Coordinate Geometry · NDA previous year questions

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2018 28 questions

Q62018 · 9 Sept 2018
Which one of the following is correct in respect of the cube roots of unity?
1. A.They are collinear
2. B.They lie on a circle of radius $3$
3. C.They form an equilateral triangle
4. D.None of the above
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Answer: C
Q302018 · 9 Sept 2018
The system of equations $2 x + y - 3 z = 5$ , $3 x - 2 y + 2 z = 5$ and $5 x - 3 y - z = 16$
1. A.is inconsistent
2. B.is consistent, with a unique solution
3. C.is consistent, with infinitely many solutions
4. D.has its solution lying along x-axis in three-dimensional space
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Answer: B
Q462018 · 9 Sept 2018
Consider the following statements: Statement I: If the line segment joining the points P $(m, n)$ and Q $(r, s)$ subtends an angle $α$ at the origin, then $cos α = \frac{m s - n r}{( m ^{2} + n ^{2} ) ( r ^{2} + s ^{2} )}$ . Statement II: In any triangle ABC, it is true that $a^{2} = b^{2} + c^{2} - 2 b c cos A$ . Which one of the following is correct in respect of the above two statements?
1. A.Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I
2. B.Both Statement I and Statement II are true, but Statement II is not the correct explanation of Statement I
3. C.Statement I is true, but Statement II is false
4. D.Statement I is false, but Statement II is true
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Answer: D
Q472018 · 9 Sept 2018
What is the area of the triangle with vertices $(x_{1}, \frac{1}{x _{1}}), (x_{2}, \frac{1}{x _{2}}), (x_{3}, \frac{1}{x _{3}})$ ?
1. A. $∣ (x_{1} - x_{2}) (x_{2} - x_{3}) (x_{3} - x_{1}) ∣$
2. B. $0$
3. C. $\frac{( x _{1} - x _{2} ) ( x _{2} - x _{3} ) ( x _{3} - x _{1} )}{x _{1} x _{2} x _{3}}$
4. D. $\frac{( x _{1} - x _{2} ) ( x _{2} - x _{3} ) ( x _{3} - x _{1} )}{2 x _{1} x _{2} x _{3}}$
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Answer: D
Q482018 · 9 Sept 2018
The angle between the two lines $l x + m y + n = 0$ and $l^{'} x + m^{'} y + n^{'} = 0$ is given by $tan^{- 1} θ$ . What is $θ$ equal to?
1. A. $\frac{l m ^{'} - l ^{'} m}{l l ^{'} - m m ^{'}}$
2. B. $\frac{l m ^{'} + l ^{'} m}{l l ^{'} + m m ^{'}}$
3. C. $\frac{l m ^{'} - l ^{'} m}{l l ^{'} + m m ^{'}}$
4. D. $\frac{l m ^{'} + l ^{'} m}{l l ^{'} - m m ^{'}}$
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Answer: C
Q492018 · 9 Sept 2018
Consider the following statements: 1. The distance between the lines $y = m x + c_{1}$ and $y = m x + c_{2}$ is $\frac{∣ c _{1} - c _{2} ∣}{1 + m ^{2}}$ . 2. The distance between the lines $a x + b y + c_{1} = 0$ and $a x + b y + c_{2} = 0$ is $\frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$ . 3. The distance between the lines $x = c_{1}$ and $x = c_{2}$ is $∣ c_{1} - c_{2} ∣$ . Which of the above statements 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
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Answer: D
Q502018 · 9 Sept 2018
What is the equation of straight line passing through the point of intersection of the lines $\frac{x}{2} + \frac{y}{3} = 1$ and $\frac{x}{3} + \frac{y}{2} = 1$ , and parallel to the line $4 x + 5 y - 6 = 0$ ?
