<p> INFOMATHS</p><p>TEST – TWO DIMENSIONAL GEOMETRY Timing : 2:00 Hrs. Date : _____ TWO DIMENSIONAL GEOMETRY 12. The angle between the pair of lines represented by 1. The area of the parallelogram formed by the lines y 2x2 – 7xy + 3y2 = 0 is : = mx, y = mx + 1, y = nx and y = nx + 1 equals: (a) 60 (b) 45 | m  n | 2 1 7  (a) (b) (c) tan   (d) 30 (m  n)2 | m  n |  6  1 1 13. The area of the triangle bounded by the straight line (c) (d) | m  n | | m  n | ax + by + c = 0, (a, b, c  0) and the co-ordinate axes is : 2. If (a, a2) falls inside the angle made by the lines 2 2 x 1 a 1 c y  x > 0 and y = 3x, x > 0, then a belongs to : (a) (b) 2 2 | bc | 2 | ab |  1  1 b2 (a) (3, ) (b)  ,3 (c) (d) 0  2  2 | ac |  1   1  14. The circumcentre of a triangle formed by the line (c)  3,  (d) 0,   2   2  xy + 2x + 2y + 4 = 0 and x + y + 2 = 0 is : 3. A straight line through the point A (3, 4) is such that (a) (-1, -1) (b) (0, -1) its intercepts between the axes is bisected at A. Its (c) (1, 1) (d) (-1, 0) equation is: 15. Distance between two parallel lines y = 2x + 7 and y (a) 3x – 4y + 7 = 0 (b) 4x + 3y = 24 = 2x + 5 is (c) 3x + 4y = 25 (d) x + y = 7 5 2 (a) units (b) units 4. The lines parallel to the x-axis and passing through 2 5 the intersection of the lines ax + 2by + 3b = 0 and bx 2 1 – 2ay – 3a = 0, where (a, b)  (0, 0) is : (c) units (d) units (a) above the x-axis at a distance of 2/3 from it 5 5 (b) above the x-axis at a distance of 3/2 from it 16. A ray of light passing through at point (1, 2) is (c) below the x-axis at a distance of 2/3 from it reflected on the x-axis is point P and passes through (d) below the x-axis at a distance of 3/2 from it the point (5, 8). Then the abscissa of the point P is : 5. If one of the lines given by 6x2 – xy + 4cy2 = 0 is 3x (a) – 3 (b) 13/3 (c) 13/5 (d) + 4y = 0, then c equals: 13/4 (a) 1 (b) –1 (c) 3 (d) –3 17. The equation of bisectors of the angles between the 6. If the sum of the slopes of the lines given by x2 – lines |x| = |y| are : 2 1 1 2cxy – 7y = 0 is four times their product, then c has (a) x =  x and x = 0 (b) x  and y  value: 2 2 (a) 1 (b) – 1 (c) 2 (d) – 2 (c) y = 0 and x = 0 (d) None of these</p><p>7. If x1, x2, x2 and y1, y2, y3 are both in G.P. with the 18. The distance of the line 2x – 3y = 4 from the point same common ratio, then the points (x1, y1), (x2, y2) (1, 1) measured parallel to the line x + y = 1 is : and (x3, y3) : 5 1 (a) 2 (b) (c) (d) 6 (a) lie on a straight line 2 2 (b) lie on an ellipse 19. The locus of the point P (x, y) satisfying the relation (c) lie on a circle (d) are vertices of a triangle (x  3)2  (y 1)2  (x  3)2  (y 1)2  6 is: 2 2 8. If the pair of lines ax + 2hxy + by + 2gx + 2fy + c = (a) straight line (b) pair of straight line 0 intersect on the y-axis, then: 2 2 2 1 2 (c) circle (d) ellipse (a) 2fgh = bg + ch (b) bg ch 20. The lines x – y – 2 = 0, x + y – 4 = 0 and x + 3y = 6 (c) abc = 2fgh (d) None of these meet in the common point : 9. The value of ‘p’ for which the equation 2 2 (a) (1, 2) (b) (2, 2) (c) (3, 1) (d) (1, 1) x + pxy + y – 5x – 7y + 6 = 0 21. The transformed equation of x2 + 6xy + 8y2 = 10 represents a pair of straight lines is :  5 2 when the axes are rotated through and angle (a) (b) 5 (c) 2 (d) 4 2 5 2 2 2 2 (a) 15x – 14xy + 3y = 20 10. The equation y – x + 2x – 1 = 0 represents: (b) 15x2 + 14xy – 3y2 = 20 (a) hyperbola (c) 15x2 + 14xy + 3y2 = 20 (b) an ellipse (d) 15x2 – 14xy – 3y2 = 20 (c) a pair of straight lines 22. The area (in square units) of the quadrilateral formed (d) a rectangular hyperbola 2 2 2 2 2 2 by two pairs of liens l x – m y – n(lx + my) = 0 and 11. For what value of ‘p’ y + xy + px – x – 2y + p = 0 l2x2 – m2y2 + n(lx – my) = 0 is : represents two straight lines: 2 2 1 1 1 n l (a) 2 (b) (c) (d) (a) (b) 3 4 2 2 | lm | | lm | INFOMATHS/MCA/MATHS TEST INFOMATHS</p><p> n n2 36. The centers of a set of circles, each of radius 3, lie on (c) (d) 2 2 2 | lm | the circle x + y = 25. The locus of any point in the 4 | lm | set is : 23. If the pair of straight lines given by (a) 4  x2 + y2  64 (b) x2 + y2  25 2 2 2 Ax + 2Hxy + By = 0, (H > AB) forms an (c) x2 + y2  25 (d) 3  x2 + y2  9 equilateral triangle with line ax + by + c = 0, then (A 37. The locus of the center of circle which touches (y – + 3B) (3A + B) is 2 2 2 2 2 2 1) + x = 1 externally and also touches x-axis is : (a) H (b) – H (c) 2H (d) 4H (a) x2 = 4y  (0, y), y < 0 24. The angle between the pair of straight lines (b) 2 2 2 2 2 2 x = y y sin  - xy sin  + x (cos  - 1) = 0 is : (c) y = 4x2     (d) 2 (a) (b) (c) (d) y = 4x  (0, y), y  R 3 4 6 2 x y 25. The distance between the pair of parallel lines 38. If the line  1 moves such that a b x2 + 4xy + 4y2 + 3x + 6y – 4 = 0 is : 1 1 1 2 1 5 5   where c is a constant, then the locus (a) 5 (b) (c) (d) a2 b2 c2 5 5 2 2 of the foot of the perpendicular from the origin to the 26. Distance between the lines 5x + 3y – 7 = 0 and 15x + line is : 9y + 14 = 0 is : (a) straight line (b) circle 35 1 35 35 (a) (b) (c) (d) (c) parabola (d) ellipse 34 3 34 3 34 2 34 39. If the circle x2 + y2 + 4x + 22y + c = 0 bisects the 27. If the slope of one of the lines given by ax2 + 2hxy + circumference of the circle x2 + y2 – 2x + 8y – d = 0, by2 = 0 is 5 times the other, then : then c + d is equal to : (a) 5h2 = ab (b) 5h2 = 9ab (a) 60 (b) 50 (c) 40 (d) 30 (c) 9h2 = 5ab (d) h2 = ab 40. The equation of the circle which passes through the 28. The gradient of one of the lines of ax2 + 2hxy + by2 = intersection of x2 + y2 + 13x – 3y = 0 and 0 is twice that of the other, then : 2x2 + 2y2 + 4x – 7y – 25 = 0 and whose center lies on (a) h2 = ab (b) h = a + b 13x + 30y = 0, is : (c) 8h2 = 9ab (d) 9h2 = 8ab (a) x2 + y2 + 30x – 13y – 25 = 0 29. Two consecutive sides of a parallelogram are 4x + (b) 4x2 + 4y2 + 30x – 13y – 25 = 0 5y = 0 and 7x + 7y = 0. One diagonal of the (c) 2x2 + 2y2 + 30x – 13y – 25 = 0 parallelogram is 11x + 7y = 9. If the other diagonal (d) x2 + y2 + 30x – 13y + 25 = 0 is ax + by + c = 0, then : 41. The polars drawn from (-1, 2) to the circles 2 2 2 2 (a) a = - 1, b = - 1, c = 2 (b) a = 1, b = - 1, c = 0 S1  x + y + 6y + 7 = 0 and S2  x + y + 6x + 1 = (c) a = - 1, b = - 1, c = 0 (d) a = 1, b = 1, c = 0 0 are : 30. The number of integral value of m, for which the x- (a) parallel (b) equal coordinate of the point of intersection of the lines 3x (c) perpendicular (d) intersecting at a point 2 2 + 4y = 0 and y = mx + 1 is also an integer is : 42. The circles ax + ay + 2g1x + 2f1y + c1 = 0 and 2 2 (a) 2 (b) 0 (c) 4 (d) 1 bx + by + 2g2x + 2f2y + c2 = 0 (a  0 and b  0) cut 31. The center of circle inscribed in square formed by orthogonally if : 2 2 the lines x – 8x + 12 = 0 and y – 14y + 45 = 0, is : (a) g1g2 + f1f2 = ac1 + bc2 (a) (4, 7) (b) (7, 4) (c) (9, 4) (d) (4, 9) (b) 2(g1g2 + f1f2) = bc1 + ac2 2 32. The tangent at (1, 7) to the curve x = y – 6 touches (c) bg1g2 + af1f2 = bc1 + ac2 2 2 the circle x + y + 16x + 12y + c = 0 at : (d) g1g2 + f1f2 = c1 + c2 (a) (6, 7) (b) (-6, 7) (c) (6, - 7) (d) (-6, -7) 43. The center of the circle x = 2 + 3cos , y = 3 sin  - 1 33. If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are is : two diameters of a circle of area 49 sq. units, the (a) (3, 3) (b) (2, -1) (c) (-2, 1) (d) (-1, 2) equation of the circle is : 44. The value of k so that x2 + y2 + kx + 4y + 2 = 0 and (a) x2 + y2 + 2x – 2y – 62 = 0 2(x2 + y2) – 4x – 3y + k = 0 cut orthogonally is : (b) x2 + y2 – 2x + 2y– 62 = 0 (a) 10/3 (b) –8/3 (c) –10/3 (d) 8/3 (c) x2 + y2 – 2x + 2y – 47 = 0 45. The two circles x2 + y2 – 2x + 6y + 6 = 0 and (d) x2 + y2 + 2x – 2y – 47 = 0 x2 + y2 – 5x + 6y + 15 = 0 : 34. If a circle passes through the point (a, b) and cuts the (a) intersect (b) are concentric circle x2 + y2 = p2 orthogonally, then the equation of (c) touch internally (d) touch externally the locus of its center is : 46. The greatest distance of the point P(10, 7) from the (a) 2ax + 2by – (a2 + b2 + p2) = 0 circle x2 + y2 – 4x – 2y – 20 = 0 is : (b) x2 + y2 – 2ax – 3by + (a2 – b2 – p2) = 0 (a) 10 (b) 15 (c) 5 (d) None of these (c) 2ax + 2by – (a2 – b2 + p2) = 0 47. The length of the common chord of the circles x2 + (d) x2 + y2 – 3ax – 4by + (a2 + b2 – p2) = 0 y2 + 2x + 3y + 1 = 0 and x2 + y2 + 4x + 3y + 2 = 0 is: 35. The center of the circle passing through (0, 0) and (1, 9 3 2 2 (a) (b) 2 2 (c) 3 2 (d) 0) and touching the circle x + y = 9 is : 2 2  1 1   1  48. The number of common tangent to the two circles (a)  ,  (b)  , 2   2 2   2  x2 + y2 – 8x + 2y = 0 and x2 + y2 – 2x – 16y + 25 = 0 is : INFOMATHS/MCA/MATHS TEST INFOMATHS</p><p>(a) 1 (b) 2 (c) 3 (d) 4 1 1 (a) (b)  (c) a (d) –a 49. If 5x – 12y + 10 = 0 and 12y – 5x + 16 = 0 are two a a tangents to a circle, then the radius of the circle is : 62. If the line touches the hyperbola x2 – (a) 1 (b) 2 (c) 4 (d) 6 2x  6y  2 50. Observe the following statements : 2y2 = 4, then the point of contact is : I. The circle x2 + y2 – 6x – 4y – 7 = 0 touches y-axis (a) (2, 6) (b) (5,2 6) II. The circle x2 + y2 + 6x + 4y – 7 = 0 touches x-axis  1 1  Which of the following is a correct statement?   (c)  ,  (d) (4, 6) (a) Both I and II are true (b) Neither I nor II is true  2 6  (c) I is true, II is false (d) I is false, II is true 63. Eccentricity of the rectangular hyperbola 51. The angle between the tangents drawn from the point 2 1 x  1 1  (1, 4) to the parabola y = 4x is : e    dx, is : 0        x x3  (a) (b) (c) (d) 6 4 3 2 1 (a) 2 (b) 2 (c) 1 (d) 52. The equation of the directrix of the parabola 2 2 y + 4y + 4x + 2 = 0 is : 64. The length of the transverse axis of the rectangular (a) x = - 1 (b) x = 1 hyperbola xy = 18 is : 3 3 (c) x   (d) x  (a) 6 (b) 12 (c) 18 (d) 9 2 2 65. The locus of the point of intersection of 53. Two common tangents to the circle x2 + y2 = 2a2 and