3D Geometry · NDA previous year questions

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?

1. A.4
2. B.3
3. C.2
4. D.1

If the direction cosines <l, m, n> of a line are connected by relation $l+2m+n=0,2l-2m+3n=0$, then what is the value of $l^2+m^2-n^2$?

1. A.$\frac{1}{101}$
2. B.$\frac{29}{101}$
3. C.$\frac{41}{101}$
4. D.$\frac{92}{101}$

Which one of the planes is parallel to the line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ ?

1. A.2x + 2y + z - 1 = 0
2. B.2x - y - 2z + 5 = 0
3. C.2x + 2y - 2z + 1 = 0
4. D.x - 2y + z - 1 = 0

What is the angle between the lines 2x = 3y = -z and 6x = -y = -4z?

1. A.0°
2. B.30°
3. C.60°
4. D.90°

What is the equation of the sphere concentric with the sphere $x^2+y^2+z^2-2x-6y-8z-5=0$ and which passes through the origin?

1. A.$x^2+y^2+z^2-2x-8z=0$
2. B.$x^2+y^2+z^2-2x-6y=0$
3. C.$x^2+y^2+z^2-6y-8z=0$
4. D.$x^2+y^2+z^2-2x-6y-8z=0$

Consider the following for the next two (02) items that follow : Let $A(1,-1,2)$ and $B(2,1,-1)$ be the end points of the diameter of the sphere $x^2+y^2+z^2+2ux+2vy+2wz-1=0$. What is $u+v+w$ equal to ?

1. A.-2
2. B.-1
3. C.1
4. D.2

Consider the following for the next two (02) items that follow : Let A(1, -1, 2) and B(2, 1, -1) be the end points of the diameter of the sphere $x^2+y^2+z^2+2ux+2vy+2wz-1=0$. $P(x,y,z)$ is any point on the sphere, then what is $P{A}^2+P{B}^2$ equal to ?

1. A.15
2. B.14
3. C.13
4. D.6.5

Consider the following for the next two (02) items that follow : Consider two lines whose direction ratios are (2, -1, 2) and (k, 3, 5). They are inclined at an angle $\frac{\pi }{4}$ What is the value of k?

1. A.4
2. B.2
3. C.1
4. D.-1

What are the direction ratios of a line which is perpendicular to both the lines ?

1. A.(1, 2, 10)
2. B.(-1, -2, 10)
3. C.(11, 12, -10)
4. D.(11, 2, -10)

Consider the following for the next two (02) items that follow : Let $\vec{a}=3\hat{i}+3\hat{j}+3\hat{k}$ and $\vec{c}=\hat{j}-\hat{k}$. Let $\vec{b}$ be such that $\vec{a}\cdot \vec{b}=27$ and $\vec{a}\times \vec{b}=9\vec{c}$ What is $\vec{b}$ equal to ?

1. A.$3\hat{i}+4\hat{j}+2\hat{k}$
2. B.$5\hat{i}+2\hat{j}+2\hat{k}$
3. C.$5\hat{i}-2\hat{j}+6\hat{k}$
4. D.$3\hat{i}+3\hat{j}+4\hat{k}$

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{6}$

Consider the equation of a sphere $x^2+y^2+z^2-4x-6y-8z-16=0$. 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 :

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?

1. A.$⟨2/3,2/3,1/3⟩$
2. B.$⟨1/3,2/3,2/3⟩$
3. C.$⟨\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}}⟩$
4. D.$⟨\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}⟩$

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?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

PQRS is a parallelogram. If $\overrightarrow{\mathrm{PR}}=\vec{\mathrm{a}}$ and $\overrightarrow{\mathrm{QS}}=\vec{\mathrm{b}}$, then what is $\overrightarrow{\mathrm{PQ}}$ equal to ?

1. A.$\vec{\mathrm{a}}+\vec{\mathrm{b}}$
2. B.$\vec{\mathrm{a}}-\vec{\mathrm{b}}$
3. C.$\frac{\vec{\mathrm{a}}+\vec{\mathrm{b}}}{2}$
4. D.$\frac{\vec{\mathrm{a}}-\vec{\mathrm{b}}}{2}$

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are θ and 2θ respectively. What is the height of the tower ?

1. A.h cos θ
2. B.h sin θ
3. C.h cos 2θ
4. D.h sin 2θ

Consider the following for the next three (03) items that follow The Plane $6x+ky+3z-12=0$ where $k\ne 0$ meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is $x^2+y^2+z^2-2x-3y-4z=0$. What is the value of k?

1. A.3
2. B.4
3. C.6
4. D.12

Consider the following for the next three (03) items that follow The Plane $6x+ky+3z-12=0$ where $k\ne 0$ meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is $x^2+y^2+z^2-2x-3y-4z=0$. If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct ?

1. A.$0<p<0.5$
2. B.$0.5<p<1$
3. C.$1<p<1.5$
4. D.$p>1.5$

Consider the following for the next three (03) items that follow The Plane $6x+ky+3z-12=0$ where $k\ne 0$ meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is $x^2+y^2+z^2-2x-3y-4z=0$. What is the equation of the line through the origin and the centre of the sphere ?

1. A.$x=y=z$
2. B.$2x=3y=4z$
3. C.$6x=3y=4z$
4. D.$6x=4y=3z$

Consider the following for the next two (2) items that follow : Let the plane $\frac{2x}{k}+\frac{2y}{3}+\frac{z}{3}=2$ pass through the point (2, 3, -6). What are the direction ratios of the normal to the plane ?

1. A.<3, 2, 1>
2. B.<2, 3, 6>
3. C.<6, 3, 2>
4. D.<1, 2, 3>

Consider the following for the next two (2) items that follow : Let the plane $\frac{2x}{k}+\frac{2y}{3}+\frac{z}{3}=2$ pass through the point (2, 3, -6). 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 ?

1. A.10
2. B.11
3. C.12
4. D.13

The locus of a point P(x, y, z) which moves in such a way that z = 7 is a

1. A.line parallel to x - axis
2. B.line parallel to y - axis
3. C.line parallel to z - axis
4. D.plane parallel to xy - plane

Consider the following statements: 1. A-line in space can have infinitely many direction ratios. 2. It is possible for certain lines that the sum of the squares of direction cosines can be equal to the sum of its direction cosines. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

The xy-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6)

1. A.internally in the ratio 2 ∶ 3
2. B.internally in the ratio 3 ∶ 2
3. C.externally in the ratio 2 ∶ 3
4. D.internally in the ratio 2 ∶ 1

The number of spheres of radius r touching the coordinate axes is

1. A.4
2. B.6
3. C.8
4. D.infinite