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INFOMATHS WORK-SHEET-3 (OLD QUESTIONS) PROBABILITY 1. All the coefficients of the equation ax2 + bx + c = 0 are determined by throwing a six-sided un-biased dice. The probability that the equation has real roots is HCU-2012 (a) 57/216 (b) 27/216 (c) 53/216 (d) 43/216 2. Suppose 4 vertical lines are drawn on a rectangular sheet of paper. We name A4 B4 3. 4. 5. A1 B1 , lines A2 B2 , A3 B3 and disjoint pairs of end points within A1 to A4 and B1 to B4 respectively without seeing how the other is marking. What is the probability that the figure thus formed has disconnected loops? HCU-2012 (a) 1/3 (b) 2/3 (c) 3/6 (d) 1/6 In a village having 5000 people, 100 people suffer from the disease Hepatitis B. It is known that the accuracy of the medical test for Hepatitis B is 90%. Suppose the medical test result comes out to be positive for Anil who belongs to the village, then what is the probability that Anil is actually having the disease. HCU-2012 (a) 0.02 (b) 0.16 (c) 0.18 (d) 0.3 Let A, B and C be the three events such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A B) = 0.08, P(A C) = 0.28, P(A B C) = 0.09. If P(A B C) 0.75, then P(B C) satisfies : PU CHD-2012 (A) P(B C) ≤0.23 (B) P(B C) ≤0.48 (C) 0.23 ≤P(B C) ≤0.48 (D) P(B C) ≤0.15 A number is chosen from each of the two sets {1, 2, 3, 4, 5, 6, 7, 8, 9} and {1, 2, 3, 4, 5, 6, 7, 8, 9}. If P denotes the probability that the sum of the two numbers be 10 and Q the probability that their sum be 8, then (P + Q) is PU CHD-2012 (B) 137 729 (C) 16 81 (D) BHU-2012 (a) 7/18 13. 14. 15. 16. 17. 18. 19. 137 81 Let P(E) denote the probability of event E. Given P(A) = 1, P(B) 1 , the values of P(A|B) and P (B|A) respectively are 2 NIMCET-2012 1 1 (a) , 4 2 7. 8. 9. 1 1 (b) , 2 4 1 (c) ,1 2 1 (d) 1, 2 A determinant is chosen at random from the set of all determinants of matrices of order 2 with elements 0 and 1 only. The probability that the determinant chosen is non-zero is NIMCET-2012 (a) 3/16 (b) 3/8 (c) 1/4 (d) None of these Coefficients of quadratic equation ax2 + bx + c = 0 are chosen by tossing three fair coins where ‘head’ means one and ‘tail’ means two. Then the probability that roots of the equation are imaginary is NIMCET-2012 (a) 7/8 (b) 5/8 (c) 3/8 (d) 1/8 A problem in Mathematics is given to three students A, B and C 20. 21. 10. (a) 11. 1 2 (b) 1 2n (c) 1 2n1 1 (a) 2 49 (b) 101 50 (c) 101 22. 23. (d) None of these One hundred identical coins each with probability P of showing up heads re tossed. If 0 < P < 1 and the probability of heads showing on 50 coins is equal to that of heads on 51 coins; then the value of P is NIMCET-2012 51 (d) 101 WORKSHEET-3 (OLD QUESTIONS ) (b) 1/3 (d) 1/5 1 1 1 , , 3 3 3 respectively they attempt independently, then the probability that the problem will solved is : BHU-2012 (a) 1/9 (b) 2/9 (c) 4/9 (d) 2/3 In a single throw with two dice, the chances of throwing eight is : BHU-2012 (a) 7/36 (b) 1/18 (c) 1/9 (d) 5/36 A single letter is selected at random from the word “probability”. The probability that it is a vowel, is : BHU-2012 (a) 3/11 (b) 4/11 (c) 2/11 (d) 0 An unprepared student takes a five question true-false exam and guesses every answer. What is the probability that the student will pass the exam if at least four correct answers is the passing grade? HCU-2011 (a) 3/16 (b) 5/32 (c) 1/32 (d) 1/8 Answer questions 17 and 18 using the following text: In a country club, 60% of the members play tennis, 40% play shuttle and 20% play both tennis and shuttle. When a member is chosen at random, What is the probability that she plays neither tennis nor shuttle? HCU-2011 (a) 0.8 (b) 0.2 (c) 0.5 (d) 0.4 If she plays tennis, what is the probability ability that she also plays shuttle? HCU-2011 (a) 2/3 (b) 2/5 (c) 1/3 (d) 1/2 If E is the event that an applicant for a home loan in employed C is the event that she possesses a car and A is the event that the loan application is approved, what does P(A|E C) represent in words? HCU-2011 (a) Probability that the loan is approved, if she is employed and possesses a car (b) Probability that the loan is approved, if she is either employed or possesses a car (c) Probability that the loan is approved, if she is neither employed nor possesses a car. (d) Probability that the loan is approved and she is employed, given that she possesses a car An anti-aircraft gun can take a maximum of four slots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth slots are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits the plane then is NIMCET-2011 (a) 0. 5 (b) 0.7235 (c) 0.6976 (d) 1.0 A random variable X has the following probability distribution x 0 1 2 3 4 5 6 7 8 7a 9a 11a 13a 15a 17a NIMCET-2011 (a) 1/81 (b) 2/82 (c) 5/81 (d) 7/81 Three coins are thrown together. The probability of getting two or more heads is BHU-2011 (a) 1/4 (b) 1/2 (c) 2/3 (d) 3/8 If four positive integers are taken at random and are multiplied together, then the probability that the last digit is 1, 3, 7 or 9 is : PU CHD-2010 (A) 24. (c) 1/6 The probability that A, B, C can solve problem is P(X = x) a 3a 5a Then the value of ‘a’ is 1 1 1 whose chances of solving it are , , respectively. If they all 2 3 4 try to solve the problem, what is the probability that the problem will be solved? NIMCET-2012, MP-2008 (a) 1/2 (b) 1/4 (c) 1/3 (d) 3/4 If a fair coin is tossed n times, then the probability that the head comes odd number of times is NIMCET-2012 Let P be a probability function on S = (l1, l2, l3, l4) such that 1 1 1 P l2 , P l3 , P l4 . Then P(l1) is 3 6 9 respectively. Suppose two players A and B join two (A) 6. the 12. 