Practice Test

Q1) D (1, 2), E (0, -1), F (2, -1) are the mid-points of the sides BC, CA and AB of the triangle ABC. Find he coordinates of the vertex A. Show Answer

Q2) Two vertices of a triangle are (3, -5) and (-7, 4). If its centroid is (2, -1), find the third vertex. Show Answer

Q3) The locus of a point such that the sum of its distances from the point(s) (0, 2) and (0, -2) is 6, is Show Answer

Q4) Find the angle between the sides BA and AC of the triangle ABC whose vertices are A(-2, 1);
B(2, 3); C(-2, -4) Show Answer

Q5) Find the equation of the line which passes through the point (3, 4) and the sum of its intercepts on the axes is 14. Show Answer

Q6) Find the equation of the line through (2, 3) so that the segment of the line intercepted between the axes is bisected at this point. Show Answer

Q7) Find the equation of the straight line parallel to 2x + 3y + 11 = 0 and which is such that the sum of its intercepts on the axes is 15. Show Answer

Q8) Find the equation of a line which divides the join of (1, 0) and (3, 0) in the ratio 2 : 1 and perpendicular to it. Show Answer

Q9) If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are vertices of a parallelogram PQRS, then Show Answer

Q10) If the algebraic sum of the perpendicular distances from the point (2, 0), (0, 2) and (1, 1) to a variable straight line be zero, then the line passes through the point Show Answer

Q11) The orthocenter of the triangle formed by the lines x = 2, y = 3 and 3 x + 2 y = 6 is at the point Show Answer

Q12) If the lines x + 2 a y + a = 0, x + 3 b y + b = 0 and x + 4 c y + c = 0 are concurrent, then a, b and c are in Show Answer

Q13) If two vertices of a triangle are (5, -1) and (-2, 3) and its orthocenter lies at the origin, then the coordinates of the third vertex are Show Answer

Q14) Length of the median from B on AC where A (-1, 3); B (1, -1); C (5, 1) is Show Answer

Q15) If three vertices of a rhombus taken in order are (2, -1), (3, 4) and (-2, 3), then fourth vertex is Show Answer

Q16) If the line y = m x meets the line x + 2 y - 1 = 0 and 2 x - y + 3 = 0 at the same point, then m is equal to Show Answer

Q17) A straight line through P (1, 2) is such that its intercept between the axes is bisected at P. Its equal to Show Answer

Q18) The equations to the sides of a triangle are x - 3 y = 0, 4 x + 3 y= 5 and 3 x + y = 0. The line 3 x - 4 y = 0 passes throguh Show Answer

Q19) The lines 2 x + y - 1 = 0, a x +3 y - 3 = 0 and 3 x +2 y - 2 = 0 are concurrent for Show Answer

Q20) A point equidistant from the lines 4 x + 3 y + 10 = 0, 5 x - 12 y + 26 = 0 and 7 x + 24 y - 50 = 0 is Show Answer

Q21) The line joining the points (-5, 6) and (2, -3) is divided by the y-axis in the ratio Show Answer

Q22) If the line passing through the point (4, 3) and (2, k) is perpendicular to the line y = 2 x + 3, then the value of k is Show Answer

Q23) The equation of a line passing through the point (1, 2) and perpendicular to the line y = 3 x + 5 y - 12 = 0 is Show Answer

Q24) If the lines:
3 y + 4 x = 1, y = x + 5 and 5 y + b x = 3,
are concurrent, the value of b = Show Answer

Q25) The equation of the line passing through the point (0, 4) and parallel to the line 3 x + 5 y + 15 = 0 is Show Answer

Q26) The centroid of a triangle whose vertices are (4, -8), (-9, 7) and (7, 9) is Show Answer

Q27) Equation of a line passing through the point (-2, -3) and perpendicular to the line x + 3 y + 5 = 0 is Show Answer

Q28) The triangle formed by the lines x + y = 0, 3x + y = 4, x - 3y = 4, is a (n) Show Answer

Q29) The lines x + 8 y + 4 = 0, x + 18 y + 6 = 0, and x + 4 c y + c = 0, are concurrent if c = Show Answer

Q30) The point A (-3, 4), B (4, 6) and C (-1, -3) form a (n) Show Answer

Q31) Area of figure enclosed by |x| + |y| = 1 is Show Answer

Q32) If P (x, y) is equidistant from (a + 1, b - 1) and (a - 1, b + 1), then which of the following is true? Show Answer

