Page 1 Test – VECTOR 3D Time :-1hr.30mins. Read carefully the instructions given below : 1. The test consists of 100 marks. 2. Q. No.1 – Q. No. 25 carries 3 marks . For every wrong answer -1 will be deducted. 3. Q.No.26 – Q. No.30 carries 5 marks each. For every wrong answer -2 will be deducted. 4. A student will get marks if he/she will answer all the correct choices to a problem, if more than one choice is the correct answer. SECTION A :- Q.1 If a line makes angles of 60° and 45° with the positive directions of the x-axis and y-axis resp. then the acute angle between the line and the z-axis is (a) 60° (b) 45° (c) 75° (d) 15° Q.2 The angle between the lines joining the points (1, 0, -3) , (2, -1, 2) and (1, 1, 1), (3, 2, 0) is (a) Cos -1 2/9 (b) 90° (c) 0° (d) Cos -1 2v2 / 9 Q.3 ABC is a triangle where A = (2, 3, 5), B = (-1, 3, 2) and C = (?, 5, µ). If the median through A is equally inclined with the axes then (a) ? = 14, µ = 20 (b) ? = 7, µ = 10 (c) ? = 7/2, µ = 5 (d) ? = 10, µ = 7 Q.4 The equation of the locus of point (1 + r/4, 1 + r/3 , 2) where r ? R, is given by (a) x – 1 = y + 1 = z – 2 (b) x – 1 = y + 1 = z – 2 (c) 4x – 3y = 7 (d) z = 2 4 3 0 3 4 0 Q.5 The angle between the lines x + 2 = y + 3 = z – 4 and x – 1 = y = z is 1 -2 1 -1 1 0 (a) p / 6 (b) p / 3 (c) p / 2 (d) 0 Q.6 The shortest distance between the line x – 3 = y = z and the y-axis is 3 0 -4 (a) 1 / 5 (b) 1 (c) 0 (d) 12 / 5 Q.7 The projection of the line segment joining the point (6, -2, 1) and the origin on the line x – 2 = y + 1 = z – 1 is 4 – 3 0 (a) 30 (b) 6 (c) 5 (d) none Q.8 If the sum of the reciprocals of the intercepts made by the plane ax + by + cz = 1 on the three axes is 1 then the plane always passes through the point (a) (2, -1, 0) (b) (1, 1, 1) (c) (-1, -1, -1) (d) (1/2, -1, 1/2) Q.9 The direction cosines of a line parallel to the planes 3x + 4y + z = 0 and x – 2y – 3z = 5 are (a) (-1, 1, -1) (b) (-1/ v3, -1/ v3, 1/ v3) (c) (-1/ v3, 1/ v3, -1/ v3) (d) none Q.10 If the line x – 1 = y + 1 = z + 1 lies in the plane 3x – 2y + 5z = 0 then ? is 1 -2 ? (a) 1 (b) – 7/5 (c) 5 / 7 (d) none Page 2 Test – VECTOR 3D Time :-1hr.30mins. Read carefully the instructions given below : 1. The test consists of 100 marks. 2. Q. No.1 – Q. No. 25 carries 3 marks . For every wrong answer -1 will be deducted. 3. Q.No.26 – Q. No.30 carries 5 marks each. For every wrong answer -2 will be deducted. 4. A student will get marks if he/she will answer all the correct choices to a problem, if more than one choice is the correct answer. SECTION A :- Q.1 If a line makes angles of 60° and 45° with the positive directions of the x-axis and y-axis resp. then the acute angle between the line and the z-axis is (a) 60° (b) 45° (c) 75° (d) 15° Q.2 The angle between the lines joining the points (1, 0, -3) , (2, -1, 2) and (1, 1, 1), (3, 2, 0) is (a) Cos -1 2/9 (b) 90° (c) 0° (d) Cos -1 2v2 / 9 Q.3 ABC is a triangle where A = (2, 3, 5), B = (-1, 3, 2) and C = (?, 5, µ). If the median through A is equally inclined with the axes then (a) ? = 