Coordinate Geometry · NDA previous year questions

Let A be a point in space such that $|\overrightarrow{OA}|=12$, where O is the origin. If $\overrightarrow{OA}$ is inclined at angles $45^{\circ }$ and $60^{\circ }$ with $x$-axis and $y$-axis respectively, then what is $\overrightarrow{OA}$ equal to?

1. A.$6\hat{i}+6\hat{j}\pm \sqrt{2}\hat{k}$
2. B.$6\hat{i}+6\sqrt{2}\hat{j}\pm 6\hat{k}$
3. C.$6\sqrt{2}\hat{i}+6\hat{j}\pm 6\hat{k}$
4. D.$3\sqrt{2}\hat{i}+3\hat{j}\pm 6\hat{k}$

What is the length of the chord of a unit circle which subtends an angle $\theta$ at the centre?

1. A.$\sin \left(\frac{\theta }{2}\right)$
2. B.$\cos \left(\frac{\theta }{2}\right)$
3. C.$2\sin \left(\frac{\theta }{2}\right)$
4. D.$2\cos \left(\frac{\theta }{2}\right)$

What is $\tan \left\{2{\tan }^{-1}\left(\frac{1}{3}\right)\right\}$ equal to?

1. A.$\frac{2}{3}$
2. B.$\frac{3}{4}$
3. C.$\frac{3}{8}$
4. D.$\frac{1}{9}$

A ladder $9\mathrm{m}$ long reaches a point $9\mathrm{m}$ below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the flagstaff is $60^{\circ }$. What is the height of the flagstaff?

1. A.$9\mathrm{m}$
2. B.$10.5\mathrm{m}$
3. C.$13.5\mathrm{m}$
4. D.$15\mathrm{m}$

The equation $ax+by+c=0$ represents a straight line

1. A.for all real numbers $a$, $b$ and $c$
2. B.only when $a\ne 0$
3. C.only when $b\ne 0$
4. D.only when at least one of $a$ and $b$ is non-zero

What is the angle between the lines $x\cos \alpha +y\sin \alpha =a$ and $x\sin \beta -y\cos \beta =a$?

1. A.$\beta -\alpha$
2. B.$\pi +\beta -\alpha$
3. C.$\frac{\pi +2\beta +2\alpha }{2}$
4. D.$\frac{\pi -2\beta +2\alpha }{2}$

If the angle between the lines joining the end points of minor axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with one of its foci is $\frac{\pi }{2}$, then what is the eccentricity of the ellipse?

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

A point on the line $\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+2}{7}$ has coordinates

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

If the points $(x,y,-3)$, $(2,0,-1)$ and $(4,2,3)$ lie on a straight line, then what are the values of $x$ and $y$ respectively?

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

A straight line passes through the point $(1,1,1)$ makes an angle $60^{\circ }$ with the positive direction of $z$-axis, and the cosine of the angles made by it with the positive directions of the $y$-axis and the $x$-axis are in the ratio $\sqrt{3}:1$. What is the acute angle between the two possible positions of the line?

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

Directions for the following three (03) items : Read the following information and answer the three items that follow : A curve $y=m{e}^{mx}$ where $m>0$ intersects $y$-axis at a point $P$. What is the slope of the curve at the point of intersection $P$?

1. A.$m$
2. B.$m^2$
3. C.$2m$
4. D.$2m^2$

Directions for the following three (03) items : Read the following information and answer the three items that follow : A curve $y=m{e}^{mx}$ where $m>0$ intersects $y$-axis at a point $P$. How much angle does the tangent at $P$ make with $y$-axis?

1. A.${\tan }^{-1}{m}^2$
2. B.${\cot }^{-1}(1+m^2)$
3. C.${\sin }^{-1}\left(\frac{1}{\sqrt{1+m^4}}\right)$
4. D.${\sec }^{-1}\sqrt{1+m^4}$

The points $(1,3)$ and $(5,1)$ are two opposite vertices of a rectangle. The other two vertices lie on the line $y=2x+c$. What is the value of c?

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

If the lines $3y+4x=1$, $y=x+5$ and $5y+bx=3$ are concurrent, then what is the value of b?

1. A.1
2. B.3
3. C.6
4. D.$\frac{1}{2}$

What is the equation of the straight line which is perpendicular to $y=x$ and passes through $(3,2)$?

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

The straight lines $x+y-4=0$, $3x+y-4=0$ and $x+3y-4=0$ form a triangle, which is

1. A.isosceles
2. B.right-angled
3. C.equilateral
4. D.scalene

A circle is drawn on the chord of a circle $x^2+y^2=a^2$ as diameter. The chord lies on the line $x+y=a$. What is the equation of the circle?

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

The circle $x^2+y^2+4x-7y+12=0$, cuts an intercept on y-axis equal to

1. A.1
2. B.3
3. C.4
4. D.7

In a circle of diameter 44 cm, the length of a chord is 22 cm. What is the length of minor arc of the chord ?

1. A.$\frac{484}{21}$ cm
2. B.$\frac{242}{21}$ cm
3. C.$\frac{121}{21}$ cm
4. D.$\frac{44}{7}$ cm

The sum of the focal distances of a point on an ellipse is constant and equal to the

1. A.length of minor axis
2. B.length of major axis
3. C.length of latus rectum
4. D.sum of the lengths of semi-major and semi-minor axes

The equation $2x^2-3y^2-6=0$ represents

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

The two parabolas $y^2=4ax$ and $x^2=4ay$ intersect

1. A.at two points on the line $y=x$
2. B.only at the origin
3. C.at three points one of which lies on $y+x=0$
4. D.only at $(4a,4a)$

The centroid of the triangle with vertices $A(2,-3,3)$, $B(5,-3,-4)$ and $C(2,-3,-2)$ is the point

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

For $r>0$, $f(r)$ is the ratio of perimeter to area of a circle of radius r. Then $f(1)+f(2)$ is equal to

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

If $f(x)=3^{1+x}$, then $f(x)f(y)f(z)$ is equal to

1. A.$f(x+y+z)$
2. B.$f(x+y+z+1)$
3. C.$f(x+y+z+2)$
4. D.$f(x+y+z+3)$