MCQs For NCERT Class 11 Mathematics Chapter 12 Introduction to Three-Dimensional Geometry

Please refer to the MCQ Questions for Class 11 Introduction to Three-Dimensional Geometry Maths with Answers. The following Introduction to Three-Dimensional Geometry Class 11 Mathematics MCQ Questions have been designed based on the latest syllabus and examination pattern for Class 11. Our experts have designed MCQ Questions for Class 11 Introduction to Three-Dimensional Geometry with Answers for all chapters in your NCERT Class 11 Mathematics book. You can access all MCQs for Class 11 Mathematics

Introduction to Three-Dimensional Geometry Class 11 MCQ Questions with Answers

See below Introduction to Three-Dimensional Geometry Class 11 Mathematics MCQ Questions, solve the questions and compare your answers with the solutions provided below.

Question. If A (6, – 3), B(-3, 5), C(4, – 2), P(a, b), then the ratio of the areas of the DPBC, DABC is
(a) | a + b |
(b) | a – b |
(c) | a + b + 2 |
(d) | a + b – 2 |

Question. The centre of a circle which passes through points (1, 1), (2, 3) and (–2, 2) is
(a) (3, 4) 
(b) (-1/14, -39/14)
(c) (-1/14, 39/14)
(d) (-39/14, 1/14)

Question. The points A(2a, 4a), B(2a, 6a) and C(2a + 3a, 5a) ,a > 0 are the vertices of
(a) an isosceles triangle
(b) a right angled triangle
(c) an acute angled triangle
(d) None of the above

Question. Two opposite vertices of a rectangle are (1, 3) and (5, 1). If the rest two vertices lie on the line y – x + l = 0, then l is equal to
(a) 1
(b) –1
(c) 2
(d) 3

Question. Find the value of x for which the points (x, – 1), (2, 1) and (4, 5) are collinear. 
(a) -1
(b) 1
(c) 2
(d) -2

Question. Area of quadrilateral whose vertices are (2, 3), (3, 4),(4, 5) and (5, 6), is equal to
(a) 0
(b) 4
(c) 6
(d) None of these

Question. If the area of the triangle with vertices (x,0), (1, 1) and (0, 2) is 4 sq units, then the value of x is
(a) -2
(b) – 4
(c) -6
(d) 8

Question. The middle point of the line segment joining (3, – 1) and (1, 1) is shifted by two units (in the sense of increasing y) perpendicular to the line segment.
Then, the coordinates of the point in the new position are
(a) (2 – √2, 2)
(b) (2, 2 – √3)
(c) (2 +√2, √3)
(d) None of these

Question. ABC is an isosceles triangle, if the coordinates of the base are B(1, 3) and C (-2, 7), the coordinates of vertex A can be
(a) (1,6)
(b) (-1/2 , 5)
(c) (5/6 , 6)
(d) (-8/ , 1/8)

Question. If an equilateral triangle has one vertex at the point (0, 0) and another at (3, 3), then the coordinates of the third vertex is
(a) (0, 2 3)
(b) (0, – 2 3)
(c) (-1, 2 3)
(d) None of these

Question. If a point is on the ZX-plane, then its coordinates will be
(a) (x, y, 0)
(b) (0, y, z)
(c) (x, 0, z)
(d) (x, y, z)

Question. X-axis is the intersection of two planes
(a) XY and XZ
(b) YZ and ZX
(c) XY and YZ
(d) None of these

Question. L is the foot of the perpendicular drawn from a point P (6, 7, 8) on the XY-plane, then the coordinates of point L are
(a) (6, 0, 0)
(b) (6, 7, 0)
(c) (6, 0, 8)
(d) None of these

Question. Which of the following statement is correct?
(a) The X-axis and Y-axis taken together determine a plane known as XY-plane
(b) The coordinates of a point in the XY-plane are of the form (0, 0, z)
(c) Coordinate planes divide the space into six octants
(d) All the above are correct

Question. The point (− 2, −3, − 4) lies in the
(a) first octant
(b) seventh octant
(c) second octant
(d) eight octant

Question. The distance between the points P(1, − 3, 4) and Q(− 4,1,2) is
(a) √5
(b) 2√5
(c) 3√5
(d) √48

Question. Coordinates of two points are A(1, 0) and B(-1, 0) and Qis a point which satisfies AQ – BQ = ± 1. The locus of the point Q is
(a) 12x² + 4y² = 3
(b) 12x² – 4y² = 3
(c) 12x² – 4y² + 3 = 0
(d) 12x² + 4y² + 3 = 0

Question. A point moves is such a way that the sum of squares of its distances from A(2, 0) and B(-2, 0) is always eaual to the square of the distance between A and B, then the locus of point P is
(a) x² + y² – 2 = 0
(b) x² + y² + 2 = 0
(c) x² + y² + 4 = 0
(d) x² + y² – 4 = 0