14, µ = 20 (b) ? = 7, µ = 10 (c) ? = 7/2, µ = 5 (d) ? = 10, µ = 7 Q.4 The equation of the locus of point (1 + r/4, 1 + r/3 , 2) where r ? R, is given by (a) x – 1 = y + 1 = z – 2 (b) x – 1 = y + 1 = z – 2 (c) 4x – 3y = 7 (d) z = 2 4 3 0 3 4 0 Q.5 The angle between the lines x + 2 = y + 3 = z – 4 and x – 1 = y = z is 1 -2 1 -1 1 0 (a) p / 6 (b) p / 3 (c) p / 2 (d) 0 Q.6 The shortest distance between the line x – 3 = y = z and the y-axis is 3 0 -4 (a) 1 / 5 (b) 1 (c) 0 (d) 12 / 5 Q.7 The projection of the line segment joining the point (6, -2, 1) and the origin on the line x – 2 = y + 1 = z – 1 is 4 – 3 0 (a) 30 (b) 6 (c) 5 (d) none Q.8 If the sum of the reciprocals of the intercepts made by the plane ax + by + cz = 1 on the three axes is 1 then the plane always passes through the point (a) (2, -1, 0) (b) (1, 1, 1) (c) (-1, -1, -1) (d) (1/2, -1, 1/2) Q.9 The direction cosines of a line parallel to the planes 3x + 4y + z = 0 and x – 2y – 3z = 5 are (a) (-1, 1, -1) (b) (-1/ v3, -1/ v3, 1/ v3) (c) (-1/ v3, 1/ v3, -1/ v3) (d) none Q.10 If the line x – 1 = y + 1 = z + 1 lies in the plane 3x – 2y + 5z = 0 then ? is 1 -2 ? (a) 1 (b) – 7/5 (c) 5 / 7 (d) none Q.11 The direction cosines of the normal to the plane 5(x - 2) = 3(y – z) are (a) 5, -3, 3 (b) 5/ v43, -3/ v43 , 3/ v43 (c) 1/2, -3/10, 3/10 (d) 1, -3/5, 3/5 Q.12 The direction cosines of the projection of the line x = y – 1 = z + 1 on the plane 2x + y – 3z = 5 are -2 1 -1 (a) 2/3, -1/3, 2/3 (b) 2/ v14, -3/ v14, 1/ v14 (c) 2/ v6, 1/ v6, 1/ v6 (d) 2/ v6, -1/ v6, 1/ v6 Q.13 Let ABC be a triangle the position vectors of whose vertices are resp. i + 2j + 4k, -2i + 2j + k and 2i + 4j – 3k. Then the triangle ABC is (a) isosceles (b) equilateral (c) right angled (d) none Q.14 One of the diagonals of a parallelepiped vector is 4j – 8k. If the two diagonals of one of its faces are 6i + 6k and 4j + 2k, then its volume is (a) 60 (b) 80 (c) 100 (d) 120 Q.15 If vectors a, b, c are unit vector such that a + b – c = 0, then the angle between vector a and b is (a) p / 6 (b) p / 3 (c) p / 2 (d) 2p / 3 Q.16 If vectors a, b, c are non-coplanar, then vector [a + 2b b + 2c c + 2a] = [a b c] (a) 3 (b) 9 (c) 8 (d) 6 Q.17 If vectors a, b, c are vectors such that vector |b| = |c|, then [(a + b) x (a + c)] x (b x c) . (b + c) (a) [a b c] (b) 0 (c) – [a b c] (d) 2 [a b c] Q.18 If vector |a . b| = v3 |a x b|, then the angle between vector a and b is (a) p / 6 (b) p / 4 (c) p / 3 (d) p / 2 Q.19 The area of the parallelogram whose diagonals are vector a – b and 3a + b, where |a| = 2, |b| = 2 and the angle between vector a and b is p / 6 is (a) 1 (b) 4 (c) 2v3 (d) 8 Q.20 If a and b are two unit vectors then the vector (a + b) x (a x b) is parallel to the vector (a) a – b (b) a + b (c) 2a – b (d) 2a + b Q.21 The vector i x (a x i) + j x (a x j) + k x (a x k) is equal to (a) 0 (b) a (c) 2a (d) none Q.22 The volume of the tetrahedron whose vertices are the points with position vectors i – 6j + 10k, -i – 3j + 7k, 5i – j + ?k and 7i – 4j + 7k is 11 cubic units if the value of ? is (a) -1 (b) 1 (c) -7 (d) 7 Q.23 If a, b, c are any three vectors, then a x (b x c) = (a x b) x c if and only if (a) b and c are collinear (b) a and c are collinear (c) a and b are collinear (d) none Q.24 The vectors AB = 3i – 2j + 2k and BC = i – 2k are the adjacent sides of a parallelogram. An angle between its diagonals is (a) p / 4 (b) p / 3 (c) 3p / 4 (d) 2p / 3 Q.25 If r.a = 0, r.b = 0 and r.c = 0 for some non-zero vectors r, then the value of [a b c] is (a) 0 (b) 1 / 2 (c) 1 (d) 2 Page 3 Test – VECTOR 3D Time :-1hr.30mins. Read carefully the instructions given below : 1. The test consists of 100 marks. 2. Q. No.1 – Q. No. 25 carries 3 marks . For every wrong answer -1 will be deducted. 3. Q.No.26 – Q. No.30 carries 5 marks each. For every wrong answer -2 will be deducted. 4. A student will get marks if he/she will answer all the correct choices to a problem, if more than one choice is the correct answer. SECTION A :- Q.1 If a line makes angles of 60° and 45° with the positive directions of the x-axis and y-axis resp. then the acute angle between the line and the z-axis is (a) 60° (b) 45° (c) 75° (d) 15° Q.2 The angle between the lines joining the points (1, 0, -3) , (2, -1, 2) and (1, 1, 1), (3, 2, 0) is (a) Cos -1 2/9 (b) 90° (c) 0° (d) Cos -1 2v2 / 9 Q.3 ABC is a triangle where A = (2, 3, 5), B = (-1, 3, 2) and C = (?, 5, µ). If the median through A is equally inclined with the axes then (a) ? = 14, µ = 20 (b) ? = 7, µ = 10 (c) ? = 7/2, µ = 5 (d) ? = 10, µ = 7 Q.4 The equation of the locus of point (1 + r/4, 1 + r/3 , 2) where r ? R, is given by (a) x – 1 = y + 1 = z – 2 (b) x – 1 = y + 1 = z – 2 (c) 4x – 3y = 7 (d) z = 2 4 3 0 3 4 0 Q.5 The angle between the lines x + 2 = y + 3 = z – 4 and x – 1 = y = z is 1 -2 1 -1 1 0 (a) p / 6 (b) p / 3 (c) p / 2 (d) 0 Q.6 The shortest distance between the line x – 3 = y = z and the y-axis is 3 0 -4 (a) 1 / 5 (b) 1 (c) 0 (d) 12 / 5 Q.7 The projection of the line segment joining the point (6, -2, 1) and the origin on the line x – 2 = y + 1 = z – 1 is 4 – 3 0 (a) 30 (b) 6 (c) 5 (d) none Q.8 If the sum of the reciprocals of the intercepts made by the plane ax + by + cz = 1 on the three axes is 1 then the plane always passes through the point (a) (2, -1, 0) (b) (1, 1, 1) (c) (-1, -1, -1) (d) (1/2, -1, 1/2) Q.9 The direction cosines of a line parallel to the planes 3x + 4y + z = 0 and x – 2y – 3z = 5 are (a) (-1, 1, -1) (b) (-1/ v3, -1/ v3, 1/ v3) (c) (-1/ v3, 1/ v3, -1/ v3) (d) none Q.10 If the line x – 1 = y + 1 = z + 1 lies in the plane 3x – 2y + 5z = 0 then ? is 1 -2 ? (a) 1 (b) – 7/5 (c) 5 / 7 (d) none Q.11 The direction cosines of the normal to the plane 5(x - 2) = 3(y – z) are (a) 5, -3, 3 (b) 5/ v43, -3/ v43 , 3/ v43 (c) 1/2, -3/10, 3/10 (d) 1, -3/5, 3/5 Q.12 The direction cosines of the projection of the line x = y – 1 = z + 1 on the plane 2x + y – 3z = 5 are -2 1 -1 (a) 2/3, -1/3, 2/3 (b) 2/ v14, -3/ v14, 1/ v14 (c) 2/ v6, 1/ v6, 1/ v6 (d) 2/ v6, -1/ v6, 1/ v6 Q.13 Let ABC be a triangle the position vectors of whose vertices are