1 8 (B) 2 7 (C) 1 625 (D) 16 625 The numbers X and Y are selected at random (without replacement) from the set (1, 2, .....3N). The probability that x2 – y2 is divisible by 3 is : PU CHD-2010 (A) 3N 1 N 1 5N 3 4N 3 (B) (C) (D) N 9N 3 3N 9N 3 1 INFOMATHS 25. 26. 27. 28. 29. Probability of happening of an event A is 0.4 Probability that in 3 independent trials, event A happens atleast once is:PU CHD-2009 (a) 0.064 (b) 0.144 (c) 0.784 (d) 0.4 A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A B) is : PU CHD-2009 (a) 3/5 (b) 0 (c) 1 (d) 1/6 India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability f India getting at least 7 points is NIMCET-2010 (a) 0.8750 (b) 0.0875 (c) 0.0625 (d) 0.0250 A coin is tossed three times The probabilities of getting head and tail alternatively is NIMCET-2010 (a) 1/11 (b) 2/3 (c) 3/4 (d) 1/4 One hundred identical coins, each with probability P of showing up a head, are tossed. If 0 < p < 1 and if the probability of heads on exactly 50 coins is equal to that of heads on exactly 51 coins then the value of p, is NIMCET-2010 (a) 30. 31. 1 2 (b) 49 50 (c) 101 101 (d) 51 101 39. 40. 41. 42. 43. 44. A dice is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of success is KIITEE-2010 (a) 8/3 (b) 3/8 (c) 4/5 (d) 5/4 If A and B are events such 3 P A B , 4 that (a) 45. 2 1 P A B , P A , then P A B is 3 4 32. 33. 34. KIITEE-2010 (a) 5/12 (b) 3/8 (c) 5/8 (d) 1/4 If A and B are any two mutually exclusive events, then P(A|AB) is equal to (PGCET– 2009) (a) P(AB) (b) P(A)/(P(A) + P(B)) (c) P(B)/P(AB) (d) None of these A man has 5 coins, two of which are double – headed, one is double – tailed and two are normal. He shuts his eyes, picks a coin at random, and tosses it. The probability that the lower face of the coin is a head is (NIMCET – 2009) (a) 1/5 (b) 2/5 (c) 3/5 (d) 4/5 A and B are independent witnesses in a case. The probability that A speaks the truth is ‘x’ and that B speaks the truth is ‘y’. If A and B agree on a certain statement, the probability that the statement is true is (NIMCET – 2009) (a) (c) 35. xy xy (1 x)(1 y ) (b) 1 x 1 y xy 1 x 1 y (d) 38. 1 x 1 y (c) 49. and P ( A ) 1 . 4 65 81 (b) 13 81 (c) 65 324 (d) 13 108 An anti aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits the plane then is (MCA : NIMCET – 2009) (a) 0.6972 (b) 0.6978 (c) 0.6976 (d) 0.6974 Let A = [2, 3, 4, …., 20, 21] number is chosen at random from the set A and it is found to be a prime number. The probability that it is more than 10 is WORKSHEET-3 (OLD QUESTIONS ) 50. (b) 13 90 (c) 19 90 45 3 2 4 90 2 (b) 1 132 (b) 1 44 (c) 5 132 (a) 2 (d) 7 132 1 1 , 3 4 and 1 . The probability that exactly one 5 5 12 (b) 7 30 MP COMBINED – 2008 (c) 13 30 (d) 3 5 Different words are written with the letters of PEACE. The probability that both E’s come together is : MP COMBINED – 2008 (a) 1/3 (b) 2/5 (c) 3/5 (d) 4/5 The probability of throwing 6 at least one in four throws of a die is: MP COMBINED – 2008 (a) 53. 1 90 4 Probabilities of three students A, B and C to pass an examination student will pass is: 52. 5 8 (d) None 210 are respectively 51. (d) Two balls are drawn at random from a bag containing 6 white, 4 red and 5 black balls. The probability that both these balls are black, is : MP COMBINED – 2008 (a) 1/21 (b) 2/15 (c) 2/21 (d) 2/35 6 boys and 6 girls sit in a row randomly. The probability that all the girls sit together is : MP COMBINED – 2008 (a) Then 1 45 A six faced die is a biased one. It is thrice more likely to show an odd number than to show an even number. It is thrown twice. The probability that the sum of the numbers in the two throws is even, is. NIMCET - 2008 (a) 4/8 (b) 5/8 (c) 6/8 (d) 7/8 A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is NIMCET - 2008 (a) 4/11 (b) 1/3 (c) 5/12 (d) None A card is drawn from a pack. The card is replaced and the pack is reshuffled. If this is done six times, the probability that 2 hearts, 2 diamonds and 2 club cards are drawn is. KIITEE – 2008 (a) xy events A and B are (NIMCET – 2009) (a) independent but not equally likely (b) mutually exclusive and independent (c) equally likely and mutually exclusive (d) equally likely but not independent. The probability that a man who is 85 yrs. old will die before attaining the age of 90 is 1/3. A1, A2, A3 and A4 are four persons who are 85 yrs. old. The probability that A1 will die before attaining the age of 90 and will be the first to die is (NIMCET – 2009) (a) 37. 47. Let A and B be two events such that 1 1 P ( A B ) , P ( A B) 6 4 36. 46. 48. xy 1 x 1 y (MCA : KIITEE – 2009) (a) 9/10 (b) 1/5 (c) 1/10 (d) None of these Find the probability that a leap year will contain either 53 Tuesday or 53 Wednesdays. HYDERABAD CENTRAL UNIVERSITY - 2009 (a) 1/5 (b) 2/5 (c) 2/3 (d) 3/7 Probability that atleast one of A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, then P(A') + P(B') is HYDERABAD CENTRAL UNIVERSITY - 2009 (a) 0.9 (b) 1.15 (c) 1.1 (d) 2 The sum of two positive real numbers is 2a. The probability that product of these two numbers is not less than 3/4 times the greatest possible product is HYDERABAD CENTRAL UNIVERSITY - 2009 (a) 1/2 (b) 1/3 (c) 1/4 (d) 9/16 If two events A and B such that P(A') = 0.3, P(B) = 0.5 and P(A B) = 0.3, then P(B/AB') is : NIMCET - 2008 (a) 1/4 (b) 3/8 (c) 1/8 (d) None A pair of unbiased dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes before 7 is. NIMCET - 2008 (a) 3/5 (b) 2/5 (c) 4/5 (d) None A letter is taken at random from the letters of the word ‘STATISTICS’ and another letter is taken at random from the letters of the word ‘ASSISTANT’. The probability that they are the same letter is. NIMCET - 2008 1 6 (b) 2 3 (c) 625 1296 (d) 671 1296 An untrue coin is such that when it is tossed the chances of appearing head is twice the chances of appearance of tail. The chance of getting head in one toss of the coin is : MP COMBINED – 2008 (a) 1/3 (b) 1/2 (c) 2/3 (d) 1 2 INFOMATHS 54. The probability of randomly chosing 3 defectless bulbs from 15 electric bulbs of which 5 bulbs are defective, is : MP COMBINED – 2008 (a) 55. 