Instructions

For the following questions answer them individually

Question 141

Each of the numbers $$x_1, x_2, ... ,x_n$$ $$(n > 4)$$, is equal to 1 or -1.
Suppose, $$x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + ....... + x_{n-3}x_{n-2} x_{n-1}x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2x_3$$ = 0, then:

Video Solution

Solution

Since each term is either 1 or -1 . To be 0 we should have even terms and with n=even , no. of terms is even .

Question 142

The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.

If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group?

Video Solution

Solution

64.75% population is below 35 years age.

The number of people aged below 35 years is 400 million.

So total number of people are, suppose x = $$\frac{400*100}{64.75}$$

Now below 15 years of age we have f/m = 0.96 so f = (m+f)*0.96/1.96 and m+f = 30*400/64.75

Solving we get, f = 30*400*0.96/(64.75*1.96) = 90.8

Hence option B .

Question 143

Sam has forgotten his friend’s seven-digit telephone number. He remembers the following:
a) the first three digits are either 635 or 674,
b) the number is odd, and
c) the number nine appears once.
If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?

Video Solution

Solution

Consider cases : 1) Last digit is 9: No. of ways in which the first 3 digits can be guessed is 2. No. of ways in which next 3 digits can be guessed is 9*9*9. So in total the number of ways of guessing = 2*9*9*9 = 1458.

2) Last digit is not 9: the number 9 can occupy any of the given position 4, 5, or 6, and there shall be an odd number at position 7.

So in total, the number of guesses = 2*3*(9*9*4) = 1944+1458 = 3402

Question 144

Let N = $$55^3 + 17^3 - 72^3$$. N is divisible by:

Video Solution

Solution

$$55^3 + 17^3 - 72^3$$ = $$(55-72)k + 17^3$$. This is divisible by 17

Remainder when $$55^3$$ is divided by 3 = 1

Remainder when $$17^3$$ is divided by 3 = -1

Remainder when $$72^3$$ is divided by 3 = 0

So, $$55^3 + 17^3 - 72^3$$ is divisible by 3

So, the answer is d) 3 and 17

Question 145

If $$x^2 + y^2 = 0.1$$ and |x-y|=0.2, then |x|+|y| is equal to:

Video Solution

Solution

$$(x - y)^2 = x^2 + y^2 - 2xy$$

$$0.04 = 0.1 - 2xy => xy = 0.03$$

So, |xy| = 0.03

$$(|x| + |y|)^2 = x^2 + y^2 + 2|xy| = 0.1 + 0.06 = 0.16$$

So, |x|+|y| = 0.4

Question 146

ABCD is a rhombus with the diagonals AC and BD intersection at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

Video Solution

Solution

The line should be parallel to AD and should be of equal distance from the origin in the third quadrant. The equation x+y = -1 satisfies all these conditions.

Question 147

Consider a circle with unit radius. There are 7 adjacent sectors, S1, S2, S3,....., S7 in the circle such that their total area is (1/8)th of the area of the circle. Further, the area of the $$j^{th}$$ sector is twice that of the $$(j-1)^{th}$$ sector, for j=2, ...... 7. What is the angle, in radians, subtended by the arc of S1 at the centre of the circle?

Video Solution

Solution

Now area of 1st sector = $$\pi * r^2 * \frac{x}{360}$$ where x - angle subtended at center

Now the next sector will have 2x as the angle, and similarly angles will be in GP with ratio = 2.

Sum of areas of all 7 sectors = $$\frac{127*x* \pi * r^2}{360}$$ which is equal to $$\frac{\pi * r^2}{8}$$

We get x = $$\frac{360}{8*127}$$

Now if converted in radians we get x = $$\pi/508$$.

Question 148

There is a vertical stack of books marked 1, 2 and 3 on Table-A, with 1 at the bottom and 3 on top. These are to be placed vertically on Table-B with 1 at the bottom and 2 on the top, by making a series of moves from one table to the other. During a move, the topmost book, or the topmost two books, or all the three, can be moved from one of the tables to the other. If there are any books on the other table, the stack being transferred should be placed, on top of the existing books, without changing the order of books in the stack that is being moved in that move. If there are no books on the other table, the stack is simply placed on the other table without disturbing the order of books in it. What is the minimum number of moves in which the above task can be accomplished?

Video Solution

Solution

For min steps 1st move would be to place book 3 from A to B

2nd move would be to place book 2 from A to B,

3rd move would be to place book 2 and 3 from B to A.

Last move would be to place books 2,3,1 from table A to Table B.

So 4 moves are atleast needed.

Question 149

The area bounded by the three curves |x+y| = 1, |x| = 1, and |y| = 1, is equal to:

Video Solution

Solution

|x| = 1 and |y| = 1 form a square of area = 2*2 = 4 sq units

|x+y| = 1 forms a set of parallel lines cutting the axes at (1,0), (0,1), (-1,0) and (0,-1). The graph is as shown:

The area bounded by the three curves is 2*2 - 1/2*1*1*2 = 4 - 1 = 3 sq units

Question 150

If the equation $$x^3 - ax^2 + bx - a = 0$$ has three real roots, then it must be the case that,

Video Solution

Solution

It can be clearly seen that if b=1 then $$x^2(x - a) + (x - a) = 0$$ an the equation gives only 1 real value of x

CAT Content

CAT 2000 Question Paper

Each of the numbers $$x_1, x_2, ... ,x_n$$ $$(n > 4)$$, is equal to 1 or -1.
Suppose, $$x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + ....... + x_{n-3}x_{n-2} x_{n-1}x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2x_3$$ = 0, then:

The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.

If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group?

Let N = $$55^3 + 17^3 - 72^3$$. N is divisible by:

If $$x^2 + y^2 = 0.1$$ and |x-y|=0.2, then |x|+|y| is equal to:

ABCD is a rhombus with the diagonals AC and BD intersection at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

The area bounded by the three curves |x+y| = 1, |x| = 1, and |y| = 1, is equal to:

If the equation $$x^3 - ax^2 + bx - a = 0$$ has three real roots, then it must be the case that,

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CAT Content

CAT 2000 Question Paper

Each of the numbers $$x_1, x_2, ... ,x_n$$ $$(n > 4)$$, is equal to 1 or -1. Suppose, $$x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + ....... + x_{n-3}x_{n-2} x_{n-1}x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2x_3$$ = 0, then:

The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million. If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group?

Let N = $$55^3 + 17^3 - 72^3$$. N is divisible by:

If $$x^2 + y^2 = 0.1$$ and |x-y|=0.2, then |x|+|y| is equal to:

ABCD is a rhombus with the diagonals AC and BD intersection at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

The area bounded by the three curves |x+y| = 1, |x| = 1, and |y| = 1, is equal to:

If the equation $$x^3 - ax^2 + bx - a = 0$$ has three real roots, then it must be the case that,

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Also Read

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Each of the numbers $$x_1, x_2, ... ,x_n$$ $$(n > 4)$$, is equal to 1 or -1.
Suppose, $$x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + ....... + x_{n-3}x_{n-2} x_{n-1}x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2x_3$$ = 0, then:

The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.

If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group?