Class 11 Mathematics Introduction To Three-Dimensional Geometry MCQs Set D

Basic Level

Question. From which of the following the distance of the point (1,2,3) is \(\sqrt{10}\)
(a) Origin
(b) \(x\)-axis
(c) \(y\)-axis
(d) \(z\)-axis
Answer: (c)

Question. If \(A(1,2,3); B(-1,-1,-1)\) be the points, then the distance \(AB\) is
(a) \(\sqrt{5}\)
(b) \(\sqrt{21}\)
(c) \(\sqrt{29}\)
(d) None of these
Answer: (c)

Question. Perpendicular distance of the point (3,4,5) from the \(y\)-axis, is
(a) \(\sqrt{34}\)
(b) \(\sqrt{41}\)
(c) 4
(d) 5
Answer: (a)

Question. Distance between the points (1,3,2) and (2,1,3) is
(a) 12
(b) \(\sqrt{12}\)
(c) \(\sqrt{6}\)
(d) 6
Answer: (c)

Question. The shortest distance of the point (a,b,c) from the \(x\)-axis is
(a) \(\sqrt{a^2+b^2}\)
(b) \(\sqrt{b^2+c^2}\)
(c) \(\sqrt{c^2+a^2}\)
(d) \(\sqrt{a^2+b^2+c^2}\)
Answer: (b)

Question. Points (1,1,1), (-2,4,1), (-1,5,5) and (2,2,5) are the vertices of
(a) Rectangle
(b) Square
(c) Parallelogram
(d) Trapezium
Answer: (c)

Question. The triangle formed by the points (0,7,10), (–1,6,6) (–4,9,6) is
(a) Equilateral
(b) Isosceles
(c) Right angled
(d) Right angled isosceles
Answer: (d)

Question. The points \(A(5,-1,1); B(7,4,7); C(1,-6,10)\) and \(D(-1,-3,4)\) are vertices of a
(a) Square
(b) Rhombus
(c) Rectangle
(d) None of these
Answer: (d)

Question. The coordinates of a point which is equidistant from the points (0,0,0), (a,0,0), (0,b,0) and (0,0,c) are given by
(a) \(\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right)\)
(b) \(\left(-\frac{a}{2}, -\frac{b}{2}, -\frac{c}{2}\right)\)
(c) \(\left(-\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right)\)
(d) \(\left(\frac{a}{2}, \frac{b}{2}, -\frac{c}{2}\right)\)
Answer: (a)

Question. If \(A(1,2,-1)\) and \(B(-1,0,1)\) are given, then the coordinates of P which divides \(AB\) externally in the ratio \(2:1\), are
(a) \(\frac{1}{3}(1, 4, -1)\)
(b) \((3, 4, -3)\)
(c) \(\frac{1}{3}(3, 4, -3)\)
(d) None of these
Answer: (b)

Question. The coordinates of the point which divides the join of the points (2, –1, 3) and (4, 3, 1) in the ratio \(3 : 4\) internally are given by
(a) \(\left(\frac{2}{7}, \frac{20}{7}, \frac{10}{7}\right)\)
(b) \(\left(\frac{15}{7}, \frac{20}{7}, \frac{3}{7}\right)\)
(c) \(\left(\frac{10}{7}, \frac{15}{7}, \frac{2}{7}\right)\)
(d) \(\left(\frac{20}{7}, \frac{5}{7}, \frac{15}{7}\right)\)
Answer: (d)

Question. Points (–2, 4, 7), (3, –6, –8) and (1, –2, –2) are
(a) Collinear
(b) Vertices of an equilateral triangle
(c) Vertices of an isosceles triangle
(d) None of these
Answer: (a)

Question. Which of the following set of points are non-collinear
(a) (1, –1, 1), (–1, 1, 1), (0, 0, 1)
(b) (1, 2, 3), (3, 2, 1), (2, 2, 2)
(c) (–2, 4, –3), (4, –3, –2), (–3, –2, 4)
(d) (2, 0, –1), (3, 2, –2), (5, 6, –4)
Answer: (c)

