Coordinate Geometry · NDA previous year questions

If A, B and C are in AP, then the straight line Ax + 2By + C = 0 will always pass through a fixed point. The fixed point is

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

If the image of the point (-4, 2) by a line mirror is (4, -2), then what is the equation of the line mirror?

1. A.y = x
2. B.y = 2x
3. C.4y = x
4. D.y = 4x

${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$ holds, when

1. A.$x\in \mathbf{R}$
2. B.$x\in \mathbf{R}-(-1,1)$ only
3. C.$x\in \mathbf{R}-\{0\}$ only
4. D.$x\in \mathbf{R}-[-1,1]$ only

A chord subtends an angle 120° at the centre of a unit circle. What is the length of the chord?

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

A particle starts from origin with a velocity (in m/s) given by the equation $\frac{\mathrm{d}x}{\mathrm{d}t}=x+1$. The time (in seconds) taken by the particle to traverse a distance of $24\mathrm{m}$ is:

1. A.$\ln 24$
2. B.$\ln 5$
3. C.$2\ln 5$
4. D.$2\ln 4$

What is ${\int }_0^a\frac{f(a-x)}{f(x)+f(a-x)}\mathrm{d}x$ equal to?

1. A.$a$
2. B.$2a$
3. C.0
4. D.$\frac{a}{2}$

What is ${\lim }_{x\to 2}\frac{x^3+x^2}{x^2+3x+2}$ equal to?

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

If ${\int }_0^a[f(x)+f(-x)]\mathrm{d}x={\int }_{-a}^{a}g(x)\mathrm{d}x$, then what is $g(x)$ equal to?

1. A.$f(x)$
2. B.$f(-x)+f(x)$
3. C.$-f(x)$
4. D.None of the above.

Consider the following statements in respect of the points (p, p - 3), (q + 3, q) and (6, 3): 1. The points lie on a straight line. 2. The points always lie in the first quadrant only for any value of p and q. 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 the acute angle between the lines x - 2 = 0 and √3x - y - 2 = 0?

1. A.0°
2. B.30°
3. C.45°
4. D.60°

The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at A(4, 2). What is the equation of the diagonal BD?

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

If any point on a hyperbola is (3tan θ, 2sec θ), then what is the eccentricity of the hyperbola?

1. A.$\frac{3}{2}$
2. B.$\frac{5}{2}$
3. C.$\frac{\sqrt{11}}{2}$
4. D.$\frac{\sqrt{13}}{2}$

Consider the following with regard to eccentricity (e) of a conic section: 1. e = 0 for circle 2. e = 1 for parabola 3. e < 1 for ellipse Which of the above statements is/are correct?

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

What is the projection of the line segment joining A(1, 7, -5) and B(-3, 4, -2) on y-axis?

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

What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)?

1. A.Zero
2. B.One
3. C.Two
4. D.Infinite

The center of the circle $(x-2a)(x-2b)+(y-2c)(y-2d)=0$ is

1. A.$(2a,2c)$
2. B.$(2b,2d)$
3. C.$(a+b,c+d)$
4. D.$(a-b,c-d)$

The point $(1,-1)$ is one of the vertices of a square. If $3x+2y=5$ is the equation of one diagonal of the square, then what is the equation of the other diagonal?

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

Let $P(x,y)$ be any point on the ellipse $25x^2+16y^2=400$. If $Q(0,3)$ and $R(0,-3)$ are two points, then what is $(PQ+PR)$ equal to?

1. A.12
2. B.10
3. C.8
4. D.6

If the circumcentre of the triangle formed by the lines $x+2=0$, $y+2=0$ and $kx+y+2=0$ is $(-1,-1)$, then what is the value of $k$?

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

In the parabola $y^2=x$, what is the length of the chord passing through the vertex and inclined to the $x$-axis at an angle $\theta$ ?

1. A.$\sin \theta \cdot {\sec }^2\theta$
2. B.$\cos \theta \cdot {\mathrm{cosec}}^2\theta$
3. C.$\cot \theta \cdot {\sec }^2\theta$
4. D.$2\tan \theta \cdot {\mathrm{cosec}}^2\theta$

Under which condition, are the points $(a,b)$, $(c,d)$ and $(a-c,b-d)$ collinear?

1. A.$ab=cd$
2. B.$ac=bd$
3. C.$ad=bc$
4. D.$abc=d$

Let ABC be a triangle. If $D(2,5)$ and $E(5,9)$ are the mid-points of the sides AB and AC respectively, then what is the length of the side BC ?

1. A.8
2. B.10
3. C.12
4. D.14

If the foot of the perpendicular drawn from the point $(0,k)$ to the line $3x-4y-5=0$ is $(3,1)$, then what is the value of $k$?

1. A.3
2. B.4
3. C.5
4. D.6

What is the obtuse angle between the lines whose slopes are $2-\sqrt{3}$ and $2+\sqrt{3}$ ?

1. A.$105^{\circ }$
2. B.$120^{\circ }$
3. C.$135^{\circ }$
4. D.$150^{\circ }$

If $3x-4y-5=0$ and $3x-4y+15=0$ are the equations of a pair of opposite sides of a square, then what is the area of the square?

1. A.4 square units
2. B.9 square units
3. C.16 square units
4. D.25 square units