3D Geometry · NDA previous year questions

For the next two (2) items that follow: The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What are the direction cosines of the line segment?

1. A.$⟨12/13,4/13,3/13⟩$
2. B.$⟨12/13,-4/13,3/13⟩$
3. C.$⟨12/13,-4/13,-3/13⟩$
4. D.$⟨-12/13,-4/13,3/13⟩$

For the next two (2) items that follow: From the point $P(3,-1,11)$, a perpendicular is drawn on the line $L$ given by the equation $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Let $Q$ be the foot of the perpendicular. What are the direction ratios of the line segment $PQ$?

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

For the next two (2) items that follow: From the point $P(3,-1,11)$, a perpendicular is drawn on the line $L$ given by the equation $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Let $Q$ be the foot of the perpendicular. What is the length of the line segment $PQ$?

1. A.$\sqrt{47}$ units
2. B.7 units
3. C.$\sqrt{53}$ units
4. D.8 units

For the next two (2) items that follow: A triangular plane $ABC$ with centroid $(1,2,3)$ cuts the coordinate axes at $A,B,C$ respectively. What are the intercepts made by the plane $ABC$ on the axes?

1. A.$3,6,9$
2. B.$1,2,3$
3. C.$1,4,9$
4. D.$2,4,6$

A triangular plane $ABC$ with centroid $(1,2,3)$ cuts the coordinate axes at $A,B,C$ respectively. What is the equation of the plane $ABC$?

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

For the next two (2) items that follow: A point $P(1,2,3)$ is one vertex of a cuboid formed by the coordinate planes and the planes passing through $P$ and parallel to the coordinate planes. What is the length of one of the diagonals of the cuboid?

1. A.$\sqrt{10}$ units
2. B.$\sqrt{14}$ units
3. C.$4$ units
4. D.$5$ units

A point $P(1,2,3)$ is one vertex of a cuboid formed by the coordinate planes and the planes passing through $P$ and parallel to the coordinate planes. What is the equation of the plane passing through $P(1,2,3)$ and parallel to $xy$-plane?

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

A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is $30^{\circ }$. What is the height of the lamp post ?

1. A.50 m
2. B.$50\sqrt{3}\text{ m}$
3. C.$\frac{50}{\sqrt{3}}\text{ m}$
4. D.100 m

Consider the function $f^{\prime \prime }(x)={\sec }^{4}x+4$ with $f(0)=0$ and $f^{\prime }(0)=0$. What is $f^{\prime }(x)$ equal to?

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

Consider the function $f^{\prime \prime }(x)={\sec }^{4}x+4$ with $f(0)=0$ and $f^{\prime }(0)=0$. What is $f(x)$ equal to?

1. A.$\frac{2\ln \sec x}{3}+\frac{{\tan }^{2}x}{6}+2x^2$
2. B.$\frac{3\ln \sec x}{2}+\frac{{\cot }^{2}x}{6}+2x^2$
3. C.$\frac{4\ln \sec x}{3}+\frac{{\sec }^{2}x}{6}+2x^2$
4. D.$\ln \sec x+\frac{{\tan }^{4}x}{12}+2x^2$

Consider $\int x{\tan }^{-1}xdx=A(x^2+1){\tan }^{-1}x+Bx+C$, where C is the constant of integration. What is the value of B?

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

What is ${\int }_0^{\pi /2}\frac{dx}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}$ equal to?

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

For the next three (03) items that follow: Consider $I={\int }_0^{\pi }\frac{xdx}{1+\sin x}$ What is $I$ equal to?

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

For the next three (03) items that follow: Consider $I={\int }_0^{\pi }\frac{xdx}{1+\sin x}$ What is ${\int }_0^{\pi }\frac{(\pi -x)dx}{1+\sin x}$ equal to?

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

For the next three (03) items that follow: Consider $I={\int }_0^{\pi }\frac{xdx}{1+\sin x}$ What is ${\int }_0^{\pi }\frac{dx}{1+\sin x}$ equal to?

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

For the next three (03) items that follow : Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$. What is the radius of the sphere?

1. A.$\sqrt{12}$
2. B.$\sqrt{14}$
3. C.12
4. D.14

For the next three (03) items that follow : Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$. What is the centre of the sphere?

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

For the next three (03) items that follow : Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$. Consider the following statements : 1. The sphere passes through the point $(0,4,0)$. 2. The point $(1,1,1)$ is at a distance of 5 unit from the centre of the 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

For the next two (02) items that follow : The line joining the points $(2,1,3)$ and $(4,-2,5)$ cuts the plane $2x+y-z=3$. Where does the line cut the plane?

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

For the next two (02) items that follow : The line joining the points $(2,1,3)$ and $(4,-2,5)$ cuts the plane $2x+y-z=3$. What is the ratio in which the plane divides the line?

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

For the next two (02) items that follow : Consider the plane passing through the points $A(2,2,1),B(3,4,2)$ and $C(7,0,6)$. Which one of the following points lies on the plane?

1. A.$(1,0,0)$
2. B.$(1,0,1)$
3. C.$(0,0,1)$
4. D.None of the above

For the next two (02) items that follow : Consider the plane passing through the points $A(2,2,1),B(3,4,2)$ and $C(7,0,6)$. What are the direction ratios of the normal to the plane?

1. A.$<1,0,1>$
2. B.$<0,1,0>$
3. C.$<1,0,-1>$
4. D.None of the above

From an aeroplane above a straight road the angles of depression of two positions at a distance 20 m apart on the road are observed to be $30^{\circ }$ and $45^{\circ }$. The height of the aeroplane above the ground is:

1. A.$10\sqrt{3}\mathrm{m}$
2. B.$10(\sqrt{3}-1)\mathrm{m}$
3. C.$10(\sqrt{3}+1)\mathrm{m}$
4. D.$20\mathrm{m}$

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 are the direction ratios of normal to plane?

1. A.$<2,3,-1>$
2. B.$<2,3,1>$
3. C.$<-1,2,3>$
4. D.None of the above

For the next three (03) items that follow: A straight line passes through $(1,-2,3)$ and perpendicular to the plane $2x+3y-z=7$. Where does the line meet the plane?

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