CAT 2008 QA Question Paper(Available):Download Solutions with Answer Key

What are the last two digits of \(7^{2008}\)?

• (1) 21
• (2) 61
• (3) 01
• (4) 41
• (5) 81

If the roots of the equation \(x^3 - ax^2 + bx - c = 0\) are three consecutive integers, then what is the smallest possible value of \(b\)?

• (1) \(-\frac{1}{\sqrt{3}}\)
• (2) \(-1\)
• (3) 0
• (4) 1
• (5) \(\frac{1}{\sqrt{3}}\)

A shop stores \(x\) kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of \(x\)?

• (1) \(2 \le x \le 6\)
• (2) \(5 \le x \le 8\)
• (3) \(9 \le x \le 12\)
• (4) \(11 \le x \le 14\)
• (5) \(13 \le x \le 18\)

What is the other root of \(f(x) = 0\)?

• (1) \(-7\)
• (2) \(-4\)
• (3) \(2\)
• (4) \(6\)
• (5) cannot be determined

What is the value of \(a + b + c\) given the above conditions?

• (1) \(9\)
• (2) \(14\)
• (3) \(13\)
• (4) \(37\)
• (5) cannot be determined

The number of common terms in the two sequences \(17, 21, 25, \ldots, 417\) and \(16, 21, 26, \ldots, 466\) is:

• (1) 78
• (2) 19
• (3) 20
• (4) 77
• (5) 22

How many integers, greater than \(999\) but not greater than \(4000\), can be formed with the digits \(0, 1, 2, 3, 4\), if repetition of digits is allowed?

• (1) 499
• (2) 500
• (3) 375
• (4) 376
• (5) 501

Neelam rides her bicycle from her house at \(A\) to her office at \(B\), taking the shortest path. The number of possible shortest paths that she can choose is:

• (1) 60
• (2) 75
• (3) 45
• (4) 90
• (5) 72

Neelam rides her bicycle from her house at \(A\) to her club at \(C\), via \(B\) taking the shortest path. The number of possible shortest paths that she can choose is:

• (1) 1170
• (2) 630
• (3) 792
• (4) 1200
• (5) 936

Let \(f(x)\) be a function satisfying \(f(x) f(y) = f(xy)\) for all real \(x, y\). If \(f(2) = 4\), then what is the value of \(f\left( \frac12 \right)\)?

• (1) 0
• (2) \(\frac14\)
• (3) \(\frac12\)
• (4) 1
• (5) cannot be determined

The seed of any positive integer \(n\) is defined as: \[ seed(n) = n, \ if n < 10 \] \[ seed(n) = seed(s(n)), \ otherwise \]
where \(s(n)\) is the sum of digits of \(n\). How many positive integers \(n\), such that \(n < 500\), will have \(seed(n) = 9\)?

• (1) 39
• (2) 72
• (3) 81
• (4) 108
• (5) 55

In \(\triangle ABC\), \(AB = 17.5\) cm, \(AC = 9\) cm. Let \(D\) be a point on \(BC\) such that \(AD \perp BC\) and \(AD = 3\) cm. What is the radius of the circumcircle of \(\triangle ABC\)?

• (1) 17.05
• (2) 27.85
• (3) 22.45
• (4) 32.25
• (5) 26.25

Consider obtuse-angled triangles with sides \(8\) cm, \(15\) cm, and \(x\) cm, where \(x\) is integer. How many such triangles exist?

• (1) 5
• (2) 21
• (3) 10
• (4) 15

Square \(ABCD\) has midpoints \(E, F, G, H\) of sides \(AB, BC, CD, DA\) respectively. Let \(L\) be the line through \(F\) and \(H\). Points \(P, Q\) are on \(L\) inside \(ABCD\) such that \(\angle APD = \angle BQC = 120^\circ\). What is the ratio of area of \(ABQCDP\) to the remaining area?

