Three Dimensional Geometry · Mathematics · NDA
MCQ (Single Correct Answer)
1
The lines 2x = 3y = $$-$$ z and 6x = $$-$$ y = $$-$$ 4z
NDA 2015 Paper 2
2
The radius of the sphere $$3{x^2} + 3{y^2} + 3{z^2} - 8x + 4y + 8z - 15 = 0$$ is
NDA 2015 Paper 2
3
The direction ratios of the line perpendicular to the lines with direction ratios < 1, $$-$$2, $$-$$2 > and < 0, 2, 1 > are
NDA 2015 Paper 2
4
What are the coordinates of the foot of the perpendicular drawn from the point (3, 5, 4) on the plane z = 0 ?
NDA 2015 Paper 2
5
The lengths of the intercepts on the coordinate axes made by the plane 5x + 2y + z $$-$$ 13 = 0 are
NDA 2015 Paper 2
6
Consider the following statements.
I. The coordinates of Q are (4, $$-$$3, $$-$$1).
II. PQ is the length more than 8 units.
III. The point (1, $$-$$1, $$-$$3) is the mid-point of the line segment PQ and lies on the given plane.
which of the above statements are correct?
I. The coordinates of Q are (4, $$-$$3, $$-$$1).
II. PQ is the length more than 8 units.
III. The point (1, $$-$$1, $$-$$3) is the mid-point of the line segment PQ and lies on the given plane.
which of the above statements are correct?
NDA 2016 Paper 2
7
Consider the following statements.
I. The direction ratios of the segment PQ are <3, $$-$$2, 2>
II. The sum of the squares of direction cosines of the line segment PQ is unity.
Which of the above statements is/are correct?
I. The direction ratios of the segment PQ are <3, $$-$$2, 2>
II. The sum of the squares of direction cosines of the line segment PQ is unity.
Which of the above statements is/are correct?
NDA 2016 Paper 2
8
What are the direction ratios of the line of intersection of the given planes?
NDA 2016 Paper 2
9
What is the equation of the line L ?
NDA 2016 Paper 2
10
What are the direction ratios of the line of intersection of the given planes?
NDA 2016 Paper 1
11
What is the equation of the plane P?
NDA 2016 Paper 1
12
If the plane P touches the sphere $${x^2} + {y^2} + {z^2} = {r^2}$$, then what is r equal to?
NDA 2016 Paper 1
13
The length of the normal from origin to the plane x + 2y $$-$$ 2z = 9 is equal to
NDA 2017 Paper 2
14
If $$\alpha$$, $$\beta$$ and $$\gamma$$ are the angles which the vector OP (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct?
1. $${\cos ^2}\alpha + {\cos ^2}\beta = {\sin ^2}\gamma $$
2. $${\sin ^2}\alpha + {\sin ^2}\beta = {\cos ^2}\gamma $$
3. $${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma = 2$$
Select the correct answer using the code given below.
1. $${\cos ^2}\alpha + {\cos ^2}\beta = {\sin ^2}\gamma $$
2. $${\sin ^2}\alpha + {\sin ^2}\beta = {\cos ^2}\gamma $$
3. $${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma = 2$$
Select the correct answer using the code given below.
NDA 2017 Paper 2
15
The point of intersection of the line joining the points ($$-$$3, 4, $$-$$8) and (5, $$-$$6, 4) with XY-plane is
NDA 2017 Paper 2
16
If the angle between the lines whose direction ratios are (2, $$-$$1, 2) and is $${\pi \over 4}$$, then the smaller value of x is
NDA 2017 Paper 2
17
A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B, and C respectively. The locus of the centre of the sphere OABC, O being the origin, is
NDA 2017 Paper 2
18
The equation of the plane passing through the line of intersection of the planes x + y + z = 1, 2x + 3y + 4z = 7, and perpendicular to the plane x $$-$$ 5y + 3z = 5 is given by
NDA 2017 Paper 2
19
A straight line with direction cosines <0, 1, 0> is
NDA 2017 Paper 1
20
(0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c) are four distinct points. What are the coordinates of the point which is equidistant from the four points?