Question. If the points (–1, 3, 2), (–4, 2, –2) and (5, 5, \(\lambda\)) are collinear, then \(\lambda =\)
(a) –10
(b) 5
(c) –5
(d) 10
Answer: (d)

Question. The area of triangle whose vertices are (1, 2, 3), (2, 5, –1) and (–1, 1, 2) is
(a) 150 sq. units
(b) 145 sq. units
(c) \(\frac{\sqrt{155}}{2}\) sq. units
(d) \(\frac{155}{2}\) sq. units
Answer: (c)

Question. Volume of a tetrahedron is K (area of one face) (length of perpendicular from the opposite vertex upon it), where K is
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{1}{6}\)
Answer: (b)

Question. A point moves so that the sum of its distances from the points (4,0,0) and (–4,0,0) remains 10. The locus of the point is
(a) \(9x^2 + 25y^2 + 25z^2 = 225\)
(b) \(9x^2 + 25y^2 - 25z^2 = 225\)
(c) \(25x^2 + 25y^2 + 9z^2 = 225\)
(d) \(9x^2 + 25y^2 + 25z^2 + 225 = 0\)
Answer: (c)

Question. If the sum of the squares of the distances of a point from the three coordinate axes be 36, then its distance from the origin is
(a) 6
(b) \(3\sqrt{2}\)
(c) \(2\sqrt{3}\)
(d) None of these
Answer: (b)

Question. All the points on the x-axis have
(a) \(x = 0\)
(b) \(y = 0\)
(c) \(x = 0, y = 0\)
(d) \(y = 0, z = 0\)
Answer: (d)

Question. The equations \(|x|=p, |y|=p, |z|=p\) in xyz space represent
(a) Cube
(b) Rhombus
(c) Sphere of radius p
(d) Point (p,p,p)
Answer: (a)

Question. The orthocentre of the triangle with vertices (1,2,3), (2,3,1) and (3,1,2) is
(a) (1, 1, 1)
(b) (2, 2, 2)
(c) (6, 6, 6)
(d) None of these
Answer: (b)

Question. If \(a+b+c = \lambda\), then circumcentre of the triangle with vertices (a,b,c); (b,c,a) and (c,a,b) is
(a) \((\lambda, \lambda, \lambda)\)
(b) \((\lambda/2, \lambda/2, \lambda/2)\)
(c) \((\lambda/3, \lambda/3, \lambda/3)\)
(d) None of these
Answer: (c)

Question. (–1,6,6), (–4,9,6) are two vertices of \(\Delta ABC\). If its centroid be (–5/3, 22/3, 22/3), then its third vertex is
(a) (0, 7, 10)
(b) (7, 0, 10)
(c) (10, 0, 7)
(d) None of these
Answer: (a)

Question. If points (2, 3, 4), (5, a, 6) and (7, 8, b) are collinear, then values of a and b are
(a) \(a=6, b=-\frac{22}{3}\)
(b) \(a=6, b=\frac{22}{3}\)
(c) \(a=\frac{22}{3}, b=6\)
(d) \(a=-\frac{22}{3}, b=-6\)
Answer: (b)

Direction cosines and Projection

Question. If a line makes angles of 30° and 45° with x-axis and y-axis, then the angle made by it with z-axis is
(a) 45°
(b) 60°
(c) 120°
(d) None of these
Answer: (b)

Question. If a straight line in space is equally inclined to the coordinate axes, the cosine of its angle of inclination to any one of the axes is
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{\sqrt{3}}\)
(d) \(\frac{1}{\sqrt{2}}\)
Answer: (c)

Question. If the length of a vector be 21 and direction ratios be 2, –3, 6, then its direction cosines are
(a) \(\frac{2}{21}, -\frac{1}{7}, \frac{2}{7}\)
(b) \(\frac{2}{7}, -\frac{3}{7}, \frac{6}{7}\)
(c) \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\)
(d) None of these
Answer: (b)

Question. If O is the origin, \(OP = 3\) with d.r.‘s –1, 2, –2 then the co-ordinates of P are
(a) (–1, 2, –2)
(b) (1, 2, 2)
(c) \(\left(-\frac{1}{9}, -\frac{2}{9}, -\frac{2}{9}\right)\)
(d) (3, 6, – 9)
Answer: (a)

