Instructions

Directions for the following three questions: Answer the questions on the basis of the information given below

In the adjoining figure I and II are circles with centres P and Q respectively, The two circles touch each other and have common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the ratio 4: 3. It is also known that the length of PO is 28 cm.

Question 1

What is the ratio of the length of PQ to that of QO?

[CAT 2004]

Video Solution

Solution

Triangles PRO and QSO are similar.

PR/QS = 4/3.

Therefore, PO/QO = 4/3 =>

Let the radius of 2 circles be 4R and 3R respectively.

28/(28-7R) = 4/3

4-R = 3

R = 1

Required ratio of PQ/QO = (4R+3R)/(28-7R) = 7R/(28-7R) = 7/21 = 1:3

Question 2

What is the radius of the circle II?
[CAT 2004]

Video Solution

Solution

Let the radius of circles I and II be 4R and 3R respectively.

Triangles PRO and QSO are similar.

PR/QS = 4/3.

Therefore, PO/QO = PR/QS

=> PO/QO = 4/3

=> 28/(28-7R) = 4/3

=> 4-R = 3

=> R = 1

Radius of smaller circle = 3R = 3

Question 3

The length of SO is

[CAT 2004]

Video Solution

Solution

Triangles PRO and QSO are similar.

PR/QS = 4/3.

Therefore, PO/QO = 4/3 =>

Let the radius of 2 circles be 4R and 3R respectively.

28/(28-7R) = 4/3

4-R = 3

R = 1

Therefore, SO = $$\sqrt{21^2 - 3^2}$$ = $$\sqrt{432}$$ = $$12 \sqrt{3}$$ cm

Instructions

Directions for the following two questions:

Answer the questions on the basis of the information given below.

$$f_1(x) = x$$ if $$0 \leq x \leq 1$$ $$f_1(x) = 1$$ if x >= 1 $$f_1(x) = 0$$ otherwise

$$f_2(x) = f_1(-x)$$ for all x

$$f_3(x) = -f_2(x)$$ for all x

$$f_4(x) = f_3(-x)$$ for all x

Question 4

How many of the following products are necessarily zero for every x:

$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$

Video Solution

Solution

Checking for different values of x . Suppose x= -0.5 we get

$$f_1(x)f_2(x) = 0*0.5 = 0$$

$$f_2(x)f_4(x) = 0.5*0 = 0$$ .

But $$f_2(x)f_3(x)$$ is not equal to zero.

Hence two functions are necessarily equal to zero and two products given above are equal to zero.

Question 5

Which of the following is necessarily true?

Video Solution

Solution

Relation between f3 and f1 would be

$$f_3(x) = -f_1(-x)$$.

Put x= -x we get

$$f_3(-x) = -f_1(x)$$ so multiply by -1 we get

$$-f_3(-x) = f_1(x)$$.

Instructions

Directions for the following two questions: Answer the questions on the basis of the information given below.

In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.

Question 6

If group B contains 23 questions, then how many questions are there in group C?

Video Solution

Solution

Group B contains 23 questions => Marks of group B = 46
Let the number of questions in A be x and in C be 77-x.
Marks of group A = x
So, x/(x+46+3*77-3x) >= 60%
=> 5x >= 3(277-2x)
=> 11x >= 831
=> x >= 75.54
=> x = 76 (min)
So, the possible number of questions in group C = 1.

Question 7

If group C contains 8 questions and group B carries at least 20% of the total marks, which of the following best describes the number of questions in group B?

Video Solution

Solution

Let the number of questions in group B be x.
So, number of questions in group A = 92-x
Marks of group B = 2x
2x/(92-x+2x+24) >= 20%
=> 10x >= 116+x
=> 9x >= 116
=> x >= 12.88
From the options, x can be 13 or 14

Instructions

For the following questions answer them individually

Question 8

Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide.

Video Solution

Solution

The relative speed is 15 km/hr = 15 km/60 min = 0.25 km/min = 250 m/min.
Therefore, one minute before they collide, they are at a distance of 250m.

Question 9

A rectangular sheet of paper, when halved by folding it at the mid point of its longer side, results in a rectangle, whose longer and shorter sides are in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is 2, what is the area of the smaller rectangle?

[CAT 2004]

Video Solution

Solution

Let the longer side of the original rectangle be x. Therefore, breadth of the modified rectangle = x/2.
=> x:2 = 2:(x/2)
=> $$x^2$$ = 8
Area of smaller rectangle = (x/2)*2 = x = $$2 \sqrt{2}$$

Question 10

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

Video Solution

Solution

Sum of the first 11 terms = 11/2 ( 2a+10d)

Sum of the first 19 terms = 19/2 (2a+18d)

=> 22a+110d = 38a+342d => 16a = -232d

=> 2a = -232/8 d = -29d

Sum of the first 30 terms = 15(2a+29d) = 0

CAT Content

CAT 2004 Question Paper

What is the ratio of the length of PQ to that of QO?

[CAT 2004]

What is the radius of the circle II?
[CAT 2004]

The length of SO is

[CAT 2004]

How many of the following products are necessarily zero for every x:

$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$

Which of the following is necessarily true?

If group B contains 23 questions, then how many questions are there in group C?

If group C contains 8 questions and group B carries at least 20% of the total marks, which of the following best describes the number of questions in group B?

Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide.

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

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CAT Content

CAT 2004 Question Paper

What is the ratio of the length of PQ to that of QO? [CAT 2004]

What is the radius of the circle II?[CAT 2004]

The length of SO is [CAT 2004]

How many of the following products are necessarily zero for every x: $$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$

Which of the following is necessarily true?

If group B contains 23 questions, then how many questions are there in group C?

If group C contains 8 questions and group B carries at least 20% of the total marks, which of the following best describes the number of questions in group B?

Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide.

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

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CAT Quant Questions | CAT Quantitative Ability

CAT DILR Questions | LRDI Questions For CAT

CAT Verbal Ability Questions | VARC Questions For CAT

Also Read

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What is the ratio of the length of PQ to that of QO?

[CAT 2004]

What is the radius of the circle II?
[CAT 2004]

The length of SO is

[CAT 2004]

How many of the following products are necessarily zero for every x:

$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$