NDA 2017 Paper 1
21
The points P(3, 2, 4), Q(4, 5, 2), R(5, 8, 0) and S(2, $$-$$1, 6) are
NDA 2017 Paper 1
22
The line passing through the points (1, 2, $$-$$1) and (3, $$-$$1, 2) meets the yz-plane at which one of the following points?
NDA 2017 Paper 1
23
Under which one of the following conditions are the lines x = ay + b; z = cy + d and x = ey + f; z = gy + h perpendicular?
NDA 2017 Paper 1
24
What is the distance of the point (2, 3, 4) from the plane 3x $$-$$ 6y + 2z + 11 = 0 ?
NDA 2018 Paper 2
25
Coordinates of the points O, P, Q and R respectively (0, 0, 0), (4, 6, 2m), (2, 0, 2n) and (2, 4, 6) L, M, N and K OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjacent sides LK and LM are each of length $$\sqrt 2 $$ ?
NDA 2018 Paper 2
26
The line $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}$$ is given by
NDA 2018 Paper 2
27
Consider the following statements
1. The angle between the planes 2x $$-$$ y + z = 1 and x + y + 2z = 3 is $${\pi \over 3}$$
2. The distance between the planes 6x $$-$$ 3y + 6z + 2 = 0 and 2x $$-$$ y + 2z + 4 = 0 is $${{10} \over 9}$$
Which of the above statement is/are correct?
1. The angle between the planes 2x $$-$$ y + z = 1 and x + y + 2z = 3 is $${\pi \over 3}$$
2. The distance between the planes 6x $$-$$ 3y + 6z + 2 = 0 and 2x $$-$$ y + 2z + 4 = 0 is $${{10} \over 9}$$
Which of the above statement is/are correct?
NDA 2018 Paper 2
28
Let the coordinates of the points A, B, C be (1, 8, 4), (0, $$-$$11, 4) and (2, $$-$$3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC ?
NDA 2018 Paper 1
29
What is the equation of the plane passing through the points ($$-$$2, 6, $$-$$6), ($$-$$3, 10, $$-$$9) and ($$-$$5, 0, $$-$$6) ?
NDA 2018 Paper 1
30
A sphere of constant radius r through the origin intersects the coordinate axes in A, B and C. What is the locus of the centroid of the $$\Delta$$ABC ?
NDA 2018 Paper 1
31
The coordinates of the vertices P, Q and R of a triangle PQR are (1, $$-$$1, 1), (3, $$-$$2, 2) and (0, 2, 6) respectively. If $$\angle$$RQP = $$\theta$$, then what is $$\angle$$PRQ equal to?
NDA 2018 Paper 1
32
If the line $${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$$ lies on the plane $$2x - 4y + z = 7$$, then what is the value of k?
NDA 2019 Paper 2
33
A point on the line $${{x - 1} \over 1} = {{y - 3} \over 2} = {{z + 2} \over 7}$$ has coordinates
NDA 2019 Paper 2
34
If the points (x, y, $$-$$3), (2, 0, $$-$$1) and C (4, 2, 3) lie on a straight line, then what are the values of x and y respectively?
NDA 2019 Paper 2
35
A straight line passes through the point (1, 1, 1) makes an angle 60$$^\circ$$ with the positive direction of Z-axis, and the cosine of the angles made by it with the positive directions of the Y-axis and the X-axis are in the ratio $$\sqrt 3 $$ : 1. What is the acute angle between the two possible positions of the line?
NDA 2019 Paper 2
36
A point on a line has coordinates (p + 1, p $$-$$ 3, $$\sqrt 2 $$p) where p is any real number. What are the direction cosines of the line?
