## Chapter 3: LINEAR EQUATIONS

### Introduction

•  (a + b)2 = a2 + 2 ab + b2
• (a - b)2 = a2 - 2ab + b2
• (a + b)3 = a3 + 3a2b + 3ab2 + b3
• (a - b)3 = a3 - 3a2b + 3 ab2 - b3
• a2 - b2 = (a - b) (a + b)
• a3 - b3 = (a - b)3 + 3 a b (a - b)

The above formulae can be used for solving giant multiplication problems.

1. 1605 * 1605 = (1605)^2 = (1600 + 5)^2 which is (a+b)^2.

### Find units digits of large exponents

Sometimes we have to calculate the powers of very large numbers like 13^35 which looks impossible but the answer is quite simple .. Recognize the cycle.

Suppose we need to find last digit of 2^ 2012

2^1 = 2

2^2= 4

2^3 = 8

2^4 = 16 = 6

2^ 5 = 32 = 2

2^ 6 = 64 =4

Hence there is a cycle of size 4 like 2-4-8-6-2-4-8-6... so all powers of 2 in multiples of 4 like 4,8,12,16... have last digit 6 since 2012 also satisfies this 2^2012  has last digit 6.

So generalizing for k > 4 if 2^k = 2^(4n+1) then last digit is 2, if 2^(4n+2) = 4, 2^(4n+3) = 8 , 2^(4n) = 6.

Similarly we can solve for all digits:

For 3 cycle is: 3-9-7-1-3-9-7-1... here too the cycle is 4 and  So generalizing for k > 4 if 3^k = 3^(4n+1) then last digit is 2, if 3^(4n+2) = 4, 3^(4n+3) = 8 , 3^(4n) = 6.

For 4 cycle is: 4-6-4-6-4-6-... hence cycle is 2 So generalizing for k > 3 if 4^k = 4^(2n+1) then last digit is 6, if 4^(2n) = 4.

Similarly we can obtain cycles for all digits till 9 any digit above this doesn't matter as 13^n is same as 3^n and 17^n is same as 7^n etc.

### Division problems

Dividend - Remainder = Divisor * Quotient.

Q. A number when divided by 342 gives a remainder 47 when the same number is divided by 19 what would be the remainder?

A.

Step 1: (number - 19) = 342 * k + 47 i.e.  number = 19*18k + 19*2+9 = 19(18k+2)+9

if this number is divided by 19 then it gives 18k+2 as quotient and 9 as remainder.

### Problems on ages

Here ages of two people shall be given as a ratio. Then their ages before x years or after x years shall be given. This ratio would be equal to a value. The question would then be to calculate their present ages.

Methodology: Assume age of the lesser entity as 'x'. The older entity become 'kx'. Then their ages 'b' years ago would be 'x-b' and 'kx-b' respectively. This would be equal to some value say 10.

Obtaining the linear equation: (x-b) + (kx-b) = 10; When we solve this we get value of x.

Q1. One year ago Jaya was four times as old as her daughter.  Six years hence, Jaya's age will exceed her daughter's age by 9 years. The ratio of the present ages of Jaya and her daughter is :
Ans: daughter's age 1 yr ago= x , Jaya's age 1 year ago = 4x
6 years hence:  (4x+7) - (x+7) = 9 , we solve this to get x;

Q2. Five years ago, the total of the ages of a father and his son was 40 years. The ratio of their present ages is 4 : 1. What is the present age of the father ?
Ans.
Present age: son = x , father = 4x;
Five years ago : sons age = x-5 , fathers age = 4x-5;
linear equation: (x-5)+(4x-5)=40;

Q. One year ago, the ratio of Gaurav’s and Sachin’s age was 6: 7 respectively. Four years hence, this ratio would become 7: 8. How old is Sachin ?
A. let the age be 'x' and 'y' respectively. x : y = 6 : 7 so 7x - 6y = 0 and
(x+4) : (y+4) = 7 : 8 so 8x+32 = 7y+28 and we get 8x-7y=-4 solving both we get x = 24, y=28.