Question. If the vertices of a triangle have integral coordinates,the triangle cannot be
(a) an equilateral triangle
(b) a right angled triangle
(c) an isosceles triangle
(d) None of these

Question. Given points are A (0, 4) and B(0, – 4), the locus of P (x, y) such that|AP – BP|= 6, is
(a) 9x² – 7y² + 63 = 0
(b) 9x² – 7y² – 63 = 0
(c) 9x² + 7y² + 63 = 0
(d) None of these

Question. The orthocentre of the triangle formed by the points (0, 0), (4, 0) and (3, 4) is
(a) (2, 0)
(b) (3/2 , 2)
(c) (3/4 , 3)
(d) (3 , 3/4)

Question. The area of the region bounded by the lines y =|x – 2|, x = 1, x = 3 and the x-axis is
(a) 1
(b) 2
(c) 3
(d) 4

Question. The straight lines x = y, x – 2y = 3 and x + 2y = – 3 form a triangle, which is
(a) isosceles
(b) equilateral
(c) right angled
(d) None of these

Question. Distance of the point (3, 4, 5) from the origin is
(a) 50
(b) 3
(c) 4
(d) 5

Question. The distance of point P(3, 4, 5) from the YZ-plane is
(a) 3 units
(b) 4 units
(c) 5 units
(d) 550 units

Question. What is the length of foot of perpendicular drawn from the point P(3, 4, 5) on Y-axis?
(a) √41
(b) √34
(c) 5
(d) None of these

Question. Points P (2, 4, 6), Q(–2, –2, –2) and R(6, 10, 14) are
(a) vertices of a triangle
(b) collinear
(c) non-collinear
(d) Both (a) and (b)

Question. The point on Y-axis which is at a distance √10 from the point (1, 2, 3), is
(a) (0, 2, 0)
(b) (0, 0, 2)
(c) (0, 0, 3)
(d) None of these

Question. The x-coordinate of the incentre of the triangle where the mid-point of the sides are (0, 1), (1, 1) and (1, 0), is
(a) 2 + √2
(b) 1 + √2
(c) 2 – √2
(d) 1 – √2

Question. The orthocentre of the triangle formed by (0, 0), (8, 0) and (4, 6) is
(a) (4,8/3)
(b) (3, 4)
(c) (4, 3)
(d) (-3, 4)

Question. If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid are
(a) (7/3 ,5/3)
(b) (5/3 ,7/3)
(c) (7, 5)
(d) None of these

Question. The incentre of the triangle formed by lines x = 0, y = 0 and 3x + 4 y = 12 , is at
(a) (1/2 ,1/2)
(b) (1, 1)
(c) (1,1/2)
(d) (1/2 ,1)

Question. If t₁ + t₂ + t₃ = -t₁ t₂ t₃=  , then orthocentre of the triangle formed by the points [at₁t₂ , a(t₁ + t₂ )], [at₂ t₃ , a(t₂ +t₃)] and[at₃ t₁ , a(t₃ + t₁ )]  ,lies on
(a) (a , 0)
(b) (- a , 0)
(c) (0, a)
(d) (0, – a)

Question. If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k is equal to 
(a) 29/5
(b) 5
(c) 6
(d) 11/5

Question. If A(2, – 3) and B(-2,1) are two vertices of a triangle and third vertex moves on the line 2x + 3 y = 9, then the locus of the centroid of the triangle is
(a) 2x – 3y = 1
(b) x – y = 1
(c) 2x + 3y = 1
(d) 2x + 3y = 3

Question. The coordinates of the circumcentre of the triangle with vertices (8, 6), (8, – 2) and (2, – 2) are
(a) (6,2/3)
(b) (8, 2)
(c) (5, -2)
(d) (5, 2)

Question. If A and B are two points having coordinates (3, 4)and (5, – 2) respectively and P is a point such that PA = PB and area of DPAB = 10 sq units, then the coordinates of P are
(a) (7, 4) and (13, 2)
(b) (7, 2) and (1, 0)
(c) (2, 7) and (4, 13)
(d) None of the above

Question. If area of triangle with vertices (0, 0), (0, 6) and (a,b) is 15 sq units, then
(a) a = ± 5, b = 5
(b) a = ±10, b = 5
(c) a = ± 5, b = 2
(d) a = ± 5, b can take any real value

Question. If the vertices P,Q, R of a ΔPQR are rational points,which of the following points of the DPQR is (are) always rational points ?
(a) Centroid
(b) Incentre
(c) Circumcentre
(d) Orthocentre
(A rational point is a point both of whose coordinates are rational numbers)

Question. ABC is a triangle with vertices A(- 1, 4), B(6, – 2) andC(- 2, 4). D, E and F are the points which divide each AB, BC and CA respectively in the ratio 3 : 1 internally. Then, the centroid of ΔDEF is
(a) (3, 6)
(b) (1, 2)
(c) (4, 8)
(d) (-3,6)