NDA 2019 Paper 2
37
The centroid of the triangle with vertices A(2, $$-$$3, 3), B(5, $$-$$3, $$-$$4) and C(2, $$-$$3, $$-$$2) is the point
NDA 2019 Paper 1
38
What is the radius of the sphere $${x^2} + {y^2} + {z^2} - 6x + 8y - 10z + 1 = 0$$ ?
NDA 2019 Paper 1
39
The equation of the plane passing through the intersection of the planes 2x + y + 2z = 9, 4x $$-$$ 5y $$-$$ 4z = 1 and the point (3, 2, 1) is
NDA 2019 Paper 1
40
The distance between the parallel planes $$4x - 2y + 4z + 9 = 0$$ and $$8x - 4y + 8z + 21 = 0$$
NDA 2019 Paper 1
41
What are the direction cosines of Z-axis?
NDA 2019 Paper 1
42
If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, then what is $u + v + w$ equal to?
NDA Mathematics 21 April 2024
43
If $\langle l, m, n \rangle$ are the direction cosines of a normal to the plane $2x − 3y + 6z + 4 = 0$, then what is the value of $49(7l^2 + m^2 − n^2)$?
NDA Mathematics 21 April 2024
44
A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is the point of intersection of line and plane?
NDA Mathematics 21 April 2024
45
If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, then what is $3p^2$ equal to?
NDA Mathematics 21 April 2024
46
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
NDA Mathematics 3 September 2023
47
Which one of the planes is parallel to the line $\rm \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ ?
NDA Mathematics 3 September 2023
48
What is the angle between the lines 2x = 3y = -z and 6x = -y = -4z?
NDA Mathematics 3 September 2023
49
What is the equation of the sphere concentric with the sphere x2 + y2 + z2 - 2x - 6y -8z - 5 = 0 and which passes through the origin?
NDA Mathematics 3 September 2023
50
A point P lies on the line joining A(1, 2, 3) and B(2, 10, 1). If z-coordinate of P is 7, what is the sum of other two coordinates?
NDA Mathematics 3 September 2023
51
What is u + v + w equal to ?
NDA Mathematics 16 April 2023
52
P(x, y, z) is any point on the sphere, then what is PA2 + PB2 equal to ?
NDA Mathematics 16 April 2023
53
What is the value of k?
NDA Mathematics 16 April 2023
54
What are the direction ratios of a line which is perpendicular to both the lines ?
NDA Mathematics 16 April 2023
55
Consider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct?
1. The points are the vertices of a rectangle ABCD.
2. The mid-point of A C is the same as that of BD.
Select the correct answer using the code given below :
NDA Mathematics 4 September 2022
56
Consider the equation of a sphere x2 + y2 + z2 − 4x − 6y − 8z − 16 = 0.
Which of the following statements is/are correct ?
1. z-axis is tangent to the sphere.
2. The centre of the sphere lies on the plane x + y + z − 9 = 0.
Select the correct answer using the code given below :
NDA Mathematics 4 September 2022
57
A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
NDA Mathematics 4 September 2022
58
Consider the following statements :
1. The direction ratios of y-axis can be <0, 4, 0>
2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>
Which of the statements given above is/are correct?
NDA Mathematics 4 September 2022
59
What is the value of k?
NDA Mathematics 10 April 2022
60
If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct ?
NDA Mathematics 10 April 2022
61
What is the equation of the line through the origin and the centre of the sphere ?
NDA Mathematics 10 April 2022
62
What are the direction ratios of the normal to the plane ?
NDA Mathematics 10 April 2022
63
If p, q and r are the intercepts made by the plane on the coordinate axes respectively, then what is (p + q + r) equal to ?
NDA Mathematics 10 April 2022
64
What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)?
NDA Mathematics 18 April 2021
65
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l4 + m4 + n4 equal to?
NDA Mathematics 18 April 2021
66
What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)?
NDA Mathematics 18 April 2021
67
The foot of the perpendicular drawn from the origin to the plane x + y + z = 3 is
NDA Mathematics 18 April 2021