# 23 days to CAT 2015 | Quick quant

We are back with another set of CAT questions. Try to solve this set in five minutes.

Q.1 A truck traveling at 70 mph uses 30% more diesel to travel a certain distance than it does when it travels at the speed of 50 mph. If the truck can travel 19.5 kilometers on a liter of diesel at 50 mph, how far can the truck travel on 10 liters of diesel at a speed of 70 mph. (CAT 2000)

Q.2 There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required? (CAT 2003)
(A) 72 (B) 90 (C) 96 (D) 144

Q.3 A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of three cases the leading digit is 1. Then M equals (CAT 2003)
(A) 31 (B) 63 (C) 75 (D) 91

Q.4 An intelligence agency forms a code of two distinct digits selected from 0, 1, 2 … 9 such that the first digit of the code is non zero. The code handwritten on a slip, can however potentially create confusion when read upside down – For example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise? (CAT 2003)
(1) 80 (2) 78 (3) 81 (4) 69

Q.5 The digits of a three digit number A are written in the reverse order to form another three digit number B. If B > A, and B – A is perfectly divisible by 7, then which of the following is necessarily true? (CAT 2005)
(1) 100 < A < 299 (2) 106 < A < 305 (3) 112 < A < 311 (4) 118 < A < 317

Solutions:

# 24 days to CAT 2015 | CAT traps I – Data Sufficiency

The CAT traps part of the 75 days to CAT 2015 series will deal with the commonly found and oft overlooked traps that can be found in entrance tests. I will be covering both generic and topic-wise traps that one can expect and be careful of while attempting the tests in the coming few posts. CAT traps is primarily an account of the silly mistakes candidates make during the high pressure scenario that is the CAT. Read more

# 25 days to CAT 2015 | Quick quant

We are back with another set of CAT questions. Try to solve this set in five minutes.

Q.1 The remainder when (15^23 + 23^23) is divided by 19 is (CAT 2004)
(A) 4 (B) 15 (C) 0 (D) 18

Q.2 If x = (16^3 + 17^3 + 18^3 + 19^3), then x divided by 70 leaves a remainder of (CAT 2004)
(A) 0 (B) 1 (C) 69 (D) 35

Q.3 Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers? (CAT 2008)
(1) 1 ≤ m ≤ 3 (2) 4 ≤ m ≤ 6 (3) 7 ≤ m ≤ 9 (4) 10 ≤ m ≤ 12 (5) 13 ≤ m ≤ 15

Q.4 In a certain examination paper, there are n questions. For j = 1, 2, 3… n, there are 2^(n-j) students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is (CAT 2003)
(1) 12 (2) 11 (3) 10 (4) 9

Q.5 The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero? (CAT 2003)
(1) 1st (2) 9th (3) 12th (4) none of these

Solutions

# 26 days to CAT 2015 | Quick quant

Five questions. Five minutes. Your time starts now!

Q. 1 What are the last two digits of 7^2008? (CAT 2008)
(1) 21 (2) 61 (3) 01 (4) 41 (5) 81

Q.2 A new flag is to be designed with six vertical stripes using some or all of the colors yellow, green, blue, and red. Then the number of ways this can be done so that no two adjacent stripes have the same color is (CAT 2004)
(1) 12 × 81 (2) 16 × 192 (3) 20 × 125 (4) 24 × 216

Q.3 Consider the set S = {2, 3, 4… 2n+1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y? (CAT 2007)
(1) 0 (2) 1 (3) n/2 (4) (n+1)/2n (5) 2008

Q.4 Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all elements of S. With how many consecutive zeroes will the product end? (CAT 2000)
(1) 1 (2) 4 (3) 5 (4) 10

Q.5 What is the sum of all two-digit numbers that give a remainder of 3 when divided by 7? (CAT 2003)
(1) 666 (2) 676 (3) 683 (4) 777