56. 57. 58. 59. 60. 3 10 64. 66. 39 256 7 12 1 12 69. 70. 71. 1 3 and the probability that neither of them occurs is 1/6. (b) 29 256 (c) 31 256 (d) (b) 11 12 (c) 1 2 (d) (b) 1 15 (c) 2 27 (d) 1 (b) 2 3 (c) 13 74. 75. 76. 1 10 (e) 16 256 (b) 1 286 (c) 37 256 Prob. of getting an odd number or a no. less than 4 in throwing a dice is : MP– 2004 (a) 1/3 (b) 2/3 (c) 1/2 (d) 3/5 Given A and B are mutually exclusive events. IFP (B) = 0. 15, P(A B) = 0.85, P(A) is equal to UPMCAT Paper – 2002 (a) 0.65 (b) 0.3 (c) 0.70 (d) N.O.T. In a pack of 52 cards, the probability of drawing at random such that it is diamond or card king is : UPMCAT Paper – 2002 (a) 1/26 (b) 4/13 (c) 3/13 (d) 1/4 Given A and B are mutually exclusive events. if: P (A B) = 0.8, P(B) = 0.2 then P(A) is equal to UPMCAT–2002 (a) 0.5 (b) 0.6 (c) 0.4 (d) N.O.T. WORKSHEET-3 (OLD QUESTIONS ) (b) 3 9 16 (c) 10 41 60 (b) 37 60 (c) 31 60 (c) P A B P A B 77. 78. (d) N.O.T. (d) N.O.T. P A B P A B (b) P A B P A B (d) None of these A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urn at random. The probability that all the balls selected are red is. NIMCET – 2008 (a) 1/3 (b) 3/10 (c) 1/8 (d) none A number x is chosen at random from (1, 2, …. 10). The probability that x satisfies the equation (x – 3) (x – 6) (x – 10) = 0 is ICET - 2007 (a) 2/5 (b) 3/5 (c) 3/10 (d) 7/10 TWO DIMENSIONAL GEOMETRY 1. Find the equation of the graph xy = 1 after a rotation of the axes by 45 degrees anti-clockwise in the new coordinate system (x', y'). HCU-2012 (a) x'2 – y'2 = 1 2 2 (b) (x' /2) - (y' /2) = 1 (c) (x'2 /2) + (y'2/2) = 1 (d) 2. 3. 1 20 28 256 2 If the events A and B are mutually exclusive then P (A B) is given by : UPMCAT Paper – 2002 (a) P(A) + P(B) (b) P(A)P(B) (c) P(A) P(B/A) (d) N.O.T. If A and B are two events, the prob. that exactly one of them, occurs in given by: UPMCAT Paper – 2002 (a) 4. 5. 1 (d) 3 (d) 1 13 If P(A' B') is equal to 19/60 then P(AB) is equal to UPMCAT Paper – 2002 (a) 37 256 5 6 Two dice are thrown once the probability of getting a sum 9 is given by : UPMCAT Paper – 2002 (a) 1/12 (b) 1/18 (c) 1/6 (d) N.O.T. In a pack of 52 cards. Two cards are drawn at random. The probability that it being club card is : UPMCAT Paper – 2002 (a) 6. 7. x' / 2 y' / 2 1 2 3 2 (B) 5 2 (C) 1 2 (D) 1 4 The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is : PU CHD-2012 (A) (1/2, 1/2) (B) (1/3, 1/3) (C) (1/4, 1/4) (D) (0, 0) The distance between the parallel lines y = 2x + 4 and 6x = 3y + 5 is PU CHD-2012, NIT-2010 (A) 17 3 8. 2 The number of points (x, y) satisfying (i) 3x - 4y = 25 and (ii) x2 + y2 25 is HCU-2012 (a) 0 (b) 1 (c) 2 (d) infinite A point P on the line 3x + 5y = 15 is equidistant from the coordinate axes. Then P can lie in HCU-2012 (a) Quadrant I only (b) Quadrant I or Quadrant III only (c) Quadrant I or Quadrant II only (d) any Quadrant A circle and a square have the same perimeter. Then HCU-2012 (a) their areas are equal (b) the area of the circle is larger (c) the area of the square is larger (d) the area of the circle is times the area of the square The eccentricity of the ellipse x2 + 4y2 + 8y – 2x + 1 = 0 is : PU CHD-2012 (A) The probability of getting atleast 6 head in 8 trials is: MP– 2004 (a) 68. 73. 67 91 The probabilities that a husband and wife will be alive 20 years from now are given by 0.8 and 0.9 respectively. What is the probability that in 20 years at least one, will be alive? Karnataka PG-CET : Paper – 2006 (a) 0.98 (b) 0.02 (c) 0.72 (d) 0.28 A bag contains 4 white and 3 black balls and a second bag contains 3 white and 3 black balls. If a ball is drawn from each of the bags, then the probability that both are of same colour is : MP Paper – 2004 3 (a) 14 67. (d) A and B play a game of dice. A throws the die first. The person who first gets a 6 is the winner. What is the probability that A wins? PUNE Paper – 2007 (a) 6/11 (b) 1/2 (c) 5/6 (d) 1/6 A player is going to play a match either in the morning or in the afternoon or in the evening all possibilities being equally likely. The probability that he wins the match is 0.6, 0.1 and 0.8 according as if the match is played in the morning, afternoon or in the evening respectively. Given that he has won the match, the probability that the match was played in the afternoon is IP Univ. Paper – 2006 (a) 65. 24 91 If two dice are tossed the probability of getting the sum at least 5 is PUNE Paper – 2007 (a) 63. (c) Then the probability of occurrence of A is. ICET – 2005 (a) 5/6 (b) 1/2 (c) 1/12 (d) 1/18 8 coins are tossed simultaneously. The probability of getting atleast six heads is ICET – 2005 (a) 62. 