resp. i + 2j + 4k, -2i + 2j + k and 2i + 4j – 3k. Then the triangle ABC is (a) isosceles (b) equilateral (c) right angled (d) none Q.14 One of the diagonals of a parallelepiped vector is 4j – 8k. If the two diagonals of one of its faces are 6i + 6k and 4j + 2k, then its volume is (a) 60 (b) 80 (c) 100 (d) 120 Q.15 If vectors a, b, c are unit vector such that a + b – c = 0, then the angle between vector a and b is (a) p / 6 (b) p / 3 (c) p / 2 (d) 2p / 3 Q.16 If vectors a, b, c are non-coplanar, then vector [a + 2b b + 2c c + 2a] = [a b c] (a) 3 (b) 9 (c) 8 (d) 6 Q.17 If vectors a, b, c are vectors such that vector |b| = |c|, then [(a + b) x (a + c)] x (b x c) . (b + c) (a) [a b c] (b) 0 (c) – [a b c] (d) 2 [a b c] Q.18 If vector |a . b| = v3 |a x b|, then the angle between vector a and b is (a) p / 6 (b) p / 4 (c) p / 3 (d) p / 2 Q.19 The area of the parallelogram whose diagonals are vector a – b and 3a + b, where |a| = 2, |b| = 2 and the angle between vector a and b is p / 6 is (a) 1 (b) 4 (c) 2v3 (d) 8 Q.20 If a and b are two unit vectors then the vector (a + b) x (a x b) is parallel to the vector (a) a – b (b) a + b (c) 2a – b (d) 2a + b Q.21 The vector i x (a x i) + j x (a x j) + k x (a x k) is equal to (a) 0 (b) a (c) 2a (d) none Q.22 The volume of the tetrahedron whose vertices are the points with position vectors i – 6j + 10k, -i – 3j + 7k, 5i – j + ?k and 7i – 4j + 7k is 11 cubic units if the value of ? is (a) -1 (b) 1 (c) -7 (d) 7 Q.23 If a, b, c are any three vectors, then a x (b x c) = (a x b) x c if and only if (a) b and c are collinear (b) a and c are collinear (c) a and b are collinear (d) none Q.24 The vectors AB = 3i – 2j + 2k and BC = i – 2k are the adjacent sides of a parallelogram. An angle between its diagonals is (a) p / 4 (b) p / 3 (c) 3p / 4 (d) 2p / 3 Q.25 If r.a = 0, r.b = 0 and r.c = 0 for some non-zero vectors r, then the value of [a b c] is (a) 0 (b) 1 / 2 (c) 1 (d) 2 Q.26 A point Q at a distance 3 from the point P(1, 1, 1) lying on the line joining the points A(0, -1, 3) and P, has the coordinates (a) (2, 3, -1) (b) (4, 7, -5) (c) (0, -1, 3) (d) (-2, -5, 7) Q.27 The direction cosines of a line passing through the origin and cutting the line x + 2 = y – 1 = z_ at Cos -1 v6/11 are 1 2 -1 (a) -1/ v11, 3/ v11, -1/ v11 (b) 1/ v11, 3/ v11, 1/ v11 (c) -1/ v6, 2/ v6, 1/ v6 (d) -3/ v11, -1/ v11, 1/ v11 Q.28 Which of the following planes intersects the planes x – y + 2z = 3 and 4x + 3y – z = 1 along the same line ? (a) 11x + 10y – 5z = 0 (b) 7x + 7y – 4z = 0 (c) 5x + 2y + z = 2 (d) none Q.29 Vectors perpendicular to i – j – k and in the plane of i + j + k and -i + j + k are (a) i + k (b) 2i + j + k (c) 3i + 2j + k (d) – 4i – 2j – 2k Q.30 Unit vectors a, b, c are coplanar. A unit vector d is perpendicular to them. If vector (a x b) x (c x d) = 1/6 i – 1/3 j+ 1/3 k and the angle between a and b is 30° then vector c is (a) (i – 2j + 2k) / 3 (b) (2i + j - k) / 3 (c) (- 2i – 2j + k) / 3 (d) (- i + 2j – 2k) / 3 Page 4 Test – VECTOR 3D Time :-1hr.30mins. Read carefully the instructions given below : 1. The test consists of 100 marks. 