3D Geometry · NDA previous year questions

A plane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation and makes an angle of elevation of $67\cdot 5^{\circ }$. What is the height of the plane above the ground ?

1. A.$10\sqrt{2+\sqrt{2}}\mathrm{km}$
2. B.$10\sqrt{2-\sqrt{2}}\mathrm{km}$
3. C.$5\sqrt{2+\sqrt{2}}\mathrm{km}$
4. D.$5\sqrt{2-\sqrt{2}}\mathrm{km}$

1. A.-1
2. B.0
3. C.1
4. D.4

1. A.$\frac{\gamma }{\alpha }$
2. B.$\frac{\gamma }{\beta }$
3. C.$\frac{2\gamma }{\alpha }$
4. D.$\frac{2\gamma }{\beta }$

Passage: The equation of the sphere $S$ is $x^2+y^2+z^2-4x-6y-12z+k=0$. Question: If the radius of the sphere $S$ is 8 units, what is the value of $k$ ?

1. A.$-15$
2. B.7
3. C.10
4. D.15

Passage: A line $L$ passing through the point $(-1,2,-3)$ is perpendicular to the plane $P$ given by $2x+3y+z+5=0$. Question: What are the direction ratios of a line $M$ parallel to the plane $P$ ?

1. A.$⟨-3,2,1⟩$
2. B.$⟨3,2,-6⟩$
3. C.$⟨1,3,2⟩$
4. D.$⟨2,2,-10⟩$

Passage: A line $L$ passing through the point $(-1,2,-3)$ is perpendicular to the plane $P$ given by $2x+3y+z+5=0$. Question: What is the equation of the line $L$ ?

1. A.$1-x=y+2=3-z$
2. B.$-(x+1)=y-2=z+3$
3. C.$3x+3=2y-4=6z+18$
4. D.$3x-3=2y+4=6z-18$

Passage: $P(-2,-3,5),Q(4,-1,5),R(6,-4,8)$ and $S(2,-6,10)$ are four points. Question: What is $(P{Q}^2+2Q{S}^2-2P{R}^2)$ equal to ?

1. A.$-1$
2. B.0
3. C.1
4. D.2

Passage: $P(-2,-3,5),Q(4,-1,5),R(6,-4,8)$ and $S(2,-6,10)$ are four points. Question: Consider the following statements : I. $PQ$ is parallel to $RS$. II. $PR$ is perpendicular to $QS$. Which of the statements given above is/are correct ?

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

For the following two (02) items: There are 8 points on a plane out of which 4 points are collinear. How many triangles can be formed by joining these points?

1. A.56
2. B.54
3. C.53
4. D.52

How many quadrilaterals can be formed by joining these points?

1. A.70
2. B.69
3. C.53
4. D.None of the above

For the following two (02) items: A plane P is parallel to the line having direction ratios $⟨1,3,2⟩$ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0 Which of the following are the direction ratios of the line of intersection of the given planes?

1. A.$⟨2,23,16⟩$
2. B.$⟨2,-23,-16⟩$
3. C.$⟨2,3,2⟩$
4. D.$⟨-1,3,-2⟩$

For the following two (02) items: A plane P is parallel to the line having direction ratios $⟨1,3,2⟩$ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0 What is the equation of the plane P?

1. A.2x - 20y + 29z + 2 = 0
2. B.2x + 3y + 2z - 4 = 0
3. C.2x - 20y + 29z - 2 = 0
4. D.x - 3y + 2z + 5 = 0

What is the radius of S?

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

On which one of the following planes does the centre of S lie?

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

What is the equation of the plane passing through the point (1,1,1) and perpendicular to the line whose direction ratios are (3,2,1)?

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

A line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes. If $\vec{a}=({\sin }^2\alpha )\hat{i}+({\sin }^2\beta )\hat{j}+({\sin }^2\gamma )\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ , then what is $\vec{a}.\vec{b}$ equal to?

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

Direction: Consider the following for the items that follow: Let $2x^2+2y^2+2z^2+3x+3y+3z-6=0$ be a sphere. What is the diameter of the sphere?

1. A.$\frac{5\sqrt{3}}{4}$
2. B.$\frac{5\sqrt{3}}{2}$
3. C.$\frac{3\sqrt{5}}{4}$
4. D.$\frac{3\sqrt{5}}{2}$

The centre of the sphere lies on the plane

1. A.2x + 2y + 2z - 3 = 0
2. B.4x + 4y + 4z - 3 = 0
3. C.4x + 8y + 8z - 15 = 0
4. D.4x + 8y + 8z + 15 = 0

Direction: Consider the following for the items that follow: Let S be the line of intersection of two planes x + y + z = 1 and 2x + 3y - 4z = 8. Which of the following are the direction ratios of S?

1. A.〈-7, -6, 1〉
2. B.〈-7, 6, 1〉
3. C.〈-6, 5, 1〉
4. D.〈6, 5, 1〉

If $⟨l,m,n⟩$ are direction cosines of S, then what is the value. of $43\left(l^2-m^2-n^2\right)$?

1. A.6
2. B.5
3. C.4
4. D.1

Direction: Consider the following for the items that follow: Let L : x + y + z + 4 = 0 = 2x - y - z + 8 be a line and P : x + 2y + 3z + 1 = 0 be a plane. What are the direction ratios of the line?

1. A.〈2, 1, -1〉
2. B.〈0, -1, 2〉
3. C.〈0, 1, -1〉
4. D.〈2, 3, -3〉

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?

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

The number of points represented by the equation $x=5$ on the $xy$-plane is

1. A.Zero
2. B.One
3. C.Two
4. D.Infinitely many

If $⟨l,m,n⟩$ are the direction cosines of a normal to the plane $2x-3y+6z+4=0$, then what is the value of $49(7l^2+m^2-n^2)$?

1. A.0
2. B.1
3. C.3
4. D.71

A line through $(1,-1,2)$ with direction ratios $⟨3,2,2⟩$ meets the plane $x+2y+3z=18$. What is the point of intersection of line and plane?

1. A.$(4,4,1)$
2. B.$(2,4,1)$
3. C.$(4,1,4)$
4. D.$(3,4,7)$