Coordinate Geometry · NDA previous year questions

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2023 30 questions

Q12023 · 3 Sept 2023
If $z \overline{z} = ∣ z + \overline{z} ∣$ , where $z = x + i y$ , $i = - 1$ , then the locus of z is a pair of:
1. A.straight lines
2. B.rectangular hyperbolas
3. C.parabolas
4. D.circles
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Answer: D
Q22023 · 3 Sept 2023
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
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Answer: C
Q32023 · 3 Sept 2023
What is the value of $12 + 5 i + 12 - 5 i$ where $i = - 1$ ?
1. A.24
2. B.25
3. C. $52$
4. D. $5 (2 - 1)$
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Answer: C
Q522023 · 3 Sept 2023
Consider the following for the next items that follow: A parabola passes through (1, 2) and satisfies the differential equation $\frac{d y}{d x} = \frac{2 y}{x}, x > 0, y > 0$ . 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. $\frac{1}{8}$
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Answer: B
Q532023 · 3 Sept 2023
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. $\frac{l n ( ab )}{4}$
2. B. $\frac{l n ( ab )}{2}$
3. C. $ln (ab)$
4. D. $2 ln (ab)$
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Answer: B
Q542023 · 3 Sept 2023
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. $ab$
2. B.ab
3. C.2ab
4. D. $\frac{ab}{2}$
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Answer: A
Q682023 · 3 Sept 2023
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. $\frac{1}{3}$
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Answer: A
Q812023 · 3 Sept 2023
What is the equation of directrix of parabola $y^{2} = 4 b x$ , 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
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Answer: B
Q822023 · 3 Sept 2023
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
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Answer: D
Q832023 · 3 Sept 2023
Given that $16 p^{2} + 49 q^{2} - 4 r^{2} - 56 pq = 0$ . Which one of the following is a point on a pair of straight lines (px + qy + r) (px + qy - r) = 0?
1. A. $(2, \frac{7}{2})$
2. B. $(2, - \frac{7}{2})$
3. C.(4, -7)
4. D.(4, 7)
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Answer: B
Q842023 · 3 Sept 2023
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. $(\frac{3}{4}, \frac{1}{4})$
2. B. $(\frac{3}{2}, \frac{1}{2})$
3. C. $(\frac{3}{4}, - \frac{1}{4})$
4. D. $(\frac{3}{2}, - \frac{1}{2})$
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Answer: B
Q852023 · 3 Sept 2023
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
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Answer: C
Q862023 · 3 Sept 2023
Consider the following statements in respect of hyperbola $\frac{x ^{2}}{c o s ^{2} θ} - \frac{y ^{2}}{s i n ^{2} θ} = 1$ : 1. The two foci are independent of $θ$ . 2. The eccentricity is $sec θ$ . 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
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Answer: C
Q872023 · 3 Sept 2023
Consider the following in respect of the circle $4 x^{2} + 4 y^{2} - 4 a x - 4 a y + a^{2} = 0$ : 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
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Answer: B
Q882023 · 3 Sept 2023
For what values of k is the line $(k - 3) x - (5 - k^{2}) y + k^{2} - 7 k + 6 = 0$ parallel to the line x + y = 1?
1. A.-1, 1
2. B.-1, 2
3. C.1, -2
4. D.2, -2
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Answer: B
Q892023 · 3 Sept 2023
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
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Answer: B
Q902023 · 3 Sept 2023
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
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Answer: C
Q962023 · 3 Sept 2023
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. $(- \frac{11}{7}, \frac{13}{7}, \frac{19}{7})$ and $(\frac{1}{7}, \frac{3}{7}, \frac{5}{7})$
2. B. $(\frac{19}{7}, - \frac{11}{7}, \frac{13}{7})$ and $(- \frac{1}{7}, \frac{3}{7}, - \frac{5}{7})$
3. C. $(\frac{13}{7}, \frac{11}{7}, \frac{19}{7})$ and $(- \frac{1}{7}, - \frac{3}{7}, \frac{5}{7})$
4. D. $(\frac{13}{7}, - \frac{11}{7}, \frac{19}{7})$ and $(\frac{1}{7}, - \frac{3}{7}, - \frac{5}{7})$
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Answer: D
Q1002023 · 3 Sept 2023
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
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Answer: A
Q1012023 · 3 Sept 2023
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
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Answer: A
Q452023 · 16 Apr 2023
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
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Answer: C
Q462023 · 16 Apr 2023
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
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Answer: D
Q472023 · 16 Apr 2023
Passage
Consider the following for the next two (02) items that follow : Given that sinx + cosx + tanx + cotx + secx + cosecx = 7
The given equation can be reduced to
1. A. $sin^{2} 2 x - 44 sin 2 x + 36 = 0$
2. B. $sin^{2} 2 x + 44 sin 2 x - 36 = 0$
3. C. $sin^{2} 2 x - 22 sin 2 x + 18 = 0$
4. D. $sin^{2} 2 x + 22 sin 2 x - 18 = 0$
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Answer: A
Q482023 · 16 Apr 2023
Passage
Consider the following for the next two (02) items that follow : Given that sinx + cosx + tanx + cotx + secx + cosecx = 7
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
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Answer: D
Q552023 · 16 Apr 2023
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
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Answer: C