Instructions

Directions for the next 3 questions:

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:

a) if f(x)=3f(-x)

b) if f(x)= -f(-x)

c) if f(x) = f(-x)

d) if 3f(x) = 6f(-x), for x >= 0

Question 121

Video Solution

Solution

In this graph, f(-1) = 1 and f(1) = 2 and since the two lines pass origin, their values increase linearly.

3f(x) = 6f(-x)

Question 122

Video Solution

Solution

In this graph, we can see that the graph on LHS of y axis is symmetrical to the graph on RHS of y axis about origin.

Hence f(x) = -f(-x)

Instructions

Directions for the next 2 questions: There are three bottles of water, A, B, C, whose capacities are 5 litres, 3 litres, and 2 litres respectively. For transferring water from one bottle to another and to drain out the bottles, there exists a piping system. The flow through these pipes is computer controlled. The computer that controls the flow through these pipes can be fed with three types of instructions, as explained below:

Initially, A is full with water, and B and C are empty.

Question 123

After executing a sequence of three instructions, bottle A contains one litre of water. The first and the third of these instructions are shown below:
First instruction: FILL (C, A)
Third instruction: FILL (C, A)
Then which of the following statements about the instruction is true?

Video Solution

Solution

After first instruction: A = 5-2 = 3L. C = 2L. In the third instruction, A yields 2L to C. It means the capacity of A before third operation was 3L. It means that C's water was either drained or transferred to B. Option B is correct.

Question 124

Consider the same sequence of three instructions ‘and the same initial state mentioned above. Three more instructions are added at the end of the above sequence to have A contain 4 litres of water. In this total sequence of six instructions, the fourth one is DRAIN (A). This is the only DRAIN instruction in the entire sequence. At the end of the execution of the above sequence, how much water (in litres) is contained in C?

Video Solution

Solution

We know that after 3 operations A contain 1 ltr and 4 th operation is Drain(a) , so 1 ltr is drained and only 4 ltrs remain in all there cylinders. Also after 6 operations A should have all the 4 ltrs in it. hence the other 2 vessels must contain 0 ltrs because there is only one drain operation in the entire sequence.

Instructions

Directions for the next 2 questions:

For a real number x, let

$$f(x) = 1/(1+x),$$ if $$x$$ is non-negative
$$f(x) = 1+x,$$ if $$x$$ is negative

$$f^n(x) = f(f^{n-1}(x)), n = 2, 3.....$$

Question 125

What is the value of the product, $$f(2) f^2(2)f^3(2) f^4(2)f^5(2)$$?

Video Solution

Solution

f(2) = 1/3

$$f^2(2)$$ = f(1/3) = 3/4

$$f^3(2)$$ = f(3/4) = 4/7

$$f^4(2)$$ = f(4/7) = 7/11

$$f^5(2)$$ = f(7/11) = 11/18

So, product = 1/18

Question 126

r is an integer 2. Then, what is the value of $$f^{r-1}(-r) + f^r(-r) + f^{r+1}(-r)$$?

Video Solution

Solution

f(-2) = -1

$$f^2(-2)$$ = f(f(-2)) = f(-1) = 0

$$f^3(-2)$$ = f(f(-2)) = f(0) = 1

So, sum = 0

Instructions

For the following questions answer them individually

Question 127

Let D be recurring decimal of the form, $$D = 0.a_1a_2a_1a_2a_1a_2...$$, where digits $$a_1$$ and $$a_2$$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?

Video Solution

Solution

Case 1: $$a_1=0$$
So, D equals $$0.0a_20a_20a_2...$$
So, 100D equals $$a_2.0a_20a_2...$$
So, 99D equals $$a_2$$

Case 2: $$a_2=0$$
So, D equals $$0.a_10a_10a_1...$$
So, 100D equals $$a_10.a_10a_1....$$
So, 99D equals $$a_10$$

So, in both the cases, 99D is an integer. From the given options, only option C satisfies this condition (198=2*99) and hence the correct answer is C.

