Coordinate Geometry · NDA previous year questions

What is the distance between the straight lines $3x+4y=9$ and $6x+8y=15$?

1. A.$3/2$
2. B.$3/10$
3. C.$6$
4. D.$5$

What is the equation of the ellipse whose vertices are $(\pm 5,0)$ and foci are at $(\pm 4,0)$?

1. A.$\frac{x^2}{25}+\frac{y^2}{9}=1$
2. B.$\frac{x^2}{16}+\frac{y^2}{9}=1$
3. C.$\frac{x^2}{25}+\frac{y^2}{16}=1$
4. D.$\frac{x^2}{9}+\frac{y^2}{25}=1$

Let the coordinates of the points A, B, C be $(1,8,4)$, $(0,-11,4)$ and $(2,-3,1)$ respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?

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

The coordinates of the vertices P, Q and R of a triangle PQR are $(1,-1,1)$, $(3,-2,2)$ and $(0,2,6)$ respectively. If $\angle RQP=\theta$, then what is $\angle PRQ$ equal to?

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

Geometrically $\mathrm{Re}\left(z^2-i\right)=2$, where $i=\sqrt{-1}$ and $\mathrm{Re}$ is the real part, represents

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

The value of the product $6^{\frac{1}{2}}\times 6^{\frac{1}{4}}\times 6^{\frac{1}{8}}\times 6^{\frac{1}{16}}\times \dots$ up to infinite terms is

1. A.6
2. B.36
3. C.216
4. D.512

The points $(a,b),(0,0),(-a,-b)$ and $(ab,b^2)$ are

1. A.the vertices of a parallelogram
2. B.the vertices of a rectangle
3. C.the vertices of a square
4. D.collinear

The angle between the lines $x+y-3=0$ and $x-y+3=0$ is $\alpha$ and the acute angle between the lines $x-\sqrt{3}y+2\sqrt{3}=0$ and $\sqrt{3}x-y+1=0$ is $\beta$. Which one of the following is correct?

1. A.$\alpha =\beta$
2. B.$\alpha >\beta$
3. C.$\alpha <\beta$
4. D.$\alpha =2\beta$

The distance of the point $(1,3)$ from the line $2x+3y=6$, measured parallel to the line $4x+y=4$, is

1. A.$\frac{5}{\sqrt{13}}\text{ units}$
2. B.$\frac{3}{\sqrt{17}}\text{ unit}$
3. C.$\sqrt{17}\text{ units}$
4. D.$\frac{\sqrt{17}}{2}\text{ units}$

The equation of a straight line which cuts off an intercept of 5 units on negative direction of $y$-axis and makes an angle $120^{\circ }$ with positive direction of $x$-axis is

1. A.$y+\sqrt{3}x+5=0$
2. B.$y-\sqrt{3}x+5=0$
3. C.$y+\sqrt{3}x-5=0$
4. D.$y-\sqrt{3}x-5=0$

The equation of the circle which passes through the points $(1,0),(0,-6)$ and $(3,4)$ is

1. A.$4x^2+4y^2+142x+47y+140=0$
2. B.$4x^2+4y^2-142x-47y+138=0$
3. C.$4x^2+4y^2-142x+47y+138=0$
4. D.$4x^2+4y^2+150x-49y+138=0$

The position of the point $(1,2)$ relative to the ellipse $2x^2+7y^2=20$ is

1. A.outside the ellipse
2. B.inside the ellipse but not at the focus
3. C.on the ellipse
4. D.at the focus

The equation of the ellipse whose centre is at origin, major axis is along $x$-axis with eccentricity $\frac{3}{4}$ and latus rectum 4 units is

1. A.$\frac{x^2}{1024}+\frac{7y^2}{64}=1$
2. B.$\frac{49x^2}{1024}+\frac{7y^2}{64}=1$
3. C.$\frac{7x^2}{1024}+\frac{49y^2}{64}=1$
4. D.$\frac{x^2}{1024}+\frac{y^2}{64}=1$

The general solution of $\frac{dy}{dx}=\frac{ax+h}{by+k}$ represents a circle only when

1. A.$a=b=0$
2. B.$a=-b\ne 0$
3. C.$a=b\ne 0,h=k$
4. D.$a=b\ne 0$

Consider the following statements : 1. $\frac{dy}{dx}$ at a point on the curve gives slope of the tangent at that point. 2. If $a(t)$ denotes acceleration of a particle, then $\int a(t)dt+c$ gives velocity of the particle. 3. If $s(t)$ gives displacement of a particle at time $t$, then $\frac{ds}{dt}$ gives its acceleration at that instant. Which of the above statements is/are correct?

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

If $y={\sec }^{-1}\left(\frac{x+1}{x-1}\right)+{\sin }^{-1}\left(\frac{x-1}{x+1}\right)$, then $\frac{dy}{dx}$ is equal to

1. A.0
2. B.1
3. C.$\frac{x-1}{x+1}$
4. D.$\frac{x+1}{x-1}$

If $l_1=\frac{d}{dx}\left(e^{\sin x}\right)$ $l_2={\lim }_{h\to 0}\frac{e^{\sin (x+h)}-e^{\sin x}}{h}$ $l_3=\int e^{\sin x}\cos xdx$ then which one of the following is correct?

1. A.$l_1\ne l_2$
2. B.$\frac{d}{dx}(l_3)=l_2$
3. C.$\int l_{3}dx=l_2$
4. D.$l_2=l_3$

The incentre of the triangle with vertices $A(1,\sqrt{3})$, $B(0,0)$ and $C(2,0)$ is

1. A.$\left(1,\frac{\sqrt{3}}{2}\right)$
2. B.$\left(\frac{2}{3},\frac{1}{\sqrt{3}}\right)$
3. C.$\left(\frac{2}{3},\frac{\sqrt{3}}{2}\right)$
4. D.$\left(1,\frac{1}{\sqrt{3}}\right)$

If the three consecutive vertices of a parallelogram are $(-2,-1)$, $(1,0)$ and $(4,3)$, then what are the coordinates of the fourth vertex?

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

What is the ratio in which the point $C\left(-\frac{2}{7},-\frac{20}{7}\right)$ divides the line joining the points $A(-2,-2)$ and $B(2,-4)$?

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

What is the equation of the straight line parallel to $2x+3y+1=0$ and passes through the point $(-1,2)$?

1. A.$2x+3y-4=0$
2. B.$2x+3y-5=0$
3. C.$x+y-1=0$
4. D.$3x-2y+7=0$

What is the acute angle between the pair of straight lines $\sqrt{2}x+\sqrt{3}y=1$ and $\sqrt{3}x+\sqrt{2}y=2$?

1. A.${\tan }^{-1}\left(\frac{1}{2\sqrt{6}}\right)$
2. B.${\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
3. C.${\tan }^{-1}(3)$
4. D.${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

If the centroid of a triangle formed by $(7,x)$, $(y,-6)$ and $(9,10)$ is $(6,3)$, then the values of $x$ and $y$ are respectively

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

The two circles $x^2+y^2=r^2$ and $x^2+y^2-10x+16=0$ intersect at two distinct points. Then which one of the following is correct?

1. A.$2<r<8$
2. B.$r=2$ or $r=8$
3. C.$r<2$
4. D.$r>2$

What is the equation of the circle which passes through the points $(3,-2)$ and $(-2,0)$ and having its centre on the line $2x-y-3=0$?

1. A.$x^2+y^2+3x+2=0$
2. B.$x^2+y^2+3x+12y+2=0$
3. C.$x^2+y^2+2x=0$
4. D.$x^2+y^2=5$