Question. The numbers 3, 4, 5 can be
(a) Direction cosines of a line
(b) Direction ratios of a line in space
(c) Coordinates of a point on the plane \(y=4, z=0\)
(d) Co-ordinates of a point on the plane \(x+y-z=0\)
Answer: (b)

Question. If l, m, n are the d.c.'s of a line, then
(a) \(l^2 + m^2 + n^2 = 0\)
(b) \(l^2 + m^2 + n^2 = 1\)
(c) \(l + m + n = 1\)
(d) \(l = m = n = 1\)
Answer: (b)

Question. If a line lies in the octant OXYZ and it makes equal angles with the axes, then
(a) \(l = m = n = \frac{1}{\sqrt{3}}\)
(b) \(l = m = n = \pm\frac{1}{\sqrt{3}}\)
(c) \(l = m = n = -\frac{1}{\sqrt{3}}\)
(d) \(l = m = n = \pm\frac{1}{\sqrt{2}}\)
Answer: (a)

Question. If a line makes equal angle with axes, then its direction ratios will be
(a) 1, 2, 3
(b) 3, 1, 2
(c) 3, 2, 1
(d) 1, 1, 1
Answer: (d)

Question. The coordinates of the point P are (x, y, z) and the direction cosines of the line OP, when O is the origin, are l, m, n. If OP = r, then
(a) \(l = x, m = y, n = z\)
(b) \(l = xr, m = yr, n = zr\)
(c) \(x = lr, y = mr, z = nr\)
(d) None of these
Answer: (c)

Question. The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the 3 concurrent edges of the cube are coordinate axes)
(a) \(\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\)
(b) –1, 1, –1
(c) 2, –2, 1
(d) 1, 1, 1
Answer: (d)

Question. If the direction ratios of a line are 1, –3, 2, then the direction cosines of the line are
(a) \(\frac{1}{\sqrt{14}}, -\frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}\)
(b) \(\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}\)
(c) \(-\frac{1}{\sqrt{14}}, \frac{3}{\sqrt{14}}, -\frac{2}{\sqrt{14}}\)
(d) \(-\frac{1}{\sqrt{14}}, -\frac{2}{\sqrt{14}}, -\frac{3}{\sqrt{14}}\)
Answer: (a)

Question. If a line make \(\alpha, \beta, \gamma\) with the positive direction of x, y and z-axis respectively. Then \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) is
(a) 1/2
(b) –1/2
(c) –1
(d) 1
Answer: (d)

Question. The direction-cosines of the line joining the points (4, 3, –5) and (–2, 1, –8) are
(a) \(\left(\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\right)\)
(b) \(\left(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\right)\)
(c) \(\left(\frac{6}{7}, \frac{3}{7}, \frac{2}{7}\right)\)
(d) None of these
Answer: (a)

Question. The direction ratios of the line joining the points (4, 3, –5) and (–2, 1, –8) are
(a) \(\left(\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\right)\)
(b) 6, 2, 3
(c) 2, 4, –13
(d) None of these
Answer: (b)

Question. The coordinates of a point P are (3, 12, 4) with respect to origin O, then the direction cosines of OP are
(a) 3, 12, 4
(b) \(\frac{1}{4}, \frac{1}{3}, \frac{1}{2}\)
(c) \(\frac{3}{13}, \frac{1}{13}, \frac{2}{13}\)
(d) \(\frac{3}{13}, \frac{12}{13}, \frac{4}{13}\)
Answer: (d)

Question. The direction cosines of a line segment AB are \(-\frac{2}{\sqrt{17}}, \frac{3}{\sqrt{17}}, -\frac{2}{\sqrt{17}}\). If \(AB = \sqrt{17}\) and the coordinates of A are (3, –6, 10), then the coordinates of B are
(a) (1, –2, 4)
(b) (2, 5, 8)
(c) (–1, 3, –8)
(d) (1, –3, 8)
Answer: (d)