Q33) For every point P(x, y, z) on the xy-plane. Show Answer

Q34) For every point (x, y, z) on the x-axis (except the origin) Show Answer

Q35) The locus of a point P(x, y, z) which moves in such a way that z = c (constant), is a Show Answer

Q36) A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of the diagonal of the parrallelopiped is Show Answer

Q37) The xy-plane divides the line joining the points (-1, 3, 4) and (2, -5, 6) in the ratio Show Answer

Q38) The points A(5, -1, 1), B(7, -4, 7), C(1, -6, 10) and D(-1, -3, 4) are the vertices of Show Answer

Q39) The equation ax + by + c = 0 represents a plane perpendicular to the Show Answer

Q40) The image of the point P(1, 3, 4) in the plane 2x - y + z + 3 = 0 is Show Answer

Q41) A, B, C are three points on the axis of x, y and z respectively at distances a, b, c from the origin O; then the coordinates of the point which is equidistant from A, B, C and O is Show Answer

Q42) The ratio in which the yz-plane divides the join of the points (-2, 4, 7) and (3, -5, 8) is Show Answer

Q43) The direction cosines of the line which is equally inclined to the axis is Show Answer

Q44) 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

Q45) The direction cosines of the line joining the points (4, 3, -5) and (-2, 1, -8) are Show Answer

Q46) If A(6, 3, 2), B(5, 1, 4), C(3, -4, 7), D(0, 2, 5) be four points, the projection of the segment CD on the line AB is Show Answer

Q47) The cosines of the angle between any two diagonals of a cube is Show Answer

Q48) A plane meets the coordinates axes in A, B, C such that the centroid of the triangle ABC is (p, q, r). The equation of the plane is Show Answer

Q49) If the centroid of the tetrahedron OABC where A, B, C are (a, 2, 3) ; (1, b, 2), (2, 1, c) respectively be (1, 2, -1), then the distance of P(a, b, c) from the origin is equal to Show Answer

Q50) The angle between the planes 2x - y + x = 6 and x + y + 2z = 7 is Show Answer

Q51) The four points (0, 4, 3), (-1, -5, -3), (-2, 2, 1) and (1, 1, -1) lie in the plane Show Answer

Q52) The equation of the plane passing through the point (-2, -2, 2) and containing the line joining the points (1, 1, 1) and (1, -1, 2) is Show Answer

Q53) The plane ax + by + cz = 1 meets the coordinates axes in A, B, C. The centroid of the triangle is Show Answer

Q54) If O is the origin and A is the point (a, b, c) then the equation of the plane through A and perpendicular to OA is given by Show Answer

Q55) The distance between the parallel planes ax + by + cz + d = 0 and ax + by + cz + d = 0 is Show Answer

Q56) The equation of the plane through the intersection of the planes x + 2y + 3z -4 = 0 and 2x + y - z + 5 = 0 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is Show Answer

Q57) If from a point P(a, b, c), perpendiculars PA, PB are drawn to yz-plane and zx-plane, then the equation of the plane OAB is Show Answer

Q58) Equation of the plane which bisects the line joining the points (3, 2, 2) and (5, 4, 6) at right angle is Show Answer

Q59) The two points (1, 1, 1) and (-3, 0, 1) with respect to the plane 3x + 4y - 12z + 13 = 0 lie on Show Answer

Q60) The coordinates of the point where the line joining the points (2, -3, 1), (3, -4, -5) cuts the plane 2x + y + z = 7 are Show Answer

Q61) 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

Q62) The projections of the directed line segment on the coordinate axes are 12, 4, 3. The direction cosines are Show Answer

Q63) The distance of the point (1, -2, 3) from the plane x - y + z = 5 measured paralleled to the line whose direction cosines are proportional to 2, 3, -6 is Show Answer

Q64) Equation of the line passing through (-1, 2, -3) and perpendicular to the plane 2x + 3y + z + 5 = 0 is Show Answer

Q65) The equation of a shpere passing through origin and the points (a, 0, 0) ; (0, b, 0) ; (0, 0, c) is Show Answer

Q66) A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of the tetrahedron OABC is Show Answer

Q67) The center of the sphere which passes through (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is Show Answer

Q68) Equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2 y + 4 z = 0 is given by Show Answer

Q69) The coordinates of the middle points of the sides of a triangle are
(4, 2), (3, 3) and (2, 2), then find the coordinates of its centroid are Show Answer

Q70) The locus of a point, whose difference of distance from point (3, 0) and (-3, 0) is 4, is Show Answer

Q71) If P (1, 2) , Q (4, 6),
R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS, then Show Answer

