Coordinate Geometry · NDA previous year questions

Consider the following statements in respect of circles $x^2+y^2-2x-2y=0$ and $x^2+y^2=1$: 1. The radius of the first circle is twice that of the second circle. 2. Both the circles pass through the origin. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Directions: For the next 3 (three) questions to follow: The vertices of a cube are (0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2), (2, 2, 0), (2, 0, 2), (0, 2, 2), (2, 2, 2) respectively. What is the angle between any two diagonals of the cube?

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

What is the angle between one of the edges of the cube and the diagonal of the cube intersecting the edge of the cube?

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

What is the angle between the diagonal of one of the faces of the cube and the diagonal of the cube intersecting the diagonal of the face of the cube?

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

What is the length of arc of a circle of radius $5\mathrm{cm}$ subtending a central angle measuring $15^{\circ }$ ?

1. A.$5\pi /12\mathrm{cm}$
2. B.$7\pi /12\mathrm{cm}$
3. C.$\pi /12\mathrm{cm}$
4. D.$\pi /5\mathrm{cm}$

What does the equation $xdy=ydx$ represent ?

1. A.A family of circles
2. B.A family of parabolas
3. C.A family of hyperbolas
4. D.A family of straight lines

The velocity $v$ of a particle at any instant $t$ moving in a straight line is given by $v=s+1$ where $s$ meter is the distance travelled in $t$ second. What is the time taken by the particle to cover a distance of 9 meter ?

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

The curve $y^2=-4ax(a>0)$ lies in :

1. A.First and fourth quadrants
2. B.First and second quadrants
3. C.Second and third quadrants
4. D.Third and fourth quadrants

The circle $x^2+y^2+4x-4y+4=0$ touches :

1. A.Only the x-axis
2. B.Only the y-axis
3. C.Both the axes
4. D.Neither of the axes

The middle point of the segment of the straight line joining the points $(p,q)$ and $(q,-p)$ is $(r/2,s/2)$. What is the length of the segment ?

1. A.$[(s^2+r^2{)}^{1/2}]/2$
2. B.$[(s^2+r^2{)}^{1/2}]/4$
3. C.$(s^2+r^2{)}^{1/2}$
4. D.$s+r$

What is the locus of a point which is equidistant from the point $(m+n,n-m)$ and the point $(m-n,n+m)$ ?

1. A.$mx=ny$
2. B.$nx=-my$
3. C.$nx=my$
4. D.$mx=-ny$

The ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$ has the same eccentricity as the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. What is the ratio of $a$ to $b$ ?

1. A.$5/13$
2. B.$13/5$
3. C.$7/8$
4. D.$8/7$

If $x$-axis is tangent to the circle $x^2+y^2+2gx+2fy+k=0$, then which one of the following is correct?

1. A.$g^2=k$
2. B.$g^2=f$
3. C.$f^2=k$
4. D.$f^2=g$

What is the sum of focal radii of any point on an ellipse equal to?

1. A.Length of latus rectum
2. B.Length of major axis
3. C.Length of minor axis
4. D.Length of semi-latus rectum

What does an equation of the first degree containing one arbitrary parameter passing through a fixed point represent?

1. A.Circle
2. B.Straight line
3. C.Parabola
4. D.Ellipse

What is the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$?

1. A.$(1,10)$
2. B.$(5,6)$
3. C.$(6,5)$
4. D.$(7,4)$

Consider the following statements: 1. The equation to a straight line parallel to the axis of $x$ is $y=d$, where $d$ is a constant. 2. The equation to the axis of $x$ is $x=0$. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

The points $(1,3,4)$, $(-1,6,10)$, $(-7,4,7)$ and $(-5,1,1)$ are the vertices of a

1. A.rhombus
2. B.rectangle
3. C.parallelogram
4. D.square

What is the eccentricity of an ellipse if its latus rectum is equal to one-half of its minor axis?

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