7 10 Probability of four digit numbers, which are divisible by three, formed out of digits 1, 2, 3, 4, 5 is : MP COMBINED – 2008 (a) 1/5 (b) 1/4 (c) 1/3 (d) 1/2 Let A and B be two events with P(A) = 1/2, P(B) = 1/3 and P(A B) = 1/4 , What is P(A B)? KARNATAKA - 2007 (a) 3/7 (b) 4/7 (c) 7/12 (d) 9/122 If three unbiased coins are tossed simultaneously then the probability of getting exactly two heads is ICET - 2007 (a) 1/8 (b) 2/8 (c) 3/8 (d) 4/8 A person gets as many rupees as the number he gets when an unbiassed 6 – faced die is thrown. If two such dice are thrown the probability of getting Rs. 10 is. ICET - 2007 (a) 1/12 (b) 5/12 (c) 13/10 (d) 19/10 Let E be the set of all integers with 1 in their units place. The probability that a number n chosen from [2, 3, 4, … 50] is an element of E is ICET - 2007 (a) 5/49 (b) 4/49 (c) 3/49 (d) 2/49 A and B independent events. The probability that both A and B occur is 61. (b) 72. (B) 1 (C) 3 5 (D) 17 5 15 The lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154 square units. Then the equation of this circle is (= 22/7) PU CHD-2012 3 INFOMATHS (A) x + y + 2x – 2y = 62 (B) x + y + 2x – 2y = 47 (C) x2 + y2 – 2x + 2y = 47 (D) x2 + y2 – 2x + 2y = 62 2 The focus of the parabola y – x – 2y + 2 = 0 is : PU CHD-2012 (A) (1/4, 0) (B) (1, 2) (C) (3/4, 1) (D) (5/4, 1) The medians of a triangle meet at (0, –3). While its two vertices are (–1, 4) and (5, 2), the third vertex is at PU CHD-2012 (A) (4, 5) (B) (–1, 2) (C) (7, 3) (D) (– 4, – 15) The area of the triangle having the vertices (4, 6), (x, 4), (6, 2) is 10 sq units. The value of x is PU CHD-2012 (A) 0 (B) 1 (C) 2 (D) 3 The position of reflection of point (4, 1) w.r.to line y = x – 1 is Pune-2012 (a) (-4, -1) (b) (1, 2) (c) (2, 3) (d) (3, 4) 6x2 + 12x + 8 – y = 0 has its standard form as? Pune-2012 2 9. 10. 11. 12. 13. 14. 15. 16. 2 2 (a) x 1 (c) x 2 2 2 18. 19. 20. x 1 y 2 2 24. 25. 319 (b) 2 183 (c) 2 (c) 2 or 3 2 3 2 (d) – 2 or 3 2 26. 27. tan 1 11 2 The equation of circle passing through (-1, 2) and concentric with x2 + y2 – 2x – 4y – 4 = 0 is : BHU-2012 (a) x2 + y2 – 2x – 4y + 1 = 0 (b) x2 + y2 – 2x – 4y + 2 = 0 (c) x2 + y2 – 2x – 4y + 4 = 0 (d) x2 + y2 – 2x – 4y + 8 = 0 The radius of the circle on which the four points of intersection of the lines (2x – y + 1) (x – 2y + 3) = 0 with the axes lie, is : BHU-2012 (b) 5 5 (c) 5 (d) 2 2 4 2 The focal distance of a point on the parabola y2 = 8x is 4. Its ordinates are : BHU-2012 (a) 1 (b) 2 (c) 3 (d) 4 The straight line x cos + y sin = p touches the ellipse x2 y 2 1 if : a 2 b2 BHU-2012 (b) p2 = a2 cos2 + b2 sin2 (d) p2 = a2 sin2 + b2 cos2 28. If the line lx + my = n touches the hyperbola x2 y 2 1 if : a 2 b2 BHU-2012 (a) a2l2 – b2m2 = n2 (c) a2l2 + b2m2 = n2 29. For the conic (b) al – bm = n (d) al + bm = n l 1 e cos , the sum of reciprocals of the r segments of any focal chord is equal to : BHU-2012 (a) l 30. 31. 3 2 (b) 2l (c) 2 32. 1 (c) l 2 (d) l The equation of tangent at (2, 2) of the curve xy2 = 4 (4 – x) is : BHU-2012 (a) x – y = 4 (b) x + y = 4 (c) x – y = 2 (d) x + y = 2 A curve given in polar form as r = a(cos() + sec ()) can be written in Cartesian form as HCU-2011 (a) x(x2 + y2) = a(2x2 + y2) Focus of the parabola x + y – 2xy – 4(x + y – 1) = 0 is NIMCET-2012 (a) (1, 1) (b) (1, 2) (c) (2, 1) (d) (0, 2) If e and er be the eccentricities of a hyperbola and its conjugate, 2 (d) 2 381 (d) 2 (b) – 2 or 2 11 (a) p2 = a2 cos2 - b2 sin2 (c) p2 = a2 sin2 - b2 cos2 The equation of the ellipse with major axis along the x-axis and passing through the points (4, 3) and (-1, 4) is NIMCET-2012 (a) 15x2 + 7y2 = 247 (b) 7x2 + 15y2 = 247 (c) 16x2 + 9y2 = 247 (d) 9x2 + 16y2 = 247 If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect orthogonally, then k is NIMCET-2012 (a) 2 of tan 1 (a) 5 (d) None of these Pune-2012 (a) inside (b) on (c) outside (d) None of these 2 The point on the curve y = 6x = x , where the tangent is parallel to x – axis is NIMCET-2012 (a) (0, 0) (b) (2, 8) (c) (6, 0) (d) (3, 9) If (4, - 3) and (-9, 7) are the two vertices of a triangle and (1, 4) is its centroid, then the area of triangle is NIMCET-2012 then 1 y a x x (b) 1 y 2 a x2 2 x (d) y = atan + x The relation that represents the shaded region in the figure given below is 1 1 r2 2 e e (a) 0 21. (b) (c) If a given point is P(10,10) and the Eq. of circle is (x – 1)2 + (y – 2)2 = 144. Where does the pt. lies 138 (a) 2 17. 1 y 2 6 1 y 1 6 2 (b) 1 NIMCET-2012 (d) None of these (c) 2 The straight line passes through the point P 2, 3 and makes an angle of 60 with the x-axis. The length of the intercept on it between the point P and the line 22. 23. x 3 y 12 is : BHU-2012 (a) 1.5 (b) 2.5 (c) 3.5 (d) 4.5 The equation of the straight line passing through the point of intersection of 4x + 3y – 8 = 0 and x + y – 1 = 0, and the point (-2, 5) is : BHU-2012 (a) 9x + 7y – 17 = 0 (b) 4x + 5y + 6 = 0 (c) 3x – 2y + 19 = 0 (d) 3x – 4y – 7 = 0 The angle between the two straight line represented by the equation 6x2 + 5xy – 4y2 + 7x + 13y – 3 = 0 is: BHU-2012 (a) tan 1 3 5 (b) tan 1 5 3 WORKSHEET-3 (OLD QUESTIONS ) 33. 34. 35. (a) y x (b) |y| |x| (c) y |x| (d) |y| x The area enclosed within the lines |x| + |y| = 1 is HCU-2011 NIMCET-2011 (a) 1 (b) 2 (c) 3 (d) 4 If 2x + 3y – 6 = 0 and 9x+ 6y – 18 = 0 cuts the axes in concyclic points, then the center of the circle is: NIMCET-2011 (a) (2, 3) (b) (3, 2) (c) (5, 5) (d) (5/2, 5/2) The number of distinct solutions (x, y) of the system of equations x2 = y2 and (x – a)2 + y2 = 1 where ‘a’ is any real number, can only be NIMCET-2011 4 INFOMATHS 36. (a) 0, 1, 2, 3, 4 or 5 (b) 0, 1 or 3 (c) 0, 1, 2 or 4 (d) 0, 2, 3 or 4 2 The vertex of parabola y − 8y +19 = 0 is 37. (a) (3, 4) (b) (4, 3) (c) (1, 3) (d) (3, 1) The eccentricity of ellipse 9x2 + 5y2 − 30y = 0 is NIMCET-2011 l on the axes, then l = 38. 