Q. Abhay’s age after six years will be three-seventh of his fathers age. Ten years ago the ratio of their ages was 1 : 5. What is Abhay’s father's age at present?
A. Abhay's age ten years ago = x, so his father is 5x. After 6 yrs from present Abhay shall be x+16 and his father shall be 5x+16 and we know that
(x+16) = 3/7 * (5x+16)

### Linear equations

Q. A man divides Rs. 8600 amongst 5 sons, 4 daughters and 2 nephews. If each daughter gets 4 times as much as nephews and each son gets 5 times as much as nephews. how much does each daughter get?
A. Let x be the share of each nephew so one daughter get 4x and son 5x.
5*(5x) + 4(4x) + 2x = 8600

Q. A man  spends 2/5 of his  salary on house rent, 3/10 of his  salary on  food and 1/8 of his salary  on  conveyance. if  he  has  Rs.1400  left  with  him, find  his  expenditure  on  food  and conveyance.
A. Salary = x ; x = 1400 + 0.4x + 0.3x+ o.125x

Q. A third of A marks in mathematics exceeds a half of his marks in English by 80. if he got 240 marks In two subjects together how many marks did he get in English?
A. x = marks in maths, y=marks in english. (x/3) - (y/2) = 80 and x + y = 240
Solving both equations gives the result.

Q. A  train  starts  full of passengers at  the  first  station  it drops 1/3 of  the passengers and takes 280 more at the second station it drops one half the new total and takes twelve more . On  arriving  at  the  third  station  it  is  found  to  have  248  passengers.  Find  the  no  of passengers in the beginning?
A. total capacity = x;
at first station : x- (x/3) + 280
at second station:  1/2{(2x/3)+280}+12
at third station: 248 = 1/2{(2x/3)+280}+12

Q. in a  office 1/3 of the workers are women ½ of the women are married and 1/3 of the married women have children if ¾ of the men are married and 2/3 of the married men have children what part of workers are without children?
A. Total workers be x. women = x/3, men = 2x/3, married women = x/3 * 1/2 = x/6;  women with children = x/6*1/3 = x/18; married men = 2x/3 * (3/4) = x/2; men with children = 2x/6 = x/3.
workers without children = x - (x/18+x/3)

Q. Village  X  has  a  population  of  68000,which  is  decreasing  at  the  rate  of  1200  per year. Village Y has a population  of 42000, which is increasing at  the  rate of 800 per year  .in how many years will  the population of  the  two villages be equal?
A. Assume 'x' as the number of years after which both populations shall be equal. So we get linear equation as: 68000 - 1200x = 42000 + 800x

Q. An employer pays Rs.20 for each day a worker works and for forfeits Rs.3 for each day is idle  at  the  end  of  sixty days  a worker  gets Rs.280  .  for how many days did  the worker remain ideal?
A. worker has 'x' working days and 'y' idle days so
20 * x - 3 * y = 280
x + y = 60

Q. When an amount was distributed among 14 boys, each of them got rs 80 more than the amount received by each boy when the same amount is distributed equally among 18 boys. What was the amount?
A. Assume total amount be 'x'.
So when 'x' is given to 14 boys each gets 'x/14' and when divided amongst 18 boys each gets 'x/18' and we know that x/14 - x/18 = 80

Q. if f(x) = log (1+x/1-x) then f(x) + f(y) is

1. f(x + y)

2. $f\frac{x+y}{1+xy}$

3. $(x+y) f \frac{1}{1+xy}$

4. $\frac{f(x) + f(y)}{1+xy}$

Ans . B

1. $f(x) + f(y) = log \frac{1+x}{1-x}$ + $log \frac{1+y}{1-y}$

2. $log \frac{(1+x)(1+y)}{(1-x)(1-y)}$

3. $log \frac{(1+x+y+xy)}{(1-x-y+xy)}$

4. $log \frac{(1+x+y+xy)}{(1+xy-(x+y))}$

5. $log \frac{1+\frac{x+y}{1+xy}}{1-\frac{x+y}{1+xy}}$

6. $f \frac{x+y}{1+xy}$

Q. Number S is obtained by squaring the sum of digits of a two-digit number D. If difference between S and D is 27, then the two-digit number D is

1. 24

2. 54

3. 34

4. 45

Ans . B

1. check choices and only choice B satisfies the condition.

2. S = (5+4)^2 = 81

3. D – S = 81 – 54 = 27. Hence, the number = 54

Q. The nth element of a series is represented as $X_n = (-1)^n * X_{n-1}$ . If $X_0$ = x and x > 0, then which of the following is always true?

