit edge. Its volume is Show Answer

Q59) Find the distance of the plane x + 2y - z = 2 from the point ( 2, -1, 3 ) as measured in the direction with DR's ( 2, 2, 1 ). Show Answer

Q60) Find the planes bisecting the actual angle between the planes x - y + 2z + 1 = 0 and 2x + y + z + 2 = 0. Show Answer

Q61) If the orthocentre and centroid of a triangle are ( -3, 5, 1 ) and ( 3, 3, -1 ) respectively, then its circumcentre is Show Answer

Q62) The distance of the point A ( -2, 3, 1 ) from the line PQ through P ( -3, 5, 2 ) which make equal angles with the axes is Show Answer

Q63) The line joining the points ( 1, 1, 2 ) and ( 3, -2, 1 ) meets the plane 3x + 2y + z = 6 at the point Show Answer

Q64) The plane passing through the point ( 5, 1, 2 ) perpendicular to the line 2 ( x - 2 ) = y - 4 = z - 5 will meet the line in the point Show Answer

Q65) The point equidistant from the point ( a, 0, 0 ), ( 0, b, 0 ), ( 0, 0, c ) and ( 0, 0, 0 ) is Show Answer

Q66) The four planes my + nz = 0, nz + lx = 0, lx + my = 0 and lx + my + nz = p from a tetrahedron whose volume is Show Answer

Q67) A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. Thought A, B and C planes are drawn parallel to the coordinate planes, then the locus of their point of intersection is Show Answer

Q68) The angles between the four diagonals of a rectangular paralleled whose edges are a, b and c are Show Answer

Q69) If the direction cosines l, m, n of a line are related by the equations l + m + n = 0, 2mn + 2ml - nl = 0, then the ordered triplet ( l, m, n ) is Show Answer

Q70) Consider the point ( 1, 3, 4 ) and the plane 2x - y + z + 3 = 0 is Show Answer

Q71) A plane passes through a fixed point ( a, b, c ) and cuts the axes in A, B and c. The locus of a point equidistant from origin A, B and C must be Show Answer

Q72) Point of intersection of the lines lies on Show Answer

Q73) Angle between the plane containing both lines and the plane 4x + y + 2z = 0 is equal to Show Answer

Q74) Statement I If centroid and circumcentre of a triangle are known its orthocentre can be found.
Statement II Centroid, orthocentre and circumcentre of a triangle are collinear. Show Answer

Q75) Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is Show Answer

Q76) An equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the origin is Show Answer

Q77) The distance of the point ( 1, -5, 9 ) from the plane x - y + z = 5 measured along a straight line x = y = z is Show Answer

Q78) The direction–ratios of the diagonal of a cube,
which joins the origin to the opposite corner are
(when the 3 concurrent edges of the cube are
coordinate axes) : Show Answer

Q79) Angle between diagonals of a cube is : Show Answer

Q80) . The point equidistant from (a, 0, 0), (0, a, 0), (0,
0, a) and (0, 0, 0) is : Show Answer

Q81) The points (5, 2, 4), (6, –1, 2) and (8, –7, k) are
collinear if k is equal to : Show Answer

Q82) The plane, which passes through the point (3,2,0)
and the line x – 3 = y – 6 = z – 4 is : Show Answer

Q83) Two systems of rectangular axes have the same
origin. If a plane cuts them at distance a, b, c and
a', b', c' from the origin, then : Show Answer

Q84) A tetrahedron has vertices 0 (0, 0, 0), A (1, 2, 1),
B (2, 1,3) and C (–1, 1,2). The angle between the
faces OAB and ABC will be : Show Answer

Q85) The image of the point (–1,3,4) in the plane:
x – 2y = 0 is : Show Answer

Q86) A line with positive direction cosines passes
through the point P (2, – 1, 2) and makes equal
angles with the coordinate axes. The line meets
the plane 2x + y + z = 9 at point Q. The length
of the line segment PQ equals: Show Answer

Q87) The distance of the point P (a,b,c) from the x–axis
is: Show Answer

Q88) The angle between two diagonals of a cube is :
Show Answer

Q89) Equation of line passing through the point (2,3,1)
and parallel to the line of intersection of the
planes x – 2y – z + 5 = 0 and x + y + 3z = 6 is: Show Answer

Q90) Foot of perpendicular drawn from the origin to
the plane 2x – 3y + 4z = 29 is: Show Answer

Q91) A line making angles 45°and 60°with the positive
direction of the axis of x and y makes with the
positive direction of z–axis, an angle of : Show Answer

Q92) If P(3,2,-4) , Q(5,4,-6) and R (9,8,-10) are collinear, then R divides P Q in the ratio Show Answer