Question 128

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

Video Solution

Solution

y=a+bx is a linear function and clearly the function shown in the table is not linear.
Lets take option B:

Suppose $$y=f(x)=a+bx+cx^2$$
f(1)=4 => a+b+c=4 -- (1)
f(2)=8 => a+2b+4c=8 -- (2)
f(3)=14 => a+3b+9c=14 -- (3)

Let us solve the equations and see if $$f(4), f(5),f(6)$$ also satisfy the given equation.

(2)-(1) => b+3c=4
(3)-(2) => b+5c=6
This gives c=1 and b=1 ==> a=2

So, $$f(x)=2+x+x^2$$
$$f(4)=2+4+4^2 = 22$$
$$f(5)=2+5+5^2 = 32$$
$$f(6)=2+6+6^2 = 44$$

As all the three also satisfy the numbers given in the table, it can be inferred that the relationship between x and y is quadratic and the correct option is option (b)

Question 129

If $$a_1 = 1$$ and $$a_{n+1} = 2a_n +5$$, n=1,2,....,then $$a_{100}$$ is equal to:

Video Solution

Solution

$$a_2 = 2*1 + 5$$
$$a_3 = 2*(2 + 5) + 5 = 2^2 + 5*2 + 5$$
$$a_4 = 2^3 + 5*2^2 + 5*2 + 5$$
...
$$a_{100} = 2^{99} + 5*(2^{98} + 2^{97} + ... + 1)$$
$$= 2^{99} + 5*1*(2^{99} - 1)/(2-1) = 2^{99} + 5*2^{99} - 5 = 6*2^{99} - 5$$

Question 130

What is the value of the following expression?

$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$

Video Solution

Solution

$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$ = 1/[(2+1)*(2-1)] + 1/[(4+1)*(4-1)] + ... + 1/[(20+1)*(20-1)]

= 1/(1*3) + 1/(3*5) + 1/(5*7) + ... + 1/(19*21)

=1/2 * ( 1/1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... +1/19 - 1/21)

=1/2 * (1 - 1/21) = 10/21

CAT Content

CAT 2000 Question Paper

After executing a sequence of three instructions, bottle A contains one litre of water. The first and the third of these instructions are shown below:
First instruction: FILL (C, A)
Third instruction: FILL (C, A)
Then which of the following statements about the instruction is true?

What is the value of the product, $$f(2) f^2(2)f^3(2) f^4(2)f^5(2)$$?

r is an integer 2. Then, what is the value of $$f^{r-1}(-r) + f^r(-r) + f^{r+1}(-r)$$?

Let D be recurring decimal of the form, $$D = 0.a_1a_2a_1a_2a_1a_2...$$, where digits $$a_1$$ and $$a_2$$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

If $$a_1 = 1$$ and $$a_{n+1} = 2a_n +5$$, n=1,2,....,then $$a_{100}$$ is equal to:

What is the value of the following expression?

$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$

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CAT Content

CAT 2000 Question Paper

After executing a sequence of three instructions, bottle A contains one litre of water. The first and the third of these instructions are shown below: First instruction: FILL (C, A) Third instruction: FILL (C, A) Then which of the following statements about the instruction is true?

What is the value of the product, $$f(2) f^2(2)f^3(2) f^4(2)f^5(2)$$?

r is an integer 2. Then, what is the value of $$f^{r-1}(-r) + f^r(-r) + f^{r+1}(-r)$$?

Let D be recurring decimal of the form, $$D = 0.a_1a_2a_1a_2a_1a_2...$$, where digits $$a_1$$ and $$a_2$$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

If $$a_1 = 1$$ and $$a_{n+1} = 2a_n +5$$, n=1,2,....,then $$a_{100}$$ is equal to:

What is the value of the following expression? $$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$

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After executing a sequence of three instructions, bottle A contains one litre of water. The first and the third of these instructions are shown below:
First instruction: FILL (C, A)
Third instruction: FILL (C, A)
Then which of the following statements about the instruction is true?

What is the value of the following expression?

$$(1/(2^2-1))+(1/(4^2-1))+(1/(6^2-1))+...+(1/(20^2-1)$$