1. $X_n$ is positive if n is even

2. $X_n$ is positive if n is odd

3. $X_n$ is negative if n is even

4. None of these

Ans . D

1. $x_0 = x, x_1 = -x, x_2 = -x$

2. $x_3 = x, x_4 = x$

3. $x_5 = -x, x_6 = -x$

4. Choices (1), (2), (3) are incorrect.

Q. If x, y and z are real numbers such that x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?

1. 5/3

2. √ 19

3. 13/3

4. none

Ans . C

1. $xy + yz + zx = 3$

2. $xy + (y + x)z = 3$

3. $xy + (y + x)(5 + x + y)= 3$

4. $xy + 5y+xy+y^2+5x+x^2+xy = 3$

5. As it is given that y is a real number, the discriminant for above equation must be greater than or equal to zero. $(x - 5)^2 - 4(x^2 - 5x + 3) \geq 0$

6. $3x^2 - 10x + 13 \geq 0$

7. $x = 1, \frac{13}{3}$

8. Largest value that x can have is $\frac{13}{3}$

Q. Neeraj has agreed to mow a lawn, which is a 20 m × 40 m rectangle. He mows it with 1 m wide strip. If Neeraj starts at one corner and mows around the lawn toward the centre, about how many times would he go round before he has mowed half the lawn?

1. 2.5

2. 3.5

3. 3.8

4. 4

Ans . C

1. Area = 40 * 20 = 800 $m^2$

2. If 3 rounds are done, area = 34 × 14 = 476 $m^2$ ⇒ Area > 3 rounds

3. if 4 rounds ⇒ Area left = 32 × 12 = 347 $m^2$

4. Hence, area should be slightly less than 4 rounds.

Q. The owner of a local jewellery store hired three watchmen to guard his diamonds, but a thief still got in and stole some diamonds. On the way out, the thief met each watchman, one at a time. To each he gave 1/2 of the diamonds he had then, and 2 more besides. He escaped with one diamond. How many did he steal originally?

1. 40

2. 36

3. 25

4. None of these

Ans . B

1. Since thief escaped with 1 diamond

2. Before 3 rd watchman he had (1 + 2) × 2 = 6 diamonds.

3. Before 2 nd watchman he had (6 + 2) × 2 = 16 diamonds

4. Before 1 st watchman he had (16 + 2) × 2 = 36 diamonds.

Q. Mayank, Mirza, Little and Jaspal bought a motorbike for Rs.60. Mayank paid one-half of the sum of the amounts paid by the other boys. Mirza paid one-third of the sum of the amounts paid by the other boys. Little paid one-fourth of the sum of the amounts paid by the other boys. How much did Jaspal have to pay?

1. Rs.15

2. Rs.13

3. Rs.17

4. None of these

Ans . B

1. Mayank paid 1/2 of the sum paid by other three.

2. Mayank paid 1/3 rd of the total amount = Rs.20

3. Similarly, Mirza paid Rs.15 and Little paid Rs.12. Remaining amount of Rs.60 – Rs.20 – Rs.15 – Rs.12 = Rs.13 is paid by Jaspal

Q. A rich merchant had collected many gold coins. He did not want anybody to know about him. One day, his wife asked, " How many gold coins do we have?" After a brief pause, he replied, "Well! if I divide the coins into two unequal numbers, then 48 times the difference between the two numbers equals the difference between the squares of the two numbers." The wife looked puzzled. Can you help the merchant's wife in finding out how many gold coins the merchant has?

1. 96

2. 53

3. 43

4. None of these

Ans . D

1. Let the number of gold coins = x + y

2. 48(x – y) = x2 - y2

3. 48(x – y) = (x – y)(x + y) ⇒ x + y = 48

4. Hence, the correct choice will be none of these.

Q. A car rental agency has the following terms. If a car is rented for 5 hr or less, then, the charge is Rs. 60 per hour or Rs. 12 per kilometre whichever is more. On the other hand, if the car is rented for more than 5 hr, the charge is Rs. 50 per hour or Rs. 7.50 per kilometre whichever is more. Akil rented a car from this agency, drove it for 30 km and ended up playing Rs. 300. For how many hours did he rent the car?

1. 4 hr

2. 5 hr

3. 6 hr

4. None of these

Ans . C

1. By trial and error:

2. 30 × 12 = 360 > 300

3. 30 × 7.5 = 225 < 300

4. 50 × 6 = 300. Hence, he rented the car for 6 hr.

Q. The number of non-negative real roots of 2x – x – 1 = 0 equals

1. 0

2. 1

3. 2

4. 3

Ans . C

1. 2x – x – 1 = 0 and so 2x – 1 = x

2. If we put x = 0, then this is satisfied and if we put x = 1, then also this is satisfied.

3. Now, if we put x = 2, the equation this is not valid.

Q. When the curves y = log10x , y=x-1 are drawn in the x-y plane, how many times do they intersect for values x ≥ 1 ?

1. Never

2. Once

3. Twice

4. More than twice

Ans . B

1. For the curves to intersect, log x = x

2. Thus log10x = 1/x or xx=10

3. This is possible for only one value of x such that 2 < x < 3.

Q. Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equations has at least one solution, such that p + q + r ≠ 0?
x+ 2y – 3z = p
2x + 6y – 11z = q
x – 2y + 7z = r

1. 5p –2q – r = 0

2. 5p + 2q + r = 0

3. 5p + 2q – r = 0

4. 5p – 2q + r = 0

Ans . A

1. It is given that p+q+r≠0 , if we consider the first option, and multiply the first equation by 5, second by –2 and third by –1, we see that the coefficients of x, y and z all add up-to zero.

