3D Geometry · NDA previous year questions

For the next three (03) items that follow: A straight line passes through $(1,-2,3)$ and perpendicular to the plane $2x+3y-z=7$. What is the image of the point $(1,-2,3)$ in the plane?

1. A.$(2,-1,5)$
2. B.$(-1,2,-3)$
3. C.$(5,4,1)$
4. D.None of the above

For the next two (02) items that follow: Consider the spheres $x^2+y^2+z^2-4y+3=0$ and $x^2+y^2+z^2+2x+4z-4=0$. What is the distance between the centres of the two spheres?

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

For the next two (02) items that follow: Consider the spheres $x^2+y^2+z^2-4y+3=0$ and $x^2+y^2+z^2+2x+4z-4=0$. Consider the following statements: 1. The two spheres intersect each other. 2. The radius of first sphere is less than that of second sphere. 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

A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. What are the direction ratios of the line?

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

What is the equation of the sphere with unit radius having centre at the origin?

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

What is the sum of the squares of direction cosines of $x$-axis?

1. A.0
2. B.$\frac{1}{3}$
3. C.1
4. D.3

The shadow of a tower standing on a level plane is found to be 50 m longer when the Sun's elevation is $30^{\circ }$ than when it is $60^{\circ }$. The height of the tower is

1. A.25 m
2. B.$25\sqrt{3}\text{ m}$
3. C.50 m
4. D.None of the above

The angle of elevation of the top of a tower of height H from the foot of another tower in the same plane is $60^{\circ }$ and the angle of elevation of the top of the second tower from the foot of the first tower is $30^{\circ }$. If h is the height of the other tower, then which one of the following is correct?

1. A.$H=2h$
2. B.$H=\sqrt{3}h$
3. C.$H=3h$
4. D.None of the above

If $\csc \theta +\cot \theta =c$, then what is $\cos \theta$ equal to?

1. A.$\frac{c}{c^2-1}$
2. B.$\frac{c}{c^2+1}$
3. C.$\frac{c^2-1}{c^2+1}$
4. D.None of the above

If $\sin \theta +2\cos \theta =1$, then what is $2\sin \theta -\cos \theta$ equal to?

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

If $A+B=90^{\circ }$, then what is $\sqrt{\sin A\sec B-\sin A\cos B}$ equal to?

1. A.$\sin A$
2. B.$\cos A$
3. C.$\tan A$
4. D.0

What is ${\tan }^{4}A-{\sec }^{4}A+{\tan }^{2}A+{\sec }^{2}A$ equal to?

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

What is the angle between the planes $2x-y-2z+1=0$ and $3x-4y+5z-3=0$?

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

If the straight line $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$ is parallel to the plane $ax+by+cz+d=0$, then which one of the following is correct?

1. A.$l+m+n=0$
2. B.$a+b+c=0$
3. C.$\frac{a}{l}+\frac{b}{m}+\frac{c}{n}=0$
4. D.$al+bm+cn=0$

The sum of the direction cosines of z-axis is

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

What is the distance between the planes $x-2y+z-1=0$ and $-3x+6y-3z+2=0$?

1. A.3 unit
2. B.1 unit
3. C.0
4. D.None of the above

If a line makes $30^{\circ }$ with the positive direction of x-axis, angle $\beta$ with the positive direction of y-axis and angle $\gamma$ with the positive direction of z-axis, then what is ${\cos }^2\beta +{\cos }^2\gamma$ equal to?

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

What should be the value of k for which the equation $3x^2+3y^2+(k+1)z^2+x-y+z=0$ represents the sphere?

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

If a line OP of length r (where 'O' is the origin) makes an angle $\alpha$ with x-axis and lies in the xz-plane, then what are the coordinates of P?

1. A.$(r\cos \alpha ,0,r\sin \alpha )$
2. B.$(0,0,r\sin \alpha )$
3. C.$(r\cos \alpha ,0,0)$
4. D.$(0,0,r\cos \alpha )$

What is the distance of the point (1, 2, 0) from yz-plane is:

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

What are the direction cosines of a line which is equally inclined to the positive directions of the axes?

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

What is the angle between the lines $\frac{x-2}{1}=\frac{y+1}{-2}=\frac{z+2}{1}$ and $\frac{x-1}{1}=\frac{2y+3}{3}=\frac{z+5}{2}$?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{6}$
4. D.None of the above

What is the equation to the plane through $(1,2,3)$ parallel to $3x+4y-5z=0$?

1. A.$3x+4y+5z+4=0$
2. B.$3x+4y-5z+14=0$
3. C.$3x+4y-5z+4=0$
4. D.$3x+4y-5z-4=0$

What are the direction ratios of the line of intersection of the planes $x=3z+4$ and $y=2z-3$?

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

What is the equation to the straight line passing through (a, b, c) and parallel to z-axis?

1. A.$\frac{x-a}{1}=\frac{y-b}{0}=\frac{z-c}{0}$
2. B.$\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}$
3. C.$\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{0}$
4. D.$\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}$