Common Admission Test (CAT) is a computer based test given in India to test Quantitative Ability, Data Interpretation, Verbal Ability and Logical Reasoning. The Indian Institutes of Management (IIMs) use the test as an important component in selecting students for the business administration programs. To get a complete idea about Common Admission Test (CAT), please refer my previous article…

Instructions:

1) The duration of this test is 50 minutes and the test is meant to be taken in one-go without any break(s).

2) This test has 25 questions. Each question carries 4 marks. Wrong answer per question will fetch -1 mark.

3) Use of slide rule, log tables and calculators is not permitted.

4) Use the blank space in the question paper for the rough work.

QUESTIONS:

1. Consider all possible seven-digit binary numbers having four 1s and three 0s. What is the sum of all such numbers?

(a) 1470 (b) 1615 (c) 1740 (d) 1825 (e) 1910

2. Which of the cones can be formed from a 252 degree sector of a circle of radius 10 by aligning the two straight sides?
(a) A cone with slant height of 10 and radius 6

(b) A cone with height of 10 and radius 6
(c) A cone with slant height of 10 and radius 7

(d) A cone with height of 10 and radius 7

(e) A cone with slant height of 10 and radius 8

3. Consider the sum of two numbers x and y whose sum ends in 9. Which of the following will necessarily end in 9?

(a) x^6 + y^6 (b) x^7 + y^7 (c) x^13 + y^13 (d) x^18 + y^18 (e) x^19 + y^19

4. Two straight roads R1 and R2 diverge from a point A at an angle of 120 degrees. Ram starts walking from point A along R1 at a uniform speed of 3km/hr. Shyam starts walking at the same time from A along R2 at a uniform speed of 2km/hr. They continue walking for 4 hours along the respective roads and reach points B and C on R1 and R2 respectively. There is a straight path connecting B and C. Then Ram returns to point A after walking along the line segments BC and CA. Shyam also returns to A after walking along line segments CB and BA. Their speeds remain unchanged. The time interval (in hours) between Ram’s and Shyam’s return to the point A is

(a) (10√19 + 26)/3 (b) (2√19 + 10)/3 (c) (√19 + 26)/3 (d) (√19 + 10)/3 (e) none of the foregoing

5. In a quadrilateral ABCD, the diagonals AC and BD intersect at O. Let OA = 2, OB = 2, OC = 3, OD = 4 and AB = 3. The area of the quadrilateral is

(a) 55/4 (b) 117/8 (c) 173/12 (d) 225/16 (e) none of the foregoing

6. For real x and y if f(x, y) = f(x+y, 1) + f(1, x+y). Then f(x, 1-x)/f(x, -x) equals

(a) ½ (b) 1 (c) 2/3 (d) 2 (e) 3/2

7. The sum of a few (more than 2 and less than 100) consecutive integers is found to be 253. There can be 2 values of the total number of terms. The positive difference between these 2 values will be

(a) 15 (b) 20 (c) 25 (d) 35 (e) 45

8. If a, b, c are greater than 1, then loga/(log(ab)) + logb/(log(bc)) + logc/(log(ca)) (is)

(a) always greater than 1

(b) always less than 2

(c) always less than 1

(d) exactly 2 of the foregoing

(e) none of the foregoing

9. Let n be number of ways in which two adjacent face of a cube can be chosen. Its faces are divided into four equal squares and let m be the number of ways in which two such adjacent squares can be chosen. Then m =

(a) 4n (b) 6n (c) 8n (d) 9n (e) 12n

10. A vertical cylinder vessel contains water in it up to height of √3 unit. The cylinder is then tilted till its axis make 30 degrees with the vertical and the level of water just covers the base of the vessel. The radius of the base of the vessel is

(a) 1 (b) 3 (c) √3/2 (d) 2/√3 (e) none of the foregoing

11. The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Then m equals

a) 150 (b) 250 (c) 350 (d) 450 (e) none of the foregoing

12. Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set {A1A2…A9}?

(a) 30 (b) 33 (c) 36 (d) 66 (e) 72

13. The volumes of sales of a product depends upon the discount x% given on its sales as (100+3x)*V/400, where V is the total volume of the product for sale. The maximum profit that can be realized on the sale of the product correspond to the which value of x?

(a) 20% (b) 25% (c) 33.33% (d) 40% (e) 50%

14. Let the system of equations iy – 1 = a(x-i) for i = 1, 2, 3 have a unique solution. Then xy equals

(a) 3 (b) 1 (c) a (d) 1/a (e) 1/3

15. A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is

(a) 21 (b) 19 (c) 18 (d) 16 (e) 15

16. Let ABC be an acute angled triangle and CD be the altitude through clip_image006. If AB = 8 and CD=6 , what is the distance between the midpoints of AD and BC?

(a) 4.5 (b) 3 (c) 5 (d) 7.5 (d) 4

17. There are 2 kinds of milk powder with different fat and protein contents. When these 2 kinds of milk powder are mixed in various proportion, the protein and fat concentration of the mixtures was found to be (5%, 8%), (6%, 6%) and (8%, x%) respectively. Then the value of x is

(a) 2 (b) 3.5 (c) 4 (d) 4.5 (e) 5

18. Two points A(x1,y1) and B(x2,y2) are chosen on the graph of f(x) = ln(x), with 0 < x1 < x2. The points C and D trisect AB, with AC < CB. Through C a horizontal line is drawn to cut the curve at E(x3,y3). If x1 = 1 and x2 = 1000, then x3 is

(a) 1/100 (b) 1/10 (c) 10 (d) 100 (e) none of the foregoing

19. The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. The maximum number of members this band can have

(a) is more than 500 (b) is less than 100 (c) is more than 100 but less than 200 (d) is more than 300 but less than 400 (e) is more than 200 but less than 300

20. A semicircle with radius R is contained in a square whose sides have length 1 unit. The maximum value of R is

(a) √2 – 1 (b) (√2 + 1)/4 (c) √3 – √2 (d) (√6 + 2)/8 (e) (√6 – √2)/2

21. A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star-polygon?

(a) 24 (b) 36 (c) 40 (d) 60 (e) none of the foregoing

22. Four friends bought a gift. Had one of them not contributed in turn, then others would have contributed w, x, y, and z fraction of money more than what they originally contributed. Then 1/(1+w) + 1/(1+x) + 1/(1+y) + 1/(1+z) (is)

(a) 4 (b) 3 (c) 2 (d) 1 (e) can not be determined

23. When one of the two circles having the ratio of their radii as x (not equal to 1) pass through the centre of other circle, the length of common chord for the two cases are found to be equal. Then x^2 + 1/x^2 =

(a) 3 (b) 4 (c) 6 (d) 8 (e) 9

24. Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

(a) 36 (b) 34 (c) 28 (d) 22 (e) 18

25. Let N be the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of N. Then which among the following about N is not true?

(a) N is divisible by 5

(b) N is divisible by exactly 2 prime numbers

(c) The positive difference between N and sum of its divisors is 205

(d) N can be written as the difference of 2 integer squares in 12 ways

(e) none of the foregoing