# 27 days to CAT 2015 | Quick quant

Q.1. The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10n, on the nth day of 2007 (n = 1, 2… 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the nth day of 2007 (n = 1, 2… 365). On which date in 2007 will the prices of these two varieties of tea be equal? (CAT 2007)
(1) May 21 (2) April 11 (3) May 20 (4) April 10 (5) June 30

Q.2 Let f(x) = max (2x + 1, 3 – 4x), where x is any real number. The minimum possible value of f(x) is (CAT 2006)
(1) 1/3 (2) 1/2 (3) 2/3 (4) 4/3 (5) 5/3

Q.3 The number of roots common between the two equations x^3 + 3x^2 + 4x + 5 = 0 and x^3 + 2x^2 + 7x + 3 = 0 is (CAT 2003)
(1) 0 (2) 1 (3) 2 (4) 3

Q.4. The remainder when 2^256 is divided by 17 is (CAT 2002)
(1) 7 (2) 13 (3) 11 (4) 1

Q.5 Consider a sequence where then nth term,
t_n=n/((n+2)) ,n=1,2,…
The value of t_3 * t_4 * t_5 * t_6 * … * t_53 equals: (CAT 2006)
(1) 2/495 (2) 2/477 (3) 12/55 (4) 1/1485 (5) 1/2970

# 28 days to CAT 2015 | Quick quant

We are back with another set. Standard rules. Solve the next five questions in five minutes. Come back and check the solutions.

Q.1 In a 4000 m race around a circular stadium having a circumference of 1000 m, the fastest runner and the slowest runner reach the same point at the end of the fifth minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race. (CAT 2003)
(1) 20 minutes (2) 15 minutes (3) 10 minutes (4) 5 minutes

Q.2 There are 6 boxes numbered 1, 2, 3, 4, 5, and 6. Each box is to be filled with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is (CAT 2003)
(1) 5 (2) 21 (3) 33 (4) 60

Q.3 Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is (CAT 2005)
(1) 4 (2) 5 (3) 6 (4) 7

Q.4 A student finds the sum of 1, 2, 3 … as his patience run out. He found the sum as 575. When the teacher declared the result wrong, the student realized that he had missed a number. What was the number that the student missed? (CAT 2002)
(1) 16 (2) 18 (3) 14 (4) 20

Q.5 Let D be a recurring decimal of the form, D = 0.a1a2a1a2a1a2…., where digits a1 and a2 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? (CAT 2000)
(1) 18 (2) 108 (3) 198 (4) 288

# 30 days to CAT 2015 | Quick quant

We’ll look at a few more problems. Hope you have gone through the earlier Quick Quant posts. We are solving questions with shortcuts and non-conventional methods. See if you can crack this set in five minutes.

Q.1 A milkman mixes 20 L of water with 80 L of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he has sold. What is the current proportion of water to milk? (CAT 2004)

(1) 2:3 (2) 1:2 (3) 1:3 (4) 3:4

Q.2 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 are they (in kms) one minute before they collide? (CAT 2004)

(1) 1/12 (2) 1/6 (3) 1/4 (4) 1/3

Q.3 In a tournament, there are n teams T1, T2… Tn with n>5. Each team consists of k players, k>3. The following pairs of teams have one player in common: T1 and T2, T2 and T3, T3 and T4… Tn-1 and Tn, Tn and T1. No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together? (CAT 2007)

(1) n(k-1) (2) k(n-1) (3) n(k-2) (4) k(k-2) (5) (n-1)(k-1)

Q.4 Len n! = 1 * 2 * 3 * 4 *… * n for integer n ≥ 1. If p = 1! + 2*2! + 3*3! + … 10*10! Then p + 2 when divided by 11! Leaves a remainder of (CAT 2005)

(1) 10 (2) 0 (3) 7 (4) 1

Q. 5 Consider a sequence of seven consecutive integers. The average of first five integers is n. The average of all seven integers is: (CAT 2000)

(1)n (2) n+1 (3) K*n, where K is a function of n (4) n+(2/7)

# 34 days to CAT 2015 | Polynomials

A polynomial is an algebraic expression consisting of many terms involving powers of the variable. The general form of the polynomial is

f(x) = a0xn + a1xn-1 + a2xn-2 + a3xn-3 + …+ an where ao, a1, a2, …. an are rational numbers and n is non-negative. The highest power of x is called the degree of the equation.

e.g. x6+ x3+1 is an equation with degree six.