Q72) The equation of the line parallel to the line 2 x + 3 y + 5 = 0 and passing through (1, 1) is Show Answer

Q73) The equation of the line perpendicular to the line 2 x + 3 y + 5 = 0 and passing through origin is Show Answer

Q74) The distance between the lines 4 x + 3 y = 11 and 8 x + 6 y = 15 is Show Answer

Q75) The equation of the straight line which is perpendicular to y = x and passes through (3, 2) will be given by Show Answer

Q76) If the points (k, 2 - k),
(-k + 1, 2 k), (-4 - k, 6 - 2 k) are collinear, then k is equal to Show Answer

Q77) The line x + y = 4 divides the line joining the points (-1, 1) and (5, 7) in the ratio Show Answer

Q78) Foot of perpendicular drawn from (0, 5) to the line 3 x - 4 y - 5 = 0 is Show Answer

Q79) The foot of the perpendicular drawn from the point (2, -1) to a straight line L is (1, 3). The equation of straight line L is Show Answer

Q80) The area of a triangle is 5 and two of its vertices are A (2, 1), B (3, - 2). Then the third vertex which lies on the line y = x + 3 is Show Answer

Q81) A point P (h, k) lies on the straight line x + y + 1 = 0 and is at a distance 5 units from the origin. If k is negative, then h is equal to Show Answer

Q82) The diagonals of a parallelogram ABCD are along the lines x + 3 y = 0 and 6 x - 2 y = 7. Then, ABCD must be a Show Answer

Q83) The equation of straight line passing through the point of intersection of the straight line 3 x - y + 2 = 0 and 5 x - 2y + 7 = 0 and having infinite slope is Show Answer

Q84) Perimeter of the quadrilateral ABCD is Show Answer

Q85) Equation of the angle bisectors of the pairs of lines P is Show Answer

Q86) Area of the triangle formed by the angle bisectors of the pair of lines P and the line L (in sq units) is Show Answer

Q87) If L' represents the line perpendicular to L passing through the point of intersection of the pair of lines P, then equation of the pair of lines representing L and L' is Show Answer

Q88) For what value of k are the two straight lines 3 x + 4 y = 1 and 4 x + 3 y + 2 k = 0 equidistant for the point (1, 1)? Show Answer

Q89) The value of k for which the lines 2 x + 3 y + a = 0 and 5 x + k y + a = 0 represent family of parallel lines is Show Answer

Q90) What is the equation of the line which passes through (4, -5) and is perpendicular to 3 x + 4 y + 5 = 0? Show Answer

Q91) If the three vertices of the parallelogram ABCD are A (1, a), B (3, a), C (2, b), then D is equal to? Show Answer

Q92) What angle does the line segment joining (5, 2) and (6, - 15) subtend at (0, 0)? Show Answer

Q93) A points P moves such that its distances from
(1, 2) and (-2, 3) are equal. Then, the locus of P is Show Answer

Q94) The equation of the locus of a point which is equidistant from the axes is Show Answer

Q95) The equation of the line, the reciprocals of whose intercepts on the axes are m and n, is given by Show Answer

Q96) What is the equation of the straight line passing through (5, -2) and
(-4, 7)? Show Answer

Q97) What is the angle between the line x + y = 1 and x - y = 1? Show Answer

Q98) What is the distance of the line 2 x + y = 2 z = 3 from the origin? Show Answer

Q99) What is the distance between the planes x - 2 y + z - 1 = 0 and - 3 x + 6 y - 3 z + 2 = 0? Show Answer

Q100) The sum of the direction cosines of Z-axis is Show Answer

Q101) What is the equation of a straight line which passes through (3, 4) and the sum of whose x and y intercepts is 14? Show Answer

Q102) A straight line passes through the points (5, 0) and (0, 3). the length of the perpendicular from the point (4, 4) on the line is Show Answer

Q103) What is the angle between the planes 2 x - y - 2 z + 1 = 0 and 3 x - 4 y + 5 z - 3 = 0? Show Answer

Q104) Two straight line paths are represented by the equations 2 x - y = 2 and -4 x + 2 y = 6. Then, the paths will Show Answer

Q105) The locus of a point equidistant from three collinear points is Show Answer

Q106) The equation to the locus of a point which is always equidistant from the points (1, 0) and (0, -2) is Show Answer

Q107) The points (5, 1), (1, -1) and (11, 4) are Show Answer

Q108) What is the perpendicular distance between the parallel lines 3 x + 4 y =9 and 9 x + 12 y + 28 = 0? Show Answer

