3D Geometry · NDA previous year questions

The two planes $ax+by+cz+d=0$ and $ax+by+cz+d_1=0$, where $d\ne d_1$, have

1. A.one point only in common
2. B.three points in common
3. C.infinite points in common
4. D.no points in common

What is the distance of the origin from the plane $2x+6y-3z+7=0$ ?

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

$PQRS$ is a parallelogram, where $\overrightarrow{PQ}=3\hat{i}+2\hat{j}-m\hat{k}$, $\overrightarrow{PS}=\hat{i}+3\hat{j}+\hat{k}$ and the area of the parallelogram is $\sqrt{90}$. What is the value of $m$?

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

If at any instant $t$, for a sphere, $r$ denotes the radius, $S$ denotes the surface area and $V$ denotes the volume, then what is $\frac{dV}{dt}$ equal to?

1. A.$\frac{1}{2}S\frac{dr}{dt}$
2. B.$\frac{1}{2}r\frac{dS}{dt}$
3. C.$r\frac{dS}{dt}$
4. D.$\frac{1}{2}{r}^2\frac{dS}{dt}$

If $f(x)=\frac{2}{3}x+\frac{3}{2},x\in R$, then what is $f^{-1}(x)$ equal to?

1. A.$\frac{3}{2}x+\frac{2}{3}$
2. B.$\frac{3}{2}x-\frac{9}{4}$
3. C.$\frac{2}{3}x-\frac{4}{9}$
4. D.$\frac{2}{3}x-\frac{2}{3}$

The planes $px+2y+2z-3=0$ and $2x-y+z+2=0$ intersect at an angle $\pi /4$. What is the value of $p^2$?

1. A.24
2. B.12
3. C.6
4. D.3

The growth of a quantity N(t) at any instant t is given by $\frac{dN(t)}{dt}=\alpha N(t)$. Given that $N(t)=c{e}^{kt}$, c is a constant. What is the value of $\alpha$?

1. A.c
2. B.k
3. C.$c+k$
4. D.$c-k$

Under what condition do the planes $bx-ay=n$, $cy-bz=l$, $az-cx=m$ intersect in a line?

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

What is the equation of the plane through z-axis and parallel to the line $\frac{x-1}{\cos \theta }=\frac{y+2}{\sin \theta }=\frac{z-3}{0}$?

1. A.$x\cot \theta +y=0$
2. B.$x\tan \theta -y=0$
3. C.$x+y\cot \theta =0$
4. D.$x-y\tan \theta =0$

If the lines $3y+4x=1$, $y=x+5$ and $5y+bx=3$ are concurrent, then what is the value of $b$?

1. A.1
2. B.3
3. C.6
4. D.0

What is the least value of $f(x)=2x^3-3x^2-12x+1$ on $[-2,2.5]$?

1. A.$-3$
2. B.8
3. C.$-19$
4. D.$-16.5$

1. A.$1\mathrm{s}$
2. B.$(\ln 10)\mathrm{s}$
3. C.$2(\ln 10)\mathrm{s}$
4. D.$10\mathrm{s}$

What is the product of the perpendiculars from the two points $(\pm \sqrt{b^2-a^2},0)$ to the line $ax\cos \varphi +by\sin \varphi =ab$ ?

1. A.$a^2$
2. B.$b^2$
3. C.$ab$
4. D.$a/b$

The direction cosines of a line are proportional to $(2,1,2)$ and the line intersects a plane perpendicularly at the point $(1,-2,4)$. What is the distance of the plane from the point $(3,2,3)$ ?

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

The foot of the perpendicular drawn from the origin to a plane is the point $(1,-3,1)$. What is the intercept cut on the $x$-axis by the plane ?

1. A.1
2. B.3
3. C.$\sqrt{11}$
4. D.11

A line makes the same angle $\alpha$ with each of the $x$ and $y$-axes. If the angle $\theta$, which it makes with the $z$-axis, is such that ${\sin }^2\theta =2{\sin }^2\alpha$, then what is the value of $\alpha$ ?

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

What is the equation of the sphere which has its centre at $(6,-1,2)$ and touches the plane $2x-y+2z-2=0$ ?

1. A.$x^2+y^2+z^2+12x-2y+4z+16=0$
2. B.$x^2+y^2+z^2+12x-2y+4z-16=0$
3. C.$x^2+y^2+z^2-12x+2y-4z+16=0$
4. D.$x^2+y^2+z^2-12x+2y-4z+25=0$

What are the direction ratios of the line determined by the planes $x-y+2z=1$ and $x+y-z=3$ ?

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

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

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

What is the equation of a plane through the $x$-axis and passing through the point $(1,2,3)$?

1. A.$x+y+z=6$
2. B.$x=1$
3. C.$y+z=5$
4. D.$z+y=1$

What is the number of planes passing through three non-collinear points?

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

What is the angle between the lines $x+z=0,y=0$ and $20x=15y=12z$?

1. A.${\cos }^{-1}(1/5)$
2. B.${\cos }^{-1}(1/7)$
3. C.${\sin }^{-1}(1/5)$
4. D.${\sin }^{-1}(1/7)$

Under what condition does the equation $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ represent a real sphere?

1. A.$u^2+v^2+w^2=d^2$
2. B.$u^2+v^2+w^2>d$
3. C.$u^2+v^2+w^2<d$
4. D.$u^2+v^2+w^2<d^2$