3D Geometry · NDA previous year questions

The equation of the plane passing through the line of intersection of the planes $x+y+z=1$, $2x+3y+4z=7$, and perpendicular to the plane $x-5y+3z=5$ is given by

1. A.$x+2y+3z-6=0$
2. B.$x+2y+3z+6=0$
3. C.$3x+4y+5z-8=0$
4. D.$3x+4y+5z+8=0$

A straight line with direction cosines $⟨0,1,0⟩$ is

1. A.parallel to $x$-axis
2. B.parallel to $y$-axis
3. C.parallel to $z$-axis
4. D.equally inclined to all the axes

The line passing through the points $(1,2,-1)$ and $(3,-1,2)$ meets the $yz$-plane at which one of the following points?

1. A.$\left(0,-\frac{7}{2},\frac{5}{2}\right)$
2. B.$\left(0,\frac{7}{2},\frac{1}{2}\right)$
3. C.$\left(0,-\frac{7}{2},-\frac{5}{2}\right)$
4. D.$\left(0,\frac{7}{2},-\frac{5}{2}\right)$

Under which one of the following conditions are the lines $x=ay+b$; $z=cy+d$ and $x=ey+f$; $z=gy+h$ perpendicular?

1. A.$ae+cg-1=0$
2. B.$ae+bf-1=0$
3. C.$ae+cg+1=0$
4. D.$ag+ce+1=0$

If $f(x)=\frac{x}{x-1}$, then what is $\frac{f(a)}{f(a+1)}$ equal to?

1. A.$f\left(-\frac{a}{a+1}\right)$
2. B.$f(a^2)$
3. C.$f\left(\frac{1}{a}\right)$
4. D.$f(-a)$

Let $f:[-6,6]\to \mathbb{R}$ be defined by $f(x)=x^2-3$. Consider the following: 1. $(f\circ f\circ f)(-1)=(f\circ f\circ f)(1)$ 2. $(f\circ f\circ f)(-1)-4(f\circ f\circ f)(1)=(f\circ f)(0)$ 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

Let $f(x)=px+q$ and $g(x)=mx+n$. Then $f(g(x))=g(f(x))$ is equivalent to

1. A.$f(p)=g(m)$
2. B.$f(q)=g(n)$
3. C.$f(n)=g(q)$
4. D.$f(m)=g(p)$

Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n ?

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

Consider the following for the next two (02) items that follow : Let Q be the image of the point P(-2, 1, -5) in the plane $3x-2y+2z+1=0$. Consider the following : 1. The coordinates of Q are (4, -3, -1). 2. PQ is of length more than 8 units. 3. The point (1, -1, -3) is the mid-point of the line segment PQ and lies on the given plane. Which of the above statements are correct?

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

Consider the following for the next two (02) items that follow : Let Q be the image of the point P(-2, 1, -5) in the plane $3x-2y+2z+1=0$. Consider the following : 1. The direction ratios of the line segment PQ are $<3,-2,2>$. 2. The sum of the squares of direction cosines of the line segment PQ is unity. 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 L passes through the point P(5, -6, 7) and is parallel to the planes $x+y+z=1$ and $2x-y-2z=3$. What are the direction ratios of the line of intersection of the given planes?

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

A line L passes through the point P(5, -6, 7) and is parallel to the planes $x+y+z=1$ and $2x-y-2z=3$. What is the equation of the line L?

1. A.$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$
2. B.$\frac{x+5}{-1}=\frac{y-6}{4}=\frac{z+7}{-3}$
3. C.$\frac{x-5}{-1}=\frac{y+6}{-4}=\frac{z-7}{3}$
4. D.$\frac{x-5}{-1}=\frac{y+6}{-4}=\frac{z-7}{-3}$

Suppose $\omega$ is a cube root of unity with $\omega \ne 1$. Suppose $P$ and $Q$ are the points on the complex plane defined by $\omega$ and ${\omega }^2$. If $O$ is the origin, then what is the angle between $OP$ and $OQ$?