perpendicular tangents to the hyperbola parabola y2 = 8ax are : x2 y2 (a) x =  (y + 2a) (b) y =  (x + 2a)  1 is : 3 1 (c) x = (y + a) (d) y =  (x + a) 2 2 2 2 2 2 (a) x + y = 2 (b) x + y = 3 54. Angle between two curves y = 4 (x + 1) and x = 4(y 2 2 2 2 + 1) is : (c) x – y = 3 (d) x + y = 4 (a) 0 (b) 90 (c) 60 (d) 30 REASONING AND QUANT 55. The point at which the line y = mx + c touches the 66. A and B together can do a piece of work in 6 days parabola y2 = 4ax is : and A alone can do it in 9 days. In how many days  a 2a   a 2a      can B alone do it? (a)  ,  (b)  ,   m2 m   m2 m  (a) 15 days (b) 16 days (c) 18 days (d) 20 days  a 2a   a 2a      67. A and B can do a piece of work in 12 days, B and C (c)  ,  (d)  ,   m2 m   m2 m  in 15 days, C and A in 20 days. How long would all 56. Area of the greatest rectangle that can be inscribed in take to finish the work together? (a) 10 days (b) 9 days x2 y2 the ellipse  1 is : 1 2 2 (c) 8 days (d) 7 days a b 2 a (a) (b) ab (c) ab (d) 2ab 68. If 3 men or 4 women can reap a field in 43 days. b How long will 7 men and 5 women take to a reap it? 57. The angle between the pair of tangents drawn from (a) 21 days (b) 16 days the point (1, 2) to the ellipse 3x2 + 2y2 = 5, is : (c) 15 days (d) 12 days (a) tan-1 (12/5) (b) tan 1(6 / 5) 69. If 12 men and 16 boys can do a piece of work in 5 days and 13 men together with 24 boys can do it in 4 (c) tan 1(12 / 5) (d) tan-1 (6/5) days. Compare the daily work done by a man with 58. If the angle between the line joining the end points that of a boy. of minor axis of an ellipse with its foci is /2, then (a) 4 : 3 (b) 2 : 1 (c) 3 : 2 (d) 2 : 5 the eccentricity of the ellipse is : 70. A certain number of men can do a work in 60 days. If there were 8 men more it could be finished in 10 1 1 3 1 (a) (b) (c) (d) days less. How many men were there in the 2 2 2 2 2 beginning? 59. Equations of the latus–rectum of the ellipse (a) 50 men (b) 45 men (c) 42 men (d) 40 men 9x2 + 4y2 – 18x – 8y – 23 = 0 are : 71. A is twice as good a workman as B; and together (a) y   5 (b) x   5 they finish a work in 14 days. In how many days can be done by each separately? (c) y 1 5 (d) x  1  5 (a) A = 42, B = 21 (b) A = 21, B = 42 60. The eccentricity of the hyperbola (c) A = 52, B = 26 (d) A = 26, B = 52 9x2 – 16y2 – 18x – 64y – 199 = 0 is : 72. A and B together can do a piece of work in 7 days. If 16 5 25 A alone does twice as much work as B in a given (a) (b) (c) (d) zero 9 4 16 time, find how long A alone would take to do the 61. What is the slope of the tangent drawn to the work? 1 hyperbola xy = a, (a  0) at the point (a, 1)? (a) 21 days (b) 10 days 2</p><p>INFOMATHS/MCA/MATHS TEST INFOMATHS</p><p>(c) 9 days (d) 12 days The team in A has beaten R, P and S, Q has beaten the 73. A can copy 80 pages in 16 hours, A and B can do teams in E, C and A. Team R is in B and the team in C is 133 pages in 19 hours. In how many hours B shall not S. copy 52 pages? 83. Where is the team Q? (a) 13 hours (b) 39 hours (a) A (b) B (c) C (d) D 1 84. Where is the team P? (c) 8 hours (d) 26 hours 2 (a) A (b) B (c) C (d) D 74. A can do a piece of work in 5 days, B in 4 days, and 85. Which team is in A? A, B and C together in 2 days, In what time could C (a) P (b) Q (c) S (d) T do it alone? 