2. Thus, 5p – 2q – r = 0 No other option satisfies this.

Q. Let g(x) = max(5 – x, x + 2). The smallest possible value of g(x) is

1. 4.0

2. 4.5

3. 1.5

4. None of the above

Ans . D

1. We can see that x + 2 is an increasing function and 5 – x is a decreasing function. This system of equation will have smallest value at the point of intersection of the two. i.e. 5 – x = x + 2 or x = 1.5.

2. Thus smallest value of g(x) = 3.5

Q. The function f(x) = |x – 2| + |2.5 – x| + |3.6 – x|, where x is a real number, attains a minimum at

1. x = 2.3

2. x = 2.5

3. x = 2.7

4. None of the above

Ans . B

1. At x = 2, f(x) = 2.1

2. At x = 2.5, f(x) = 1.6

3. At x = 3.6, f(x) = 2.7.Hence, at x = 2.5, f(x) will be minimum.

Q. How many three digit positive integers, with digits x, y and z in the hundred's, ten's and unit's place respectively exist such that x < y, z < y and x ≠ 0?

1. 245

2. 285

3. 240

4. 320

Ans . C

1. If y = 2 (it cannot be 0 or 1), then x can take 1 value and z can take 2 values.

2. Thus with y = 2, a total of 1 × 2 = 2 numbers can be formed. With y = 3, 2 × 3 = 6 numbers can be formed. Similarly checking for all values of y from 2 to 9 and adding up we get the answer as 240.

### CAT Problems

Q.

1. 99/100
2. 1/100
3. 100/101
4. 101/102

Ans:c

Q.

1. 8/(1-x8)
2. 4x/(1+x2)
3. 4/(1-x6)
4. 4/(1+x4)

Ans:a

Q.Let a, b be any positive integers and x = 0 or 1, then

1. ax*b(1-x) = ax+(1-x)b
2. ax*b(1-x) = bx+(1-x)a
3. none of these
4. ax*b(1-x) = a(1-x)*bx

Ans:a

Q.If n is any positive integer, then (n3 – n) is divisible

1. Always by 12
2. Never by 12
3. Always by 6
4. Never by 6

Ans.c

Q.value of (1-d3)/(1-d)

1. > 1 if d > –1
2. > 3 if d > 1
3. > 2 if 0 <d < 0.5
4. < 2 if d < –2

Ans.b

Q.I brought 30 books on Mathematics, Physics, and Chemistry, priced at Rs.17, Rs.19, and Rs.23 per book respectively, for distribution among poor students of Standard X of a school. The physics books were more in number than the Mathematics books but less than the Chemistry books, the difference being more than one. The total cost amounted to Rs.620. How many books on Mathematics, Physics, and Chemistry could have been bought respectively?

1. 5, 8, 17
2. 5, 12, 13
3. 5, 10, 15
4. 5, 6, 19

Ans.c

Q.The last time Rahul bought Diwali cards, he found that the four types of cards that he liked were priced Rs.2.00, Rs.3.50, Rs.4.50 and Rs.5.00 each. As Rahul wanted 30 cards, he took five each of two kinds and ten each of the other two, putting down the exact number of 10 rupees notes on the payment counter. How many notes did Rahul give?

1. 8
2. 9
3. 10
4. 11

Ans.d

Q.A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?

1. y = (2x+3)/(3x+4)
2. y = (2x+3)/(3x-2)
3. y = (3x+4)/(4x-5)
4. y = none

Ans.b

Q.What is the value of k for which the following system of equations has no solution: 2x – 8y = 3 and kx + 4y = 10

1. -2
2. 1
3. -1
4. 2

Ans.c

Q.How many 3-digit even numbers can you form such that if one of the digits is 5 then the following digit must be 7?

1. 5
2. 405
3. 365
4. 495

Ans.a

Q.A lord got an order from a garment manufacturer for 480 Denim Shirts. He brought 12 sewing machines and appointed some expert tailors to do the job. However, many didn’t report to duty. As a result, each of those who did, had to stitch 32 more shirts than originally planned by Alord, with equal distribution of work. How many tailors had been appointed earlier and how many had not reported for work?