Q93) Let A (1,-1, 2) and B (2, 3, -1) be two points. if a point P divides A B internally in the ratio 2 : 3, then the position vector of P is Show Answer

Q94) The locus of a point which moves so that the difference of the squares of its distances from two given
points is constant, is a Show Answer

Q95) The x y - plane divides the line joining the points (-1, 3, 4) and (2, -5, 6) Show Answer

Q96) A parallelepiped is formed by planes drown through the points (2, 3, 5) and (5, 9, 7) parallel to the
coordinate planes. The length of a diagonal of the parallelepiped is Show Answer

Q97) If direction cosines of two lines are proportional to (2, 3, −6) and (3, −4, 5), then acute angle between
them is Show Answer

Q98) The perpendicular distance of the point (6,5,8) from y-axis is Show Answer

Q99) The points A (4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are Show Answer

Q100) The distance of point A (-2,3,1) from the line PQ through P(-3,5,2) which make equal angles with the axes is Show Answer

Q101) If the foot of the perpendicular from (0,0,0) to a plane is (1,2,2) then the equation of the plane is Show Answer

Q102) A plane pass through a fixed point (p,q) and cut the axes in A, B, C. Then the locus of the centre of the sphere OABC is Show Answer

Q103) The ratio in which
yz - plane divides the line segment joining (-3, 4, -2) and (2,1,3) is Show Answer

Q104) 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 Show Answer

Q105) The 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 Show Answer

Q106) If the foot of the perpendicular from the origin to a plane is (a,b,c) , then equation of the plane is Show Answer

Q107) If (2,-1,3) is the foot of the perpendicular drown from the origin to the plane, then the equation of the plane is Show Answer

Q108) If for a plane, the intercepts on the coordinate axes 8,4,4 then the length of the perpendicular from the origin on the plane is Show Answer

Q109) A line passes through two points A (2,-3,-1) and B ( 8,-1,2). The coordinates of a point on this line at a distance of 14 units from A are Show Answer

Q110) The perimeter of the triangle with verities at (1,0,0) . (0,1,0) and (0,0,1) is Show Answer

Q111) A point on x - axis which is equidistance from both the points (1,2,3) and (3,5,-2) is Show Answer

Q112) Let (3,4,-1) and (-1,2,3) are the end points of a diameter of sphere. Then the radius of the sphere is equal to Show Answer

Q113) The points (5,-4,2), 94,-3,1) (7,-6,4) and (8,-7,5) are the vertices of Show Answer

Q114) If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1,2,4) , then the equation of the plane is Show Answer

Q115) A point on XOZ - plane divides the join of (5,-3,-2) and (1,2,-2) at Show Answer

Q116) A plane makes intercepts -6,3,4 upon coordinate axes. Then, length of perpendicular from origin on it is Show Answer

Q117) The equation of the plane which bisects the line joining (2,3,4) and (6,7,8), is Show Answer

Q118) The distance between the points (1,4,5) and (2,2,3) is Show Answer

Q119) The equation of the plane through the points (1,2,3) , (-1,4,2) and (3,1,1) is Show Answer

Q120) The equation of the sphere touching the three coordinate planes is Show Answer

Q121) A mirror and a source of light are situated at the origin O and 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 proportional to 1,-1,1, then direction cosines of the reflected ray are Show Answer

Q122) The points A (-1,3,0), B(2,2,1) & C(1,1,3) determine a plane. The distance from plane to point D (5,7,8) is Show Answer

Q123) Let a plane passes through the point P ( -1,-1,1) and also passes through a line joining the points Q(0,1,1)Q (0,1,1,) and R( 0,0,2). then the distance of the plane from the point (0,0,0) is Show Answer

Q124) The direction cosines of the line passing through P(2,3,-1) and the origin are Show Answer

Q125) The equation to the straight line passing through the points (4,-5,-2) and (-1,5,3) is Show Answer

Q126) If for a plane, intercepts on coordinate axes are 8,4,4, then length of perpendicular from origin to plane is Show Answer

Q127) The equation of the plane passing through the mid-point of the line segment of join of the points P(1,2,3) and Q(3,4,5) and perpendicular to it is Show Answer

Q128) If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1,2,4) then the equation of the plane is Show Answer

Q129) The equation of the plane passing through the midpoint of the line of join of the points (1,2,3) and (3,4,5) and perpendicular to it is Show Answer

Q130) The cosine of the angle A of the triangle with verities A (1,-1,2), B(6,11,2), C(1,2,6) is Show Answer

Q131) The angle between a line whose direction ratios are in the ratio 2 : 2 : 1 and a line joining (3,1,4) to (7,2,12) is Show Answer

Q132) The coordinates of the foot of the perpendicular drawn from the point A(1,0,3) to the join of the points B (4,7,1) and C ( 3,5,3) are Show Answer