Question. If (0, 1) is the orthocentre and (2, 3) is the centroid of a triangle. Then, its circumcentre is
(a) (3, 2)
(b) (1, 0)
(c) (4, 3)
(d) (3, 4)

Question. The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (- 2,6). The third vertex is
(a) (0, 0)
(b) (4, 7)
(c) (7, 4)
(d) (7, 7)

Question. The locus of a point P which moves such that 2 PA = 3 PB, where coordinates of points A and Bare (0, 0) and (4, – 3), is
(a) 5x² – 5y² – 72x + 54y + 225 = 0
(b) 5x² + 5y² – 72x + 54y + 225 = 0
(c) 5x² + 5y² + 72x – 54y + 225 = 0
(d) 5x² + 5y² – 72x – 54y – 225 = 0

Question. The locus of a point whose difference of distance from points (3, 0) and (- 3,0) is 4, is
(a) x²/4 – y²/5 = 1
(b) x²/5 – y²/4 = 1
(c) x²/2 – y²/3 = 1
(d) x²/3 – y²/2 = 1

Question. The locus of a point whose difference of distance from points (3, 0) and (- 3,0) is 4, is
(a) x²/4 – y²/5 = 1
(b) x²/5 – y²/4 = 1
(c) x²/2 – y²/3 = 1
(d) x²/3 – y²/2 = 1

Question. What is the equation of the locus of a point which moves such that 4 times its distance from the x-axis is the square of its distance from the origin ?
(a) x² + y² – 4y = 0
(b) x² +  y² – 4|y| = 0
(c) x² +  y² – 4x = 0
(d) x² +  y² – 4|x| = 0

Question. A rod of length l slides with its ends on two perpendicular lines. The locus of a point which divides it in the ratio 1 : 2, is
(a) 36x² + 9y² = 4I² 
(b) 36x² + 9y² =  I² 
(c) 9x² + 36y² = 4I² 
(d) 9x² + 36y² = 4I² 

Question. A point moves in such a way that the sum of its distances from two fixed points (ae, 0) and (-ae, 0) is 2a. Then, the locus of the points is
(a) x²/a² + y²/a2(I-e²) = 1
(b) x²/a² – y²/a2(I-e²) = 1
(c) x²/a²(I-e2) + y²/a² = 1
(d) None of these

Question. If two points A(a, 0) and B(-a, 0) are stationary and if ∠A – ∠B = q in ΔABC, the locus of C is
(a) x² + y² + 2xy tanθ = a²
(b) x² – y² + 2xy tanθ = a²
(c) x² + y² + 2xy cotθ = a²
(d) x² – y² + 2xy cotθ = a²

Question. If A(cos a, sin a), B(sin a, – cos a), C(1, 2) are the vertices of a DABC, then the locus of centroid of triangle is
(a) x² + y² + 2x – 4y + 1= 0
(b) 3x² + y² – 2x – 4y + 1= 0
(c) x² + y² – 2x – 4y + 3= 0
(d) None of the above

Question. The coordinate axes rotated through an angle 135°. If the coordinates of a point P in the new system are known to be (4, – 3), then the coordinates of P in the original system are
(a) (1/√2 . 7/√2)
(b) (1/√2 . -7/√2)
(c) (-1/√2 . -7/√2)
(d) (-1/√2 . 7/√2)

Question. If two vertices of a triangle are (- 2,3) and (5, – 1). Orthocentre lies at the origin and centroid on the line x + y = 7, then the third vertex lies at
(a) (7, 4)
(b) (8, 14)
(c) (12, 21)
(d) None of these

Question. If two vertices of an equilateral triangle are (0, 0) and (3,3 3), then the third vertex is
(a) (3, – 3)
(b) (-3,3)
(c) (-3,3 3)
(d) None of these

Question. If a point P(4, 3) is rotated through an angle 45° in anti-clockwise direction about origin, then coordinates of P in new position are
(a) (1/√2 . 7/√2)
(b) (-7/√2 . -1/√2)
(c) (-1/√2 . 7/√2)
(d) (1/√2 . -7/√2)

Question. If origin is shifted to (7, – 4), then point (4, 5) shifted to
(a) (-3,9)
(b) (3, 9)
(c) (11, 1)
(d) None of these

Question. If the axes are rotated through an angle 60°, the coordinates of a point in the new systemare (2, – √3), then its original coordinates are
(a) (5/3 . √2/3)
(b) (-5/3 . √2/3)
(c) (5/2 . √3/2)
(d) (-5/2 . √3/2)

Question. Without change of axes the origin is shifted to (h, k), then from the equation x² + y² – 4x + 6y – 7 = 0 the terms containing linear powers are missing. Then,point (h, k) is
(a) (3, 2)
(b) (–3, 2)
(c) (2, –3)
(d) (–2, –3)