1. 12, 4
2. 10, 3
3. 10, 4
4. None of these

Ans.c

Q.Iqbal dealt some cards to Mushtaq and himself from a full pack of playing cards and laid the rest aside. Iqbal then said to Mushtaq. “If you give me a certain number of your cards, I will have four times as many cards as you will have. If I give you the same number of cards, I will have thrice as many cards as you will have “. Of the given choices, which could represent the number of cards with Iqbal?

1. 9
2. 31
3. 12
4. 35

Ans.b

Q.Three times the first of three consecutive odd positive integers is 3 more than twice the third. What is the third integer?

1. 15
2. 9
3. 11
4. 5

Ans.a

Q.273 – 2 72 – 2 71 is the same as

1. 2 69
2. 2 70
3. 2 71
4. 2 72

Ans.c

Q.The number of integers n satisfying –n + 2 ≥ 0 and 2n ≥ 4 is

1. 0
2. 1
3. 2
4. 3

Ans.b

Q.The sum of two integers is 10 and the sum of their reciprocals is 5/12. Then the larger of these integers is

1. 2
2. 4
3. 6
4. 8

Ans.c

Q.If y = f(x) and f(x) = (1–x) / (1 + x), which of the following is true?

1. f(2x) = f(x) – 1
2. x = f(2y) – 1
3. f(1/x) = f(x)
4. x = f(y)

Ans.d

Q.A player rolls a die and receives the same number of rupees as the number of dots on the face that turns up. What should the player pay for each roll if he wants to make a profit of one rupee per throw of the die in the long run?

1. Rs. 2.50
2. Rs. 2
3. Rs.3.50
4. Rs. 4

Ans.a

Q.Let Y = minimum of {(x + 2), (3 – x)}. What is the maximum value of Y for 0≤x≤1?

1. 1.0
2. 1.5
3. 3.1
4. 2.5

Ans.d

Q.x, y and z are three positive integers such that x > y > z. Which of the following is closest to the product xyz?

1. (x – 1)yz
2. x(y – 1)z
3. xy(z – 1)
4. x(y + 1)z

Ans.a

Q.What is the greatest power of 5 which can divide 80! exactly.

1. 16
2. 20
3. 19
4. None of these

Ans.c

Q.A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the digits of both the numbers and carried out the multiplication and found that the product was exactly the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in the description of the exercise. Which one is that?

1. 14, 22
2. 13, 62
3. 19, 33
4. 42, 28

Ans.b

Q.Find the minimum integral value of n such that the division 55n/124 leaves no remainder.

1. 124
2. 123
3. 31
4. 62

Ans.a

Q.Let k be a positive integer such that k + 4 is divisible by 7. Then the smallest positive integer n, greater than 2, such that k + 2n is divisible by 7 equals

1. 9
2. 7
3. 5
4. 3

Ans.a

Q.A calculator has two memory buttons, A and B. Value 1 is initially stored in both memory locations. The following sequence of steps is carried out five times:
multiply A to B
store the result in A
What is the value stored in memory location A after this procedure?

1. 120
2. 450
3. 720
4. 250

Ans.c

Q.A one rupee coin is placed on a table. The maximum number of similar one rupee coins which can be placed on the table, around it, with each one of them touching it and only two others is

1. 8
2. 6
3. 10
4. 4

Ans.b

Q.In Sivakasi, each boy’s quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?

1. 200
2. 150
3. 125
4. 175

Ans.b

Q.In a six-node network, two nodes are connected to all the other nodes. Of the remaining four, each is connected to four nodes. What is the total number of links in the network?

1. 13
2. 15
3. 7
4. 26

Ans.a

Q.If x is a positive integer such that 2x + 12 is perfectly divisible by x, then the number of possible values of x is

1. 2
2. 5
3. 6
4. 12

Ans.c

Q.A man starting at a point walks one km east, then two km north, then one km east, then one km north, then one km east and then one km north to arrive at the destination. What is the shortest distance from the starting point to the destination?

1. 2√2km
2. 7 km
3. 3√2km
4. 5 km

Ans.d

Q.An outgoing batch of students wants to gift PA system worth Rs.4200 to their school. If the teachers offer to pay 50% more than the students, and an external benefactor gives three times teachers’ contribution, how much should the teachers donate?

1. 600
2. 840
3. 900
4. 1200

Ans.c

Q.A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p2 – 1) is

1. never divisible by 6
2. always divisible by 6, and may or may not be divisible by 12.
3. always divisible by 12, and may or may not be divisible by 24.
4. always divisible by 24.