Q133) A line passes through the points (6,-7,-1) and (2,-3,1). The direction cosines of the line so directed that the angle made by it with the positive direction of x-axis is acute, are Show Answer

Q134) The equation to the plane through the points (2,3,1) and (4,-5,3) parallel to x-axis is Show Answer

Q135) If P be the point (2,6,3), then equation of the plane through P at right angle right angle to OP, O being the origin, is Show Answer

Q136) What are the D R 's of vector parallel to (2,-1,1) and (3,4,-1) ? Show Answer

Q137) A sphere of constant radius 2 k passes through the origin and meets the axes in A, B, C. The locus of the centroid of the tetrahedron ABC is Show Answer

Q138) The direction cosines of line which is perpendicular to lines whose direction cosines are proportional to (1,-1,2) and (2,1,-1) are Show Answer

Q139) The line segment adjoininh the points A, B makes projection 1,4,3 on x,y,z - axes respectively. Then, the direction cosines of AB are Show Answer

Q140) Cosine of the angle between two diagonals of cube is equal to Show Answer

Q141) The equation of the plane which bisects the line joining (2,3,4) and (6,7,8) is Show Answer

Q142) If a line lies in the octant OXYZ and it makes equal angles with the axes, then Show Answer

Q143) The points A (5,-1,1) , B (7,-4,7), C (1,-6,10) and D (-1,-3,4) are vertices of a Show Answer

Q144) If direction cosines of two lines are proportional to (2,3,-6) & (3,-4,5) then acute angle between them is Show Answer

Q145) Let (3,4,-1) and (-1,2,3) are end points of a diameter of sphere. Then, radius of the sphere is equal to Show Answer

Q146) Let O be the origin and P be the point at a distance 3 units from origin. If direction ratios of OP are (1,-2,-2), then coordinates of P is given by Show Answer

Q147) The triangle formed by the points (0,7,10), (-1,6,6), (-4,9,6) is Show Answer

Q148) The area of triangle whose vertices are (1,2,3), (2,5,-1) and (-1,1,2) is Show Answer

Q149) The projection of line joining points (3,4,5) & ( 4,6,3) on line joining points (-1, 2, 4) and (1, 0, 5) is Show Answer

Q150) The equation of the through the point (2,3,1) and (4,-5,3) and parallel to x-axis is Show Answer

Q151) If A,B,C, D are the points (2,3,-1), (3,5,-3), (1,2,3), (3,5,7) respectively, then angle between AB & CD is Show Answer

Q152) The equation of the plane through the points (1,2,3), (-1,4,2) and (3,1,1) is Show Answer

Q153) The point in the xy-plane which is equidistant from the points (2,0,3), (0,3,2) and (0,0,1) is Show Answer

Q154) Let P (-7,1,-5) be a point on a plane & let 0 be origin. If OP is normal to plane, then equation of plane is Show Answer

Q155) The line drown from (4,-1,2) the point (-3,2,3) meets a plane at right angle at the point (-10,5,4), then the equation of plane is Show Answer

Q156) The projection of a directed line segment on coordinate axes are 12, 4, 3. The D C's of the line are Show Answer

Q157) Coordinates of foot of perpendicular drawn from point P(1,0,3) to join of points A (4,7,1) & B (3,5,3) is Show Answer

Q158) Equation of a line passing through (-1,2,-3) & perpendicular to the plane 2x + 3y + z + 5 = 0 is Show Answer

Q159) The equation of the straight line passing through the points (4,-5,-2) and (-1,5,3) is Show Answer

Q160) The length of the perpendicular from the origin to the plane 3x + 4y + 12z = 52 is Show Answer

Q161) The smallest radius of the sphere passing through (1,0,0), (0,1,0) and (0,0,1) is Show Answer

Q162) Foot of the perpendicular from B(-2,1,4) to the plane is (3,1,2). Then, the equation of the plane is Show Answer

Q163) The center of sphere passes through four points (0,0,0), (0,2,0), (1,0,0) and (0,0,4) is Show Answer

Q164) A variable plane moves so that sum of the reciprocals of its intercepts on the coordinate axes is 1/2 Ten, the plane passes through Show Answer

Q165) Which of the following is an equation of a sphere ? Show Answer

Q166) A point moves such that the sum of its distance from points (4,0,0) and (-4,0,0) is 10, then the locus of the point is Show Answer

Q167) A plane makes intercepts 3 & 4 respectively on z-axis & x-axis. If plane is parallel to y-axis, then its equation is Show Answer

Q168) If x coordinate of a point P of line joining points Q (2,2,1) and R (5,2,-2) is 4, then z coordinate of P is Show Answer