CAT 2003

 

1) If three positive real numbers x,y,z satisfy y-x = z-y and xyz = 4, then what is the minimum possible value of y?

 

 

 

 

2) An intelligence agency forms a code of two distinct digits selected from 0,1,2,…….,9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion when and read upside down – for example, the code 91 may appear as 16. how many codes are there for which no such confusion can arise?

  1. 80
  2. 78
  3. 71
  4. 69

3) Consider two different cloth-cutting processes. In the first one, n circular cloth pieces are cut from a square cloth piece of side a in the following steps: the original square of side a is divided into n smaller squares, not necessarily of the same size; then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from the square of side a and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total area of scrap cloth generated in the former to that in the latter is:

 

 

 

4) In the figure below (not drawn to scale), rectangle ABCD. Is inscribed in the circle with centre at O. The length of side AB is greater than that of side BC. The ratio of the area of the circle to the area of the rectangle ABCD is π : √3. The line segment DE intersects AB at E such that angle ODC = angle ADE. What is the ratio AE : AD ?

  1. 1 : √3
  2. 1 : √2
  3. 1 : 2√3
  4. 1 : 2

5) There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lies are required?

  1. 72
  2. 90
  3. 96
  4. 144

6) In the figure (not drawn to scale) given, P is a point on AB such that AP:PB = 4:3. PQ is parallel to AC and QD is parallel to CP. In triangle ARC, ARC = 90°, and in triangle PQS, PSQ = 90°. The length of QS is 6 cms. What is ratio AP:PD?

  1. 10 : 3
  2. 2 : 1
  3. 7 : 3
  4. 8 : 3

7) A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower The top of the tower is observed from the car and, in the process, it takes 10 minutes for the angle of elevation to change from 45° to 60°. After how much more time will this car reach the base of the tower?

  1. 5 (√3 + 1)
  2. 6 (√3 + √2)
  3. 7 (√3 – 1)
  4. 8 (√3 – 2)

8) In the figure (not drawn to scale) given below, if AD =CD = BC, and angle BCE = 96°, how much is angle DBC?

  1. 32°
  2. 84°
  3. 64°
  4. Cannot be determined

9) If both a and b belong to the set {1,2,3,4}, then the number of equations of the form ax² + bx + 1 = 0 having real roots is

  1. 10
  2. 7
  3. 6
  4. 12

10) If   then a possible value of x is given by

  1. 10
  2. 1/100
  3. 1/1000
  4. None of these

11) What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?

  1. 666
  2. 676
  3. 683
  4. 777

12) In the figure below (not drawn to scale). A,B and C are there points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. if angle ATC = 30° and ACT = 50°, then the angle BOA is:

  1. 100°
  2. 150°
  3. 80°
  4. Not possible to determine

13) What is the sum of ‘n’ terms in the series: log n + log (m² / n) + log (m³ / n²) + log (m4/n3) + …. 14) Let S1 be a square of side a. Another square S2 is formed by joining the mid-points of the sides of S1. The same process is applied to S2 to form yet another square S3, and so on. If A1, A2, A3,……be the areas and P1, P2, P3,……..be the perimeters of S1, S2, S3,……., respectively, then the ratio 15) The infinite sum 1 + 4/7 + 9/72 + 16/73 + 25/74 + … equals

  1. 27/14
  2. 21/13
  3. 49/27
  4. 256/147

16) Consider the sets T= {n, n+1, n+2, n+3, n+4}, where n = 1,2,3,…96. How many of these sets contain 6 or any integral multiple thereof (i.e., any of the numbers 6,12,18,…) ?

  1. 80
  2. 81
  3. 82
  4. 83

17) Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?

  1. 1/3
  2. 1/2
  3. 2/3
  4. 5/6

18) The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3 + 2x2 + 7x + 3 = 0 is:

  1. 0
  2. 1
  3. 2
  4. 3

19) A real number x satisfying (1 – 1/n) < x ≤ (3+1/n), for every positive integer n, is best described by:

  1. 1 < x < 4
  2. 1 < x ≤ 3
  3. 0 < x ≤ 4
  4. 1 ≤ x ≤ 3

DIRECTIONS for questions 20 to 22: Answer the questions on the basis of the information given below. The seven basic symbols in a certain numeral system and their respective values are as follows: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1000 In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur contiguously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + l = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then, the smaller value is subtracted from the larger. For example, XLVI = (50 – 10) + 5 + 1 = 46. 20) The value of the numeral MDCCLXXXVII is:

  1. 1687
  2. 1787
  3. 1887
  4. 1987

21) The value of the numeral MCMXCIX is:

  1. 1999
  2. 1899
  3. 1989
  4. 1889

22) Which of the following can represent the numeral for 1995?

  • A.)  MCMLXXV
  • B.)  MCMXCV
  • C.)  MVD
  • D.)  MVM
  1. Only (A) and (B)
  2. Only (C) and (D)
  3. Only (B) and (D)
  4. Only (D)

DIRECTIONS for questions 23 to 25: Answer the questions on the basis of the tables given below. Two binary operations @ and * are defined over the set {a, e, f, g, h} as per the following tables:

Thus, according to the first table f @ g =a, while according to the second table g*h = f, and so on. 23) What is the smallest positive integer n such that gn = e ?

  1. 4
  2. 5
  3. 2
  4. 3

24) Upon simplification, f @ [ f * { f @ ( f * f ) } ] equals:

  1. e
  2. f
  3. g
  4. h

 

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