**Practice Test – I **

1. If the straight line mx – y = 1 + 2x intersects the circle x^{2} + y^{2} = 1 at least at one point, then the set of values of m is

(A) [-4/3 , 0]

(B) [-4/3 , 4/3]

(C) [0 , 4/3]

(D) all of these

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2. Circles are drawn having the sides of triangle ABC as their diameters. Radical centre of the circles is the

(A) Circumcentre of triangle ABC

(B) In centre of triangle ABC

(C) Orthocentre of triangle ABC

(D) Centroid of triangle ABC

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3. The circle described on the line joining the points (0, 1), (a, b) as diameter cuts the x-axis at points whose abscissae are roots of the equation

(A) x^{2} + ax + b = 0

(B) x^{2} – ax + b = 0

(C) x^{2} + ax – b = 0

(D) x^{2} – ax – b = 0

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4. The straight line y = m x + c cuts the circle x^{2} + y^{2} = a^{2} at real points if

(A) √[a^{2}(1 + m^{2})] ≤|c|

(B) √[a^{2}(1- m^{2})] ≤|c|

(C) √[a^{2}(1 + m^{2})] ≥|c|

(D) √[a^{2}(1 – m^{2})] ≥|c|

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5. The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x^{2} + y^{2} = 9 is

(A) (3/2 , 1/2)

(B) (1/2 , 3/2)

(C) (1/2 ,1/2)

(D) (1/2 , -√2)

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6. If circles are drawn on the sides of the triangle formed by the lines x = 0, y = 0 and x + y = 2, as diameters, then the radical centre of three circles is

(A) (0, 0)

(B) (1, 1)

(C) (√2 , 1)

(D) None of these

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7. The length of the chord cut off by y = 2x + 1 from the circle x^{2} + y^{2} = 2 is

(A) 5/6

(B) 6/5

(C) 6/√5

(D) √5/6

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8. The coordinates of middle point of the chord 2x – 5y + 18 = 0 cut off by the circle x^{2} + y^{2} – 6x + 2y – 54 = 0 is

(A) (1, 4)

(B) (–4, 2)

(C) (4, 1)

(D) (6, 6)

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9. The locus of the point, such that tangents drawn from it to the circle x^{2} + y^{2} – 6x – 8y = 0 are perpendicular to each other, is

(A) x^{2} + y^{2} – 6x – 8y –25 = 0

(B) x^{2} + y^{2} + 6x – 8y – 25 = 0

(C) x^{2} + y^{2} – 6x – 8y + 25 = 0

(D) none of these

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10. The equation of circle touching the line 2x + 3y + 1 = 0 at the point (1, -1) and passing through the focus of the parabola y^{2} = 4x is

(A) 3x^{2} + 3y^{2} – 8x + 3y + 5 = 0

(B) 3x^{2} + 3y^{2} + 8x – 3y + 5 = 0

(C) x^{2} + y^{2} – 3x + y + 6 = 0

(D) none of these

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11. The value of k for which two tangents can be drawn from (k , k) to the circle x^{2} + y^{2} + 2x + 2y – 16 = 0 is

(A) k ∈ R^{+}

(B) k ∈ R^{–}

(C) k ∈ ( -∞ , -4) ∪ ( 2, ∞)

(D) k ∈ ( 0, 1]

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12. The lines 3x – 4y + λ = 0 and 6x – 8y + μ = 0 are tangents to the same circle. The radius of the circle is

(A) | 2λ-μ |/20

(B) | 2μ-λ |/20

(C) | 2λ+μ |/20

(D) none of these

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13. The locus of the centres of the circles passing through the origin and intersecting the fixed circle x^{2}+ y^{2} – 5x + 3y – 1 = 0 orthogonally is

(A) a straight line of slope 3/5

(B) a circle

(C) a pair of straight line

(D) none of these

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14. If the line y – mx + m – 1 = 0 cuts the circle x^{2} + y^{2} – 4x – 4y + 4 = 0 at two real points, then m belongs to

(A) [1, 1]

(B) [-2, 2]

(C) (-∞, ∞)

(D) [-4, 4]

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15. The equation of the circle with centre on the x-axis and touching the line 3x + 4y – 11 = 0 at the point (1, 2) is

(A) x^{2} + y^{2} – x – 4 = 0

(B) x^{2} + y^{2} + 2x – 7 = 0

(C) x^{2} + y^{2} + x – 6 = 0

(D) none of these

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16. Equation of a circle with centre (4, 3) touching the circle x^{2} + y^{2} = 1 is

