Coordinate Geometry · NDA previous year questions

Passage: Let $X(a,p),Y(b,q)$ and $Z(c,r)$ be the points such that $a,b$ and $c$ are in AP. Question: If $p,q$ and $r$ are in AP, then the points $X,Y$ and $Z$ are

1. A.on a straight line
2. B.on a circle
3. C.on a parabola
4. D.the vertices of a triangle

Passage: The equation of a straight line which passes through the origin and makes an angle $\alpha$ ($0\le \alpha \le 90^{\circ }$) with the line $L:x+\sqrt{3}y+3\sqrt{3}=0$ is $x-\sqrt{3}y=0$. Question: What is the angle made by the line $L$ with positive direction of $y$-axis ?

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

Passage: The equation of a straight line which passes through the origin and makes an angle $\alpha$ ($0\le \alpha \le 90^{\circ }$) with the line $L:x+\sqrt{3}y+3\sqrt{3}=0$ is $x-\sqrt{3}y=0$. Question: What is the value of $\alpha$ ?

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

Passage: Consider the points $A(0,2),B(2,3),C(4,5)$ and $D(0,k)$. Question: If a circle is drawn through $A,B$ and $D$, then what is the diameter of the circle ?

1. A.$3\sqrt{10}$
2. B.$5\sqrt{10}$
3. C.$3\sqrt{12}$
4. D.$5\sqrt{12}$

Passage: Consider the points $A(0,2),B(2,3),C(4,5)$ and $D(0,k)$. Question: If the points lie on a circle, then what is/are the possible value(s) of $k$ ?

1. A.2 only
2. B.5 only
3. C.2, 17
4. D.5, 17

What is the length of the chord of a unit circle which subtends at the centre of the circle an angle of $45^{\circ }$ ?

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

Passage: Let $k$ be the area between the curve $y=\sin x$ and x-axis in the interval $\left[0,\frac{\pi }{4}\right]$. Question: What is the area between the curve $y=\cos x$ and the x-axis in the interval $\left[\frac{\pi }{4},\frac{\pi }{2}\right]$ ?

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

Passage: Let $k$ be the area between the curve $y=\sin x$ and x-axis in the interval $\left[0,\frac{\pi }{4}\right]$. Question: What is the area between the curve $y=\sin x$ and the x-axis in the interval $\left[\frac{\pi }{4},\frac{\pi }{2}\right]$ ?

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

For the following two (02) items: Let A(1, - 1, 0) B(- 2, 1, 8) and C(- 1, 2, 7) are three consecutive vertices of a parallelogram ABCD. What is the fourth vertex D?

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

1. A.26/77
2. B.27/77
3. C.82/237
4. D.83 / (237)

For different values of m, the equation 4y=mx-m+ 2 represents

1. A.parallel lines
2. B.concurrent lines
3. C.lines at a fixed distance from the origin of coordinates
4. D.the same line

The equation of the locus of a point equidistant from the points (a, b) and (c, d) is (a - c)  x + (b - d)  y + k = 0 What is the value of k?

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

Consider the following statements in respect of the equation $x^2+3y=0$ I. The equation represents the equation to parabola that opens upwards. II. The axis of the parabola is x = 0 III. The equation of the latus rectum is 4y - 3 = 0 How many of the statements given above are correct?

1. A.None
2. B.One
3. C.Two
4. D.All three

What is the sum of the intercepts of the line $\frac{x}{a^2}+\frac{y}{b^2}=\frac{2}{a^2+b^2}$ on the coordinate axes?

1. A.2
2. B.1
3. C.1/2
4. D.$a^2+b^2$

Which one of the following is the perpendicular form of the straight line $\sqrt{3}x+2y=7$ ?

1. A.$y=-\frac{\sqrt{3}}{2}x+\frac{7}{2}$
2. B.$\frac{x}{(\frac{7}{\sqrt{3}})}+\frac{y}{(\frac{7}{2})}=1$
3. C.$\frac{\sqrt{3}}{\sqrt{7}}x+\frac{2}{\sqrt{7}}y=\sqrt{7}$
4. D.$\frac{\sqrt{3}}{\sqrt{7}}x+\frac{2}{\sqrt{7}}y=7$

If the vertices Band D of a square ABCD are (2, 3) and (4, 1) respectively, then what is the area of the square?

1. A.2 square units
2. B.3 square units
3. C.4 square units
4. D.8 square units

The circle $x^2+y^2-2kx-2ky+k^2=0$ touches the x-axis at P and y-axis at Q. What is PQ equal to?

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

What is the distance between the foci of the hyperbola $x^2-4y^2=1$

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

Let $\vec{p}=\vec{a}-\vec{b},\vec{q}=\vec{a}+\vec{b}$. If $|\vec{a}|=|\vec{b}|=2$ and $\vec{a}\cdot \vec{b}=2$, then what is the value of $|\vec{p}\times \vec{q}|$?

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

What is PN equal to?

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

What is MN equal to?

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

A man at M, standing $100\mathrm{m}$ away from the base (P) of a chimney of height $50\mathrm{m}$, observes the angle of elevation of the highest point (Q) of the smoke to be $45^{\circ }$. The highest point of the chimney is at R. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?

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

If k is a root of $x^2-4x+1=0$, then what is ${\tan }^{-1}k+{\tan }^{-1}\frac{1}{k}$ equal to?

1. A.-$\pi /2$
2. B.0
3. C.$\pi /4$
4. D.$\pi /2$

If ${\tan }^{-1}k+{\tan }^{-1}\frac{1}{2}=\frac{\pi }{4}$, then what is the value of k?

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

Under what condition will the lines $m^{2}x+ny-1=0$ and $n^{2}x-my+2=0$ be perpendicular?

1. A.mn-1=0
2. B.mn + 1 = 0
3. C.m + n = 0
4. D.m - n = 0