Coordinate Geometry · NDA previous year questions

The line x + y = 4 cuts the line joining P(-1, 1) and Q(5, 7) at R. What is PR ∶ RQ equal to ?

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

What is the sum of the intercepts of the line whose perpendicular distance from origin is 4 units and the angle which the normal makes with positive direction of x-axis is 15°?

1. A.8
2. B.4√6
3. C.8√6
4. D.16

If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?

1. A.$\left(-\frac{11}{7},\frac{13}{7},\frac{19}{7}\right)$ and $\left(\frac{1}{7},\frac{3}{7},\frac{5}{7}\right)$
2. B.$\left(\frac{19}{7},-\frac{11}{7},\frac{13}{7}\right)$ and $\left(-\frac{1}{7},\frac{3}{7},-\frac{5}{7}\right)$
3. C.$\left(\frac{13}{7},\frac{11}{7},\frac{19}{7}\right)$ and $\left(-\frac{1}{7},-\frac{3}{7},\frac{5}{7}\right)$
4. D.$\left(\frac{13}{7},-\frac{11}{7},\frac{19}{7}\right)$ and $\left(\frac{1}{7},-\frac{3}{7},-\frac{5}{7}\right)$

A point P lies on the line joining A(1, 2, 3) and B(2, 10, 1). If z-coordinate of P is 7, what is the sum of other two coordinates?

1. A.-15
2. B.-13
3. C.-11
4. D.-9

The sum of deviations of n numbers from 10 and 20 are p and q respectively. If $(p-q{)}^2=10000$, then what is the value of n?

1. A.10
2. B.20
3. C.50
4. D.100

Consider the following for the next two (02) items that follow : In a triangle POR, P is the largest angle and $\cos P=\frac{1}{3}$. Further the in-circle of the triangle touches the sides PQ, QR and RP at N, L and M respectively such that the lengths PN, QL and RM are n, $n+2$, $n+4$ respectively where n is an integer. What is the value of n ?

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

Consider the following for the next two (02) items that follow : In a triangle POR, P is the largest angle and $\cos P=\frac{1}{3}$. Further the in-circle of the triangle touches the sides PQ, QR and RP at N, L and M respectively such that the lengths PN, QL and RM are n, $n+2$, $n+4$ respectively where n is an integer. What is the length of the smallest side ?

1. A.12
2. B.14
3. C.16
4. D.18

The given equation can be reduced to

1. A.${\sin }^{2}2x-44\sin 2x+36=0$
2. B.${\sin }^{2}2x+44\sin 2x-36=0$
3. C.${\sin }^{2}2x-22\sin 2x+18=0$
4. D.${\sin }^{2}2x+22\sin 2x-18=0$

If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?

1. A.0
2. B.14
3. C.21
4. D.28

Consider the following for the next two (02) items that follow : A quadrilateral is formed by the lines x = 0, y = 0, x + y = 1 and 6x + y = 3. What is the equation of diagonal through origin ?

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

What is the equation of other diagonal ?

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

Consider the following for the next two (02) items that follow : $P(x,y)$ is any point on the ellipse $x^2+4y^2=1$. Let E, F be the foci of the ellipse. What is PE + PF equal to ?

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

Consider the following points : 1. $\left(\frac{\sqrt{3}}{2},0\right)$ 2. $\left(\frac{\sqrt{3}}{2},\frac{1}{4}\right)$ 3. $\left(\frac{\sqrt{3}}{2},-\frac{1}{4}\right)$ Which of the above points lie on latus rectum of ellipse ?

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

Consider the following for the next two (02) items that follow : The line $y=x$ partitions the circle $(x-a{)}^2+y^2=a^2$ in two segments. What is the area of minor segment ?

1. A.$\frac{(\pi -2)a^2}{4}$
2. B.$\frac{(\pi -1)a^2}{4}$
3. C.$\frac{(\pi -2)a^2}{2}$
4. D.$\frac{(\pi -1)a^2}{2}$

Consider the following for the next two (02) items that follow : The line $y=x$ partitions the circle $(x-a{)}^2+y^2=a^2$ in two segments. What is the area of major segment ?

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

What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?

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

Consider the following statements in respect of the line passing through origin and inclining at an angle of $75^{\circ }$ with the positive direction of x-axis : 1. The line passes through the point $\left(1,\frac{1}{2-\sqrt{3}}\right)$. 2. The line entirely lies in first and third quadrants. 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

If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line?

1. A.3x + 4y − 25 = 0
2. B.4x + 3y − 24 = 0
3. C.4x − 3y = 0
4. D.3x − 4y + 7 = 0

The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid- point of AB is at the origin and B lies above the origin. What is the equation of line passing through (8, 0) and parallel to the side AC ?

1. A.$x-\sqrt{3}y-8=0$
2. B.$x+\sqrt{3}y-8=0$
3. C.$\sqrt{3}x+y-8\sqrt{3}=0$
4. D.$\sqrt{3}x-y-8\sqrt{3}=0$

The centre of the circle passing through the origin and making positive intercepts 4 and 6 on the coordinate axes, lies on the line

1. A.2x − y + 1 = 0
2. B.3x − 2y − 1 = 0
3. C.3x − 4y + 6 = 0
4. D.2x + 3y − 26 = 0

The centre of an ellipse is at (0, 0), major axis is on the y-axis. If the ellipse passes through (3, 2) and (1, 6), then what is its eccentricity ?

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

An equilateral triangle is inscribed in a parabola $x^2=\sqrt{3}y$ where one vertex of the triangle is at the vertex of the parabola. If p is the length of side of the triangle and q is the length of the latus rectum, then which one of the following is correct ?

1. A.$p=q$
2. B.$p=\sqrt{3}q$
3. C.$p=2\sqrt{3}q$
4. D.$2\sqrt{3}p=q$

Consider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct? 1. The points are the vertices of a rectangle ABCD. 2. The mid-point of A C is the same as that of BD. 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 does the equation $x\frac{\mathrm{d}y}{\mathrm{d}x}-2y=0$ represent ?

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

If the points with coordinates $(-5,0)$, $(5p^2,10p)$ and $(5q^2,10q)$ are collinear, then what is the value of pq where $p\ne q$ ?

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