39. 1 3 (b) 2 3 (c) 4 (d) 1 4 Point A is a + 2b, P is a and P divides AB in the ratio of 2 : 3. The position vector of B is BHU-2011 (a) 2a – b (b) b – 2a (c) a – 3b (d) b If the position vectors of A and B are a and b respectively, then the position vector of a point P which divides AB in the ratio 1 : 2 is BHU-2011 ab 3 a 2b (c) 3 b 2a 3 b 2a (d) 3 (a) 40. 3 51. 53. y = be-x/a at the point 41. 42. BHU-2011 (a) where it crosses the y-axis (b) where it crosses the x-axis (c) (0, 0) (d) (1, 1) Every homogeneous equation of second degree in x and y represent a pair of lines BHU-2011 (a) parallel to x-axis (b) perpendicular to y-axis (c) through the origin (d) parallel to y-axis The difference of the focal distances of any point on the hyperbola x2 y 2 1 is a 2 b2 54. 55. 56. (a) a (b) 2a (c) b (d) 2b If in ellipse the length of latusrectum is equal to half of major axis, then eccentricity of the ellipse is BHU-2011 (a) 44. 45. 46. 3 2 (b) 1 (c) 2 2 (d) 1 3 2 An equilateral triangle is inscribed in the parabola y = 4ax whose vertex is at the vertex of the parabola. The length of its side is BHU-2011 (a) a 3 (b) 2a 3 (c) 4a 3 (d) 8a 3 Two circles x2 + y2 = 5 and x2 + y2 – 6x + 8 = 0 are given. Then the equation of the circle through their point of intersection and the point (1, 1) is BHU-2011 (a) 7x2 + 7y2 – 18x + 4 = 0 (b) x2 + y2 – 3x + 1 = 0 (c) x2 + y2 – 4x + 2 = 0 (d) x2 + y2 – 5x + 3 = 0 The equation x 2 57. (a) (c) 4 3 , 5 5 1 5 , 5 4 (b) (d) 3 1 , 5 5 2 1 , 5 5 48. The angle between the asymptotes of the hyperbola 27x2 – 9y2 = 24 is NIMCET-2010 (a) 60 (b) 120 (c) 30 (d) 150 WORKSHEET-3 (OLD QUESTIONS ) NIMCET-2010 (a) 0 (b) 1 (c) -1 (d) 2 The number of integral values of m for which the x coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is KIITEE-2010 (a) 2 (b) 0 (c) 4 (d) 1 The pair of straight lines joining the origin to the common point of x2 + y2 = 4 and y = 3x + c perpendicular if c2 is equal to KIITEE-2010 (a) 20 (b) 13 (c) 1/5 (d) 1 Intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation of the circle on AB as diameter is KIITEE-2010 (a) x2 + y2 + x – y = 0 (b) x2 + y2 – x + y = 0 (c) x2 + y2 + x + y = 0 (d) x2 + y2 – x – y = 0 The locus of a point which moves such that the tangents from it to the two circles x2 + y2 – 5x – 3 = 0 and 3x2 + 3y2 + 2x + 4y – 6 = 0 are equal is KIITEE-2010 (a) 2x2 + 2y2 + 7x + 4y – 3 = 0 (b) 17x + 4y + 3 = 0 (c) 4x2 + 4y2 – 3x + 4y – 9 = 0 (d) 13x – 4y + 15 = 0 If a 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y2 = 4ax and x2 = 4ay, then KIITEE-2010 (a) d2 + (3b – 2c)2 = 0 (b) d2 + (3b + 2c)2 = 0 (c) d2 + (2b – 3c)2 = 0 (d) d2 + (2b + 3c)2 = 0 The distances from the foci of P (a, b) on the ellipse KIITEE-2010 (b) 4 5 a 5 (d) None of these The locus of a point P(,) moving under the condition that the x2 y 2 1 is a 2 b2 KIITEE-2010 (a) an ellipse (c) a parabola 58. (b) a circle (d) a hyperbola It the foci of the ellipse x2 y 2 1 144 81 25 represents a 47. is line y = x + is a tangent to the hyperbola 4 y 2 4 xy 4 x 2 y 1 BHU-2011 (a) straight line (b) circle (c) parabola (d) pair of lines The coordinates of the orthocenter of the triangle formed by the lines 2x2 – 2y2 + 3xy + 3x + y + 1 = 0 and 3x + 2y + 1 = 0 are BHU-2011 2 p b c a q c = 0, then the value of a b r x2 y 2 1 are 9 25 5 (a) 4 b 4 4 (c) 5 b 5 BHU-2011 43. 2 p q r + p a q b r c 52. x y 1 touches the curve a b (B) a b (D) N.O.T 50. If a p, b q, c r and (b) The straight line NIMCET-2010 (A) a2 + b2 (C) (a2 +b2)2 NIMCET-2011 (a) X2 Y2 1 intercepts equal length a 2 b2 49. If any tangent to the ellipse 59. 60. 62. and the hyperbola coincide, then the value of b2 is KIITEE-2010 (a) 3 (b) 16 (c) 9 (d) 12 The medians of a triangle meet at (0, - 3) and two vertices are at (1, 4) and (5, 2). Then the third vertex is at KIITEE-2010 (a) (4, 15) (b) (-4, 15) (c) (-4, 15) (d) (4, -15) The length of the perpendicular drawn from the point (3, - 2) on the line 5x – 12y – 9 = 0 is PGCET-2010 (a) 61. x2 y 2 1 25 b 2 28 13 (b) 31 13 (c) 30 10 (d) None of these If the lines x – 6y + a = 0, 2x + 3y + 4 = 0 and x + 4y + 1 = 0 are concurrent, then the value of ‘a’ is PGCET-2010 (a) 4 (b) 8 (c) 5 (d) 6 2 the angle between the lines represented by x + 3xy + 2y2 = 0 is PGCET-2010 (a) tan-1(2/3) (b) tan-1(1/3) 5 INFOMATHS 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. (c) tan-1(3/2) (d) None of these If the circle 9x2 + 9y2 = 16 cuts the x-axis at (a, 0) and (-a, 0), then a is PGCET-2010 (a) 2/3 (b) 3/4 (c) 1/4 (d) 4/3 The length of the perpendicular drawn from the point (1, 1) on the 15x + 8y + 45 = 0 is (PGCET paper – 2009) (a) 3 (b) 4 (c) 5 (d) 2 The equation of the line passing through the point of intersection 2x – y + 5 = 0 and x + y + 1 = 0 and the point (5, - 2) is (PGCET paper – 2009) (a) 3x + 7y – 1 = 0 (b) x + 2y + 1 = 0 (c) 5x + 6y + 3 = 0 (d) None of these The point of intersection of the lines represented by 2x2 – 9xy + 4y2 = 0 is (PGCET paper – 2009) (a) (0, 0) (b) (0, 1) (c) (1, 0) (d) (1, 1) 2 2 If y = x + c is a tangent to the circle x + y = 8, then c is (PGCET paper – 2009) (a) 3 (b) 2 (c) 4 (d) 1 The equation of the parabola whose vertex is (1, 1) and focus is (4, 1) is (PGCET paper – 2009) (a) (y – 1)2 = 12(x – 1) (b) (y – 2)2 = 13(x – 2) (c) (y – 1)2 = 10(x + 1) (d) None of these If the distance of any point (x, y) from the origin is defined as d(x, y)= max (|x|, |y|), then the locus of the point (x, y) where d(x, y) = 1 is MCA : NIMCET – 2009, KIITEE-2010 (a) a square of area 1 sq. unit (b) a circle of radius 1 (c) a triangle (d) a square of area 4 sq. units Let ABC be an isosceles triangle with AB = BC. If base BC is parallel to x-axis and m1, m2 are slopes of medians drawn through the angular points B and C, then (MCA : NIMCET – 2009) (a) m1m2 = - 1 (b) m1 + m2 = 0 (c) m1m2 = 2 (d) (m1 – m2)2 + 2m1m2=0 The straight lines (c) 76. 