1. A.$60^{\circ }$
2. B.$90^{\circ }$
3. C.$120^{\circ }$
4. D.$150^{\circ }$

Suppose ${\omega }_1$ and ${\omega }_2$ are two distinct cube roots of unity different from 1. Then what is $({\omega }_1-{\omega }_2{)}^2$ equal to?

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

For the next three (3) items that follow: A plane $P$ passes through the line of intersection of the planes $2x-y+3z=2,x+y-z=1$ and the point $(1,0,1)$. What are the direction ratios of the line of intersection of the given planes?

1. A.$⟨2,-5,-3⟩$
2. B.$⟨1,-5,-3⟩$
3. C.$⟨2,5,3⟩$
4. D.$⟨1,3,5⟩$

For the next three (3) items that follow: A plane $P$ passes through the line of intersection of the planes $2x-y+3z=2,x+y-z=1$ and the point $(1,0,1)$. What is the equation of the plane $P$?

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

For the next three (3) items that follow: A plane $P$ passes through the line of intersection of the planes $2x-y+3z=2,x+y-z=1$ and the point $(1,0,1)$. If the plane $P$ touches the sphere $x^2+y^2+z^2=r^2$, then what is $r$ equal to?

1. A.$\frac{2}{\sqrt{29}}$
2. B.$\frac{4}{\sqrt{29}}$
3. C.$\frac{5}{\sqrt{29}}$
4. D.1

If $$f(x_1)-f(x_2)=f\left(\frac{x_1-x_2}{1-x_1{x}_2}\right)$$ for $x_1,x_2\in (-1,1)$, then what is $f(x)$ equal to?

1. A.$\ln \left(\frac{1-x}{1+x}\right)$
2. B.$\ln \left(\frac{2+x}{1-x}\right)$
3. C.${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$
4. D.${\tan }^{-1}\left(\frac{1+x}{1-x}\right)$

The radius of the sphere $3x^2+3y^2+3z^2-8x+4y+8z-15=0$ is

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

The direction ratios of the line perpendicular to the lines with direction ratios $⟨1,-2,-2⟩$ and $⟨0,2,1⟩$ are

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

What are the co-ordinates of the foot of the perpendicular drawn from the point $(3,5,4)$ on the plane $z=0$?

1. A.$(0,5,4)$
2. B.$(3,5,0)$
3. C.$(3,0,4)$
4. D.$(0,0,4)$

The lengths of the intercepts on the co-ordinate axes made by the plane $5x+2y+z-13=0$ are

1. A.5, 2, 1 unit
2. B.$\frac{13}{5},\frac{13}{2},13\text{ unit}$
3. C.$\frac{5}{13},\frac{2}{13},\frac{1}{13}\text{ unit}$
4. D.1, 2, 5 unit

If $g(x)=\frac{1}{f(x)}$ and $f(x)=x,x\ne 0$, then which one of the following is correct?

1. A.$f(f(f(g(g(f(x)))))))=g(g(f(g(f(x)))))$
2. B.$f(f(g(g(g(f(x)))))))=g(g(f(g(f(x)))))$
3. C.$f(g(f(g(g(f(g(x))))))))=g(g(f(g(f(x)))))$
4. D.$f(f(f(g(g(f(x)))))))=f(f(f(g(f(x)))))$

The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances $49\mathrm{m}$ and $36\mathrm{m}$ are $43^{\circ }$ and $47^{\circ }$ respectively. What is the height of the tower?

1. A.$40\mathrm{m}$
2. B.$42\mathrm{m}$
3. C.$45\mathrm{m}$
4. D.$47\mathrm{m}$

In a triangle $ABC$, $a=(1+\sqrt{3})\mathrm{cm}$, $b=2\mathrm{cm}$ and angle $C=60^{\circ }$. Then the other two angles are

1. A.$45^{\circ }\text{ and }75^{\circ }$
2. B.$30^{\circ }\text{ and }90^{\circ }$
3. C.$105^{\circ }\text{ and }15^{\circ }$
4. D.$100^{\circ }\text{ and }20^{\circ }$