1. A. $20 x + 25 y - 54 = 0$
2. B. $25 x + 20 y - 54 = 0$
3. C. $4 x + 5 y - 54 = 0$
4. D. $4 x + 5 y - 45 = 0$
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Answer: A
Q512018 · 9 Sept 2018
The equation of a circle whose end points of a diameter are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is
1. A. $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = x^{2} + y^{2}$
2. B. $(x - x_{1})^{2} + (y - y_{1})^{2} = x_{2} y_{2}$
3. C. $x^{2} + y^{2} + 2 x_{1} x_{2} + 2 y_{1} y_{2} = 0$
4. D. $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$
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Answer: D
Q522018 · 9 Sept 2018
If y-axis touches the circle $x^{2} + y^{2} + g x + f y + \frac{c}{4} = 0$ , then the normal at this point intersects the circle at the point
1. A. $(- \frac{g}{2}, - \frac{f}{2})$
2. B. $(- g, - \frac{f}{2})$
3. C. $(- \frac{g}{2}, f)$
4. D. $(- g, - f)$
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Answer: B
Q532018 · 9 Sept 2018
The second degree equation $x^{2} + 4 y^{2} - 2 x - 4 y + 2 = 0$ represents
1. A.A point
2. B.An ellipse of semi-major axis 1
3. C.An ellipse with eccentricity $\frac{3}{2}$
4. D.None of the above
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Answer: A
Q552018 · 9 Sept 2018
Coordinates of the points O, P, Q and R are respectively $(0, 0, 0)$ , $(4, 6, 2 m)$ , $(2, 0, 2 n)$ and $(2, 4, 6)$ . Let L, M, N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjacent sides LK and LM are each of length $2$ . What are the values of m and n respectively?
1. A.6, 2
2. B.1, 3
3. C.3, 1
4. D.None of the above
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Answer: C
Q562018 · 9 Sept 2018
The line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ is given by
1. A. $x + y + z = 6$ , $x + 2 y - 3 z = - 4$
2. B. $x + 2 y - 2 z = - 1$ , $4 x + 4 y - 5 z - 3 = 0$
3. C. $3 x + 2 y - 3 z = 0$ , $3 x - 6 y + 3 z = - 2$
4. D. $3 x + 2 y - 3 z = - 2$ , $3 x - 6 y + 3 z = 0$
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Answer: D
Q712018 · 9 Sept 2018
Let the slope of the curve $y = cos^{- 1} (sin x)$ be $tan θ$ . Then the value of $θ$ in the interval $(0, π)$ is
1. A. $\frac{π}{6}$
2. B. $\frac{3 π}{4}$
3. C. $\frac{π}{4}$
4. D. $\frac{π}{2}$
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Answer: B, C
Q342018 · 22 Apr 2018
ABC is a triangle inscribed in a circle with centre O. Let $α = ∠ B A C$ , where $4 5^{\circ} < α < 9 0^{\circ}$ . Let $β = ∠ B O C$ . Which one of the following is correct?
1. A. $cos β = \frac{1 - t a n ^{2} α}{1 + t a n ^{2} α}$
2. B. $cos β = \frac{1 + t a n ^{2} α}{1 - t a n ^{2} α}$
3. C. $cos β = \frac{2 t a n α}{1 + t a n ^{2} α}$
4. D. $sin β = 2 sin^{2} α$
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Answer: A
Q352018 · 22 Apr 2018
If $\frac{s i n ( x + y )}{s i n ( x - y )} = \frac{a + b}{a - b}$ , then what is $\frac{t a n x}{t a n y}$ equal to?
1. A. $\frac{a}{b}$
2. B. $\frac{b}{a}$
3. C. $\frac{a + b}{a - b}$
4. D. $\frac{a - b}{a + b}$
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Answer: A
Q362018 · 22 Apr 2018
If $sin α + sin β = 0 = cos α + cos β$ , where $0 < β < α < 2 π$ , then which one of the following is correct?
