Coordinate Geometry · NDA previous year questions

If two sides of a square lie on the lines $2x+y-3=0$ and $4x+2y+5=0$, then what is the area of the square in square units?

1. A.6.05
2. B.6.15
3. C.6.25
4. D.6.35

ABC is a triangle with A(3, 5). The mid-points of sides AB, AC are at (-1, 2), (6, 4) respectively. What are the coordinates of centroid of the triangle ABC?

1. A.$\left(\frac{8}{3},\frac{11}{3}\right)$
2. B.$\left(\frac{7}{3},\frac{7}{3}\right)$
3. C.$\left(2,\frac{8}{3}\right)$
4. D.$\left(\frac{8}{3},2\right)$

ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x - y - 3 = 0 and x + y - 5 = 0. If θ is one of the equal angles, then what is cotθ equal to?

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

In the parabola $y^2=8x$, the focal distance of a point P lying on it is 8 units. Which of the following statements is/are correct? 1. The coordinates of $P$ can be $(6,4\sqrt{3})$. 2. The perpendicular distance of $P$ from the directrix of parabola is 8 units. Select the correct answer using the code given below:

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

What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90°?

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

Let $\vec{a}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If $\vec{a}\times (\vec{b}\times \vec{a})=\alpha \hat{i}-\beta \hat{j}+\gamma \hat{k}$, then what is the value of $\alpha +\beta +\gamma$?

1. A.8
2. B.7
3. C.6
4. D.1

Consider the following for the next two (02) items that follow: The circle $x^2+y^2-2x=0$ is partitioned by line $y=x$ in two segments. Let $A_1,A_2$ be the areas of major and minor segments respectively. What is the value of $A_1$?

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

Consider the following for the next two (02) items that follow: The circle $x^2+y^2-2x=0$ is partitioned by line $y=x$ in two segments. Let $A_1,A_2$ be the areas of major and minor segments respectively. What is the value of $\frac{2(A_1+A_2)}{A_1-3A_2}$ ?

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

Consider the following for the next two (02) items that follow : Let $3f(x)+f\left(\frac{1}{x}\right)=\frac{1}{x}+1$ What is $f(x)$ equal to?

1. A.$\frac{1}{8x}-\frac{x}{8}+\frac{1}{4}$
2. B.$\frac{3}{8x}-\frac{x}{8}+\frac{3}{4}$
3. C.$\frac{3}{8x}+\frac{x}{8}+\frac{1}{4}$
4. D.$\frac{3}{8x}-\frac{x}{8}+\frac{1}{4}$

Consider the following for the next two (02) items that follow: Let $3f(x)+f\left(\frac{1}{x}\right)=\frac{1}{x}+1$ What is $8{\int }_1^{2}f(x)dx$ equal to?

1. A.$\ln (8\sqrt{e})$
2. B.$\ln (4\sqrt{e})$
3. C.$\ln 2$
4. D.$\ln 2-1$

If $z\overline{z}=|z+\overline{z}|$, where $z=x+iy$, $i=\sqrt{-1}$, then the locus of z is a pair of:

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

If 1! + 3! + 5! + 7! + ... + 199! is divided by 24, what is the remainder?

1. A.3
2. B.6
3. C.7
4. D.9

What is the value of $\sqrt{12+5i}+\sqrt{12-5i}$ where $i=\sqrt{-1}$ ?

1. A.24
2. B.25
3. C.$5\sqrt{2}$
4. D.$5(\sqrt{2}-1)$

Consider the following for the next items that follow: A parabola passes through (1, 2) and satisfies the differential equation $\frac{dy}{dx}=\frac{2y}{x},x>0,y>0$. What is the length of latus rectum of the parabola?

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

Consider the following for the next items that follow: Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and $g(x)=x-1$. What is ${\lim }_{x\to 1}\frac{f(x)-1}{g(x)}$ equal to?

1. A.$\frac{\ln (ab)}{4}$
2. B.$\frac{\ln (ab)}{2}$
3. C.$\ln (ab)$
4. D.$2\ln (ab)$

Consider the following for the next items that follow: Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and $g(x)=x-1$. What is ${\lim }_{x\to 1}f(x{)}^{\frac{1}{g(x)}}$ equal to?

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

What is the area between the curve f(x) = x |x| and x-axis for x = [-1, 1]?

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

What is the equation of directrix of parabola $y^2=4bx$, where $b<0$ and $b^2+b-2=0$?

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

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

1. A.lying on the same circle
2. B.vertices of a square
3. C.vertices of a parallelogram that is not a square
4. D.collinear

Given that $16p^2+49q^2-4r^2-56pq=0$. Which one of the following is a point on a pair of straight lines (px + qy + r) (px + qy - r) = 0?

1. A.$\left(2,\frac{7}{2}\right)$
2. B.$\left(2,-\frac{7}{2}\right)$
3. C.(4, -7)
4. D.(4, 7)

If 3x + y - 5 = 0 is the equation of a chord of the circle $x^2+y^2-25=0$, then what are the coordinates of the mid-point of the chord ?

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

Consider the following in respect of the equation $\frac{x^2}{24-k}+\frac{y^2}{k-16}=2$. 1. The equation represents an ellipse if k = 19. 2. The equation represents a hyperbola if k = 12. 3. The equation represents a circle if k = 20. How many of the statements given above are correct?

1. A.Only one
2. B.Only two
3. C.All three
4. D.None

Consider the following statements in respect of hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$: 1. The two foci are independent of $\theta$. 2. The eccentricity is $\sec \theta$. 3. The distance between the two foci is 2 units. How many of the statements given above are correct?

1. A.Only one
2. B.Only two
3. C.All three
4. D.None

Consider the following in respect of the circle $4x^2+4y^2-4ax-4ay+a^2=0$: 1. The circle touches both the axes. 2. The diameter of the circle is 2a. 3. The centre of the circle lies on the line x + y = a. How many of the statements given above are correct?

1. A.Only one
2. B.Only two
3. C.All three
4. D.None

For what values of k is the line $(k-3)x-(5-k^2)y+k^2-7k+6=0$ parallel to the line x + y = 1?

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