If the degree is 1, then the polynomial is referred to as a linear equation. Similarly, if the degree is 2, then the polynomial is referred to as a quadratic equation.

The value of x for which the polynomial f(x) reduces to zero is called the root of the equation. Graphically, it is the point at which the graph of f(x) cuts the X-axis.

Properties of roots

1. The number of roots will depend on the degree of the equation. A polynomial of the nth degree will have n roots. For example, the equation x4 + 4x2+2 =0 has 4 roots. Note that the roots can be either real or imaginary. The total number of roots will be 4.

2. Imaginary roots occur in pairs. Thus, if 5+3i is one root of f(x), then 5-3i will also be a root of f(x).

3. Descartes Rule: This rule is used to find the maximum number of positive and negative real roots of a polynomial. Note that it does not provide the exact number of positive and negative real roots.

The number of positive roots of a polynomial is either equal to the number of sign changes between consecutive coefficients or it is less than it by an even number.

e.g. f(x) = x5 - x4 + 3x3 + 9x2 - x + 5. Notice that the sign changes from x5 to x4, from x4 to 3x3, from 9x2 to x and again from x to 5. Therefore, there are 4 sign changes. Hence, 4 is the maximum possible number of positive roots.

As per Descartes Rule, the number of positive roots could be 4 or 2 (=4-2) or 0 (=2-2). Basically, the number has to be counted down by 2 from the maximum possible number of roots.

Hence for f(x) = x5 - x4 + 3x3 + 9x2 - x + 5, there are either 4, 2 or 0 positive roots.

Similarly for negative roots, we need to check the number of sign changes for f(-x).

f(-x) = -x5 - x4 - 3x3 + 9x2 + x + 5

We can see that there is only 1 sign change (from 3x3 to 9x2). Hence the maximum possible number of negative real roots is 1. In this case, we don’t need to count down by 2 as doing so will result in a negative number. Hence, we know that f(x) contains only 1 negative root.

Summing up, f(x) has 4,2 or 0 positive roots and exactly 1 negative root.

1. Corollaries of Descartes Rule:

a. If the coefficients of f(x) are all positive then the equation has no positive root. This is because there are no sign changes if all coefficients are positive.

b. If the coefficients of even powers of x are all of one sign and the coefficients of odd powers of x are all of opposite signs then the equation has no negative real root. Again, this follows from the sign change rule of Descartes.

c. If the equation contains only even powers of x and all the coefficients are of the same sign, then the equation will have no real roots.

d. If the equation contains only odd powers of x and all the coefficients are of the same sign, then the equation will only have 0 as a real root.

5. Sum and product of roots:

For a polynomial f(x) = a0xn + a1xn-1 + a2xn-2 + a3xn-3 + …+ an, let α1, α2, α3…. αn be the roots.

Then,

Sum of the roots (α12+ α3+ …+αn) = – a1 / a0

Sum of the products of the roots taken two at a time (α1α2+ α2α3+….) = a2/a0

Sum of the product of the roots taken three at a time (α1α2 α3+ α2α3 α4+….)  = – a3 / a0

…..

Product of the roots (α1 α2 α3…. αn ) = (-1)n an / a0

Let’s look at a few examples:

Q.1. If the equation x3 – ax2 + bx – a =0 has three real roots, then it must be the case that (CAT 2000)

a. b=1     b. b≠1       c. a=1         d. a≠1

Answer:  We can write the above equation as x2 (x – a) + bx – a =0

If b = 1 then we can take (x-a) as common.