77. 78. 2 ,0 3 7 2 , 3 3 (b) x 2 x 2 x 3y (d) None of these (d) x y3 2 0 25 0 2 as a diameter is HYDERABAD CENTRAL UNIVERSITY - 2009 (a) (b) (c) (d) 81. 82. 21 0 2 25 x 2 x 2 3x y 0 2 25 x 2 x 2 x 3y 0 2 25 x 2 x 2 3x y 0 2 x x 3x y 2 2 Loci of a point equidistant to (2, 0) and x = - 2 is HYDERABAD CENTRAL UNIVERSITY - 2009 (a) y2 = 8x (b) y2 = 4x (c) x2 = 2y (d) x2 = 16y Given two fixed points A(-3, 0) and B(3, 0) with AB = 6, the equation of the locus of point P which moves such that PA + PB = 8 is HYDERABAD CENTRAL UNIVERSITY - 2009 x y 1 8 6 x y (c) 1 7 16 (a) 83. (b) (d) x2 y2 1 16 9 x2 y2 1 16 7 If y = mx bisects the angle between the lines x2 (tan2 + cos2) + 2xy tan - y2 sin2 = 0 when = /3, then the value of 3m 2 4m (a) 1 is (b) NIMCET - 2008 1 (c) 3 (d) 7 3 3 If a, b, c are the roots of the equation x3 – 3px2 + 3qx – 1 = 0, then the centroid of the triangle with vertices 1 c, c 85. 86. 1 1 a, , b, a b is at the point (a) (p, q) and NIMCET - 2008 (b) p q , 3 3 (c) (p + q, p – q) (d) (3p, 3q) Equation of the common tangent touching the circle (x – 3)2 + y2 = 2 9 and the parabola y = 4x above the x – axis is NIMCET - 2008 (a) 3 y 3x 1 (b) 3 y ( x 3) (c) 3y x 3 (d) 3 y (3x 1) 87. The coordinates of a point on the line x + y = 3 such that the point is at equal distances from the lines |x| = |y| are KIITEE - 2008 (a) (3, 0) (b) (-3, 0) (c) (0, - 3) (d) None Lines are drawn through the point P (-2, -3) to meet the circle x2 + y2 – 2x – 10y + 1 = 0. The length of the line segment PA, A being the point on the circle where the line meets the circle is. KIITEE - 2008 88. (a) 4 3 (b) 16 (c) 48 (d) None If the common chord of the circles x2 + (y - )2 = 16 and x2 + y2 = 16 subtend a right angle at the origin then is equal to. MCA : KIITEE - 2008 89. (a) 4 2 (b) 4 2 (c) 4 (d) 8 The equation of any tangent to the circle x2 + y2 – 2x + 4y – 4 = 0 is KIITEE - 2008 7 2 , 3 3 Two distinct chords drawn from the point (p, q) on the circle x2 + y2 = px + qy, where pq 0 are bisected by the x-axis then (MCA : KIITEE - 2009) (a) |p| = |q| (b) p2 = 8q2 (c) p2 < 8q2 (d) p2 > 8q2 The length of the latus rectum of the parabola x = ay2 + by + c is (MCA : KIITEE - 2009) (a) a/4 (b) 1/4a (c) 1/a (d) a/3 The equation of the tangent to the x2 – 2y2 = 18 which is perpendicular to the line x – y = 0 (MCA : KIITEE - 2009) (a) x + y = 3 (b) x + y =3/2 (c) x + y + 2 = 0 79. 80. 84. x y x y 1 k and , k 0 meet a b a b k on (MCA : NIMCET – 2009) (a) a parabola (b) an ellipse (c) a hyperbola (d) a circle The equation of the line segment AB is y = x, if A and B lie on the same side of the line mirror 2x – y = 1 the image of AB has the equation (MCA : KIITEE - 2009) (a) 7x – y = 6 (b) x + y = 2 (c) 8x + y = 9 (d) None of these The point (-1, 1) and (1, -1) are symmetrical about the line (MCA : KIITEE - 2009) (a) y + x = 0 (b) y = x (c) x + y = 1 (d) None of these The product of perpendiculars drawn from the point (1, 2) to the pair of lines x2 + 4xy + y2 = 0 is (MCA : KIITEE - 2009) (a) 9/4 (b) 9/16 (c) 3/4 (d) None of these The centroid of the triangle whose three sides are given by the combined equation (x2 + 7xy + 2y2) (y – 1) = 0 is (MCA : KIITEE - 2009) (a) HYDERABAD CENTRAL UNIVERSITY - 2009 (a) 2, 2 (b) 4,2 2 (c) 2 2 ,4 (d) 4 2 ,4 The equation of the circle having the chord x – y = 1 of the circle (a) y mx 3 1 m 2 (b) y mx 2 3 1 m 2 (c) y mx m 2 3 1 m 2 (d) None The sides of the rectangle of the greatest area that can be inscribed in the ellipse x2 + 2y2 = 8, are given by WORKSHEET-3 (OLD QUESTIONS ) 6 INFOMATHS 90. The equation of the circle whose two diameters are 2x – 3y + 12 = 0 and x + 4y – 5 = 0 and the area of which is 154 sq. units, will be : 91. 92. 22 7 107. MP COMBINED - 2008 (a) x2 + y2 + 6x – 4y + 36 = 0 (b) x2 + y2 + 3x – 2y + 18 = 0 (c) x2 + y2 – 6x + 4y + 36 = 0 (d) x2 + y2 + 6x – 4y – 36 = 0 The circle x2 + y2 – 2x + 2y + 1 = 0 touches: MP COMBINED - 2008 (a) Only x-axis (b) Only y-axis (c) Both the axes (d) None of the axes If the line hx + ky = 1 touches the circle (x y ) 2 2 1 a2 108. 109. , 110. then the locus of the point (h, k) will be: 93. 94. 95. 96. 97. 98. 99. (a) x2 + y2 = a2 MP COMBINED - 2008 (b) x2 + y2 = 2a2 (c) x2 + y2 = 1 (d) x2 y2 a2 2 Equation of the circle concentric to the circle x2 + y2 – x + 2y + 7 = 0 and passing through (-1, -2) will be: MP COMBINED - 2008 (a) x2 + y2 + x + 2y = 0 (b) x2 + y2 – x + 2y + 2 = 0 (c) 2(x2 + y2) – x + 2y = 0 (d) x2 + y2 – x + 2y – 2 = 0 For the circle x2 + y2 – 4x + 2y + 6 = 0, the equation of the diameter passing through the origin is: MP COMBINED - 2008 (a) x – 2y = 0 (b) x + 2y = 0 (c) 2x – y = 0 (d) 2x + y = 0 The circle x2 + y2 + 2ax – a2 = 0: (MP COMBINED – 2008) (a) touches x – axis (b) touches y – axis (c) touches both the axis (d) intersects both the axes The circles x2 + y2 + 2g1x + f1y + c1 = 0 and x2 + y2 + g2x + 2f2y + c2 = 0 cut each other orthogonally, then : (MP COMBINED – 2008) (a) 2g1g2 + 2f1f2 = c1 + c2 (b) g1g2 + f1f2 = c1 + c2 (c) g1g2 + f1f2 = 2(c1 + c2) (d) g1g2 + f1f2 + c1 + c2 = 0 If the straight line 3x + 4y = touches the parabola