2. Q. No.1 – Q. No. 25 carries 3 marks . For every wrong answer -1 will be deducted. 3. Q.No.26 – Q. No.30 carries 5 marks each. For every wrong answer -2 will be deducted. 4. A student will get marks if he/she will answer all the correct choices to a problem, if more than one choice is the correct answer. SECTION A :- Q.1 If a line makes angles of 60° and 45° with the positive directions of the x-axis and y-axis resp. then the acute angle between the line and the z-axis is (a) 60° (b) 45° (c) 75° (d) 15° Q.2 The angle between the lines joining the points (1, 0, -3) , (2, -1, 2) and (1, 1, 1), (3, 2, 0) is (a) Cos -1 2/9 (b) 90° (c) 0° (d) Cos -1 2v2 / 9 Q.3 ABC is a triangle where A = (2, 3, 5), B = (-1, 3, 2) and C = (?, 5, µ). If the median through A is equally inclined with the axes then (a) ? = 14, µ = 20 (b) ? = 7, µ = 10 (c) ? = 7/2, µ = 5 (d) ? = 10, µ = 7 Q.4 The equation of the locus of point (1 + r/4, 1 + r/3 , 2) where r ? R, is given by (a) x – 1 = y + 1 = z – 2 (b) x – 1 = y + 1 = z – 2 (c) 4x – 3y = 7 (d) z = 2 4 3 0 3 4 0 Q.5 The angle between the lines x + 2 = y + 3 = z – 4 and x – 1 = y = z is 1 -2 1 -1 1 0 (a) p / 6 (b) p / 3 (c) p / 2 (d) 0 Q.6 The shortest distance between the line x – 3 = y = z and the y-axis is 3 0 -4 (a) 1 / 5 (b) 1 (c) 0 (d) 12 / 5 Q.7 The projection of the line segment joining the point (6, -2, 1) and the origin on the line x – 2 = y + 1 = z – 1 is 4 – 3 0 (a) 30 (b) 6 (c) 5 (d) none Q.8 If the sum of the reciprocals of the intercepts made by the plane ax + by + cz = 1 on the three axes is 1 then the plane always passes through the point (a) (2, -1, 0) (b) (1, 1, 1) (c) (-1, -1, -1) (d) (1/2, -1, 1/2) Q.9 The direction cosines of a line parallel to the planes 3x + 4y + z = 0 and x – 2y – 3z = 5 are (a) (-1, 1, -1) (b) (-1/ v3, -1/ v3, 1/ v3) (c) (-1/ v3, 1/ v3, -1/ v3) (d) none Q.10 If the line x – 1 = y + 1 = z + 1 lies in the plane 3x – 2y + 5z = 0 then ? is 1 -2 ? (a) 1 (b) – 7/5 (c) 5 / 7 (d) none Q.11 The direction cosines of the normal to the plane 5(x - 2) = 3(y – z) are (a) 5, -3, 3 (b) 5/ v43, -3/ v43 , 3/ v43 (c) 1/2, -3/10, 3/10 (d) 1, -3/5, 3/5 Q.12 The direction cosines of the projection of the line x = y – 1 = z + 1 on the plane 2x + y – 3z = 5 are -2 1 -1 (a) 2/3, -1/3, 2/3 (b) 2/ v14, -3/ v14, 1/ v14 (c) 2/ v6, 1/ v6, 1/ v6 (d) 2/ v6, -1/ v6, 1/ v6 Q.13 Let ABC be a triangle the position vectors of whose vertices are resp. i + 2j + 4k, -2i + 2j + k and 2i + 4j – 3k. Then the triangle ABC is (a) isosceles (b) equilateral (c) right angled (d) none Q.14 One of the diagonals of a parallelepiped vector is 4j – 8k. If the two diagonals of one of its faces are 6i + 6k and 4j + 2k, then its volume is (a) 60 (b) 80 (c) 100 (d) 120 Q.15 If vectors a, b, c are unit vector such that a + b – c = 0, then the angle between vector a and b is (a) p / 6 (b) p / 3 (c) p / 2 (d) 2p / 3 Q.16 If vectors a, b, c are non-coplanar, then vector [a + 2b b + 2c c + 2a] = [a b c] (a) 3 (b) 9 (c) 