Ans.d

Q.If 8 + 12 = 2, 7 + 14 = 3, then 10 + 18 = ?

1. 10
2. 4
3. 6
4. 18

Ans.a

Q.What is the distance between the points A(3, 8) and B(–2, –7)?

1. 5√2
2. 5
3. 5√10
4. 10√ 2

Ans.c

Q.Two oranges, three bananas and four apples cost Rs.15. Three oranges, two bananas and one apple cost Rs 10. I bought 3 oranges, 3 bananas and 3 apples. How much did I pay?

1. 10
2. 8
3. 15
4. cant say

Ans.c

Q.From each of the two given numbers, half the smaller number is subtracted. Of the resulting numbers the larger one is three times as large as the smaller. What is the ratio of the two numbers?

1. 2 : 1
2. 3 : 1
3. 3 : 2
4. None

Ans.a

Q.Eighty five children went to an amusement park where they could ride on the merry – go round, roller coaster, and Ferris wheel. It was known that 20 of them took all three rides, and 55 of them took at least two of the three rides. Each ride cost Re.1, and the total receipt of the amusement park was Rs.145.How many children did not try any of the rides?

1. 5
2. 10
3. 15
4. 20

Ans.c

Q.Eighty five children went to an amusement park where they could ride on the merry – go round, roller coaster, and Ferris wheel. It was known that 20 of them took all three rides, and 55 of them took at least two of the three rides. Each ride cost Re.1, and the total receipt of the amusement park was Rs.145.How many children took exactly one ride?

1. 5
2. 10
3. 15
4. 20

Ans.c

Q.John bought five mangoes and ten oranges together for forty rupees. Subsequently, he returned one mango and got two oranges in exchange. The price of an orange would be

1. 1
2. 2
3. 3
4. 4

Ans.b

Q.The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is

1. 26
2. 18
3. 31
4. none

Ans.a

Q.let un+1 = 2un+1 (n=0,1,2) and u0=0. then u10 is nearest to

1. 1023
2. 2047
3. 4095
4. 8195

Ans.a

Q.Let x < 0.50, 0 < y < 1, z > 1. Given a set of numbers, the middle number, when they are arranged in ascending order, is called the median. So the median of the numbers x, y and z would be

1. less than one
2. between 0 and 1
3. greater than 1
4. cannot say

Ans.b

Q.If log7 log5 ( √x+5 + √x ) = 0 ; find x

1. 1
2. 0
3. 2
4. none

Ans.d

Q.if a+b+c=0 where a≠b≠c then a2/(2a2+bc) + b2/(2b2+ac) + c2/(2c2+ab) is equal to

1. zero
2. 1
3. -1
4. abc

Ans.b

Q.If one root of x2+px+12=0 is 4, while the equation x2 - 7x+q = 0 has equal roots, then the value of q is

1. 49/4
2. 4/49
3. 4
4. 1/4

Ans.a

If md (x) = | x |,
mn (x, y) = minimum of x and y and
Ma( a, b, c, ...)= maximum of (a, b, c)

Q.Value of Ma ( md(a) , mn (md(b),a), mn (ab , md(ac)) ) where a = -2, b = -3, c = 4 is

1. 2
2. 6
3. 8
4. -2

Ans.b

Q.Given that a > b then the relation Ma[md(a), mn(a,b)] = mn[a,md(Ma(a,b))]does not hold if

1. a < 0, b < 0
2. a > 0, b > 0
3. a > 0, b < 0,|a| < |b|
4. a > 0, b < 0,|a| > |b|

Ans.a

Q.It takes the pendulum of a clock 7 seconds to strike 4 o’clock. How much time will it take to strike 11 o’clock?

1. 18 seconds
2. 20 seconds
3. 19.25 seconds
4. 23.33 seconds

Ans.d

Q.What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?

1. 264
2. 259
3. 269
4. none

Ans.b

Q.56-1 is divisible by

1. 13
2. 31
3. 5
4. none

Ans.b

Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

1. Rs.60
2. Rs.34
3. Rs.66
4. Rs.28

Ans.c

Q.Who started with the lowest amount?

1. Suvarna
2. Tara
3. Uma
4. Vibha

Ans.d

Q.Who started with the highest amount?

1. Suvarna
2. Tara
3. Uma
4. Vibha

Ans.a

Q.What was the amount with Uma at the end of the second round?

1. 36
2. 72
3. 16
4. none

Ans.b

Q.72 hens cost Rs.__ 96.7__. Then what does each hen cost, where two digits in place of ‘__’ are not visible or are written in illegible hand?