(A) x^{2} + y^{2} – 8x – 6y – 9 = 0

(B) x^{2} + y^{2} – 8x – 6y + 11 = 0

(C) x^{2} + y^{2} – 8x – 6y – 11 = 0

(D) x^{2} + y^{2} – 8x – 6y + 9 = 0

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17. The co-ordinates of the point on the circle x^{2} + y^{2} – 12x – 4y + 30 = 0 which is farthest from the origin are

(A) (9, 3)

(B) (8,5)

(C) (12, 4)

(D) none of these

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18. The locus of a point from which the length of tangents to the circles x^{2} + y^{2} = 4 and 2(x^{2} + y^{2}) – 10x + 3y – 2 = 0 are equal, is

(A) a straight line inclined at π/4 with the line joining the centres of the circles

(B) a circle

(C) an ellipse

(D) a straight line perpendicular to the line joining the centres of the circles

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19. The centre of the circle inscribed in a square formed by the lines x^{2} – 8x + 12 = 0 and y^{2} – 14y + 45 = 0 is

(A) (4, 7)

(B) (7, 4)

(C) (9, 4)

(D) (4, 9)

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20. If from any point on the circle x^{2} + y^{2} = a^{2} tangents are drawn to the circle x^{2} + y^{2} = a^{2} sin^{2} α , then the angle between the tangents, is

(A) α/2

(B) α

(C) 2α

(D) 4α

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21. Equation of chord AB of circle x^{2} + y^{2} = 2 passing through P(2 , 2) such that PB/PA = 3, is given by

(A) x = 3 y

(B) x = y

(C) y – 2 = √3 (x – 2)

(D) none of these

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22. Four distinct points (2K , 3K), (1 , 0), (0 , 1) and (0 , 0) lie on a circle when

(A) all values of K are integral

(B) 0 < K < 1

(C) K < 0

(D) For two values of K

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23. A line is drawn through a fixed point P (a,b) to cut the circle x^{2}+y^{2} = r^{2} at A and B. Then PA.PB is equal to

(A) (α + β)^{2} – r^{2}

(B) α^{2} + β^{2} – r^{2}

(C) (α – β)^{2} + r^{2}

(D) none of these

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24. If the tangent to the circle x ^{2} + y^{2} = 5 at the point (1, -2) also touches the circle x^{2} + y^{2} – 8x + 6y + 20 = 0, then its point of contact is

(A) (3 , 1)

(B) (-3 , 1)

(C) (3 , -1)

(D) (-3 , -1)

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25. The line 4 x + 3y –4 = 0 divides the circumference of the circle centred at (5 , 3) in the ratio 1 : 2. Then the equation of the circle is

(A) x^{2} + y^{2} –10x – 6y – 66 = 0

(B) x^{2} + y^{2} –10x – 6y +100 = 0

(C) x^{2} + y^{2} –10x – 6y + 66 = 0

(D) none of these

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26. The maximum distance of the point (4, 4) from the circle x^{2} + y^{2} – 2x – 15 = 0 is

(A) 10

(B) 9

(C) 5

(D) none of these

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27. If the circle x^{2} + y^{2} + 4x + 22y + l = 0 bisects the circumference of the circle x^{2} + y^{2} – 2x + 8y – m = 0 , then l + m is equal to

(A) 60

(B) 50

(C) 40

(D) 56

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28. The value(s) of m for which the line y = mx lies wholly outside the circle x^{2} + y^{2} –2x – 4y +1 = 0, is(are)

(A) m ∈ ( -4/3, 0)

(B) m ∈ ( -4/3, 0]

(C) m ∈ (0, 4/3)

(D) none of these

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29. A, B, C, D are the points of intersection with the co-ordinate axes of the lines ax + by = ab and bx + ay = ab then

(A) A, B, C, D are concyclic

(B) A, B, C, D forms a parallelogram

(C) A, B, C, D forms a rhombus

(D) none of these

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30. The length of the tangent from any point on the circle 15x^{2} + 15y^{2} – 48x + 64y = 0 to the two circles 5x^{2} + 5y^{2} – 24x + 32y + 75 = 0 and 5x^{2} + 5y^{2} – 48x + 64y + 300 = 0 are in the ratio of

(A) 1 : 2

(B) 2 : 3

(C) 3 : 4

(D) none of these

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**21. B 22. D 23. B 24. C 25. A 26. B 27. B 28. A 29. A 30. A**