y2 = 12x then value of is (MCA : MP COMBINED – 2008) (a) 16 (b) 9 (c) – 12 (d) – 16 For the parabola y2 = 14x, the tangent parallel to the line x + y + 7 = 0 is : (MCA : MP COMBINED – 2008) (a) x + y + 14 = 0 (b) x + y + 1 = 0 (c) 2(x + y) + 7 = 0 (d) x + y = 0 Eccentricity of the ellipse 9x2 + 5y2 – 30y = 0 is : (MP COMBINED – 2008) (a) 1/3 (b) 2/3 (c) 4/9 (d) 5/9 2 100. For the ellipse 2 x y 1 , S1 and S2 are two foci then for 64 36 any point P lying on the ellipse S1P + S2P equals: (MCA : MP COMBINED – 2008) (a) 6 (b) 8 (c) 12 (d) 16 101. The coordinates of the foci of the hyperbola 9x2 – 16y2 = 144 are: (MCA : MP COMBINED – 2008) (a) (0, 4) (b) ( 4, 0) (c) (0, 5) (d) ( 5, 0) 102. The lengths of transverse and conjugate axes of the hyperbola x2 2y2 – 2x + 8y – 1 = 0 will be respectively: (MCA : MP COMBINED – 2008) (a) 2 3 ,2 6 (b) (c) 4 3 ,4 6 (d) 111. 112. 113. 114. 115. 116. 117. 118. 119. KARNATAKA - 2007 (a) +1/2 (b) –1/2 (c) 0 (d) +1/2 or – 1/2 The point of intersection of lines (i) x + 2y + 3 = 0 and (ii) 3x + 4y + 7 = 0 is KARNATAKA - 2007 (a) (1, 1) (b) (1, - 1) (c) (-1, 1) (d) (-1, -1) The acute angle between the lines (i) 2x – y + 13 = 0 and (ii) 2x – 6y + 7 = 0 KARNATAKA - 2007 (a) 0 (b) 30 (c) 45 (d) 60 If the points (k, - 3), (2, - 5) and (-1, -8) are collinear then K = ICET - 2007 (a) 0 (b) 4 (c) – 2 (d) – 3 The equation of the line with slope -3/4 and y – intercept 2 is ICET - 2007 (a) 3x + 4y = 8 (b) 3x + 4y + 8 = 0 (c) 4x + 3y = 2 (d) 3x + 4y = 4 If the lines x + 2y + 1 = 0, x + 3y + 1 = 0 and x + 4y + 1 = 0 pass through a point then a + = ICET - 2007 (a) (b) 2 (c) 1/ (d) 1/2 Equation of the line passing through the point (2, 3) and perpendicular to the segment joining the points (1, 2) – (1, 5) is ICET – 2005 (a) 2x – 3y – 13 = 0 (b) 2x – 3y – 9 = 0 (c) 2x – 3y – 11 = 0 (d) 2x – 3y – 7 = 0 The two sides forming the right angle of the triangle whose area is 24 sq. cm. are in the ratio 3:4. Then the length of the hypotenuse (in cm) is ICET – 2005 (a) 12 (b) 10 (c) 8 (d) 5 The equation of the circle passing through the origin and making intercepts of 4 and 3 or OX and OY respectively is ICET – 2005 (a) x2 + y2 – 3x – 4y = 0 (b) x2 + y2 + 4x + 3y = 0 (c) x2 + y2 + 3x + 4y = 0 (d) x2 + y2 – 4x – 3y = 0 The equation of the straight line which cuts off equal intercepts from the axis and passes through the point (1, - 2) is ICET – 2005 (a) 2x + 2y + 1 = 0 (b) x + y + 1 = 0 (c) x + y – 1 = 0 (d) 2x + 2y – 1 = 0 If the lines 2x + 3y = 6, 8x – 9y + 4 = 0, ax + 6y = 13 are concurrent, then a = ICET – 2005 (a) 3 (b) – 3 (c) – 5 (d) 5 The points of concurrence of medians of a triangle is ICET – 2005 (a) incentre (b) orthocenter (c) centroid (d) circumcentre If (0, 0), (2, 2) and (0, a) form a right angled isosceles triangle, then a = ICET – 2005 (a) 4 (b) – 4 (c) 3 (d) – 3 The area of the largest rectangle, whose sides are parallel to the coordinate axes, that can be inscribed in the ellipse x2 y 2 1 25 9 (a) 10 (b) 20 (c) 30 (d) 20 5 (e) 20 6 120. The orthocenter of the triangle determined by the lines 6x2 + 5xy – 6y2 – 29x + 2y + 28 = 0 and 11x – 2y – 7 = 0 is IP Univ. Paper – 2006 (a) (-4, 5) (b) (4, 4) (c) (6, 7} (d) (2, 1) (e) (-1, 3) 121. a, b, c R. if 2a + 36 + 4c = 0, then the line ax + by + c = 0 (a) 3 6 1 1 3, 6 2 2 103. For the given equation x2 + y2 – 4x + 6y – 12 = 0, the centre of the circle is KARNATAKA - 2007 (a) (-2, 3) (b) (-3, 2) (c) (3, - 2) (d) (2, - 3) 104. The circumference of the circle x2 + y2 + 2x + 6y – 12 = 0 the centre of the circle is KARNATAKA - 2007 (a) 2 (b) 8 (c) 3 (d) None 105. The locus of a point which moves in a plane such that its distance from a fixed point is equal to its distance from a fixed line is. KARNATAKA - 2007 (a) Parabola (b) Hyperbola (c) Ellipse (d) Circle 106. In parabola y2 = 4kx, if the length of Latus Rectum is 2 then k is WORKSHEET-3 (OLD QUESTIONS ) IP Univ. Paper – 2006 (c) 1 3 , 3 5 5 3 , 2 7 (b) (d) 2 3 , 3 4 1 3 , 7 11 (e) 1 3 , 2 4 122. The distance of the point (x, y) form y-axis is Karnataka PG-CET : Paper 2006 (a) x (b) y (c) |x| (d) |y| 123. If the lines 4x + 3y = 1, y = x + 5 and 5y + bx = 3 are concurrent, then the value of b is Karnataka PG-CET : Paper 2006 (a) 1 (b) 3 (c) 6 (d) 0 124. The system of equations x + y = 2 and 2x + 2y = 3 has Karnataka PG-CET : Paper 2006 (a) No solution (b) a unique solution (c) finitely many solutions (d) infinitely many solutions 125. The radius of the circle 16x2 + 16y2 = 8x + 32y – 257 = 0 Karnataka PG-CET : Paper 2006 7 INFOMATHS (a) 8 (b) 6 (c) 15 (d) None of these 126. Axis of the parabola x2 – 3y – 6x + 6 = 0 is Karnataka PG-CET : Paper 2006 (a) x = - 3 (b) y = - 1 (c) x = 3 (d) y = 1 127. The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is Karnataka PG-CET : 2006 (a) a circle (b) a straight line (c) a hyperbola (d) an ellipse 128. The Eccentricity of a rectangular hyperbola is : MP : MCA Paper – 2003 (a) 3 (b) 2 (c) 3 / 2 (d) 2 129. From a point (x1, y1) two tangent can be drawn on circle x2 + y2 = a2 if: MP : MCA Paper – 2003 (a) x12 y12 a 2 0 (b) x12 y12 a 2 0 (c) x12 y12 a 2 0 (d) None of these 130. The sum of the distance of a point on the ellipse x2 y 2 1 to a 2 b2 its foci is equal to : MP : MCA– 2003 (a) semi major axis (b) major axis (c) semi minor axis (d) minor axis 131. The foci of hyperbola 9x2 – 25y2 + 54x + 50y – 169 = 0 is MP – 2003 (a) (-3, 1) (c) 3 (b) 34,1 2 3 34,1 (d) None of these 2 132. If two circles x + y + 2g1x + 2f1y + c1 = 0, x2 + y2 + 2g2n + 2f2y + c2 = 0 will cut each