CAT 2004 Question Paper

Instructions

For the following questions answer them individually

Question 21

Let $$C$$ be a circle with centre $$P_0$$ and $$AB$$ be a diameter of $$C$$. Suppose $$P_1$$ is the mid point of the line segment $$P_0B$$,$$P_2$$ is the mid point of the line segment $$P_1B$$ and so on. Let $$C_1,C_2,C_3,...$$ be circles with diameters $$P_0P_1, P_1P_2, P_2P_3...$$ respectively. Suppose the circles $$C_1, C_2, C_3,...$$ are all shaded. The ratio of the area of the unshaded portion of $$C$$ to that of the original circle is

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Question 22

Consider the sequence of numbers $$a_1, a_2, a_3$$....... to infinity where $$a_1 = 81.33$$ and $$a_2 = -19$$ and $$a_j = a_{j-1} - a_{j-2}$$ for $$j > 3$$. What is the sum of the first 6002 terms of this sequence?

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Question 23

A sprinter starts running on a circular path of radius r metres. Her average speed (in metres/minute) is $$\pi$$ r during the first 30 seconds, $$\pi$$ r/2 during next one minute, $$\pi$$ r/4 during next 2 minutes, $$\pi$$ r/8 during next 4 minutes, and so on. What is the ratio of the time taken for the nth round to that for the previous round?

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Question 24

Let $$u = ({\log_2 x})^2 - 6 {\log_2 x} + 12$$ where x is a real number. Then the equation $$x^u = 256$$, has

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Question 25

If the lengths of diagonals DF, AG and CE of the cube shown in the adjoining figure are equal to the three sides of a triangle, then the radius of the circle circumscribing that triangle will be?

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Question 26

In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how many distinct routes can a car reach point B from point A?

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Question 27

On a semicircle with diameter AD, chord BC is parallel to the diameter. Further; AD and BC are separated by 2cm, while AD has length 8. What is the length of BC?

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Question 28

A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?

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Question 29

In the adjoining figure, chord ED is parallel to the diameter AC of the circle. If angle CBE = 65┬░, then what is the value of angle DEC?

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Question 30

The remainder, when $$(15^{23} + 23^{23})$$ is divided by 19, is

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