3D Geometry · NDA previous year questions

What is ${\lim }_{x\to 0}\frac{\sqrt{1+x}-1}{x}$ equal to?

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

What is ${\lim }_{x\to 0}\frac{2(1-\cos x)}{x^2}$ equal to?

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

Consider the following : 1. ${\lim }_{x\to 0}\frac{1}{x}$ exists. 2. ${\lim }_{x\to 0}{e}^{\frac{1}{x}}$ does not exist. Which of the above is/are correct?

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

If $x^m+y^m=1$ such that $\frac{dy}{dx}=-\frac{x}{y}$, then what should be the value of m?

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

What is the cosine of angle between the planes $x+y+z+1=0$ and $2x-2y+2z+1=0$?

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

If $A=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -1\\ 1& 2\end{array}\right]$, then what is $B^{-1}{A}^{-1}$ equal to?

1. A.$\left[\begin{array}{cc}1& -3\\ -1& 2\end{array}\right]$
2. B.$\left[\begin{array}{cc}-1& 3\\ 1& -2\end{array}\right]$
3. C.$\left[\begin{array}{cc}-1& 3\\ 1& 2\end{array}\right]$
4. D.$\left[\begin{array}{cc}-1& -3\\ 1& 2\end{array}\right]$

What is the sum of the squares of direction cosines of the line joining the points $(1,2,-3)$ and $(-2,3,1)$?

1. A.0
2. B.1
3. C.3
4. D.$\frac{2}{\sqrt{26}}$

What is the diameter of the sphere $x^2+y^2+z^2-4x+6y-8z-7=0$?

1. A.4 units
2. B.5 units
3. C.6 units
4. D.12 units

What are the direction ratios of normal to the plane $2x-y+2z+1=0$?

1. A.$⟨2,1,2⟩$
2. B.$⟨1,-\frac{1}{2},1⟩$
3. C.$⟨1,-2,1⟩$
4. D.None of the above

What is the percentage of persons who read all the three papers?

1. A.20%
2. B.25%
3. C.30%
4. D.40%

What is the percentage of persons who read only two papers?

1. A.19%
2. B.31%
3. C.44%
4. D.None of the above

What is the percentage of persons who read only one paper?

1. A.38%
2. B.48%
3. C.51%
4. D.None of the above

What is the percentage of persons who read only A but neither B nor C?

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

What is the locus of points of intersection of a sphere and a plane?

1. A.Circle
2. B.Ellipse
3. C.Parabola
4. D.Hyperbola

What is the angle between two planes $2x-y+z=4$ and $x+y+2z=6$?

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

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point P on the plane, the angle of elevation of the bottom of the flag staff is $\beta$ and that of the top is $\alpha$. What is the height of the tower ?

1. A.$\frac{h\tan \beta }{\tan \alpha -\tan \beta }$
2. B.$\frac{h\tan \beta }{\tan \alpha +\tan \beta }$
3. C.$\frac{h\cos \beta }{\cos \alpha -\cos \beta }$
4. D.$\frac{h}{\cos (\alpha -\beta )}$

1. A.p is one of the cube roots of unity
2. B.q is one of the cube roots of unity
3. C.$\frac{p}{q}$ is one of the cube roots of unity
4. D.None of the above

An aeroplane flying at a height of 300 m above the ground passes vertically above another plane at an instant when the angles of elevation of two planes from the same point on the ground are $60^{\circ }$ and $45^{\circ }$ respectively. What is the height of the lower plane from the ground ?

1. A.50 m
2. B.$\frac{100}{\sqrt{3}}$ m
3. C.$100\sqrt{3}$ m
4. D.$150(\sqrt{3}+1)$ m

What is the equation of the plane passing through the point $(1,-1,-1)$ and perpendicular to each of the planes $x-2y-8z=0$ and $2x+5y-z=0$?

1. A.$7x-3y+2z=14$
2. B.$2x+5y-3z=12$
3. C.$x-7y+3z=4$
4. D.$14x-5y+3z=16$

What is the value of $\tan 15^{\circ }\cdot \tan 195^{\circ }$?

1. A.$7-4\sqrt{3}$
2. B.$7+4\sqrt{3}$
3. C.$7+2\sqrt{3}$
4. D.$7+6\sqrt{3}$

The equation to sphere passing through origin and the points $(-1,0,0)$, $(0,-2,0)$ and $(0,0,-3)$ is $x^2+y^2+z^2+f(x,y,z)=0$. What is $f(x,y,z)$ equal to?

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

What is the angle between the lines whose direction cosines are proportional to $⟨2,3,4⟩$ and $⟨1,-2,1⟩$ respectively?

1. A.$90^{\circ }$
2. B.$60^{\circ }$
3. C.$45^{\circ }$
4. D.$30^{\circ }$

What is the radius of the sphere $x^2+y^2+z^2-x-y-z=0$ ?

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

Which one of the following points lies on the plane $2x+3y-6z=21$ ?

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

What is the acute angle between the planes $x+y+2z=3$ and $-2x+y-z=11$?

1. A.$\pi /5$
2. B.$\pi /4$
3. C.$\pi /6$
4. D.$\pi /3$