Coordinate Geometry · NDA previous year questions

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $(3\sqrt{5},1)$ and the length of its latus rectum is $\frac{4}{3}$ units. The length of the conjugate axis is

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

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 is the length of the line segment?

1. A.19 units
2. B.17 units
3. C.15 units
4. D.13 units

For the next three (3) items that follow : Consider the parametric equation $x=\frac{a(1-t^2)}{1+t^2}$, $y=\frac{2at}{1+t^2}$ What does the equation represent?

1. A.It represents a circle of diameter $a$
2. B.It represents a circle of radius $a$
3. C.It represents a parabola
4. D.None of the above

Consider the parametric equation $x=\frac{a(1-t^2)}{1+t^2}$, $y=\frac{2at}{1+t^2}$ What is $\frac{dy}{dx}$ equal to?

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

Consider the parametric equation $x=\frac{a(1-t^2)}{1+t^2}$, $y=\frac{2at}{1+t^2}$ What is $\frac{d^{2}y}{d{x}^2}$ equal to?

1. A.$\frac{a^2}{y^2}$
2. B.$\frac{a^2}{x^2}$
3. C.$-\frac{a^2}{x^2}$
4. D.$-\frac{a^2}{y^3}$

For the next two (2) items that follow : Given that $\frac{d}{dx}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=Ax+B$. What is the value of $A$?

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

Given that $\frac{d}{dx}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=Ax+B$. What is the value of $B$?

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

$A(3,4)$ and $B(5,-2)$ are two points and P is a point such that PA = PB. If the area of triangle PAB is 10 square unit, what are the coordinates of P?

1. A.$(1,0)$ only
2. B.$(7,2)$ only
3. C.$(1,0)$ or $(7,2)$
4. D.Neither $(1,0)$ nor $(7,2)$

What is the product of the perpendiculars drawn from the points $(\pm \sqrt{a^2-b^2},0)$ upon the line $bx\cos \alpha +ay\sin \alpha =ab$?

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

Which one of the following is correct in respect of the equations $\frac{x-1}{2}=\frac{y-2}{3}$ and $2x+3y=5$?

1. A.They represent two lines which are parallel.
2. B.They represent two lines which are perpendicular.
3. C.They represent two lines which are neither parallel nor perpendicular.
4. D.The first equation does not represent a line.

For the next two (02) items that follow : Consider the circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$. What is the distance between the centres of the two circles?

1. A.$\sqrt{a^2+b^2}$
2. B.$a^2+b^2$
3. C.$a+b$
4. D.$2(a+b)$

For the next two (02) items that follow : Consider the circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$. The two circles touch each other if

1. A.$c=\sqrt{a^2+b^2}$
2. B.$\frac{1}{c}=\frac{1}{a^2}+\frac{1}{b^2}$
3. C.$c=\frac{1}{a^2}+\frac{1}{b^2}$
4. D.$c=\frac{1}{a^2+b^2}$

For the next three (03) items that follow : The line $2y=3x+12$ cuts the parabola $4y=3x^2$. Where does the line cut the parabola?

1. A.At $(-2,3)$ only
2. B.At $(4,12)$ only
3. C.At both $(-2,3)$ and $(4,12)$
4. D.Neither at $(-2,3)$ nor at $(4,12)$

What is the length of the latus rectum of the ellipse $25x^2+16y^2=400$?

1. A.25/2
2. B.25/4
3. C.16/5
4. D.32/5

If $f(x)=ax+b$ and $g(x)=cx+d$ such that $f(g(x))=g(f(x))$, then which one of the following is correct ?

1. A.$f(c)=g(a)$
2. B.$f(a)=g(c)$
3. C.$f(c)=g(d)$
4. D.$f(d)=g(b)$

What is the area of the parabola $y^2=4bx$ bounded by its latus rectum?

1. A.$2b^2/3\text{ square unit}$
2. B.$4b^2/3\text{ square unit}$
3. C.$b^2\text{ square unit}$
4. D.$8b^2/3\text{ square unit}$

What is the area of the triangle OAB where O is the origin, $\overrightarrow{OA}=3\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{OB}=2\hat{i}+\hat{j}-3\hat{k}$?

1. A.$5\sqrt{6}\text{ square unit}$
2. B.$\frac{5\sqrt{6}}{2}\text{ square unit}$
3. C.$\sqrt{6}\text{ square unit}$
4. D.$\sqrt{30}\text{ square unit}$

Let $z=x+iy$ where $x,y$ are real variables and $i=\sqrt{-1}$. If $|2z-1|=|z-2|$, then the point $z$ describes:

1. A.A circle
2. B.An ellipse
3. C.A hyperbola
4. D.A parabola

If $|z+\bar{z}|=|z-\bar{z}|$, then the locus of $z$ is:

1. A.A pair of straight lines
2. B.A line
3. C.A set of four straight lines
4. D.A circle

Consider the following statements: 1. If $3\theta$ is an acute angle such that $\sin 3\theta =\cos 2\theta$, then the measurement of $\theta$ in radian equals to $\frac{\pi }{10}$. 2. One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle. 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

Consider the following statements: 1. ${\tan }^{-1}1+{\tan }^{-1}(0.5)=\pi /2$ 2. ${\sin }^{-1}(1/3)+{\cos }^{-1}(1/3)=\pi /2$ 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

What is ${\sin }^{-1}\sin \frac{3\pi }{5}$ equal to?

1. A.$\frac{3\pi }{5}$
2. B.$\frac{2\pi }{5}$
3. C.$\frac{\pi }{5}$
4. D.None of the above

If $x$ and $y$ are positive and $xy>1$, then what is ${\tan }^{-1}x+{\tan }^{-1}y$ equal to?

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

What is the equation of the line through $(1,2)$ so that the segment of the line intercepted between the axes is bisected at this point?

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

What is the equation of straight line passing through the point $(4,3)$ and making equal intercepts on the coordinate axes?

1. A.$x+y=7$
2. B.$3x+4y=7$
3. C.$x-y=1$
4. D.None of the above