8 (d) 6 Q.17 If vectors a, b, c are vectors such that vector |b| = |c|, then [(a + b) x (a + c)] x (b x c) . (b + c) (a) [a b c] (b) 0 (c) – [a b c] (d) 2 [a b c] Q.18 If vector |a . b| = v3 |a x b|, then the angle between vector a and b is (a) p / 6 (b) p / 4 (c) p / 3 (d) p / 2 Q.19 The area of the parallelogram whose diagonals are vector a – b and 3a + b, where |a| = 2, |b| = 2 and the angle between vector a and b is p / 6 is (a) 1 (b) 4 (c) 2v3 (d) 8 Q.20 If a and b are two unit vectors then the vector (a + b) x (a x b) is parallel to the vector (a) a – b (b) a + b (c) 2a – b (d) 2a + b Q.21 The vector i x (a x i) + j x (a x j) + k x (a x k) is equal to (a) 0 (b) a (c) 2a (d) none Q.22 The volume of the tetrahedron whose vertices are the points with position vectors i – 6j + 10k, -i – 3j + 7k, 5i – j + ?k and 7i – 4j + 7k is 11 cubic units if the value of ? is (a) -1 (b) 1 (c) -7 (d) 7 Q.23 If a, b, c are any three vectors, then a x (b x c) = (a x b) x c if and only if (a) b and c are collinear (b) a and c are collinear (c) a and b are collinear (d) none Q.24 The vectors AB = 3i – 2j + 2k and BC = i – 2k are the adjacent sides of a parallelogram. An angle between its diagonals is (a) p / 4 (b) p / 3 (c) 3p / 4 (d) 2p / 3 Q.25 If r.a = 0, r.b = 0 and r.c = 0 for some non-zero vectors r, then the value of [a b c] is (a) 0 (b) 1 / 2 (c) 1 (d) 2 Q.26 A point Q at a distance 3 from the point P(1, 1, 1) lying on the line joining the points A(0, -1, 3) and P, has the coordinates (a) (2, 3, -1) (b) (4, 7, -5) (c) (0, -1, 3) (d) (-2, -5, 7) Q.27 The direction cosines of a line passing through the origin and cutting the line x + 2 = y – 1 = z_ at Cos -1 v6/11 are 1 2 -1 (a) -1/ v11, 3/ v11, -1/ v11 (b) 1/ v11, 3/ v11, 1/ v11 (c) -1/ v6, 2/ v6, 1/ v6 (d) -3/ v11, -1/ v11, 1/ v11 Q.28 Which of the following planes intersects the planes x – y + 2z = 3 and 4x + 3y – z = 1 along the same line ? (a) 11x + 10y – 5z = 0 (b) 7x + 7y – 4z = 0 (c) 5x + 2y + z = 2 (d) none Q.29 Vectors perpendicular to i – j – k and in the plane of i + j + k and -i + j + k are (a) i + k (b) 2i + j + k (c) 3i + 2j + k (d) – 4i – 2j – 2k Q.30 Unit vectors a, b, c are coplanar. A unit vector d is perpendicular to them. If vector (a x b) x (c x d) = 1/6 i – 1/3 j+ 1/3 k and the angle between a and b is 30° then vector c is (a) (i – 2j + 2k) / 3 (b) (2i + j - k) / 3 (c) (- 2i – 2j + k) / 3 (d) (- i + 2j – 2k) / 3 Test – VECTOR 3D Time :-1hr.30mins. Read carefully the instructions given below : 5. The test consists of 100 marks. 6. Q. No.1 – Q. No. 25 are single choice correct answer carries 3 marks . For every wrong answer -1 will be deducted. 7. Q.No.26 – Q. No.30 are more than one choice as correct answer carries 5 marks each. For every wrong answer -2 will be deducted. 8. Students should write the correct answer on the back of their O.R.S. ANSWERS SECTION A :- 1.A 2.D 3.B 4.B 5.A 6.D 7.B 8.B 9.C 10.B 11.B 12.D 13.C 14.D 15.D 16.B 17.B 18.A 19.B 20.A 21.C 22.B,D 23.B 24.A,C 25.A 26.A,C 27.A,B,D 28.A 29.B,D 30.A,DRead More