1. Rs.3.23
2. Rs.5.11
3. Rs.5.51
4. Rs.7.22

Ans.c

Q.A person who has a certain amount with him goes to market. He can buy 50 oranges or 40 mangoes. He retains 10% of the amount for taxi fares and buys 20 mangoes and of the balance he purchases oranges. Number of oranges he can purchase is

1. 36
2. 40
3. 15
4. 20

Ans.d

Q.The value of (553 + 453) / (552 - 55 * 45 + 452)

1. 100
2. 105
3. 125
4. 75

Ans.a

Q.For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true?

1. It is even
2. Divisible by 3
3. Divisible by the sum of the square of first n natural numbers
4. Never divisible by 237

Ans.d

Q.One root of x2 + kx – 8 = 0 is square of the other. Then the value of k is

1. 2
2. 8
3. -8
4. -2

Ans.d

Q.Two positive integers differ by 4 and sum of their reciprocals is 10/21. Then one of the numbers is

1. 3
2. 1
3. 5
4. 21

Ans.a

Q.What is the value of m which satisfies 3m2 – 21m + 30 < 0?

1. m < 2 or m > 5
2. m > 2
3. 2 < m < 5
4. Both (a) and (c)

Ans.c

Q.Largest value of min(2 + x2 , 6 – 3x), when x > 0, i

1. 1
2. 2
3. 3
4. 4

Ans.c

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

Q.Find the value of me(a + mo(le(a, b)); mo(a + me(mo(a), mo(b))), at a = –2 and b = –3.

1. 1
2. 0
3. 5
4. 3

Ans.a

Q.Which of the following must always be correct for a, b > 0?

1. mo(le(a, b)) ≥ (me(mo(a), mo(b))
2. mo(le(a, b)) > (me(mo(a), mo(b))
3. mo(le(a, b)) < (le(mo(a), mo(b))
4. mo(le(a,b)) = le(mo(a), mo(b))

Ans.d

Q.For what values of 'a' is me(a2 – 3a, a – 3) > 0?

1. a > 3
2. 0 < a < 3
3. a < 0
4. a = 3

Ans.b

Q.For what values of 'a' is le(a2 – 3a, a – 3) > 0?

1. a > 3
2. 0 < a < 3
3. a < 0
4. both b and c

Ans.d

Q.If log2 log7 (x2-x+37)= 1, then what could be the value of 'x'?

1. 3

2. 5

3. 4

4. none

Ans.c

Q.If n is an integer, how many values of n will give an integral value of (16n2+7n+6)/n ?

1. 2

2. 3

3. 4

4. none

Ans.d

Q.A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number. If his answer differed from the actual one by 770, find the number.

1. 1584

2. 2520

3. 1728

4. 1656

Ans.a

Q.P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

1. 20

2. 65

3. 16

4. 35

Ans.d

Q.If the roots x1 and x2 of the quadratic equation x2 - 2x + c = 0 also satisfy the equation 7x2 - 4x1 = 47 then which of the following is true?

1. c=-15

2. x1 = -5, x2 = -3

3. x1 = 4.5, x2 = -2.5

4. none

Ans.a

Q.If m and n are integers divisible by 5, which of the following is not necessarily true?

1. m – n is divisible by 5

2. m2 – n2 is divisible by 25

3. m + n is divisible by 10

4. None of these

Ans.c

Q.Which of the following is true?

1. 732 = (73)2

2. 732 > (73)2

3. 732 < (73)2

4. none of these

Ans.b

Q.P, Q and R are three consecutive odd numbers in ascending order. If the value of three times P is 3 less than two times R, find the value of R.

1. 5

2. 7

3. 9

4. 11

Ans.c

For these questions the following functions have been defined.
la(x,y,z) = min (x+y,y+z)
le(x,y,z) = max (x-y,y-z)
ma(x,y,z) = 1/2 [la(x,y,z) + le(x,y,z)]

Q.Given that x > y > z > 0 Which of the following is necessarily true?

1. la(x,y,z) < le(x,y,z)

2. la(x,y,z) > ma(x,y,z)

3. le(x,y,z) > ma(x,y,z)

4. none

Ans.b

Q.what is the value of ma (10,4 , le ( la (10,5,3),5,3))

1. 7

2. 6.5

3. 8

4. 7.5

Ans.b

Q.ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

1. 9

2. 7

3. 4

4. 2

Ans.c

Q. n3 is odd. Which of the following statement(s) is(are) true? I. n is odd. II. n2 is odd. III. n is even.

1. I only

2. II only

3. I and II

4. I and III

Ans . c

Production pattern for number of units (in cubic feet) per day.For a truck that can carry 2,000 cubic ft, hiring cost per day is Rs. 1,000. Storing cost per cubic feet is Rs. 5 per day.