other and satisfies g1 g 2 f1 f 2 c1 c2 2 MP : MCA Paper – 2003 (a) π/3 (b) π/2 (c) 3π/2 (d) π/4 133. Two circles x2 + y2 + 2gx + c = 0 and x2 + y2 + 2fy + c = 0 touch each other, then : MP :– 2003 (a) g2 + f2 = c3 (b) g2 + f2 = c (c) c(g2 + f2) = g2f2 (d) g2 + f2c = g2f2 134. S1 = x2 + y2 – 4x – 6y + 10 = 0 S2 = x2 + y2 – 2x – 2y – 4 Angle between S1 and S2 is UPMCAT : paper – 2002 (a) 90 (b) 60 (c) 45 (d) None of these 135. The equation of line passing through the intersection of lines 5x – 6y – 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to 3x – 5y + 27 = 0 is : UPMCAT :– 2002 (a) 5x + 3y + 10 = 0 (b) 5x + 3y + 21 = 0 (c) 5x + 3y + 18 = 0 (d) 5x + 3y + 8 = 0 136. The area of triangle formed by y = m1x + c, y = m2x + c2 and y axis is : UPMCAT : paper – 2002 1 c1 c2 2 m1 m2 2 1 c1 c2 (c) 2 m1 m2 2 (a) 1 c1 c2 2 m1 m2 2 1 c2 c1 (d) 2 m1 m2 2 (b) (a) (c) (b) 13 26 , 5 5 (d) None of these 138. The area of the region bonded by the curve y = x2 and the line y = x is UPMCAT : paper – 2002 1 Sq U 64 1 (c) Sq U 6 (a) (b) 1 Sq U 12 (d) N.O.T. 139. If 4x2 + 9y2 + 12xy + 6x ….. + 9y – 4 = 0 represents two parallel lines then the distance between. The lines is: UPMCAT:– 2002 WORKSHEET-3 (OLD QUESTIONS ) 4 (b) 13 5 (c) 13 (d) None of these 13 140. If (± 3, 0) be focus of ellipse and semi major axis is 6. Then equ. of ellipse is: UPMCAT :– 2002 (a) (c) x2 y 2 1 36 45 x2 y 2 1 36 27 (b) x2 y 2 1 27 36 (d) None of these 141. If 2x2 – 5xy + 2y2 – 3x + 1 = 0, represents pairs of lines, then the angle between the lines is : UPMCAT : paper – 2002 (a) tan-1 (2/3) (b) tan-1 (4/3) -1 (c) tan (3/4) (d) None of these 142. The condition that eqa. ax2 + by2 + 2gx + 2fy + 2hxy + c = 0 represents a pair of the line is (i) a h g h b f g f 0 c (ii) abc + 2fgh – af2 – bg2 – ch2 = 0 (iii) af + bg + ch = 0 (iv) af2 = bg2 ; h2 = ab UPMCAT:– 2002 (a) i, ii (b) ii, iv (c) i, iv (d) i, ii and iv 143. A ellipse has e 1 , directrix is x + 6 = 0, and has a focus at (0, 2 0) then the eqn. of ellipse is: UPMCAT :– 2002 (a) 3x2 + 4y2 + 12x – 36 = 0 (b) 3x2 + 4y2 – 12x + 36 = 0 (c) 3x2 + 4y2 – 12x – 36 = 0 (d) None of these 144. The eqn. of the ellipse has its centre at (1, 2), a focus at (6, 2) and passing through the point (4, 6) : UPMCAT :– 2002 (a) (c) x 1 2 45 x 1 20 2 y 2 2 1 (b) 20 y 2 x 1 45 2 y 2 20 2 1 2 1 (d) None of these 45 145. The tangents of the circle x2 + y2 = 4 at the points A and B meet at P(-4, 0). The area of the quadrilateral PAOB where O is the origin is. KIITEE - 2008 (a) 4 (b) 6 2 (c) 3 (d) None 146. The x2 + y2 + 2x = 0, R touches the parabola y2 = 4x externally. Then KIITEE - 2008 (a) > 1 (b) < 0 (c) > 0 (d) None 147. A point P on the ellipse 137. Reflection of the point P(1, 2) in x + 2y + 4 = 0 is UPMCAT : paper – 2002 13 26 , 5 5 13 26 , 5 5 3 (a) 8 x2 y2 1 25 9 has the eccentric angle . The sum of the distance of P from the two foci is. KIITEE - 2008 (a) 10 (b) 6 148. For the hyperbola (c) 5 x2 cos (d) 3 y2 sin 2 following remains constant when varies? 2 1 which of the MCA : KIITEE - 2008 (a) directrix (b) eccentricity (c) abscissae of foci (d) abscissae of vertices 149. The sum of the intercepts made on the axes of co-ordinates by any tangent to the curve x y 2 is equal to KIITEE - 2008 (a) 4 (b) 8 (c) 2 (d) None 150. If the focus and directrix of a parabola are (-sin , cos ) and x cos + y sin = p respectively, then length of the latus rectum will be: (MP COMBINED – 2008) (a) 2p (b) 4p (c) p2 (d) p(cos – sin ) 151. The distance between the two focii of a hyperbola H is 12. The distance between the two directories of hyperbola H is 3. The acute angle between the asymptotes of H in degrees is 8 INFOMATHS (a) 30 (b) 40 IP Univ. Paper – 2006 (d) 60 (e) 70 (c) 45 152. L1 || L2. Slope of L1 = 9. Also L3 || L4. Slope of L4 these lines touch the ellipse 1 . All 25 x2 y 2 1 . The area of the 25 9 parallelogram determine by these lines is IP University : Paper - 2006 (a) 21 (b) 28 (c) 40 (d) 56 (e) 60 153. If P, Q, R, S are four distinct collinear points such that PR PS RP RQ is k , then, the value of . RQ SQ PS QS 1 k (a) 1 l (c) 1 k (b) 1 k 1 k 1 k (d) 1 k 1 k (a) (c) IP University : Paper - 2006 2 156. The limiting points of the system of coaxial circles of which two members are x2 + y2 + 2x + 4y + 7 = 0 and x2 + y2 + 5x + y + 4 = 0 is: MP : MCA Paper – 2003 (a) (-2, 1) and (0, - 3) (b) (2, 1) and (0, 3) (c) (4, 1) and (0, 6) (d) None of these 157. The length of common chord of the circles (x – a)2 + y2 = a2 and x2 + (y – b)2 = b2 is : MP : MCA Paper – 2003 2 2 (e) N.O.T. (a) 154. P moves on the line y = 3x + 10. Q moves on the parabola y = 24x. The shortest value of the segment PQ is IP University - 2006 7 12 8 (b) (c) 10 7 2 6 (d) (c) (e) 6 15 155. The line 2x + y – 1 = 0 cuts the curve 5x2 + xy – y2 – 3x – y + 1 = 0 at P and Q. O is the origin. The acute angle between the lines OP and OQ is IP University - 2006 (a) 7 (b) 6 (c) 5 (d) 4 WORKSHEET-3 (OLD QUESTIONS ) (e) (b) 2ab ab a b2 2 (d) None of these a2 b2 158. An arch way is in the shape of a semi ellipse. The road level being the major axis. If the breadth of the road is 30 metres and the height of the arch is 6m at a distance of 2 metre from the side, then find the greatest height of the arch. MP : MCA Paper – 2003 2 (a) 2 a 2 b2 25 14 14 25 14 7 m (b) m (d) 45 14 14 45 14 7 m m 159. The locus midpoint of a chord of the circle x2 + y2 = 4, which subtend angle 90 at the centre. UPMCAT : paper – 2002 (a) x + y + 3 = 0 (b) x2 + y2 = 0 (c) x + y + 2 = 0 (d) x2 + y2 = 2 3 9