86. A road network has parallel roads, which are (a) 20 days (b) 18 days equidistant from each other and running north-south (c) 15 days (d) 12 days or east-west only. The road junctions A, B, C, H and 75. B can do a piece of work in 6 hours, B and C do it in X are such that A is east of B and west of C. H is 2 south-west of C and south-east of B. B is south-east 4 hours and A, B and C in 2 hours. In how many of X. Which of the junctions are the farthest south 3 and the farthest east? hours can A and B do that work? (a) H, B (b) H, C (c) C, H (d) B, H 1 (a) 11 hours (b) 6 hours 87. Four players A, B, C and D have to form into two 7 pairs, however, no pair can play together more than 3 3 seven times in a row. A and B have played seven (c) 2 hours (d) 3 hours 7 7 games in a row. C and D have three in a row. C does Questions 76 and 77 are based on the following: not want to work with A. Who would partner with Six friends A, B, C, D, E and F are sitting round a B? hexagonal table. F, who is sitting exactly opposite A, is to (a) A (b) D be immediate right of B. D is between A and B and is (c) C (d) Cannot be determined exactly opposite to C. 76. Who are sitting next to A? Questions 88 to 92 are based on the following: (a) D and E (b) D and F (c) C and E (d) B and D All the roads of a city are either perpendicular or parallel 77. Who is sitting opposite to B? to one another. The roads are all straight. Roads A, B, C, (a) A (b) C (c) E (d) F D and E are parallel to one another. Roads F, G, H, I, J, K, 78. Two taps A and B can fill a tank in 10 hours and 15 L and M are parallel to one another. hours and 15 hours respectively. If both the taps are Road A is 1 km east of road B. opened together, the tank will be full in ? Road B is ½ km west of road C. (a) 8 hrs (b) 6 hrs (c) 5 hrs (d) 10 hrs Road D is 1 km west of road E. 79. Bucket p has thrice the capacity as bucket Q. It takes Road G is ½ km south of road H. 60 turns for bucket P to fill the empty drum. How Road I is 1 km north of road J. many turns it will take for both the buckets p and q Road K is ½ km north of road L. having each turn together to fill the empty drum: Road K is 1 km south of road M. (a) 40 (b) 35 (c) 45 (d) 50 88. Which of the following is necessarily true? 80. Two pipes A and B fill a tank in 15 hours and 20 (a) E and B intersect hours respectively while a third pipe c can empty the (b) D is 2 km west of B full tank in 25 hours. All the three pipes are opened (c) D is at least 2 km west of A in the beginning, after 10 hours, c is closed. In how (d) M is 1.5 km north of L much time, will the tank be full ? ans:12 hrs 89. If E is between B and C, which of the following (a) 12 hrs (b) 10 hrs (c) 15 hrs (d) 20 hrs is false? 81. Two pipes A and B and can fill a cistern in 12 (a) D is 2 km west of A minutes and 15 minutes respectively but a third pipe (b) C is less than 1.5 km from D c can empty the full tank in 6 mintues. A and B are (c) Distance from E to B added to distance of E to C kept open for 5 minutes in the beginning and then c is ½ km is also opened. In what time is the cistern emptied (d) E is less than 1 km from A ….. 