Question. If the distance between the points (a, 0, 1) and (0, 1, 2) is √27, then the value of a is
(a) 5
(b) ± 5
(c) − 5
(d) None of these

Question. If P(− 2, 3, 5),Q(1,2, 3) and R(7, 0, − 1), then
(a) P, Qand R are collinear
(b) P, Qand R are non-collinear
(c) PR =4PQ
(d) PQ + QR = 2PR

Question. Let the opposite angular points of a square be (3, 4) and (1, –1). Then, the coordinates of the remaining angular points are
(a) (9/2 , 1/2) and (-1/2 , 5/2)
(b) (9/2 ,-1/2) and (-1/2 , 5/2)
(c) (-9/2 ,1/2) and (-1/2 , 5/2)
d) None of these 

Question. If the points (a₁ , b₂ ) , (a₂ , b₂)  and (a₃ , b₃ ) are collinear,then lines a_ix + b_iy + 1 = 0 for i = 1,2,3, are
(a) concurrent
(b) identical
(c) parallel
(d) None of these

Question. 

0, then the points (x₁ , y₁ ) , (x₂ , y₂) and (x₃ , y₃) are
(a) vertices of an equilateral triangle
(b) vertices of a right angled triangle
(c) vertices of an isosceles triangle
(d) None of the above

Question. The length of altitude through A of the ΔABC, where A º (- 3,0), B º (4, – 1), C º (5,2), is
(a) 2/√10
(b) 4/√10
(c) 11/√10
(d) 22/√10

Question . An equilateral triangle has each side equal to a. If the coordinates of its vertices are (x₁ , y₁), (x₂ , y₂) and (x₃ , y₃ ) , then the square of the determinant

equals to
(a) 3a⁴ 
(b) 3/4a⁴
(c) 4a⁴
(d) None of these

Question. If the point A (3, 2, 2) and B (5, 5, 4) are equidistant from P, which is on X-axis, then the coordinates of P are

Question. The points (5, –1, 1), (7, – 4, 7), (1, – 6, 10) and (−1, − 3, 4) are
(a) the vertices of a rectangle
(b) the vertices of a square
(c) the vertices of a rhombus
(d) None of these

Question. The coordinates of the point R, which divides the segment joining P(x₁ , y₁ , z₁ ) and Q (x₂ , y₂ , z₂ ) internally in the ratio k : 1, are

Question. The coordinate of the point P which divides the line joining the points A(−2, 0, 6) and B(10, − 6, − 12) internally in the ratio 5 : 1.
(a) (8, 5, 9) (b)
(–8, 5, 9)
(c) (8, –5, –9)
(d) None of these

Question. The ratio in which the line joining (2, 4, 5) and (3, 5, –4) is divided by the YZ-plane, is
(a) 2 : 3
(b) 3 : 2
(c) –2 : 3
(d) 4 : –3

Question. A point on XOZ-plane divides the join of (5, –3, –2) and (1, 2, –2) at

Question. The centroid of the triangle, if the mid-point of the sides of triangles are D(1, 2, –3), E(3, 0, 1) and F(–1, 1, –4), is
(a) (1, 2, –1)
(b) (1, 1, –2)
(c) (0, 1, –2)
(d) None of these

Case Based MCQs

Four students in traditional dresses represent four states of India, standing at the points represented by O(0, 0, 0), A(a, 0, 0), B(0, b, 0)C(0, 0, c). If a girl representing BHARATMATA be placed in such a way that she is equidistant from the four students, then answer the following questions which are based on above it.

Question. x-coordinate of girl representing BHARATMATA is
(a) a
(b) a/2
(c) a/3
(d) a/4

Question. y-coordinate of girl representing BHARATMATA is
(a) a
(b) b/2
(c) 2b
(d) 3b

Question. z-coordinate of girl representing BHARATMATA is
(a) b
(b) c
(c) c/2
(d) 2c

Question. Which concept is used for finding the coordinates of point?
(a) Distance formula
(b) Section formula
(c) Mid-point formula
(d) All of these

Question. Which of the following is coordinates of origin point?
(a) (0, 0, 0)
(b) (0, b, 0)
(c) (a, 0, 0)
(d) (0, 0, c)

Abhay and Birsen are roaming in a circular ground. At a moment their positions are A(2,1, − 3) and B(5, − 8, 3). And the points A and B are the vertices of the diameter Using above information, answer the following.

Question. The coordinates of the centre is

Question. The coordinates of the point which divides the line AB into the ratio 3 :1 internally, is

Question. The value of the diameter is
(a) 2√14
(b) √14
(c) √129
(d) 3√14

Question. The value of the radius is
(a) √14
(b) √14/2
(c) √129/2
(d) 3√14/2