1. A. $α = π - β$
2. B. $α = π + β$
3. C. $α = 2 π - β$
4. D. $2 α = π + 2 β$
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Answer: B
Q372018 · 22 Apr 2018
Suppose $cos A$ is given. If only one value of $cos (\frac{A}{2})$ is possible, then $A$ must be
1. A.An odd multiple of $9 0^{\circ}$
2. B.A multiple of $9 0^{\circ}$
3. C.An odd multiple of $18 0^{\circ}$
4. D.A multiple of $18 0^{\circ}$
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Answer: C
Q382018 · 22 Apr 2018
If $cos α + cos β + cos γ = 0$ , where $0 < α \leq \frac{π}{2}, 0 < β \leq \frac{π}{2}, 0 < γ \leq \frac{π}{2}$ , then what is the value of $sin α + sin β + sin γ$ ?
1. A. $0$
2. B. $3$
3. C. $\frac{5 2}{2}$
4. D. $\frac{3 2}{2}$
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Answer: B
Q482018 · 22 Apr 2018
What is the angle between the straight lines $(m^{2} - mn) y = (mn + n^{2}) x + n^{3}$ and $(mn + m^{2}) y = (mn - n^{2}) x + m^{3}$ , where $m > n$ ?
1. A. $tan^{- 1} (\frac{2 mn}{m ^{2} + n ^{2}})$
2. B. $tan^{- 1} (\frac{4 m ^{2} n ^{2}}{m ^{4} - n ^{4}})$
3. C. $tan^{- 1} (\frac{4 m ^{2} n ^{2}}{m ^{4} + n ^{4}})$
4. D. $4 5^{\circ}$
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Answer: B
Q492018 · 22 Apr 2018
What is the equation of the straight line cutting off an intercept 2 from the negative direction of y-axis and inclined at $3 0^{\circ}$ with the positive direction of x-axis?
1. A. $x - 23 y - 32 = 0$
2. B. $x + 23 y - 32 = 0$
3. C. $x + 3 y - 23 = 0$
4. D. $x - 3 y - 23 = 0$
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Answer: D
Q512018 · 22 Apr 2018
Consider the following statements: 1. The length $p$ of the perpendicular from the origin to the line $a x + b y = c$ satisfies the relation $p^{2} = \frac{c ^{2}}{a ^{2} + b ^{2}}$ . 2. The length $p$ of the perpendicular from the origin to the line $\frac{x}{a} + \frac{y}{b} = 1$ satisfies the relation $\frac{1}{p ^{2}} = \frac{1}{a ^{2}} + \frac{1}{b ^{2}}$ . 3. The length $p$ of the perpendicular from the origin to the line $y = m x + c$ satisfies the relation $\frac{1}{p ^{2}} = \frac{1 + m ^{2} + c ^{2}}{c ^{2}}$ . Which of the above is/are correct?
1. A.1, 2 and 3
2. B.1 only
3. C.1 and 2 only
4. D.2 only
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Answer: C
Q522018 · 22 Apr 2018
What is the equation of the straight line passing through the point $(2, 3)$ and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis?
1. A. $2 x + y = 5$
2. B. $2 x + y = 7$
3. C. $x + 2 y = 7$
4. D. $2 x - y = 1$
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Answer: B
Q532018 · 22 Apr 2018
The perpendiculars that fall from any point of the straight line $2 x + 11 y = 5$ upon the two straight lines $24 x + 7 y = 20$ and $4 x - 3 y = 2$ are
1. A.12 and 4 respectively
2. B.11 and 5 respectively
3. C.Equal to each other
4. D.Not equal to each other
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Answer: C
Q542018 · 22 Apr 2018
The equation of the line, when the portion of it intercepted between the axes is divided by the point $(2, 3)$ in the ratio of $3 : 2$ , is
1. A.Either $x + y = 4$ or $9 x + y = 12$
2. B.Either $x + y = 5$ or $4 x + 9 y = 30$
3. C.Either $x + y = 4$ or $x + 9 y = 12$
4. D.Either $x + y = 5$ or $9 x + 4 y = 30$
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Answer: D
Q552018 · 22 Apr 2018
What is the distance between the straight lines $3 x + 4 y = 9$ and $6 x + 8 y = 15$ ?
1. A. $3/2$
2. B. $3/10$
3. C. $6$
4. D. $5$
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Answer: B