Q109) If p, q, r and s be the distances from origin of the points (2, 6),
(3, 4), (4, 5) and (-2, 5) respectively. Which one of the following is a whole number? Show Answer

Q110) From the point (4, 3) a perpendicular is dropped on the X-axis as well as on the Y-axis. If the lengths of perpendiculars are p and q respectively, then which one of the following is correct? Show Answer

Q111) The line y = 0 divides the line joining the points
(3, -5) and (-4, 7) in the ratio Show Answer

Q112) What is the equation to the plane through (1, 2, 3) parallel to 3 x + 4 y - 5 z = 0? Show Answer

Q113) What is the distance between the lines 3 x + 4 y = 9 a 6 x + 8 y = 18? Show Answer

Q114) What is the equation of line passing through (0, 1) and making an angle with the Y-axis equal to the inclination of the line x - y = 4 with X-axis? Show Answer

Q115) What is the perpendicular distance of the point (x, y) from X-axis? Show Answer

Q116) What is the cosine of angle between the planes x + y + z + 1 = 0 and 2 x - 2 y + 2 z + 1 = 0? Show Answer

Q117) For what value of k, are the lines x + 2 y - 9 = 0 and k x + 4 y + 5 = 0 parallel? Show Answer

Q118) What are the coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y - 11 = 0? Show Answer

Q119) If (p, q) is the point on the X-axis equidistant from the point (1, 2) and (2, 3) then which one of the following is correct? Show Answer

Q120) What is the locus of a point which moves equidistant from the coordinate axes? Show Answer

Q121) The line m x + n y = 1 passes through the points (1, 2) and (2, 1). What is the value of m? Show Answer

Q122) What is the equation of the line joining the origin with the point of intersection of the lines 4 x + 3 y = 12 and 3 x + 4 y = 12? Show Answer

Q123) What is the equation of the line passing through (2, -3) and parallel to Y-axis? Show Answer

Q124) What is the locus of the point which is at a distance 8 units to the left of Y-axis? Show Answer

Q125) If (a, 0), (0, b) and (1, 1) are collinear, what is (a + b - a b) equal to? Show Answer

Q126) Two straight lines x - 3 y - 2 = 0 and 2 x - 6 y - 6 = 0 Show Answer

Q127) If the area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq unit, then what is the value of k? Show Answer

Q128) If the lines 3 y + 4 x = 1, y = x + 5 and 5 y + b x = 3 are concurrent, then what is the value of b? Show Answer

Q129) If (-5, 4) divides the line segment between the coordinate axes in the ratio 1 : 2, then what is its equation? Show Answer

Q130) What is the image of the point (1, 2) on the line 3 x + 4 y - 1 = 0? Show Answer

Q131) What is the locus of a point which is equidistant from the point (m + n, n - m) and the point (m - n, n + m)? Show Answer

Q132) The middle point of the segment of the straight line joining the points (p, q) and (q, -p) is (r/2, s/2). What is the length of the segment? Show Answer

Q133) What is the area of the triangle formed by the line y - x = 0, y + x = 0, x = c ? Show Answer

Q134) What does an equation of the first degree containing one arbitrary parameter passing through a fixed point represent? Show Answer

Q135) What is the foot of the perpendicular from the point (2, 3) on the line x + y - 11 = 0? Show Answer

Q136) The ratio in which the line joining (2, 4, 5), (3, 5, -4) is divided by the YZ - plane is Show Answer

Q137) The equation of the line through the point (2, 3, -5) and equally inclined to the axes are Show Answer

Q138) Consider the points (a - 1, a, a + 1), (a, a + 1, a - 1) and (a + 1, a - 1, a).
I. These points always form the vertices of an equilateral triangle for any real value of a.
II. The area of the triangle formed by these points is independent of a.
Which of the above statements is/are correct? Show Answer

Q139) The equation of the plane passing through the point (-2, -2, 2) and containing the line joining the points (1, 1, 1) and (1, -1, 2) is Show Answer

Q140) If from a point P (a, b, c) perpendiculars PA, PB are drawn to YZ-plane and ZX-plane, then the equation of the plane OAB is Show Answer

Q141) Equation of the plane perpendicular to the plane x - 2y + 5z + 1 = 0 which passes through the points (2, -3, 1) and (-1, 1, -7) is given by Show Answer

Q142) Equation of the plane through P (2, 3, -1) at right angle to OP is Show Answer

Q143) Equation of the plane that passes through the point (2, -3, 1) and is perpendicular to the line joining the points (3, 4, -1) and (2, -1, 5) is given by Show Answer