90. If road E is between B and C, then the distance (a) 50 mints (b) 45 mints between A and D is (c) 35 mints (d) 30 mints (a) Less than 1 km 82. Three pipes A, B and C can fill a tank in 6 hours. (b) Between 1 km and 1 ½ km After working at it together for 2hours, C is closed (c) Between 1 ½ km and 2 km and A and B can fill the remaining part in 7 hours. (d) More than 2 The number of hours taken by C alone to fill the 91. Which of the following possibilities would make cistern is: some two roads coincide? (a) 20 (b) 15 (c) 14 (d) 19 (a) L is 1/2 km north of I (b) C is 1 km west of D Questions 83 to 85 are based on the following: (c) I is ½ km north of K Cricket clubs in five towns A, B, C, D and E have one (d) E and B are ½ km apart team each named P, Q, R, S and T, not necessarily in the same order. </p><p>INFOMATHS/MCA/MATHS TEST INFOMATHS</p><p>92. If K is parallel to I and if K is 1/2 km south of J and 1 km north of G, which of the following two roads Questions 97 to 100 are based on the following: would be ½ km apart? Six members of a family A, B, C, D, E and F are (a) I and K (b) J and G Psychologist, Manager, Advocate, Jeweller, Doctor and (c) I and G (d) J and K Engineer but not necessarily in the same order. Doctor is the grandfather of F, who is a psychologist Questions 93 to 96 are based in the following: Manager D is married to A Five boys A, B, C, D, E and Five girls P, Q, R, S, T are C, who is a Jeweller, is married to Advocate standing in two rows facing each other not necessarily in B is the mother of F and E the order. E is not at any ends. C is to the immediate right There are two married couples in the family of B and D is to the immediate left of A, who is facing P. 97. What is the profession of A? There are as many girls between P and Q as between R (a) Manager (b) Engineer and S. A is second to the left of B. S and R are not facing (c) Cannot be determined (d) N.O.T either B or D. 98. What is the profession of E? 93. Which pair of boys are standing at the ends of the (a) Manager (b) Engineer row? (c) Doctor (d) None of these (a) C and D (b) C and B (c) D and B (d) N.O.T 99. How is A related to E? 94. Which of the following is definitely true? (a) Grandmother (b) Wife (a) C is third to the right of D (c) Grandfather (d) None of these (b) D is facing P 100. How many male members are there in the family? (c) C is facing S (d) None of these (a) Two (b) Three 95. Who is standing to the immediate right of A? (c) Four (d) Cannot be determined (a) E (b) C (c) D (d) B 96. Who is facing B? (a) R (b) S (c) Q (d) T MY PERFORMANCE ANALYSIS No. of Questions No. of Right No. of Wrong Net Score Percentage Accuracy Attempted (=A) Responses (=R) Responses (=W) (NS = R – 0.25W) (= PA= 100R/A)</p><p>INFOMATHS/MCA/MATHS TEST INFOMATHS</p><p>37. Two Dimensional Geometry-2016 New 1 2 3 4 5 6 7 8 9 10 D B B D D C A A A C 11 12 13 14 15 16 17 18 19 20 C B B A C C C A B C 21 22 23 24 25 26 27 28 29 30 C A D D A C B C B A 31 32 33 34 35 36 37 38 39 40 A D C A B A A B B B 41 42 43 44 45 46 47 48 49 50 D B B A C B B B A B 51 52 53 54 55 56 57 58 59 60 C D B B B D C B C B 61 62 63 64 65 66 67 68 69 70 B D B B A C A D B D 71 72 73 74 75 76 77 78 79 80 B B D A D A C B C A 81 82 83 84 85 86 87 88 89 90 B C D C D B C D A C 91 92 93 94 95 96 97 98 99 100 C D A D A C D B C D</p><p>INFOMATHS/MCA/MATHS TEST</p>