Q. If all the units should be sent to the market, then on which days should the trucks be hired to minimize the cost?

1. 2nd, 4th, 6th, 7th

2. 7th

3. 2nd, 4th, 5th, 7th

4. None of these

Ans . c

Q. If the storage cost is reduced to Re. 0.80 per cubic feet per day, then on which day(s), should the truck be hired?

1. 4th

2. 7th

3. 4th and 7th

4. None of these

Ans . b

Q. (BE)2 = MPB, where B, E, M and P are distinct integers. Then M =

1. 2

2. 3

3. 9

4. none

Ans . b

Q. Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

1. 19800

2. 41976

3. 32976

4. None of these

Ans . c

Q. A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.

1. 5

2. 4

3. 1

4. cant say

Ans . a

Q. A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively. How many integers between 0 and 100 belong to set A?

1. 0

2. 1

3. 2

4. cant say

Ans . b

A, B, C and D collected one-rupee coins following the given pattern.
Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q. The maximum number of coins collected by any one of them cannot exceed

1. 64

2. 36

3. 54

4. None of these

Ans . a

Q. If A collected 54 coins, then the difference in the number of coins between the one who collected maximum number of coins and the one who collected the second highest number of coins must be at least

1. 12

2. 24

3. 30

4. None of these

Ans . c

Q. If A collected 54 coins and B collected two more coins than twice the number of coins collected by C, then the number of coins collected by B could be

1. 28

2. 20

3. 26

4. 22

Ans . d

Q. The number of positive integer valued pairs (x, y) satisfying 4x – 17y = 1 and x ≤ 1000 is

1. 59

2. 57

3. 55

4. 58

Ans . A

Q. Let a, b, c be distinct digits. Consider a two-digit number 'ab' and a three-digit number 'ccb', both defined under the usual decimal number system, if (ab)2 = ccb > 300, then the value of b is

1. 1

2. 0

3. 5

4. 6

Ans . A

Q. The remainder when 784 is divided by 342 is

1. 0

2. 1

3. 49

4. 341

Ans . B

Q. If |r-6| = 11 and |2q-12| = 8 what is the minimum possible value of q/r ?

1. -2/5

2. 2/17

3. 10/17

4. none of these

Ans . D

Q. If n = x + 1 where x is the product of four consecutive positive integers, then which of the following is/are true?
A. n is odd
B. n is prime
C. n is a perfect square

1. A and C only

2. A and B only

3. A only

4. None of these

Ans . A

Q. If n2 = 12345678987654321, what is n?

1. 12344321

2. 1235789

3. 111111111

4. 11111111

Ans . C

Ten coins are distributed among four people P, Q, R and S such that one of them gets one coin, another gets two coins, the third gets three coins and the fourth gets four coins. It is known that Q gets more coins than P, and S gets fewer coins than R.

Q. If the number of coins distributed to Q is twice the number distributed to P, then which one of the following is necessarily true?

1. R gets an even number of coins.

2. R gets an odd number of coins.

3. S gets an even number of coins.

4. S gets an odd number of coins.

Ans . D

Q. If R gets at least two more coins than S, then which one of the following is necessarily true?

1. Q gets at least two more coins than S.

2. Q gets more coins than S.

3. P gets more coins than S.

4. P and Q together get at least five coins.

Ans . B

Q. If Q gets fewer coins than R, then which one of the following is not necessarily true?

1. P and Q together get at least four coins.

2. Q and S together get at least four coins.

3. R and S together get at least five coins.

4. P and R together get at least five coins.

Ans . A

A young girl Roopa leaves home with x flowers, goes to the bank of a nearby river. On the bank of the river, there are four places of worship, standing in a row. She dips all the x flowers into the river. The number of flowers doubles. Then she enters the first place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the second place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the third place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the fourth place of worship, offers y flowers to the deity. Now she is left with no flowers in hand.

Q. If Roopa leaves home with 30 flowers, the number of flowers she offers to each deity is

1. 30

2. 31

3. 32

4. 33

Ans . C

Q. The minimum number of flowers that could be offered to each deity is

1. 0

2. 15

3. 16

4. CANT SAY

Ans . C

Q. The minimum number of flowers with which Roopa leaves home is

1. 16

2. 15

3. 0

4. can't say

Ans . B

### Quiz

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