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

Video Solution

Solution

Radius of the circles $$C_1, C_2, C_3,...$$ would be in GP with (R/4),(R/8),(R/16) and so on. Radii of circles are in the ratio 1:4.

Ratio of unshaded region to the ratio of original circle = 1-$$\frac{Ratio\ of\ shaded\ region}{Ratio\ of\ original\ circle}$$

= 1-$$\frac{\pi r^2/16 + \pi r^2/64+.....}{\pi r^2}$$ = 1-$$\frac{1/16}{(1-1/4)}$$ = 1- 1/12 = $$\frac{11}{12}$$ = 11:12

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 $$j\ge3$$. What is the sum of the first 6002 terms of this sequence?

Video Solution

Solution

According to given conditions the terms are 81.33, -19, -100.33, -81.33, 19, 100.33, 81.33,-19,.. Hence the series repeats after every 6 terms . Also summation of these 6 terms is 0. Hence summation is 60002 terms will we sum of first 2 terms which is 62.33.

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?

Video Solution

Solution

Let radius be 1 units and p = 3.14 or $$\pi$$ . So circumference is $$2*\pi$$.

According to given condition distance covered in first 1/2 mins = $$\pi$$/2 km, distance covered in next 1 min = $$\pi$$/2 km, distance covered in next 2 mins = $$\pi/2$$ km and finally distance covered in next 4 minutes = $$\pi/2$$ km.

Time taken to cover first round = 1/2 + 1 + 2 + 4 = 7.5 minutes.

Now time taken to cover $$\pi/2$$ is in GP.

For the second round the time taken is = 8+16+32+64 = 120

Ratio = 120/7.5 = 16

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

Video Solution

Solution

$$x^u = 256$$

Taking log to the base 2 on both the sides,

$$u * \log_{2}{x} = \log_{2}{256}$$

=>$$[({\log_2 x})^2 - 6 {\log_2 x} + 12] * \log_{2}{x} = 8$$

$$(log_2 x)^3 - 6(log_2 x)^2 + 12log_2 x = 8$$

Let $$log_2 x = t$$

$$t^3 - 6t^2 +12t - 8 = 0$$

$$(t-2)^3 = 0$$

Therefore, $$log_2 x = 2$$

=> $$x = 4$$ is the only solution

Hence, option B is the correct answer.

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?

Video Solution

Solution

Consider side of the cube as x.

So diagonal will be of length $$\sqrt{3}$$ * x.

Now if diagonals are side of equilateral triangle we get area = 3*$$\sqrt{3}*x^2$$ /4 .

Also in a triangle

4 * Area * R = Product of sides

4* 3*$$\sqrt{3}*x^2$$ /4 * R = .3*$$\sqrt{3}*x^3$$

R = x

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?

Video Solution

Solution

The person has to take 3 steps north and 4 steps west, in whatever way he travels.

Total steps = 7, 3 north and 4 west.

Number of ways = 7!/(4!3!) = 35

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?

Video Solution

Solution

In the triangle BEO, BE = $$\sqrt{4^2-2^2}$$ = $$2\sqrt3$$

==> BC = $$4\sqrt3$$

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?

Video Solution

Solution

Using pythagoras we can find diagonal of small square shown in fig. = 2*$$\sqrt2$$ . Now this is equal to 2 + r + $$\sqrt2$$ *r. Equating we get r = $$6- 4 \sqrt{2}$$

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?

Video Solution

Solution

If EBC = 65 then EOC = 130 then OEC = OCE = 25. NOw since OC and ED are parallel we have OCE = OED = 25. Hence option D.

Question 30

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

Video Solution

Solution

The remainder when $$15^{23}$$ is divided by 19 equals $$(-4)^{23}$$
The remainder when $$23^{23}$$ is divided by 19 equals $$4^{23}$$
So, the sum of the two equals$$(-4)^{23}+(4)^{23}=0$$

CAT Content

CAT 2004 Question Paper

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 $$j\ge3$$. What is the sum of the first 6002 terms of this sequence?

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

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?

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?

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?

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?

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?

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

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CAT Content

CAT 2004 Question Paper

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 $$j\ge3$$. What is the sum of the first 6002 terms of this sequence?

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

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?

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?

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?

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?

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?

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

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