• (1) \(\frac{4\sqrt{2}}{3}\)
• (2) \(2 + \sqrt{3}\)
• (3) \(\frac{10 - 3\sqrt{3}}{9}\)
• (4) \(1 + \frac1{\sqrt{3}}\)
• (5) \(2\sqrt{3} - 1\)

What is the number of distinct terms in the expansion of \((a + b + c)^{20}\)?

• (1) 231
• (2) 242
• (3) 243
• (4) 210

Which of the following cannot be true?

• (1) At least two horses finished before Spotted
• (2) Red finished last
• (3) There were three horses between Black and Spotted
• (4) There were three horses between White and Red
• (5) Grey came in second

Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?

• (1) Spotted came in first
• (2) Red finished last
• (3) White came in second
• (4) Black came in second
• (5) There was one horse between Black and White

What is the number of matches played by the champion?


A. The entry list for the tournament consists of 83 players.

B. The champion received one bye.

• (1) If A alone but not B alone is sufficient
• (2) If B alone but not A alone is sufficient
• (3) If both A and B together are sufficient
• (4) If A alone is sufficient and B alone is sufficient
• (5) If not even A and B together are sufficient

If the number of players in the first round was between 65 and 128, what is the exact value of \(n\)?


A. Exactly one player received a bye in the entire tournament.

B. One player received a bye while moving on to the fourth round from the third round.

• (1) If A alone but not B alone is sufficient
• (2) If B alone but not A alone is sufficient
• (3) If both A and B together are sufficient
• (4) If A alone is sufficient and B alone is sufficient
• (5) If not even A and B together are sufficient

Two circles, both of radii \(1\) cm, intersect such that the circumference of each one passes through the centre of the other. What is the area (in sq. cm.) of the intersecting region?

• (1) \(\frac{\pi}{3} - \frac{\sqrt{3}}{4}\)
• (2) \(\frac{2\pi}{3} + \frac{\sqrt{3}}{2}\)
• (3) \(\frac{4\pi}{3} - \frac{\sqrt{3}}{2}\)
• (4) \(\frac{4\pi}{3} + \frac{\sqrt{3}}{2}\)
• (5) \(\frac{2\pi}{3} - \frac{\sqrt{3}}{2}\)

Rahim drives from city A to station C at \(70\) km/h. Train leaves city B (500 km south of A) at 8:00 am toward C at \(50\) km/h. C is located between S and SW of A with AC at \(30^\circ\) to AB. Rahim must reach C at least 15 minutes before train. Latest time to leave A?

• (1) 6:15 am
• (2) 6:30 am
• (3) 6:45 am
• (4) 7:00 am
• (5) 7:15 am

Three consecutive positive integers are raised to the first, second, and third powers respectively and added. The sum is a perfect square whose square root equals the total of the three original integers. Which range best describes the minimum integer \(m\) of these three?

• (1) \(1 \le m \le 3\)
• (2) \(4 \le m \le 6\)
• (3) \(7 \le m \le 9\)
• (4) \(10 \le m \le 12\)
• (5) \(13 \le m \le 15\)

Find: \(\sum_{k=1}^{2007} \sqrt{1 + \frac{1}{k^2} + \frac{1}{(k+1)^2}}\)

• (1) \(2008 - \frac{1}{2008}\)
• (2) \(2007 - \frac{1}{2007}\)
• (3) \(2007 - \frac{1}{2008}\)
• (4) \(2008 - \frac{1}{2007}\)
• (5) \(2008 - \frac{1}{2009}\)

A right circular cone has base radius \(4\) cm and height \(10\) cm. A cylinder is to be placed inside the cone with one flat surface resting on the base of the cone. Find the largest possible total surface area (sq. cm) of the cylinder.

• (1) \(\frac{100\pi}{3}\)
• (2) \(80\pi\)
• (3) \(\frac{120\pi}{7}\)
• (4) \(\frac{130\pi}{9}\)
• (5) \(\frac{110\pi}{7}\)