Let’s take b=1 and check.

x2 (x – a) + x – a =0

(x-a) (x2+1)=0

Hence, x= a is one root. The other roots are x2 =-1 => the other two roots are imaginary.

But the question says that the equation has three real roots. Hence, the assumption that b= 1 is wrong. Hence, b≠1 which is option b.

Q.2. The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3 + 2x2 + 7x + 3 = 0 is (CAT 2003)

a. 0      b. 1      c. 2      d. 3

Answer: Equate both the equations to find the intersection points.

x3 + 3x2 + 4x + 5 = x3 + 2x2 + 7x + 3

=> x2-3x+2=0

=> x=2 or x=1

However, don’t make the mistake of taking them for the common roots. These are just the intersection points. We need to check if any of these is a root of both equations.

Substituting x=2 in the first equation will give us some positive value and hence it is not equal to 0. Hence, 2 cannot be a root of the first equation.

Similarly, substituting x=1 in the first equation will give us some positive value and hence it is not equal to 0. Hence, 1 cannot be a root of the first equation.

Hence, neither of the intersection points are roots. Hence, there is no root common to both the equations. Hence option a is the answer.

Here’s one for you to solve:

Q.3. If the roots of the equation x3 – ax2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b? (CAT 2008)

a. -1/√3       b. –1    c. 0     d. 1     e. 1/√3

# 35 days to CAT 2015 | Quick quant

Another set of five questions from quant. See if you can crack this set in five minutes.

Q.1 Arun, Barun and Kiranmala start from the same place and travel in the same direction at speeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barun and Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmala start? (CAT 2006)
(1) 3
(2) 3.5
(3) 4
(4) 4.5
(5) 5

Q.2 A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x? (CAT 2008)
(1) 2 ≤ x ≤ 6
(2) 5 ≤ x ≤ 8
(3) 9 ≤ x ≤ 12
(4) 11≤ x ≤ 14
(5) 13 ≤ x ≤ 18

Q.3 Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 m. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is (CAT 2005)
(1) 3√2
(2) 1 + π
(3) 4 π/3
(4) 5

Q.4 Number S is equal to the square of the sum of the digits of a 2 digit number D. If the difference between S and D is 27 then D is (CAT 2002)
(1) 32
(2) 54
(3) 64
(4) 52

Q.5 The sum of four consecutive two digit odd numbers when divided by 10 becomes a perfect square. Which of the following can possibly be one of these four numbers? (CAT 2006)
(1) 21
(2) 25
(3) 41
(4) 67
(5) 73

# 36 days to CAT 2015 | Quick quant

Hope you liked the first article on Quick Quant. Have a look at these five questions and see if you can crack this set in five minutes.

Q.1 Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose? (CAT 2005)
(1) 5
(2) 10
(3) 9
(4) 15

Q.2 The number of positive integers n in the range 12 ≤ n ≤ 40 such that the product (n-1)(n-2)…3.2.1 is not divisible by n is (CAT 2003)
(1) 5
(2) 7
(3) 13
(4) 14

Q.3 Let g(x) = max (5 – x, x + 2). The smallest possible value of g(x) is (CAT 2003)
(1) 4.0
(2) 4.5
(3) 1.5
(4) none of these

Q.4 If pqr = 1 then 1/(1+p+q^(-1) ) + 1/(1+q+r^(-1) ) + 1/(1+r+p^(-1) ) is equivalent to (CAT 2002)
(1) p+q+r
(2) 1/(p+q+r)
(3) 1
(4) p^(-1)+q^(-1)+r^(-1)

Q.5 In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. If was found that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is (CAT 2005)
(1) 200
(2) 216
(3) 235
(4) 256

These questions can be solved using simple logic and conceptual understanding. We’ll go through each of them now.  Read more