Q144) Direction cosines of the line which is perpendicular to the lines whose direction ratios are 1, -1, 2 and 2, 1, -1 are given by Show Answer

Q145) What is the sum of the squares of direction cosines of X-axis? Show Answer

Q146) What is the equation of the sphere with unit radius having center at the origin? Show Answer

Q147) What is the distance between the planes x - 2y + z - 1 = 0 and -3x + 6y - 3z + 2 = 0? Show Answer

Q148) What is the angle between the panes 2x - y - 2z + 1 = 0 and 3x - 4y + 5z - 3 = 0? Show Answer

Q149) What are the direction cosines of a line which is equally inclined to the positive directions of the axes? Show Answer

Q150) What is the distance of the point (1, 2, 0) from YZ-plane is Show Answer

Q151) What is the equation to the plane through (1, 2, 3) parallel to 3x + 4y - 5z = 0? Show Answer

Q152) What are the direction ratios of the line of intersection of the planes x = 3z + 4 and y = 2z -3? Show Answer

Q153) What is the equation of the straight line passing through (a, b, c) and parallel to Z-axis? Show Answer

Q154) What is the direction ratios of normal to the plane 2x - y + 2z + 1 = 0? Show Answer

Q155) What is the sum of the squares of direction cosines of the line joining the points (1, 2, -3) and (-2, 3, 1)? Show Answer

Q156) What is the cosine of angle between the planes x + y + z + 1 = 0 and 2x - 2y + 2z + 1 = 0? Show Answer

Q157) What is the acute angle between the planes x + y + 2z = 3 and -2x + y - z = 11? Show Answer

Q158) Which one of the following points lies on the plane 2x + 3y - 6z = 21? Show Answer

Q159) What is the angle between the lines whose direction cosines are proportional to (2, 3, 4) and (1, -2, 1) respectively? Show Answer

Q160) What is the distance of the origin from the plane 2x + 6y - 3z + 7 = 0? Show Answer

Q161) Under what condition do the planes bx - ay = n, cy - bz = l, az - cx = m intersect in a line? Show Answer

Q162) What is the angle between any two diagonals of the cube? Show Answer

Q163) What is the angle between one of the edges of the cube and the diagonal of the cube intersecting the edge of the cube? Show Answer

Q164) What is the angle between the diagonal of one of the faces of the cube and the diagonal of the cube intersecting the diagonal of the face of the cube? Show Answer

Q165) If the line through the points A (k, 1, -1) and B (2k, 0, 2) is perpendicular to the line through the points B and C (2 + 2k, k, 1), then what is the value of k ? Show Answer

Q166) The direction cosines of a line are proportional to (2, 1, 2) and the line intersects a plane perpendicularly at the point (1, -2, 4). What is the distance of the plane from the points (3, 2, 3)? Show Answer

Q167) The foot of the perpendicular drawn from the origin to a plane is the point (1, -3, 1). What is the intercept cut on the X-axis by the plane? Show Answer

Q168) What are the direction ratios of the line determined by the planes x - y + 2z = 1 and x + y - z = 3? Show Answer

Q169) What is the equation of the sphere which has its center at (6, -1, 2) and touches the plane 2x - y + 2z - 2 = 0? Show Answer

Q170) The points (1, 3, 4), (-1, 6, 10), (-7, 4, 7) and (-5, 1, 1) are the vertices of a Show Answer

Q171) What is the angle between the lines x + z = 0, y = 0 and 20x = 15y = 12z ? Show Answer

Q172) What is the number of planes passing through three non-collilnear points? Show Answer

Q173) What is the angle between the planes 2x - y + z = 6 and x + y + 2z = 3? Show Answer

Q174) What is the equation of a plane through the X-axis and passing through the point (1, 2, 3)? Show Answer

Q175) Curve of intersection of two sphere is Show Answer

Q176) What are the coordinates of the point equidistant from the four points (0, 0, 0), (2, 0, 0), (0, 4, 0), (0, 0, 6) ? Show Answer

Q177) The equation by + cz +d =0 represents a plane parallel to which one of the following? Show Answer

Q178) Which one of the following planes is normal to the plane 3x + y + z = 5? Show Answer

Q179) The line passing through (1, 2, 3) and having direction ratios given by < 1, 2, 3 > cuts the X-axis at a distance k from origin. What is the value of k? Show Answer

Q180) Which one of the following planes contains the Z-axis? Show Answer

Q181) Let O (0, 0, 0), P (3, 4, 5), Q (m, n, r) and R (1, 1, 1) be the vertices of a parallelogram taken in order. What is the value of m + n + r? Show Answer